ABSTRACT
Title of Dissertation:ANALOGIES AS CATEGORIZATION
PHENOMENA: STUDIES FROM
SCIENTIFIC DISCOURSE
Leslie Jill Atkins, Ph.D., 2004
Dissertation Directed By:Associate Professor David Hammer,
Department of Curriculum and Instruction,
Department of Physics
Studies on the role of analogies in science classrooms have tended to focus on
analogies that come from the teacher or curriculum, and not the analogies that
students generate.  Such studies are derivative of an educational system that values
content knowledge over scientific creativity, and derivative of a model of teaching in
which the teacher?s role is to convey content knowledge.  This dissertation begins
with the contention that science classrooms should encourage scientific thinking and
one role of the teacher is to model that behavior and identify and encourage it in her
students.  One element of scientific thinking is analogy.  This dissertation focuses on
student-generated analogies in science, and offers a model for understanding these.  I
provide evidence that generated analogies are assertions of categorization, and the
base of an analogy is the constructed prototype of an ad hoc category.  Drawing from
research on categorization, I argue that generated analogies are based in schemas and
cognitive models.  This model allows for a clear distinction between analogy and
literal similarity; prior to this research analogy has been considered to exist on a
spectrum of similarity, differing from literal similarity to the degree that structural
relations hold but features do not.  I argue for a definition in which generated
analogies are an assertion of an unexpected categorization: that is, they are asserted as
contradictions to an expected schema.
ANALOGIES AS CATEGORIZATION PHENOMENA: STUDIES FROM
SCIENTIFIC DISCOURSE
By
Leslie Jill Atkins
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park, in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2004
Advisory Committee:
Associate Professor David Hammer, Chair
Dr. Andy Elby
Associate Professor Sarah Eno
Professor Edward F. Redish
Assistant Professor Emily van Zee
? Copyright by
Leslie Jill Atkins
2004
ii
Dedication
For women in physics.
?As I say, I can speak only for myself,
but as soon as I got here the rules became different.
They didn?t apply to me any more, or to anyone else except a distant runt,
almost invisible in its litter.  So how was
I to know who to stand up to, when to turn abrasive, when all things
nestled,
equidistant, all hearts were charming, and it was good to be natural and
sincere??
School was over,
not just for that day but forever and for seasons to come.
The reason was that the truth was just average
on the iniquity scale, and nobody wanted to get involved?
You see we all thought the ride would be lovely
and worth the trip, which it was, but now we cannot go anywhere
having already been everywhere. No, do you
understand how realistic it all is??
And so we faced the new day
like a pilgrim who sees the end of his journey
deferred forever.
Who could predict where we would be led, to what
extremes of aloneness?  Yet the horizon is civil.
-Ashbery, Girls on the Run
iii
Acknowledgements
The idea behind this thesis is that the theories we develop with our science are
analogies to the stories that we tell with our lives.  And so this dissertation holds a mirror
to my life and reflects its stories ? the stories that have brought me here and brought
about this work.  And those stories have as much to do with community and friendship as
they do physics and education.  I would like to acknowledge all of my friends, in
particular Dorothy, Kathryn, Noam and Sam, for conversation about the things that
matter, and my Seattle roommates ? especially Amber, Laura, Manu and Sam ? for
creating a true home.  I would like to thank Jerry Seidler, in whose lab I discovered that I
did not want to do experimental physics and I was given the freedom and support to
decide that.  I thank Stamatis Vokos, the deus ex machina of my story, and Joe Redish,
both of whom took risks for me and I am grateful for and buoyed by their trust.  Graduate
students Matty, Paul and Rosemary provided hours of critique and conversation in
helping me hone the details of this thesis and kept me smiling through the massive
frustrations.  And friends outside of graduate school, Wendy and Anne in particular,
reminded me of life beyond academia, while the Elliott family gave me hope that
academia could be everything I wanted it to be.  Were it not for my education at the
Governor?s School of North Carolina, both as a student and a teacher, I would never have
seen myself as someone with a story to tell or a theory to share.  The work that happens
there is incredible and profound.  Thank you to Mrs. Liz Woolard, my brilliant high
school physics teacher.  And to Janet Coffey who is going to be an amazing professor and
advisor and was so encouraging ? especially in the final stretch.  The teachers from SIPS
iv
I cannot thank enough; without their careful attention to students and remarkable work as
teachers there would be no data and no thesis.  For all of the Physics Education Research
Group, past and present: Renee Michelle, Tom, Rosemary, Matty, Paul, Ray, Tim, Paul,
Jonathan, Rebecca, Loucas, Rachel, Laura, David, Joe, Andy, Emily and David.  You are
the greatest evidence of my thesis that science is influenced by the stories you tell with
the rest of your life ? the stories you have given me have created the science in this
thesis.  These are stories of authenticity, theoretical inquiry, identity, frames and
epistemology, of course, but also stories of communication, big hearts, laughter,
dedication, teaching, birthday cakes, cicada roasts and community.  And with all my
heart a thank you to David Hammer, whose attention to and excitement about students?
ideas, which I first witnessed with regard to elementary students, applies to graduate
students as well.  He has given me the voice that I had hoped graduate school would and
feared that it wouldn?t.
For Richard, you know my whole story ? told over lunches in the physics
courtyard, driving across the country, along the West Coast Trail, in Chamonix, Bear
Lake, Mazama, and, too many times, over the phone late at night.  I am so lucky.  You're
for me.
And finally, for my family: mom, dad, Sara and Ben.  Thank you.  Your love is
tremendous.
v
Table of Contents
Dedication ??????????????????...????????ii
Acknowledgements ???????????????????????.iii
Table of Contents ???????????????...????????.v
Chapter 1: ???????????????????????????1
Introduction ????????????????????...????...1
A brief history of analogy research ????????????????4
Major themes ????????????????????????...5
Chapter overview ???????????????????????.6
Chapter two: Review of the literature ???????????????6
Chapter three: Origins of the study and methodological considerations ???.7
Chapter four: Phenomenological coherence ?????????????7
Chapter five: The ontology of mind ???????????????...8
Chapter six: Analogies in the history of science ???????????..8
Chapter seven: Implications for instruction ?????????????.8
Chapter eight: Summary and directions for future research ???????..9
Appendix A ? J: Transcripts 1 ? 10 ????????????????10
Appendix K: Young children?s analogies ?????????????...10
Chapter 2: Review of the Literature ?????????????????12
Introduction and overview ???????????????????...12
Analogies ??????????????????????????.15
Analogies in philosophy and the philosophy of science ????????...15
Analogies in cognitive science ?????????????????...17
Analogies in linguistics ????????????????????.19
Analogies in science and science education ?????????????20
Summary of analogy research ??????????????????23
Categorization ????????????????????????..24
Overview ?????????????????????????.24
Categorization and fuzzy sets ??????????????????25
Rosch and prototype theory ??????????????????...27
Properties of categories ????????????????????.29
Idealized cognitive models ???????????????????31
Metonymy, cognitive models and phenomenological-primitives ?????..33
Summary of categorization research ???????????????...35
Analogies as categorization ???????????????????.35
Metaphors as category-inclusion statements ????????????...35
Idealized cognitive models and lexical networks ???????????.38
Reconciling categorization and structure mapping views of analogies ???..40
Conclusion ?????????????????????????...40
Chapter 3: Origins of the Study & Methodological Considerations ????...43
Origins of the study ??????????????????????..43
History of research on analogies ?????????????????..45
vi
Past research ????????????????????????45
Limitations of past methodology ?????????????????48
Recent approaches to the study of analogical reasoning ????????.52
Methodology ?????????????????????????54
A note on the data ??????????????????????55
The case study methodology ??????????????????.55
Summary ??????????????????????????..57
Chapter 4: Phenomenological Coherence ??????????????...58
Introduction ?????????????????????????..58
Models of analogy from the literature ???????????????58
Analogies as categorization ??????????????????...60
Multiple analogies ???????????????????????62
Multiple analogies: Example 1 ?????????????????..62
Multiple analogies: Example 2 68
Chains of analogies ????????????????????.??.71
Chains of analogies: Example 1 ?????????????????.72
Chains of analogies: Example 2 75
Chains of analogies: Example 3 ?????????????????.76
Construction of the base ????????????????????...80
Depiction of the base in research on analogies ????????????80
Prototypes in categorization ??????????????????..81
Construction of the prototype ??????????????????82
Construction of the base in student-generated analogies ????????..83
Near and far transfer analogies ??????????????????86
Research on prototypes and research on transfer ???????????.86
Examples from student-generated analogies ????????????...87
Variable representation of the base ????????????????..88
Variable representation: Example 1 ????????????????89
Variable representation: Example 2 92
Previous claims of analogy as categorization ????????????..97
Analogies as negative assertions ?????????????????.98
Violations of expected schemas ?????????????????.98
Examples from student-generated analogies: Analogies as negative assertion ?99
Conclusion ?????????????????????????..101
Chapter 5: The Ontology of Mind ?????????????????. 105
Introduction ?????????????????????????. 105
History of ontology of mind and description of the chapter ??????... 105
Section 1: A theoretical account of the ontology of mind ???????.. 107
Structure-mapping ?????????????????????... 107
The arguments for variability in conceptual representations ??????? 109
Structures in a manifold ontology of mind ?????????????. 113
Interlude: A distinction between similarity and analogy ???????? 117
The conflation of similarity and analogy in past definitions ???????117
A proposed definition of analogy as a change of schema ???????? 122
Section 2:  The base of generated analogies as representations of a schema . 125
vii
Schemas and p-prims in the cup/water analogies ..??????????.. 125
The Beanbag Analogies ???????????????????... 132
Styrofoam and Ice Skating ?...????????????????? 136
Analogies regarding a quantum mechanics problem ?????????... 142
The ontology of authenticity ??????????????????. 144
Conclusion ?????????????????????????.. 146
Chapter 6: Analogies in the History of Science ????????????. 148
Introduction ?????????????????????????. 148
History and philosophy of science ????????????????.. 151
Comparative literature ????????????????????...  154
Membranes ????????????????????????. 155
Networks ????????????????????????? 158
Luca Turin: Analogies involving scent ??????????????.. 162
Cognitive science ??????????????????????? 164
Conclusion ?????????????????????????.. 168
Chapter 7: Implications for Instruction ???????????????.. 170
Introduction ?????????????????????????. 170
A critique of standard analogy use in classrooms ??????????.. 171
Implications for instruction ???????????????????. 177
Students ought to generate their own analogies ???????????.. 177
Expect variability and multiple analogies ?????????????.. 180
A reconsideration of the idea of transfer ?????????????? 182
An example of these implications for instruction in practice ??????. 184
Examples from transcripts ??????????????????? 184
Examples from the literature ??????????????????. 185
An example from my own teaching ???????????????... 186
Finding analogies in the classroom ????????????????. 193
The National Science Education Standards ????????????? 195
The case for diversity in science ?????????????????. 196
Conclusion ?????????????????????????.. 197
Chapter 8: Summary & Directions for Future Research ????????? 198
Introduction ?????????????????????????. 198
Summary ??????????????????????????. 199
Negative analogies ??????????????????????.. 200
Literary analysis and differ?nce ????????????????? 200
Negative analogies ?????????????????????.. 202
Negative analogies in physics ?????????????????... 204
Negative analogies as a caveat ?????????????????.. 205
Directions for future research regarding analogies and mind ??????. 206
The changing schema ????????????????????... 206
Conceptual blending ????????????????????? 207
The implications of technology on science ????????????? 207
Science in the absence of analogy ????????????????. 208
Embodied cognition and analogies in science ???????????? 209
Analogies as a tool for exploring categorization ???????????. 210
viii
Network theory and analog ??????????????????... 210
Directions for future research on the implications for instruction ????.. 212
Questions regarding student epistemology ?????????????. 212
The design of learning environments???????????????.. 212
Concluding thoughts ?????????????????????... 213
Appendix A ? J: Transcripts ???????????????????.. 215
Appendix K: Young children?s analogies ??????????????.. 292
Glossary???????????????????????????..302
Bibliography??????????????????????????304
Curriculum Vitae????????????????????????.312
1
Chapter 1:  Introduction and Major Themes
Introduction
Metaphor is the currency of knowledge.  I have spent my life learning incredible
amounts of disparate, disconnected, obscure, useless pieces of knowledge, and
they have turned out to be, almost all of them, extremely useful.  Why?  Because
there is no such thing as disconnected facts.  There is only complex structure.
And both to explain complex structure to others and, perhaps more important ?
this is forgotten, usually ? to understand them oneself, one needs better
metaphors.  If I was able to understand this, it was because my chaotic accrual of
information simply gave me better metaphors than anyone else?
Translate a concept from its field for use to where it is unknown, and it is
always fresh and powerful.  In buying outside, you are doing intellectual
arbitrage.  The rate limiting step in this is your willingness to continuously
translate, to force strange languages to be yours, to live in between, to be
everywhere and nowhere.
-Burr, 2002
Many scientific theories evolve from analogy ? noticing links others have missed
or relating a fact that others have not noticed.  Luca Turin (Burr, 2002) related the
mechanism for smell to electron tunneling spectroscopy.  The arguments and evidence
for his model came not from chemistry or biology, but perfumery and jet fuel technology.
Einstein was a patent clerk in an era when trains were fast becoming a primary means of
transportation.  Many patents of this time concerned synchronizing clocks, and Galison
has proposed that considering this problem in the context of what Einstein knew from
physics led him to his theory of relativity (Galison, 2003).  Faraday?s research notes
express deep analogical reasoning: thinking of electro-magnetism in terms of lines in
space (the ?field concept?) (Nersessian, 1985).  Maxwell related magnetism to vortices
and ?idle wheels? (Nersessian, 2002).  Kosterlitz and Thouless (1973) related 2-
2
dimensional phase transitions to topology.  The scientific literature is overrun with
breakthroughs that developed from analogical reasoning ? relating seemingly unrelated
topics that, once related, establish a new research paradigm.
The use of analogies in science is not restricted to insights from creative
scientists, but is part of the regular patter of students and instructors when discussing
scientific ideas.  In the following transcript, undergraduates in a conceptual physics
course have been exploring static electricity in conductors and in insulators.  The students
have worked in small groups, discussing the differences between what it?s like for a
charge in metal versus Styrofoam, and are now reporting and discussing their conclusions
with the instructor (transcript 1, lines 41 ? 107):
Christie:  Like they were just saying in metal it?s [the charge is] always
moving.  So if it?s always moving it has more room to move and
that would mean to say that the molecules are, like, less tightly
packed together or less dense.  And we were thinking of Styrofoam
as more dense then ?  I?m just trying to figure out first if that?s
right and how it relates.
Lea:  Alright I, going on that idea.  I don?t agree with you saying that the
Styrofoam is more dense I think it's less dense.  And so that?s why
the charges get caught up in it.  'Cause it's like ? like cotton.  And
the, the pan is more dense and so they?re able to slide across it like
they can ice skate across it easier.  So that?s why they move around
more 'cause it?s more dense so they can slide across it more and the
Styrofoam is less dense and so they get stuck in it.  Like they can?t
move as much.
Instructor: ? I?m thinking of like, pouring water into a sponge versus pouring
water onto a hard surface.  Like this sponge is actually less dense
and there?s room for it to absorb the water and the you know if you
pour it onto something hard there?s no room for it to absorb.
Anna: You?re saying that the charge is, like, on top of the metal?  Like on
the outside?
Lea: Yeah.  Hold on-I mean-I don?t-
Anna:  I think it?s like made up of it-like, they?re electrons.
Lea:  Like, I don?t know.  But it?s definitely a lot smoother, like, and
they?re they?re denser and so they can move around more freely.
3
Paul:  I just want to- I know there are people here- I just want to clarify
Lea and Anna your question, your question was ?Is charge moving
on the surface as opposed to moving inside.?  Right? And so this
would be like are the fish swimming in the middle of the fishbowl
or are they sort of somehow stuck to the edge of the fishbowl?
Lydia: Alright, well I was going to say it makes sense to me if the pie
plate is more dense, then it would be easier for them to move
within it.  But I do think that it?s inside of the metal not outside.
Because it?s harder- I mean, if there?s more space to travel like in
the foam, you can?t get from one place to another easily but it?s all
real close together so it can sort of hop along inside.
Paul:  Oh so it?s like stepping stones?
Lydia: Kind of.
Paul: Like in the Styrofoam it?s really far [Lydia: Like yeah.] to the next
stepping stone and so I can?t get there I?m stuck here.  [Lydia:
Right] But in the metal the stones are really close together and so
then I can kind of just walk across.
Notice the analogies drawn in the brief transcript above: the motion of electrons in metal
is like ice-skating, the metal is like a countertop or closely spaced stepping-stones.
Styrofoam is like a sponge or cotton.  What cognitive work does this do for the students?
Why are these analogies chosen?  How do they influence the resulting theories the
students develop?  What do these analogies reveal about the way the students
conceptualize the world?
There is no shortage of research to turn to in addressing these questions and
seeking to understand analogies in science.  Research from education, cognitive science,
linguistics, anthropology, and the history and philosophy of science all have something to
say about the analogies that the scientists and students are employing.  This dissertation
draws from all of these fields, together with analogies generated by students in science
classrooms and study groups, in constructing a model for generated analogies in science
and explores the implications of this model on science instruction.
A brief history of analogy research
4
What cognitive capabilities underlie our fundamental human achievements?
Although a complete answer remains elusive, one basic component is a special
kind of symbolic ability ? the ability to pick out patterns, to identify recurrences
of these patterns despite variation in the elements that compose them, to form
concepts that abstract and reify these patterns, and to express these concepts in
language.  Analogy, in its most general sense, is this ability to think about
relational patterns.
-Holyoak, Gentner and Kokinov, 2001
Analogies, long recognized as being far more than figurative, expressive forms of
speech, have frequently been used as tools for teaching students about science (Clement,
1993, Gick and Holyoak, 1983) and the benefits of this has been explored extensively.
The story that this research tells suggests that the base of the analogy ? the sponge, say,
or the stepping-stones in the transcript above ? is the cognitive foundation for the
analogy.  It is from this base that students extrapolate a structure and map onto a target.
But there are several things problematic with this story when applying it to
student-generated analogies in classrooms.  First, the story contradicts other findings
from education that inform us that we do not have single, unitary representations of
concepts in mind that can be mapped onto others.  Second, it is a model built from studies
regarding how students interpret and understand analogies that they are given ? or even
how we, as teachers, come to understand their analogies ? but not the things students are
doing when they assert analogies.  If generating, critiquing and working with analogies
are part of the practice of science we want students to learn, we need to better understand
moments like the one above, in which students generate analogies and the analogies
introduced by the instructor are responsive to student ideas rather than content goals.
Because of the focus on the interpretation of analogies less well known, researched and
5
understood are the ways in which analogies are constructed by students seeking to
explain the world in a scientific way (May, Hammer and Roy, 2004).
Major themes
The story I want to tell, using analogies generated in student scientific discourse
and in science as it is practiced, is one of analogies as assertions of categorization ? in
particular, a categorization that is unexpected.  Categories, as I will explain in the
following chapter, stem from our cognitive models of the world, built from schemas.
Lea?s analogy to ice-skating (presented at the beginning of this chapter) is, first, an
acknowledgement that density typically makes motion difficult.  Her analogy is asserting
that another manner of categorization exists for dense media: there are those that, by
virtue of their density, make motion easier.  The interpretation of analogies as assertions
of categorization echoes Eva Feder Kittay's  (1997) comments with regard to metaphor ?
made in the field of linguistics, outside of physics and education.
If metaphors do not report an antecedent similarity, but instead create the
similarity, they do so by dislodging some items from familiar classifications and
regrouping them with items that normally belong to different, even disjoint
categories.  So dislodging and regrouping items or subclassifications not only
creates a new category, but also disrupts normal classifications.
This thesis will outline the reasoning behind the ?normal classifications? that we have,
based on Lakoff?s research in idealized cognitive models and diSessa?s theory of
phenomenological primitives.  Furthermore, the bases of analogies are explained in this
context as arising from the categories that are constructed from cognitive models and are
the ad hoc, constructed prototypes of these categories.
The ontology of mind implicit in this theory is compared with other theories of
analogy, in particular structure-mapping (Gentner, 1983).  I argue for a manifold, small
6
scale, ?knowledge-in-pieces? ontology of mind and support this argument with data from
other findings in cognitive science, linguistics and education.
Finally, this thesis has strong implications for instruction, education research and
cognitive science, starting from the premise that generating analogies should be an
important part of the science classroom ? not as a tool for acquiring content knowledge
but as a goal of a science education because it is, in part, what science is.  Second, the
focus on interpreting analogies and lack of attention to generated analogies is challenged:
such a focus misses the cognitive flexibility and utility of most analogies, as analogies are
surely generated more often and with greater effect than they are interpreted.  Third, the
ideas underlying ?transfer? and ?misconceptions?-based curricula are called into
question.  Finally, I caution research ? in particular education research ? against ?in
vitro? studies that are not balanced by ?in vivo? studies of cognition in the wild.
Applying findings generated in laboratories that limit cognitive variability and lack the
context of a classroom to educational and other ?real world? scenarios can result in a
detrimental shift of focus from students? abilities and the variability of student reasoning
to a focus on a unitary conception of students? reasoning and abilities.
Chapter overview
Chapter Two: Review of the literature
In the following chapter, I will outline the existing research on analogies and
categorization.  My emphasis is on the characteristics of generated analogies and
categorization, and not by what mechanism the mind creates schemas and their associated
categories.  I present research from cognitive science and linguistics on analogies and
metaphors and the role that analogies have played in science education.  Then I outline
7
the research on categorization and the interplay between categories and idealized
cognitive models.  Research from physics education on phenomenological primitives is
then related to the idealized cognitive model.  Finally, past research relating analogies to
categorization is reviewed.
Chapter Three: Methodological Considerations
This chapter is dedicated to outlining the history of research in analogical
reasoning and situating my methodology within this history.  In particular I focus on the
methodology of past research and the kind of information that this methodology affords.
The limitations of laboratory-based studies are addressed along with the philosophy of
mind and of causality that are inherent to these studies.  I then overview more recent
studies of analogy and the more qualitative methodologies of these along with the
advantages and limitations of these approaches.  I then explain the approach that I take,
the philosophy behind this approach and the information that can and cannot be gleaned
from this research.
Chapter Four: The phenomenological evidence
A distinction has been made in the literature between behavior ? the observable
part of cognition ? and cognitive structure ? the underlying ontology of mind responsible
for that behavior.  Chapters four and five address these two sides of cognition, chapter
four focusing on cognitive behavior and five on the underlying cognitive structure.  In
this chapter I first review past research on categorization, focusing on the observable
structure of categories, and contrast that with existing research on analogies.  I then
present transcripts from students in science classrooms, study groups, research groups
and faculty and present an argument for the interpretation of analogies in a categorization
8
framework, basing this interpretation on the behavior.  These claims will be contrasted
with other models of analogy, in particular structure-mapping and transfer. The analogies
discussed here will be explored again in the following chapter for their implications on
cognitive structure.
Chapter Five: Cognitive structure and analogies
Having presented phenomenological evidence for analogies as categorization, in
this chapter I focus on the underlying cognitive structure that can account for that
behavior and is consistent with a categorization interpretation of student-generated
analogies.  I begin this chapter with a review of the different approaches that have been
taken in the past to cognitive structure in both education and cognitive science and
present arguments in favor of a manifold ontology of mind.  Analogies presented in the
previous chapter are then revisited for their consistency with this manifold ontology.
Chapter Six: Analogies in the history of science
Though this thesis does not focus on the history of science and the evolution of
scientific theories, several findings from the history of science can be brought to bear in
defense of my claims.  In this chapter, I briefly outline research in the history of science
and cite several theories and ideas that have evolved via analogy.  These ideas
demonstrate that important concepts in science arose from schemas provided by changes
in our political systems and our technology.  I outline these theories and their
development and show how it is consistent with a categorization model of analogy.
Finally, I show how a study in the history and philosophy of science has changed the
definition of a concept and how this new definition, which is rooted in the idea of
9
conceptual change via physical analogy, is consistent with the categorization framework
of generated analogies.
Chapter Seven: Implications for instruction.
There are three main points that I would like to make with regard to science
instruction with this thesis.  The first point reflects my motivation behind studying
student-generated analogies: the use of analogies and analogical reasoning is, in large
part, what it means to do science.  Content knowledge of the science disciplines is
important, but even more so is the ability to think scientifically: to be able to understand
that content knowledge as it relates to theories and experiences that you have, to be able
to create your own models using analogies to concepts that you understand and find
familiar, to understand the implications of these models and negotiate these analogies: in
essence, to create your own knowledge from your own experience in a scientific manner.
If we accept this first claim, that science instruction should emphasize how to create and
negotiate analogies, our conception of what analogies are matters and will influence the
way in which a teacher identifies and responds to analogical reasoning in the classroom.
The second point, then, is that analogies are not indicative of a fixed representation of a
particular concept.  The mind is far more fluid and complex than that.  Understanding
analogies as assertions of categorization requires that instruction be sensitive to this
variability, for categories are inherently variable.  And third, by understanding analogies
as assertions of categorization, questions of transfer ? the holy grail of education ?
become not questions of near and far transfer, but of prototypical and aprototypical
analogies.  How this translates into practice will be explored in further detail in this
chapter.
10
Chapter Eight: Directions for future research and conclusion.
While I make claims about what analogies assert, I have not come to any
conclusions about how this happens.  What habits of mind and what structure of
education and environment can encourage this kind of creative re-categorization of
concepts?  Nor do I explore theories that are not built from analogy or have no obvious
analog or the extremely significant role that community and dialog plays in both
constructing and negotiating an analogy.  In the final chapter, I summarize my
conclusions and introduce these questions and suggest directions for future research on
these topics.
Appendix A-J: Transcripts
Included in the appendix are full transcripts from conversations in which the
analogies presented in the dissertation are taken. These transcripts are referenced in the
chapters, so the reader may find the entirety of the transcript that I had available.  In some
cases, these transcripts span entire class-periods of discussion; in other cases, only a short
piece of that conversation exists.
Appendix K: Young Children?s Analogies and Transcripts
Chapters 4 and 5 presented several analogies from student conversations in
science, and chapter 6 details analogies by professionals in science and related fields.  In
the appendix I address analogies that were not included in previous chapters, particularly
those asserted by younger students.  These students are able to generate analogies (for
example, magnets are like clay) but lack explicit awareness of that analogy.  These
students are confused by the mapping to the target, or confuse the assertion with literal
class inclusion (so that magnets are not like clay but are clay).  This confusion is
11
consistent with findings from Karmiloff-Smith (1992) on representational redescription
and is predicted by Glucksberg?s et al. (1997) dual reference theory.  Structure-mapping
and other theories that limit analogies to pairwise analyses of the target and base would
not predict and cannot account for the confusion young students show between
statements of class-inclusion and statements of superordinate-class-inclusion.   This
chapter presents theories and findings of representational redescription, dual reference,
semantic fields and polysemy and argues that, in light of those findings, the mistakes and
confusions that young students display with analogies are consistent with a categorization
interpretation of analogy.
12
Chapter 2: Review of the Literature
Introduction and Overview
In the latter half of the 20
th
 century research in two fields, categorization and
analogy, has challenged long held classical views of these topics.  Up until the nineteenth
century, discussions on analogy, and in particular metaphor, were insistent on one theme:
analogies are decorations of speech; they do not contribute to the cognitive meaning of
the discourse, but instead lend it color, vividness, emotional impact, or accessibility.
Thus it was characteristic of the Enlightenment philosophers and their predecessors, such
as Hobbes and Locke, to insist that though philosophers may sometimes have good
reason to communicate their thoughts with metaphors, they should do their thinking
entirely without metaphors.  Only by using nothing but unambiguous, literal language
could knowledge be gained and communicated properly (Hobbes, 1651):
To conclude, the light of humane minds is perspicuous words, but by exact
definitions first snuffed, and purged from ambiguity; reason is the pace; increase
of science, the way; and the benefit of mankind, the end. And, on the contrary,
metaphors, and senseless and ambiguous words are like ignes fatui; and reasoning
upon them is wandering amongst innumerable absurdities; and their end,
contention and sedition, or contempt.
Similarly, John Locke, in the Essay Concerning Human Understanding, criticized
imprecise and ambiguous ?civil? language and proposed a proper and well-defined
philosophical language that ?may serve to convey the precise notions of things, and to
express in general propositions certain and undoubted truths, which the mind may rest
upon and be satisfied with in its search after true knowledge? (Locke, 1686).
What characterizes almost all theories of metaphor from the time of the
13
Romantics up through the twentieth century is the rejection of this theme.  Metaphors, it
has been argued, are not cognitively dispensable decorations.  They contribute to the
cognitive meaning of our discourse and they are indispensable, not only to philosophical
discourse, but to ordinary, and even scientific discourse.  Nietzsche went so far as to
argue that all speech is metaphorical and truth is ?a mobile army of metaphors?
(Nietzsche, 1873).  Lakoff and Johnson?s seminal work, Metaphors We Live By (1980),
begins with the statement of traditional interpretations of metaphor and summarily reject
this interpretation:
Metaphor is for most people a device of the poetic imagination and the rhetorical
flourish ? a matter of extraordinary rather than ordinary language? We have
found, on the contrary, that metaphor is pervasive in everyday life, not just in
language but in thought and action.  Our ordinary conceptual system, in terms of
which we both think and act, is fundamentally metaphorical.  (p3)
This shift in understanding of the importance of analogies ? from ornamental to
fundamental ? has resulted in the emergence of research in the structure and
comprehension of analogy.  The results of this research will be addressed later in this
chapter.
Categorization, too, has experienced a dramatic reinterpretation in the 20
th
century.  As Lakoff summarizes in Women, Fire and Dangerous Things: What
Categories Reveal about the Mind (p 6):
From the time of Aristotle to the later work of Wittgenstein, categories were
thought [to] be well understood and unproblematic.  They were assumed to be
abstract containers, with things either inside or outside the category.  Things were
assumed to be in the same category if and only if they had certain properties in
common.  And the properties they had in common were taken as defining the
category.
This classical theory was not the result of empirical study.  It was not even a
subject of major debate.  It was a philosophical position arrived at on the basis of
14
a priori speculation.  Over the centuries it simply became part of the background
assumptions taken for granted in most scholarly disciplines.  In fact, until very
recently, the classical theory of categories was not even thought of as a theory.
Current theories on categorization have shifted from this Aristotelian view towards what
has become known as the ?prototype? view, the importance of which was first established
by Rosch (1973).  This shift in our understanding of categorization can be traced to the
Whorfian hypothesis, which claims that language is not merely a medium for the
expression of our thoughts but that ?linguistic patterns themselves determine what the
individual perceives in this world and how he thinks about it.  Since these patterns vary
widely, the modes of thinking and perceiving in groups utilizing different linguistic
systems will result in basically different world views? (Fearing, 1954).  Rosch?s tests of
the Whorfian hypothesis led her to study the nature of color categories among speakers of
languages without a blue-green color distinction.  In these studies, she identified structure
within categories, with some members of a category being seen as more or less
prototypical than others.  This recognition, with its associated research paradigm,
launched a field of study into categorization that has come to be characterized as the
?prototype? view.  More recent developments in this field have extended the types of
categories studied from ?common, or stable, categories to ad hoc categories? (Shen,
1992) ? from Rosch?s study of color categories to Barsalou?s study of ?things to take
from your house during a fire.?  The results of this research and its implications on
cognitive structure will be discussed later in this chapter.
It has not gone unnoticed that these two research programs, analogy and
categorization, are both studies of similarity and each may have insights that might
inform the other, but as late as 1992 it was remarked:  ?Despite the obvious affinity of
15
these two fields of research, the link between them has received little attention in
cognitive psychology or in other disciplines? (Shen, 1992).  Shen and others (most
notably Glucksberg and Keysar), have, during the 1990?s, dedicated study to linking
these two research enterprises.  However, the emphasis of this research has been on
metaphors and their interpretation, developing ?a coherent and unified framework by
assuming that metaphor interpretation is, in fact, a process of ad hoc category formation?
(Shen 1992), and ?a basis for a theory of metaphor comprehension, and also clarifies why
people use metaphors instead of similes? (Glucksberg and Keysar, 1990).  It is the aim of
this thesis, in the context of scientific discourse, to extend the link between metaphor and
categorization to include analogical statements and to argue that the statement of an
analogy ? not merely its comprehension ? is an assertion of categorization.  Below I will
outline the existing research on analogies and categorization, along with the research that
attempts to link these two fields of study, in more depth.
Analogies
Analogies in philosophy and the philosophy of science
The modern views of analogy can be traced to the influences of philosophers Max
Black (1962) and Mary Hesse (1966).  Black?s work built on that of Ivon Richards?
(1936) work in the field of rhetoric, who proposed a set of useful terms for talking about
metaphors (the ?topic? or ?tenor,? the ?vehicle,? and the ?ground?) and a theory of how
metaphors function.  This theory, called the tensive view, emphasized the conceptual
incompatibility, or ?tension,? between the terms (the topic and the vehicle) in a metaphor
(Ortony, 1993).  Black?s interpretation of metaphor, referred to as the interaction model,
claims that metaphor is a cognitively irreducible phenomenon that works not at the level
16
of word combination, but much deeper, arising out of the interactions between the
conceptual structures underlying words.  In comparing two concepts, the significance of
one concept is not merely projected onto another, but the two interact, altering the
perception of both concepts.  Explaining this theory in metaphorical terms, Black offers
the following analogy (1962):
Suppose I look at the night sky through a piece of heavily smoked glass on which
certain lines have been left clear. Then I shall see only the stars that can be made
to lie on the lines previously prepared on the screen, and the stars I do see will be
seen as organized by the screen's structure. We can think of the metaphor as such
as screen and the system of 'associated commonplaces' of the focal world as the
network of lines upon the screen.
In his theory, Black claims that the following are true of all metaphors (synopsis from
Ortony, 1993):
? Metaphors consist of primary and secondary components. (In the statement ?X is
like Y? X is referred to as the primary and Y the secondary.  Other literature refers
to these as the tenor, vehicle, target, etc., and there is no standard convention
across disciplines.  I will use base and target, recognizing that these make
implications about the nature of analogical reasoning.)
? The significance of the base subject is not so much as a ?thing? as it is a system.
? The associated implications of the base are projected onto the target.
? This projection selects, organizes, emphasizes and suppresses features of the
target component.  Through this interaction of the two subjects there is a selection
of properties, an implication on the target, and, reciprocally, an implication back
on the secondary.
This interpretation of metaphor, which is still central to many theories of metaphor, was
incorporated four years later by Mary Hesse, a philosopher of science, whose ?treatise on
analogy in science argued that analogies are powerful forces in discovery and conceptual
change? (Holyoak, Gentner and Kokinov, 2001).  This treatise, Models and Analogies in
Science (1966), argued that ?the deductive model of scientific explanation should be
modified and supplemented by a view of theoretical explanation as metaphoric
17
redescription of the domain of the explanandum.? Additionally, she coins three new
terms in analogical reasoning: the positive, negative and neutral analogies.  In positing an
analogy, ?positive analogies? are known to transfer from the base to the target, ?neutral
analogies? are possible elements of the base that are present in the target, and ?negative
analogies? are elements that do not transfer.
Analogies in cognitive science
In the 1980?s, concurrent with Lakoff and Johnson?s work on metaphor, there was
a shift in research from four-term analogies (of the type found in standardized tests ?
a:b::c:d) to more complex analogies, such as those found in science and language.
According to Holyoak, Gentner and Kokinov (2001), who were themselves pivotal in this
shift,
This exploration led to a more general focus on the role of experience in
reasoning and the relationships among reasoning, learning, and memory, giving
rise to an approach termed ? case-based? reasoning (e.g., Kolodner 1993).  In
contrast to rule-based approaches to reasoning (the approach that was dominant in
AI at the time), case-based reasoning emphasized the usefulness of retrieving and
adapting cases or analogs stored in long term memory when deriving solutions to
novel problems.
This work continued to adopt Black?s model of metaphor, most significantly the system
of relations in the base of the analogy.  Among the best-known models developed at this
time is Gentner?s theory of structure-mapping (1983).  In this model
the central idea is that an analogy is a mapping of knowledge from one domain
(the base) to another (the target) such that a system of relations that holds among
the base objects also holds among the target objects.  In interpreting an analogy,
people seek to put the objects of the base in one-to-one correspondence with the
objects of the target so as to obtain the maximal structural match? Thus, an
analogy is a way of aligning and focusing on relational commonalities
independently of the objects in which those relations are embedded. (Gentner and
Jeziorski, 1993.)
18
From this model a computational model, the Structure Mapping Engine, was developed
by Falkenhainer, Forbus and Gentner (1990).  Holyoak and Thagard (1989) have a
similar (ACME) program for analogies that is instead based on a connectionist network,
but the underlying model of analogy is the same.  It is important to note that these models
will take two systems (for example, the solar system and the Rutherford model of the
atom) and, as Falkenhainer, Forbus and Gentner (1990) describe, construct ?matching
algorithms consistent with [the] theory.?
In contrast to these models are those posited by Douglas Hofstadter and the Fluid
Analogies Research Group (FARG) at Indiana University, beginning with Melanie
Mitchell?s (1990) dissertation.  Their computational models do not interpret a given
analogy, but generate novel analogies.  These models, Copycat, LetterSpirit and Metacat,
are based on the thesis
that mental activity consists of many tiny independent events and that the seeming
unity of a human mind is merely a consequence of the regularity of the statistics
of such large collections of events. Thus the metaphor of the ?intelligent ant
colony? and the image of ?active concepts??have inspired our models for over
two decades. The models all involve the nondeterministic interaction of many tiny
events that take place in simulated parallel. The models also feature both long-
term and short-term memories, the former of which houses permanent concepts,
and the latter of which is like a stage on which temporary mental structures are
built, modified, and eventually razed. Events in each memory profoundly
influence the multiple tiny parallel processes, and in that way, each memory
affects what goes on in the other.  (Hofstadter, 2004)
This thesis concerns itself with the creation of analogies, as opposed to their
interpretation.  I will show in later chapters that structure-mapping cannot account for
analogy creation, and argue that the Hofstadter?s interpretation of cognition as ?many
tiny independent events? is consistent with modern views on categorization, which are
consistent with observed properties of student-generated analogies in science.
19
Analogies in linguistics
Concurrent with this shift in cognitive science (and, in particular, artificial
intelligence) from rule-based to case-based reasoning, the field of linguistics began a
departure from traditional Chomskian emphasis on linguistic competence towards ?an
increasing concern with linguistic performance and pragmatics? (Ortony, 1993).
Representative of this shift is Lakoff and Johnson?s Metaphors We Live By.  In this book
the authors argue that our ideas are not only referred to linguistically through metaphors
(defined by the authors as ?understanding and experiencing one kind of thing in terms of
another?) but also actually conceptualized in metaphorical terms.  The choice of a
secondary subject in these metaphors is one that allows the speaker to conceptualize ?the
nonphysical in terms of the physical ? that is, we conceptualize the less clearly delineated
in terms of the more clearly delineated? the prime candidates for concepts that are
understood directly are the simple spatial concepts, such as UP.?  This claim is evident in
physics, where we speak of high and low potentials and energy and conceptualize atomic
forces as wells.
As our choice of metaphor will reflect our conceptualization of the phenomena,
our metaphorical choice may change depending on the context of the problem.
Consider the following statements, ?Light consists of particles? which apparently
contradicts ?light consists of waves,? but both are taken as true by physicists
relative to which aspects of light are picked out by different experiments.  (Lakoff
and Johnson, 1980)
There is enormous utility in being able to conceive of light in different ways ? to
categorize light as a particle or as a wave.  As Lakoff and Johnson state (1980),
understanding our experiences in terms of objects and substances allows us to
pick out parts of our experience and treat them as discrete entities or substances of
a uniform kind.  Once we can identify our experiences as entities or substances,
we can refer to them, categorize them, group them, and quantify them ? and, by
20
this means, reason about them.  When things are not clearly discrete or bounded,
we still categorize them as such, e.g. mountains, street corners, hedges, etc.
That is, there is a relationship between these categories and metaphorical language: our
language, which Lakoff and Johnson demonstrate is profoundly metaphorical, allows us
to categorize phenomena.
Analogies in general, and metaphors in particular, are a topic of widespread
interest and debate in linguistics at this time.  Some of these debates will be further
reviewed in later chapters.  For a more complete review, see The Analogical Mind
(Gentner, Holyoak and Kokinov, 2001).
Analogies in science and science education
?I think it would also be practical to design a curriculum based on an inquiry
into the use of metaphor.  Unless I am sorely mistaken, metaphor is at present
rarely approached in school except by English teachers during lessons in poetry.
This strikes me as absurd, since I do not see how it is possible for a subject to be
understood in the absence of any insight into the metaphors on which it is
constructed.  All subjects are based on powerful metaphors that direct and
organize the way we will do our thinking.
-Neil Postman, Conscientious Objections:
Stirring up Trouble about Language,
Technology, and Education
A significant feature of the existing research on the role of analogies in instruction
is that the focus has been on analogies drawn by the teacher and explained to the
students.  Research paradigms for constructing and testing models of analogical
reasoning have similarly focused on the comprehension of analogies that have been
created by the researcher.  Very little work has looked at generated analogies.  This
paradigm is indicative of a tradition of science education in which the students are
interpreting instruction from the teacher, and not discussing and debating their own views
of science.  As reforms in science education call for greater attention to student ideas and
21
student reasoning, the existing theories on analogy interpretation are of less use and there
is a greater demand for understanding analogies as they are generated.  Here I provide a
brief overview on the research on the role of analogies in the classroom, the role of
analogies in science, and the disconnect between what scientists do and what is attended
to in classrooms.
In a pioneering work on scientific analogies in instruction, Flowing Waters or
Teeming Crowds: Mental Models of Electricity (Gentner and Gentner, 1983), students
were given instruction in circuits using one of two different analogies: one visualized the
flow of current as a crowd of electrons, one a flowing water-like substance.  (Experts use
both models to represent different features of current.)  Tests given post-instruction were
designed to test whether analogy is an important source of insight (the Generative
Analogy hypothesis) by ?asking whether truly different inferences in a given target
domain are engendered by different analogies? (Gentner and Gentner, 1983).  The
answers that students gave were overwhelmingly representative of the analogical model
they had been taught.
Clement?s work in physics education has focused extensively on analogical
reasoning in the physics classroom, with attention primarily on explicit analogies
generated by teachers and interpreted by the students.  In Clement (1992), he describes
?bridging analogies.?  These analogies are instantiated by the teacher and are used to
motivate the students to see a likeness between two phenomena (say, the force provided
by a spring supporting a book and a similar force from a table supporting a book) by
providing intermediate analogies (a tight spring, a flexible board).  The idea that a student
will recognize ?A is like B? and ?B is like C? but not recognize that ?A is like C? without
22
explicit instruction supports a model of categorization in which ?family resemblance? to
the category prototype defines the membership of the category.  In this way, prior
research on analogical reasoning can facilitate the link between analogies and
categorization.  In the following section I will provide an overview on research into
categorization.
More recent research on the role of analogies in physics continue in the paradigm
of instruction by analogy:
Taber, in When the Analogy Breaks Down: Modeling the Atom on the Solar
System (2001), notes that:  ?Analogy is one of the most potent tools in a teacher's
repertoire and has been recognized as a common feature of quality science teaching.?  He
voices concerns regarding the utility of a solar-system model of the electron and the
students? lack of knowledge regarding the solar system in drawing such an analogy.
Mould (1998) is one of many instructional analogies that are frequently
introduced in Physics Education.  In this article, he suggests the use of an analogy to
resolve the puzzle of the lost energy when two capacitors are joined together by
comparing this scenario to the concept of water flowing between two tanks.
Harrison and Treagust (1999) have explored the relationship between the
analogies we tell our students and the models students construct from these analogies,
arguing that ?students do not interpret scientific analogical models in the way we
intended? and investigate potential factors in the consistency of analogy use in models of
atoms.
Notable exceptions to investigations on teacher-generated, didactic analogies
include Duit, Roth, Komorek and Wilbers (2001) who ?studied analogy generation and
23
development and analogical reasoning with 25 German tenth graders in a physics class.
Results show the advantages and disadvantages of using analogies in promoting
conceptual change and as a teaching technique.?  Their focus, however, was on the utility
of analogy in conceptual change and not the generation of analogy as a goal in itself.
And Yerrick, Doster, Nugent, Parke and Crawley (2003) who investigated the role of
analogies in pre-service teachers? conversations and argue for analogy as part of
preservice teachers? conceptual development.
Turning to the role of analogy in science, and not just the science classroom,
Dunbar (2000, 2001) has conducted research on ?in vivo analogies,? going to research
groups and listening to the ways in which they use analogies and analogical reasoning in
their work and discourse.  Analogies, he finds, are ?frequent in science and in all aspects
of human thinking.?  They are ubiquitous and crucial to the ways experts reason about
science.  One goal of science education could be considered to develop habits and skills
that scientists employ in scientific reasoning.  The focus in the literature on analogies as a
means of arriving at correct conceptual understanding ignores the import of developing
analogical reasoning as a skill to be mastered in and of itself.  As May, Hammer and Roy
(2004) noted, ?inasmuch as expertise at inquiry supports? students and scientists
developing conceptual understanding, young children?s development of understanding
and abilities for analogical reasoning will serve them better than learning the content
knowledge of an expert.?
Summary of analogy research
The main theme from this research that I will address is that the majority of
research on analogies, and educational analogies in particular, has concerned how people
24
interpret analogies and not on the creation of analogies and its implications on mental
models of concepts.  If analogical reasoning is an important feature of scientific
reasoning, then one goal of science education should entail fostering the use of analogies
and developing students? abilities in this realm.  As educational researchers,
understanding analogical reasoning, what assertions analogies make, what they reveal
about the mind, are of importance ? not because such findings allow us to better convey
content, but because they inform us about a skill that is, in itself, one we should foster in
our students.
Categorization
Overview
Categorization is not a matter to be taken lightly. There is nothing more basic than
categorization to our thought, perception, action, and speech, Every time we see
something as a kind of thing, for example, a tree, we are categorizing. Whenever
we reason about kinds of things ? chairs, nations, illnesses, emotions, any kind of
thing at all ? we are employing categories. Whenever we intentionally perform
any kind of action, say something as mundane as writing with a pencil,
hammering with a hammer, or ironing clothes, we are using categories.
-George Lakoff,
from Women, Fire and Dangerous Things
In 1987, George Lakoff published Women, Fire, and Dangerous Things: What
Categories Reveal about the Mind.  This work thoroughly summarizes the research on
categorization up to that time and interprets the philosophical implications of these
findings.  Rosch, who tied together existing research on categorization and created an
experimental paradigm for investigating categorization, began from the opposite end ?
she received her undergraduate training in philosophy and brought this to bear on
psychology.  Her honors thesis was on Wittgenstein?s 1953 treatise, Philosophical
25
Investigations (Scharmer, 1999), a work that is credited with ?the first major crack in the
classical theory [of categorization]? (Lakoff, p16).
The classical view of categories held that there were rules of membership and if
an item met these rules then it was a member of the category (or, there are properties that
define a category and all members must share these properties).  In this view categories,
therefore, were seen as binary with no internal structure: an item either was or was not a
member of that category, and the research paradigm for categorization was to define the
rules or properties.  Wittgenstein?s work addressed the fact that certain categories, such
as ?game,? do not fit this classical description.  There is no one property that all games
share, but rather there are family resemblances between games, so that ?chess and Go
both involve long term strategy? chess and poker both involve competition.  Poker and
old maid are both card games? (Lakoff, 1987 p. 16).  Additionally, these categories had
no fixed boundary ? they could be extended and redefined as new games are introduced,
or as previous games shift their context.  About this time, a similar recognition of
inconsistencies in our conception of categories occurred in mathematics.
Categorization and fuzzy sets
In mathematics, the analog of a linguistic category is the set.  In classical set
theory an instance either belongs to a set or it does not; there is no middle ground.  An
item?s membership in a set is determined by whether or not it has the properties that
define that set, called the class intension.  For example, if the set is the class of all objects
that are green, the intension of that set is the property green.  This clean definition of the
set allows for simple rules to be associated with them.  The set of objects that are either
green or square is the union of the sets of green objects and square objects.  The set of all
26
things that are green and square is the intersection of the sets of green objects and square
objects.  However, a complication associated with set theory occurs at the border between
the set and non-set.  For an intension that splits the set of all objects in two, such as
greenness, items will either belong to the set of green objects, or to the set of non-green
objects, but there is an ambiguity for items that falls at this boundary between green and
not-green.  Applying this simple mathematical structure of sets to human-created
categories does not work.  There is logical inconsistency in assigning such items to both
sets and to neither set.  And the commonly accepted solution to this problem is the ?law
of excluded middle,? which simply forbids there being an object that falls exactly at the
border of set and not-set.  Zadeh (1965) saw this solution as indicative of the fact that
classical set theory does not capture the way in which we experience the world.  Things
come in gradations: some animals are birds, and some animals are mammals, while some,
like the platypus, with its bill, webbed feet and fur, seem to fall somewhere in between.
Statements are not always either true or false.  The standard solution to the
incompatibility of set-theory sets and human categories is to claim that mathematics, with
its rigid structure and well-defined systems, does not apply to many real-world problems.
Zadeh's solution was to allow for gradual sets and ?fuzzy logic.?  The amendment he
made to standard sets was to allow membership in a set to be a non-binary concept, and
then extend the operations on ordinary sets to account for this allowance.  If an element x
has membership in set A with value v and in set B with a value w, then the operations on
sets are adjusted in the following way:
Intersection: The value of x in A?B is the minimum of v and w.
Union: The value of x in A?B is the maximum of v and w.
27
Complement: The value of x in the complement of A is 1-v.
This theory allows for a category of ?P and not P? ? for example, the category defined by
?an apple that is not an apple.?  The crabapple and Adam?s apple are both judged to be
members of this category.  Zadeh?s fuzzy set theory reflected a change in the conception
of categories that had been developing in anthropology.
Rosch and prototype theory
About the same time, anthropologists, influenced by the Whorfian hypothesis
(1956), assumed that color categories were arbitrary and different languages could carve
the color spectrum in different, arbitrary ways.  However by the late 1960?s it was found
that, though different languages do vary in the number and kinds of colors they name,
there is regularity in color categories among different cultures (Berlin and Kay, 1969).
Speakers of different languages that disagree on color category boundaries will agree on
which colors were good examples of these categories.  Rosch, who was conducting
research on the Dani in New Guinea, began looking at their color categories.  She depicts
the prior research on categorization in this way:
When psychologists did research on concept learning, they used artificial
concepts and sets of artificial stimuli that were constructed so that they formed
little micro-worlds in which those prevailing beliefs about the nature of categories
were already built in. Then they?d do their learning experiments. But what they
found out in terms of the nature of categories was already a foregone conclusion
because that was what they had already built into it (Scharmer, 1999).
Rosch argued that, because of the way the perceptual system works, certain areas in the
color space are more salient than others, and that those salient colors are first noticed,
most easily remembered, and become prototypes around which color categories form in
cultures. The Dani had only two basic color terms, dark and light, making this culture
28
ideal for testing the hypothesis.  By teaching them novel color categories, structured
around natural and unnatural color schemes, Rosch found that the Dani remembered the
hypothesized ?universal prototype? colors better than other colors, and it was much easier
to learn categories structured around those colors than categories structured some other
way.  Further research extended this to shapes and other categories, and supported her
thesis that
categories form around and (or) are mentally represented by salient or information
rich or highly imaginable stimuli which become prototypes for the category.
Other items are judged in relation to these prototypes; that's the way they form
gradients of category membership.  There don't need to be any attributes which all
category members have in common, no defining attributes, and category
boundaries don't need to be definite (Scharmer, 1999).
In the research that followed, Rosch (1975) showed that certain category members were
judged to be better examples of the category than others.  A robin is judged to be a better
example of the set birds than a penguin, although, strictly speaking, both are birds.  This
psychological rendition of Zadeh?s fuzzy set theory established fuzzy logic as an
important component of AI research.  In addition, Rosch?s research established
experimental paradigms for investigating categorization, attending to features such as the
following (using the category ?bird? for illustration):
? Direct rating. (How birdlike is this?)
? Reaction time.  (Show a picture and ask: Is this a bird?)
? Producing an example. (Draw a bird.)
? Asymmetry of similarity. (Are ducks like robins?  Are robins like ducks?)
? Asymmetry of generalization. (Robins get the flu; do ducks?  Ducks get the flu;
do robins?)
29
In this paradigm, a prototype will receive a high rating, low reaction time, and resembles
the example produced.  From this research, properties of categories were determined.
These are detailed in the following section.
Properties of categories
It was readily apparent that the structure of real categories, as researched by
Rosch and others, was not consistent with the classical view.  Categories, which in the
classical view were devoid of any internal structure, were shown to possess both
centrality gradience (the idea that some categories have members that, though clearly
within the category boundaries, are more or less representative of the category) and
membership gradience (the idea that some categories have degrees of membership, so
that the distinction between member and non-member, the category boundary, is not
clear).  Members of many categories were related to one another in a ?family
resemblance? manner, so that no one property is common among all members.  The
structure of categories leads to ?a basic psychological asymmetry: the less prototypical
category member is conceived of as closer (i.e., more similar) to the more prototypical
member than vice versa? (Shen, 1999).  People will more readily compare a non-
prototype to a prototype and will more likely generalize from the prototype to the non-
prototype than vice versa (Rips, 1975).   There is a primary level of categories, known as
the basic level, that are ?primary with respect to the following factors: gestalt perception,
image formation, motor movement, knowledge organization, ease of cognitive
processing, and ease of linguistic expression? (Lakoff, 1980).  The most central members
of a category can function as metonyms for that category ? a property common to many
languages.  For example, American Sign Language has no sign for the category jewelry
30
and in ASL this category is referred to by listing the prototypical members (Newport and
Bellugi, 1978 p 62); the Hopi call all trees ?cottonwood,? the name of the most common
deciduous tree in their habitat (Trager, 1936-9); Shoshoni speakers refer to large birds in
general as well as to eagles themselves as eagles (Hage and Miller, 1976).
From this data, initial theories on categorization argued for interpreting category
membership and structure as degree of similarity to the category prototype.  However,
further work showed that categories are not defined solely by family resemblance to a
prototype, but have an intellectual and ecological basis.  Barsalou?s (1983) studies of ?ad
hoc? categories, categories that cannot be interpreted as fixed cognitive structures, such
as ?foods not to eat when on a diet,? or ?things to do at a convention,? found that
members of these categories retain the graded structure and typicality effects that Rosch
found.  However, these categories did not necessarily show a family resemblance to the
prototype.  Instead, Barsalou argues, these categories are goal-oriented; a chocolate cake
has little resemblance to peanut brittle, but abstaining from these satisfies the goal of
eating as few calories as possible.  Similarly, research was beginning to reveal that
similarity alone could not account for even the more typical, stable categories.  For
example, ?the claim that something is a dog does more than assert some degree of
similarity to a prototype; it also appeals to our underlying intuitions and beliefs about the
nature of animals.  The effect of these beliefs is to make some similarities between
objects decisive and others simply irrelevant? (Neisser, 1987 p3).  The claim that
similarity alone is an explanation, according to Goodman (1972), is ?a pretender, an
imposter, a quack.  [Similarity] has, indeed, its place and its uses, but is more often found
where it does not belong, professing powers it does not possess.?  In Murphy and
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Medin?s (1985) paper, ?The Role of Theories in Conceptual Coherence? they argue
against similarity arguments, claiming that
Current ideas, maxims, and theories concerning the structure of concepts are
insufficient to provide an account of conceptual coherence.  All such accounts
rely directly or indirectly on the notion of similarity, and we argue that the notion
of similarity relationships is not sufficiently constraining to determine which
concepts will be coherent or meaningful.  These approaches are inadequate, in
part, because they fail to represent intra- and inter-concept relations and more
general world knowledge.  We propose a different approach in which attention is
focused on people?s theories about the world.
The argument entails that
categorization assumes a (folk) theory on the part of the person who is engaged in
that particular cognitive process.  This theory ?guides? him in selecting the
relevant features and the relevant feature correlations; in other words, noticing
features and feature correlations is not an ?objective? process based on similarity,
but is instead theory-dependent.?  (Shen, 1992)
The effect of these findings is to make some similarities between objects decisive and
others irrelevant.  With these and other findings, Rosch eventually
came to the conclusion that prototype effects, defined operationally by
experiment, underdetermined mental representations.  The effects constrained the
possibilities for what representations might be, but there was no one-to-one
correspondence between the effects and mental representations.  The effects had
?sources,? but one could not determine the sources given the effects.  (Lakoff p
43)
An alternative to prototype theory is described below.
Idealized cognitive models
As a theory to explain the prototypes that Rosch first documented, Lakoff, in
Women, Fire and Dangerous Things claims that ?prototype effects result from the nature
of cognitive models, which can be viewed as ?theories? of some subject matter.?  (p. 45)
Lakoff terms these models ?idealized cognitive models,? or ICMs, and suggests that, to
the degree to which the model does not represent reality, these ICMs will lead to
32
categorization and prototype effects.  A classic example from linguistic research of
categorization, prototype effects and gradience of membership in a category is the term
bachelor (Fillmore, 1982).  While most people will define a bachelor as an unmarried
adult male, certain unmarried adult males are not representative members of the category
of bachelors.  Lakoff (1987) argues that
bachelor is defined with respect to an ICM in which there is a human society with
(typically monogamous) marriage, and a typical marriageable age.  The ideal
model says nothing about the existence of priests, ?long-term unmarried
couplings,? homosexuality, Moslems who are permitted four wives and have only
three, etc.  With respect to this ICM, a bachelor is simply an unmarried adult
man.
This idealized model, however, does not fit the world very precisely.  It is oversimplified
in its background assumptions. The person referred to deviates from prototypical
bachelorhood if either the ICM fails to fit the world perfectly or the person referred to
deviates from being an unmarried adult male.
Under this account bachelor is not a graded category.  It is an all-or-none concept
relative to the appropriate ICM.  The ICM characterizes representative bachelors.
One kind of gradience arises from the degree to which to ungraded ICM fits our
knowledge (or assumptions) about the world.  (Lakoff, 1987)
ICM?s can closely match the world, in which case the categories that you developed to
create this ICM are robust categories with little gradience.  The idea of the ICM has been
further parsed and is perhaps best represented by schema theory.  Schemas are short
?scripts? or stories that we have about the world and the way it works: event schemas that
are abstracted from our experience of certain events, image schemas that provide
structure for conceptualizations ?  ?schemas of intermediate abstractions [between mental
images in abstract propositions] that are readily imagined? (Palmer, 1996 p. 66) ?  and
proposition schemas: abstractions that act as models of thought and behavior and specify
33
?concepts and the relations which hold among them.?  (Quinn, 1987)  It is only within a
particular schema that a category is meaningful, and these categories become les
meaningful and exhibit a graded structure to the degree that the schema in which they are
defined does not apply.  A claim of analogies as assertions of categorization then entails
analogies as instantiations of particular schemas.
Metonymy, cognitive models and phenomenological-primitives
Phenomenological primitives (p-prims) (diSessa 1993) were developed to address
the ?preconceptions? of students in problem solving in physics.  They are ?the intuitive
equivalent of physics laws; they may explain other phenomena, but are not themselves
explained with the knowledge system.?  As defined by diSessa, p-prims are ?cued to an
active state on the basis of perceived configurations, which are themselves previously
activated knowledge structures.?  In this way p-prims are elements within larger models.
P-prims ?often originate as minimal abstractions of common phenomena,? and are
?nearly minimal memory elements, evoked as a whole.?  By way of example, consider
one class of p-prims: the ?constraint cluster.?  This class includes bouncing, supporting,
guiding, clamping, and carrying.  These p-prims are not fundamental for a physicist (all
can be explained in terms of forces) but are often elicited in conversations with students
as explanations for physical behavior.  The p-prims have a ?schematization? such as, for
the ?supporting? p-prim, ??strong? or stable underlying object keeps overlaying and
touching object in place.?  (diSessa, 1993 p. 216)
That a p-prim has a full schematization but is often represented only partially by a
particularly salient feature of the schema (e.g., ?supporting? as an explanation entails two
objects, one of which is strong or stable and underlies another object which touches it) is
34
indicative of the schematization of a p-prim being an idealized cognitive model.  This
single salient feature of the schema is what Lakoff refers to as a metonym for the
idealized cognitive model.  Lakoff presents the following example to explain metonymy
is the context of ICMs (Lakoff, 1987):
A linguist who does fieldwork on Ojibwa, a Native American language... asked
speakers of Ojibwa who had come to a party how they got there.  He got answers
like the following (translated into English):
-I started to come.
-I stepped into a canoe.
-I got into a car.
He figured out what was going on when he read Schank and Abelson's Scripts,
Plans, Goals, and Understanding (1977).  Going somewhere in a vehicle involves
a structured scenario (or, in our terms, and ICM)... In Ojibwa it is conventional to
use the embarcation point of an ICM.
That is to say, the embarcation point is a metonym for the entire structured scenario, or
ICM.  English, too, uses a point of the journey to refer to the whole.  Typical English
responses to the question of ?How did you get here?? may be: ?I have a car,? or ?I
biked.?  Neither comment conveys the entire journey, but chooses one part to represent
the whole.  This is only possible because we have a model for the journey, and one part
can, metonymically, elicit the whole.  In the same way, phenomenological primitives as
explanations for physical phenomena are only possible because of a larger schema.
The claim I will make, that the p-prim and schema precedes or is in some way
more fundamental than the analogy itself, is echoed in diSessa?s studies involving the
?Montessori bell conundrum.?  In this problem, students are presented with bells made of
the same material, same length, same height, but varying widths.  Almost without
exception students predict (erroneously) that the thicker bells will have a lower pitch.
DiSessa reports:
35
Although most subjects were ready with analogies ? church bells compared with
jingle bells, xylophones, musical instruments of various sizes ? I was struck that
some initially could not produce any example of the phenomenon they identified
to be at the root of the situation.  This, along with the rapidity and expressed
certainty of responses, heightened my confidence that a p-prim (or several) was at
stake rather than analogy.
In following chapters, I will present data that argue that generated analogies stem from  a
set of schemas or p-prims.
Summary of categorization research
The main points I will be taking from categorization research are the established
properties of categories (including graded structure, asymmetry of generalization,
prototypes, family resemblance, and metonymy), that categories can be considered ad
hoc constructions (and still retain the characteristics of a category), that theories,
expectations and goals underlie our construction of some categories, and that graded
structure in categories is a reflection on the degree to which these theories match the
?real? world.
Analogies as Categorization
Metaphors as category-inclusion statements
Consider the following statements, one typically considered categorization and
the second analogy:
1. This ball bearing is a mass.
2. In circuit with an inductor, capacitor and resistor, the inductor is like a mass.
The first is a statement that the ball bearing is a member of the category ?mass.?  The
second, however, is not a statement that the inductor is a member of that category, but, I
argue, by drawing the analogy we are suggesting that the inductor and that the members
36
of the ?mass? category share some categorical grouping in common.  One could
characterize that category ?things which slow the rate of change? (this characterization is
referred to as a ?ground? in a metaphorical comparison).  This category is rarely referred
to and therefore is not a stable category with its own name, but more of the ad hoc type
category that Barsalou (1983) introduced ? a spontaneously constructed category that is
structured by theories or goals.  Glucksberg and Keyser (1990) were the first to identify
this relationship and have argued for interpreting metaphorical assertions as categorical
assertions, claiming that
When people use metaphors, they are saying exactly what they mean.  When, for
example, someone says that ?Sam is a pig,? that is precisely what is meant; that
the person designated by the name ?Sam? is a member of the superordinate
category referred to by the word ?pig.?
Glucksberg and Keyser argue that the choice of the secondary subject in a metaphor (pig
in ?Sam is a pig?) reflects a tendency of languages to have names for basic level objects
but not for superordinate categories.  Such examples can be seen in the English language,
as in the aphorisms ?Boys will be boys? or ?Cambodia has become Vietnam?s Vietnam?
(or, as mentioned recently in the 2004 election, ?Florida does not want to be the next
Florida).  These expressions use a ?single referring expression? in two distinct ways, to
allude to the entity itself and to refer to the category? that this entity has come to
exemplify? (Glucksberg and Keyser, 1990 p. 411).  (For example, ?eagle? in Shoshoni
and ?cottonwood? in Hopi, as mentioned in a previous section.)  When the base of an
analogy (termed the vehicle in the context of metaphor) is used as both an exemplar and
as an ad hoc name for a category, Glucksberg et al. (1997) call this linguistic move ?dual
reference.?  As an example of dual reference,
37
the phrase ?a responsibility is a shackle? can be used to refer to the concrete,
physical device that is made of metal, often has chains, can be locked around
someone?s arms and legs, and so forth, and it can also be sued to refer to the
abstract, general category of constraining entities.  We refer to such abstract,
general concepts as attributive categories.  (Glucksberg et al, 1997 p. 334)
The authors claim that ?nouns can be used to make dual reference whenever a
superordinate category has not been lexicalized and a category exemplar is available that
is prototypical of that category.?
Glucksberg and Keysar continue with the assertion that metaphor is not a literal
comparison, and must be considered property attributions that extend or create categories.
As an example, they contrast the literal comparison with the metaphorical:
?Copper is like tin?? cannot be paraphrased as category assertions and still make
sense, for example, ? ?Copper is tin.?
Thus the paradox: Two unlike things compared can be paraphrased as a
categorical assertion, whereas two like things compared cannot.  This paradox
may hold the key to a fundamental difference between literal and metaphoric
comparisons.  We argue that metaphors are not understood as comparisons, but
rather as property attributions that either extend old categories or create new ones.
I hold that the difference in these statements is a result of what Roger Brown originally
described: ?Metaphor differs from other superordinate-subordinate relations in that the
superordinate is not given a name of its own.  Instead, the name of one subordinate is
extended to the other? (Brown, 1958).   ?Tin? is not a name for the category of which
they are members (owing to the fact that this category has a name and that tin is not a
prototypical member) and ?jail? is (because this category, as it is less stable, has no name
and is referred to by its most prototypical member).  Furthermore, placing tin and copper
in the same category is expected and does not violate any previously held ontology or
necessitate the construction of a new, ad hoc category, unlike placing jobs and jails into
the same category.  And, as Barsalou (1983) has shown, ad hoc categories have the same
38
structure and properties as traditional categories.  The difference, then, between a
metaphorical comparison and a literal one may have cognitive importance in terms of
stability of the category and conceptual coherence, but not in terms of the conceptual
structure.  Shen?s (1992) Metaphor and Categories makes similar claims as Glucksberg
and Keysar.  He argues that ?in interpreting a metaphorical comparison, an ad hoc
category is constructed so that the two metaphorical terms are conceived of as its
members,? and that the secondary term in a metaphorical statement is typically ?a
prototypical member of that category.?  Shen, too, makes a distinction between literal
comparisons and metaphorical comparisons, claiming that a literal comparison is
indicative of a ?common? category, while  ad hoc categories are represented by
metaphorical comparisons.
Traditionally, metaphors have been interpreted as statements of similitude and not
categorization because of the assumption that words (such as ?pig? when claim that the
person ?Sam is a pig?) refer to specific taxonomy.  Studies on how our minds perceive of
words and how these perceptions are related to taxonomic definitions versus
metaphorical relationships are detailed in the following section.
Idealized cognitive models and lexical networks
Eve Sweetser?s 1984 tests of the definition of ?lie? (as in ?to tell a lie?).  She
points out that, while most people define a ?lie? as ?a false statement,? in practice:
A consistent pattern was found: falsity of belief is the most important element of
the prototype of lie, intended deception the next most important element, and
factual falsity is the least important.
These findings are consistent with the idea that humans conceive of the world using
idealized cognitive models and that the imperfection of these models results in a graded
39
structure of categories.  In the ?lie? example the ICM being employed is a model of
communication, as studied by Grice (1975).  Grice removed the study of language from
its Chomskian position of mathematical clarity and tied it to the study of communication
arguing that, in order to understand the way language works, one must understand
expectations that exist in communication.  These expectations (the idealized cognitive
model of communication) entail clarity, truth, information, and relevance on behalf of the
speaker and influence our expectations for the definitions of words.
If words, such as ?lie? are not used in a manner consistent with the definition one
would find in a dictionary, how are words represented in the mind?  Recent
psycholinguistic theory has suggested that the mental lexicon, instead of being organized
in a dictionary-style, is far more like a thesaurus.  That is to say, the way our minds
perceive of words is not so much as obeying rigid definitions with propositional structure,
but rather the meaning of one word is tied to a network of related words ? words that
have appeared in similar contexts, words that have appeared in context with that word,
and words that have related meanings.  Computationally generated lexical networks have
been developed to represent the lexical network of the English language as expressed in
dictionaries (one example is the well-known Wordnet (Fellbaum, 1998)).  These
thesaurus-like structures link words in definitions into a network using various
algorithms.  Gaume et al. (2002), building on categorization research that they summarize
as establishing the ?conceptual flexibility? as opposed to ?rigid and discontinuous
categories,? argue that words themselves constitute categories and contend that these
lexical networks weave a ?mental lexicon distributed around metaphoric poles.?  In this
regard, dictionary definitions of terms such as ?lie,? ?pig,? and ?boys,? especially in the
40
contexts noted above, can be expected to fall short of the full meaning of these terms as
used in regular language.  When these terms are expressed in a lexical-network sense,
they can be viewed as representing categories of characteristics or qualities, rendering the
metaphorical statements, such as ?he?s a pig? and ?boys will be boys,? as assertions of
class-inclusion as Glucksberg and Keysar (1992) have argued.
Reconciling categorization and structure-mapping views of analogies
To address the ideas raised by Glucksberg and Keysar and reconcile these with
Gentner?s structure-mapping theory of analogy, Bowdle and Gentner describe a ?path in
figurative language comprehension? that claims that there is a shift in the method of
comparison in figurative language.   Novel metaphors, they claim, are interpreted via a
structure-mapping mechanism, while conventionalized metaphors ? words that have an
original meaning that is different from an often-used meaning (e.g., ?roadblock? or
?goldmine?) ? are interpreted as categorization.  The issue of comprehension of metaphor
is not at the heart of this thesis: what someone means when they make a novel analogical
statement has to do with the creation of an analogy.  It is reasonable, however, to expect
that understanding a conventional metaphor might be a similar process to creating your
own metaphor: in both cases the categorical commonalities (the ground of the category)
is known, while interpreting a novel metaphor would require a ?search? of possible
meanings of the secondary subject that are being implied.
Conclusion
Words, such as ?lie? and ?pig,? have definitions that exist in a dictionary ?  stable
definitions that people will readily agree on as sensible.  These definitions possess a rigid
propositional structure, so that ?pig? is thought to mean ?pink animal with a snout? and
41
?lie? is defined as ?an untrue statement.?  But these definitions are only valid when you
accept a certain model about the ways in which we communicate.  When you take into
account that communication is not so straightforward as our idealized cognitive models
of language assume, pinning down the exact description or definition of a word, such as
?lie? or ?pig,? is a much more difficult endeavor ? more context dependent and slippery.
Lexical maps, which link related terms into a network, more closely approximate the
lexical structure (or dictionary) of our minds.  Accepting this about language has
implications for the interpretation of analogical statements: traditionally, an analogical
statement is not literal because the base of the analogy is not intended to be interpreted
literally.  Instead, I argue that analogical statements are assertions of categorization, and
the difference between analogy and traditional categorization is that analogies violate an
expected ontology and may even necessitate the construction of a new, ad hoc category.
To say that ?Sam is a pig? demands that you consider the nodes in the ?pig? lexical
network that could possibly relate to Sam (including muddy, slovenly, lazy, messy) ? just
as when one says ?I eat pig? (typically viewed as a non-metaphorical statement, but one
which calls up the pork chop, ham, and sausage aspects of the pig lexical network).  The
difference lies in that ?pig? is defined in the cognitive model that allows for this animal
as distinct from others and ?Sam is a pig? requires you to ?turn off? some parts of the
network.
In the following chapters I will argue that generated analogies in science are
statements of category-inclusion that violate the expected categorization of the target.
When a student asserts that a cup of water is like a cat in a basket, she is constructing an
ad hoc category, this category is intimately tied to a theory and cognitive model, is
42
derivative of this model, and that the secondary item in the analogy is a prototype of this
ad hoc category.  Furthermore, these categories and their structure are indicative of an
underlying idealized cognitive model ? ones that are often metonymized by
phenomenological primitives.  This interpretation of analogies is responsive to findings
from cognitive science, linguistics and education that argue for a manifold ontology of
mind.
43
Chapter 3:  Origins of the Study & Methodological
Considerations
Origins of the Study
The following passages are taken from a ?Science Talk? (Gallas, 1995) in 5
th
grade classroom in rural Maryland.  It is early November and the science resource
teacher, Bruce Booher, has come to lead a discussion.  The students have worked with
Mr. Booher in the past and this format of science instruction is not unfamiliar to them.
The question that he has chosen comes from an experiment suggested on a NASA
website regarding zero gravity.  The students are posed the following question (NASA,
1999): a cup full of water is inverted on a cookie tray and the tray is rapidly pulled out
from underneath the cup (see Fig. 3.1).  What happens to the cup-water system?  The
students will later observe that the water does not leave the cup as it falls to the ground ?
the cup falls at the same rate as the water and the water will only spill out once it reaches
the ground.  Most students, however, believe the water will ?go everywhere,? ?spill,? or
?splash? as the tray is removed and report as much.  Their answers do not give any
rationale or mechanism by which this will happen.  However, one student predicted the
correct outcome and explained her prediction with a spontaneous analogy (transcript 2,
lines 8 ? 36).
44
Fig. 3.1: The Experiment: A tray pulled out from under a cup
Teacher: ?let?s see what some? I see a lot of other hands up.  Um,
Miranda?
Miranda: I predict that when it falls off it?s going to stay in the cup
until it gets down to the floor and then it?ll splash.
Teacher: So you have a prediction that when I slide it off of the tray
the water is going to stay in the cup.  Now that?s very
different from what they?re saying.
Miranda: ?Cause at home when I have like something in a basket and
when I go like that real quick [student swings arm around,
miming that the basket is swung overhead and quickly
pulled down] it stays in.  So when ? and when I pull it
down like this [motions pulling basket straight down] like
upside down on the way down it stays in until it gets to the
bottom and then it comes out.
Teacher: So you?re using now this example of something that you?ve
done at home where you have an object in a bucket ?  or a
basket, you said ?  and what do you do? You ?
Miranda: I go like this and then I pull it down and it stays at the top
until I stop and then it comes out.  [Motions swinging
overhead and pulling down, lifts hands to show that it stays
at the top of the basket.]
Teacher: So you swing this ? what?s in the basket?  What object is in
the basket?
Miranda: Sometimes I put like, like a little toy cat that I?m playing
roller-coaster with and put it in there and I pull it down and
it stays in the back [motions that the cat is up at the top]
until I stop and then it comes out.
Wholly aside from the fact that this analogy leads to the correct prediction, this analogy
shows the beginnings of deep scientific reasoning.  As I will show in later chapters,
45
mechanistic analogies to phenomena with which you have experience are ubiquitous in
the scientific literature.  Furthermore, once this analogy is brought up the tone of the
conversation changed; student hands shot up, and mechanistic reasoning and scientific
explanations were brought up.  In an attempt to understand this moment ? what, exactly,
was the significance of Miranda?s analogy, what are the assertions that it makes, the
cognitive work that it does, why it was brought up, how it was negotiated, how the other
students react to and negotiate the analogy ? there are few models of analogical reasoning
to turn to, fewer still on spontaneously-generated analogies; studies on analogical
reasoning suggest that comments like Miranda?s should be rare, are indicative of
expertise, and more likely with prompting by the teacher.  Furthermore, most models of
analogy are stem from variable process/regularity approaches to explanation and not
causal or meaning-based models. In this chapter I will outline previous studies on
analogical reasoning, their philosophical assumptions, the strengths of these approaches
and their shortcomings.  I then outline the tradition in which my own study is based and
explain the theoretical nature of this dissertation and its methodology.
History of Research on Analogies
Past research
Owing, perhaps, to the behaviorist tradition in psychology, the success of
experimental methods in more objective sciences, or stemming from the more recent
analogy of the mind to a computer with its fixed rules and stability, studies in learning,
particularly in cognitive science, have tended to focus on quantitative studies in
laboratory settings.  The history of analogical reasoning is no exception.
46
The most well known example of experiments on analogical reasoning is that by
Gick and Holyoak (1980 p 307-308).  In their study, participants were presented with
Duncker?s (1945) ?radiation problem.?  The problem begins with an anecdote:
A general wishes to capture a fortress located in the center of a country. There are
many roads radiating outward from the fortress. All have been mined so that
while small groups of men can pass over the roads safely, a large force will
detonate the mines. A full-scale direct attack is therefore impossible. The
general?s solution is to divide his army into small groups, send each group to the
head of a different road, and have the groups converge simultaneously on the
fortress.
The students were told to memorize the above passage and then asked to solve the
following problem:
You are a doctor faced with a patient who has a malignant tumor in his stomach.
It is impossible to operate on the patient, but unless the tumor is destroyed the
patient will die. There is a kind of ray that may be used to destroy the tumor. If
the rays reach the tumor all at once and with sufficiently high intensity, the tumor
will be destroyed, but surrounding tissue may be damaged as well. At lower
intensities the rays are harmless to healthy tissue, but they will not affect the
tumor either. What type of procedure might be used to destroy the tumor with the
rays, and at the same time avoid destroying the healthy tissue?
Few students were able to solve the problem without explicit instruction to use the story
that they had memorized.  Once prompted, over 90% of students could solve the problem
correctly using the convergence principles from the fortress story.  This study has been
widely cited as evidence that students find analogical reasoning difficult and that transfer
is a rare phenomenon.  Furthermore, it established a paradigm for investigating analogical
reasoning: present participants in the study with two analogically similar scenarios and
see if they notice/draw the intended analogy.
This methodology has been reiterated in future studies of analogical reasoning
with similar results.  In one study, Gentner, Ratterman and Forbus (1993) gave subjects a
47
series of stories to read and followed up one week later with another set of stories.  Some
stories were structurally similar to stories in the first set, while others shared superficial
features ? about a similar topic or involving a similar kind of animal, say.  The subjects
were then asked which of the original stories these reminded them of and they chose the
stories that shared superficial features.  Again, the authors argued that subjects are not
drawing analogies between deep structural similarities.  However, Markman and Gentner
(1993) report a study in which they ask participants to look at two pictures and find the
?same thing? in the two photos; they found in this case that items that occupied similar
roles were more often chosen as similar than superficially similar items.
Novick (1988) reports a similar finding in the context of mathematics to Markman
and Gentner?s findings ? that analogies based on structure are possible ? but that such
analogies are more likely if the participant has sufficient expertise in the subject.  (Which
echoes findings by Chi, Feltovich and Glaser (1981) in which expert physicists tend to
group problems based on principles rather than surface similarities, while novice physics
students grouped the same problems using superficial characteristics.)
These studies, and those with similar methodologies, are powerful for their clear
findings; they tell us quite definitively that in these scenarios, under these circumstances,
subjects behave in this way.  Indeed, this methodology, in the tradition of scientific based
reasoning (SBR), is hailed in a National Research Council report, Scientific Research in
Education (2002).
48
Limitations of past methodology
What, then, are the limitations of such research for understanding spontaneously
generated analogies in science?  While it has its strengths of clear and exact findings, I
claim that such focused, constrained, laboratory-based studies (1) establish too strong
constraints on what counts as analogical reasoning,  (2) strip analogies of the crucial
context in which they are constructed, and (3) fail to capture a causal explanation of
analogical reasoning.  I detail these below, and then introduce a qualitative alternative to
such studies of analogical reasoning.  I conclude by introducing the methodology
employed in this thesis.
(1) Constraints
Consider the story by Hammer et al.?s (Hammer, Elby, Scherr, Redish, 2004) of a
student discussing the size of mirror necessary to see your entire body in it.  This student,
Sherry, determines that
you need a mirror the same size as your body, because your whole body has to be
able to fit in it.  Other students in her group used ideas about reflection to argue
that the mirror would need to be half that size, but Sherry defended her reasoning.
The next week she told her group about a discovery at home: She owns a mirror
roughly half her height, and it shows a reflection of her whole body.  She had
known the answer to the question all along ? she saw it every day.
In their article, this anecdote is brought up to call into question the notion of transfer.  I
mention it here to suggest that even when students are not drawing appropriate analogies
? such as between the tumor and the fortress in Gick and Holyoak?s? they may be
drawing deep analogies nonetheless.  Sherry, the authors suggest, may have been drawing
a tacit analogy to doors or paintings instead of the more correct analogy to windows.  The
participants in Gick and Holyoak?s study may be drawing analogies to bullets, sound in
49
rooms, or light coming in through windows (it certainly seems just as bright by the
window as it does in the center of the room).  Gick and Holyoak may claim that
participants did not draw the desired analogy between the tumor and the fortress, but they
may not claim that students did not draw an analogy.
(2) Lack of context
Similarly, Gentner, Ratterman and Forbus?s findings demonstrate that similarity ?
or, more accurately, reminding ? is based initially on superficial characteristics when
done in the context of a laboratory, but these stories lacked context and immediacy.
Recently I was explaining to a friend the story of Moneyball (Lewis, 2003), in which a
baseball manager ignores baseball?s cultural wisdom of choosing draft picks in favor of
statistical analysis by Harvard graduates.  My friend replied that he was reading a similar
story about Bj?rk in the New Yorker (Ross, 2004) and commented that this seems to be a
trend in nonfiction writing ? to tell the story of someone who has an entirely different
perspective on the ?system? and revolutionizes it despite naysayers to great success.  In
this real-life context, comparing Moneyball to superficially similar stories (say, ?Casey at
the Bat,? Thayer, 1997) would have been unnatural and bizarre, while comparing
Moneyball to Bj?rk was creative and insightful, but also natural and a logical
conversational step.  In a laboratory setting, with few stories to choose from and no
reason for the stories to be told in the first place, our cognitive behavior can be unnatural
and bizarre.  It is important, then, to pair our laboratory based studies of cognition with
studies ?in the wild.?
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Why has it required conscious and deliberate effort to integrate context into
studies of learning and cognition?  As Lemke explains:
We blame the early Moderns of Rene Descartes? 17th-century Europe for
cleaving Mind from Body and Society from Nature (e.g. Shapin & Schaffer 1985,
Latour 1993). From them we inherited a chain ?  cognition in the mind, mind ?in?
a material brain, brain in a mindless body, body in a natural environment separate
from society, society made up of persons not bodies, persons defined by cultures,
cultures created by minds ? a chain that binds us still and runs us ?round and
?round in ever smaller circles.
We rebel, we transgress. We want the freedom to construct a materiality of
mind, an intelligence of the body. We want meaning to arise from material
processes and Culture to be once again a part of Nature. We want to re-situate
cognition in a larger meaning-making system of which our bodies and brains are
only one part. We are willing to pay the price, to abdicate our Lordship over
Creation, to become partners rather than over-seers. Creation, after all, has been
getting pretty unruly anyway.
That is, Cartesian dualism, which proved so powerful for the traditionally ?hard?
sciences, has left its legacy in the behavioral sciences which may not be appropriate: it
places meaning outside the scope of scientific research, separating questions of meaning
? which are intimately tied to context ? from questions of science.
Since the initial studies of analogy reported above, which occurred in the early
1980?s, researchers in cognitive science and education have become increasingly
concerned with the assumptions noted by Lemke that laboratory-based studies make on
the nature of cognition.  Gibson (1979), studying perception, emphasized the need to
study vision in terms of people behaving in the real world performing meaningful tasks
rather than subjects responding to the artificial and acontextual conditions of the
laboratory.  Lave, continuing this paradigm into the learning sciences, performed a series
of classic studies (Lave 1988, Lave & Wenger 1991) observing people ?  tailors,
midwives, and dieters ?  in real-world settings as they engaged in problem-solving.  She
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found that their strategies in these immediate, concrete, specific, and meaning-rich
situations differed from the disconnected problems of school or the tasks posed in a
psychology lab.  From these studies, she coined the term ?situated cognition? to describe
cognition as a ?nexus of relations between the mind at work and the world in which it
works? (Lave, 1988 p. 1), and any claims about how cognition ?works? must take into
account the world in which it is working.
(3) Causal explanation
A final concern for laboratory-based studies of analogical reasoning comes from
Maxwell?s (2004) concerns about the types of claims these studies can make about
causation.  That is to say, he finds fault with what causation means when arrived at from
a laboratory-based, quantitative study.  Concerned with the emphasis on scientific based
research (SBR) in the National Research Council?s Scientific Research in Education
(2002) report, Maxwell traces the philosophical traditions of SBR and argues that it
assumes a ?regularity? view of causation.  The regularity view is
based on an analysis of the contribution of differences in values of particular
variables to differences in other variables. The comparison of conditions or
groups in which the presumed causal factor takes different values, while other
factors are held constant or statistically controlled, is central to this approach to
causation.
He further argues that ?the central manifestation of the regularity view in the NRC (2002)
report is its presentation of causality as primarily pertaining to whether x caused y, rather
than how it did so? (Maxwell, 2004 pp. 125?129).  This claim is certainly true in the
studies of analogical reasoning reported above: these experiments cite circumstances
under which deep analogies are or are not drawn (such as: expertise is important, deep
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analogies are rare in the context of comparing two items), but make no claims as to why
analogies are drawn, what cognitive work these analogies do, the purpose that analogies
serve and why students fail to draw analogies in certain situations.
These three concerns ? the constraints that laboratory based research places on
analogical reasoning, the claims of situated cognition that demand ecologically valid
research, and failure of variance theory research to arrive at meaning or mechanism ?
have motivated researchers to perform qualitative studies of analogical reasoning.  These
studies are discussed below.
Recent approaches to the study of analogical reasoning
Noting the criticisms of laboratory-based research on cognitive processes that
claim ?what we know of cognition is based on arbitrary tasks bearing little relationship to
the cognitive processes that occur in naturalistic settings? (Dunbar and Blanchette, p
334), and in the tradition of situated cognition research methodology, Kevin Dunbar and
Isabelle Blanchette investigated the use of analogy in natural contexts.  They explain,
?we wanted to discover what similarities people note and under what circumstances their
reasoning is based on superficial or structural similarities.?  They began by video- and
audio-taping molecular biology and immunology labs and analyzing the types and
frequency of analogy generation in these conversations.
A similar approach was taken in a study of generated analogies in 6
th
 grade
mathematics classrooms.  A recent article by Richland, Holyoak and Stigler (2004)
reports an in-vivo study of analogies with their research in eighth grade mathematics
53
classrooms.  As with Dunbar?s methodology, Richland et al. construct a coding scheme
and deep analysis of the kinds of analogies that are generated in the classroom.
The analyses performed on the data from these studies, though situated in a
naturalistic context and with strong qualitative components, also continue a quantitative
paradigm.  These and other studies (for example, Pittman, 1999 and VanLehn, 1998) look
at the kinds of analogies, the frequency of these analogies, rely on a large corpus of data
to draw statistical claims about what happened ? again ?primarily pertaining to whether x
caused y, rather than how it did so? (Maxwell 2004) or, perhaps more accurately in these
cases, if x occurred and not why.  To get at the meaning behind analogical reasoning ?
that is, what are the assertions that analogies make, what cognitive work do they do, how
might the mind be organized to allow for this ? I rely not on statistical analyses of data or
rigorous coding of transcripts (which, of course, are invaluable tools of the educational
researcher and a crucial component of many important methodologies), but instead a
variety of approaches (primarily phenomenology and case study) and on the philosophy
of causality behind process theory.
An alternative to the regularity view (which Maxwell notes as characteristic of
variance theory) is that of ?process theory? (Maxwell, 2004):
Process theory? deals with events and the processes that connect them; it is
based on an analysis of the causal processes by which some events influence
others. It is fundamentally different from variance theory as a way of thinking
about scientific explanation? (p. 5)
A realist, process-oriented approach to explanation is compatible with, and
facilitates, the key strengths of qualitative research. In particular, it recognizes the
reality and importance of meaning, as well as of physical and behavioral
phenomena, as having explanatory significance, and the essentially interpretive
nature of our understanding of the former. It also recognizes the explanatory
importance of the context of the phenomena studied, and does so in a way that
54
does not simply reduce this context to a set of ?extraneous variables.? It relies
fundamentally on an understanding of the processes by which an event or
situation occurs, rather than simply a comparison of situations involving the
presence and absence of the presumed cause. Finally, it legitimates a concern with
understanding particular situations and events, rather than addressing only
general patterns.  (p. 8)
Maxwell then argues that qualitative research methods, because of their strengths at
?identifying causality in particular cases, the importance of context as integral to causal
processes, and the role of meaning and interpretive understanding in causal explanation?
are a crucial element in education research.
It is in this tradition and due to these concerns that I choose to approach
understanding student-generated analogies as they occur in classroom discussions.
Methodology
This thesis is a theoretical account of generated analogies in science, which began
with an insight into Miranda?s generated analogy above; namely that the thing she seems
to be making an assertion of categorization rather than a direct one-to-one mapping of
this cup to that experience of twirling her basket.  Following this insight, I developed an
account of generated analogies as assertions of categorization using the events in this
classroom research from categorization.  This account was then compared with generated
analogies in other contexts: different classrooms of various ages, historical accounts of
analogy use in science, research group meetings and informal conversations.  In these
analyses, I compared my story of categorization to the existing accounts of analogy
(primarily structure mapping), and I will argue that a categorization perspective is more
55
successful and generative than the others.  Further implications of this perspective may
then be tested in a more quantitative methodology.
A note on the data
None of the data presented in this thesis was collected for the purpose of studying
analogies in science.  While collecting data for a project on student inquiry in physical
science, and knowing that I was interested in studying analogies in science (because of
my own propensity for them), Miranda?s particularly powerful student-generated analogy
struck me.  At the same time, I was reading a book by Lakoff (1987) ? Women, Fire and
Dangerous Things: What Categories Reveal About the Mind.  With this one example of
student-generated analogies and this one perspective on cognition, I began to develop a
rough theoretical perspective for understanding student-generated analogies in science as
assertions of categorization around this first analogy.  I then turned to a corpus of data
from student inquiry into physical science, research group meetings and classrooms to
check against my account of analogical reasoning.
The case study methodology
Such a study is indicative of the case study tradition.  The case study primarily
addresses the how and the why research questions ? in my case, why analogies are
generated, how they help students and the cognitive work that they do.  The case study
aims to provide a detailed description and analysis of the observed case.  It acknowledges
the importance of studying the phenomenon as a whole and does not consist of a linear
model of inquiry, noting that ?there are complex relationships within phenomena, [and]
taking them apart may result in losing some of their important aspects.?  Additionally, the
56
?case study is also naturalistic in the sense that it studies cases in their physical context,
in which the researcher is also interested? the researcher has limited or in some cases no
control over the case of study.  Case study method also requires the study of a
contemporary phenomenon or situation within its real life context? (Louca, 2004, p. 35).
Furthermore it is the evaluation of a single case that Maxwell (2004) and
Davidson (1967) argue may be a more powerful method for arriving at causality than the
variant theory approaches.  Davidson notes that we  ?can infer cause in single
experiments . . . [and that] providing them with conceptual help in doing so is a virtue,
not a vice; failing to do so is a major flaw in a theory of cause-probing methods?
(Davidson, 1967, p. 465).
One key component of the case study, indeed of all qualitative research, is
triangulation of information in which ?researchers make use of multiple and different
sources, methods, investigators and theories to provide corroborating evidence.
Typically this process involves corroborating evidence from different sources to shed
light on a theme or perspective? (Creswell p 202).  As I continued taping student
conversations in physical science, which were replete with spontaneously generated
analogies, I had further data to bring into and try out against my theoretical framework.
This data primarily comes from elementary school classrooms throughout Maryland,
ranging from second through sixth grade.  The teachers in this study were part of the
project Case Studies in Elementary Science and are particularly skilled at listening and
attending to student ideas in science, which was significant in providing me with student-
generated analogies.  In addition, I used data from studies on high school and
57
undergraduate physics students ? data initially acquired for other research purposes ?
along with tapes from research group meetings and faculty conversations.  And finally,
the framework was compared to analogies by scientists, as reported in studies from the
history and philosophy of science, comparative literature and popular non-fiction.  The
analysis began with a phenomenological analysis, which will be explained in the
following chapter and then a more ontological, causal analysis, which I will describe in
Chapter 5.
Summary
This is a theoretical dissertation, one that argues for a particular perspective on
student-generated analogies and the ontology of mind.  As such, quantitative methods ?
even those that are typically used in more qualitative research (coding, for example) ? are
not employed here.  That is not to say that these are not relevant to a framework of
spontaneously-generated analogies.  However, as Maxwell (2004) argues,
I would argue that strictly experimental designs, with no qualitative components,
are a comparatively powerful method for understanding only when three
conditions obtain. First, there should be a well-developed theory that informs the
intervention and research design and allows interpretation of the experimental
results (Bernard, 2000, pp. 55?56)? Second, the causal process investigated
should be manipulable, fairly straightforward and simple? Third, the situation
should not be conducive to the direct investigation of causal processes.
This thesis aims to begin with the first of these criteria by understanding the why of
student-generated analogies, providing a framework of categorization and consistent with
a manifold ontology of mind.  Once established, the ability to manipulate the causal
processes and whether or not these may be investigated directly may then be asked.
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Chapter 4:  Phenomenological Coherence
Introduction
The central thesis of this dissertation is that student generated analogies in science
can best be interpreted as assertions of categorization.  This description of analogy began
with an analysis of a single classroom, first introduced in the previous chapter.  In this
chapter, I will highlight particular phenomenological aspects of this and other analogies
and the negotiation of these analogies.  I will explore the consistency of these aspects
with a categorization framework.  Alternative theories of analogical reasoning will be
contrasted with a categorization description.  In following chapter, I will address the
implications that these phenomenological properties have on cognitive structure and how
a theory of cognitive structure that consists of a manifold ontology of mind can, in turn,
provide a more formal definition of analogy and account for the phenomenology
described here.  The phenomenological features that will be detailed below are: multiple
analogies that serve to enumerate a category, multiple analogies that serve to analogy
?hop,? far-transfer analogies introduced before near-transfer analogies, constructing the
base of an analogy rather than recalling the base from memory, a variable representation
of that base, and analogies as offering an alternative to another way of understanding this
phenomenon.
Models of analogy from the literature
To understand these phenomenological aspects of generated analogies, one might
first turn to an established model of analogy.  As noted in Chapter 2, these models, first
59
developed in the 1970?s, have evolved to the commonly accepted models of structure-
mapping (Gentner, 1983) and MAC/FAC (Gentner and Forbus, 1991).  The initial
theories of analogy, commonly referred to as feature-matching theories, were based on
similarities between features, properties and behaviors of the primary and secondary
subjects of the analogy (Johnson and Malgady, 1980, Miller 1979, and Tversky 1977).  It
has since been widely recognized that the claim of ?similarity? is vague and
underdetermines the correspondences and nature of analogical reasoning (Lakoff 1987).
Similar claims were made in categorization: while members of a category were assumed
to have certain features in common, defining these features was problematic.  Some
features, such as ?seat? for the category of objects called ?chairs? appear to have names
that showed them not to be meaningful prior to the knowledge of the object as a chair.
?Large? for the object ?piano? has meaning only in relation to categorization of the object
in terms of a superordinate category.  And ?you eat on it? for the object ?table? is a
functional attribute that requires knowledge about humans, their activities and the world
(Rosch 1978).  Similarly, analogies are not based on superficial attributes and feature-
matching between subjects, but apply to a more abstract structure of the subjects.
Gentner (1983) developed a theory, structure-mapping, to address the fact that analogies
are not feature comparisons, but much more structural.
Structure-mapping theory argues that interpreting an analogy involves both
alignment and projection.  The process is described in Bowdle and Gentner (1999):
Structure-mapping theory assumes that interpreting a metaphor involves two
interrelated mechanisms: alignment and projection.  The alignments process
operates in a local-to-global fashion to create a maximal structurally consistent
match between two representations that observes one-to-one mapping and parallel
connectivity (Falkenhainer, Forbus and Gentner, 1989).  That is, each object of
60
one representation can be placed in correspondence with at most one object of the
other representation, and arguments of aligned relations are themselves aligned.
A further constraint on the alignment process is systematicity:  Alignments that
form deeply interconnected structures, in which higher-order relations constrain
lower-order relations, are preferred over less systematic sets of commonalities.
Once a structurally consistent match between the target and base domains has
been found, further predicates from the base that are connected to the common
system can be projected to the target as candidate inferences.
There are several shortcomings of structure-mapping theory when trying to understand
student-generated analogies.  First, such a model is designed to explain ?interpreting an
analogy? and not the process by which that analogy was created.
1
  Additionally, while
structure-mapping can illustrate what an analogy is, it is not clear why a student would
map knowledge from one domain onto another, under what circumstances analogies are
generated, or how the analogy will evolve in the classroom.  Structure-mapping is a
powerful model for how an analogy, once introduced and understood may be formalized
and used to draw further inferences, but it is not a model for how analogies are generated
and the kind of work this generation does.
Analogies as categorization
To address the phenomena mentioned above and understand the role of generated
analogies, I will argue for a categorization framework; that is, I assert that the role of
Miranda?s analogy between a falling cup of water and a toy cat swinging in a basket (first
mentioned in chapter 3 and explored in detail below) is not to establish a one-to-one
mapping between this particular cup of water and a particular instance of swinging a
                                                  
1
 In some references, structure-mapping is portrayed not as a model for how we interpret
analogy, but generate (Falkenhainer, Forbus and Gentner, 1989 p 2): ?Structure-mapping
decomposes analogical processing into three stages?: 1. Access: Given a current target
situation~retrieve from long-term memory another description, the base, which is
analogous or similar to the target.?
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basket overhead.  Rather, the cat/basket serves to represent a more abstracted, general
category ? that of, perhaps, containers that do not spill their contents when overturned
(though certainly not so well-defined in Miranda?s mind, and lacking the propositional
structure that such a characterization implies).  By constructing this category, she has
introduced an alternative cognitive model, allowing for a new set of causal mechanisms
to be explored.  As this category is negotiated, adapted and understood, additional
analogies are introduced as a means of negotiating and understanding this category.  This
idea echoes and expands upon claims made in cognitive science regarding the
interpretation of metaphor.  When the base of an analogy (termed the vehicle in the
context of metaphor) is used as both an exemplar and as an ad hoc name for a category,
Glucksberg et al. (1997) call this linguistic move ?dual reference.?  As an example of
dual reference, the phrase ?a responsibility is a shackle? can be used to refer to the
concrete,
physical device that is made of metal, often has chains, can be locked around
someone?s arms and legs, and so forth, and it can also be used to refer to the
abstract, general category of constraining entities.  We refer to such abstract,
general concepts as attributive categories.  (Glucksberg et al 1997 p 334)
The authors claim that ?nouns can be used to make dual reference whenever a
superordinate category has not been lexicalized and a category exemplar is available that
is prototypical of that category.?  I will expand upon this idea to include more than nouns
in metaphorical scenarios and focusing on the generation of analogies
Again, in this chapter I will focus on the phenomenological aspects of generated
analogies in science ? in particular, the pattern of multiple analogies, chains of analogies,
near and far transfer analogies, analogy bases constructed as opposed to recalled, and the
62
variability of representation of the base in such analogies.  In the following chapter I will
focus on the underlying cognitive structure, cognitive models and ontology of mind that
is consistent with this analysis.
Multiple Analogies
The first phenomenological aspect of student-generated analogies that I present is
that of multiple analogies.  In each of the following transcripts, an analogy is generated
and then, in the negotiation of that analogy, further analogies are brought up.  None of the
following analogies is at odds with the initial analogy, nor are they extensions or
modifications of that analogy, rather they are consistent with the initial analogy and, I
argue, serve to aid in understanding the category ? the kind of thing ? that the initial
analogy asserts.  Below I present these multiple analogies and demonstrate the
consistency between multiple analogies and a categorization framework of student-
generated analogies in science.  This chapter consists primarily of kind of checklist of
phenomena that student-generated analogies have in common with categorization; in the
following chapter I will account for this correlation in a more theoretical manner by
addressing a theory of mind that can account for and explain these phenomena.
Multiple analogies: Example 1
In Chapter 3, I first introduced the following analogy from a 5
th
 grade classroom
in a rural Maryland public school.  In this transcript, the students have been visited by the
science resource teacher and posed the following question (NASA, 1999): a cup full of
water is inverted on a cookie tray and the tray is rapidly pulled out from underneath the
cup (see Fig. 4.1).  What happens to the cup-water system?  The students will later
observe
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Fig. 4.1: The Experiment: A tray pulled out from under a cup
that the water does not leave the cup as it falls to the ground ? the cup falls at the same
rate as the water and the water will only spill out once it reaches the ground.  However,
perhaps not surprisingly, the students initially predict that the water will spill from the
cup or spread across the tray.  A student then offers a different prediction and introduces
the analogy (transcript 2, lines 8 ? 36) that the cup of water is like a cat in a basket and
will not spill when overturned
The claim that I will continue to make throughout this dissertation is that
Miranda, with her analogy, is making the assertion that the cup of water belongs to a
category or class of phenomena that is typified by the toy cat swinging in the basket,
rather than mapping the structure of swinging a cat in a basket to the phenomenon of the
overturned cup of water.  Consistent with this story are the multiple analogies that ensue.
Following Miranda?s introduction of the analogy to a toy cat swinging in a basket, the
following analogies are introduced, consistent with Miranda?s analogy: throwing buckets
of water (transcript 2, lines 134 - 144)
Teacher:  It could turn sideways like that.  And that would make a difference.
Okay let?s get ? a lot of you have been very patient.  Cody?
Cody:  Um because when I was um having bucket full of water and I
swing it around and then when I throw it the bucket of water still
stays in there- the water, and? Yeah and then when I throw it the
bucket of water still stays until it hits something.
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throwing a bag of Halloween candy (transcript 2, lines 179 ? 189) and tossing dice in a
hat,
Isaac:  Um I pre- I don?t- um I agree with Miranda but I don?t think air
has anything to do with it.  Because um yesterday at Trick-at-Treat
I had like a bunch of candy and I swung it around that?s like when
I was bored and stuff ?
Teacher:In your bag?
Isaac: Yeah.  And none of the candy came out.  I like kept on swinging it
and also when me and Johnny play monopoly there?s like this little
hat that we play with when we roll the dice [Teacher: Mmm hmm.]
and like we always put the dice in and flip it back to each other
with the dice in it and we always catch it and it stays and the dice
stay in.
and twirling Easter candy (transcript 2, lines 200 ? 207)
Teacher:?Alexandra?
Alexandra: Um when Miranda said how when she dropped the cat in a um
basket- I?ve done that with um my Easter candy but with more
candy in it and when I turned it over when I got up here and it
dropped it all went everywhere.
Teacher: But when you were swinging it, it didn?t fall out until you got up
here and then stopped and then it all fell out.  [Alexandra nods.]   
In the structure-mapping model of analogy, the role of multiple analogies is not clear: if
students understand the target and base and their relationship to one another, the analogy
has served its purpose and reiterations of this structure-mapping with additional analogies
should not be necessary.  However, multiple analogies are consistent with categorization,
as categories typically consist of multiple members and these serve to better define and
negotiate the category they are constructing.  Researchers have shown, perhaps not
surprisingly, that students? abilities to categorize properly are greatly enhanced when
multiple members of a category are shown (Namy and Gentner, 1999) and these multiple
analogies can be understood to be negotiations of the category Miranda is asserting.
65
What is surprising about these multiple analogies is that, although they are
consistent with Miranda?s cat/basket analogy, they are not always introduced by students
who agree with Miranda?s prediction: neither Cody, who introduced the bucket of water
analogy, nor Alexandra, who brought up the Easter candy analogy, believe that the water
will stay in the cup, and Isaac agrees with Miranda?s prediction but, without a rationale
for why, disagrees with the suggested mechanism of this prediction.  These students are
not discussing the ?mapping? of a base onto a target: they are not relating their analogies
to the cup of water, they have not put items in a one-to-one alignment or made candidate
inferences, as structure-mapping suggests.  Rather they are exploring the phenomenon of
the toy cat in the basket in its own right by mentioning other members of the category
that it represents ? that is, other items that do not spill from their containers ? as a means
of negotiating the category to which the base of the analogy belongs.
A particularly telling moment in the multiple analogies as categorization story is
when Alexandra claims (transcript 2, lines 200 ? 207):
Alexandra: Um when Miranda said how when she dropped the cat in a um
basket- I?ve done that with um my Easter candy but with more
candy in it and when I turned it over when I got up here and it
dropped it all went everywhere.
Teacher: But when you were swinging it, it didn?t fall out until you got up
here and then stopped and then it all fell out.  [Alexandra nods.]
What does Alexandra mean when she claims to have ?done that??  What is that?
No one ever explicitly discusses the abstracted ?category? of ?overturned containers that
do not spill their contents?? and yet that cannot refer to the concrete example that
Miranda mentioned: Alexandra does not claim to have spun a basket overhead.  Instead,
Alexandra, indeed the whole class, recognizes that Miranda is using the toy cat in a
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basket as an instance of a more general category, one that Alexandra represents with her
Easter candy and refers to by her claim to have done ?that.?  In another use of ?that? to
refer to the class of phenomena that her initial analogy constructs, Miranda, in line 55
(transcript 2), says:
Miranda: And it?ll be the same thing with the water the air will push the
water up until it falls down and then it will go everywhere.
Because when it comes down the air is pushing upwards and [it
keeps/I keep?] the water in there- because I?ve also done that in the
bathtub when you?ve got your cup, I?ll like I?ll fill it with water put
my hand and drop it the water stays in until it hits the bathtub and
then it goes everywhere.
To claim ?I have also done that? signifies there is a ?that? to refer to ? the category of
phenomena to which the cat/basket, cup/water and Easter candy/basket belong.  The use
of ?that? is often indicative of categorization: imagine that you are telling someone about
training for and running a marathon and she replies ?I?ve done that.?  One would assume
she means she has run a marathon ? not the same marathon as you, but rather recognizes
the (slightly) more abstract category of marathons in general.  For a more abstract case,
consider the example from the previous chapter, in which a friend and I were discussing
Moneyball (a story about baseball management) and he mentions that he is reading a
similar story and brings up Bj?rk (an Icelandic pop star).  Though I don?t have a
transcript of our conversation, one can imagine saying, ?That?s a really popular story to
tell these days.?  In this case, ?that? would mean an abstracted type of story ? a category
to which Moneyball and Bj?rk belong.  Miranda?s analogy, then, is to instantiate this
category of containers that do not spill when overturned ? construct it as a category and
assert that the cup of water is a member.  The other students understand this, come up
with other members of the category that Miranda has constructed, and debate whether or
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not the cup of water is, indeed, a member of this category and Alexandra?s comment to
say that she has ?done that with um my Easter candy.?
If the use of the word ?that? seems scant evidence for categorization to be at play,
this is only due to the fact that categorization is so ubiquitous it frequently escapes our
notice.  Every conceivable noun describes a category, and every time I identify a window
as being, in fact, a window, I am making a categorical assertion.  Verbs, too, can be seen
as categories; the claim ?I leap? makes a categorical assertion about the kind of activity I
am doing, ignores the subtle differences between this leap and other acts of leaping ? just
as running-a-marathon or swinging-something-overhead-so-that-its-contents-do-not-
immediately-fall-out can be a category.  While running-a-marathon is a relatively exact
category (in this context), the nuances of many categories can be quite complex.
Consider all of the ways in which ?leap? can be used: ?a leap of faith,? ?look before you
leap,? a ?flying leap,? or ?leap frog.?  Or, as Hofstadter (2004, p. 505) explains,
such lists go on and on virtually forever, and yet the amazing fact is that few
people have any inkling of the vastness of their mental lexicons (see Becker
1975).  To be sure, most adults use their vast mental lexicons with great
virtuosity, but they have stunningly little explicit awareness of what they are
doing.
And just as lexical items and phrases describe categories, there are categories for which
we have no simple labels.  As with the Moneyball/Bj?rk analogies, there are ?stories
about someone who has an entirely different perspective on the ?system? and
revolutionizes it despite naysayers to great success.?  And in these categories, multiple
members serve to negotiate and define the category.  As a second example of the role of
multiple analogies as defining and negotiating a category, I present another transcript of
student conversations of science.
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Multiple analogies: Example 2
The following transcript, first introduced in Chapter 1, is from an undergraduate
physics course for elementary education majors, Inquiry into Physical Science.  The
students have been investigating the electrical properties of Styrofoam and metal, and
noticed that Styrofoam is easy to charge (simply rub with wool) and metal is not, but
metal can easily give you a shock and there are ways to charge metal once you have a
charged object.  They have been asked to explain the differences between these two
materials by describing ?what life is like? for a charge in each.  In doing so, multiple
analogies are introduced.  (In this transcript, by class convention, the two types of
charges are referred to as ?top? and ?bottom.?)  Hana begins the discussion with an
analogy between the charges and fish (Transcript 1, lines 1 ? 17):
Hana: I kind of see the charge in metal as like, fish in a fish bowl?  Like
they never really stop moving, they?re always kind of floating
around wherever they kind of feel like going and that?s just how I
see it in my head, like them always moving around.  And I don?t
know what hap- I don?t know how to describe it really I don?t
really know what happens once another charge is brought closer
then.
Instructor: Does this make sense to you then?
Hana: Yeah.
Instructor:  So this is ? so this is like two kinds of fish. [Hana: Yeah.]  And in
metal they can move around.  They?re kind of stuck inside the
bowl, but within the bowl they can move around.
Hana: But I also think that they can leave the bowl at some point because
?
Instructor: Well we get shocked right?
 This idea of continual motion of the charges is mentioned in another analogy (Transcript
1, lines 20 ? 33):
Kelli: That same idea I was thinking except more like ping pong balls
that bounce all around and that?s why if there?s top and bottom
charges they?re moving around a lot and they?re kind of attracting
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and repelling and attracting and repelling each other the tops and
bottoms that go all over the place ? but once the extra bottom
charge is added it?s almost trying to like reneutralize itself and the
tops are attracting to the extra bottoms.  And then they?re trying to
kick out the other extra bottoms so they can get back into their
whole little [Student: Balance.] balance.  Bouncing around.
Instructor:  So this- I think what you?re offering is an explanation of why I get
a shock.  Is that ? am I wrong?
Kelli:  No, you?re not.
In both analogies, the students are, first, introducing an imagistic analogy that describes
the motion of the charges and then, in the case of Kelli?s analogy, explaining how,
mechanistically, this is consistent with observed phenomena.  However, a student notes
the following (transcript 1, lines 46 ? 52):
Christie:  We were thinking that ? like they were saying that in metal it?s
always moving, so if it?s always moving it has more room to move
and that would mean to say that the molecules are less tightly
packed together or less dense and we were thinking of Styrofoam
as more dense than ?  I?m just trying to figure out first if that?s
right and how it relates.
A second group of multiple analogies stems from Christie?s comment, ?If it?s always
moving it has more room to move and that would mean to say that the molecules are less
tightly packed together or less dense and we were thinking of Styrofoam as more
dense...?.  The students are all in agreement that charge seems to move more freely in
metal than in Styrofoam (as in the analogies to fish in a fish bowl or ping pong balls
bouncing all around), but this seems to imply that there is ?more room to move? ?
meaning that metal should be less dense than Styrofoam.  Lea, picking up on the fact that
metal is more dense than Styrofoam, constructs an analogy that incorporates both: the
Styrofoam is like cotton while the metal is like ice skating, in which the density of the
medium allows for easy movement (transcript 1, lines 53 ? 60).  This category, media
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whose density allows for movement, is non-intuitive: crowds are hard to walk through,
syrup is hard to walk through.  In understanding and negotiating this category, other
analogies are mentioned: a sponge, whose lack of density is a trap for water versus the
dense countertop that allows water to flow easily (transcript 1, lines 61 ? 66):
Instructor: Lea I want to add ? I think you?re sort of what I when I hear you
talk I?m thinking of like, pouring water into a sponge versus
pouring water onto a hard surface. [Lea: Yeah.]  Like this sponge
is actually less dense and there?s room for it to absorb the water
and the you know if you pour it onto something hard there?s no
room for it to absorb.
And later another analogy, consistent with the previous two is mentioned ? stepping
stones that, when far apart, inhibit easy passage but when densely packed are easy to
negotiate (transcript 1, lines 113 ? 121).
Lydia: I was going to say I think the pie plate is more dense but I do think
that it?s inside not outside because if there?s more space to travel
then the molecules can?t get from one space to another easily but
it?s all real close together so it can sort of hop along inside.
Instructor:  Oh so it?s like stepping stones [Lydia: Kind of.] like in the
Styrofoam it?s really far to the next stepping stone so it?s like can?t
get there I?m stuck here. [Lydia: Right] but in the metal the stones
are really close together so I can kind of walk across.  [Lydia:
Yeah.]
I will revisit this analogy in the following section for the manner in which it differs from
those that came before.  In the above transcript there are several analogies mentioned:
- charges are like fish in a fishbowl (lines 1-4)
- charges are like ping-pong balls (lines 20-24)
- metal is like ice-skating (lines 56-57)
- Styrofoam is like a sponge (line 63)
- charge flow is like steam escaping a shower (lines 98-104)
- metal is like a set of closely spaced stepping stones (line 120)
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Some of these analogies are mentioned relatively independently, but, as in the case of
Miranda?s analogy and the multiple analogies around it, some analogies in the transcript
above further develop a category that the original analogy has constructed.
Chains of Analogies
A second manner in which multiple analogies are used is not to flesh out a
category, but as a means of ?stepping? from one analogy to another.  In this section, I will
present examples of analogies that are not consistent with one another but can each be
seen to be a slight ?tweak? on the one that came before.  Such analogies are consistent
with findings from categorization ? in particular family resemblance, described below.
It is important to note that research indicates categories do not have a fixed
representation or set of criteria for membership.  Our categories are constantly reshuffled
and recast depending on the context.  At times, the concept of leaping is very much
related to movement, at other times to lack of support, and no dictionary definition can
capture the full character of the way this word is used in everyday language.  As noted in
chapter two, ?there is no one property that all games share, but rather there are family
resemblances between games, so that ?chess and Go both involve long term strategy?
chess and poker both involve competition.  Poker and old maid are both card games?
(Lakoff, 1987 p. 16).  Similarly, ?leap-frog? and a ?flying leap? indicate a kind of
forward motion off of the ground, while a ?flying leap? and ?look before you leap? are
situations in which you lack support.  This family resemblance is indicative of two
things: one, as mentioned above, that no one characteristic defines a category, and two,
there is a hierarchy of categorization.  Rosch referred to the most accessible level ? the
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easiest to learn, recognize, recall, and shortest to name ? the basic level and underlying
this level are more fine grained categorizations: such as games dividing into card games.
If analogies, then, are assertions of categorization, we might expect multiple
analogies as a way of negotiating the category, and chains of analogies (as with Go,
chess, poker and old maid) as family resemblances are followed within a category,
subdividing that category into a finer grained one.  The analogies in the previous section
are relatively consistent with one another as a particular category becomes increasingly
defined.  In this section, I present analogies that follow chains of ?family resemblance? to
tease apart distinctions and more narrowly define a particular categorization.
Chains of analogies: Example 1
The first transcript below is from a research group meeting of the Physics
Education Research Group.  Paul, a graduate student, is interested in authentic classroom
activities and is discussing his definition of authentic.  Key to this definition is that
authenticity is a property not only of the activity but also but also in the way that the
students relate to that activity and the coherence of this to the scientific community of
practice (that is, do the students know what they?re doing?  Would scientists agree?).
This is at odds with definitions of ?authentic? activities that situate authenticity as a
property of the curriculum itself.  In this transcript, analogies to authenticity are
suggested, and multiple analogies are pursued.  These analogies are both along the lines
of honing in on and refining a particular analogy (akin to the multiple analogies above)
and ?family resemblance? analogies that are more ?horizontal? than ?vertical? chains.
The transcript begins with David recapping Paul?s concerns with the standard definition
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of authentic curricula, and Rachel?s introduction of an analogy between ?authentic? and
?fun? (transcript 3, lines 1 ? 9):
David:  It?s not a responsive definition of authentic.  Authentic is defined
pre-experience. And so what your [Paul?s] sense is what?s going to
be authentic is about watching the student and and what is
authentic for this group ? may be different from what?s authentic
science for this group.  And you don?t like defining authenticity in
a way that isn?t responsive.  So the content isn?t responsive but
also the sense of what is authentic isn?t responsive.
Rachel:  So ontologically authenticity is like fun.  Which would be ?
David: Oh that?s great.
Andy: Oh that is beautiful!
The analogy is understood by the group and further explained in lines 17 ? 29 (transcript
3):
Andy: ? it emerges from an activity but it?s really ultimately lives inside ?
Rachel: Right and I mean you could say ? I mean you couldn?t look at a
thing on paper and declare that it was fun until you could see
people do it and see them have fun.
David: And it may be fun for some people and not fun for other people.
Andy: You could ? an experienced teacher could make guesses about
what?s more likely to result in fun blah blah blah.
Rachel:  Sure, sure.  But really ultimately you don?t know until after
[inaudible]?
David:  Or anyone ? what what my kids think of as fun might not be the
same as what Rachel thinks of as fun [trails off].
However, a weakness of the analogy is identified: Paul has argued that authenticity must
not only be recognized by the students? relationship to the curriculum, but a
(hypothetical) community of practice must have a similar understanding of the
relationship between the students and this curriculum.  This weakness is identified and
alternative analogies are proposed (transcript 3, lines 38 ? 70):
Leslie:  Is there a community of fun practice? [Laughter.]
Rachel:  Or norms?  [Laughter.]
Leslie:  Like with the community of practice the scientist is someone
who?s outside deciding whether or not it was science but with fun
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there isn?t ? so it?s a negative analogy ? but with a community ?
yeah there?s no community of practice.
David: I don?t know what you mean ?  there?s no authentic community of
practice?
Leslie:  You only have to ask a person who?s having fun if they?re having
fun.  But this (definition of authenticity) implies that you have to
ask the scientists whether they?re doing science.
David:  Ahhh ? right.  Gotcha.
Andy:  Not only does it have to be meaningful it has to be meaningful in
the right way ? but ?  yeah I?m having fun but it?s ? you know ?
low-brow fun instead of highbrow fun.  Guffaw guffaw!  Gotcha.
[Laughter.]
David:  So can we patch that?  Is there, is there another?
Rachel:  Good clean fun?...
Leslie:  I have a multiple analogy if that?s okay?  I?m thinking it?s more
like worship ? like, you know if you?re worshipping but a religion
is going to also decide if what you did was worship.
Andy: Oooh.
David: Right.  Right that?s good.
Andy:  It?s gotta pass both tests.  That is good.  Good clean fun works, too.
You decide if it?s fun, I decide if it?s good and clean!
[Laughter.]
Leslie: Or pornography?
The analogies above construct a particular ontology (as Rachel identifies) ? or category ?
of adjectives: those that are not inherent properties of the object they describe, but, like
?fun,? are a measure of both the activity and the participant.  This does not quite describe
the sense in which Paul is speaking of authenticity.  In his description of an authentic
activity, Paul also relies on the coherence of the students? understanding of the activity
with the community of practice?s understanding of that activity.  To patch the analogy,
two alternative analogies are suggested.  One is a more ?vertical? analogy: replacing fun
with ?good clean fun? (line 60).  I say vertical because this is similar to looking at games
and then honing in on ?card games? because the category ?games? is too vague and lacks
the detail that ?card games? has.  An alternative ?patch? is to move more ?horizontally? ?
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as if moving from ?games of chance? to ?card games? as Leslie (I) does by suggesting
?worship? or ?pornography? as alternative analogies.
The multiple analogies presented above do not serve the same role as the multiple
analogies presented in the previous section.  Instead of serving to negotiate and
understand the category that the analogy constructs, they alter this analogy ? however
they do so in a manner consistent with categorization.  One possible ?tweak? to a
category is to further parse it ? categories have hierarchies (as with ?chair? which is a
base level category ? and ?easy chair? or ?desk chair? or ?recliner,? which are
subdivisions of this category).  Taking the idea of ?fun? into ?good clean fun? is such a
move.  An alternative move is one of ?family resemblance? in which categories are
related to one another within a hierarchy (as with ?easy chair? and ?desk chair?) ? it is
this move that Leslie makes in moving from ?fun? to ?worship.?
Chains of analogies: Example 2
Furthermore, these analogies, unlike the multiple analogies that are consonant
with the cat in the basket, are in response to something problematic ? a small piece of the
story doesn?t work out.  This is the story in the final analogy presented in the analogies
regarding charges in metal (transcript I, lines 113 ? 121) in which the metal is like
closely-spaced stepping stones.  Implicit in the analogies that the other students have
suggested is that the charge will then travel on the outside of the metal: ice-skaters travel
on top of ice and water travels on countertops ? implying charge will travel on the outside
of metal.  This is very much a structure-mapping story, in which the structure of the base
makes implications for the target: items in the base and the target are placed in a one-to-
one alignment and inferences are projected from that alignment.  Charge traveling on the
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outside of the metal is an inference that structure-mapping can predict and explain.  In
this regard, structure-mapping tells a story that is an important piece of the work that
analogies do in science classrooms and in science, and address a part of analogies that I
do not: how they are used to draw inferences and establish a sense of mechanism.  My
claims are not regarding Lydia?s interpretation and understanding of these implications,
but how they in turn construct new analogies.  This implication is one that Lydia
challenges.  (I note that)  The ?tweak? to the analogies is then suggested by the instructor,
who moves from a countertop analogy to a stepping-stones analogy.
Chains of analogies: Example 3
The following transcript from a third grade classroom presents another example
of this ?family resemblance? property of multiple analogies.  In this classroom, the
teacher, Trisha Kagey, has asked the students: if you are running with a beanbag and
want it to fall on an X, should you release it before, when, or after you reach the X?  In
the discussion that follows, several analogies are introduced, many of which are slight
modifications of the one that preceded it.  Below I present transcripts from the chain of
analogies:
- the beanbag is like a baseball bat (line 59)
- it is like a leaf being dropped from a bus (line 74)
- it is like a rock being dropped from a bus or a bike (lines 36 and 76)
- the leaf is like a feather, and a rock is different (line 136)
- the rock is like a tree and yet made from the things that are different (188)
The first analogy is mentioned (in transcript 4, lines 35 ? 46) after Adam argues for his
prediction with Newton?s Laws.  I present it here for completeness: this analogy is not a
part of the ?chain? of analogies that ensue but the teacher?s instructions and the student?s
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analogy may play a role in the other students? expectations about what kind of knowledge
to bring to bear on the question:
Teacher: ?but trying to just explain it to someone using your experiences ?
see if you can use it that way.    Explain to ? like you?re explaining
to a kindergartner.  They?re not going to be able to understand that
law.
Adam:  Like um ? if ? like if something ? if you?re riding your bike, um ?
it?s in motion.  And you?re going to keep going until you get
stopped by like ?  um, a rock or something.  And, and ? or going
uphill.  And so if you?re on a bike, and you get ? you can get
stopped by something else, like a rock or something.
Teacher:  So if we?re thinking of your analogy to a bike, or your explanation
with a bike, what?s stopping it, and this is ?
Adam:  Um, no.  Well, in the situation of dropping the beanbag.  Like, um,
it?s thing is the ground, and because the beanbag is running against
the ground, um ?  it?s getting slower.  Because like the beanbag is
um ?  getting ?  I don?t know how to explain this.
Connor, in lines 55 ? 62 (transcript 4), does not address Adam?s prediction and
explanation, but instead offers his own analogy:
Connor:  I would think the bean bag would ? might fall behind where you
want it to fall because when I put ? when I played baseball ? they
always said don?t throw the bat because it might hit the catcher and
not one of the um person because we?re using metal bats, and ? so
we drop it, you drop it and then you ? . Well, when I drop it, it
usually swings backwards; it wouldn?t be behind the plate instead
of the front of the plate.
Connor?s analogy changes as he considers it further (transcript 4, lines 71 ? 78):
Teacher:  Why do you think it fell behind?
Connor:  Well actually it didn?t mostly.  It got on the side or in front because
?  well because you?re supposed to drop it because you don?t need
a bat while you?re running the bases.  Once you drop it, I?m just
thinking also, what Adam is ?  well a bus ?  well if you were on a
bus and you had uh, this little leaf that you found, and the window
was open, and you drop it, it will go ?  it?ll be going backwards.
It is interesting to note that the original analogy that Connor introduced should lead him
to predict that the beanbag falls ahead of you when you drop it ? and the analogy he then
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offers that is consistent with his prediction is a scenario that he has not likely done.  This
phenomenological aspect of analogies will be explored later in this chapter.  In terms of
the chain of analogies that is being constructed, Connor has moved from the very similar
running-drop to dropping from a bus.  In lines 83 ? 94 (transcript 4), another student
challenges Connor?s analogy and he tweaks it further:
Lauren:  Because I think that?s cause ?  you?re talking about a leaf that?s
falling?  That?s because the ?  it?s sort of ?  the bus is going back,
so it?s making like the air move.  And the leaves are really, really
light, so the reason they are going backward is because ? .  Um,
well it?s going so fast ?  a bus is like going so fast that it?s
probably making the air go that way.  So that way the leaves are
going that way.
[Many talking in disagreement.]
Connor:  What if you did it with a rock?  The same thing will probably
happen with a rock.  Because you are probably like a bus, that you
make the air come ?  no one moves, don?t you notice that um,
objects like in cars or something ?  when you?re going really fast
on your bike that are ?  that um, you sometimes, [inaudible] and
leaves your ankle on your back step and actually move.
The distinction between the ?little leaf? and the rock are considered in lines 146 ? 150
(transcript 4), and an analogy between the leaf and a feather are drawn:
Kamran:  But, if you ?  cause you know a feather is ?  it, it, it goes with the
air just simply its a light.  That?s why ?  same with a leaf.  A leaf is
very light.  And if you ?  a leaf falls [Inaudible.] goes to air.   It
doesn?t go, ?leaf ?  boom.?  It doesn?t go like that.
And then (transcript 4, lines 188 - 190) the idea of the leaf is tweaked into considering the
weightiness of a tree:
Kamran:  Yeah, because the weight pulls it right down.  If a tree ?  it?s
heavy, and it?s heavier than all the leaves it has, so the leaves will
make it fly.
Through these analogies, we see the students following a chain of reasoning that begins
with a particular phenomenon ? that of the running drop ? and steps through a series of
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family resemblances to arrive at two subcategories of the running drop phenomena.
Kamran?s comment (line 188) explores some confusion in the distinction in these
categories, saying ?the weight pulls it right down.  If a tree ? it?s heavy, and it?s heavier
than all the leaves it has, so the leaves will make it fly ? flies here.  And the tree, it will
just go down.?
These two sections have demonstrated that a feature of student-generated
analogies in science is multiple analogies.  At times these analogies that come up in
student conversations are independent of one another, but often they are related and this
property is one that is neither predicted nor explained by current models of analogy.  The
predominant model of analogies ? in particular analogies in science ? is that of structure-
mapping.  This model is admittedly designed to explain analogies that have been
constructed and the role that they then play.  However, as shown above, it cannot account
for multiple analogies.  Categorization, which inherently involves multiple members and
has been demonstrated to relate items not through any set of formal rules but rather
through family resemblance, more coherently captures the manner in which
spontaneously generated analogies in science are constructed and negotiated.  In the
following two sections, I will focus on the choice of the base for the analogies that
students generate: first focusing on the base as a construction as opposed to a
recollection, and then considering the role that similarity to the target of the analogy
playas in the selection/construction of this base.
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Construction of the base
Depiction of the base in research on analogies
Analogies are often depicted as a mapping from a well-understood base to a less-
understood target, and implicit in this depiction is that the base is something that was
previously known, experienced, considered, and then, during the act of the analogy,
recalled.  The following quotes (with italics added) about analogy reveal this assumption,
in which the analog (or base) is ?retrieved,? ?stored in memory,? ?accessed,? and ?in
memory,? but not ?constructed? or ?created?:
? ?A theory of analogical reasoning must explain how an analog is retrieved.?
(Vosniadu and Ortony, 1989 p 7)
? ?Given a current target situation, retrieve from long-term memory another
description, the base, which is analogous or similar to the target.?
(Falkenhainer, Forbus and Gentner, 1989 p 2)
? ?Mental experience is full of moments in which a current situation reminds us
of some prior experience stored in memory?  (Gentner, 1989 p 199) and,
? ?the chronological first step in an experiential learning sequence is accessing
the potential analog.? (p 200)
? ?Analogical problem solving involves three steps, each of which raises
difficult theoretical problems.  The first step involves accessing a plausibly
useful analog in memory?? (Holyoak and Thagard, 1989 p 242)
Not all researchers describe analogy in this way.  Anderson and Thompson (1989, p 267)
note that the base of an analogy can come from someone?s ?own past? the behavior of
another?or it might come from adapting an example given in a textbook.  The source for
the analogy can be either an explicit experience or a generic or schemalike
representation.?  However, explicit reference to generated analogies as having a base that
is constructed rather than recalled is rare.
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Prototypes in categorization
Categorization research has a similar history, beginning with theories of category
representation as static things stored in memory and recalled for the purposes of judging
category membership.  Though there is no single theory of prototypes and the graded
structure of categories, it was originally argued that a category was represented by a
single representation, which was an amalgam of all exemplars stored in memory.
Judging whether or not a new item belongs in a category involves comparing it to this
amalgam.  And this involves recalling all known instances of the category to construct a
?typical? exemplar that is ?a unified representation rather than separate representations
for each member.?  (Murphy, 2002 p 42)  To address concerns with this initial theory, the
idea of ?feature combination? was added, in which features that are averaged together are
first combined, so that ?
people keep track not just of individual features but configurations of two or more
features.  For example, perhaps people notice how often bears have claws and eat
garbage, or have fur and are white ? that is, combinations of two features.
(Murphy 2002 p. 45)
Concerns with the computational demands these theories make on memory and recall,
schema were introduced.  A schema is ?a structured representation that divides up the
properties of an item into dimensions (usually called slots) and values on those
dimensions (fillers of the slots).  Features of exemplars were then stored with a given
weight into particular slots.  An alternative to this (typically called the ?prototype view?)
is the exemplar view, in which every instance in memory is used in the construction of a
category rather than the average of these.  While both the prototype and the exemplar
view proved generative and had great explanatory power, their representation of mind
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was problematic and the categories they sought to describe proved to be quite narrow, as
Barsalou?s research in ad hoc categories and point-of-view categories shows.
Construction of the prototype
Barsalou?s (1983) studies of ?ad hoc? categories addressed categories that cannot
be interpreted as fixed cognitive structures, such as ?foods not to eat when on a diet,? or
?things to do at a convention.?  These categories, which maintained the
phenomenological properties of more conventional categories (such as a graded
structure), could not be explained simply using the exemplar and prototype theories.
Surely, Barsalou argued, one does not store information on such detailed categories but
rather constructs it.  In later research (Barsalou, 1987) extended this idea, asking
participants to take the point of view of a professor, Chinese person, or ?redneck? in
judging membership in categories.  Again, these categories displayed a
phenomenological consistency with conventional categories but, again, could not be
stored representations.  Barsalou argues that ?rather than being retrieved as static units
from memory to represent categories, concepts originate in a highly flexible process that
retrieves generic and episodic information from long-term memory to construct
temporary constructs in working memory?  (Barsalou 1987).  This is not meant to imply
that there is not stable knowledge in long term memory, but rather that the concepts in
working memory ? the ideas that are pondered, discussed, articulated and reasoned with ?
are temporary constructs and, as such, sensitive to context and goals and inherently
unstable.  This interpretation of categorization Murphy (2002) contrasts with the
prototype and exemplar views, referring to it as the ?knowledge view,? in which ?part of
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categorization and other conceptual processes may be a reasoning process that infers
properties or constructs explanations from general knowledge.?  (Murphy, 2002, p 60-61)
It is this picture that is missing from most analogy research.  In part, this is
because the focus of analogy research on how people understand a given analogy ? in
which case the base of the analogy is present, and one must recall information about that
base.  It is also due in part, perhaps, to the analogy of the mind to a computer in which
context and knowledge construction is irrelevant.  However, when paying attention to the
analogies that students create when discussing scientific ideas, these analogies are not
always drawn to a well-understood base.  Rather, the base may be a construction, as with
our representations of categories.
Construction of the base in student-generated analogies
In the transcripts above, bases that are clearly not recalled from memory are most
profound in the beanbag transcript.  In line 64 of this transcript, Connor has predicted that
the beanbag will fall backwards and has drawn an analogy to dropping a baseball bat.
When the teacher asks ?Why do you think it fell behind??  He, surprisingly, notes
?Actually it didn?t mostly.?  And then he selects a more appropriate analogy:  ?if you
were on a bus and you had uh, this little leaf that you found, and the window was open,
and you drop it, it will go ?  it?ll be going backwards.?  While it is possible that he has
dropped a ?little leaf? from a bus window, he does not claim to have done so and it is
reasonable to imagine he hasn?t.  And in line 91, his suggested analogy ?What if you did
it with a rock?? is based more on a sense of theory than a past experience.  Instead of
recalling this experience from memory, he is constructing it from ?a reasoning process
that infers properties or constructs explanations from general knowledge? (Murphy, 2002
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p 61) ? as has been argued happens in the construction of representations of categories.
Kamran, in line 154 (transcript 4), makes this searching for a representation from
principles (rather than from memory) clear as he reasons through possible analogies: ?A
rock is different, a rock has ?  it?s also like, it?s solid, but it?s not that a leaf isn?t solid, or
a feather isn?t solid.  A feather ?  but you have to ?  it?s very small, and it?s very like thin,
so you kind of say like solid.  But anything hollow, like if you have a paper box??
A second example of such construction of the base of the analogy comes from the
Physics 115 course.  Students were asked to make sense of several phenomena of circuits.
In one course, an analogy was suggested: imagine a vacuum cleaner sucking up beads,
and the light bulbs were like small filters in the tube of the vacuum cleaner.  In another
course, the analogy was drawn to a hose that is already full of water (so that it takes no
time for the bulbs to light) that has small holes in it (which represented the light bulb).  In
each case, the students are seeking to explain their reasoning via an analogy that they
have constructed rather than one they are simply recalling.  Again, this underscores the
fact that students are not mapping from a well-understood base onto a less-understood
target.  The consistency of the constructed bases of analogies will be explored in the
following chapter, in which I will show that these analogies are constructed from the
schemas or p-prims that the students have activated; for now it is important to note that,
consistent with the spontaneous construction of categories, the base of an analogy can be
a spontaneous construction.
A final example of the constructed base of analogies comes from two
undergraduate students who are trying to solve a problem in their quantum mechanics
homework assignment.  In this problem, they are asked to find the total angular
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momentum, which the first student, Ben, believes you can answer from first principles.
Anselm argues that understanding a simple case does not necessarily mean that you will
be able to solve the more difficult problem, and explains this using a constructed (as
opposed to a recalled) analogy (transcript 5, lines 15 ? 27)
2
.
Ben:We should be able to figure this out from today?s lecture.
Anselm:No you shouldn?t.
Ben: He?s gonna explain in detail probably Wednesday how you
actually get to J [the total angular momentum].
Anselm:But see you?re doing the wrong thing.  ?Cause you?re assuming
that if you have the example: ?suppose there?s a charge here,
what?s the electric field due to it??  You can figure out ? suppose
you have Bugs Bunny, and he?s charged, what?s the electric field
around his ears?  Alright?  Because you have a simple example
when they?re both the same, you?re not going to be able to figure
out exactly what you?re supposed to do when the rules weren?t the
same.
I will revisit this transcript below for its evidence of a variable representation of the base
of the analogy ? here I would just like to note that Anselm?s analogy, ?suppose you have
Bugs Bunny, and he?s charged, what?s the electric field around his ears,? is not a problem
that these students have been assigned in the past and tried to solve.  Rather, Anselm is
constructing this as a representation of a category of problems that cannot be solved from
first principles.  The category is an ad hoc construction and its representation, Bugs
Bunny?s electrically charged ears, are similarly an ad hoc construction.  Constructed, I
argue, by categorizational reasoning; and clearly not a map from a known base to an
unknown target.
                                                  
2
 Ray Hodges, who was a TA in the room with the students at the time, aided in this
interpretation of these comments.
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Near and far transfer analogies
Research on prototypes and research on transfer
The initial research on categorization established a graded structure of categories,
in which some members are recalled more quickly and with greater frequency than
others, are judged to be better exemplars of the category and are recognized as category
members more rapidly than others.  Theories behind these effects are varied and will be
explored in the following chapter but, as explained in the above section, it is not due to
simple feature matching, but seems to be rooted in reasoning processes and explanations.
Analogy research has shown that near-transfer analogies are far more easy to
achieve than far transfer analogies ? where near-transfer refers to analogies that are
within-domain or have similar features, while far-transfer analogies are those that bear
little superficial resemblance to one another.  As explained in the previous chapter,
however, this may be an effect of the particular style of research being conducted and the
lack of meaningful context in which the analogies are situated.  And, contrary to this
research, ?far transfer? analogies do occur in student discourse and are not uncommon.
If one interprets analogies as categorization, then perhaps the ideas of near and far
transfer are not relevant.  Instead, one would expect the base of the analogy to be, instead
of superficially near, prototypical of the category.  In this section I will present evidence
that the base of analogies are not chosen for the similarity of features or ?nearness.?
Arguing that the choice is, instead, prototypical requires appeals to theoretical
considerations that will be addressed in the next chapter.
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Examples from student-generated analogies
Returning to the cup/water and cat/basket set of analogies, Miranda (line 59)
claims to have done something that is quite similar to the case in question (involving
cups, water, and dropping), but her initial analogy to explain her reasoning came from a
much less similar experience.  In line 25, to explain why she believes water will not spill
from the cup as it falls, Miranda offers the analogy of spinning her toy cat overhead in a
basket.  Later, in line 59, Miranda says: ?I?ve also done that in the bathtub when you?ve
got your cup, I?ll like I?ll fill it with water put my hand and drop it the water stays in until
it hits the bathtub and then it goes everywhere.?  If the base of the analogy is arrived at
through a process of retrieval from memory, as many models of analogy imply, one
would expect near-transfer before far-transfer, as the similarity of features would be key
in retrieving the analogy.  If, instead, the base of an analogy is a construction from this
categorization, one would expect a base that is prototypical of the category it is asserting;
near and far transfer are not significant questions in this framework.  Rather, that the
cat/basket analogy is chosen first, is much more convincing and is referred to repeatedly
in the classroom can be understood by its prototypicality (or centrality) in the category it
serves to describe.  Additionally, concerns that Miranda is ?making up? the analogy to
the cup in the bathtub (this concern has been voiced in discussions with others regarding
the analogy) are not important: given that Miranda chooses these two analogies,
regardless of whether they are experienced or fabricated, she does so in this order and the
class responds to them in this way.
Miranda serves as a particularly powerful example because she refers to two
analogies and chooses the ?further? analogy first and it is this analogy that holds sway in
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the classroom and is repeatedly referred to by the teacher and students.  Far transfer
analogies are not at all uncommon.  Other examples from analogies above are:
? comparing a cup of water to a toy cat in a basket
? envisioning electrical charges in metal as fish in a fish bowl
? drawing an analogy between motion of charged particles in Styrofoam and the
motion of water through a sponge
? drawing an analogy between motion of charged particles in metal and using
stepping stones
? comparing dropping keys on an x to hitting a rock on your bike, and
? explaining your approach to solving a total angular momentum problem to
solving a problem of the electric field around Bugs Bunny?s ears.
While many of the above analogies are creative and intriguing, none are outside of the
bounds of ?normal? conversation in science, and in many cases a superficially ?closer?
analogy could seem too close and a more strange statement to make than the far transfer
analogies.  In the next chapter I will argue that these analogies ? the ?far? analogies ? are
appropriate and, indeed, expected over near transfer analogies because of their
relationship to the category that they serve to represent.
Variable representation of the base
In an effort to understand analogies, the majority of research from cognitive
science and education has focused on the comprehension of analogies provided by the
researcher or teacher, or the application of a desired analogy (for example, in Holyoak
and Thagard?s study of transfer).  Such studies limit the variability of representations of
concepts that researchers can observe.  The study presented here originates in research on
student inquiry in science classrooms.  These classrooms place a premium on student
reasoning and explanation of ideas, and, as such, allow for student-generated analogies
(which are far more rare in a classroom with a focus on content goals over process goals).
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In this section I will present two different kinds of analogies: one set of analogies
are presented to explain different ideas about the mechanisms of a scientific
phenomenon; the second set are analogies used to explain the speakers? conceptions of
the epistemological type of scientific phenomena ? that is, what is the nature of
knowledge that applies to this phenomenon.  In each case, the representation of the
concept to which the analogy is drawn is chosen from among many possible
representations, and categorization, because of its fluid nature and flexibility, can explain
this choice of representation.
Variable representation: Example 1
In the following transcript, there are multiple possible representations that the
base can take, and the one that is taken depends on the conceptions of the target.  While
not entirely in discordance with structure-mapping (the process of alignment could be
considered choosing the representation), a categorization framework shifts the
importance and nature of analogy; when someone says ?a is like b? they are not saying:
?the structure of this one thing, b, has a lot in common with the structure of this other
thing, a.?  Instead, the assertion of analogy is more akin to ?a belongs to a class of things
typified by b ? it?s the same kind of thing.?  Where a ?kind of thing? defines a (often ad
hoc) category and more may be brought to bear on a than just the relationships that exist
in b.
This first transcript below is of non-science faculty at the Governor?s School of
North Carolina.  In the lounge of the faculty dorm, they are discussing what happens to a
rock in space as it receives energy from the sun: will it heat up indefinitely or only to a
certain point?  And if it only heats up to a certain point, why does it stop there?  This rock
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is referred to as ?David? because of a previous conversation about the differences
between humans and statues (namely, Michelangelo?s ?David?) in space.  Marc is content
to say that David will reach a certain temperature and stay there, and explains this using
an analogy.  Note that Marc prefers a ?water? analogy of light, and is able to take on a
?money? analogy of light that is contradictory to Vic?s ?money? analogy of light.  This
demonstrates that, one, the choice of base in an analogy is one of ontology and, two, the
base can shift representation according to the ontology ascribed to the target (transcript 6,
lines 471 ? 498).
Marc:  Okay? let?s ?  let?s say David is in a shadow, right?  Okay- he
enters the sun.  The sun bombards him with all this energy right?
So in a second it?s now at 5 degrees.  Can it radiate heat that fast?
No.  So in the next second it?s 10 degrees.  It?s now radiating a
little bit more heat but there?s more energy coming in.  So it gets to
15 degrees.  But at some point it?s radiating enough heat to
stabilize at 20 degrees.
Steve: But why?
Cameron: What?
Marc: Or let?s think of think of like a, think of like a basin, ok?  Think of
a tub.  With the drain open, okay?  The drain is open.  Now if I
open the spigot [Uh huh.] ? if I open it too slow then the tub
doesn?t fill.  But if I open the spigot fast enough there?s water
filling up the bottom, and yet some is also draining out, right?
[Right.]  If I open the spigot up fast enough it doesn?t matter if the
bottom is open, the top will overflow but at some point if I reach
the right point the tub could stay at a certain level, even if water?s
going in and water?s going out, right?  If they came in at the same
rate [Steve: Right, but ? ] the tub would fill up.
Steve: But that?s ? what?s David?s drain?
Vic:  That?s ? this is my question.  What?s David?s drain?
Tom:  Well, what?s the sun?s drain?  The sun is clearly radiating heat and
energy.
Marc:  Yeah I mean that?s just the ?
Steve:  The sun is radiating light.
The conversation continues, and over the next three minutes Marc suggests that things
radiate ? it?s just what they do; like a drain and like the sun, the rock gives off ?radiation
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and stuff.?  To explain why we can no longer see the radiation when the rock re-radiates,
Marc offers that the light changed from a visible form to a non-visible form.  But how
?ROYGBIV? (the colors of the rainbow) become ?infra-ROY? bothers Vic.  The analogy
of light as water and a rock as a tub contrasts with the analogy that Vic invokes to point
out a problem in Marc?s re-radiation explanation in the following transcript (transcript 6,
lines 1038 ? 1054).
Vic: Wait wait wait ? every ? everybody is payin? me money.
Everybody is paying me money in different forms ? in dollars,
five-dollar bills, twenty- dollar bills. [Marc: Okay.] I?m savin? all
of my one dollar bills that I don?t give away ? I do not spend any
of my dollar bills on anything ever.  Which means that I am
gradually accumulating one-dollar bills ? even if I?m spending it in
fives and tens and twenties.  So what do I do when I end up with a
thousand dollars in one-dollar bills that I don?t know what to do
with?
Marc: I?m gonna change that analogy [Others: Groan.] ? or I could keep
it!  I could keep it! Okay?  I?ll keep it.  Fine ? you know what I?m
gonna do with those one dollar bills?
Vic:Tell me.
Marc:Well ? those dollar bills become ?  you, you spend 50 cents of it in
terms of heat and you throw the other 50 cents of it away but we
can?t see those 50 cents because we?re only attuned ?
Tom: You?re losing the analogy.
The choice of this analogy is a largely ontological choice: Marc has a ?water-like?
ontology of light and one can imagine that, just as turning a cup of water into two half-
cups of water needs no mechanistic explanation, turning red (a high energy wavelength)
into infrared (a lower energy wavelength) does not require further explanation ? it
happens ?by the virtue of your existence.?  Marc is making a claim about the kind of
thing that light is ? it belongs to the class of things that water belongs to.  This class could
be described as a conserved quantity that does not come in discrete chunks and flows
easily from one ?container? to another.  Vic has a different conception about the kind of
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thing that light is ? one that is organized in discrete quantities and matches the category
that currency might belong to: net wealth does not count quarters as different from dollar
bills, but in terms of actual objects, they are physically different and do not ?mutate? into
one another.  That this analogy is a statement of categorization (the kind of thing that
light is) is evident in the objection that Tom believes Marc ?loses the analogy? when he
violates this ontology.  Marc, though he wants to ?change that analogy? is able to take it
on by representing money in as a ?fluid? ontology.  It is reasonable that money could be
conceptualized in the manner Marc intends: when conceiving of someone?s bank balance,
it makes sense to think of money as belonging to the ontology of fluids: if I deposit a
quarter my bank will certainly allow me to withdraw a penny.  And gas stations routinely
charge to the tenth of a cent (or at least to the nine-tenths!).  Structure-mapping and other
interpretations of analogies that assume a particular conception to the base of an analogy
miss the point that the claims are being made to signify a class of objects, and that a base
and target can have multiple representations and belong to several different classes (or
categories).
Variable representation: Example set 2
In this section I present three transcripts: the first two are conventional analogies
? the ?tree in the forest? conundrum (albeit with slightly different features), and ?apples
and oranges? ? and the third is a novel analogy regarding Bugs Bunny and electric fields.
The first two analogies are intended to establish the categorization model and provide a
means of interpreting the third analogy: if the first two are instances of categorization, as
I believe is apparent, then so must be the third.  And, as with the variable interpretations
of ?money? (as fluid and divisible or ?hard? currency), the base of these analogies may
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be interpreted in many ways.  Furthermore, to echo a prior claim, the third analogy is
novel and cannot be argued to be a well-understood base that is recalled from memory
but rather is constructed on the fly to represent an ad hoc category.
The first transcript is from a conversation in a 5
th
 grade classroom.  The class has
been discussing light and heat and the relationship between the two.  After reasoning that
light ?contains? heat because sunlight feels warm, Lisa notes that the light from overhead
fluorescent lights does not make you hot (transcript 7, lines 86 ? 102).
Lisa: I think that sometimes, well, most of the times, light is not always
containing heat.  Like, like this light up here, it?s not con ? it?s not,
its not ?
Dashawn:Burning?
Lisa: Yeah, like making you hot.
Kyle: Yeah it?s not making anything hot.
Anna: But it's just ? what if you go up there and touch it?  It would ? ?
Brian: That's because your finger is an object.  When it hits something it?s
hot.
Teacher:Oh, I see.  So you get, there's a reaction when you touch light.
Brian: But it's also a question like, um, if a door slams ?  if a locker door
slams and no one?s around to hear it, does it make a noise?
Because you don't know if ?  if you don't touch it and the light is
making heat and making the air hot.  You won't know.
A variant on the standard philosophical question ?if a tree falls in the forest and no one
hears it, does it make a sound?? is raised here as an analogy to explain that not only does
light need to ?hit something? to make it hot, but ?it?s also a question? of if, in the absence
of a measurement, heat may not be a meaningful concept.  It is an argument about the
kind of thing that heat is.  This analogy could be interpreted in a structure-mapping
framework in that it maps elements from the secondary onto the primary; one could align
the locker with the light, the slamming with the light hitting a finger, and the heat with
the sound.  But categorization highlights a very different aspect of this analogical
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reasoning, namely the ontological goal of expressing the kind of thing that heat is.  The
analogy is not drawn for the purpose of comparing two items and making a projection,
but to make a sophisticated claim about the nature of light and heat.
An epistemological analogy is drawn in the following transcript.  The two
students are undergraduate quantum mechanics students working on a problem set
together.  They encounter one problem (determining ?J,? the total angular momentum)
and have trouble making headway from first principles so they try to work backwards
from the answer.  Ben can kluge together numbers that are present in the problem to
arrive at the answer, but the way in which he assimilates these numbers is nonsensical, as
he notes in the transcript below (transcript 5, lines 65 ? 74).
Ben:All right, look at this ? look at this.  If you take all the positives
and add them together, you get eight-ninths.
Anselm:Oh, oh.
Ben:You take the negative, you get one-ninth.
Anselm: Yeah that?s ?
Ben: But you?re mixing apples and oranges.  It?s dumb!
Anselm: Yeah that?s so messed up, yeah that?s not the answer.  If I just
ignore the fact that I?m in the three-halves one-half and I?m in the
one-half one-half and I just add them all together?
Here the assertion of ?you?re mixing apples and oranges? is clearly not a matter of
structure-mapping, but categorization.  The ?apples? are not aligned to a specific element
present in the physics problem.  Rather, ?apples and oranges? has come to represent a
category of dissimilar things erroneously compared, and Ben?s statement is a categorical
assertion of the type of thing that he was doing.  This claim is not new.  Glucksberg and
Keysar (1990) argued that the interpretation of a metaphorical statement was a process of
categorization.  As a means of accounting for this theory of metaphor interpretation as
categorization, it has been argued (Gibbs, 1992, Bowdle and Gentner, 1999) that
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conventional metaphors, such as apples-and-oranges, are interpreted as categories, but
that interpreting novel metaphors is structure-mapping.  In accordance with Bowdle and
Gentner?s research and the categorization theory of metaphor, the creation and assertion
of the conventional metaphor ?you?re mixing apples and oranges? seems clearly an
instance of categorization and not structure-mapping.  However, I claim that categorical
assertions are also made with novel analogies.  Bowdle and Gentner (1999) have argued
against that, at least in the interpretation of novel analogies, but perhaps generated novel
analogies could be considered assertions of categorization.  In the following transcript a
novel analogy is used in a similar way to the ?apples and oranges? and the ?tree in a
forest? conventional analogies.  The base of the analogy is clearly novel, not well-
understood, and invented on the fly, but its role in the analogy is no different from that of
the conventional analogies.  The elements of the base of the analogy do not clearly map
onto elements in the target, and the claim posed by the analogy is not instantiated by
projection but by categorization.  Again, a categorization model of analogy is far more
capable of understanding the role of the base than a structure-mapping model.
The students in the previous transcript continue to work on the problem of total
angular momentum (transcript 5, lines 15 ? 26).  Ben believes that they should be able to
determine the answer because they know the constituent angular momenta and have some
background in adding these to find the total.  Anselm argues that this does not necessarily
mean they can solve this more complicated problem, and draws a novel analogy to
explain himself, that of the problem first introduced above of the electric field around
Bugs Bunny?s ears.  Anselm?s analogy is to say that this problem of adding angular
momenta is like finding the field around an oddly shaped object ? not because of any
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similar structures (the charge and Bugs Bunny don?t correspond to any particular items in
the angular momentum case) ? but because you cannot always find the answer to
complex problems using knowledge about simple problems.  Knowing the field of an
electron doesn?t mean that you know the field around an oddly shaped object, like the
ears of Bugs Bunny.  In this analogy, the structure of the problem (calculating total
angular momentum from components) is only very weakly similar to the structure in the
Bugs Bunny scenario (determining the field around an oddly shaped object); the
similarity, and thus the analogy, lies in the fact that they are a similar type of problem.
Furthermore, the base here is created ?on the fly? ? the idea that we have a stored
representation of Bugs Bunny as an odd shape for the electric field to take seems highly
unlikely.  Far more plausible is the spontaneous construction of an ad hoc category of
things that are more complex than their simple parts.  The analogy is not from a well-
understood base to a poorly-understood target, but instead the base is constructed
spontaneously to represent an ad hoc category ? that of problems you cannot solve from
first principles ?  and asserts membership in this category.  In defense of structure-
mapping, it could be argued that the elements being aligned were not the particular
elements of the problem at hand (the charge, the Bunny, and the constituent angular
momenta), but rather the problems themselves are elements in a larger structure.  But
such a claim would bring us back to the problem of representation.  The Bugs Bunny
analogy can be represented as a structure in itself, or as an element in a larger structure,
and choosing which representation is the one to use is a problem that structure-mapping
does not address.
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Previous Claims of Analogy as Categorization
The claim of analogy as a categorization phenomenon is not new.  The greatest
proponents of this theory are Glucksberg and Keysar (1990).  Why have their arguments
failed and what does this dissertation bring to bear that others have not?  First, I argue
that past claims are overwhelmingly with regard to the interpretation of analogies and the
claim I am making is about the assertions made by a generated analogy.  Interpreting
analogies allows for a much more narrow range of representations than generating
analogies does.  Second, as noted by Gentner, Bowdle Wolff and Boronat (2001) the
?category-based approach is ?localist:? it assumes a metaphor conveys a categorical
relation between a particular pair of terms.  Thus this approach addresses single
metaphors and not extended systems of metaphors.?
As demonstrated above, I am not making ?localist? claims with regard to
analogies.  Rather, I will argue in the following chapter that analogies assert categories
based on schemas: categories are defined only within a particular cognitive model, and
this cognitive model carries with it a quite extended schema.  Previous arguments for a
categorization model of analogy used a far more classical model of categorization.  The
following section addresses a final phenomenological property of analogies ? one that is
not particularly demonstrative of the categorization framework at first glance, but within
a particular ontology of mind, together with a current understanding of categorization, is
particularly revealing of the reasons why we use analogies and the cognitive work that
they do and is strongly supportive of a categorization framework
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Analogies as Negative Assertions
In this final section on phenomenology, I demonstrate that analogies are asserted
as an alternative to another way of understanding a particular phenomenon.  In part, this
is a definition of analogy that distinguishes analogy from other forms of similarity, but it
is a definition that arose from the data.
In observing what seemed to be analogies and an effort to understand the
cognitive work that these insightful moments of analogy did for the students, it became
clear that what I understood to be analogies are the statements of similarity that are an
unexpected similarity.  This is a matter of convention, but one that, as I will demonstrate
in the following chapter, is a rather powerful convention that is consistent with a
particular ontology of mind and understanding of student reasoning.
Violations of expected schemas
In Women, Fire and Dangerous Things, Lakoff (1987), who argues that our
categories derive from and are defined within cognitive models, notes that one indication
of cognitive models is the use the term ?but? ? as in, ?she?s a mother but she has a job.?
Such a statement makes far more sense than ?she?s a mother but she doesn?t have a job?
? which sounds strange.  He argues that, to understand these statements and why one
sounds so strange, we must turn to the idea of the cognitive model.  ?Mother? exists in a
complex model of nurturance and work, and ?working mother? is defined relative to this
model (Lakoff, 1987 p 80):
A working mother is not simply a mother who happens to be working.  The
category working mother is defined in contrast to the stereotypical housewife-
mother.  The housewife=mother stereotype arises from a stereotypical view of
nuturance, which is associated with the nurturance model.  According to the
stereotypical view, mothers who do not stay at home all day with their children
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cannot properly nurture them.  There is also a stereotypical view of work,
according to which it is done away from the home, and housework and child-
rearing don?t count.  This is the stereotype that the bumper sticker ?Every Mother
Is a Working Mother? is meant to counter.
The housewife-mother stereotype is therefore defined relative to the
nurturance model of motherhood.  This may be obvious, but it is not a trivial fact.
I mention this because the analogies that have been presented in this chapter all have this
?but? quality to them.  As in, ?the cup is overturned but it doesn?t spill? ? which makes
far more sense than ?the cup is overturned but it spills.?  As Lakoff says, this may be
obvious, but it is not trivial.  In the following chapter I will explore why this is not trivial
? how it can inform us about cognitive structure and our cognitive models of the world,
and then I will revisit this again in considering what it means for education.  Here I
present it as a final phenomenological property of analogies ? student generated analogies
in science are ?defined in contrast? to expectations, just as ?working mother? is defined.
Examples from student-generated analogies: Analogies as negative assertions
The analogies presented in this chapter are as follows:
? an overturned cup of water does not spill like other overturned cups of water, but
keeps the water inside, like a toy cat swung overhead in a basket (transcript II),
? density of an object enables/enhances the motion of charged particles as opposed
to hampering it, just like a countertop lets water flow while a sponge doesn?t, or
stepping stones must be closely spaced to allow you to step (transcript I),
? ?authenticity? is not a property of an activity, but is a more ?interactive? adjective
? one that is not solely an attribute of curriculum, but arises from the interaction
of the student and community with that curriculum, just as worship is an activity
that requires a practitioner and a community of practice (transcript 3)
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? dropping a beanbag is not like (at second glance) dropping a bat, but can be
understood as dropping a leaf from a bus, or
? it is not like dropping a leaf from a bus, it is like dropping a rock from that bus
(transcript 4),
? a quantum mechanics problem is not one to be solved from first principles, just
like you could not use a simple field equation to find the field around Bugs
Bunny?s ears (transcript 5),
? a rock is not just a container of heat but also gives off heat, like sinks not only
hold water but also have a drain (transcript 6),
? light cannot be received at one frequency and given off in a different frequency,
just as one cannot be received a quarter and turn it into dimes and nickels, or
? light can change from red to infrared, just as one can have a dollar?s worth of
wealth and spend only fifty cents (transcript 6),
? in order to determine if light contains heat you must put your finger in the light,
just like the question of sound in the absence of a listener (a door slams and no
one hears it ? transcript 7), and
? numbers normally can be added together or multiplied unproblematically, but
when these numbers mean something (as in a quantum mechanics problem)
adding these numbers might be akin to mixing apples and oranges ? it doesn?t
make sense (transcript 5).
In each of these analogies, they are (perhaps implicitly) not only a claim of similarity ?
the cup of water is like the cat in the basket ? but equally, if not more significantly, they
are a claim of dissimilarity.  They arise as contradictions to what is expected.  Numbers
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usually can be added unproblematically (of course, this is only true in a mathematics
classroom and rarely, if ever, true when applied to life); overturned things usually spill;
objects are usually seen as generators of energy or receivers of energy but not both ? not
a tub with a drain; whether or not something happened is not usually contingent upon
someone being there to observe it.  The way in which this plays into a categorization
framework will be explored in greater detail in the following chapter.  For now, I just
would like to note that this property of analogies is not explained by structure-mapping
and other theories of analogies.  When it is accounted for, it is added on in an ad hoc
manner ? by assuming that context or goals is significant and needs to be accounted for ?
but accounted for in a somehow distinct way (Gentner and Markman, 1997).  In their
model of analogy, this phenomenology is not inherent to the analogy, but part of the
context.  In the following chapter, I will argue that it is fundamental to analogy and to the
role that analogy plays.  Furthermore, it allows for a definition of analogy that
distinguishes analogy from routine categorization and similarity.
Conclusion
This chapter, which details the similarities between student-generated analogies
and properties of categorization, may seem to disregard the adage: ?Correlation does not
imply causation.?  Categorization and analogy are both related to similarity, so they
should have some phenomena in common ? but that does not imply that they are the same
thing, arise from the same cognitive mechanism.  The following chapter is designed to
address causation by introducing an ontology of mind that can account analogies as
arising from the same cognitive mechanism as categorization ? the only distinction being
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that analogies offer an alternative categorization to what is expected.  The correlation
between analogies and categorization are summarized below.
Analogies that students spontaneously generate in science classrooms are often
presented in multiples: analogies that are all members of a more general class of
phenomena, in which the generated analogies are in agreement with one another, or
analogies that are tweaks from one to the next, representing a class of phenomena that
bear a family resemblance to one another.  These analogies are often ?far transfer?
analogies that bear little superficial resemblance to their target.  They may be constructed
on the fly as opposed to recalled from past experience.  And the base of the analogy may
have a variable representation that changes as the analogy is negotiated.  These
phenomenological properties of student-generated analogies reflect properties that are
representative of categorization phenomena: categories have multiple members and are
often related not by strict rules or similarity of features, but a family resemblance that
links the various members of the category.  Categories are represented by a category
prototype, which can account for the prevalence of ?far transfer? analogies.  They are not
stored representations that are recalled, but rather are constructed in a flexible process.
Gibbs, in his article refuting Glucksberg and Keysar?s (1990) theory of analogy as
a categorization phenomenon, claims (Gibbs, 1992):
Most metaphorical expressions instantiate, sometimes in spectacular ways,
preexisting metaphorical mappings in long-term memory whereby knowledge
from a target domain is partially understood in terms of a dissimilar source
domain? Metaphors do not simply arise out of temporary, ad hoc categorization
processes perhaps to meet particular communicative purposes.  Instead, metaphor
is a fundamental characteristic of how people categorize and makes (sic) sense of
their experience.  Verbal metaphors?reflect particular instantiations of
metaphorical categorization schemes in long-term memory.
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These claims take into account the systematicity of metaphor and polysemy (one word
with multiple, related meanings) and are designed to address the more common
metaphorical expressions and not spontaneously generated ones.
However, I would like to argue that these claims cannot be extended to
spontaneously generated analogies.  In light of the above analogies from scientific
discourse, such as Bugs Bunny, light-as-currency, and dropping rocks from buses, these
claims of metaphorical expressions as ?preexisting metaphorical mappings? are clearly
not true of analogies in general, and particularly not true of student-generated analogies
in science, in which they are often investigating phenomena for which they have limited
experience and hence no established metaphorical mapping.
Perhaps this inconsistency between the findings from studies of metaphor and the
study presented here of student-generated analogies in science is due to inappropriately
conflating the two ? established metaphor and spontaneously generated analogy.
Metaphors, such as the parallels between the ways in which we discuss arguments and
the ways in which we discuss war, are ubiquitous in the English culture and apparently
?preexisting,? while a student inventing a language for discussing quantum mechanics or
falling cups of water must be somehow distinct from using a common language to discuss
arguments.  However, if I conflate these here ? metaphorical expressions and student
generated analogies ? it is because the literature is unclear on the division between
metaphorical expressions that are pre-existing and those that are more akin to student-
generated analogies in science.  At best, the literature virtually defines analogy to be
structure-mapping and puts metaphor on a distinct footing because of this definition ? but
such a definition rules out many of the instances of analogy that are discussed here.  This
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does not negate the fact that metaphor and analogy are a fundamental characteristic of
how people categorize and make sense of experience ? but such an idea need not imply
that metaphors and analogies (that is, the mapping of two domains) and their associated
categories (that is, the set of phenomena that are consistent with a particular analogy) are
fixed representations.  Instead, I will argue in the following chapter that what is stored in
long-term memory are particular schemas that may be combined in any number of ways
and give rise to what appear, at times, to be stable categories.
In a purely phenomenological sense, the properties of analogies outlined above
are consistent with properties of categorization.  Categorization does not require that
there be stored representations of concepts or categories that we recall, but rather that
?concepts originate in a highly flexible process that retrieves generic and episodic
information from long-term memory to construct temporary constructs in working
memory?  (Barsalou 1987).  In the following chapter, I will outline an ontology of mind
that is consistent with the findings from categorization and can account for the analogies
described here.
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Chapter 5:  The Ontology of Mind
Introduction
This chapter takes the properties of student-generated analogies in science that
were detailed in the previous chapter and introduces a theory of mind that can account for
these, explaining how a particular ontology of mind can begin to account for findings in
categorization and details why these findings should be expected to apply to analogies.  I
begin with an introduction to the idea behind ?ontology of mind:? what are the things that
researchers have attributed to the mind and what does research show to be the
fundamental building blocks of thought?  I first review research in cognitive science and
education that treats the mind as ?having? representations for concepts, and then
introduce challenges to this ontology of mind.  I introduce an alternative to this ontology,
in which smaller schemas are the things the mind ?has? and these building blocks are put
together into larger models that are in turn used to construct representations for concepts.
The consistency of this understanding of the mind with categorization and, in turn,
analogies is then explored.
History of Ontology of Mind and Description of the Chapter
Perhaps drawing from an analogy to computers, concepts have long been treated
in cognitive science and education as internal mental representations that are then acted
on by computational processes.  Such research would say that a student ?has? a concept
or ?has? a misconception.  This assumption of concepts as stable representations and its
implications on the ontology of concepts in the mind has been called into question in the
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last decade in several fields, most notably by a paradigm in education of situated
cognition.  Situated cognition claims that knowledge is intrinsically situated, ?being in
part a product of the activity, context, and culture in which it is developed and used?
(Brown, Collins and Duguid, 1989) and, as such, one cannot discuss what ?thing? a
student knows ? the very ontology of knowledge as thing is what they call into question.
Despite these concerns about mental representations, the most widely accepted and used
model of analogy is Gentner?s 1983 structure-mapping theory, a theory that ascribes
representations to concepts and then acts on these.  But there are multiple ways in which
we represent the concepts used in an analogy as demonstrated in the previous chapter.
?Money? can be pictured as a fluid kind of substance or a hard currency kind of
substance.  ?Apples and oranges? can refer to apples and oranges or to a more general
category, just as ?bugs bunny?s electrically charged ears? can mean just that, or can mean
a strange shape with an intractable solution.  As such, mapping the structure of a concept
in an analogy must first entail creating a representation that can be mapped, a process
that structure-mapping does not explicate.
If the mind does not have static, unitary representations for concepts, what is the
ontology of mind?  Alternative to the unitary ontology of mind that is inherent to many
models of analogy is a manifold ontology, as expressed by schema theory, idealized
cognitive models and phenomenological primitives.  These theories have been employed
in explaining the graded structure of categories.  As an alternative to structure-mapping, I
argue that categorization, in the modern, non-classical sense (arising from cognitive
models), more effectively describes analogical assertions ? not only because of the
phenomenological similarities between categorization and student-generated analogies,
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but also because of the ontology of mind implicit in a categorization framework of
analogy.
This chapter is divided into two parts.  In the first, a theoretical account and
literature review, I will sketch the basic idea behind structure-mapping, highlighting the
assumptions that it makes about the representation of concepts in mind ? assumptions of
the ontology of mind.  I will contrast these assumptions with concerns from cognitive
science, linguistics and education that argue against stable, large-scale structures of mind,
and detail the alternative theories that account for this manifold ontology.  I will then in
the second part turn to student-generated analogies in science and show how these
theories, in particular phenomenological primitives and idealized cognitive models, are
consistent with these analogies.
Section 1: A theoretical account of the ontology of mind
Structure-mapping
As explained in the previous chapter, structure-mapping theory argues that
interpreting an analogy involves both alignment and projection.  The process is described
in Bowdle and Gentner (1999):
Structure-mapping theory assumes that interpreting a metaphor involves two
interrelated mechanisms: alignment and projection.  The alignments process
operates in a local-to-global fashion to create a maximal structurally consistent
match between two representations that observes one-to-one mapping and parallel
connectivity (Falkenhainer, Forbus and Gentner, 1989).  That is, each object of
one representation can be placed in correspondence with at most one object of the
other representation, and arguments of aligned relations are themselves aligned.
A further constraint on the alignment process is systematicity:  Alignments that
form deeply interconnected structures, in which higher-order relations constrain
lower-order relations, are preferred over less systematic sets of commonalities.
Once a structurally consistent match between the target and base domains has
been found, further predicates from the base that are connected to the common
system can be projected to the target as candidate inferences.
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The claims from this theory that I intend to highlight are the single representation of a
structure that is derivative of the base, and one-to-one alignment of objects in that
representation.  In particular, this theory assumes that concepts have representations that
are then operated on; in a sense, the base of the analogy is primary and the structure ? a
structure ? ?belongs? to that base.  The variability and the stability of this structure, the
representation, are not explicitly addressed in the original theory, nor is it necessary to
assume that the structure is a stable and invariant property of the base ? however, the
associated computational model of structure-mapping, the Structure Mapping Engine
(Falkenhainer, Forbus and Gentner, 1989) consistently presents these representations as
unitary cognitive structures belonging to the base.  But just as research from
categorization reveals that the concept of a leap has no simple propositional structure and
no unitary cognitive representation as a categorical construct, neither will ?leap? or other
concepts have a single representation and structure when applied in an analogical
construction.  Defining the representation of a concept for mapping in a structure-
mapping theory is not a simple act of recall as the theory implies ? it is not our concepts
that have stored representations that we simply recall, but rather our schemas that do or
do not apply to concepts.  Failing to address this presumes either that there are stable,
unitary representations or that the retrieval of one representation from the manifold that
exist is not a crucial element to analogy ? both of these assumptions are challenged
below, first in a review of the literature on variability of conceptual representation and
then I turn to student-generated analogies and evidence against stable, unitary
representations of concepts.
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The arguments for variability in conceptual representations
Research from psycholinguistics, education and categorization has identified
manifold representations of concepts and argued that when concepts are manifested as
singular, stable structures this does not imply a unitary structure to concepts in the mind.
These findings are further detailed below.
Psycholinguistics
Recent psycholinguistic theory has suggested that the mental lexicon, instead of
being organized in a dictionary-style, is far more like a thesaurus.  That is to say, the way
our minds represent words is not so much as obeying rigid definitions with propositional
structure, but rather the meaning of one word is tied to a network of related words ?
words that have appeared in similar contexts, words that have appeared in context with
that word, and words that have related meanings.  Computationally generated lexical
networks have been developed to represent the lexical network of the English language
(one example is the well-known Wordnet, by Fellbaum, 1998).  These thesaurus-like
structures link words in definitions into a network using various algorithms.  Gaume et al.
(2002), building on categorization research that they summarize as establishing the
?conceptual flexibility? as opposed to ?rigid and discontinuous categories,? argue that
words themselves constitute categories and contend that these lexical networks weave a
?mental lexicon distributed around metaphoric poles.?  Amin (2001), in a cognitive
linguistics study of heat, makes a similar claim about certain conceptualizations ?as
dynamic constructions at the moment of use,? finding that ?a stable assignment for the
ontology of ?heat? is absent from the layperson?s core understanding, but rather emerges
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in specific explanatory contexts.?  (p 38)  Quinn (1987) has reported similar findings with
regard to the concept of marriage as having conceptual flexibility:
Quinn (1987) has found, in studying conversations about marriage in minute
detail, that each spouse in a marriage has multiple, and often contradictory,
understandings of what marriage is.  But it is common in a discussion of marriage
for a spouse to shift mid-sentence to a different understanding which is
inconsistent with the one they sentence started out with.  (Lakoff, 1987 p 215)
As with claims from situated cognition, these findings counter the relatively frequent
assumption in cognitive science that there are single, fixed mental representations of
concepts.  In drawing analogies between a base and target, assuming a single
representation of that base will fall short of explaining the power and feat of the
analogical mapping, as demonstrated in the transcripts presented in the previous chapter.
Students are able to shift representation of concepts used in an analogy and choose to
represent a concept as an epistemological (as with ?Bugs Bunny? in transcript 5),
ontological (as with ?authenticity? in transcript 3) or mechanistic (as with ?ice skating? in
transcript 1) statement.  Categorization, in which categories arise from cognitive models,
as I explain below, allows for this flexibility and accounts for the nature of analogical
reasoning that students display.
Education
In addressing student difficulties in physics, several researchers refer to common
?misconceptions? that students have and propose curriculum to address these (e.g.,
Doran, 1972; Caramazza, McCloskey and Green, 1981; Griffiths and Preston, 1992;
Brown, 1992).  Implicit in these statements, and their associated curriculum, is that there
are stable, consistent conceptions students have that educators can find and change.
However, Taber, in a study of students? conceptions of bonding in chemistry (Taber,
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2000), found that students employed different representations to the same concept on
different occasions.  He reports that the idea that students cognitively ascribe a particular
structure to concepts is a reflection of ?the researchers? conceptualizations ? explicit or
tacit ? about the nature of cognitive structure.?  This claim echoes Barsalou?s criticism of
the paradigm in categorization research that aimed to define the structure that students
ascribe to categories (Barsalou, 1987):
When investigators use linguistic analysis to determine prototypes, definitions,
and idealized cognitive models, they appear to assume that there are invariant
concepts in long-term memory that need to be fully characterized.
Similar to Barsalou?s findings (if you look for variability in representations of categories
you will find it) Taber found that the idea that students hold a particular conception of a
scientific phenomenon is flawed.  An individual learner can simultaneously ?hold in
cognitive structure several alternative stable and coherent explanatory schemes that are
applied to the same concept area?  (Taber, 2000).  A theory of mind to account for these
findings involves schema theory, detailed below.  I first return to the evidence from
categorization research that tells a similar story of variability, and then introduce schema
theory, which can account for these findings and can account for the graded structure of
categories.
Categorization
Rosch (1973) established that human categories were not, as one might assume,
simple ?containers? of which an exemplar was either a member or not.  Rather, categories
exhibit a graded structure with some members being judged more prototypical of the
category than others, and a gradience in membership, so that the distinction between a
category member and a non-member is not clear.  Continued research by Rosch and
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others was designed to determine the structure of these categories and the origins of that
structure.  Barsalou?s initial research on categorization looked at ad hoc categories (such
as ?foods not to eat on a diet? or ?items to take from your house in a fire?) and showed
that these categories, though certainly not stable categories that are represented
cognitively prior to their construction, still possessed the graded structure found in
?common? categories.  His continued research looked into the stability of this graded
structure (Barsalou, 1987).  He asked participants to judge the typicality of category
members from their own point of view and the point of view of others (such as a
professor?s point of view) and analyzed between- and within-subject reliability of
categorizational structure.  The findings point to significant variability in graded
structure: participants were able to judge the typicality from others? points of view
(occasionally with stunning accuracy).  Additionally, the within-subject judgments of
typicality varied (with moderately typical category exemplars changing rating the most).
Graded structure, Barsalou concludes, is ?a highly flexible and unstable phenomenon.?
Context, linguistic context, point of view, and other factors affect the typicality assigned
to category exemplars.  Surely, he argues, people do not possess representations in long-
term memory of how a professor would assign structure to the category of dinner foods.
The implication on the structure of categories, Barsalou argues, is that ?rather
than being retrieved as static units from memory to represent categories, concepts
originate in a highly flexible process that retrieves generic and episodic information from
long-term memory to construct temporary constructs in working memory?  (Barsalou
1987).  This is not meant to imply that there is not stable knowledge in long term
memory, but rather that the concepts in working memory ? the ideas that are pondered,
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discussed, articulated and reasoned with ? are temporary constructs and, as such,
sensitive to context and goals and inherently unstable.  Furthermore, prototypes of
categories are not stable properties of categories ? these are not the elements that are
stored in mind and organize categorization.  They arise from cognitive processes on more
stable units.  But what are the stable units that minds have, then?
Structures in a manifold ontology of mind
Resources
Stemming from findings regarding the variability of reasoning in science,
Hammer (2004) has argued against a research paradigm in science education that focuses
on student misconceptions and its implicit assumptions on the nature of concepts and
mind.
What sorts of things do we attribute to students? minds?  It has become
conventional to speak and think in terms of conceptions, na?ve theories, and
stages of development.  These are all attributions of stable properties, and they
account well for patterns that can occur in student reasoning.  They do not
account well, however, for the variability and multiple patterns illustrated
[elsewhere].
As an alternative to the unitary conception of mind, he offers what is termed the
?resource model? as a more fruitful ontology of mind with multiple, fine-grained
cognitive resources that are or not activated.  Different conceptions of marriage, as
mentioned above, could be considered different resources for understanding this concept
that are activated at different moments.  This is not to say that students cannot or should
not have a robust, stable representation of a particular concept, but rather, as noted by
Hammer: ?Ontology need not recapitulate phenomenology?  The cognitive objects we
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attribute to minds need not align closely with the ideas and behaviors we hope students to
transfer.?  (Hammer, et al 2004).
Idealized Cognitive Models, Schemas, and P-prims
Lakoff (1987) has proposed that categories are derivative of idealized cognitive
models of the world (ICMs), ?which van be viewed as ?theories? of some subject matter.?
(Lakoff, 1987 p. 45).  These ?theories? can be parsed into various schemas or short
?scripts? that we have about the world and the way it works: event schemas that are
abstracted from our experience of certain events, image schemas that provide structure
for conceptualizations ?  ?schemas of intermediate abstractions [between mental images
in abstract propositions] that are readily imagined? (Palmer, 1996 p. 66) ?  and
proposition schemas: abstractions that act as models of thought and behavior and specify
?concepts and the relations which hold among them.?  (Quinn 1987)
Schema theory holds that we possess a pattern of associations in our mind that
lead to locally coherent ways of understanding and negotiating our world.  As described
by Rumelhart (1981), schemas are
the fundamental elements upon which all information processing depends.
Schema[s] are employed in the process of interpreting sensory data, ? I
retrieving information from memory, in organizing actions, in determining goalas,
? in allocating resources, and generally in guiding the flow of processing in the
system?[Schemas represent knowledge] about ? objects, situations, events,
sequences of events, actions, and sequences of actions.
And schemas are distinguished from the more generic ?models? in Redish (2003, p 13):
I follow the notation of D?Andrade and call such a pattern a schema if it is a
?bounded, distinct, unitary representation? that is not too large to hold in working
memory.  I call a pattern a (mental) model if it consists of ?an interrelated set of
elements which fit together to represent something.  Typically one uses a model to
reason with or calculate from by mentally manipulating the parts of the model in
order to solve some problem.?  (D?Andrade, 1995 p 151)
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This is to say: what we do have in our minds are short scripts, sequences, or stories that
can be combined to create models.  These are ?bounded, distinct [and] unitary,? unlike
the phenomena that they may describe. Taber (2000) noted that an individual learner can
simultaneously ?hold in cognitive structure several alternative stable and coherent
explanatory schemes that are applied to the same concept area,? namely in the concept of
bonding; these coherent explanatory schemes are what I mean by a locally coherent
structure ? bounded, distinct, and unitary scripts for understanding bonding ? while
bonding is understood with a manifold set of schemas.  (Of course, part of science
involves reconciling competing schemas and placing them within a larger explanatory
framework that accounts for both ? it is in this that Hammer (2004) notes, ?the cognitive
objects we attribute to minds need not align closely with the ideas and behaviors we hope
students to transfer.?)
It is only within our schemas that categories are defined and meaningful.  As
noted in the literature review, the question ?is the Pope a bachelor?? is a confusing
question.  By all definitions of ?bachelor? the answer is yes, but no one would ever refer
to the Pope as a bachelor.  Lakoff explains this paradox with an appeal to cognitive
models: ?bachelor? is defined and meaningful only within a cognitive model (a sets of
schemas) of society that has marriage and the schemas associated with marriage ? and
these schemas that are activated do not take into account our schemas involving clergy.
Therefore this category ? bachelors ? becomes less meaningful and exhibits a graded
structure to the degree that the schema in which it is defined does not apply (as in the
case of the Pope).  And prototypes of our categories arise from the concretization of these
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cognitive models.  It is not the exemplars that organize our categories, but the schemas
and compilations of those schemas into larger cognitive models that organize (and at
times construct) exemplars into categories.  When discussing prototypes in this
dissertation I am not referring to static exemplars around which our categories are
organized, but ad hoc constructions and recollections that are organized by the schemas,
cognitive models and resources that are activated.
This differs greatly from the representation of concepts from structure mapping
and the related computational model.  For even if the model took into account a
variability of representations, the model still attributes these representations as stored
properties of the base and not more abstract, general schemas.
1
In the model of analogies as assertions of categorization, then, there is implicitly
some underlying schema involved.  In physics education research, a set of simple,
                                                  
1
 For example, noting the lack of variability for concepts described in the Structure
Mapping Engine, the following suggestion for incorporating multiple representations is
introduced (Falkenhainer, Forbus and Gentner, 1989, p 39):
The SME algorithm is of necessity sensitive to the detailed form of the
representation, since we are forbidding domain-specific inference in the matching
process. Existing Al systems rarely have more than one or two distinct ways to
describe any particular situation or theory. But as our programs grow more
complex (or as we consider modeling the range and depth of human knowledge)
the number of structurally distinct representations for the same situation is likely
to increase.  For example, a story might be represented at the highest level by a
simple classification (GREEK-TRAGEDY) at an intermediate level by
relationships involving the major characters (i.e., (CAUSE (MELTING WAX)
FALL)),and at the lowest level by something like conceptual dependencies.  An
engineer's knowledge of a calculator might include its functional description, the
algorithms it uses, and the axioms of arithmetic expressed in set theory. Unless
there is some window of overlap between the levels of description for base and
target, no analogy will be found.
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primitive schemas that students use has been identified.  These, as noted in chapter two,
are phenomenological primitives.  From chapter two (p. 32, this document),
They are ?the intuitive equivalent of physics laws; they may explain other
phenomena, but are not themselves explained with the knowledge system.?  As
defined by diSessa, p-prims are ?cued to an active state on the basis of perceived
configurations, which are themselves previously activated knowledge structures.?
In this way p-prims are elements within larger models.  P-prims ?often originate
as minimal abstractions of common phenomena,? and are ?nearly minimal
memory elements, evoked as a whole.?  By way of example, consider one class of
p-prims: the ?constraint cluster.?  This class includes bouncing, supporting,
guiding, clamping, and carrying.  These p-prims are not fundamental for a
physicist (all can be explained in terms of forces) but are often elicited in
conversations with students as explanations for physical behavior.  The p-prims
have a ?schematization? such as, for the ?supporting? p-prim, ??strong? or stable
underlying object keeps overlaying and touching object in place.?  (diSessa, 1993
p. 216)
Many of the analogies that students express can be shown to have their origin in
phenomenological primitives.  Below, I will show that the analogies presented in chapter
two for their phenomenological similarity to categorization can be understood by this
ontology of mind: they are based in particular schemas and the role of the analogy is to
move the target of the analogy from one locally coherent structure to another.  First I
would like to address a point raised in the previous chapter: analogies as negative
assertions, and the distinction between similarity and analogy.
Interlude: A distinction between similarity and analogy
The conflation of similarity and analogy in past definitions
If this is how the mind works ? it has stored schemas that become activated and
put together in a variety of ways, and these schemas are responsible for our categories,
and, as Lakoff (1987) noted, everything is an act of categorization:
Every time we see something as a kind of thing, for example, a tree, we are
categorizing. Whenever we reason about kinds of things ? chairs, nations,
illnesses, emotions, any kind of thing at all ? we are employing categories.
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Whenever we intentionally perform any kind of action, say something as
mundane as writing with a pencil, hammering with a hammer, or ironing clothes,
we are using categories.
then perhaps structure mapping is just a way of detecting a schema that applies.  That is
to say, structure-mapping involves abstracting a structure from the base that you then
map onto the target, and, given that the mind has stored schemas, perhaps ?structure? is
simply another word for ?schema.? The schema is the structure and this is mapped onto a
new scenario in what we see as analogy, and then the only piece lacking from the
structure-mapping story is how that particular schema is arrived at, given the manifold
that exist.  But if this is the case, then every act of categorization becomes an act of
analogy, as Hofstader (2003, p 506) believes to be the case:
The triggering of prior mental categories by some kind of input ? whether sensory
or more abstract ? is, I insist, an act of analogy-making.  Why is this?  Because
whenever a set of incoming stimuli activates one or more mental categories, some
amount of slippage must occur (no instance of a category ever being precisely
identical to a prior instance).  Categories are quintessentially fluid entities; they
adapt to a set of incoming stimuli and try to align themselves with it.  The process
of inexact matching between prior categories and new things being perceived
(whether those ? things? are physical objects or bite-size events or grand sagas) is
analogy-making par excellence.  How could anyone deny this?  After all, it is the
mental mapping onto each other of two entities ? one old and sound asleep in the
recess of long-term memory, the other new and gaily dancing on the mind?s
center stage ? that in fact differ from each other in a myriad of ways.
Consider the diagram below, first presented to me by Redish (research group meeting,
2004) and constructed by Edward Adelson. When participants are shown this diagram
and asked which square is darker, A or B, everyone will claim that A is the darker square.
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Fig. 5.1
In fact, the squares are the same shade ? but even knowing this it is hard (if not
impossible) to convince your mind otherwise.  We cannot help but run the ?schema? that
helps us judge the relative brightness of objects.  And so, in a sense, we are mapping all
of our prior experiences with intensity, shades and shadows onto this new experience.
According to structure mapping, this is analogy ? we have mapped the structure of prior
experience (things in the shade are lighter than they appear) onto this new experience (B
is in the shade, it must be lighter); according to Lakoff, this is categorization (we are
using a cognitive model and placing this picture into that model); according to
Hofstadter, these ? categorization and analogy ? are one and the same.  A similar claim
can be made of the students in transcript 2 that believe this new cup of water will spill:
they are assuming that this cup of water is like all other cups of water ? it will spill when
overturned.  They are mapping a structure or schema involving cups of water onto this
new cup of water.  But there is something that feels different about Miranda claiming that
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a cup of water is like a toy cat in a basket and the other students who are implicitly
mapping their prior experience with cups and water onto this instance of a cup and water.
This distinction between analogy and more ?knee-jerk? categorization/
identification of schema is addressed by diSessa (1993), in studying the ?Montessori bell
conundrum.?  In this problem, students are presented with bells made of the same
material, same length, same height, but varying widths.  Almost without exception
students predict (erroneously) that the thicker bells will have a lower pitch.  DiSessa
reports:
Although most subjects were ready with analogies ? church bells compared with
jingle bells, xylophones, musical instruments of various sizes ? I was struck that
some initially could not produce any example of the phenomenon they identified
to be at the root of the situation.  This, along with the rapidity and expressed
certainty of responses, heightened my confidence that a p-prim (or several) was at
stake rather than analogy.  (diSessa, 1993)
That is, students are able to make a prediction for the Montessori-bell conundrum without
any explicit reference to an analogous case.  Many students are able to construct, post-
hoc, an analogy to explain their reasoning, but some cannot ? suggesting that the
prediction for these bells was not made with any explicit analogical reasoning between
this set of bells and other sets of bells or instruments.  DiSessa, accordingly, distinguishes
this automatic assumption/prediction from analogy.
This distinction between analogy and p-prim that diSessa makes is not consistent
with structure-mapping or other accounts of analogy: while diSessa claims that a p-prim
is not an instance of analogy, Gentner?s description of structure-mapping and
Hofstadter?s account of the ubiquity of analogy construct a definition of analogy in which
any kind of similarity is analogy.  And perhaps it is quite fruitful not to distinguish
instances of p-prims and schemas from analogy.  This is Hofstadter?s approach, and
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surely understanding the more routine acts of categorization ? how we recognize an ?a?
in handwriting that we have never seen before, for example ? can shed light on how we
make the more creative feats of imagining a cup of water to be like a toy cat in a basket.
I would like to distinguish the more creative aspects of analogy, those that are
powerful for their ability to shift from one locally coherent structure to another, from the
more routine kinds of activities that our minds undertake automatically and without
cognitive effort.  This is a piece that is missing from structure-mapping and is
acknowledged by its authors, who note that the Structure Mapping Engine (a
computational model of structure mapping) finds literal similarity to be the best possible
match when determining the soundness of an analogy: when comparing relational
structures and disregarding surface features, the literally similar structures will, naturally,
be the strongest possible match (Gentner, 1989).  Or, as Gentner and Markman note
(1997, p 48), ? this contrast between analogy and literal similarity is in fact a continuum,
not a dichotomy.  Yet it is an important continuum psychologically, because overall
similarity comparisons are far easier to notice and map than analogical comparisons,
especially for novices.?  However, other researchers have found criticism in this
continuum account of analogy, in particular the lack of attention to goals and context.  As
Holyoak (1985, p 74-75) notes, ?even objects that Gentner would term ?literally similar?
can be analogically related if a goal is apparent.?  Commenting on this criticism, Gentner
notes ?since this is essentially a question of terminology, it may be undecidable?
(Gentner, 1989 p 220).
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A proposed definition of analogy as a change of schema
It is a question of terminology and a matter of choice as to whether or not this is
what we mean by analogy and if we would like analogy to exist on a continuum from
similarity or be somehow distinct, but I would like to focus on those analogies that take
us from one schema to another and take deliberate cognitive effort, and contrast those
with more routine categorization that happens cognitively automatically.  Both are acts of
categorization, but one, analogy, I will use to mean a recategorization.  In this way,
analogy is so powerful because of what Koestler has identified as the ?essence of
creativity:? being able to view a situation or an object from two different frames of
reference, or two ?unrelated matrices of thought? (Koestler, 1964).  Or, as Chi (1997)
clarifies, ?the essence of creativity is? re-representing an entity or a situation from one
?ontological? tree of concepts and categories to another ontological tree of concepts and
categories.?  This recategorization is more profound than considering a person to be both
a daughter and a sister and a chef ? that is, it is not simply choosing one of a myriad of
schemas that apply to a phenomenon, but instead invokes a schema that is at odds with
the alternative category.  For example, considering a cup of water to be unlike most cups
of water that do spill, and more like a toy cat in a basket that doesn?t.
Using a dial ammeter to measure current, undergraduate students in my Physics
115 course measured the current coming from the battery in the following circuits:
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Fig. 5.2: Circuit set-ups for measuring current from battery
Slightly more current leaves the battery for the circuit on the left.  Though the
measurements were initially taken for another purpose, I later referred to them as a
counter-argument to a student?s claim that the battery always puts out the same amount of
current.  But the students claimed ? erroneously ? that the ammeter readings gave the
same value.  It?s understandable ? the dials are difficult to read precisely and students
often round ? but that alone doesn?t explain it, as students will often argue over
insignificant differences in readings and each lab group reported the same findings: the
ammeters read the same value for the current leaving the battery.  Only one student
recorded ?100+? and ?100?? because of the discrepancy in the ammeter readings.
Perhaps the students had expectations about what these numbers should be ? but this
expectation did not come from their knowledge of circuits or experience with ammeters
(none had extensive experience with either), but from experiences with phenomena in
which the output from the source is not mitigated by the consumer ? like rain, perhaps.
This can also help explain why students, when handed a bulb, battery and wire, often first
try to light the bulb in this manner:
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Fig. 5.3: Typical arrangement tried by students
The battery is a source of energy and the light is a consumer of that energy, and the
students assume that, as with every other source of energy, you need a path from that
source to the consumer.  You rarely (if ever) need a path back.  The students? attempts
make sense ? they are matching a pattern they have observed before with other sources of
energy and mapping it onto a new case.  But is this analogy?  Again, Hofstadter (2002)
argues that it is, and structure mapping would say that either this is analogy, or that
structure mapping requires an explicit reference to the base ? a reasonable argument but
also somewhat post hoc.
I argue that this is not analogy by definition, because in the absence of such a
definition we risk turning everything into analogy.  Analogy is a deliberate cognitive step
that involves a negative assertion, claiming that this source of energy (the battery) is not
like other sources of energy.
In this chapter, I am providing the sketch of an underlying cognitive mechanism
to account for the phenomenology described in the previous chapter.  What are the pieces
of mind that can explain student-generated analogies?  The themes of findings on the
ontology of mind that will be revisited below in the context of student-generated
analogies is as follows: if anything is stably stored in the mind in invariant ?chunks? it is
not large-scale theories or concepts, but much smaller scripts and stories that provide a
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kind of ?alphabet? of sorts for constructing theories.  This suggests that while certain
schemas may be associated with a particular analogical base, it is more accurate to
consider the schema as the fundamental cognitive unit.  Or, structure mapping implies
that an analogical base ?has? ? rather objectively ? a particular structure.  Rather, the
theories of schemas, p-prims and ICMs suggest that a particular schema or p-prim is
primary and this schema becomes concrete by constructing a base.  The base is a
representation of a particular schema and is understood and interpreted within that
schema.
Beginning, again, with Miranda?s analogy of the toy cat in a basket, below I will
outline the schemas associated with the analogies that students generate and argue that
these analogies are particularly relevant (or prototypical) of that schema.
Section 2:  The base of generated analogies as representations of a schema
The analogies presented in the previous chapter for the phenomenological
similarities between categorization and student-generated analogies, are presented and
analyzed below for the schema from which they derive.  The categorical assertions that
the analogies make are consistent with these schemas, and the ontology of mind implicit
in a categorization framework of analogy is consistent with findings outlined above.
Schemas and p-prims in the cup/water analogies
The transcript from a 5
th
 grade class, discussing what happens to a cup of water as
it falls, was first introduced for the phenomenological evidence in support of
categorization; in the exchange surrounding this analogy there are multiple analogies
presented, the analogy that is first presented is ?far? transfer, and it is presented in
opposition to the schema other students implicitly apply.  I present these analogies in this
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chapter to make an argument for the reasoning behind these analogies, for the schema in
which they are based and how the analogies may be viewed as prototypical or particularly
characteristic of that schema.
Having been asked to predict what will happen when an overturned cup of water
is dropped from a cookie sheet, a student predicts that the water will not fall out until it
hits the ground and uses the analogy to a toy cat being swung overhead to explain her
prediction (transcript 2, lines 13 ? 26), claiming ?I pull it down and it stays in the back
[motions that the cat is up at the top] until I stop and then it comes out.?
Past models of analogy argue for a one-to-one alignment of objects in the target
and base of the analogy that, once made, allow for candidate inferences to be drawn (in
this case the candidate inference is that this cup will not spill).  Instead, I posit that the
function of Miranda?s analogy was, first, to identify a different story ? one about
swinging baskets that, when overturned, don?t spill ? and used this event schema to
reclassify this cup as a different kind of thing.  It isn?t like most overturned cups, she
argues ? it belongs to a class of phenomena that is typified by an overturned, swinging
basket.  By understanding the schema Miranda is employing, this analogy is not so much
an instance of ?far transfer? but instead prototypical of a particular event schema with
which Miranda is familiar.  The very idea of ?far transfer? ? which has been the focus of
many research studies ? may not be the most meaningful concept in studying learning.
Far more relevant in drawing analogies is recognizing that the base of an analogy
is the concretization of an appropriate schema, and that base is a prototype ? that is, the
spontaneous construction of a representation for that schema.  The cat/basket analogy is
like an airport hub: we arrive here before we arrive at nearby towns ? it?s an entry point
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to a certain area.  Our minds don?t detect similarity solely by matching features.  Just as
travel distance is measured along roads from airports, and not as the crow flies, it does
not make sense to discuss ?near? and ?far? analogies when you are restricted to moving
through the cognitive map along particular schemas and via particular patterns of
activations.  You arrive at the airport before you arrive at closer towns ? it takes less time
for me to get from DC to Seattle than to get to Mazama, a mountain town east of Seattle.
Schema
The theoretical and empirical support for this interpretation of Miranda?s analogy
comes respectively from research in physics education on phenomenological primitives
(p-prims) and two phenomena surrounding this analogy: Miranda?s gesture during her
explanation, and Miranda?s choice of base in the analogy.  P-prims ?often originate as
minimal abstractions of common phenomena,? and are ?nearly minimal memory
elements, evoked as a whole? (diSessa, 1993).  By way of example, consider a
particularly relevant class of p-prims (relevant to the transcript above): the ?constraint
cluster.?  This class includes bouncing, supporting, guiding, clamping, and carrying.
These p-prims are not fundamental for a physicist (all can be explained in terms of
forces) but are often elicited in conversations with students as explanations for physical
behavior.  The p-prims have a ?schematization? such as, for the ?supporting? p-prim,
??strong? or stable underlying object keeps overlaying and touching object in place.?
(diSessa, 2003 p 216)
Miranda is employing the ?carrying? p-prim (and its associated schematization)
together with noting the upside-down container.  Her gestures, in particular, are
indicative of this p-prim.  With her explanation she begins by miming holding a right-
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side-up basket, arm at side, and then swung overhead.  She repeats this motion again with
each successive explanation: the toy cat is held by a right-side-up basket, swung
overhead, and held in the bottom of the basket as she pulls down.  When she stops pulling
(that is to say, she stops the ?carrying? part of her actions) the cat falls out.
Prototype
Prototypes are, as defined by Rosch, the first members of a category that you
recall, are quickly recognized as members of that category, and can be used to generalize
about other category members.  With respect to the ontology of mind presented in this
chapter, prototypes are not primary, nor do they serve as mental representations of
categories; rather, schemas are primary and may be put together together to create a
larger scenario ? such as the falling cup of water ? from these schemas we construct
categories, and it is through the concretization of these schemas and categories that we
arrive at prototypes.  They are constructed on the fly from the schema (or bundle of
schemas) that are activated.  In terms of understanding Miranda?s analogy, it seems likely
that the base of the analogy (the cat/basket) is prototypical (in that it is drawn first, used
to generalize, and easy to learn).  Miranda has a set of schemas that have been activated ?
perhaps carrying and overturned (and even, perhaps, things that are surprising ? one can
imagine Miranda thinking: ?He wouldn?t ask this question unless the answer was
something interesting and weird.?) ? and she makes these schemas concrete and, hence,
relatively stable by the construction and assertion of this base, which is latched onto not
because of its similarity to the target of the analogy but because it is the most immediate
and unproblematic representation of these activated schemas.  Miranda, in fact, has
experience with more ?similar? members of this category but constructs an analogy to the
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cat/basket first.  She refers to the cat/basket as a ?rollercoaster? game but does not use a
rollercoaster as her analogy, and then claims that the cup of water is like an instance
when she dropped a cup of water in her bathtub (transcript 2, lines 55 ? 61).
As noted in the chapter on the phenomenology, previous models of analogy in
which objects are placed in a one-to-one alignment would predict that Miranda should
first draw the bathtub analogy: the features are closer and more easily mapped.  In a
categorization model in which the category emerges from the associated schema, the
swinging basket is naturally a more readily available analogy, as it is, for Miranda, a
more prototypical/accessible a case for the schema associated with carrying things
upside-down.  Cups with water are often in a different schema ? the water frequently
spills from overturned cups ? while the toy cat rarely falls from an overturned basket in
this scenario.  With the rollercoaster scenario ?carrying? would be a less relevant p-prim
? ?guiding? as a p-prim would be more applicable (and, indeed, has been identified as the
p-prim used in explaining the motion of a train on its tracks).
Multiple analogies
As the conversation continues, students in this classroom present multiple
analogies relating to the toy cat in the basket to a bucket of water, tossing Halloween
candy, throwing a hat with dice inside, and a basket of Easter candy (transcript 2, lines
137, 181, 184, and 201).  Not all of the students that introduce these analogies agree with
Miranda that the water will stay in the cup ? however, they are able to understand the
schema that Miranda has identified and are able to identify other members of the
categories constructed by this schema ? ones, perhaps, more prototypical for them and
more obvious members of a category associated with the ?carrying? p-prim (the dice are
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carried by the hat and the water carried by the bucket).  In this way, the above listing of
analogies can be seen as ?fleshing out? the category in order to better categorize the
novel water-cup system.
A recategorization of the target
A final claim I would like to make regarding this transcript is that the students are
recategorizing the cup/water with the analogies they generate ? an idea first introduced in
chapter 4 and revisited here for the ontological and theoretical aspects of this claim.  This
requires that the cup/water was originally an element in a different schema, and the role
of the analogy is not merely to map a new schema onto this existing problem, but to
change the schema in which it is understood and defined.  Why is this distinction
important?  First, it acknowledges that we understand and categorize objects and
phenomena by an identification with a schema that we have in mind and that it is
impossible not to do this: this is just how the mind works.  If structure-mapping is this,
the identification of a schema to apply to a given scenario, then, as noted above,
everything is structure-mapping and it is not a reasonable account of analogy as distinct
from routine similarity and categorization.  Second, it acknowledges that the schema (or
structure or p-prim) is primary: we schematize and categorize without awareness or
careful consideration, but automatically.
If schemas are primary, it makes sense to think of the base of an analogy as
representing a member of a category that the schema defines and not the other way
around.  That is, we do not move from the particular to the abstract when generating an
analogy (though this may be a more reasonable idea in interpreting an analogy).  We
search for an analogy to explain, concretize and help us understand a schema we have
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already identified.  Third, we only search for this analogy when the schema is at odds
with a more expected schema ? if the schema is expected, as with water spilling from a
cup, it is often so obvious as to be invisible.  (One can imagine that most students in the
class are at a loss as to what to explain when asked what they think will happen to the
overturned cup of water.)
This is consistent with diSessa?s (1993) set of heuristics for identifying p-prims.
Foremost among these is the ?principle of obviousness.?  As diSessa state (p 121):
The familiarity and unproblematic nature of some physical events needs
explanation.  In the present context, this usually means they need a p-prim to
attach to them.  In general, p-prims establish abstract classes of unproblematic
happenings.  This is the opposite of misconceptions research strategy, which
never analyzes ?correct? intuitions.  The principle of obviousness gains
explanatory power in conjunction with the principle of invariance; having
understood p-prims underlying common events, we may be able to understand
subjects? reactions to uncommon events using those same p-prims.
Just as misconceptions research does not analyze ?correct? intuitions, neither does most
analogy research, in particular structure-mapping, address the incredible ubiquity of
?structure mapping? in everyday occurrences that we do not consider to be analogy.
There is an obvious and expected answer to ?what will happen to the water when you
drop the cup.?  Most students assume that the water will spill and splash ? that they do
not explain this further is evidence that a p-prim is at play, in that it seems ?obvious? and
?impenetrable.?  Even Miranda, who predicts the water will stay in the cup, expresses
surprise and fascination when the experiment is performed and the water stays in.
Indeed, even to a trained physicist the result is eye-catching.  But the brain has a way of
understanding this, has a schema in place for this, and the role of the analogy is to
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identify an alternative schema and place this cup of water in a category associated with
that schema.
The Beanbag Analogies
The analogies presented in this section were introduced in the last chapter for the
phenomenological property that the base of the analogies are often constructed or
invented, rather than recalled.  Here I argue that the three students each are operating
under three different schemas.  As such, the analogies they choose differ and are
prototypical of the schema they are employing.  In the following transcript from a third
grade class, the teacher has told the students that she is going to run while holding a
beanbag that she wants to drop onto an ?X? marked on the classroom floor.  Should she
drop the beanbag before, when, or after she reaches the ?X??  Adam, one of the first
students to speak, addresses Newton?s Laws.  The teacher (Trisha Kagey) asks for
explanation.  The following statements incite multiple spontaneous analogies, comparing
the beanbag to a bike, a bat, a leaf, a rock and a feather (transcript 3, lines 35 ? 40, 55 ?
73, 83 ? 87, 154 ? 159 and 188 ? 190):
Adam: ? if you?re riding your bike, um ? it?s in motion.  And you?re
going to keep going until you get stopped by like ? um, a rock or
something ? or going uphill?
Connor: I would think the bean bag would ? might fall behind where you
want it to fall because when I put ?  when I played baseball ?  they
always said don?t throw the bat because it might hit the catcher and
not one of the um person because we?re using metal bats, and ?  so
we drop it, you drop it and then you ? . Well, when I drop it, it
usually swings backwards; it wouldn?t be behind the plate instead
of the front of the plate?
Teacher:  Why do you think it fell behind?
Connor:  Well actually it didn?t mostly.  It got on the side or in front because
? well because you?re supposed to drop it because you don?t need a
bat while you?re running the bases.  Once you drop it, I?m just
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thinking also, what Adam is ?  well a bus ?  well if you were on a
bus and you had uh, this little leaf that you found, and the window
was open, and you drop it, it will go ?  it?ll be going backwards?
Lauren:  Because I think that?s cause ? you?re talking about a leaf that?s
falling?  That?s because the ? it?s sort of ? the bus is going back, so
it?s making like the air move.  And the leaves are really, really
light, so the reason they are going backward is because ? Um, well
it?s going so fast?
Connor:What if you did it with a rock?  The same thing will probably
happen with a rock.  Because you are probably like a bus, that you
make the air come ?
Kamran:? A rock is different, a rock has ? it?s also like, it?s solid, but it?s
not that a leaf isn?t solid, or a feather isn?t solid.  A feather ? but
you have to ? it?s very small, and it?s very like thin, so you kind of
say like solid. But anything hollow, like if you have a paper box?
Kamran:  Yeah, because the weight pulls it right down.  If a tree ?  it?s
heavy, and it?s heavier than all the leaves it has, so the leaves will
make it fly ?  fly here.  And the tree, it will just go down.
In this segment, the beanbag drop is compared to a bike, a baseball bat, a leaf and a rock.
The person doing the running drop is compared to the bike, the bus and a person.  These
items are not chosen arbitrarily, but because of a particular schema the students have.
Schema 1: A Newtonian Schema
Before discussing his analogy, Adam says ?it will instead of going straight down,
it will go um ?  it will go in front because it?s not stopped yet.  But when it hits the
ground, because there is friction on the ground ? there is more friction on the ground than
in the air it will get stopped and land somewhere around there (the X).?  It is only after
the teacher asks Adam to explain it more clearly (?like you?re explaining it to a
Kindergartener?) that Adam, after some pause, comes up with his analogy.  This schema,
which is consistent with Newton?s Laws (objects in motion stay in motion), is not one
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with many members that would be well known to a young student ? when walking,
swimming, or pushing something, for example, you naturally slow down.  A prototype is
not a category member that is necessarily seen frequently but one that is seen frequently
as a member of the category.  ?People?s perceptions of how frequently exemplars
instantiate their category, rather than people?s familiarity with exemplars, appears to be
the measure of frequency that is most central to graded structure? (Barsalou 1987).  The
bike is an appropriate and, for a child, prototypical instance of this schema, as it tends to
keep rolling when on a flat surface ? unlike when walking or running.
Schema 2: Intrinsic Motion
The analogies introduced by Connor and Kamran are rooted in a schema different
from Adam?s and more consistent with a phenomenological primitive.  First consider the
evolution of Connor?s analogies.  Connor jumps from analogy to analogy ? first the
baseball bat, which he discards when he analyzes it further, then the bus with a leaf,
which evolves to the bus with a rock when he is pressed by Lauren.  These are chosen
because he believes that you will need to release the bean-bag after passing the X.  This
belief is evidence of a schema in which dropped items will be pushed backwards ? and he
is able to invoke two analogies as evidence.  The prototypicality of these will be
discussed in further detail in the next section, but a brief sketch is provided here.  When
switching to the bus analogy, there is no reason for the bus with a rock to be any more
salient and immediate an analogy than the bus with a leaf unless Connor has a schema in
mind already: one involving the ?wind? pushing the object back, in which case a leaf is
much more prototypical in this scenario than a rock, even though the rock has features
(heaviness and irregular shape) that would make it much closer to the beanbag. Lauren
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picks up on Connor?s selection of analogy as representing a category to which the keys
would not belong, as they are heavy, and a person would not belong, as people are slower
than buses.  She suggests that ?the leaves are really, really light, so the reason they are
going backward is because, um, well it?s going so fast ? a bus is like going so fast that
it?s probably making the air go that way.?  Only then does Connor move his analogy
?closer? to the beanbag by posing the question ?what if you did it with a rock??  It is
Kamran who identifies the two categories that are invoked by these students? schemas.  I
would like to suggest that the students are employing the p-prim of ?intrinsic or
spontaneous motion? (diSessa 1993), which has a schematization of ?especially heavy or
large things resist motion.?   Such a p-prim creates a dichotomy of objects that do and
objects that don?t resist motion (Adam?s schema of ?an object in motion will stay in
motion? has no such dichotomy in Newtonian physics, but in ?real world? physics there
seem to be objects that obey this law and ones that don?t).  Kamran, a particularly vocal
student who seems to voice his thought process, grapples with the categories that have
been invoked by Connor?s analogies: he wants to say that the rock and the leaf belong to
different classes of objects, and yet he knows that they are both solids.
An illuminating comment is made when Kamran states that ?a rock is different?
but it?s not that a leaf isn?t solid, or a feather isn?t solid.?  In this passage, we see Kamran
negotiating the two categories that have been instantiated by Connor?s schema, trying to
match these categories with known categories (solids) but coming up empty: ?A rock is
different, a rock has ? it?s also like, it?s solid, but it?s not that a leaf isn?t solid, or a
feather isn?t solid.?  He struggles to identify the ways in which these items could belong
in different categories by identifying multiple members of those categories: a feather and
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a paper box as belonging to the category for which leaves are a prototype.  But again he
is stumped by the tree ? ?heavier than all the leaves it has? ? as an object in the category
for which the rock is prototypical, yet constructed of items for which the feather is
prototpyical
2
.  Kamran, because of his transparent thought process, demonstrates that
these analogies are clearly defining certain categories and the task the students have is to
determine to which category the keys belong.
The choice of base in these analogies is discussed in the following section, in
which I address prototypes of ?composite? categories, such as things that you drop and
items from which you drop them (running and bats, buses and rocks, etc).
Styrofoam and ice-skating
The following set of analogies is taken from an undergraduate physics course.
This course is a laboratory based conceptual physics course at the University of Maryland
in which the students have no textbook and work in small groups and as a class to
understand physical phenomena.  In the following transcript, the students have worked in
small groups, discussing the differences between what it?s like for a charge in metal
versus Styrofoam, and are now reporting and discussing their conclusions.  The following
analogies were drawn below were presented for their evidence of multiple analogies.  I
return to these here and identify the role that schemas play, and the reschematization that
the analogies perform (transcript 1, lines 1 ? 107).  These analogies are assertions of
categorization, and hence derivative of a schema, the base a prototype of the categories
                                                  
2
 An alternative interpretation of this statement is that the tree shows how weight is the
significant factor ? ?It has leaves that want to make it fly, but it doesn?t because the tree
is so heavy.?
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defined within and organized by that schema, and multiple analogies as a way of
negotiating that category.
Schemas
For a person, motion through something dense ? a dense crowd, say, or a densely
furnished room ? is usually difficult.  Motion is generally easiest in a medium that is not
very dense.  No one addresses this explicitly with analogy, again indicative of what
diSessa has called the ?principle of obviousness.?  However, students have noticed that
charges appear to move easily in metals and are somehow ?stuck? in the less dense
Styrofoam and recognize the paradox.
This paradox demands that one categorize the motion of charges in a way that
violates some expectations regarding motion.  As such, it requires explanation and
students are able to identify a schema in which this is not a paradox.  By setting up an
analogy the students are able to recategorize the metal and Styrofoam in a way consistent
with schemas already at their disposal.  Initially Christie suggests that the metal must be
less dense ? but, I contend, what Christie means by suggesting that metal is less dense
than Styrofoam is that in a metal the motion of charges is easy, and so the metal belongs
to a class of objects that allow for motion ? a class often typified by low-density places.
Lea?s analogy identifies a case in which density enables rather than inhibits movement:
ice-skating.
Multiple Analogies
In the above section, the phenomenological primitive of ?intrinsic or spontaneous
motion? (diSessa 1993) has a schematization of ?especially heavy or large things resist
motion? that establishes two categories (things that resist motion and things that do not).
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Similarly, the schema of density enabling motion creates two categories: items that are
dense enough to enable motion and items that are too not-dense and prohibit motion.  The
instructor, identifying this second category in this schema, finds a prototype: the sponge.
Though not dense, it will not allow water to pass through easily.  And Lea mentions
water on a countertop and then the instructor makes an analogy to stepping-stones ?
analogies which, when viewed as an assertion of categorization, identify more members
of the category typified by ice-skating.
Prototypicality
Claims of prototypicality are difficult to make: without doing an explicit study of
the graded structure of the ad hoc category ?media whose density enables motion,?
determining the prototypicality of a statement can only be inferred.  However, some
principles of prototypes can aid us in determining or arguing for the prototypicality of a
particular base of an analogy.  Rosch (1976) reports that
prototypes appear to be just those members of a category which most reflect the
redundancy structure of the category as a whole.  Categories form to maximize
the information rich clusters of attributes in the environment and, thus, the cue
validity of the attributes of categories.  Prototypes of categories appear to form in
such a manner as to maximize the clusters and cue validity within categories.
For this reason, the prototypical bird would be one that is small, has feathers and wings
and can fly.  But defining prototypicality in a category that is not taxonomic, but instead
one that is either goal based (as ?foods to eat on a diet? for example) or otherwise ad hoc,
such as ?a medium for which density enables mobility,? is difficult.  Though ad hoc
categories have a graded structure, ?there appears to be a large class of determinants that
is impossible to specify completely and that depends to some extent on the category and
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on the context in which it is perceived? (Barsalou, 1987 p. 104).  Factors determining the
graded structure include:
? Central tendency ? for example, the features of birds mentioned above.
? Similarity to ideals associated with that category ?where ideals are properties that
exemplars should have if they are to best serve goals associated with their
category.?  (Barsalou, 1987 p 105)  For example, low-calorie foods for the
category of things to eat when on a diet.
? How frequently it is perceived as instantiating its category.  ?People?s perceptions
of how frequently exemplars instantiate their category, rather than people?s
familiarity with exemplars, appears to be the measure of frequency that is most
central to graded structure.?  (Barsalou 1987)
And yet these arguments still beg the question: why is a leaf falling from a bus (Analogy
II) prototypical?  Surely this scenario is infrequently, if ever, perceived.  And what is
meant by ?central tendency? is difficult to ascertain in a relatively composite category
for, as noted by Lakoff (1987 b ?Cognitive Models and Prototype Theory?), a good
example of a striped apple is neither a good example of striped things nor a good example
of apples, and a small galaxy is not the intersection of prototypically small things and
galaxies.  Similarly, ?things that fall behind you when dropped? is not the intersection of
things that you drop and things that fall at an angle, while ice is not a good example of
dense media and skating is not prototypical of motion.
To explain these as prototypes for the categories they instantiate, the claim of
?similarity to ideals associated with that category? is more informative.  However,
defining those ideals requires an appeal to idealized cognitive models, schemas and p-
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prims.  In particular, I argue that the schemas that are being employed in the bus-leaf
analogy and the ice-skating analogy are built from multiple p-prims.  These p-prims are
put together in a larger composite schema that is then concretized by the base of the
analogy.  The prototype structure ? the fact that something is identified most quickly and
is a better exemplar of the category ? need not imply that the prototype is a permanent
fixture stored in memory as a representation of a schema, but rather that, under these
circumstances and in this set of activated schemas, this particular representation of the
schema is accessed most easily.
Miranda?s analogy hinged on a single p-prim (namely carrying), but her selection
for the base of her analogy arose from this schema (carrying) together with the upside-
down property of the cup and even, perhaps, the idea that ?something weird? would
happen.  The analogies presented in the past two transcripts are also based on phenomena
more complex than a single p-prim.  Consider the schema that Connor may have: I posit
that Connor has two stories or schemas that this beanbag-running-drop invokes: one is
simply the schema of running and dropping something, for which baseball and the bat is
quite prototypical and is tied to a second schema of what happens after that drop occurs.
This second schema is something like ?things in motion are pushed backwards by the
wind.?  This schema, contrary to Adam?s schema of ?things in motion stay in motion
until stopped,? may entail the p-prims (all quotations from diSessa 1993) ?intrinsic or
spontaneous resistance? (with a schematization of ?especially heavy or large things resist
motion?), and ?force as mover? (here the ?wind? force moving the leaf).  Such a schema
would explain why Connor rejects his initial analogy (it is derivative of a different
schema than addresses the question at hand, namely things that you drop as you run) and
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explains his choice of objects for the second analogy he chooses.  The leaf is prototypical
? readily activated within this schema ? in that it is consistently a member of object that
are affected by wind, and the bus is prototypical in that it consistently creates this ?wind?
force (or, as Lauren says, ?well it?s going so fast ?  a bus is like going so fast that it?s
probably making the air go that way?) and is an object with which students are quite
familiar.
The analogies surrounding the motion of a charged particle in metal (an aluminum
pie plate) hinge on the idea that density is in some way enabling motion.  One implication
from the initial analogies (ice skating, a countertop and a sponge) suggest to one student
that the charges are moving across the top of the metal (transcript 1, lines 83 ? 86), which
she finds problematic because charges should be throughout the metal (?it?s made up of
it?):
Anna:So you?re saying the charge is like on top of the metal?  Like on
the outside?
Lea: Yes.
Anna:It?s like made up of it ? like, they?re electrons.
Lydia agrees with the student above and suggests that the electrons, instead of ?skating?
across the solid metal, are hopping from one molecule to the next.  Paul then concretizes
the schema, selecting an analogy to stepping-stones (transcript 1, lines 113 ? 121):
Lydia: I was going to say I think the pie plate is more dense but I do think
that it?s inside not outside because if there?s more space to travel
then the molecules can?t get from one space to another easily but
it?s all [inaudible].
Instructor:  Oh so it?s like stepping stones [Lydia: Kind of.] like in the
Styrofoam it?s really far to the next stepping stone so it?s like can?t
get there I?m stuck here. [Lydia: Right] but in the metal the stones
are really close together so I can kind of walk across.  [Lydia:
Yeah.]
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The base of the analogy comes from the schema, and ? just as happens in the beanbag
analogies ? as that schema changes, so do the analogies.
Analogies regarding a quantum mechanics problem
The following transcript, presented in the previous chapter, follows two
undergraduate physics majors working on a homework assignment on angular
momentum in quantum mechanics.  The students have had instruction on how to arrive at
the quantum numbers S and L but are asked to find the square of the total angular
momentum, (J)
2
, which is (S + L)
2
, or (S)
2
+2(S?L)+(L)
2
.  The students know how to find
(S)
2
 and (L)
2
 but not 2(S?L).  They have a solution set from another student that provides
the answer but not the steps to arrive at that answer. 
 
There are two analogies presented in
the transcript, both an expression of the kind of problem they are solving: how to
approach the problem and how to understand the quantities in the problem. The first
(transcript 5, Lines 10 ? 27):
Anselm: ?Cause you?re assuming that if you have the example, suppose
there?s a charge here, what?s the electric field due to it?  You can
figure out, suppose you have Bugs Bunny, and he?s charged,
what?s the electric field around his ears?  All right.  Because you
have a simple example when they?re both the same, you?re not
going to be able to figure out exactly what you?re supposed to do
when the rules weren?t the same.  Cause now it?s fixed.
And then, (transcript 5, lines 64 ? 80):
Ben:  But you?re mixing apples and oranges.  It?s dumb!
Anselm:  Yeah that?s so messed up, yeah that?s not the answer.  If I just
ignore the fact that I?m in the three-halves one-half and I?m in the
one-half one-half and I just add them all together,
Ben:  I once had a professor tell me that um, well if you got the right
answer, you certainly know how to do the problem.  I had to
convince him no sir, you can jiggle these numbers any way you
want.  And come up with the right answer if you know the right
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answer in advance.  Of course we?re not sure that this is the right
answer.
Reschematization of the base
What kind of analogies should one expect in discussions of quantum mechanics?
What schemas are available to understand this branch of physics that has no obvious
analogs?  In this discussion between two students, their analogies are with respect to
mathematics and problem-solving strategies, not concerning the nature of quantum
mechanics.  While this transcript is not long enough to draw any meaningful information
regarding analogies in quantum mechanics, it is interesting to note that the schemas
introduced here are not regarding wavefunctions or expectation values, but instead related
to problem solving and epistemology.
The first analogy is a response by Anselm to Ben?s claim ?We should be able to
figure this out from today?s lecture.?  Ben believes they should be able to solve the
problem with information from the day?s lecture on simple quantum numbers.  Anselm
offers an analogous case and then provides the abstract schema from which that case was
derived: ?Suppose you have Bugs Bunny, and he?s charged, what?s the electric field
around his ears?  All right.  Because you have a simple example when they?re both the
same, you?re not going to be able to figure out exactly what you?re supposed to do when
the rules weren?t the same.?  It is clear that Anselm?s analogy is a reschematization ? this
problem is not one that can be solved from simple principles but requires more
sophisticated tools.
The second analogy is, again, a reschematization.  The students know what the
correct numerical answer is and are trying different combinations of numbers in the
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problem to arrive at that answer.  When dealing with numbers in many (if not most)
mathematics courses, there are rarely rules about which numbers can be combined and in
what order.  In situations when numbers have physical meaning ? as in this physics
problem ? there are rules about what kinds of numbers may be added and in what way.
These rules have been temporarily ignored to find a pattern by which the students may
arrive at the correct answer, but when the answer is arrived at, Ben notes the need to
reschematize the problem from one of combining numbers without physical meaning to
recognizing the meaning (or lack thereof) behind the math, claiming: ?But you?re mixing
apples and oranges.  It?s dumb!?  He then tells a story (in what could be interpreted to be
a multiple analogy) that relates this idea again and reiterates the reschematization ?
beginning with the professor?s claim that ?if you got the right answer, you certainly know
how to do the problem? and then contrasting that with the story: ?you can jiggle these
numbers any way you want.  And come up with the right answer if you know the right
answer in advance.?  In telling the story he tells the more abstract schemas that apply.
And again, each schema is locally coherent ? it is a routine that makes sense in a limited
set of problems.
The ontology of authenticity
Reschematization of the base
This analogy, from a Physics Education Research Group research meeting, was
presented previously for the chain of analogies it presents.  The meeting is being run by
Paul who is trying out a definition of ?authentic? in the context of classroom activities.
He has chosen to define authenticity as not only a property of the activity, but also
relating to the students? interaction with that activity and the (science) community of
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practice?s judgment of the activity.  (For example, how would the students characterize
the reason for what they?re doing?  Would scientists agree with that reason?)  Not only is
this is at odds with common definitions of ?authentic? curricula as a property of the
curriculum itself, but it is contrary to a cognitive model of attributes in which a property
is a property of something: defined relatively objectively and inherent to that something.
Most adjectives or properties belong to this kind of ontology: if I claim that a car is fast,
red, and Japanese, for example, there is an objective measure of the truth of that claim
that is independent of culture, personality, or me.  Anyone else looking at that car will
agree that it is fast, red and Japanese.  If I have an authentic pearl earring, authentic is
used in the same way: an objective measure and a property of the pearl, independent of
context.  Rachel asserts an analogy to make explicit claims on the ontology of
authenticity: ontologically, authenticity is not like fast, red or Japanese.  She finds an
attribute (fun) that is easily understood as not being inherent in an activity.  (Transcript 3,
lines 9 ? 29).
There is an entire story, or schema or cognitive model, associated with ?fun,? and
?fun? occupies a role in this story.  Similarly, authenticity, Paul argues, has a similar
story and occupies the same role in that story.  This ?role? is a category ? one generated
by the schema.  However, there is a fundamental difference between the story that you
tell for ?fun? and that Paul is trying to tell about ?authenticity? ? there is a community of
practice argument.  Leslie (I) introduces this question ? first identifying the schema and
then moving towards finding the analogy (transcript 3, lines 38 ? 79).
Rachel prefaces her analogy by claiming that the analogy will be one of ontology.
The ?work? that this analogy does for the group is to say: what you?re doing with
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authenticity is not new ? we have a way of thinking about this.  You (Paul) are placing
?authenticity? into an existing ontology ? one that is characterized by ?fun.?  Leslie?s
(my) concern was that the ontology was not entirely consistent with Paul?s definition.
Whereas the ?fun? of an activity can be determined solely by the person doing the
activity, Paul relates authenticity to a community of practice ? so that a scientific
community must agree with the student?s judgment of an activity.  Attempts to ?patch?
the analogy by considering ?good, clean fun? instead of just ?fun? are an attempt to
change the ontology of that base.  ?Worship? was chosen by Leslie because of its
relationship to both an individual and a community.
Conclusion
How is a concept represented in memory?  Is it represented by the kind of
knowledge it is?  The way that you find its solution?  By the ontology of the items in the
concept?  Is there a stable representation of ?apples? as ?the thing that can?t be compared
to oranges??  A representation of Bugs Bunny as ?a strange shape for which the electric
field would be difficult to construct??  Is money represented as currency or wealth?  If we
ascribe to concepts a particular representation in memory, then Bugs Bunny would have
to have such an attribute (or, at the very least, would have to be connected to these ideas)
and Marc and Vic would have to have different concepts of the ontology of money.
Theories of analogy that attribute stable representations to concepts must account for the
overwhelming number of features that are part of the representation of a concept.
As mentioned previously, to account for the nature of categories, in particular
their graded structure, Lakoff (1987) has proposed that categories are defined within
particular idealized cognitive models (ICMs) of how the world works.  It is only within a
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particular ICM (or schema) that a category is meaningful, and these categories become
les meaningful and exhibit a graded structure to the degree that the schema in which they
are defined does not apply.  Categories, then, arise from schemas which are activated or
not, applicable or not, depending on context.  Because of the variety of schemas and the
variety of ways they may be combined, categories can have a flexible structure, and
members of categories can shift their membership.  Categories need not have a fixed
representation, but arise from the particular schemas and resources that are activated by
the context.
As an alternative to structure-mapping and other theories of analogies that require
unitary representations of concepts, I posit that analogies are assertions of categorization.
Instead of ?preexisting metaphorical mappings? (Gibbs, 1992) analogies instantiate
preexisting schemas and their associated categories.  Categorization does not require that
there be stored representations of concepts or categories, but that ?concepts originate in a
highly flexible process that retrieves generic and episodic information from long-term
memory to construct temporary constructs in working memory?  (Barsalou 1987).
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Chapter 6:  Analogies in the History of Science
Introduction
The past two chapters are the bulk of the dissertation ? starting with the properties
of generated analogies in science (that is, the phenomenology) and then addressing an
ontology of mind to account for these properties.  In this chapter, I introduce the ways in
which analogies have historically been used in science and explore the consistency
between analogies in science and the model of analogies introduced in the previous
chapters.  These analogies are detailed in works of comparative literature, popularized
science, cognitive science and the history of science.   I begin with analogies from
physics and how the role of physical analogy ? namely Maxwell?s analogy between
magnetism and gears ? is understood in the philosophy of science.  Then I turn to biology
and the idea that the theories introduced by science are often brought about by the
recognition or activation of a schema.  That is, identifying and projecting schemas ? one
important part of analogy ? is how science happens.  However, schema projection is not
the whole story ? often the projection of a schema is implicit, and it is the deliberate use
of a contradictory schema that I define to be analogy.  I then consider the deliberate use
of analogy and the ways in which this has come to be understood by historians of science
and cognitive scientists.
In Metaphors We Live By (1980) Lakoff and Johnson argue that metaphors
structure our thoughts and influence our conceptions of the world, and that the role of
metaphor in scientific thinking provides one of the best illustrations of this principle:
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?Formal scientific theories are attempts to consistently extend a set of ontological and
structural metaphors.?  This claim that scientific theories are extensions of the metaphors
that we employ in other areas is nothing new.  In the 1850?s, with the development of the
telegraph and its influence on theories of the nervous system, many scientists recognized,
and at least one scientist argued against, the epistemological value of analogy in science.
As noted by Laura Otis (2002), Claude Bernard (1858) criticized the analogy between
nerves and the telegraph, claiming, ?people?s ?knowledge? of the nervous system had
consisted largely of a series of comparisons, ?the expression of a way of seeing meant to
explain the facts.?  Priding himself on his empiricism, Bernard mistrusted analogy as a
means of constructing knowledge.?  This concern ? that analogies (a ?way of seeing?)
masquerade as understanding (?explain the facts?) ? was echoed in a cognitive science
course at the University of Maryland.  The professor noted that cognitive scientists have
employed the idea of ?outshining? to explain phenomena in which stimuli we usually
attend to are ignored in the presence of other stimuli, analogous to the way in which the
sun ?outshines? the stars during the day.  Star?s light is not weakened by the sun?s light
but ?outshone? so that they are not seen.  Outshining in the solar sense, he argued, has an
understood mechanism, while outshining in the cognitive science sense cannot
realistically have the same mechanism, but no mechanism is proposed.  In this case, the
analogy masks the lack of understanding.  What is missing from this analogy?  What role
does ?outshining? serve?  Why mention it at all if it does not contribute to the
understanding?  What, exactly, is the epistemological value of analogy?
First, I have argued, the importance of an analogy is its assertion of an unexpected
categorization, meaning that you are identifying an alternative cognitive model for this
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scenario.  In the case of ?outshining,? the analogy functions, as all analogies do, to
reschematize the phenomenon.  For outshining, the reschematization is changing the
cognitive model from one in which stimuli are responded to on the basis of their absolute
value, and instead responded to only by their relative value.  Such analogies in science,
because they are an assertion of categorization, and therefore shift the cognitive model
applied in understanding the target, is what Koestler has identified as the ?essence of
creativity:? being able to view a situation or an object from two different frames of
reference, or two ?unrelated matrices of thought? (Koestler, 1964).  Or, as Chi (1997)
states, ?the essence of creativity is? re-representing an entity or a situation from one
?ontological? tree of concepts and categories to another ontological tree of concepts and
categories.?  (Although I differ from Chi in the robustness and structure of those ?trees? ?
these activated schemas may be put together in a wholly new way to understand the new
phenomenon.)
Second, re-categorizing the target of the analogy is extremely powerful: not just
because of the inferences one may draw, but because it affords one a new language.
Once the target is understood through the lens of a different (and possibly more
appropriate) cognitive model, the target may now be referred to using the language and
categories defined within the new cognitive model.  It is within this new framework that a
structure-mapping process may take place to construct a mechanism ? a crucial and
necessary step for employing the analogy ? but the primary role of analogy is the
entrance into that cognitive model ? to ?tie it down? in a concrete way ? and access to a
new language with new categories.  This new framework is similar to what Otis (2002)
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refers to as a new image (her use of the term ?image? is not in the sense of image-schema
or image-representation):
Metaphors provoke and give birth to new images.  By establishing and reinforcing
connections, they encourage us to see in new ways.  While Bernard is correct that
assertions of likeness alter the way we see, Lakoff and Johnson are equally correct
in claiming that ?much of cultural change arises from the introduction of new
metaphorical concepts and the loss of old ones.?  Alterations in the way we see
can be extremely productive.
Because of this process of theory building via analogy, our scientific theories are often an
extension of the stories that our lives tell: through our political systems, technology, and
experiences.
This chapter is not intended to be an in-depth analysis of the evolution of theories
in the history of science.  Instead I draw from the work of others regarding the history of
science and explore the consistency between their ideas and my definition of analogy.  I
will explain how several theories that are clearly based in analogy and detailed in
comparative literature (Otis, discussing Ram?n y Cajal), popularized science (Burr,
writing about Turin), cognitive science (Gentner on Kepler), and the history of science
(Nersessian on Maxwell) are consistent with a category framework of analogy.  This is
not to say that all scientific theories evolve via analogy to known systems; rather, I claim
that the process of analogy is prevalent in the scientific community and the manner in
which analogies are used is consistent with a categorization framework.
History and Philosophy of Science
I begin with examples in the history and philosophy of science from physics.  As
noted in the first chapter, it is not hard to find instances of analogy in the creation of
theories in physics.  Einstein, who had a background in physics, was working as a patent
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clerk in Switzerland during an era of train travel (Galison, 2003).  A great deal of patents
at that time involved synchronizing clocks from one station to the next ? and Einstein?s
special theory of relativity was often described (by Einstein and others) as a question of
synchronizing clocks on a train.  Kosterlitz and Thouless (1973) drew an analogy
between order parameters and their associated phases on the one hand and topology on
the other.  Nancy Nersessian, a philosopher of science, has studied what is referred to as
?the method of physical analogy.?  She studies model-based reasoning, such as the type
employed by Maxwell in determining the electromagnetic field equations.  Maxwell
constructed a model of electromagnetism in materials that consisted of vorticies that
created a series of interlocking gears ? ?idle gears.?
Nersessian has developed a hypothesis regarding the manner in which mental-
modeling works and how it is employed by scientists.  Her mental-modeling hypothesis is
that
In certain problem-solving tasks humans reason by constructing an internal iconic
model of the situations, events, and processes that in dynamic cases can be
manipulated through simulation.  In constructing a model, information in various
formats, including linguistic, formulaic, and imagistic, where the latter is taken
here to include various perceptual modalities, can be used.
A question that is often asked of model-based reasoning is that, given that the model is an
existing and understood model, how can model-based reasoning ?be generative of
conceptual change in science??  To this Nersessian responds that it ?requires a
fundamental revision of the understanding of concepts, conceptual structures, conceptual
change, and reasoning.?  To address this concern, Nersessian argues, we must revise the
notion of a concept:
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A basic ingredient of the revision is to view the representation of a concept as
providing sets of constraints for generating members of classes of models.
Concept formation and change is a process of generating new, and modifying
existing, constraints.  This is accomplished through iteratively constructing
models embodying specific constraints until a model of the same type with respect
to the salient constraints of the phenomena under investigation, the ?target?
phenomena, is achieved. (Nersessian, 2002, p 139)
I would like to stress from this the claim that the representation of a concept is a set of
constraints for generating new members of classes of models.   The set of constraints is
identified by finding analogous cases (in the case of Nersessian?s studies, these analogous
cases are models) and a case is deemed analogous by being one of the ?same type with
respect to the salient constraints of the phenomena.?  Mental-models, then, which are
prevalent in the development of scientific theories, operate by being members of a class ?
a prototype of a category, I argue ? that are useful for their ability to determine the set of
constraints necessary for the representation of a concept.  Though the final negotiation of
the set of constraints may be a structure-mapping process between the particular base of
the analogy (or model in the method of physical analogy), the primary role of analogy ?
the assertion made when the analogy first is introduced ? is one of categorization
1
.
Again, the categorization is indicative of a particular cognitive model and that model
creates a language for discussing the concept.  This differs from structure-mapping
theories of analogy in that the significance is not tied to the base in particular except for
its role in being a prototype for a category invoked by a common schema. Nersessian
(2002, p. 138) writes,
                                                  
1
 Another possible interpretation, which I will not go into in detail here, is that this
category is a wholly new category that represents a ?blended space? as detailed by
Fauconnier and Turner (1994).  This idea will be explored in the conclusion as a possible
direction for further research.
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In model-based reasoning processes, a central objective is to create a model that is
of the same kind with respect to salient dimensions of the target phenomena one is
trying to represent.  Thus, although an instance of a model is specific, inferences
made with it in a reasoning process are generic.  In viewing a model generically,
one takes it as representing features, such as structure and behaviors, common to
members of a class of phenomena.  The relation between the generic model and
the specific instantiation is similar to the type-token distinction used in logic.
Generality in representation is achieved by interpreting the components of the
representation as referring to object, property, relation, or behavior types rather
than tokens of these.  One cannot draw or imagine a ?triangle in general? but only
some specific instance of a triangle? In considering the behavior of a physical
system such as a spring, again one often draws or imagines a specific
representation? That is, the reasoning context demands that the interpretation of
the specific spring be generic.?
The point that I take from this for my thesis is that the base of an analogy is more abstract
than a particular analogy appears.  Maxwell?s analogies of idle gears was no more tied to
that particular representation of idle gears than claiming two ideas are ?apples and
oranges? is tied to any particular instance of comparing apples and oranges.  It is simply a
prototypical instance of a class of phenomena that share a place within a certain cognitive
model.  Similarly, Miranda?s analogy to the toy cat was only one salient example of the
story (or cognitive model or phenomenological primitive) of carrying.  The base of the
analogy refers to a category ? ?one takes it as representing features, such as structure and
behaviors, common to members of a class of phenomena? (emphasis added).  And the
base in particular is chosen only because we cannot imagine categories in general or
reason about them generically but must choose a single representation to reason with.
This representation is what researchers in categorization have termed the category
prototype.
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Comparative Literature
I now turn from physics to biology: first findings from comparative literature and
then a reiteration of these findings from studies on scent.  The following ideas from
comparative literature are by a professor who has a background in biology.  The idea that
her findings underscore is that our science evolves from the cognitive models that we
have in mind, as provided by our culture and technology.
Membranes
Otis (1999), a professor of comparative literature, was pursuing the concept of
identity as scientist-authors in the 1800?s defined it.  In particular, she explored how ?the
changing understandings of personal and national identity encouraged people of the
1830?s to see living things as associations of independent units? (Otis, 2001).  Both the
political climate of colonialism and the scientific studies of cells created the ideas of
entities with porous but definite membrane borders.  In her work Membranes (Otis,
1999), she writes:
In their respective languages, all of these physician-authors confront their
cultures? demands for borders, and they express and challenge them through
common metaphors and maneuvers.  This coincidence suggests that imperialistic
culture, which offers the same metaphors to scientists and novelists, shapes both
biology and literature by shaping the language through which they express
themselves? the relationship between literature and science is one of mutual
feedback and suggestibilty, each contributing to and drawing up on the ?cultural
medium? out of which it grows.  Culture, however, does not ?determine? science
or literature any more than science and literature determine culture: personal
vision persists, despite all indoctrination and all scientific training.  (p. 3)
One particularly illuminating story that Otis tells is that of Ram?n y Cajal, the Nobel
Prize winning biologist who determined that the nervous system is made of discrete cells
and is not a continuous net-like structure, as it appears.  She questions why this had not
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been determined before ? Golgi had invented the necessary techniques years before
Ram?n y Cajal employed it in studies of nerves ? and she questions ?what was it, I
wondered, that drove Ram?n y Cajal to keep looking, determined to resolve boundaries
between cells when there appeared to be no boundaries at all??  The answer, she
determines, is that the cultural medium of the time was creating a particular vision:
Many factors besides the essential technical ones affect what one sees under a
microscope, or at least the way one describes it.  It has been proposed, for
instance, that late 18th-century German philosophy, with its stress on individual
perception, inspired people in many fields to conceive of life in terms of
independent living units (Rothfield, 97).  How might politics and culture shaped
cell theory?? Cell theory relies on the ability to perceive borders? Germ
theory? encourages one to think in terms of ?inside? and ?outside.?  If one
believes that invisible germs, spread by human contact, can make one sick, one
becomes more and more anxious about penetration and about any connection with
other people ? the same anxieties inspired by imperialism?  (Otis 1999 p 5).
That is to say, the cultural medium in which science exists strongly influences the way in
which data is seen: the existing stories the culture has constructed provide expectations
for the data.  Her study on the connection between national agenda, scientific theory and
literature, compels Otis to claim that ?the division between the humanities and the natural
sciences [is] another boundary arbitrarily drawn.  Scholars on both sides of the line want
to answer the same questions, and we express ourselves through metaphors provided by a
common culture.?  It is no secret, particularly within qualitative studies, that our culture
biases our interpretation of data.  What Otis demands that we recognize is that our culture
influences our hard sciences as well as our art, and it does it in the same way.  This
echoes a comment by Robert Irwin, an abstract artist who was paired with a physicist in
?cultural exchange? experiment.  Initially both artist and scientist were pessimistic on the
merits of this pairing, but quickly found that they were both addressing common
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questions with their work and found collaboration easy.  Irwin, in his biography,
comments:
I really feel that there is a kind of dialog of immanence.  That certain questions
become demanding and potentially answerable at a certain point in time, and that
everyone involved on a particular level of asking questions, whether he is a
physicist or a philosopher or an artist, is essentially involved in the same
questions.  They are universal in that sense.  And although we may use different
methods to come at them, even different thought forms in terms of how we deal
with them ?  and we will eventually use a different methodology in terms of how
we innovate them ?  still, really those questions are happening at the same
moment in time.  So that when we find these so-called accidental
interrelationships between art and science, I don't think they're accidental at all.
                                                                                    -Robert Irwin
This ?dialog of immanence? I believe is related to what Otis refers to as the ?cultural
medium.?   The ?certain questions? that become demanding are those that our paradigms
provide us.  And those paradigms are not so local as Kuhn (1970) suggests in his
Structure of Scientific Revolutions ? they can be broad cultural paradigms.  They are the
cognitive models supplied by our daily experience in our culture.  These models (stories
or schemas) are responsible for our categories, which in turn allow us to be creative and
reconceptualize our science, art, medicine, political systems and economies.  If creativity
is a shifting of categorization (?re-representing an entity or a situation from one
?ontological? tree of concepts and categories to another ontological tree of concepts and
categories? Chi, 1997), and analogies are assertions of that unexpected categorization,
then it is changes in our cultural medium
2
 that provides us with the new categories,
                                                  
2
 It is not, of course, such a strong dichotomy: culture versus science.  Art, technology
and science are all contributing to the cultural climate.  The point that Otis is making,
though, is that these parts of culture are far more intertwined than previously thought.
Culture, as defined by the political climate in particular, influences science ? hard science
?  far more than one would think and the way it does it is by giving us new schemas and
the new language associated with those.
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derivative of new cognitive models, in the first place that allow us to be creative.  These
?accidental interrelationships? (Wechsler/Irwin), are not accidental at all, but are due to
the ?metaphors our culture provides us? (Otis, above).  And these metaphors are not
simply local structures that we map but are creating new ontologies, new categories,
derivative of new cognitive models.
So far, this points to the following: we see ? in our data and in our art ? what we
expect to see and those expectations come from the cognitive models we have developed
from our experiences and our culture.  We understand and categorize phenomena based
on the cognitive models that we have in place.  But this is only part of the story.  For this
still begs the question: is this analogy?  Insofar as I have defined it, unless there is an
explicit negative assertion, where the mind considers two possible schemas ? each
coherent within its own framework, but only one of which is possible ? then this is not
analogy.  Ram?n y Cajal?s insistence on looking for membranes may be such an instance,
but it is not clear from the story provided.  It is doubtful that he understood his research
in the larger cultural paradigm and deliberately chose a schema of boundaries over the
continuum model.
Rather, this insistence on borders and boundaries in what are otherwise
continuous, unbounded regions, sets up a particular resource and this resource is
continually activated by the culture.  It was, perhaps, never a deliberate cognitive
?choice? to activate it.  For a case that is more clearly one of analogy, consider the role of
networks and the telegraph on understanding the nervous system.
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Networks
Otis? first study, a study on identity, led her to a realization that the concept of
membranes pervaded literature and science at a time when the political climate of
imperialism demanded the idea of borders.  In a second study, Otis (2001) explores the
idea of communication and traces the idea of networks, again looking at the nervous
system and its theoretical development as consistent with the telegraph.  Just as McLuhan
professed, ?the medium is the message,? so Otis (2001) finds that the message cannot be
?abstracted from the medium that transmits it.?  In particular, the advent of electronic
communication has forced a reconceptualization of our selves:
Since the late 1840?s, electronic communications networks have changed the way
we see our bodies, our neighbors, and the world.  For a century and a half, these
networks have suggested webs, leading their users to think as though they were
part of a net.  Between 1845 and 1895, the development of the telegraph
transformed people?s understanding of communication and, with it, their notion of
their relation to others.  As the telegraph affected language, Carey argues, it
?changed the forms of social relations mediated by language? (Carey, James, 1989
p 210).  The telegraph became ?a thing to think with,? shaping the thoughts that it
wired.  (Carey p 204)
In the seventeenth century it was thought that muscular motion, determined in the brain,
was mediated by pressures in a nervous fluid.  As Otis (2001, p. 14) notes: ?observing the
brain?s ventricles filled with cerebrospinal fluid, the earliest anatomists envisioned the
nerves as a kind of circulatory system, drawing inferences about their structure and
function by comparing them to a system whose structure and function were more
obvious.?  Though criticisms of this mechanism were made and scientists (namely
Galvani) argued for replacing the hydraulic model of the nervous system with an
electronic model, this alternative was not adopted until after the advent of the telegraph.
As Lenoir (1993) argues, scientists? familiarity with electrical circuits affects ?not just the
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way they performed their experiments but the way they conceived of the nervous system
itself.?  That is, the technology allowed for a change in theory ? and not because of the
ability to make new measurements (telegraphs as a technology are not a tool for
discovery in that sense), nor because of the change in theory regarding telegraphs
(electricity was understood well enough to created the telegraph but was not applied to
nerves until after the telegraph was invented), but because of a cognitive model that it
afforded, allowing scientists to see and categorize phenomena that they had no way of
understanding otherwise.
In this century, the nervous system is referred to ?in the language of the
cybernetic web? (Otis, 2001).  Indeed, even the idea of where thought occurs has moved
from the brain to this interconnected web in the paradigm of distributed cognition
(Brown, Collins, and Duguid, 1989) ? an idea that one could argue finds its origins in the
transmission and growth of knowledge enabled by the internet.  This relationship
between technology and theory ? in which technology is not only created by changing
theories, but often vice versa ? can offer an explanation for Robert Irwin?s observation
that artists and scientists seem to address questions that become ?potentially answerable
at a certain point in time.?  These questions arise because, as Otis claims, ?the
technological metaphors affected? decisions about which phenomena to study and what
experiments to perform.? (p 3)  Not only because or technology affects what we can
study, but what we choose to study and how we represent and understand our findings.
Some ideas seep into the consciousness of our culture ? through our language and
technology ? so that we cannot help but use these ideas as a lens on other phenomena.
Some ideas require an explicit cognitive work ? it is a lens you must ?put on? to see
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phenomena in a new way.  With Ram?n y Cajal?s work on cell membranes, the lens was
perhaps unconscious.  But with the electrical properties of the nervous system the
analogies were explicit (Dubois-Reymond, 1868 p 97):
just as little telegraph wires, do the nerves betray by any external symptom that
any or what news is speeding along them; and, like those wires, in order to be fit
for service, they must be entire.  But, unlike those wires, they do not, once cut,
recover their conducting power when their ends are caused to meet again.
Again, this explicit use of analogy reiterates the manner in which certain implicit
metaphors and categorization differ from deliberate analogy.  Though many researchers
place analogy on a continuum that includes rather mundane instances of similarity and
categorization (Gentner and Markman,1997; Hofstadter, 2001), I claim (or more
accurately define) that analogy differs from categorization and routine similarity in that
analogy requires a reschematization of the target while categorization does not.  In the
following quote, Otis (2001) notes the problematic distinction between category and
analogy:
In his Foreward to Kittler?s Discourse Networks, David E. Wellbery declares that
?in its nervous system, the body itself is a medial apparatus (xiv).?  He means, of
course, that the nervous system is like an electronic medium ? or does he?  If they
perform the same functions, are nerves like cables or are they identical, members
of the same functional category?  Metaphors elide likeness, masking a key
epistemological link.  But what is the epistemological value of metaphor?  What
does one gain by saying that one thing is like another?? It could be objected that
nerves are alive and thus inherently different from any sort of technological
apparatus.  Since the early nineteenth century, though, drawing a distinction
between organic and technological systems has grown increasingly problematic.
In this paragraph, Otis notes that the claim of analogical likeness is not so far from the
claim of category inclusion.  The reason one may wish to say that nerves are ?like? an
electronic medium is because of nerves have this quality of being alive ?  and so
couching this claim in analogy form (?nerves are like cables?) recognizes that there is a
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violation of expected categorization: living things are not machines.  But increasingly this
distinction between organic and technological is problematic ? as we begin to understand
the human body in terms of machines, this idea may shift from an assertion of analogy
(shifting the schematization of the concept) to more routine categorization (in which the
?machine? category no longer differentiates living things from non-living things), in
which someone may claim that the body is a medial apparatus ? or even simply use the
language of telegraphs to speak of the nervous system ? without there being the tone of
analogy present.
Below I present a more modern example of technology (the scanning electron
microscope) influencing science by providing a schema by which we understand scent,
and the way in which the previous understanding of scent had been a rather implicit
application of a schema that, in terms of predictions and an understanding of olfaction,
was not generative.  I then return to theories in physics with an analysis of a historical
analogy relating the sun?s relationship to the motion of the planets to the sun?s light.
Luca Turin: Analogies involving scent
The mystery of our sense of smell stems from the puzzling ability to smell
everything instantaneously.  In order to smell something, we take particles of it into our
nose and these are detected by the olfactory system.  However, as noted by Burr (2002),
our other systems that detect particles that have come into our body, the digestive system
and the immune system, have evolved so that they can either work instantaneously on a
limited number of molecules (the digestive system) or can handle a myriad of molecules
but take a significant amount of time (the immune system).  This stems from the role of
enzymes ? the body either has enzymes on hand, perfectly manufactured to bond to the
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molecule or it must create the enzymes.  The digestive system has evolved to have
enzymes ready to digest a limited number of molecules and does so immediately, while
the immune system has to make them ? a powerful ability that enables us to fight off
diseases that we have never in our evolutionary history seen before, but an ability that
takes time, hence the difference in response times.  The paradox of smell is that the
olfactory system can smell a manmade molecule that has never been smelled before and
can do so instantaneously.  Perhaps unwittingly, prior to Turin?s work on smell, scientists
were assuming that smell worked according to the same principles as the digestive and
immune systems ? it is only because of this parallel that one should be surprised that the
olfactory system can handle novel molecules instantaneously (had they assumed smell
worked according to principles similar to sight, where we can see shapes we have never
seen before, then this property of smell would not be surprising).
Turin?s idea was that smell was not appropriately schematized ? in a sense, the
scientists were asking if the Pope is a bachelor.  The questions they asked about the sense
of smell were difficult to answer because they were not appropriate.  Questions asked of
enzymes, of shape and timing were not the right questions ? they were questions
provided from the schemas associated with the immune system and digestive system, a
schema that does not fit the olfactory system ? but with no alternative schema, these were
the only questions available to ask.  Just as the invention of the telegraph inspired theory
of the nervous system, recent developments in microscopy in physics provided an
analogy for Turin, a way of reschematizing smell.  Turin proposed that smell works
according to the same principles as a tunneling microscope: the molecule providing an
electrical connection and the strength of that connection related information regarding the
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structure of the molecule.  In this schema, the shape of the molecule is not important,
enzymes do not factor into the process of smelling, and a question of how we can smell
novel molecules instantaneously is not an issue.  As with Otis?s findings in membranes
and networks, it was changes in technology that enabled changes in theory: not because
the technology was a necessary element in discovery but because the technology created
a new schema and allowed the reschematization of smell.
Cognitive Science
Gentner?s (2002) structure mapping theory has been to understand developments
in the history of science.  In particular, she has looked at the analogies of Johannes
Kepler.  Kepler, born in the 1500?s, inherited an astronomy of spheres in which planets
circumnavigated the sun by the will of souls (later translated to angels or virtues or
spirits).  However, there were regularities and features in the data of the motions of the
planets that required explanation.  In seeking to understand why the planets that were
further from the sun moved more slowly, he argued (Kepler, 1596, p199):
?one of two conclusions must be reached: either the moving souls are weaker the
further they are from the Sun; or, there is a single moving soul in the center of all
the spheres, this is, in the Sun, and it impels each body more strongly in
proportion to how near it is.
That is, the sun is responsible for the motion of the planets and the closer you are to the
sun the more it can make you move ? the sun is transmitting a motive power through
space to the planets and this power is closer when one is closer to the sun.  To make sense
of this argument, he appealed to analogy:
I shall propose to the reader the clearly authentic example of light, since it also
makes its nest in the sun?Who, I ask, will say that light is something material?
Nevertheless, it carries out its operations with respect to place, suffers alteration,
is reflected and refracted, and assumes quantities so as to be dense or rare, and to
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be capable of being taken as a surface wherever it falls upon something
illuminable.  Now just as it is said in optics, that light does not exist in the
intermediate space between the source and the illuminable, this is equally true of
the motive power (Kepler, 1609/1992, p 383)
To understand the role that analogy plays for Kepler and ?knowledge change? (p 28),
Gentner (2002) argues that there are ?at least six ways in which the process of analogical
comparison can lead to knowledge change?
1. highlighting and schema abstraction ? extracting common systems from
representations?
2. projection of candidate inferences?
3. noticing alignable differences ? becoming aware of contrasts on dimensions
that are present in both analogs?
4. re-representation ? altering one or both representations so as to improve the
match?
5. incremental analogizing?, and last, the rarest of these,
6. re-structuring ? altering the domain structure of one domain in terms of the
other?
Gentner then discusses how Kepler used analogy in these ways to effect knowledge
change.  However, at the end of the article she notes:  ?I have focused here on the use of
analogies in online thought ? that is, the processes of analogical reasoning once one has
both analogs in mind.  But it is obviously crucial to ask how potential analogs come to
mind.?  As noted in chapter 5, and echoed by the findings above by Otis, I argue that the
analog (that is, the base of the analogy) is not what is primary: the analogy does not first
spring to mind and then induce a ?schema abstraction? (Gentner, 2002).  Rather the story
(or schema, p-prim, or idealized cognitive model) is primary and this story is what is first
accessed.  Once accessed or identified, the analog is constructed or identified as a
?prototypical? member of the category that this model defines ? concretizing or unitizing
these schemas entailed in the cognitive model.  For clarification and further explanation,
consider the following visual ?toy? (Fig. 7.1).
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Recognizing the irony in making an analogy about analogies to clarify my points,
I would like to argue two points: first, that it is having a schema for rabbits that allows
you to re-represent this drawing and second, that the schema, and not any particular
representation of rabbits, is primary.
Fig. 7.1
When you shift the representation of this drawing from, say, the duck to the rabbit you
can only do so because you know what rabbits look like.  You have a ?schema? in mind
that allows for long ears and tiny nose, and though the rabbit pictured above may not look
exactly like any rabbit you have ever seen, you can quickly recognize elements of the
?rabbit story? that apply here.  For Kepler, if the sun did not seem to give off any kind of
energy, or if the intensity of light did not decrease with distance, I would like to suggest
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that not only would there have been no analog for Kepler to draw but that even
identifying this ?story? in the case of planetary motion would have been exceedingly
improbable: we only see the rabbit because we have seen other rabbits ? because our
mind knows what rabbit ears look like.  Kepler?s mind knew properties of light ? and so
was attuned to recognizing when things decay with distance, that travel through space
without being seen and have an effect on objects.  This echoes the answer Otis gives to
her question, ?what was it, I wondered, that drove Ram?n y Cajal to keep looking,
determined to resolve boundaries between cells when there appeared to be no boundaries
at all??  That is, the culture had created a story of boundaries and membranes, so that this
story could be identified in other places.  
This idea ? that our abstract ideas stem from our experiences in the world, has also
been argued by Lakoff and Nunez (2001) in Where Mathematics Comes From, in which
they argue that math is not the abstract ideal that we imagine it to be, but arises from
metaphors to physical experience.
When you think about it, it seems obvious: The only mathematical ideas that human
beings can have are ideas that the human brain allows.  We know a lot about what
human ideas are like from research in Cognitive Science.  Most ideas are
unconscious, and that is no less true of mathematical ideas.  Abstract ideas, for the
most part, arise via conceptual metaphor-a mechanism for projecting embodied
(that is, sensory-motor) reasoning to abstract reasoning.  (Lakoff and Nunez, 2001 p
xxi)
Mathematics, as with science, does not simply evolve from the data or from the numbers,
but from the ?ideas that the human brain allows.?  And these ideas that the human brain
allows are the ideas that it has acquired from the surrounding culture and from sensory-
motor experience ? they provide us with schemas that we may then project onto new
experiences.
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Structure-mapping suggests that, to re-represent the drawing as a rabbit, one must
first imagine a rabbit and then align the features of that rabbit with the features in this
drawing.  The single representation of a rabbit is primary and is used, together with the
rabbit in the picture, to access the more abstract schema.  It suggests that we might have
to consider a limitless possibility of things ? rabbits, cats, dogs, tables ? and one by one
consider the ways in which the structures of these various potential analogs align with the
target.  Instead, I find far more plausible that the rabbit ?schema,? rather than a particular
representation of a rabbit, is primary.  We access the analog via the schema, rather than
the other way around.  (Another point to note is that you are not drawing an analogy
between the rabbit and the duck but rather re-representing the drawing, understanding
this ?stick-out piece? as ears instead of bill.)
Conclusion
The impetus for my thesis and the bulk of my support comes from analysis of
student-generated analogies in science.  However, studies in the history and philosophy
of science, comparative literature and cognitive science have developed stories of theory
development that are consistent with my claims that derive from student reasoning.
These claims, consistent with student-generated analogies, are also consistent with
analogies from philosophy and biology.  In particular, the ways in which
? analogies are derivative of the schemas provided by the surrounding
culture,
? these schemas are primary and are used to access or construct analogs
? the likeness expressed by analogy is a class-inclusion statement,
? concepts are defined as constraints for class-inclusion, and
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? the specific is used to represent the general
are all consistent with generated analogies as assertions of categorization.
In the appendix, I turn from expert scientists? analogies to young students?
analogies and find patterns that are reminiscent of the findings reported here.  I will
return to the ideas from this chapter in the following chapter in which I consider the
implications for instruction: if this is what science looks like, and if this is how scientists
construct theory, there are profound implications on how students should be taught.
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Chapter 7:  Implications for Instruction
How we understand the mind matters? it matters for what we value in
ourselves and others ? for education, for research, for the way we set up
human institutions, and most important for what counts as a humane way
to live and act? Our ideas about what people can learn and should be
learning, as well as what they should be doing with what they learn,
depend on our concept of learning itself.  It is important that we have
discovered that learning for the most part is neither rote learning nor the
learning of mechanical procedures.  It is important that we have
discovered that rational thought goes well beyond the literal and the
mechanical.
-Lakoff, 1987 (Preface)
Introduction
The above quote is from a 1987 publication by George Lakoff on categorization.
He claims: ?it is important that we have discovered that learning for the most part is
neither rote learning nor the learning of mechanical procedures? that rational thought
goes well beyond the literal and the mechanical.?  My claims with respect to analogies
echo the findings that Lakoff hails in the preface above.  In the 17 years since this
publication, how has instruction responded to these important discoveries?  What would a
response to such findings look like in practice?  How should they be incorporated in the
classroom?  In this chapter, I begin with a critique of a relatively standard approach to
analogy-use in the classroom.  I then highlight three important implications that this
thesis, with its focus on student-generated analogies and the categorization interpretation
of these analogies, has for instruction.  The first is that student-generated analogies ought
to be a goal of science education.  The second relates to an appreciation and
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understanding of the manifold nature of students? minds.  And finally, when conceiving
of analogies as a form of categorization, questions and goals regarding transfer change.  I
then present examples from analogies in this dissertation, from the literature and from my
own teaching that illustrate these implications in practice.  The relationship of these
implications to the National Science Education Standards and to calls for a greater
diversity in science will be explored.
A critique of standard analogy use in classrooms
Lulis, Evens and Michael (2004) report on ?How Human Tutors Employ Analogy
to Facilitate Understanding? in the context of medical school students receiving tutoring
on the heart and its baroreceptor reflex.  The tutors in this study are referred to as ?expert
tutors? and their practices are being studied for the creation of an electronic tutoring
system based on the computational model provided by the Structure Mapping Engine
(Falkenhainer, Forbus, and Gentner, 1986).  I present this here as an example of what is
considered (by some) to be best practices for analogies in education and offer a critique
of these practices.  All transcript quotes provided below are from Lulis, Evens and
Michael (2004).  Figure 7.1 (from Scott and Schactman, 2004) below provides a
schematic of the heart to help the reader understand the conversation:
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Fig. 7.1
Student:  If I make an analogy of you try to fill a sink with water and you ?
Tutor:Try to fill a balloon with water, since that?s what we?re dealing
with, a distensible object.
Student: Okay.
Following this, Lulis et al write:
The following structures underlie what the tutor does (as discussed in Lulis & Evens,
2003; Lulis, Evens and Michael, 2003):
Structure for the balloon:
? fill a balloon with water
? it will be distend
? the pressure in the balloon increases as it distends
Structure for the heart
? fill the right atrium
? the right atrium will distend
? the pressure will increase as it distends
The above example demonstrated the effectiveness of the accepted structure mapping
approach of connecting new knowledge to knowledge already understood by the
student.  As a result, the student develops a better understanding of the new concept
(Gentner, 1983, 1988; Goldblum, 2001; Holyoak and Thagard, 1995).
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This is the first analogy mentioned in the article and it is applauded by the authors
as effective because of its ability to connect knowledge about the heart to ?knowledge
already understood by the student.?  The veracity of this claim is dubious ? the student
has not demonstrated any knowledge of pressure in balloons ? but the greater point I
would like to argue against is the idea of what learning is that is presented in this brief
transcript.  The tutoring offered in this transcript values content knowledge over the
ability to think scientifically: the role of the teacher is to ?correct misconceptions? (Lulis,
Evens and Michael, 2004).  The primary goal of instruction was to ?develop a better
understanding of the new concept? and not to increase the student?s ability to create his
own models, evaluate them on his own for their goodness of fit and recommend new
models if the fit is poor.  The student has constructed a model of the heart as a sink and is
immediately cut-off and told, instead, to model the heart as a balloon.  This is deemed
effective because of the balloon?s structural map to the heart.  I argue that this may be
counter productive instruction because the student is actively discouraged from pursuing
his own analogy.  In fact, depending on how his analogy played out, it could be seen as
more deeply structural than the tutor?s as it did not rely on superficial similarities, such as
distending.  The pressure at the tub?s outlet will increase with increasing volume of fluid.
This relation between volume and pressure, in fact, is not altered by the tutor?s
prompting.  That the student has not changed his ?misconceptions? and is still not
creating the analogy that the tutor wills is evident in the following transcript from a later
session with the tutor as he states that size will:
Tutor: We can look at the central blood chamber that means the big veins
and the atria together as though they were an elastic chamber.  Is
that not correct?
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Student:Yeah, and the heart is the pump.
Tutor:Well, let?s stick to this elastic chamber and look at it first more or
less in isolation.  If you have an elastic chamber what are the
things that determine the pressure inside that chamber?
Student:Size.
Tutor: No.
Student:I mean if you ? .  I mean ? .  Area is one but I gather for the heart ?
Tutor:Area of what?
Student:Area that ? I mean if you want to know what the pressure is of a
gas or well liquids aren?t that ? .  We?re not talking about gas,
we?re talking about liquids.  And liquids aren?t affected by size
much because you can?t compress the molecules that much.
Tutor:Oh, you mean the volume occupied by the liquid, expansion and
condensation of the liquid.  No.  That?s not an issue.
Student: No, because we?re talking about liquids and liquids aren?t affected.
Like with gas, besides the container matters a lot ?
Tutor:Let?s throw away this atria central venous system and take instead
something inanimate elastic stretcher, say like a balloon.  Right?
What determines the pressure inside the balloon?
In this transcript the student again asserts an analogy the heart is like a pump, and is cut-
off from continuing with this analogy.  Instead, the tutor wants to focus on the ?elastic
chamber?-like properties of the veins and atria and why it is best to consider the atria as
elastic chambers and not a pump are never made clear.  The tutor is not taking advantage
of the student?s analogy to try to understand the student?s thinking.  Even to the objective
of content understanding, the student?s analogy can give the tutor information to help
with diagnosis of student difficulties
When the student is asked what affects the pressure in the atria (the upper
chambers of the heart), he offers size ? which is reasonable and predictable from a sink
analogy of the heart.  However the tutor clearly has other factors in mind, and flatly
responds ?No? to the student?s ideas.  But the student, having never understood the
failure of his heart-as-sink model or heart-as-pump model, is then clueless as to the
?correct? answer that the tutor desires and seems baffled.  Similarly, the tutor
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misunderstands the student?s reasoning, perhaps interpreting ?size? as being not the
volume of blood in the vein, but, perhaps, the volume occupied by a fixed amount of
blood.  The tutor cuts off the student and provides an analogy to balloons ? which are, of
course, filled with the very thing the student just determined as an important factor: gas
instead of liquid.
And finally, there is a student-generated analogy in which the student proposes
that the heart is like a traffic cop.  The following session was an online tutoring session
and the transcript is a written transcript between the two:
Student:  Would it be a reasonable analogy to look at the heart like a traffic
cop?
If it slows down the rate of blood flow (lets fewer cars through)
then there will be a backup behind it (a backflow of blood prior to
the heart, and therefore an increase in CVP [central venous
pressure]) and fewer cars coming through (less blood coming out
of the heart and therefore a decrease in MAP [mean arterial
pressure])
Tutor:The analogy is OK.
But just as traffic jam does not occur because cars back up, the
increase in CVP caused by a fall in CO [cardiac output] is not the
result of blood BACKING UP.
Everything goes in one direction.
Student: Well, slowing down would be a better way to put it, then.
Tutor:Yes.
A traffic jam caused by everybody piling into the same area at
once.
The authors (Lulis, et al) describe the passage above as follows:
?? an analogy proposed by the student between the heart and a traffic cop.  The
mapping between these analogs is not correct; the tutor proposes a more suitable
analogy between the heart and a traffic jam.?
But in what way was the analog ?not correct??  Clearly the tutor found something wrong.
The tutor mistook a ?back up? for ?backing up? ? but slowing down for the traffic cop
and creating a ?back up? never necessitate a physical reverse of direction.  To ?correct?
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the student?s misunderstood idea, the tutor recommends thinking of the heart as a traffic
jam.
In summary, the tutor has a concept that he/she would like expressed by the
student, and this tutor has a particular analogy in mind for conveying this concept.  When
the student proposes alternative analogies (which may not even be incorrect), the tutor,
concerned with establishing the correct scientific answer in the student and not concerned
with fostering the scientific reasoning abilities of the student, cuts off analogy generation.
While clearly this is not effective in fostering scientific creativity, it also is not clear that
it dislodges any supposed misconception that the student has.
Furthermore, the interpretation of the authors, who apply a rigorous structure-
mapping technique to analyzing these analogies, suggests that the heart cannot be
conceived simultaneously and fruitfully as both a pump, elastic chamber, sink, traffic jam
and traffic cop.  By their interpretations, the structure implicit in these models is either
right or wrong, a misconception or a correct idea.  The task of the instructor is to give the
student the correct analogy for the purpose of correct conceptual understanding.
I would like to note that the instruction described above for implementation in an
electronic tutoring system is described not by the creators and proponents of structure-
mapping, but is instead one interpretation of how structure-mapping may be employed in
the classroom.  A more fruitful manner in which to employ the concepts behind structure-
mapping is case-based reasoning, and the merits of this approach are detailed by Gentner,
Loewenstein and Thompson (2003).  The approach described in that article focuses on
ways in which one may foster analogical retrieval and transfer in the students, as opposed
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to content knowledge.  Yet one message is still the same: the students must be told the
analogical cases in the first place before abstracting from these cases on their own.
As an alternative to instruction by analogy, with the focus on misconceptions and
correct ideas, I argue instead for instruction that fosters analogy and encourages the
students to create their own analogies and determine the merits of them.  With an
appreciation for the role of chains of analogies (analogy ?hopping?), the initial analogy?s
merits are not in the conceptual correctness in a structure-mapping sense, but instead the
analogy is meaningful for the activation of resources: schemas and cognitive models that
may help them negotiate a more meaningful analogy and understand the ways in which
various analogies are applicable, so that the heart can be conceived of as a pump,
chamber, sink, traffic jam and traffic cop.
Implications for instruction
Students ought to generate their own analogies
Studies on creativity ? creativity in general and creativity in science ? have shown
that
the essence of creativity is to be able to view a situation or an object from two
different frames of reference, or two ?unrelated matrices of thought? (Koestler,
1964).  This is sometimes referred to as restructuring.  Restructuring, thus, is
often viewed as being able to see a problem in a ?new way? that is fundamentally
different.  However, defining creativity in this way merely begs the question of
what constitutes a ?new way,? a ?different frame of reference,? or ?an unrelated
matrix of thought???  A new perspective is defined here as re-representing an
entity or a situation from one ?ontological? tree of concepts and categories to
another ontological tree of concepts and categories?
  -Chi (1997)
In this thesis, I have argued that analogies are assertions of categorization that violate
expected categorization.  In short, analogies are powerful precisely because they do this
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thing that has been identified as the essence of creativity.  It is not a leap, then, to suggest
that a goal of education should be to encourage and foster creativity by teaching students
to generate and elaborate analogies.
Furthermore, using analogies is part of what it means to do science.  ?In studies of
the work in molecular biology laboratories, Kevin Dunbar found that the most creative
and productive labs showed a high frequency in the generation of analogies and the
sustained collaborative elaboration of analogies? (Kittay, 1997 p. 400).  In particular,
Dunbar found that when formulating hypotheses scientists rely on ?far transfer?
analogies: analogies that rely on structural and not superficial similarities.  Given that
doing science, then, entails generating and elaborating analogies, science classrooms
should encourage the generation and elaboration of analogies.
Of course, analogies are frequently used in the classroom.   In a recent article by
Richland, Holyoak and Stigler (2004), they report on an in-vivo study of analogies in
eighth grade mathematics classrooms.  As with Dunbar?s finding, Richland et al found
that analogical reasoning is quite prevalent and even far transfer analogies are not
uncommon.  However, it was not the students but the teachers who generated the vast
majority of analogies in their study, and these analogies were aimed at achieving goals
relating to conceptual understanding: getting students to solve problems correctly or
understand why they are applying a certain method.  These analogies, then, are not
indicative of students? use of analogies in the classroom.  Even more troublesome is that
the concerns raised in Richland, Holyoak and Stigler?s (2004) article were not regarding
this fact.  Instead, the concerns questioned whether or not students were able to interpret
the teachers? far-transfer analogies.  Teacher-generated analogies are important and can
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play a role in encouraging student-generated analogies; but they should be responsive to
the students? ideas and not solely for the purpose of imparting knowledge.
These findings and the concerns are representative of the pervasive focus on the
facts of science as opposed to the creative inquiry that is involved in doing science, a
focus reiterated in classrooms by a reliance on textbooks and testing.  The question that
tests and textbooks answer is: how can I get my students to know x, and how can I
determine whether or not they have learned x?  Similarly, cognitive science literature on
analogies (such as Gentner & Gentner, 1983; Clement, 1983; Glucksberg and Keysar,
1990; Gibbs, 1992) consistently focuses on participants? abilities and processes regarding
the interpretation of analogies and only very rarely (Hofstadter, et al, 1995) the
generation of these analogies.  The question that is unanswered by these textbooks, tests
and studies is: how can I help my students learn how to learn?  How can I evaluate
whether or not they will be creative scientists?  How can I help them to create their own
theories?
Of course, one might expect that presenting students with carefully constructed
analogies will aid them in constructing their own analogies, and so the teacher, as
presenter of analogy, is modeling a behavior for the students that they may in turn adopt.
But such analogies are not responsive to the schemas that the students have and may
seem disconnected from their own lives.  Furthermore, simply modeling behavior for
students without giving them time to practice that skill on their own will not always result
in the students adopting that behavior.  There must be space for the creation and
elaboration of analogies by the students.  I will detail in a later section what this might
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look like in practice.  First I highlight implications from my research on the interpretation
of student-generated analogies.
Expect variability and multiple analogies
To borrow again from Lakoff?s quote that introduces this chapter, ?It is important
that we have discovered that learning for the most part is neither rote learning nor the
learning of mechanical procedures.  It is important that we have discovered that rational
thought goes well beyond the literal and the mechanical.?  The discoveries that Lakoff
refers to are discoveries in linguistics in general, and categorization in particular.  To this,
I would like to add that it is important that we have discovered that children ? even young
children ? can and do create significant, structural, ?far transfer? analogies.  It is
important to recognize the ability of students to shift representation of the base in
analogies.  It is important to recognize that analogies often appear in multiples, that they
have a strong similarity to categorization, that they can be used to make claims of
structure, of epistemology, and of ontology.
In what way are these findings important for education?  Primarily it is an
implication for education research.  First, if we value the generation of analogies it is
important to understand the cognitive work that analogies do.  Far from what structure-
mapping suggests, generated analogies are not simple pairwise projections from a base to
a target and learning, as Lakoff identifies, ?is not the learning of mechanical procedures.?
That is to say, applying the algorithm of a structure map is not the heart of analogy and
while it may promote content knowledge acquisition, it does not necessarily promote
creative and insightful reschematization.  The structure-mapping application of analogy
limits the power of an analogy ? it can only hold particular inferences, and those
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inferences must come from a structure that exists in a stable, fixed representation in the
single base.  Analogies are powerful not because of a projection from a single known
phenomenon (the base) onto an unknown or misunderstood phenomenon (the target), but
because of their ability to completely change the categorization of the target.  And
categories, because of their basis in a particular cognitive model, can make powerful
claims on the target.  These claims are not limited to what the base alone conveys, but,
more abstractly, what the category and its associated cognitive model imply.
Furthermore, because of this it is not imperative ? or even reasonable to expect ? that
students reason with conceptually (that is, structurally) isomorphic analogies.  Rather the
analogies may enable other students to access alternative models and follow chains of
analogies to arrive at an appropriate understanding.
A second claim is that, if we value analogy generation, we ought to know where
to look for it.  Young children have been described as unlikely to create analogies of ?far
transfer? (for example, Carver, Price and Wilken, 2000).  I will take issue with the
concept of ?far transfer? in a later section, but for now it is important to note that second
graders were able to compare magnets to clay and electricity.  Fifth graders compared a
swimming pool to space, a cup of water to a toy cat swinging in a basket, and running
with keys to falling off of your bike.
There is not, then, a particular age in schoolchildren when one should not expect
and encourage analogy generation from students.  The sophistication and facility with
analogy surely increases with age, but analogy ? even far transfer analogy ? is prevalent
among students from at least first grade on.
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A third claim that has implication for instruction is the claim that this research
makes on the ontology of mind.  Not only are the concepts that are being employed in
analogy the concretization of a set of activated schemas, but the concepts themselves are
variable.  A concept can be employed in an analogy for its epistemology, its physical
behavior, or its general ontology.  A concept can shift representation, as with money
being used to represent the ?hard? ontology of currency or the ?fluid? ontology of net
wealth.  For this reason, science education researchers should recognize the variability of
student reasoning: the base of an analogy is not a single, unitary concept that is fixed in
the student?s mind, but highly variable.  Teachers need to allow students to express their
senses of a concept, to identify the cognitive models they are employing in defining this
concept and allow for a shift in this representation.  (Clarifications of these ideas are
provided in the section below on these implications in practice.  Significant to this claim
is that the initial analogy does not need to be conceptually correct to be generative of
meaningful science ? conceptually and epistemologically.)
A reconsideration of the idea of transfer
Finally I would like to call into question the ideas behind ?transfer? ? a holy grail
of education.  Transfer is described as ?the ability to extend what has been learned in one
context to new contexts? (Bransford, Brown, & Cocking, 1999).  Laboratory-based
studies that address whether or not students transfer a particular technique or theory from
one domain onto another have demonstrated that far transfer ? transfer in which there are
few superficial similarities ? is difficult and rare.  ?Near? transfer, in which there are
more feature similarities between the base and the target, are more common but not
consistently achieved either (Holyoak and Thagard, 1989).  In the study reported above
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by Richland, Holyoak and Stigler (2004), it was found that, unlike in the cognitive
science laboratory settings, analogies teachers present in classrooms are often far transfer
analogies.  Dunbar?s research in science research groups found this as well ? though the
majority of analogies were within-domain, far-transfer analogies played a significant role.
Similarly, my findings in science classrooms and discussions show that instances of
transfer, as measured by analogy use, are not difficult to find, with far transfer
demonstrated across many different ages of students.  But I would like to call into
question the very idea of far transfer as a meaningful distinction.  In a category
framework of analogy, one would not expect far transfer to be as difficult if the base of
the analogy is a category prototype.  Furthermore, near transfer, because it does not
require a re-categorization of the target, is not an analogy in the sense that it does not
shift ontology ? it is routine categorization.  Again, this is a choice of definition but one
that can provide a more formal model for what is meant by transfer.  Perhaps the idea
behind transfer that we should be focusing on as educators and researchers is the ability
for students to draw analogies that re-categorize the target from an expected or automatic
categorization to a novel one, the ability to make inferences with this new categorization,
and the ability of students to identify meaningful and prototypical choices for the bases of
their analogies.  Focusing on whether or not the base of the analogy shares superficial
features with the target misses the point of analogy: the selection of a prototypical
member of a category to serve as a base in expressing a reclassification of the target.
When a student is asked what will happen to keys if you drop them while running, is the
analogy of dropping a rock from a bus ?near? or ?far? transfer?  How would that compare
to, say, a student drawing an analogy to walking and dropping coins?  The features of the
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second analogy are certainly ?nearer? to the base, and dropping coins while walking
surely happens far more frequently than rocks are dropped from buses.  But in
discussions with students the first analogy (or similar analogies) is far more frequent than
the second (which I have never heard).  How, then, can we claim that far transfer is hard
and near transfer easy?  Instead, I argue that transfer is better understood in terms of a
change in cognitive model, with prototypical instances as more accessible instances of a
particular cognitive model.
Implications for instruction in practice
Examples from transcripts
If these implications are taken into account in a classroom, what will that look
like?  The transcripts peppered throughout this dissertation prior to this chapter give an
idea.  Throughout this dissertation are analogies in classrooms that are constructed by
students and by teachers who are responsive to those students.  To detail analogy
generation by an instructor that is done in a responsive manner, consider the transcript
presented in Chapter 1 and contrast the use of analogy here with that by the tutor at the
introduction of this chapter (transcript 1, lines 53 ? 71):
Lea:  ? the pan is more dense so they?re able to slide across it like they
can ice skate across the [inaudible] here.  So that?s why they move
around more ?cause it?s more dense so they can slide across it more
and the Styrofoam is less dense and so they get stuck in it.  Like so
they can?t move as much.
Instructor: Lea I want to add ? I think you?re sort of what I when I hear you
talk I?m thinking of like, pouring water into a sponge versus
pouring water onto a hard surface. [Lea: Yeah.]  Like this sponge
is actually less dense and there?s room for it to absorb the water
and the you know if you pour it onto something hard there?s no
room for it to absorb.  But Christie ? I mean this is an interesting
thing you guys are both thinking that density is important but one
of you is thinking that more density means one thing and one of
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you is thinking more density means the other thing.  Is that is that ?
am I right?  [Christie: Yeah.]
And in lines 113 ? 121:
Lydia: I was going to say I think the pie plate is more dense but I do think
that it?s inside not outside because if there?s more space to travel
then the molecules can?t get from one space to another easily but
it?s all [inaudible].
Instructor:  Oh so it?s like stepping stones [Lydia: Kind of.] like in the
Styrofoam it?s really far to the next stepping stone so it?s like can?t
get there I?m stuck here. [Lydia: Right] but in the metal the stones
are really close together so I can kind of walk across.  [Lydia:
Yeah.]
In addition to creating a classroom in which students are encouraged to construct their
own models and explain these with analogy, the instructor constructs analogies as well.
However, rather than constructing analogies for the motion of charged particles and
presenting them to the class, the instructor constructs analogies that are responsive to the
ideas from the students ? elaborations on their ideas (?oh, so it?s like stepping stones?) as
opposed to contradictions to their ideas or even unrelated to their ideas (as was the case in
the tutor?s analogies presented above in the previous section).
Examples from the literature
Another example of responsive use of analogy in the classroom has been detailed
in May, Hammer and Roy (2004).  The class is considering how earthquakes happen and
one student constructs a lava/pressure model of earthquake:
Skander: That?s what I mean. A rock could ? like, the volcano is this big
[motions with hands] and you?re on this side of the ground, a rock
could go in, and pretend like, pretend the lava is water and the
giant rock is a cube [Teacher: okay] it goes up and since it?s
blocked the ground has to shake which causes it to crack open so it
it?ll actually like go up farther.
Teacher: Okay, so you?re ?
Skander: So it?s like you?re actually flooding the cup of the water.
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Teacher: And so the rock acting as the ice cube is flooding the lava so it has
to come up and go out?
Skander: It doesn?t have to, it just makes the ground come, it just needs
space to go up. It?s just causing it to shake and crack open.
Although this analogy is suggesting a mechanism that is incorrect for understanding
earthquakes, the teacher does not contradict Skander (as the tutor does at the beginning of
this chapter) or call on a different student, but allows the student to continue with the
generation of analogy.  In the following section I follow an example from my own
teaching, in which I detail how incorrect analogies can play out in the classroom.
An example from my own teaching
Here I present an overview of a week-long conversation in a high school science
class regarding why the sky is blue.  This conversation was not recorded; data comes
from notes and photographs of the blackboard, where ideas were collected.  I was a co-
instructor of this course and I did not refrain from interjecting my own analogies for the
students.  The class is at a state funded summer program for ?academically and
intellectually gifted? high school seniors.  There are 27 students in this class.  Most of
these students are from public schools in small towns and rural areas.  At the time of this
conversation, the students had been in school for four weeks.  A typical day began with
either a ?Fermi Question? (a question requiring students to answer a question numerically
for which there was not enough information to determine the exact answer) or a ?Science
Talk? question (a more conceptual type question, generally about a physics concept).  By
the end of the summer students pose and answer their own questions as a class.  In the
beginning of the fifth week, the question was raised, ?Why is the sky blue, and why does
it get darker the further up you go (like in an airplane)??  In groups of 4 they addressed
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this question, and, as the teacher, I instructed them: ?Don?t just say ?because of the
atmosphere.?  Be sure you say how you think the atmosphere creates blue.?  (Not a direct
quote.)  As they worked in groups, my co-teacher and I circulated around, talking with
each table and having them sketch their ideas on dry-erase boards to share with the class.
After about 20 minutes, I addressed the class as a whole.  I had noticed that some tables
were thinking of the atmosphere as a filter, and others as a prism.  I asked if all of the
groups had one of these ideas, or were there more?  Between the six tables, there were the
following ideas:
? The atmosphere is undergoing atomic emission.
? The atmosphere is like a prism: different angles to that prism see different colors.
? The thickness of the atmosphere somehow matters.
? Energy loss in the atmosphere creates a reddening of the light.
(The ?filter? theory was discarded early on by a table that recognized that this would
block out every color except blue.)  In addition to these initial, whole-group theories,
other theories were proposed by individuals:
? Thin film interference in the layers of atmosphere preferentially select blue light.
? Distance from the sun matters.  (A correlation between planet colors and the order
of colors in the rainbow, the class never seriously entertained this theory.)
? Reflection from the blue ocean creates a predominantly blue sky.  (Quickly
discarded this theory.)
Perhaps because of the charisma of the proponent of the theory, the ?atomic emission
theory? was adopted early on as a theory to consider in depth.  The students recognized
that this theory had to account for red sunsets and red pollution, green flashes and green
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tornado skies.  It had to account for the colors of skies on other planets (both of my
classes that addressed this question became very focused on the color of other
atmospheres) and for the sailors? aphorism: ?Red skies at morning, sailors take warning;
red skies at night, sailors delight.?  One student noted that skies seem a more deep or
crisp blue in the winter, another that dawn was not as red as dusk.  After lunch one day,
students came in to tell me that the sky is light blue ? almost white ? close to the sun and
a darker blue away from the sun (as drawn below).
Fig. 7.2: Description of shades of blue in the sky
In negotiating these constraints, the students could incorporate some ideas from
atomic emission, but not all.  The theory grew to be an atomic emission theory combined
with a weather component to explain other colors and a filter component to block
ultraviolet light.  A question that was could not be addressed by this theory concerned
why white paper does not look blue.  After too many ad hoc components were added on
to the theory, the students as a class decided it could not be correct.  (Questions and
comments addressing the ?atomic emission theory? are in the figure below, a snapshot of
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one blackboard.  All student comments and questions were noted for the whole class on
the blackboard, in addition, many students took notes and represented their ideas for the
class on their group?s dry-erase boards.)
Fig. 7.3: Blackboard notes of class discussion
We then revisited theories they had mentioned on the first day of discussion.  In
writing them on the board, I asked one student about his thin-film theory.  He commented
that it was ?stupid? and we shouldn?t consider it.  Knowing that this student had been
questioning in the past how bubbles get their oil-slick-like colors, I pressed him for his
rationale in creating the theory.  The bubbles were mentioned and I pointed out, using his
experience with bubbles, that really what all of these theories have in common is that
they are a way of using white light to get colored light.  Bubble ?juice? itself is clear and
has no color, nor does a prism or neon gas (a model for atomic emission) or ?red shifted?
white light, and white light bouncing off of a blue surface will then appear blue, even on
?colorless? white.  All of these systems have no apparent inherent color and yet they all
have a mechanism by which they attain color.  More importantly, the students recognized
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that these mechanisms were inherently different, so that a prism model was not the same
as an interference model or an emission model or a red shift model.
At this point I note several ways in which I, as a teacher, was incorporating the
implications for instruction mentioned above: my rationale in structuring the course in
this way and my interpretation of the students? comments.
First, I know why the sky is blue and could easily have delivered a 10-minute
lecture on the topic, expeditiously explaining why the sky is blue ? incorporating
analogies if helpful.  (Although some questions they brought up are ones I had never
considered and would have been hard pressed to answer!)  Moreover, I know that the
students do not know enough physics to correctly answer the question of why the sky is
blue.  The pattern of dipole radiation that is responsible for our blue sky is beyond the
scope of a high school physics class, although the basic story of scattering is one that they
could reason about and understand.  My reason for asking the students to develop their
own models reflects my contention that education should encourage students to learn how
to create their own models and draw their own analogies.  I also believe that it is possible
to address this question scientifically even if one does not know enough science to arrive
at the ?correct? answer ? for though Newton did not know enough about light to
understand reflection he addressed these questions in a scientific manner and arrived at
questions that proved important to later research.  At any point, I could have posed
questions to them that would have poked holes in particular theories, but I chose to allow
the students to find these themselves, as part of what it means to do science.
Second, I did not assume that the analogies arose from a pairwise analysis of
target and base: initial analogies were identifying ways in which one can get color from
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colorless things; the analogy springs to mind before the structures (in a ?structure
mapping? sense) can be evaluated and checked; this evaluation was done in large part as
an entire class and was not part of the analogy process for any single individual.  These
analogies, the initial models proposed by the students, echo the analogy process that
Miranda used (in Chapter 4).  In the same way that the overturned cup of water is not
always like an overturned cup of water ? at times it is much more like a toy cat swinging
in a basket, so too is air not always like air: at times it is more like a prism, a neon sign,
or a hologram.  An understanding of the analogical base as a fixed structure cannot
expect and understand the generativity of other analogies.  Expecting the student might
draw a series of analogies, finding her way from that first idea to another and to another,
a teacher would not be so concerned that the first analogy be correct.  If we think of
analogies in target-base pairs, that first mapping would be critical.  But if we think of
analogies as activation of different sets of resources, that first analogy could lead to a
second.   In fact, few students knew how a prism worked ? the deep structure of the
analogy they were proposing, or details about atomic emission.  Discussions of how these
things worked were figured out as a class at times, with a few brief (five minute) lectures
from me.  These explanations and discussions evolved into new models based on new
analogies: an ant/giraffe model of light moving through media that explained why red is
less impeded than blue, a ?marching band? model of refraction to explain why light
bends, and a trombone/piano model to explain the difference between a ?chord? of colors
and a pure color.  From this arose new questions: is our blue sky an ?average? color or a
pure color?  Shouldn?t we see a green sky at times if our atmosphere is a prism?  If blue
is ?impeded? in the atmosphere and so gets to us later, how does that affect the colors that
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we see?  Representations of air, prisms and bubbles are all variable, so that they may
belong to similar categories (things that are colorless and yet have color) and distinct
categories.
Following negotiations of several candidate theories over a five-day period, the
class settled on a ?prism? model for creating blue skies.  They still had a two questions:
why we don?t see a green sky between the blue day and the red night, and by what
mechanism do long wavelengths bend less than short ones (we had an ?ant/giraffe?
analogy, but it seemed more mnemonic than a model).  But these questions aside, the
class was relatively confident in their model and asked for the ?right answer.?  I then read
them a passage from the New York Times? Science Tuesday section.  (Coincidentally,
this article on colors in nature was published the day before I had promised to tell them
?the answer.?)  The passage reads as follows (Angier, 2004):
Another type of structural color results from the incoherent scattering of light,
also known as Tyndall or Rayleigh scattering. The sky is the most renowned
example of such scattering at work. Sunlight and its complete spectrum of
radiation falls on the atmosphere, a diffuse wilderness of particles. Most of the
light rays are too wide of wale to be impeded on their journey earthward, but blue
light is so short-waved that it invariably meets a molecule it cannot ignore and is
scattered across the sky.
Following our five-day discussion on why the sky is blue, this passage (?the answer?)
was met with puzzlement.  The students were upset that it said nothing about green, or
why sunsets are red.  The phrase ?too wide of wale to be impeded? was written off as
utterly incomprehensible ? how can something be too wide to be impeded?  Length, not
width, was the determining factor in our discussions of blue skies.  In our discussion
following the reading, the students determined that the passage was vacuous.  In
particular, there was no reason for them to believe this model over any other model; the
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explanation did not address any of the questions that they still had; the explanation did
not express why other models fail; and it never explained that the scattered light had to
eventually reach your eye to be seen ? a hang-up that had troubled many students in our
discussions.  Rather, it seemed to suggest that if there is blue light scattered among the
atmosphere we will necessarily see that atmosphere as blue.
The blue-sky explanation from the New York Times still uses analogy, albeit
quite loosely, as a tool for conceptual understanding.  Describing a photon as ?wide? is a
metaphor: photons do not have much physical extent in the direction perpendicular to
their propagation.  But I am not primarily arguing against teaching via analogy because
the analogy that was provided by the article was provided was not one that the students
could make sense of; rather, by having the students build their own models by drawing
from analogies to systems with which they were familiar, and then exploring why they hit
on these models and negotiating the implications of these models, created a far richer
understanding of the material, developed in them a sense of what it is like to ?do?
science, and allowed the students to quite realistically model the interactions and
discussions one would find in a research group.
1
Finding analogies in the classroom
What kinds of discussion topics foster analogy generation?  As noted earlier in the
thesis, they are negative assertions ? such as a cup of water that doesn?t spill, colorless air
that looks a particular color, or physics problems that cannot be solved from first
principles.
                                                  
1
 There is not an explanation that is detailed and coherent enough about why the sky is
blue that will enable students to understand why other models fail.  Nor can a detailed
explanation help students learn how to create theories and models on their own.
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Lakoff (1987) notes that ?but? can often be used to identify cognitive models.
For example, in the phrase ?She?s a mother, but she works,? the ?but? acknowledges that
there is something about ?motherhood? that is (culturally, at least) at odds with having a
paying job.  Similarly, all of the topics above have that ?but? quality to them (I wrote
them in a way to highlight this aspect, of course).  In some way these scenarios do not
have the expected outcome, meaning that the topic is in some way at odds with an
idealized cognitive model.  And so, in order to make sense of the scenario in a scientific
way (meaning coherent with known phenomena and experience), the topic must be
situated in a different cognitive model.  And, since cognitive models give rise to
categories, to situate the topic in a new and appropriate cognitive model this topic must
be granted membership into a new category.  The category has a prototype and this
prototype is selected via an analogy to demonstrate the new category and its associated
cognitive model.
Many, but not all, of the topics above were chosen for their ability to be reasoned
about in terms of ?tangible? ideas, which is a part of analogy
2
.  Another element of
analogy is the change of categorization.  Science, and physics in particular, is rife with
paradoxes
3
, and these paradoxes can be resolved within science.  One tactic I have in
teaching is what I refer to as the ?what?s weird.?  When a student performs an experiment
or makes an observation and thinks, ?that?s weird,? I tell them to stop right there and ask:
                                                  
2
 In my class, we were having a discussion of Kuhn?s Structure of Scientific Revolutions
and by modeling theory building we modeled paradigm shifts.  The focus was not on
analogies (though doubtless my interest in them fostered their construction).
3
 My graduate quantum mechanics professor, Dr. David Boulware, lectured one day on
the redundancy of the phrase ?apparent paradox.?  In physics, he claimed, all paradoxes
are only apparent and can be resolved.
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why is this at odds with what you would expect?  Identifying the story that you expected
to take place and recognizing the rationale for that story allows you to understand the
schema that you were applying and other examples of that schema.  Then the students are
asked to ?make sense of it.?  How can you tell this story in a way that the initially
unexpected outcome makes sense?  In doing this, analogies are frequently generated.
The National Science Education Standards
Another question I would like to raise and answer in this chapter is whether these
implications for instruction are consistent with the National Science Education Standards
(NRC, 1998).  In fact, the Standards make no explicit reference to encouraging students
to generate and critique their own analogies.  However, analogy generation and use is a
powerful tool for addressing many of the goals set forward in the National Science
Education Standards, in particular those involving the ?abilities necessary to do scientific
inquiry.?  These are:
? Identify questions and concepts that guide scientific investigations
? Design and conduct scientific investigations
? Use technology and mathematics to improve investigations and communications
? Formulate and revise scientific explanations and models using logic and evidence
? Recognize and analyze alternative explanations and models
? Communicate and defend a scientific argument
As I have argued, student-generated analogies address these goals: they are a key part of
how science happens; they often arise when experimental results clash with expectations
? a fruitful place to identify questions and guide investigation; they can be the basis for
conducting an investigation and constructing a model; the evaluation of analogy is deeply
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tied to logic and evidence; and analogies are routinely used by scientists in
communicating and defending their scientific arguments.
The case for diversity in science
A final claim that I would like to make is more an implication for science ? the
enterprise and industry ? than science education, though the two are, of course, related
and to change the enterprise and industry of science one must have educated scientists to
effect that change.  But I hasten to add that the causes for such a lack of diversity among
scientists are perhaps only related to education insofar as education is influenced by the
culture of science and the culture at large.  Therefore these implications should be not
only for instruction but an argument to the culture of science, as any lasting change must
come from the source.
As noted in the chapter addressing the history of science, our discoveries ? the
things we pay attention to, notice, expect and explain ? are not recognized objectively.
Rather, the theories we create with our science are analogies to the stories we tell with
our lives.  Though the technology and techniques necessary to identify cellular structure ?
membranes ? in the nervous system had existed for years, they were not discovered until
the surrounding culture developed a story of membrane-like countries ? dividing up the
landscape into distinct separate self-contained units.  Though electrical phenomena had
been studied for years, the nervous system itself could not be understood until the
telegraph was invented and gave us the causal story behind impulses in nerves.  To
follow an analogy presented in chapter 6, ?rabbit ears? in a drawing of a duck are
recognized only by people who have seen rabbits.  The conclusion we must reach, then, is
that people with different stories will notice different things, make different discoveries
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and have different theories about these discoveries.  Their analogies will be different
because their stories are different.  And surely a diversity of discoveries and theories are a
benefit to the enterprise of science.  The more that the culture of science, and the culture
at large, allow for a diversity of scientists, engage with and allow a diversity of
interpretations and theories, the far richer our science will become.
Conclusion
?The essence of creativity the essence of creativity is to be able to view a situation
or an object from two different frames of reference, or two ?unrelated matrices of
thought? (Koestler, 1964).?  And analogy, as this thesis has argued, is an assertion of
categorization that places the target in a new or unexpected category.  Instruction that
fosters analogy generation, then, is instruction that fosters creativity.  Furthermore,
analogy generation is a crucial part of what it means to do science.  This cannot be taught
by the careful selection and presentation of well-vetted analogies, but instead by
encouraging students to generate and negotiate their own.  Classrooms that focus on
student reasoning and incorporate the creation of analogy rather than teaching by analogy
encourage students to be creative and scientific.  In doing so, many of the benchmarks in
the National Science Education Standards will be addressed.  Furthermore, an
understanding and appreciation for the role of analogies in science, and the basis of these
analogies in the schemas that we bring to our lab benches, can only call for a greater
diversity ? cultural and gender ? in our scientific community.
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Chapter 8:  Summary & Directions for Future Research
Introduction
You fight your superficiality, your shallowness, so as to try to come at people
without unreal expectations, without an overload of bias or hope or arrogance, as
untanklike as you can be, sans cannon and machine guns and steel plating half a
foot think; you come at them unmenacingly on your own ten toes instead of
tearing up the turf with your caterpillar treads, take them on with an open mind, as
equals, man to man as we used to say, and yet you never fail to get them wrong?
The fact remains that getting people right is not what living is all about anyway.
It?s getting them wrong that is living, getting them wrong, and wrong and wrong
and then, on careful reconsideration, getting them wrong again.  That?s how we
know we?re alive: we?re wrong.
-Philip Roth, American Pastoral
When I began my study on analogies, I was interested in a study on negative
analogies: that is, the places where our analogies are wrong, when the target and the base
of our analogies don?t quite match, where these mismatches occur, and what that tells us.
I was asking this question having recently sat in on a course in literary theory ? a field
where scholars have argued that we define our terms and our stories negatively: by what
they are not instead of by what they are ? and noticing parallels between this idea, based
in literature, and my experience in the lab and as a physics teacher.  However, the lack of
information on student-generated analogies shifted my focus from negative analogies to
analogies in general, establishing what analogies are and what kinds of claims they make.
Following a summary below, I begin the consideration of directions for future research
with a discussion on the literature of differ?nce and the relationship between this idea and
the idea of a negative analogy.  I then present a consistency between these ideas of
difference and negative analogy and the framework behind student-generated analogies
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presented in this dissertation.  Finally, I conclude with suggestions of a direction for
future research ? first with respect to negative analogies and then analogies, categories
and science education.
Summary
In this dissertation, I began with the argument that student-generated analogies
phenomenologically have the properties that have been recognized in categorization
research: multiple members, family resemblance, constructed bases with multiple
possible representations, and far transfer analogies.  Furthermore, these properties are
neither predicted nor explained by models of analogy that focus on recall of a base and
then an extrapolation of a structure or schema from that base.  When considering an
ontology of mind that allows for this phenomenology, I have argued that a schema-
theoretical model of mind accounts for these properties: it attributes schemas (p-prims,
scripts, and cognitive models) as the building blocks of mind that become activated and
then select and construct concretizations.  It has been argued (i.e., Lakoff 1987) that our
categories are defined within and derivative of our schemas, and I argue that the base of
an analogy is such a category ? a representation, constructed in the moment, of a set of
schemas.  The assertion of an analogy is an assertion that the target of the analogy
belongs to the category represented by the base, as opposed to simply a match between
the target and the base.  Under this ontology, it becomes possible to sketch a definition of
analogy that differs quite fundamentally from similarity ? a definition lacking from other
models of analogy.  Analogy in this definition becomes a deliberate cognitive move that
acknowledges the presence and possibility of two distinct cognitive models and
privileges one over the other.
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In considering the directions for future research, I would like to begin by
considering this final point above: analogy as a negative assertion and a move from
something ? the cup of water is not like other cups of water, a beanbag dropping is not
like throwing a rock from a bus, motion of electrons in a dense media is not difficult like
the motion of a person through a dense crowd, and solving angular momentum problems
in physics is not something to be done from first principles.  This idea of defining and
understanding concepts by what they are not echoes claims from literary analysis, which
in turn raises questions for future research.
Negative analogies
Literary analysis and differ?nce
At the beginning of the 20
th
 century literary analysts were dedicated to identifying
similarities between texts ? this script that defined a story.  Their work focused on
determining the story on which all stories were based ? thereby reducing all particular
stories to the general.  Jung (1969) explored the relationships between all myths and
rituals.  Psychologists (among them Fromm, 1957) developed a theory of fairytale in
which all fairytales are variations on a single fairytale theme.  But in the 1950?s Roland
Barthes began to stress the importance of recognizing how things are different ? that the
theme is not the meaning, but the variation on that theme. Barthes summarizes the task of
analysts of narrative in reducing all stories to one model as a:
task as exhausting as it is ultimately undesirable, for the text thereby loses its
difference.  This difference is not, obviously, some complete irreducible quality
(according to a mythic view of literary creation), it is not what designates the
individuality of each text, what names, signs, finishes off each work with a
flourish; on the contrary, it is a difference which does not stop and which is
articulated upon the infinity of texts, of languages, of systems: a difference of
which each text is the return.  A choice must then be made: either to place all texts
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in a demonstrative oscillation, equalizing them under the scrutiny of an in-
different science, forcing them to rejoin, inductively, the Copy from which we
will then make them derive; or else to restore each text, not to its individuality,
but to its function, making it cohere, even before we talk about it, by the infinite
paradigm of difference, subjecting it from the outset to a basic typology, to an
evaluation?. (Barthes/Miller, 1991)
Barthes theory stems from a tradition in semantic theory developed some 40 years earlier.
In the 1910?s Ferdinand de Saussure developed a theory of ?differ?nce? whereby words
(or rather ?signs? ? the sum total of information from gesture, word, tone, etc.) get their
meaning not from a simple one-to-one relationship with the external world, but ?one unit
has value within the system because it is not some other unit within the system? (Tyson,
1999).  Concepts are defined in opposition to other concepts.
Laura Otis, whose work was reviewed when discussing the history of science,
comments on noticing the following parallels between Saussure and science, retelling her
experience in a lab that studied visual perception:
I was a biochemist, a mere visitor to the lab, but I learned an important lesson that
night.  The eye, and the regions of the brain that interpret visual information,
respond only to changes, to borders between light and dark.  There are cells that
fire only when a bar of light moves horizontally, and cells that fire only when it
moves vertically? there are no cells that respond to a uniformly illuminated
screen, with no movement, no edges, and no borders?
Fresh from the lab, I learned the same lesson in Jonathan Culler?s
introductory course on literary theory.  Explaining Saussure?s idea of how words
were paired with objects, Culler proposed that we define concepts not based on
what they are, but on what they are not.  When defining something, we typically
compare it to something similar and then, like the eye, focus on the way it differs
from the concepts most closely related to it.  A cow, for instance, has four legs
like a horse, but it is fatter.  There is no natural match between a word and the
thing it represents; no positive assertion of a thing?s identity, just as there will be
no firing in response to a blank screen, even when it is brightly lit.  Like our
visual system, we create meaning only through the differences we perceive and
the boundaries we believe are present.  (Otis, 1999 p. 1?2)
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Otis, a professor of comparative literature, presents this story at the beginning of her
work on membranes to note two things: one, she is interested in boundaries ? this book
explores the concept of membranes in political and scientific climates, and, two, that the
boundary ?between the humanities and the natural sciences as a another boundary
arbitrarily drawn.?  (Otis, p. 3.)  I present it here to revisit the idea of the negative
assertions that are implicit to analogy.
Negative analogies
I have argued that analogies involve more than just similarity, but also
dissimilarity ? they dislodge the target from its expected categorization and schema and
place it in another one.  The claims of Barthes and Saussure, and reiterated by Otis,
suggest that perhaps the most significant function of the analogy is the ?dislodging,?
negative part of analogies and the contrast the analogy highlights: what is crucial is not
the claim that a is like b but instead, implicit in this claim, that a is not like c.  However,
the idea of differ?nce also suggests that there is something negative inherent in the ?a is
like b? claim ? for Barthes claims that ?[reducing all stories to one model is a] task as
exhausting as it is ultimately undesirable, for the text thereby loses its difference.?  It is
this idea that I suggest as a significant area for future research.
When drawing analogies between two cases, the significance comes not only from
the similarities between the two compared cases and the difference between the target
case?s expected and asserted schemas, but also the differences between the two compared
cases.  The value assigned by the analogy also comes from the difference between the
items that are asserted to be members of the same category.  It illustrates what is essential
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to the category as well as identifying what is unique about the phenomena.  Furthermore,
as Barthes (1991) notes, this difference is
not what designates the individuality of each text, what names, signs, finishes off
each work with a flourish; on the contrary, it is a difference which does not stop
and which is articulated upon the infinity of texts, of languages, of systems: a
difference of which each text is the return.
By paying attention to how our target differs from the base, and doing so in a dynamic
way, we set up a dynamic science ? one that evolves, ?making it cohere, even before we
talk about it, by the infinite paradigm of difference, subjecting it from the outset to a
basic typology, to an evaluation?? (Barthes/Miller, 1991).
Without negative analogies we?re just doing the same thing over and over ? we
have a set of solved problems and the business of science is to recognize how new
phenomena fit within these solved problems, but taking into account the negative
analogies, we?re changing ? adapting our categories, extending and limiting them.  Below
I present a brief sketch of research and questions concerning negative analogies ? that is,
the places where analogies are not correct but the analogy overall is still accepted and the
base is understood within this new cognitive model.  These analogies shift our
understanding not only of the target ? because it is placed into a new, unexpected schema
? but also the base of the analogy, because elements that seemed inherent to that base and
fundamental to the schema become tangential.
At the time that I first came across these readings by Barthes and Saussure, I had
just left a condensed matter laboratory, where I spent most of my time writing computer
programs to interpret data on the structures of sandpiles (Atkins, et al. 2001).  Much of
programming ? at least in the sciences ? involves taking a piece of code that is in some
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fundamental way the code that you need (usually from Numerical Recipes in C), but not
quite, and modifying it.  The original programs often bear very little resemblance to the
final program ? but the core of the idea remains.  How do programmers choose a
particular structure for their programs?  How do they analyze the strengths of these
particular lines of code, know what to modify and what to keep?
Rachel Scherr, as part of her doctoral work on relativity, created a set of tutorials
that leads students to arrive at the relativity of simultaneity.  The context of the tutorial is
two relativisitcally related reference frames, each of which is observing a single tape
player in a scenario that is contrived to give students the following options: either the
tape player is seen by one observer to play and another observer to remain off, or
simultaneity is relative.  As Scherr  (conversation) claims: ?If you have to choose, I?d
much rather give up simultaneity than allow the tape player to both play and not play.?  It
is this determining of what is essential that lies at the heart of much of scientific
discovery ? do we ?give up? simultaneity in order to preserve causality?  Which is more
fundamental to our science if we can?t have both?  Do we allow popes to be bachelors or
do we refine our definition of bachelor?  Which is the more primary?  Are quasi-crystals
crystals?  Are viruses organisms?
Negative analogies in physics
In ?Analogy as the Central Motor of Discovery in Physics,? a talk given by
Hofstadter at the Ohio State University (2003), Hofstadter argues that analogies are a
driving mechanism for discovery in science.  He details the progression of ideas from a
literal field with hills and valleys, to assigning the gravitational potential to every point in
space ? a scalar field.  And from there scientists tried to draw the analogy from the
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gravitational field to the electric field, and eventually ran into a problem with magnetism:
what could they ?give up? or ?tweak? about the scalar potential and still maintain its
more important underlying structure (as a thing you differentiate to find the force)? ? the
answer was to recast the scalar field as a vector field.  Much of science consists of trying
to extend a known law into a new area where it was not meant to be applied ? drawing
analogies between pieces and ?tweaking? a few things.  These tweaks ? that the electric
field is like the gravitational field but different, or that in relativistic scenarios we must
abandon simultaneity ? are the differ?nce introduced by Saussure and expanded upon by
Barthes.  They argue that in drawing an analogy between the two fields, the relevance is
not in the ways in which the two are similar ? that overarching schema in which both are
understood ? but the places where they are different.  This idea has been termed the
?negative analogy? in the philosophy of science.
Negative analogies as a caveat
The negative analogy ? the pieces of the base that do not transfer to the target ?
was first introduced by Hesse in 1966, but has not been widely recognized since.  One
reference can be found in ?The Metaphorical Transfer of Models? by P.B. Sloep (1997).
In this article, Darwin?s natural selection is seen as an analogy to the selective breeding
of gardeners, and the negative analogies are as crucial as the positive analogies:
?natural selection and artificial selection differ too.  It is not for nothing that we
said in the above ?it looks as if someone selected them.?  While the proto-fantails
were hand-picked by their breeders, in natural populations selection results from
natural processes such as the struggle for existence.  Here we have a difference or
a negative analogy, as Hesse would call it: a human selecting agent versus natural
processes.  Negative analogies are as essential an ingredient of metaphorical
transfer as positive analogies.  A metaphor without them would cease to be a
metaphor ? i.e. an explicitly non-literal referring term ? it would be the ?real
thing? ? i.e. a literal referring term ? and we would end up with an identity
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relation rather than one of analogy.  At the same time, negative analogies embody
the limitations of a metaphor? if negative analogies are not recognized for what
they are, mistaken inferences loom large.
It is true that negative analogies are a crucial element of science ? but not solely because
?if not recognized for what they are, mistaken inferences loom large.?  I don?t want to
place a caveat against negative analogies ? there is the danger in reading this passage and
believing negative analogies to be no more than pitfalls, limitations and places for error.
This interpretation of negative analogies is widespread (i.e., Gentner and Gentner, 1983;
Clement, 1987; and Lulis, Evens and Michael, 2004).  David Brookes (2003, 2004) has
argued that scientists are inconsistent with their approaches to problems: speaking of
quantum mechanics in the Bohmian sense one moment and then the Schrodinger model
the next.  We talk about heat as though it is a fluid, but treat it mathematically as a
process.  This is cited as a concern, but I believe that many and conflicting
representations of phenomena may be a strength if appreciated and understood for what
they are.
Meaning arises in these negative analogies ? the places where the representations
are not accurate (can lead to incorrect predictions) and where they conflict ? and this may
be a strength of analogies.  Meaning comes not only from the patterns and similarities
between things but also the differences.  It is in the tweaks ? the fact that the magnetic
field is a vector, that heat seems like a fluid and isn?t, that we think of quantum
mechanics in a Bohmian sense but also don?t ? that we find the science.
Directions for future research regarding analogies and mind
The changing schema
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A direction for future research is to investigate what happens to the schemas
through the use of analogy?  How does the analogy change the understanding not only of
the target, but also of the schema for which the base is a representation?  For instances in
which the schema is ?tweaked? in constructing analogies, how do we determine when it
is a ?variation on a theme? and when it is a new theme altogether?  Is this a continuum or
is there a clean delineation between including the target in an established and stable
schema and changing that schema slightly to accommodate the target?
Conceptual blending
A possible extension of these questions ? concerning how the negative analogies
between the target and the base affect our understanding of the schema that applies ?
relates to work on conceptual blending.  Fauconnier and Turner (1994) note the following
regarding conceptual blending (also known as conceptual integration):
? Mental spaces are small conceptual packets constructed as we think and
talk, for purposes of local understanding and action.  They are
interconnected, and can be modified as thought and discourse unfold.
? In blending, structure from two input spaces is projected to a third space
(the ?blend?).
? The blend inherits partial structure from the input spaces, and has
emergent structure of its own.
These ideas are related to and could possibly provide a partial answer to the questions
raised above regarding how our schemas change to accommodate new phenomena.
Constructed bases (as opposed to recalled ones) may arise from the piecing together of
schemas that are not typically associated with one another.
The implications of technology on science
Another interesting question is to explore the connection between technology and
theory.  One interesting implication from Otis? research is that not only does our science
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create new technology, but technology creates new science: not only because of the
technological affordances, but the introduction of new schemas and new language.
Scientists had no accurate way of envisioning the nervous system prior to the invention of
the telegraph ? they borrowed language from hydraulics but it was of limited use.  There
was no way of understanding the olfactory system prior to the scanning tunneling
microscope ? scientists spoke in terms of enzymes because that was the paradigm in other
biological processes.  Is this a rare phenomenon, or can most of our scientific theories
and conceptual revolutions be, at least in part, attributed to changes in schemas that were
brought about by new technologies?
Science in the absence of analogy
Related to this question are questions concerning systems for which there are no
?good? analogies.  In particular, I am interested in the way that quantum mechanics is
taught and the way in which it is conceptualized.  There are few, if any, systems that are
analogous to quantum mechanics: most seemingly analogous systems incorporate a
?hidden variables? component.  What are the analogies that people use to understand
quantum mechanics?  What schemas do people tap into to understand and come to grips
with this strange phenomenon?  Above I suggested that quantum mechanics is perhaps
not an analogy to physical phenomena but to mathematical ones.  Although I later began
to conceptualize probability amplitudes by comparing them to electromagnetic waves, it
was initially an understanding of linear algebra that first informed and explained quantum
mechanics.  (My undergraduate professor for mathematical methods for physicists
claimed that all of physics is linear algebra.)  Of course, analogies from physical to
mathematical systems beg the question as to whether or not mathematics is understood
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via analogies to the physical world.  Lakoff and Nunez, in the book Where Mathematics
Comes From, argue that math, far from being an abstract and objective field, is intimately
tied to analogies to the physical world:
When you think about it, it seems obvious: The only mathematical ideas that human
beings can have are ideas that the human brain allows.  We know a lot about what
human ideas are like from research in Cognitive Science.  Most ideas are
unconscious, and that is no less true of mathematical ideas.  Abstract ideas, for the
most part, arise via conceptual metaphor-a mechanism for projecting embodied
(that is, sensory-motor) reasoning to abstract reasoning.
This book argues that conceptual metaphor plays a central, defining role in
mathematical ideas within the cognitive unconscious ? from arithmetic and algebra
to sets and logic to infinity in all of its forms.  (Lakoff and Nunez, 2001 preface)
Embodied cognition and analogies in science
In the above passage is a final question for generated analogies in science ? that of
embodied cognition.  The idea of embodied cognition is that ?human ideas are, to a large
extent, grounded in sensory-motor experience.  Abstract human ideas make use of
precisely formulatable cognitive mechanisms such as conceptual metaphors that import
modes of reasoning from sensory-motor experience.?  (Lakoff and Nunez, 2001, xxi).
That is, when you ask someone what a triangle is, the definition they immediately turn to
is not mathematical formalism, but a process by which you draw that triangle ? often
involving gestures.
For abstract concepts like ?what is truth??  Barsalou and Wiemer-Hastings (2004)
have shown that definitions often involve scenarios: ?It?s when you??.  The relationship
between analogies, categories, gesture and embodied cognition is interesting.  The
schemas that I have argued as fundamental are abstractions from concrete, embodied
experience and analogies between concrete experiences, I have argued, are mediated by
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this abstract schema.  Perhaps at some deeper level, these schemas are represented as
sensory-motor patterns, which are in turn associated with objects existing in the world,
and the choice of base for our analogies may be related to this sensory-motor experience.
This relationship has not been explored in this thesis but is a logical extension of this
work.
Analogies as a tool for exploring categorization
Having established that analogies are assertions of categorization, this research
can inform many of the open questions in categorization.  Current questions involve the
structure of categories and distinctions between taxonomic categories and other forms of
categories.  The vast majority of investigations of the mind?s organization start with a
prompt for the participants of the study, such as asking for a list of trees, ?things to take
from a house during a fire,? or even to guess at a professor?s judgment of the typicality of
a certain type of bird.  Such studies miss the everyday acts of categorization that happen
without awareness or effort.  Students in a class discussing why this sky is blue came up
with many different analogies: the sky is like a bubble, a prism, or a neon sign.  But when
these students were informed by the instructor that their analogies all were ways of
getting color from a colorless thing, they expressed surprise.  And had they been given
the prompt, ?what are ways to get color from colorless things,? it is not clear that the
analogies ? or members of this category ? would have been identical to the analogies they
constructed without such a prompt.  This suggests an alternative and less ?invasive? or
contrived manner of arriving at the organization of concepts by studying spontaneously-
generated analogies.
Network theory and analogy
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A final direction for this research is to use analogies as a means of understanding
the cognitive map as a conceptual network.  Lexical maps have been constructed
(Fellbaum, 2003 and others), that relate our semantic terms in a network ? thesaurus-like
or internet-like in its construction.  In these, words are nodes and linked to one another,
and the activation of a node activates those that are linked to it.  A similar construction
has been considered for the linking and activtion of concepts (Collins and Loftus, 1975) ?
but this would beg the question of what is a node?  As Hofstadter notes (2001) a concept
can be quite large and particular, as in ?a strange shape that the electrical charge may take
that cannot be then be solved from first principles.?  And yet our large concepts seem to
be pieced together from smaller schemas.  How are these concepts and schemas
organized in the mind?  What size pieces are fundamental ? or is that question even
meaningful?
 Below I turn to questions for the implications of this research on instruction.
Directions for future research on the implications for instruction
While I make claims about what analogies assert and the cognitive mechanisms
involved, I have not come to any conclusions about how this happens.  How does
Miranda make this incredible leap from seeing a cup of water to thinking of a cat in a
basket ? why were these schemas activated for her?  Of course the experience with the
schemas involved was necessary, just as Luca Turin had to know about scanning electron
microscopes before he could draw an analogy between this and scent, but what habits of
mind and what structure of education can encourage this kind of creative re-
categorization of concepts?
Questions regarding student epistemology
212
As a first pass at a partial answer to this question, I would like to suggest that
students must know that they should generate analogies as part of what it means to do
science.  That is, it requires a change of students? epistemological stance towards science.
This is an argument that was discussed in the chapter regarding implications for
instruction.  But it leaves open the question as to how to change student epistemologies.  I
have found ? anecdotally ? that telling students to use analogies often results in
superficial analogies: temperature equilibrating is analogized to a chameleon changing
color, or an electrical circuit is imagined to be like a cow?s digestive tract (both analogies
have occurred in Maryland physics classrooms).  These students, when told to draw an
analogy, are not choosing a story that makes sense to them could apply here.  I don?t
quite know what they?re doing with the analogies they construct ? and they don?t quite
know why I am asking them to construct.  In trying to understand these moments, I?ve
wondered if these strange analogies can be understood in the context of a teaching
philosophy from The Inner Game of Tennis (Gallwey, 1997).  This sports psychology
book argues that too explicit of instruction can (in the context of tennis) lead to unnatural
and unfavorable results; telling someone where the ball should hit the racket, say,
decontextualizes what should be one piece of an integrated whole.  So perhaps an explicit
focus on analogies prevents the natural evolution of an analogy ? as response to an
unexpected result and stemming from activated schemas.  This is just speculation at this
point.  But it suggests a possible starting point for how we can design curriculum and
learning spaces in a way that encourages analogy generation.
The design of learning environments
213
An additional, related question is whether the use of analogy (or lack thereof) a
question of expectations, epistemology, or is it a domain-general ability?  That is, in
designing our learning environments to encourage analogy, should we be addressing
students? ideas about what kind of knowledge they should bring to bear?  Should we try
to encourage the activation of multiple, contradictory schemas?  Or is it a general facility
with or predisposition towards analogies ? does a facility with analogies in, say,
literature, translate to an increased use of analogy in science?
Concluding thoughts
Before my work on analogies, I was involved in curriculum development
surrounding wave phenomena.  In watching students interact with this and similar
curricula I was dismayed that, though they eventually understood the concepts that the
curriculum was addressing, the students never approached new topics in a creative,
scientific, sophisticated way.  They were better engineers in the end ? they could apply
algorithms that the curriculum had carefully developed, but they weren?t better scientists.
It seemed they were at a loss when learning new topics and constructing novel ideas.  A
student came to class one day wearing a t-shirt from a punk rock band.  It read: You can
lead a man to reason, but you can?t make him think.  I wanted my work to address this.
To begin to think of the classroom as more than a place to lead our students to reason, but
as a place where deep scientific thinking occurs.  I think this thesis is a start.  It highlights
one significant component of scientific thinking ? analogy.  I claim that analogy is the
ability to consider alternative models, deliberately overriding cognitive knee-jerk
reactions to phenomena by tapping into alternative models and representing the
categories that these models construct.
214
In Drawing on the Right Side of the Brain (Edwards, 1989), readers are instructed
to turn a Picasso sketch upside down and then try to reproduce the upside-down drawing.
Results are phenomenal ? even people who have trouble drawing stick figures are often
create remarkably accurate reproductions.  We cannot help but see a nose when a right-
side-up nose is drawn before us, but by turning it over we can begin to see the lines and
curves as lines and curves and override our assumptions about how noses are shaped or
where eyes go.  Analogies allow us to do a similar thing ? they demand that we turn the
pictures upside down and dislodge the cognitive models that we were applying.
Analogies allow us to stop seeing the nose as a nose ? we stop seeing the cup as a cup
and instead pay attention to the way in which it as a basket, how metal is a set of
stepping-stones and quantum mechanics problems are like Bugs Bunny?s ears.
215
Appendix A
Transcript 1
This transcript is from an undergraduate course in physics, Physics 115: Inquiry
into Physical Science.  The students have been investigating electrical phenomena using
Styrofoam (an insulator that charges easily when rubbed with wool) and a metal pie plate
(a conductor).  They began their study of charges using transparent ?scotch? tape: when
two pieces of tape are put one on top of the other and peeled from a table top, they each
get an excess of different types of electrical charge (positive and negative) which the
students have termed ?top? and ?bottom.?  Prior to the discussion below, the students
have been asked why Styrofoam and metal have such different electrical properties and
have worked in small groups to address this question.  They are now presenting their
ideas to the rest of the class:
Hana: I kind of see the charge in metal as like, fish in a fish1
bowl?  Like they never really stop moving, they?re2
always kind of floating around wherever they kind of feel3
like going and that?s just how I see it in my head, like4
them always moving around.  And I don?t know what5
hap- I don?t know how to describe it really I don?t really6
know what happens once another charge is brought closer7
then.8
Instructor:  Does this make sense to you then?9
Hana: Yeah.10
Instructor:  So this is ? so this is like two kinds of fish. [Hana: Yeah.]1
And in metal they can move around.  They?re kind of12
stuck inside the bowl, but within the bowl they can move13
around.14
Hana: But I also think that they can leave the bowl at some15
point because?16
Instructor: Well we get shocked right?  I mean that?s?okay.    I?m a17
charge I?m in the pie plate, what am I doin?? [Hana:18
Movin?.]  Movin? I?m movin?.19
Kelli: That same idea I was thinking except more like ping pong20
balls that bounce all around and that?s why if there?s top21
216
and bottom charges they?re moving around a lot and2
they?re kind of attracting and repelling and attracting and23
repelling each other the tops and bottoms that go all over24
the place?but once the extra bottom charge is added it?s25
almost trying to like reneutralize itself and the tops are26
attracting to the extra bottoms.  And then they?re trying to27
kick out the other extra bottoms so they can get back into28
their whole little [Student: Balance.] balance.  Bouncing29
around.30
Instructor:  So this- I think what you?re offering is an explanation of31
why I get a shock.  Is that?am I wrong?32
Kelli:  No, you?re not.3
Instructor:  So you?re, you?re thinking that if there?s extra bottom34
charges in there it?s like they want to get out because it?s35
unbalanced.  And it has to do with them just kind of all36
bouncing around like ping-pong balls if you?ve watched37
the lottery drawing.  Alright.  Okay.  Terianna?38
Terianna:  So- are you saying that they?so she?s saying?are you39
saying that there?s a little [inaudible] that charge moves40
throughout the pie plate?41
Kelli:  I think I think so.42
Instructor:  Okay?so are you agreeing with that picture that I drew43
down there?okay so it really is going with always4
moving.45
Christie:  We were thinking that?like they were saying that in46
metal it?s always moving, so if it?s always moving it has47
more room to move and that would mean to say that the48
molecules are less tightly packed together or less dense49
and we were thinking of Styrofoam as more dense than?50
I?m just trying to figure out first if that?s right and how it51
relates.52
Lea:  I don?t agree with you saying that the Styrofoam is more53
dense because, so that?s why the charges get caught up in54
it because like?[inaudible] pan is more dense so they?re5
able to slide across it like they can ice skate across the56
[inaudible] here.  So that?s why they move around more57
?cause it?s more dense so they can slide across it more58
and the Styrofoam is less dense and so they get stuck in59
it.  Like so they can?t move as much.60
Instructor: Lea I want to add?I think you?re sort of what I when I61
hear you talk I?m thinking of like, pouring water into a62
sponge versus pouring water onto a hard surface. [Lea:63
Yeah.]  Like this sponge is actually less dense and there?s64
room for it to absorb the water and the you know if you65
pour it onto something hard there?s no room for it to6
217
absorb.  But Christie?I mean this is an interesting thing67
you guys are both thinking that density is important but68
one of you is thinking that more density means one thing69
and one of you is thinking more density means the other70
thing.  Is that is that?am I right?  [Christie: Yeah.]71
Lea:Yeah, I have a question about the sponge- like the charge72
instead of being able to move freely past it and they get73
kind of stuck in it like water and the pan is like the74
countertop and you pour water on it it?s going to slide all75
around and stuff but the charges can move more freely on76
the pan there?s probably? I think that maybe this has7
more charge in it because there?s more places for the78
charge to leave than the pan does, but there?s more79
potential charge in this than the pie pan.  [Unknown: Like80
it can hold more charge.] Yeah it can hold more charge81
but they can?t move around as freely.82
Student:So you?re saying the charge is like [inaudible] out on the83
metal?  Like on the outside?84
Lea: Yes.85
Student:It?s like made up of it?like, they?re electrons.86
Lea:  Yeah?like I don?t know but it?s like definitely a lot87
smoother.  They?re, they?re denser they can move around8
more freely like.89
Instructor:Hana?90
Hana:  As, as far as like air is concerned?air moves from high91
pressure areas to low pressure areas and so like I don?t92
know I don?t know if that?s like a completely separate93
idea?94
Instructor: But you?re offering this as a kind of ?maybe this explains95
why there?s charge pressure?there?s high charge96
pressure and low charge pressure.?97
Hana: Kind of but I know that I know that high?like when98
you?re in the shower and the shower doors close and it9
gets all steamy as soon you open the door you feel like10
the cold air feels like it?s rushing in?  Because inside the101
shower it?s low pressure and then once the door opens all102
that high pressure kind of like rolls right into the into the103
low pressure area.104
Instructor:  I just want to- I know- there are people here- I just want105
to clarify Lea and Hana your question your question was106
is charge moving on the surface as opposed to moving107
inside and so this would be like are the fish swimming in108
the middle of the fishbowl or are they somehow sort of109
stuck on the edge of the fishbowl.  Is that sort of what10
you?re ? [yeah I disagree with her idea] okay.11
218
Lea:  I really don?t know I was just trying to?12
Lydia: I was going to say I think the pie plate is more dense but I13
do think that it?s inside not outside because if there?s14
more space to travel then the molecules can?t get from15
one space to another easily but it?s all [inaudible].16
Instructor:  Oh so it?s like stepping stones [Lydia: Kind of.] like in17
the Styrofoam it?s really far to the next stepping stone so18
it?s like can?t get there I?m stuck here. [Lydia: Right] but19
in the metal the stones are really close together so I can120
kind of walk across.  [Lydia: Yeah.]121
219
Appendix B
Transcript 2
The following analogy is from a 5
th
 grade classroom in a rural Maryland public
school.  In this transcript, the students have been visited by the science resource teacher
and posed the following question (NASA, 1999): a cup full of water is inverted on a
cookie tray and the tray is rapidly pulled out from underneath the cup (see Fig. 2).  What
happens to the cup-water system?  The students will later observe that the water does not
Fig. 2: The Experiment: A tray pulled out from under a cup
leave the cup as it falls to the ground?the cup falls at the same rate as the water and the
water will only spill out once it reaches the ground.
Teacher: So you?re predicting as I slide the cup off it?s also going1
to go all over the tray, too.2
Gabrielle: Like, it?ll spread and then it will fall.3
Teacher: It?s- the water is going to spread and then it?s going to4
fall down.  Okay now I might come back to some of these5
ideas that you?ve had but let?s see what some ? I see a lot6
of other hands up.  Um, Miranda?7
Miranda: I predict that when it falls off it?s going to stay in the cup8
until it gets down to the floor and then it?ll splash.9
Teacher: So you have a prediction that when I slide it off of the10
tray the water is going to stay in the cup.  Now that?s very1
different from what they?re saying.12
Miranda: ?Cause at home when I have like something in a basket13
and when I go like that real quick it stays in.  So when ?14
and when I pull it down like this like upside down on the15
220
way down it stays in until it gets to the bottom and then it16
comes out.17
Teacher: So you?re using now this example of something that18
you?ve done at home where you have an object in a19
bucket- or a basket- you said and what do you do? You?20
Miranda: I go like this and then I pull it down and it stays at the top21
until I stop and then it comes out.2
Teacher: So you swing this ? what?s in the basket?  What object is23
in the basket?24
Miranda: Sometimes I put like like a little toy cat that I?m playing25
?roller-coaster? with and put it in there and I pull it down26
[Teacher: Is that right?] and it stays in the back until I27
stop and then it comes out.28
Teacher: So you swing this basket like this?do you, do you do it29
quickly or slowly?  Or.30
Miranda: I do it quickly like that.31
Teacher: You do it like that and then pull it down [Miranda: Mm32
hmm.] and the cat stays in the basket [Miranda: Until I3
stop.] even when you have it upside down like this and34
pulling it down and when you stop35
Miranda: It comes out.36
Teacher: The cat comes out.  Okay now has anybody else- want,37
can relate to this also?  Looks like a lot of you can.  Let?s38
hear some of your ideas.  Let me come over to ? thank39
you Alyssa.40
Alyssa:  Um.  Um what she?s also talking about it?s the air- it?s41
like pushing the cat up against the the bottom of the42
basket which is holding it back from going out.43
Teacher: Okay, so now you?re- and this is where I was going to go,4
too, after what you said is, um, explain how the cat would45
stay in there and you?re thinking is, Alyssa, is that the46
air?47
Alyssa:  ?is pushing the cat towards the bottom.48
Teacher: Okay, so so the air is pushing the cat towards the bottom49
of the basket keeping it inside the basket.  When?now50
when I?m thinking of that, and maybe some of the others51
can help me with this, but I can picture the air when you52
pull it down being forced up against the cat?is that what53
you?re talking about?  Miranda?54
Miranda: And it?ll be the same thing with the water the air will5
push the water up until it falls down and then it will go56
everywhere.  Because when it comes down the air is57
pushing upwards and [it keeps/I keep?] the water in there-58
because I?ve also done that in the bathtub when you?ve59
got your cup, I?ll like I?ll fill it with water put my hand60
221
and drop it the water stays in until it hits the bathtub and61
then it goes everywhere.62
Teacher: So your thinking is then this air is keeping the water63
inside the cup.  Alyssa your thinking is that the air keeps64
the cat inside that basket that Miranda was describing.65
You?re shaking your head ?no? Gabrielle?6
Gabrielle: Yeah because like when you put like a toy cat in a basket,67
that?s only that one small thing but you now have a cup68
full of water, so how would the air keep it up if the cup is69
already full?70
Teacher: I?m I?m not sure I follow what you just described, you71
said the cat is just a small [Gabrielle: Toy.] toy so?72
Gabrielle: So and it?s only one thing.73
Teacher: Okay wait a minute?so the cat?s only one thing but the74
water is?75
Gabrielle: So you you?re going to have the cup full so how would76
the air like keep it up if the cup is full?7
Teacher: Oh- so her did you hear what she just said Miranda?78
[Miranda: Mm hmm.]  She?s saying that if I fill the cup-79
if I fill this cup- you, you?re saying if I fill the cup to the80
very top there?s no room for the air to keep the water in81
there.  Okay well, I I have a question for you then82
Gabrielle?what?s your prediction if I fill this cup all the83
way to the top, turn it over, slide the cup off what will the84
water do?  I filled it all the way to the top, turned it over.85
Gabrielle:  How far do you now have the cup up?86
Teacher: How far in the air?87
Gabrielle: No to the edge.8
Teacher: Oh to the edge?  I ? I was probably, I don?t know, as89
close to the edge as I can get it.90
Gabrielle: Well, I think it?s91
Teacher: And then I?ll do this.  Or?92
Gabrielle: Oh my prediction is that um, what Alexandra said that the93
cup slides off then it?ll just go down?94
Teacher: Okay so you?re still with your original prediction where95
you think the water is going to go everywhere.96
Gabrielle: Yeah but it?s?I think it, what it?s going to do, is the97
cup?ll fall off and some of the water will splash into the98
bucket.  I don?t think it?s going to go?9
Teacher: I?m sorry say that one more time it?s going to do what?10
It?s going to?101
Gabrielle:  I think when the cup slides off that the water will go102
down to the bucket and make a splash.103
Teacher:  So it?s not going to stay in the cup like Miranda?s104
predicting.105
222
Gabrielle: Right.106
Teacher:  Okay what about if I only fill the cup up halfway?  Now107
we?ve got room for this idea we?ve got room for the air.108
What do you predict will happen?  And I slide it off of109
there.  Maybe I?ll just push it off of there like that.10
[Teacher pushes empty cup off and it tumbles in the air11
on its way down.]12
Gabrielle:  Well, see, the cup like turned a little bit, it like went side13
a little bit so um.14
Teacher:  Well what do you predict the water will do if I only fill it15
half way?  Is it going to do the same thing as if I fill it all16
the way?17
Gabrielle:  Um? yes it?s going to do the same thing.18
Teacher:  It?s going to do the same thing.  [Gabrielle: Yes.]  So you19
don?t think air has anything to do with it then?  Keeping120
the water in the cup?121
Gabrielle:  No because like when you?re turning the basket then you12
kind of keep it straight when you?re holding the handle123
like that.  When you drop the cup the cup can just zoom124
all directions too.125
Teacher:  Oh I see what you?re saying?when you?re holding the126
handle on a on a basket or a bucket you?re kind of127
keeping?128
Gabrielle:  You?re keeping it straight.129
Teacher:  But your thinking is when it falls like this it could130
tumble?it?s going to move?there could be some131
movement in the cup.132
Gabrielle:  It?ll turn sideways.13
Teacher:  It could turn sideways like that.  And that would make a134
difference.  Okay let?s get ? a lot of you have been very135
patient.  Cody?136
Cody:  Um because when I was um having bucket full of water137
and I swing it around and then when I throw it the bucket138
of water still stays in there- the water, and139
Teacher: You?re going around throwing buckets of water?140
Cody: Yeah-they?re made out of plastic.141
Teacher:  When you when you swing that bucket of water around?142
Cody: Yeah and then when I throw it the bucket of water still143
stays until it hits something.14
Teacher:  So the water stays inside the bucket even when you when145
you?ve let it go the water stays inside of there.  Well what146
do you predict will happen when I do this with the cup of147
water?148
Cody: It will turn before it hits.149
Teacher: What will turn, the cup?150
223
Cody: It- it would, the water would go out a little bit and then it151
would splash.152
Teacher:  When I do this, what will the water do?153
Cody: Um the water ? if you pushed it hard enough the water154
would go out flat a little bit and then fall.15
Teacher: So if I push it hard enough?well that might not be a156
good example.  But if I push it hard enough you?re saying157
that water will stay there for a moment- it will stay flat is158
what you said- and then what would happen?159
Cody: Then it will fall down.160
Teacher: And then the water?s going to fall down.  Out of the cup?161
The water will fall out of the cup?  Or?162
Cody:  The water would splash out before it- after it gets out of163
the cup and then fall down.164
Teacher: So it?s going to it?s going to go like this, water?s going to165
stay flat for a split second?16
Cody: And then the cup and the water will fall down.167
Teacher:  Will the water stay in the cup as it?s falling?  The water?s168
going to fall out of the cup.  But you just said that when169
you swing the bucket, though, that the water stays in170
there.171
Cody: But when you push on the flat then um?   It?ll turn172
around?173
Teacher: So we?re still thinking about this pushing it off of here,174
though?that175
Cody:[Inaudible.]176
Teacher: So it forces the cup to to uh turn.  Isaac?17
Isaac:  Um I pre- I don?t- um I agree with Miranda but I don?t178
think air has anything to do with it.  Because um179
yesterday at Trick-at-Treat I had like a bunch of candy180
and I swung it around that?s like when I was bored and181
stuff?182
Teacher:In your bag?183
Isaac: Yeah.  And none of the candy came out.  I like kept on184
swinging it and also when me and Johnny play monopoly185
there?s like this little hat that we play with when we roll186
the dice [Teacher: Mmm hmm.] and like we always put187
the dice in and flip it back to each other with the dice in it18
and we always catch it and it stays and the dice stay in.189
Teacher: And you?re in agreement with Miranda that the water will190
stay in the cup however you don?t think air has anything191
to do with it.192
Isaac: No.193
224
Teacher: Have you given any thoughts as to why the water will194
stay in the cup or what what uh what will cause the water195
to stay in there?196
Isaac: I just predict that cause of experiments?197
[a few minutes of conversation regarding what happens198
before the cup falls]19
Teacher:?Alexandra?20
Alexandra: Um when Miranda said how when she dropped the cat in201
a um basket- I?ve done that with um my Easter candy but202
with more candy in it and when I turned it over when I203
got up here and it dropped it all went everywhere.204
Teacher: But when you were swinging it, it didn?t fall out until you205
got up here and then stopped and then it all fell out.206
[Alexandra nods.]207
Teacher: Isaac?208
Isaac: I don?t really agree with Dillon because the only reason209
it?ll fall out is, um, if the board is crooked and the cup is210
straight and also if the cup has a crack in it.21
Teacher: The cup is pretty good- I mean it doesn?t have any cracks212
in it, and although the cookie tray it looks a little? pretty213
straight.  It?s hard for you to tell I know.  Um.  Let me214
just summarize, really, for my own benefit and perhaps215
for yours as well.  And then we?ll come to you Gabrielle.216
Teacher:  We have a couple of different ideas.  At least three I217
think.  We have Dillon and his group who are thinking218
that ? or I?m sorry predicting ? that when I turn the cup219
over the water is going to go everywhere.  We have20
another group, I think it might be I would call the21
Amanda group who is predicting that water?s going to22
stay that it will stay in the cup even when I turn it over23
and the water will stay in the cup when I push it off of the24
tray.  Ok?  And then we have some of you- Alexandra25
who had originally predicted and Gabrielle that water is26
going to fall out of the cup when I push it off of the tray.27
Are there any other ideas, other than those three?28
Ethan: Are we going to be able to try it out.29
Teacher: We are going to be able to try it out.230
Student 1: Today?231
Teacher: Today.232
Student 2:  We always do it fifteen minutes before we?re done.23
Teacher: Well now I want to hear your talking and your thinking234
about this now.  Gabrielle, you had something to share go235
ahead.236
Gabrielle: I was going to say that I think the water will fall out once237
it leaves the board.238
225
Teacher: Soon as I do this the water?s going to.239
Gabrielle: Yeah because see when you put the board240
Teacher: The Cookie Tray?241
Gabrielle: Yeah.242
Teacher: Please use the correct scientific terms, ok, this is a243
?Cookie Tray.?  I?m just kidding.24
Gabrielle: When you put the cup on the tray and then when you go245
to slide it off some, some of it like that- not all of it246
comes off at once.  Some of it comes off a little bit so I247
think when, when a little bit is off the water will just fall248
down because there?s a crack.249
Teacher: Oh.  Let, let me repeat what I heard you say and then I?ll250
get to you Alexandra.  It gets to this point some of the251
water is going to come out.252
Gabrielle:  Yeah because, um, that part has a gap in it so I think the253
water?s going to fall out.254
Teacher:  Do you see what she?s talking about you have this little25
bit?256
226
Appendix C
Transcript 3
The first transcript below is from a research group meeting of the Physics
Education Research Group.  Paul, a graduate student, is interested in authentic classroom
activities and is discussing his definition of authentic.  Key to this definition is that
authenticity is a property not only of the activity but also but also in the way that the
students relate to that activity and the coherence of this to the scientific community of
practice (that is, do the students know what they?re doing?  Would scientists agree?).
This is at odds with definitions of ?authentic? activities that situate authenticity as a
property of the curriculum itself.  The transcript begins with David summarizing the
concerns with respect to prior definitions of authentic.
David:  It?s not a responsive definition of authentic.  Authentic is1
defined pre-experience. And so what your [Paul?s] sense2
is what?s going to be authentic is about watching the3
student and and what is authentic for this group?may be4
different from what?s authentic science for this group.5
And you don?t like defining authenticity in a way that6
isn?t responsive.  So the content isn?t responsive but also7
the sense of what is authentic isn?t responsive.8
Rachel:  So ontologically authenticity is like fun.  Which would9
be?10
David: Oh that?s great.1
Andy: Oh that is beautiful!12
[Laughter.]13
Rachel: ?because you can?14
David: There?s an analogy for you Leslie. [Leslie: Yeah.] Are15
you taping this? [Leslie: It?s being taped.] Alright.16
Andy: ? it emerges from an activity but it?s really ultimately17
lives inside?18
Rachel: Right and I mean you could say?I mean you couldn?t19
look at a thing on paper and declare that it was fun until20
you could see people do it and see them have fun.21
David: And it may be fun for some people and not fun for other2
people.23
227
Andy: You could ? an experienced teacher could make guesses24
about what?s more likely to result in fun blah blah blah.25
Rachel:  Sure, sure.  But really ultimately you don?t know until26
after [inaudible].27
David:  Or anyone?what what my kids think of as fun might not28
be the same as what Rachel thinks of as fun [trails off].29
I?m sorry I?m just struck I?m I?m thinking of Leslie?s30
stuff for a moment?what?s cool is how powerful it is to31
connect that question?how powerful it is what you just32
did. That when you say it?s like ?fun? boy that brings in a3
lot of stuff to help me understand the kind of claim that34
that we?re batting around.35
[Undiscernable/is that right? Comments.]36
Paul: Ten minutes?we could break early?37
Leslie:  Is there a community of fun practice?38
[Laughter.]39
Rachel:  Or norms?  [Laughter.]40
Leslie:  Like with the community of practice the scientist is41
someone who?s outside deciding whether or not it was42
science but with fun there isn?t ? so it?s a negative43
analogy?but with a community...  yeah there?s no4
community of practice.45
David: I don?t know what you mean? there?s no authentic46
community of practice?47
Leslie:  You only have to ask a person who?s having fun if48
they?re having fun.  But this (definition of authenticity)49
implies that you have to ask the scientists whether they?re50
doing science.51
David:  Ahhh?right.  Gotcha.52
Andy:  Not only does it have to be meaningful it has to be53
meaningful in the right way?but? yeah I?m having fun54
but it?s ? you know?low-brow fun instead of highbrow5
fun.  Guffaw guffaw!  Gotcha.56
[Laughter.]57
David:  So can we patch that?  Is there, is there another58
Matty:  [Indiscernible.  Overlap?like.]59
Rachel:  Good clean fun?60
[Laughter.]61
Matty:  Like I?m wondering you know?what can be considered62
?fun? by a group of people you know there?s like overlap63
like a culture who considers certain kinds of things fun64
and there?s like overlap between the adult (?)65
individuals?same with like if you consider that like the6
community of fun practice, like the community of67
science, I think individually everyone would consider...68
228
Leslie:  I have a multiple analogy if that?s okay?  I?m thinking it?s69
more like worship?like, you know if you?re worshipping70
but a religion is going to also decide if what you did was71
worship.72
Andy: Oooh.73
David: Right.  Right that?s good.74
Andy:  It?s gotta pass both tests.  That is good.  Good clean fun75
works, too.  You decide if it?s fun, I decide if it?s good76
and clean!7
[Laughter.]78
Leslie: Or pornography?79
[Laughter.]80
David:  Paul wants to stop.  We?re done.81
229
Appendix D
Transcript 4
The following transcript is from a third grade classroom in a suburban Maryland
public school.  In this classroom, the teacher has asked the students: if you are running
with a beanbag and want it to fall on an X, should you release it before, when, or after
you reach the X?  This  question is particularly interesting because all three answers are
plausible and can be argued with analogies to past experiences.
Camille:  I think? I think that you?re gonna drop it before.1
Teacher:  Why?2
Camille:  Because it? you just keep on running and then? I don?t3
know.4
Teacher:  Why would it not drop straight down?5
Camille:  Um? I?m just thinking that maybe if you?re running it6
might just go back, back? I mean forward when?7
Teacher:  Because I am running?  And it will go forward with me8
because I?m running?9
Camille:  Yeah.10
Teacher:  How many people think it will be forward? it will go1
forward.  I  need to drop it like somewhere around here.12
Way before hand.  Adam, that?s what you think?  When13
you back up, can you help me please.14
Adam:  Um, I think, because one of Isaac Newton?s rule of15
physics is a body in motions tend to stay? a body in16
motion tends to stay in motion until stopped, or17
something like that.  And that, um? when you?re18
running and you let go, if you let go before, it will? it19
will instead of going straight down, it will go um? it20
will go in front because it?s not stopped yet.  But when it21
hits the ground, because there is friction on the ground?2
there is more friction on the ground than in the air it will23
get stopped and land somewhere around there.24
Teacher:  You had a really um? a really good idea that I want you25
to really kind of explain it to some people that may not26
understand your law of motion and may not understand27
the word friction.  So I need you to try to restate your28
idea, and try to think about not even using that law as29
an? as an example, but trying to just explain it to30
someone using your experiences? see if you can use it31
230
that way.    Explain to? like you?re explaining to a32
kindergartner.  They?re not going to be able to understand3
that law.34
Adam:  Like um?if?like if something?if you?re riding your35
bike, um?it?s in motion.  And you?re going to keep36
going until you get stopped by like? um, a rock or37
something.  And, and?or going uphill.  And so if you?re38
on a bike, and you get?you can get stopped by39
something else, like a rock or something.40
Teacher:  So if we?re thinking of your analogy to a bike, or your41
explanation with a bike, what?s stopping it, and this is?42
Adam:  Um, no.  Well, in the situation of dropping the beanbag.43
Like, um, it?s thing is the ground, and because the4
beanbag is running against the ground, um? it?s getting45
slower.  Because like the beanbag is um? getting? I46
don?t know how to explain this.47
Teacher:  Well, I?m gonna ask you a question, I want you to think48
about it for a second, and then I am going to come back49
to you.  Okay?  I want you to think about whether you50
think the beanbag?s in motion.  Because you know I?m in51
motion: my body?s moving, but is the beanbag in motion?52
I want you to think about that question and I?m going to53
come to you in just a minute.  Okay. Um? Connor?54
Connor:  I would think the bean bag would?might fall behind5
where you want it to fall because when I put?when I56
played baseball?they always said don?t throw the bat57
because it might hit the catcher and not one of the um58
person because we?re using metal bats, and?so we drop59
it, you drop it and then you?. Well, when I drop it, it60
usually swings backwards; it wouldn?t be behind the plate61
instead of the front of the plate.62
Teacher:  So you?re saying you have to drop it like somewhere over63
there, right? in order to get it to fall over there?64
Connor:  Probably.65
Teacher:  Cause of baseball.6
Connor:  Yes.67
Teacher:  And you saw that it usually fell behind.68
Connor:  Yes.69
Teacher:  Why do you think it fell behind?70
Connor:  Well actually it didn?t mostly.  It got on the side or in71
front because? well because you?re supposed to drop it72
because you don?t need a bat while you?re running the73
bases.  Once you drop it, I?m just thinking also, what74
Adam is? well a bus? well if you were on a bus and75
you had uh, this little leaf that you found, and the window76
231
was open, and you drop it, it will go? it?ll be going7
backwards.78
Lauren:  That?s because?79
Teacher: That?s okay, you can interrupt him if you want to talk80
about his idea.81
Connor:  It might go backwards.82
Lauren:  Because I think that?s cause? you?re talking about a leaf83
that?s falling?  That?s because the? it?s sort of? the bus84
is going back, so it?s making like the air move.  And the85
leaves are really, really light, so the reason they are going86
backward is because?.  Um, well it?s going so fast? a87
bus is like going so fast that it?s probably making the air8
go that way.  So that way the leaves are going that way.89
[Many talking in disagreement.]90
Connor:  What if you did it with a rock?  The same thing will91
probably happen with a rock.  Because you are probably92
like a bus, that you make the air come? no one moves,93
don?t you notice that um, objects like in cars or94
something? when you?re going really fast on your bike95
that are? that um, you sometimes, [inaudible] and leaves96
your ankle on your back step and actually move.97
Val: We?re talking about a bean bag.98
Teacher:  What do you think is the key to that?  His question is, is9
about the beanbag.  So, um, since he?s thinking that10
maybe the weight matters? what I am going to do is just101
pass this around, so you can touch it and think about that,102
um? if you think the weight matters, might help you to103
answer the question?104
Kamran:  Well, I think that?105
Teacher:  You can?t interrupt somebody.  You can share ideas, but106
you can?t interrupt somebody as they?re sharing.  Okay107
you have to make sure he?s finished.  And when108
Connor?s finished he?ll let you know by calling on109
another person. Okay?10
Connor:  ? but sometimes, yes I do believe that it might be about11
the weight also.  So the heavier it is, probably it would12
land where you want it to.  So say you had like a rock13
besides that beanbag, it would probably land where you14
wanted it to, because it would probably be heavier than15
the beanbag.16
[Interruption as another teacher asks an unrelated17
question]18
Teacher:  Connor has this idea that the depend? it depends on how19
heavy the object is that we?re dropped.  So I passed120
around the bean bag.  And um? that might be a good121
232
idea for Kamran to talk about, so Kamran is going to talk12
about that now.  The weight and whether or not that123
matters.124
Kamran:  Yeah I think weight matters, because when Newton125
discovered gravity, gravity it?s, it?s has heavy enough to126
come straight down.  If you?re moving and I?m not127
talking about beanbag, and I?m gonna put a pencil here128
and pretend that?s my mark.  And then I ?m going to129
move, but then I?m gonna drop down it and see what130
happens. But? like?131
Teacher:  I don?t want you test it.  We?re talking about it first.132
We?re not going to test it, okay?13
Kamran:  Yeah, okay. But, but, I think that it, it, it matters on134
weight.  A bus? you?re on a bus and the bus is the135
motion.  Now if you drop a feather, that feather is going136
to go back.  Also the bus in motion is producing like a137
kind of wind.138
Teacher:  Yeah, mmhmm.139
Kamran:  It?s giving air.  Yeah.  So the air is getting pushed into?140
it?s pushing the air back.  Um? this is a drawing, this is141
the air, this is the bus.  The bus is going? the air is going142
back.  And if one of the windows, a feather comes the air143
is pushing it away.14
Teacher:  Mmhmm. Okay, okay.145
Kamran:  But, if you? cause you know a feather is? it, it, it goes146
with the air just simply its a light.  That?s why? same147
with a leaf.  A leaf is very light.  And if you? a leaf falls148
[Inaudible.] goes to air.   It doesn?t go, ?leaf? boom.?  It149
doesn?t go like that.150
Teacher:  Okay, so?151
Kamran:  But a rock?152
Teacher:  Yeah?153
Kamran:  A rock is different, a rock has? it?s also like, it?s solid,154
but it?s not that a leaf isn?t solid, or a feather isn?t solid.15
A feather? but you have to? it?s very small, and it?s156
very like thin, so you kind of say like solid.  But anything157
hollow, like if you have a paper box, that people would158
watch [Inaudible.]?159
Teacher:  Now we have a beanbag here.  So since we have a160
beanbag, and you know that it?s not in a bus, we?re just161
driving? I mean we?re just running.  What do you think162
is gonna happen?163
Kamran:  This?164
Teacher:  Cause you have that weight in your hand so you have a165
good idea.16
233
Kamran:  Um?. now this is hard, because somehow you can see167
it?s heavy, and somehow you can say it?s very light.168
Kamran:  It?s hard, that?s hard to say.169
Teacher:  So Cameron.  So if you had to? get to share your idea170
right now what would it be.  Where do you think you171
think you need to stand to get it to hit right there.172
Kamran:  I think you have to put it right there.  Probably? we?ll it173
depends on what? here, you?re dropping it? you?re174
dropping it here, here?175
Teacher:  Right by my side? just drop it? I?m just dropping, not176
throwing.  Just drop it.17
Kamran:  Yeah I think you have to be on, in order to? you have to178
be on this directly, and the middle of your shoe has to be179
on the line in order to make it drop?180
Teacher:  Okay and you? and you?181
Kamran:  Into it also, also it?s the most of minute [Inaudible.], if182
you?re going to see your shoe going on this [inaudible].183
Teacher:  It will go to the left.184
Kamran:  Yes.185
Teacher:  Um? Now you?re reason for having it hit right there,186
does that have something to do with the weight?187
Kamran:  Yeah, because the weight pulls it right down.  If a tree?18
it?s heavy, and it?s heavier than all the leaves it has, so189
the leaves will make it fly.190
[Laughter.]191
Kamran:  ? flies here.  And the tree, it will just go down.192
Teacher:  Okay, can you share? can you call on somebody that?s193
going to agree or disagree with you now.194
Kamran:  Um?Kathryn.195
Kathryn:  Um I agree with what you?re saying and all [Inaudible.]...196
Teacher:  I?m sorry, I can?t? I can?t go on cause I can?t hear you197
very well.198
Kathryn:  Um, uh? when you told me the first time, I didn?t really19
get it, so I [Inaudible.]?.20
Teacher:  That?s okay.201
Kathryn:  Can  redo it?202
Teacher:  Um, don?t worry about that now.203
Kathryn:  Okay.  I think that if you want it to land? well if you?re204
riding on a bus and you drop the rock down, it?s heavier205
than the wind, so I think it?ll go straight down.206
Teacher:  So you think that it needs to be heavier than the wind.207
Kathryn:  Yes, cause it?s going to go straight down.  A beanbag and208
the rug?if it?s heavy and um? I would say you would209
have to be in the middle of the line, because210
[inaudible]? if you?re running it?ll go all the way behind21
234
you.  But if it?s heavy enough, it?ll just drown you have212
to be in the middle of the line.213
Teacher:  Okay, I am going to ask you to stop right now.214
235
Appendix E
Transcript 5
The following transcript follows two undergraduate physics majors working on a
homework assignment on angular momentum in quantum mechanics.  The students have
had instruction on how to arrive at the quantum numbers S and L but are asked to find the
square of the total angular momentum, J
2
, which is (S + L)
2
, or S
2
+2SL+L
2
.  The students
know how to find S
2
 and L
2
 but not 2SL.  They have a solution set from another student
that provides the answer but not the steps to arrive at that answer.
Ben:  That?s what we?re looking for in the first place.  You1
don?t think that?s it?2
Anselm:  Well J^2 is L^2 plus S^2 plus 2SL.  And SL is3
presumably the thing that could take on separate values4
within the probabilities.5
Ben:  It doesn?t talk about J^2 though does it.  [unclear]6
formula [unclear].  Although we should be able to figure7
it out cause it said the rules are the same whether you use8
spin or angular momentum.9
Anselm:  Yeah.  That?s fine if you know the rule.  But we don?t10
know what S times L really does.  Cause SL doesn?t1
really have a quantum number.  Is it the square root of S12
and the square root of little-L, and then you take that13
times that plus one?14
Ben:  We should be able to figure this out from today?s lecture.15
Anselm:  No you shouldn?t.16
Ben:  He?s gonna explain in detail probably Wednesday how17
you actually get to J.18
Anselm:  But see you?re doing the wrong thing.  Cause you?re19
assuming that if you have the example, suppose there?s a20
charge here, what?s the electric field due to it?  You can21
figure out, suppose you have Bugs Bunny, and he?s2
charged, what?s the electric field around his ears?  All23
right.  Because you have a simple example when they?re24
both the same, you?re not going to be able to figure out25
exactly what you?re supposed to do when the rules26
weren?t the same.  Cause now it?s fixed.27
236
Ben:  What other choice do we have?28
Anselm:  So like there?s no like S, S+1, it?s more like the square29
root of little-L times the square root of little-S maybe?  I30
don?t know.31
Ben:  Well what other choice do we have?32
Anselm:  Cry?3
Ben:  How else - Yeah.  Okay, let?s cry, I?ll go cry.  Now let?s34
go on to B.  ...I don?t see anything that [I think?s gonna35
help us].  And I feel like the book is just pathetic in this36
regard.  It doesn?t give us any help at all.  I don?t think it37
does, let me look back maybe there was something in it38
that helps.  Combining spin and [angular momentum].39
Anselm:  Does your thing have the answers for this problem or40
not?41
Ben:  Yeah.42
Anselm:  So are there multiple J-squareds then?43
Ben:  Yes there are.4
Anselm:  Are they the ones we found?45
Ben:  No they?re not.46
Anselm:  All right.  What are they?  Let?s work from the answer.47
Ben:  Eight-ninths and one-ninth.48
Anselm:  That?s the probabilities?49
Ben:  Uh-huh.50
Anselm:  Eight-ninths and one-ninth.  All right.  And what51
are?what are the values?52
Ben:  Fifteen whatevers in the, whatever we came up with.  The53
values are the same.54
Anselm:  We didn?t come up with that.5
Ben:  Yes we did.56
Anselm:  Well yeah we did.57
Ben:  These are the values.58
Anselm:  Yeah we did come up with the values.  Um, but it?s eight-59
ninths and one-ninth?  And now you say60
Ben:  If you want to believe that.61
Anselm:  And now you say - You said at one point you had eight-62
ninths and one-ninth, where?d you get eight-ninths and63
one-ninth.  What did you do.64
Ben:  All right, look at this look at this.  If you take all the65
positives and add them together, you get eight-ninths.6
Anselm:  Oh, oh.67
Ben:  You take the negative, you get one-ninth.68
Anselm:  Yeah that?s69
Ben:  But you?re mixing apples and oranges.  It?s dumb!70
Anselm:  Yeah that?s so messed up, yeah that?s not the answer.  If I71
just ignore the fact that I?m in the three-halves one-half72
237
and I?m in the one-half one-half and I just add them all73
together,74
Ben:  I once had a professor tell me that um, well if you got the75
right answer, you certainly know how to do the problem.76
I had to convince him no sir, you can jiggle these7
numbers any way you want.  And come up with the right78
answer if you know the right answer in advance.  Of79
course we?re not sure that this is the right answer.80
Anselm:  It?s been pretty good.  Except for h-bar [seize two].  All81
right so somehow we need to get eight-ninths and one-82
ninth?83
Ben:  Yeah.  Somehow.84
Anselm:  [muttering - reading?]  If we were somehow free to swap85
around some of these square root of two over threes, then86
we could make it.87
Ben:  No they came from the table.8
Anselm:  Yeah, but I?m saying if we were.89
Ben:  [shakes head]90
Anselm:  I know we?re not.91
Ben:  Wishful thinking won?t help.92
Anselm:  We would have to?yeah somehow it would only be the93
square root of two-thirds would have to multiply both of94
these.95
Ben:  We can always [unclear] from class.  You made me late96
for class this morning.97
Anselm:  I!  You were the one, excuse me,98
Ben:  You insisted on going over problem 37 and9
Anselm:  My last semester of college I?ve never been late to class10
before and it?s all your fault.  Ruined my record, man.101
238
Appendix F
Transcript 6
The transcript below is from a discussion that faculty members at the Governor?s
School of North Carolina had in the lounge of the faculty dorm.  One faculty member,
Steve, asked what I was teaching in my class and mentioned that students wondered if
humans would ?blow up? in space or not.  The faculty began discussing this question and
began to tease apart different factors by considering what would happen to a statue in
space (this statue was Michelangelo?s David and references to David in the text are
regarding that discussion).  This discussion then turned to whether or not a human would
freeze while in space or heat up from the sun.  This lasted much of the afternoon and we
agreed to continue on the following Sunday.  They began the discussion the next Sunday
by wondering how the Earth heats up and cools off, which turned to a question about
seasons: are they due to the elliptical orbit or due to the tilt of the Earth?  The transcript
below starts with that question.  (?Leslie? in this transcript is me?the author.) Leslie: 
Okay, so where are we now on the ellipse question?
Tom: I think this whole, if the sun is not at the center of the1
ellipse but is one of the foci of the ellipse [which is true] I2
think this whole business thing3
?: Is all about the atmosphere filtering4
Marc: Un-unless you?unless we could measure and discover5
that, one hemisphere?s average temperature is different6
than the others7
Joel:  Well but the earth is much fatter in the middle, you know8
what I mean?  So the part that?s that?s facing the sun, the9
closest part to the sun the on that?s getting the most direct10
rays is always the equator.  Because that?s the way it1
works.  And that?s the nature of the angle of of the?12
239
Marc:  That that?s shape?13
Katie: Mmm hmm has to be, that?s what makes the equator14
Cameron: The equator?s not always the closest part to the sun15
Marc:  Right but it?s always getting the most direct sun.16
Cameron: Not always.17
Marc: No no it?s true.18
Cameron:  That?s what the tropics are about.19
Tom:  Yeah that would only be true if?20
Steve: That?s that?s ? it?s some sort of area?21
Marc:  Right right I see.2
Katie:  You?re right, but still that area?the belt.  The fat-belt.23
Leslie: So why does the tilt matter?24
Steve: Well they think it?s ?cause the (laughter) ? well here?s the25
thing, now I have a problem for all you people you and26
your filter (laughter) is that that?s fine I?ll accept your27
filter all if you can explain to me why even when the28
earth is demonstrably further from the sun we don?t have29
any differences in temperature between north and south30
pole?31
Tom: Why does angle matter more than distance?32
Vic: Or why is distance irrelevant?3
Steve: Or- right, why ? how can distance be irrelevant if the34
earth is much further from the sun at some points in its35
orbit and the average like if pick some spot latitude and36
long and like halve the southern and northern37
counterparts for it--?38
Vic:  But wait wait wait--39
Steve: But if it?s going that much further farther away why is the40
temperature not that much different?41
Vic:  But wait wait-- I have a thought right ? this is not a42
question of like, that we can, like if there was a fire in43
that fireplace and the farther back I move from the4
fireplace the drop in temperature? there?s atmosphere45
involved.  The heat?s traveling in a different way.  So that46
it?s not a question of, like, radiant heat.  The heat that47
we?re getting from the sun is a question of light, right?48
Katie:  Yeah because that once we get into space it?s cold out49
there.50
Vic: Right.  Heat?s not traveling through space in the way that51
radiant heat does.52
Marc:  How does heat traveling through space?53
Katie:  As light.54
Vic:  As light.  As energy.  And and then the atmosphere bends5
the light.56
240
Katie:  Once it hits something [Tom: that makes sense to me]57
that gets [Marc: heated up].  It?s like  And then it can?t58
get back out.59
Amelia: Right.  It?s trapped.60
Vic: Because of the atmosphere.61
Katie:  Once it turns from- -I don?t really understand this but it62
seems like it turns from light to heat but it?s all energy63
but it?s64
Vic: We?re into the angle theory.65
Marc: It turns from energy to heat.  Not necessarily to light to6
heat because there are other forms of non-visible light.67
Amelia:  Okay.  So therefore when the68
Katie:  Energy in the form of light69
Marc: Right gamma and ? ?which is nonvisible light.70
Infrared.71
Amelia:  Which would might wonder why there wasn?t necessarily72
why there wouldn?t be a huge different in temperatures in73
the northern and southern hemispheres.74
Cameron: So are you saying?75
?: Yeah?76
Cameron: So are you saying that the angle doesn?t matter?7
Steve:  No, I?m just saying that if ? if we don?t think distance78
does matter then we have to be able to explain why79
distance doesn?t matter.  And why--80
Amelia:He?s tough?he doesn?t let you get away with anything.81
Steve:  ?even though the earth is much further from the sun at82
some points in its orbit summer temp maybe is?83
Marc: But we don?t know about that ? it?s possible that there is84
a different average temperature.85
Steve: Well but you were just saying that you didn?t think there86
was.87
Marc:  Well I asked and someone said: did you read the8
Antarctica book?  And she said??I don?t deny that it can89
get pretty damn cold down there?it?s just the question is90
what?s the difference.?91
Vic:  Well it- the other question is then why is then the92
different planets does it seem to get colder the farther93
away from the sun?  Right?like Pluto is like frozen94
solid.95
Steve:  Is that just because they have different atmospheres?96
Vic:  That?s exactly what I?m wondering is that is it a question97
of distance?98
Marc:  Well clearly it has something to do with distance because9
we don?t feel the heat from other stars?I mean distance10
has something to do with it.101
241
Tom: What?s the temperature on Jupiter- I mean it seems the102
gas?103
Vic: What?s the temp on Jupiter [singing]104
Steve:  What?s the moon?s temper?what?s the moon?s105
atmosphere like.106
Marc: There is no atmosphere.107
Vic: Right.108
Steve: that?s not true?there?s atmosphere on the moon.109
Marc:  There?s no atmosphere there.10
Vic:  A really reall? it doesn?t enough gravitational force all11
on its own to have significant atmosphere.  It has very12
very thin atmosphere.13
Cameron:  So what?s the difference between temperature on the14
moon between day and night?15
Marc: Oh it?s huge!  It?s enormously huge.  [Why?] There?s no16
atmosphere to trap in the energy?it?s like the greenhouse17
effect.18
Tom:  You can?t have temperature without an atmosphere?19
Steve:  But I thought we couldn?t have heat without [couldn?t120
have what?] heat without atmosphere.121
Marc:  No, see?I?m not sure I ever signed up for that.12
Marc:  No there is such a thing as radiant heat?that?s what the123
sun?124
Steve:  But if it?s in a vacuum, I learned last week that space is a125
vacuum.126
Vic:  That?s what I hear.127
Katie: I learned that science doesn?t suck.128
Cameron: But if you put something on there, like a thermometer,129
which is measuring the temperature, it?s hotter in the day130
because it?s absorbing the131
Vic:  light and rays and things.  Right ?cause it?then there?s132
no atmosphere at all to filter it and it?s just absorbing13
everything.134
Marc: Vacuum means, by the way there?s no? I don?t know135
first of all , I was going to say something that there?s no136
mass but these electromagnetic waves they don?t have137
any mass really so you can still call something a vacuum.138
I don?t know we?re getting technical?there?s a lot of139
energy in space!  It?s filled with energy.140
Steve: Right?right, but that doesn?t make it heat.141
Tom: Yeah weren?t we talking about if you?re just completely142
in outer space, not exploding (laughter) it?s we decided it143
was meaningless to talk about heat isn?t that right?  And14
so presumably isn?t that the same thing if you?ve got a145
planet without an atmosphere?  Then it?s just an object in146
242
space?  That?s just like you standing there in your space147
suit trying to measure the temperature.148
Marc: but it?s HOT on the sunny side of the moon?it?s hot.149
Tom: It is?150
Marc: yeah.151
Steve:  So the moon has enough atmosphere to create heat.152
Cameron:  So I think what we?ve learned here is that, [Katie (to153
Marc?): How do you know it?s hot?] uh, is that the154
atmosphere helps, like, mitigate temperature change.15
?: The surface itself?  We can do that also with looking at156
other planets, right.157
Tom: Well but then it?s reflected too, right?158
Marc: Say again (to Katie?)?159
Katie: Like if we know what the atmosphere is like on other160
planets and they have--161
Marc: Like Venus for example, Venus is very hot162
Amelia: We are mostly water and it does reflect light which163
bounces back and forth.164
Marc: Yeah yeah?the atmosphere reflects most of the light?165
Katie:  Yeah if it were too hot up in here we would be just16
Marc:We would fry.167
Amelia: Well the oceans would evaporate.168
Katie: yeah that?s I?m trying to say.169
Marc: The oceans would boil the four horsemen?170
Steve: By the way when we?re done with this I have to tell a171
story about what the priest said at mass today?it was172
nothing about prostitutes or anything like it was last year.173
Amelia: Did he talk about the four horsemen apocalypse?174
Steve:  we talked a little bit about the apocalypse? it was175
actually pretty good.  [Conversation turns to discussion of176
weekly happenings.]17
Vic: [Yelling to get attention] Heat of the sun and the moon178
and the stars!179
Steve:  Alright.  Heat of the sun and the moon and the stars.180
Leslie:  Okay?put a thermometer.  It?s gonna do?181
Katie:  I?m pretty sure that the mercury?s gonna leak out.182
Amelia:  And then it?ll float in these little silver balls.183
Tom:  How does a thermometer work again?184
Katie:  That?s pressure ? a thermometer has to do with pressure185
so it wouldn?t do anything in a vacuum.186
Marc: No a thermometer isn?t pressure- barometer is pressure.187
Steve:  How do you get the mercury to move?18
Marc:  Okay- what happens is, your particles have to interact189
?cause I remember as a kid reading somewhere that if you190
could actually get to the center of the sun it?s so gaseous191
243
the particles are so disparate that you actually freeze to192
death because there wouldn?t be much interaction193
between you and the particles of the sun.194
Steve:  You?d need a hell of a space suit to get there though.195
Marc:  That?s what I?m talking about.  And you be blind by all196
that light.  But actually there?s no particle interaction so197
I?m assuming that- I?m imagining that if you take a198
thermometer and put it into the vacuum of space there are19
very few molecules to interact with your thermometer20
and that?s why it?s cold out there.  So the question201
is?and I remember this, this is a question in202
science?how does energy travel?  That?s ? they tried to203
prove there was an ether in space!  [right]  And they204
found out there?s no ether, so how does energy travel in205
space?  [right]  It?s electromagnetic waves.206
Amelia: What do what do we call this stuff?207
Katie:  I wish you were a little smarter.208
Steve:  I do too.209
Amelia:  Most of space is that stuff and there?s that other stuff.210
Steve:  But- alright so like thermometers in space?21
Leslie:  No no?just a thermometer on earth, how would that212
work?213
Steve:  Okay?214
Tom:  What is it about mercury, what happens to it?215
Marc:  well it expands.  But it?has to be hitting the bulb, I mean216
like the heat that the gas- the air. [it?s what heat it]217
yeah?what?218
Tom:  Heat is like the ?219
Marc:  Heat is a measure of the amount energy?the motion of20
molecules.21
Tom: There has there has there have to be molecules moving to22
measure heat.  That?s right.23
Steve:  But how do they?wait a minute?look.24
Katie:  So you go into space where there?s a vacuum and there?s25
no molecules out there to interact with the little bulb at26
the bottom the mercury inside, it?s not gonna measure27
anything.28
Marc:  That?s right.  And that?s why it?s cold in space.  Because29
your body doesn?t feel any of those molecules either.230
Steve: Wait a minute.  Because sometimes you have [Katie: I?m231
afraid we?re guilty of reification but ok] but sometimes232
you have thermometers that are in the sun and in the23
shade right next to each other and the sun one is much234
hotter.235
244
Marc:  That?s because the air surrounding the hotter is moving236
faster and in the shade.237
Amelia:  And if the vacuum has no air.238
Marc:  You?re right, then you wouldn?t measure anything.239
We?re agreeing.240
Steve:  Does it seem like Weitz agrees with everyone?241
Katie:  Nuh uh he?s been fighting with you all afternoon.242
Cameron:  Even when you agree.243
Amelia:  Yeah- he even says for you to shut up (?)24
Leslie:  Okay guys I?ve got a thermometer in space, one is in a245
box and one is facing the sun so one can?t see the sun and246
one can.  Are they gonna measure the same non-247
temperature?248
Cameron:  What?s in the box?249
Leslie:  The shade.250
Marc: Well is it the vacuum of space?251
Steve: Lava.252
Leslie:  Still in the vacuum of space.253
Katie:  Wrapped in leather.254
Marc:  No no wait a second but you?re right though because um25
for example, solar panels that face the sun?like if you256
had a satellite with a solar panel and it?s facing the sun257
the carbon fibers, everything?s expanding.  That?s how258
they sometimes lose these satellites because?so the259
energy260
Amelia:  Marc talks too much.261
Katie:  is he like this in every class?262
[Laughter, jokes about Mr. wizard and kids.]263
Marc:  So it must be - it must be- a temperature.  Well, there?s264
heat in space.  I mean your satellite.265
Vic: The object is absorbing the energy by the light.26
Marc:  So would the thermometer absorb the energy?267
Vic: Why not?268
Marc:  Well then- then we have to change our answer then the269
thermometer would rec??270
Vic: I was going to say that all along?I don?t know what you271
guys think (?)272
Katie: I?m not saying that? it seems like the way the273
thermometer works though it seems like it wouldn?t274
register.  It might275
Tom:  I don?t quite understand how a thermometer works276
Katie: Well?27
Tom:  ?Cause we?re talking about heat.278
Katie: Weitz?279
245
Marc: What if the what if the what if the energy that?s ? the280
gamma ray and the infrared and whatever?what if all of281
that electromagnetic energy excited the mercury in the282
ball and it expanded?  Then it might measure something.283
Katie:  IF that?s how a thermometer works then, ok.284
Steve:  How do we?is that how a thermometer works?285
Vic:  Is there a reason that a thermometer can?t absorb energy?286
Marc:  Is there a reason why?  That was your question?  [right] is287
there a reason why it might not absorb?28
Vic:  And then there?s the question of whether or not.289
Steve: What?s the intake on a thermometer?290
Marc:  It?s a metal bulb ? it?s just291
Steve: It?s a metal bulb that?s exposed.  And like the292
thermometer could be put in our mouth.293
Katie:  A thermometer that you put in your mouth?it reacts294
differently than a thermometer that you have outside295
because if I if I am right the thermometer I have when296
I?m sick it does not measure how cold my house is.  Am I297
makin any sense?298
Vic:  I dunno.29
Katie:  When you when you don?t have it in your mouth and it?s30
not actually touching anything301
Marc: Right then it yeah302
Katie: It might be warm or cold but it doesn?t say my house is ?303
[yeah]304
Amelia:  How many ways are there to measure heat?305
Marc:  But but mercury thermometer does?sure it does.  You306
shake it out and?307
Leslie:  Amelia just asked?308
Marc:  Oh.  What was your question?309
Amelia:  How many ways are there to measure heat?310
Cameron:  That?s only one kind of thermometer, right?31
Katie:  That?s what I mean that?s what I?m tryin to say.  The312
difference-313
Cameron:  You?re saying something different.314
Amelia:  ?Cause that?s not what they use in space to measure315
temperatures like on the moon.  What?why are we even316
talking about this?317
Vic:  Are we completely distracted?318
Katie: She asked us what a thermometer would do in space.319
Amelia:  But she didn?t say what kind of thermometer.320
Marc: But in order to measure heat you have to have something321
reacting with the environment.  [ok] so if it?s not mercury32
it?s? silicon in a chip or something- something has to be323
246
reacting with the environment and my question is [well324
my] is there anything in space to react with.325
Amelia:  But temperature is measured on the moon is it not?326
Marc: That?s what I?m- I agree with you.327
Amelia:  So how do they do it?  She?s not going to tell us, I?m328
sure.  How- how is temperature measured on the moon?329
If it?s been done then how?30
Marc:  But the question is why is it so- why is it so cold? It?s31
measured some way!  It?s measured some way?let?s, I32
can agree it?s measured some way but take that33
measuring system and put it in space.  Is there anything34
that it can react with?  Is there any particles in space?35
wait, what?s our question again?36
Leslie:  We put the mercury thing up there?a normal thing up37
there?it?s gonna have a reading on it.  Is that reading38
going to tell you the temperature or is it just?it doesn?t39
work any more?340
Marc:  Well it it should, I mean?it?ll be very very cold but341
there?ll be some heat there.342
Steve: Mmmm.  I disagree.343
Katie:  The question, I think, is will the thermometer measure34
that heat?does it work in space?345
Marc: There?there aren?t many particles for it to interact with346
which is why it?s so cold I guess.347
Steve:  It?s a vacuum though.348
Marc:  There aren?t any particles though.349
Steve:  But the moon has? if it?s in space?350
Marc:  The moon is a rock?it interacts with the gamma rays351
[in background Amelia, dark matter?]352
Steve:  Neil Armstrong bounces.  On the moon.353
Vic: That?s correct?but yeah that?s gravity and not354
necessarily the atmosphere.35
[Amelia, in background: ?only this much about dark356
matter?and if they don?t know and by they I mean the357
scientists, how are we supposed to know.]358
Steve:  If we didn?t have gravity we?d ____ on the moon.359
Marc: That?s correct?so you might, one could argue that the360
moon has one sixth the atmosphere since it has one sixth361
the gravity.362
Leslie: That was Amelia?s? I didn?t hear your last comment.363
Amelia:  Oh I was talking about?I mean, isn?t most of space dark364
matter and then some of space is dark energy, right?  Or365
something like that?36
Leslie:  We don?t know.367
247
Amelia:  We don?t know?like we don?t even know what dark368
energy is and barely know what dark matter is, and so if369
scientists who study it don?t know then how are we370
supposed to know what?s happening.371
Leslie:  Well the thing is either our theories are wrong or there?s372
mass we can?t find, you take your pick.373
Amelia: But what I?m saying is, nobody knows374
Leslie:  Right.  Yeah.375
Marc:  I still think it has to do with the amount of particles to376
interact with?in space there are few particles so there?s37
very little heat cause you?re not feeling it it?s not?well378
not very little, yeah? but on the moon it?s a big rock so379
it?s taking all of this energy and it?s interacting with all of380
this energy that?s coming at it.381
?:  Something about atmosphere382
Leslie:  Well what?s so special about the atmosphere that helps it?383
Tom:  Well there- there wouldn?t be a temperature around the384
moon but the moon itself?385
Marc: The moon itself has a temperature.386
Vic:  Wait wait wait?no no?this is387
Katie:  You could plunge a thermometer into the moon.  Like a38
turkey [laughter] into one of the crevasses of the moon.389
Marc: Up the moon?s butt.  And baste the moon a little bit.390
[break]391
Vic: The problem is the problem is?the lack of significant392
atmosphere on the moon means that one side is going to393
be very hot and the other side is going to be very cold.394
Tom: Because the atmosphere tends to preserve the395
temperature.396
Vic: Right.  The atmosphere is going to stabilize it the earth is397
within a certain temperature.  And then there are other398
weather patterns that happen in the atmosphere that might39
move that energy around.40
Marc:  It also serves to reflect.401
Steve: I think an atmosphere matters because radiant light hits402
the one sixth, whatever, atmosphere of the moon there are403
particles to  hit? that that light can, I don?t know,404
whatever, I?m making it up not? triggers molecules to405
move, creates heat.406
Marc:  but that?s not the only source of heat that heats the moon.407
The moon itself absorbs a lot of the radiation that the sun408
is giving off.409
Steve: Right but that?s the way it?410
Marc: the ground absorbs radiation.41
?: Why does it need to be atmosphere?412
248
Steve:  Well I guess- the reason I?m thinking atmosphere matters413
is because if we don?t have atmosphere?if you don?t414
have like ? spaces of vacuum [the rock] right right or415
where you have.  So a rock-lava-David? David? is in416
space? does he get hot?417
Marc: Yeah.418
Tom: David can have a temperature because because he is419
made up of molecules which are420
Steve:  So if David passes in back of, like, Venus de Milo, and is421
in the shadow for like a little bit and then moves back42
into the sun, David warms up?423
Marc, Tom: Yeah.  Pretty much.424
Tom: That makes sense to me.425
Marc: That makes perfect sense.426
Leslie:  Does he keep warming up indefinitely?427
Steve: Right.428
Marc: No because there?s finite amount of?he would reach a429
stable point.430
Leslie: Why?431
Vic: Yeah- why?432
Vic: There?s no way for you to give off that heat?there?s no43
way for you to radiate that heat.434
Marc: No but you do- but you do radiate the heat.435
Tom: you don?t heat up indefinitely436
[Many voices.]437
Marc: Because all matter would evaporate.438
Steve:  David explodes eventually.439
Marc:  Ok, yeah- eventually the earth explodes?but that?s not40
what we?re talking about.41
Vic: The earth explodes?!42
Steve: Yeah- that?s what the priest was talking about.43
Marc: That?s only if we sleep with prostitutes.44
Katie: The priest said the Earth was gonna explode?45
Steve:  You shouldn?t believe Pentecostals.46
[Discussion of a recent field trip to a church.]47
Leslie:  Well is this what?s causing global warming?just that the48
sun keeps shining on the earth?49
Marc: No it?s the thickening of the atmosphere that?s trapping450
more and more.451
Cameron:  But the energy can?t get out.452
Vic: Yeah why would why would- that?s would, I guess that?s453
true, so why would the energy have to be spent as heat?454
Like radiated as heat from the David floating in space?45
Or could he not give off his heat?456
Cameron:  So why can?t we radiate heat out?457
249
Marc: Well we do radiate heat out but the increasing density of458
carbon dioxide is trapping ?459
Vic: We?re not worried about the Earth we?re worried about460
David.461
Cameron:  I?m talking about a guy floating in space.462
Tom: Oh so global warming?just forget about that.463
Steve: For the time being.464
Marc: David does radiate heat.  David does radiate heat.465
Amelia:  But only for a while, right?  Is he radiating it as fast as he46
[?] it?467
Marc:  Once he reaches a stable point?468
Katie: you can?t play that card [?]469
Marc: why?470
Marc:  Okay? let?s? let?s say David is in a shadow, right?471
Okay- he enters the sun.  The sun bombards him with all472
this energy right?  So in a second it?s now at 5 degrees.473
Can it radiate heat that fast?  No.  So in the next second474
it?s 10 degrees.  It?s now radiating a little bit more heat475
but there?s more energy coming in.  So it gets to 15476
degrees.  But at some point it?s radiating enough heat to47
stabilize at 20 degrees.478
Steve: But why?479
Cameron: What?480
Marc: Or let?s think of think of like a, think of like a basin, ok?481
Think of a tub.  With the drain open, okay?  The drain is482
open.  Now if I open the spigot [Uh huh.] ? if I open it483
too slow then the tub doesn?t fill.  But if I open the spigot484
fast enough there?s water filling up the bottom, and yet485
some is also draining out, right?  [Right.]  If I open the486
spigot up fast enough it doesn?t matter if the bottom is487
open, the top will overflow but at some point if I reach48
the right point the tub could stay at a certain level, even if489
water?s going in and water?s going out, right?  If they490
came in at the same rate [Steve: Right, but?] the tub491
would fill up.492
Steve: But that?s ? what?s David?s drain?493
Vic:  That?s?this is my question.  What?s David?s drain?494
Tom:  Well, what?s the sun?s drain?  The sun is clearly radiating495
heat and energy.496
Marc:  Yeah I mean that?s just the?497
Steve:  The sun is radiating light.498
Vic:  I want to?49
[Many voices.]50
Marc: And gamma rays (in response to Steve) and x-rays and all501
these other things.502
250
[Amelia squeals to imitate being bombarded with rays.]503
Steve: The sun is giving off light rays.504
Marc:  It?s giving off a whole?505
Cameron:  Why is the sun so special that something else can?t give506
off energy in whatever form?507
Steve:  Well the sun?s giving off light?508
Vic: so it?s generating509
Marc: and radiation and stuff.510
Cameron: So it?s all about the light.51
Steve:  yeah right I think it?s about light.512
Cameron: that?s the only kind of energy.513
Vic: no no there?s radiations and stuff.514
Marc: let?s say the spectrum the electromagnetic spectrum.515
Leslie:  If you say light you don?t necessarily have to mean516
light?517
Steve: I just mean like, rays.518
Vic: Radiations.519
Steve: So I thought that we were saying that ? or y?all were520
saying that the Earth is warm North v. South we have521
seasons because of the way that light interacts with the52
atmosphere, right?  And filters it at different angles.523
Marc: The atmosphere will reflect?reflect not absorb?will524
reflect more sunlight depending on the angle at which it?s525
tilted towards the sun.526
Steve:If the earth didn?t have an atmosphere.527
Marc: Far less light- far less energy- would be reflected and the528
earth would be a lot hotter.529
Vic: Oh oh oh oh oh oh oh oh .  Oh oh oh .530
Marc: The Oceans would boil the apocalypse.531
Tom: If the earth did not have an atmosphere there would be no532
seasons, right?  Because.53
Marc:  No no?yeah yeah yeah.534
Vic: I had just had an I had an idea well- or at least about why535
distance is at least partly significant in terms of the536
amount of heat that an object absorbs, right??that ? that537
if you?ve got a source of of the radiation stuff, then it all538
goes back to surface area again.  So that if you?re Pluto539
and your?re this far away, the amount of surface area that540
you have exposed in terms of this circle that?s radiating541
heat?the closer that you get the more of your surface542
area is going to get sort of?543
Marc: This has to do with size?Pluto?s much smaller than the54
Earth.545
Leslie:  Vic, do you think Marc understood you?546
Marc: I?m sorry.547
251
Vic: I?I don?t think he did.548
Leslie:  Marc do you think you understood Vic?549
Marc:  I thought I did, but given his pause I don?t think that I50
did.51
Vic: Alright?if we can go back to the moon, right? [ok]  If if52
the moon gets in the way of the sun and it blocks out the53
entire sun it can only do that because of the distance54
because of the relative distance between the Earth and55
stuff.56
Marc: And the relative size.57
Vic: Right.  So that- if the moon were closer [to what?] to the58
sun they would block out less of the sun? or or more!59
[laughter.]  Or more! Or less!560
Leslie: No wait?more or less?which one?  Both?561
Steve:  It would block out less of the sun.562
Marc: Think about putting your hand on a screen?making a563
shadow puppet.  The closer you get to the light source?564
Amelia: The smaller it gets.565
Marc: The bigger the puppet.56
Amelia:No.567
Tom: The closer you get to the source of the light?568
Marc: My god we?re getting? I don?t know now.569
Vic:  Right right?570
Amelia: Oh the source!  I?m sorry I?m sorry.571
Vic:  Closer to the light you?re absorbing more of the light.572
Marc:That?s right.573
Vic: Your hand hasn?t changed size?now the closer it gets to574
the screen the more it absorbs.  That?s exactly what I?m575
saying ? so the clue (?) this hand puppet far away576
absorbing a little bit and that ? Mars is the one that?s57
closest to the sun so it?s absorbing more of the thing578
Marc: Over those huge distances it makes a difference.579
Vic: Regardless of how?yeah?regardless of how much580
surface area.581
Marc: That?s right, but then over such large distances distance582
makes a difference but given just the tilt of the earth I?m583
thinking that the tilt has much more to do with it.584
Vic: No I?m just talking about why- why distance is still585
significant.586
Marc: Well of course distance is absolutely significant.587
Cameron:  Distance is not significant it?s the amount of diffusion58
that happens within that distance.589
Vic: Okay.590
Leslie:  What do you mean ?amount of diffusion??591
252
Cameron: Like?when my hand is closer to the projector?with the592
projector the lens makes the light go like this?so it?s593
more diffuse when you get out towards the edges and the594
band, width is narrower when you?re closer to it.  [Marc:595
radiates]596
Leslie: So it gets weaker as you go out.597
Cameron: It?s the same amount of energy but it?s spread over?598
Leslie:  Is it solely because of what Vic said?is it just because59
it?s going over more area or is it also just losing energy as60
it travels?601
Cameron:  It?s probably losing energy as it travels, too.602
Vic: Wait though it?603
Cameron: Well, it?s gonna hit other things.604
?: You?ve got (?) atmosphere.605
Vic: Yeah ok sure.606
Vic:  I?m thinking in terms of stars and stuff?607
Marc: Light weakens the further away you are, that?s why we608
have brighter stars? well one of the reasons why we609
have brighter stars and dimmer stars is that the energy610
that the light has decreases.61
Leslie:  But is that because it?s just traveling further or is that612
because ? well, how should it decay if it?s just because of613
distance?614
Tom: Inverse square, right? Because it?s?uh?615
Marc: Because you have three dimensions.616
[Laughter.]617
Leslie: [To the tape recorder.]  Okay?Tom is taking his hands618
and flinging them apart over and over again.619
Tom: Don?t you see?  Can?t you see?620
Marc: I gotta listen to these tapes.621
Marc: what question are we pondering?62
Leslie: I ? I happen to know how fast light decays?and if you623
could tell me what it would be if it were only due to624
area?that it takes up?that the area that the light?s going625
into is getting bigger?626
Tom: Right right right!  That?s it the area is getting bigger, so627
like?damn.628
Marc: So it would have to illuminate an ever greater source of629
area so each individual point gets less.630
Cameron:  That?s what I was talking about earlier.631
Leslie: That is?so I?m wondering why Tom thought it would be632
distance squared.  So if I?m close to the sun it?ll be x63
bright , if I?m twice as far away it will be one fourth.634
Marc: [Sighs.]635
Vic:  So that- you win.  It is distance.636
253
Marc:  Look the distance between earth and Pluto has to do with637
distance but the difference between winter and summer638
has to do with tilt, not distance.639
Steve: But tilt is insufficient by itself.  Tilt ?640
Marc:  I mean ap?I mean angle of incident energy.  That is641
correct.642
Marc:  So then I can guess maybe distance has a component.643
Steve:  I?m troubled by David.  I want to know why he radiates64
heat.645
Vic:  You mean how he gives off heat?646
Steve:  I want to know why he gets warm and why he can647
radiate.648
Marc: He reflects?I?m not sure that he radiates.  Well?649
Tom: He will reflect.650
Marc: He?ll reflect him because we?ll be able to see him.651
Steve: But where will it go?652
Marc: That?s how we see it.  We see it because he reflects.653
Steve: But it?s not light it?s heat.. it?s not heat it?s light654
Leslie: Does it get hotter at all?  Does he absorb any of it?65
Marc:  He absorbs it?that?s what gives him a temperature that656
we can measure.657
Leslie:  Ok?why does he stop?  Or is he always going to keep on658
absorbing it.659
Steve: Right. Because there?s no way?there?s nowhere for60
that?61
Marc: His temperature will change depending on his relation to62
the energy source.  If he?s moving towards the sun he?s63
getting hotter if he?s?64
Vic: Right but to borrow your bathtub metaphor right if he65
keeps?if he keeps absorbing energy without being able66
to?67
Steve: He?s going to overflow.68
Vic: He?s going to overflow and blowup.69
Marc:But he is?he is ? I could imagine?I could imagine670
perhaps-- tiny thing, MAYBE, blowing up.  But a certain671
size would be able to reflect that at a good enough rate672
that it would be able to stabilize.673
Steve:  But if he gets hot?I?ll buy that he reflects?but if he also674
gets hot?either there?s some way that he gives off heat675
or676
Cameron: I say that he?s radiating heat.67
Steve: But how can he radiate heat?678
Cameron: How can he not radiate heat?679
Steve: There?s no where for the molecules to go (?)680
254
Marc: Ok but this is very tricky.  Electromagnetic radiation681
travels in a vacuum.  How does that happen?  I don?t682
know.  But it happens.  Now if we want to sit here and683
think?I mean that?s684
Steve: That?s what happens the light hits David?on David?it,685
the light becomes heat and then it eventually becomes686
something other than heat and leaves David?687
Marc: Ah- no energy hits him.  It?s absorbed.  It?s also reflected.68
And as it?s reflected it takes some of it?s energy with it in689
the form of a different type of energy because ?690
Vic: Right right right ok?again what we keep getting at or I691
think that what my trouble is with Steve here is again692
Steve: Not with Steve, with my concept693
Vic: I?m sorry?your concept.  I am in agreement with your694
concept.695
Steve: I mean I understand having problems but like let?s696
maintain boundaries (laughter).697
Marc: So what happens is?this is the sun coming in and it?s698
very wavy ? it looks like a nice ruffles potato chip,69
right?it comes and it hits the David and some of it is70
absorbed so its molecule are beating right and some of it701
is reflected in the form of something less wavy.702
Cameron: The problem is, I think, the distinction between energy703
and heat.704
Steve: No, that?s not the problem.705
Tom: But I still don?t understand (Steve: Tom is with me) so706
that the energy from the sun is always striking David and707
some of that is being reflected.708
Marc: That is correct.709
Steve: So David is accumulating energy.710
Marc: At a certain rate-- in one second it gets a certain amount71
of energy in the second second it gets the same amount of712
energy713
[Many voices.]714
Steve: So if it?s always in the sun it?s going to continue to715
collect infinite amounts of energy?716
Tom: It would end up start reflecting as much energy as?717
Marc: Think of it- think of it like a rhythm, ok?  Boom boom718
boom boom right so- it gets energy and it reflects and it719
gets energy and it reflects?so it?s constant?it?s just its720
just it?s just stable.721
Steve: It?s not stable!  [many voices]72
Marc: Then it would not it would not it would not! you?re723
thinking of something that?heat is measured by ticking,724
right? Tick tick tick ok?  what you?re saying is the sun is725
255
hitting it and it?s constantly accelerating because it?s726
constantly getting new energy, right? Tick-tic-tick-727
tktktktktk! Pow!  Right- that?s not what?s728
happening?it?s just tick, and some energy comes in just729
as it?s about to slow down it gets new energy so it?it?s730
that energy that keeps it at that constant rate.731
Steve: I don?t buy equilibrium.732
Marc: Uh uh?73
Steve:  I reject that.  I reject the assumption of equilibrium.734
Vic: Because then David would never ever change735
temperature.736
Marc: It would if it got closer or farther away- lets say that it?s737
traveling.738
Leslie: Let?s keep David still!  Let?s keep David still for now.739
Marc: David is still- David is not moving.  Energy comes in?it740
vibrates, right?741
Leslie: So did David heat up or ?742
Marc: It did just heat up, ok? ? now what keeps it going?  It?s743
the next calorie calorie of energy that keeps it74
going?without that calorie of energy it cools down.745
Some of it some of it set the molecule in motion and746
some of it reflected away.  And then the next calorie747
comes?748
Steve: Wait wait?right before that last- the second burst of749
energy it was at zero again?750
Marc: Well maybe.751
Steve: David?s back to zero.?what you?re saying is energy752
hits?one?half leaves half [half stays] stays?[that?s753
right] and then sort of withers.  [then meanwhile the next754
one comes in].  I reject that too but I understand your75
logic.756
Marc: At least I?ve been heard.757
Leslie: In your system is energy conserved?758
Marc: Yes?because some of the energy has been absorbed and759
some has been reflected and that hopefully should equal760
the same amount of energy coming in.761
Leslie:  Ok?the energy that?s absorbed is it lost forever or does762
it somehow escape back into the universe?763
Marc:  Ah?is it absorbed forever?  Well it?s stored in the764
temperature of this thing!  If I could take that and heat765
water with it, I?d have recovered the energy.76
Vic: Huh?767
Marc: That?s how we?this thing has energy- it?s like it?s like768
it?s like a computer ? it?s memory.  It?s storage.769
Steve: Does that mean it will heat up?70
256
Marc: No ? because it?s not, it?s not, the rate of the vibration of71
the molecule is not increasing infinitely it?s just getting72
energy.  Without that energy73
Steve: Molecules get energy and stop, molecules get energy and74
stop.75
Tom: And why do they stop?76
Marc: Well they reach a stable?you?re assuming that it?s77
preordained that this one , I mean, if the energy?s coming78
in too quickly it will, I mean?it?s not preordained that79
it?s going to tick like this I mean maybe it will tick like780
this?maybe it?s really oneoneoneone? but eventually781
it?s going to be reflecting it at a stable rate.782
Cameron: I don?t think reflecting is the right word.783
Marc: or radiating.784
Cameron: It?s getting rid of energy somehow and part of your785
problem is that you?re assuming is that it?s a linear- it786
keeps getting more heat and it keeps reflect?or getting787
rid of , or excuse me energy, keeps absorbing more78
energy and keeps reflecting energy at the same rate.789
Steve: I don?t understand what?790
[Many voices]791
[Marc simulates being pushed like a pendulum.]792
Marc: Just give me a push that way ok?  Good.  Now I jumped793
all the way here.794
Leslie: Now wait?if you start with some energy, why did you795
slow down and stop over there?796
Marc: Well, ?cause I don?t know?molecular forces? because797
that?s the only amount of energy he gave me!  So come at798
me again?79
Tom: Wait no no no?you slowed down and stopped because80
you were running into something.801
Marc: I was running into something the the the?David has802
intermolecular forces, ok?  So it has?the weak force the803
strong force it has these intermolecular forces, it pulls it804
back right?  So push me again.805
Tom: SURE! [Laughter.]806
Marc:  And I don?t mind?right and if he punches me harder I807
go farther but if he?s punching me at a constant rate I will808
only go the same distance every time, I won?t eventually809
go out the door unless he starts punching me harder?but810
I?m not accumulating energy here I?m going to the same81
spot every time.812
Leslie: So you?re not accumulating energy.813
Marc: That?s right?I?m not accumulating energy.814
Leslie: Then where is the energy going?815
257
Marc: What do you mean?  He?s just sending it out and I?m816
absorbing it and I?m going back and he?s sending out the817
same amount818
Steve: But if you absorb it that means that you keep it [Marc:819
keep it]820
Marc: Well the energy?s going into my running back and forth821
ok?  this is where the energy?s going it?s kinetic energy!82
I?m now running back and forth, right?  But I?m going to823
get tired unless he pushes me again I?m not going to have824
the energy anymore?825
Steve: But the energy doesn?t?I?m no physicist?but I don?t826
think energy just disappears like that.827
Marc: Where is it disappearing? it?s stored in kinetic energy?if828
I?m on a treadmill I can turn a turbine.  If you put me on a829
treadmill you?re using my running energy.830
[Voices.]831
Marc: Wait wait wait- stop a second!  This is where the energy832
is- it?s kinetic energy this is usable energy!83
Amelia: But Marc if you?re still the energy?it?s still in you.834
Marc: I don?t know where that?in the example he?s pushing835
me right?  And you want to836
Leslie: Let?s think of when you turn around?are you moving?837
Marc: That?s potential energy?right?  I?m about to go838
BACK?if you could stop me here you could put a839
treadmill?I have the ability to work the treadmill.840
Vic: I?m having serious problems ? reconciling the metaphor841
has taken on levels of complexity.842
Marc: When you stretch a rubber band is there energy in that843
rubber band?  Yes?it?s the amount of energy84
[Jokes and laughter.]845
Steve: Marc I think you might sit down.  [To Leslie] Can you846
give us a little bit of guidance.847
Leslie:Um?I I sense that the rest of the group disagrees with848
him (Marc).849
Amelia: I don?t.850
Cameron: I agreed with Marc too until he started talking about851
potential and kinetic energy and then I wasn?t buying852
that.853
Vic: I was on the fence.854
Steve: I was so not on the fence.85
Leslie: Can you illustrate in a way that makes sense to856
Marc?and Marc you?re going to have to listen?[I?m857
listenin!] and I want you to be able to repeat Steve?s858
argument back to him, not in Steve?s words, in your own859
words.  [Ok]860
258
Steve: So?David?s in space David is hit by energy [x] x861
amount of energy.  Some of that energy is reflected off862
for some other?in the form of light back somewhere863
else.  The rest of it hits David and is absorbed by David864
and David?s molecules start to move more quickly.865
Tom: So at some point- perhaps when David?s molecules start86
moving quickly enough, he actually starts to radiate867
energy in a different way.868
Steve: But then the question becomes: if he?s, so if only a part of869
that energy is what?s reflected off in the form of light (a870
few jokes) um.. but if the light continues to hit and only a871
part of it goes off then [explodes or melts] is heating up,872
right?  And then my question is, and it?s really just a873
question, because David?s in space, so he?s not874
surrounded by molecules that could be heated, so they?d875
be, so he?d be able to radiate, Marc, how does he, how876
does this rock have the capacity to radiate anything if87
there?s nothing except for light, space and David?there?s878
nothing for him to radiate for heat to be transferred out.879
So?I think, the way I see it is that the light comes in,80
part that?s reflected off, the rest of it sticks in David, and81
because energy doesn?t disappear David just continues to82
heat up and maybe explode.  I don?t know?83
Cameron: What happens when he explodes?84
Steve: I think when he explodes, then there are smaller chunks85
of David.86
Leslie: The moon, so far, has not exploded.87
Tom: I mean things don?t explode so we?re missing something.88
Steve: Right, but there?s somewhere?if David is really in89
space, and so he?s surrounded by a vacuum and it?s a890
perfect vacuum, then there?s no other molecules891
surrounding him that can carry radiant heat.892
Cameron: Ok- I have a question.  Marc said that radiant heat is not893
the only way it can expel energy.894
Steve: Right- exactly--- no and that?s what I?m looking for, is I895
just need some help on where.896
Marc: Let me ask you a question?how did that energy get to897
David?David?s in a vacuum?how did that energy get to898
David in the first place?89
Steve:  I?m not fully convinced that David even heats up because90
he?s hit.  I?ll buy that he?s hit by energy and I?ll buy that901
he sends some of it off [that?s fine] but then does he not902
absorb any of it?  And if he does then903
Marc: He does?he absorbs it and he heats up?he goes from904
some temperature to a higher temperature.905
259
Steve: What?s to stop him from heating?906
Marc: He heats to the point that he heats.907
Steve: That?s teleological.908
Marc: No- but it happens? [tape flips.]909
910
Tom:And the light?David?s not going to produce it in the91
same way.912
Marc: No but he?s going to radiate it in the same way.913
Marc: Yeah.914
Tom: What?s the difference?915
Steve: Why916
Marc: One is generating?the other is, the other is? look. Let?s917
Tom: Well what are we?one is generating?918
Marc: Let?s talk radia? let?s talk energy.  Let?s not talk heat or919
light.  The sun is generating energy.  How does that920
energy travel?  IT travels in a vacuum.  Why can?t the921
energy? David IS generating energy.  He?s generating92
energy because his  molecules ? it?s a different form923
Steve: Only if he?s absorbing the energy.924
Marc:  That?s right?he absorbs it925
Steve: That?s an assumption we?re making.926
Marc: Right, but I?m saying he absorbs it.  We know the earth927
absorbs it, we know the moon absorbs it.  We know that928
molecules in a vacuum absorb energy that?s being thrown929
at them.  That?s how the atmosphere heats up that?s how930
the earth heats up?931
Steve: So only if??932
Leslie:  So let?s say David?s green.  [Ok ok David?s green.] What93
does that mean. Let?s assume it.934
Steve: So he?s absorbs he?s absorbing green935
Marc: He?s reflecting green936
Steve: Oh reflecting green and absorbing everything else.937
Leslie: Everything else- ok.  so now this this is just easier to talk938
about?there?s the green light that?s reflected we don?t939
have to worry about that anymore.  What happens to all940
the other light that he?s absorbing?941
Marc: It?s absorbed by David.942
Leslie: And then what?  He just keeps heating up each time more943
comes in?94
Vic: Ok here?s the thing?945
Marc: You chose a stable color?green is a manifestation of946
stable energy.  He is green degrees warm.  Right?947
Vic: Huh?948
Marc: He will only be green degrees warm.949
260
Leslie: [Points to a green shoe.]  Is his shoe green degrees warm?950
The sole of Cameron?s shoe?951
Marc: It?s a metaphor?it?s a metaphor?they?re always green.952
Tom: It?s the same temperature as the rug (in the room).953
Marc: Well you know what it?s probably not the same954
temperature have you ever had a black surface and a95
white surface?  The black surface is hotter than the white.956
Steve: Only?.957
[Laughter.]958
Amelia: Black cars are hotter than white cars?that?s why people959
who live in deserts drive white cars.960
Tom: We?re talking about things in an atmosphere.  We?re also961
talking about black which absorbs other forms of energy.962
Marc: Yeah- but the analogy, you can see the963
analogy?different colors.964
Tom: I can?t?black, black is like all of the colors, white is965
none of the colors.96
Marc: Ok so green is some of the colors so I can imagine that967
green would be warmer than white.968
[More about colors.]969
970
Leslie: The question about green was to differentiate between971
reflected light, transmitted light, created light, the green972
that we see from David is reflected light ? it never973
did?David never had to worry about it.  He just threw it974
away as soon as it came.  The rest of the light gets in975
somehow and is no longer visible [right].  Ok.976
Vic: It can not be being reflected.97
Leslie: Right?and if it?s absorbed?for a car, does that make it978
hotter?979
Tom, Vic: Sure, yeah.980
Leslie:  Ok?so it?s going to make David hotter.981
Marc: Yeah.982
Leslie: Okay.983
Tom:  But it has to go somewhere because David does not984
Katie: heat985
up indefinitely986
Tom:987
does not  heat up indefinitely unless you [?]98
Marc: But it goes in the form of green light!  That?s where it989
goes!90
Leslie: No?we?ve already taken care of the green light?it?s not91
like he gets brighter as he heats up, it?s still just plain92
old?93
261
Tom: And the light is traveling?there are all these different94
frequencies of light coming at David.95
Marc: That?s right.96
Tom: So where.97
Steve: So every light ? he?s heating up with ROY BIV and98
giving away green, right?  So he?s accumulating ROY99
and BIV.100
Marc: Right.  Ok101
Steve: And he?s accumulating it and accumulating it and102
accumulating it103
Tom: He has to get rid of it some way we just don?t know.104
Vic: So at some point does David become?105
Marc: No but he gives it away in other forms of energy?heat-106
well infrared? Infrared primarily.107
Vic: Wait wait wait you cant you can?t you can?t change ROY108
to infra-ROY.109
Marc: Of course?(pauses)1010
Vic: You can?t you can?t give you can?t change ROY into101
infra-ROY1012
Marc: Well why not?1013
Vic: How does he do it!?1014
Marc: Because it?s ju- because as the rock absorbs the energy1015
the waves slow down ? it?s my frequency? it?s this1016
[does the motion, hitting thing]?the rock absorbs the ba-1017
ba-ba-ba-ba and it absorbs some of it so now it?s just?it1018
isn?t visible anymore our eyes just aren?t able?if you1019
wear infrared goggles.1020
Vic:  I?m sorry I think the problem is that?that in addition to1021
ROY and BIV, we?re already assuming ultraviolet and102
infrared, and let?s say that he gives away all those things,1023
too, in addition to green.  So he?s gotten rid of all of the1024
infra and all of the ultra and all of the gamma1025
Marc: And you?re asking me where does it go?  It goes in the1026
form of that?you can?t dismiss that!1027
Vic: I?m NOT dismissing that!  At all ? what I?m1028
suggesting is that we still have ROY and BIV that we-1029
that he?s not changing it into something else?1030
Marc: But he is?1031
[Voices.]1032
Leslie: Can you come up with another way of explaining ? I?I103
see the discrepancy, and I don?t think you understand the1034
discrepancy and I?m wondering if there?s a way you1035
might explain the discrepancy better?I have analogy I1036
can use ?1037
262
Vic: Wait wait wait?every?everybody is payin? me money.1038
Everybody is paying me money in different forms?in1039
dollars, five dollar bills, twenty dollar bills. [ok] I?m1040
savin? all of my 1 dollar bills that I give away?I do not1041
spend any of my dollar bills on anything ever.  Which1042
means that I am gradually accumulating 1 dollar1043
bills?even if I?m spending it in 5s and 10s and 20s.  so104
what do I do when I end up with 1000 dollars in one1045
dollar bills that I don?t know what to do with?!1046
Marc: I?m gonna change that analogy?or I could keep it I1047
could keep it! Ok?  I?ll keep it fine?you know what I?m1048
gonna do with those one dollar bills? [tell me]1049
Well?those dollar bills become? you you spend 501050
cents of it in terms of heat and you throw the other 501051
cents of it away but we can?t see those 50 cents because1052
we?re only attuned?1053
Tom: You?re losing the analogy.1054
Leslie: The question is: how did that dollar turn into 50 cents.  If105
the only thing you can throw away is coins, what1056
happened to the dollar bills.1057
Vic: I didn?t cut them into the shape of coins and pass them1058
out?I still have I have 1 dollar bills that I cannot break, I1059
cannot change at the bank because there?s no freakin?1060
bank.1061
Marc: But I?ll tell you what happens.  If you really were [?1062
[mocking: I?ll tell you what happens?] if you really really1063
really were ??if you  (someone enters and we applaud)1064
you?d be drownin? in one dollar bills.1065
Vic: This is our point!106
Marc: But my  point is that you can?t stop converting those 11067
dollar bills to 50 cents!  By the virtue of your existence it1068
means you must be (?)1069
Amelia: What happened to exploding in space?1070
Marc: What no no no?o1071
Vic: Our question fundamentally is?is?how in what way did1072
I change my dollars to 50 cents.1073
Marc: I?ll tell you ?1074
Then I interject, Steve notes that it isn?t fun to ?make stuff up? anymore and it feels like
an argument and not a discussion.  We take a break and don?t resume.
263
Appendix G
Transcript 7
The fifth grade classroom in this transcript has been working with solar ovens
over a series of weeks.  In this transcript, they are discussing the relationship between
light and heat.
Student: That you can only heat so it travels that you can?t see1
traveling because it travels and then it attracts the to black2
and it would catch to black it would like to stay onto3
black I guess because it's black.  (Students giggling)4
Student: I think I think I disagree with what I believe in that uh the5
light was exactly the uh?6
Teacher:  So you disagree from what you said earlier, you mean?7
Student: Yeah.8
Teacher: Okay.9
David: because um I I think that the air is like heat because like a10
heater uses the air, it heats it up and the air travels around1
the house to heat up the whole house.12
Student: I dis13
Teacher: And how is that different from what you said before?14
David: Because I said that the heat is like15
Wasolla: I disagree with David because16
Teacher: By the way David, I think that so cool that that you17
thought something else and your ideas changed and you18
were able to express that.  That's really ? that's19
neat?that that I like the way you?re thinking.20
Wasolla: I disagree with David what he said about air being heat.21
Because air can sometimes be cold.2
Student: But23
Student:No but24
Student: Yeah but heater?we have a heater25
Student: Not but that26
Student: Heater, we have a heater27
Student: That?s that?s28
Student: We have a heater thing in our house and you can turn up29
the temperature.30
Student: Yeah, that?s what everybody has31
Student: I. I didn?t say that.32
Teacher: Let?s listen to David one more time, Listen up.3
David: Wasolla, I didn?t say that the air is not?is always hot.  I34
just said air is hot sometimes, so it could turn cold.35
264
Teacher: What made you change your idea?36
David: Because I thought of uh my dad working on the heater.37
Because my dad does heat.  Heaters and air38
conditioners?and then I thought of the word in from the39
air conditioner?air? And then I thought that if air can be40
cold from an air conditioner, then why can?t a heater use41
air to make it hot?42
Teacher: Well the heater uses air and air is what makes the heat?43
David: Yeah.4
Student: I thought is was always gas.45
David: Yeah, because there?s like vents everywhere.46
Teacher:So what do you think it is in the air that is making heat?47
Student: Hmmm.48
David: I think it is the light.  I think the light is like heating the49
air up.50
Student: I always thought it was like a gas or fire or something.51
Teacher: What?52
Student: The heat53
Teacher: The heat, the gas or the fire?54
Student: Yeah, not the light because if you have the dryer my5
mom always tells me to keep stuff away from the dryer56
because fire goes through there and it might catch on fire57
or something.58
Teacher: Hmmm.59
Brian: When I was up in New England, on our big ski trip thing,60
um I slept um on the floor ?cuz there were only two61
bedrooms.  And I was right near the heater and my mom62
told me not to put anything near it or like it was an63
electric heater down on the bottom and my mom told me64
not to put anything near it or inside of the because it?ll65
catch on fire.6
Teacher:Hmmm67
Student: If you put paper near it, you know it can like burn it.68
Teacher: Ok, so there's something going on that can do that.  Um,69
let's get back to let's let's continue where we?re going70
with this idea of light and heat and how heat is created71
and so forth.  I like what you what you're thinking.72
Student: I think that light, this is um I'm kind of changing my73
mind, I think light is kind of light a comet it's going so74
fast like is burning up into heat it's like going so fast it's75
turning into heat.  So when by the time it gets down to76
earth its heat and you sometimes it's heat and not um7
Teacher: What is that what you're saying?78
Student: I think that?s kind of like what Seong is saying.79
Teacher: kinda like what Seong was saying, yeah.80
265
Student:[Unclear.]81
Teacher: How many people think that you get the heat when light82
hits something, like causes heat.83
Student: I do.  I do.  Me.  I do.  [Students mumbling].84
Teacher:What do you think?  Why do you think that?85
Lisa: I think that sometimes, well, most of the times, light is86
not always containing heat.  Like, like this light up here,87
it?s not con?it?s not, its not ?8
Dashawn:Burning?89
Lisa: Yeah, like making you hot.90
Kyle: Yeah it?s not making anything hot.91
Anna: But it's just?what if you go up there and touch it?  It92
would??93
Brian: That's because your finger is an object.  When it hits94
something it?s hot.95
Teacher:Oh, I see.  So you get, there's a reaction when you touch96
light.97
Brian: But it's also a question like, um, if a door slams? if a98
locker door slams and no one?s around to hear it, does it9
make a noise? Because you don't know if? if you don't10
touch it and the light is making heat and making the air101
hot.  You won't know.102
David: Yeah well how is, fine then put it this way how does the103
fire makes you hot when you're not touching it?104
Brian: Because it's warm!105
Student: Because that's a fire.106
David: But it's still, it's  the light source.107
Student: Heat heat.108
Brian: But when you say a lightbulb, a lightbulb doesn't really109
run on fire.  It runs on electricity.10
David: I know!  I thought it  [?] but it?s still a light source.11
Student: I?its because its wood.  Wood catches on fire like paper12
so when you eh?wood?if woods are made out of tree13
trunks.14
Student: Woods?  Woods?15
Student: Wood.  [?]  Tree trunks16
Teacher: Like a piece of wood.17
Student: Yeah, um, it?s made into paper sometimes, and when you18
put that together with like newspaper and you light it, um19
it forms a fire.120
Teacher: So anytime you have light, do you have heat?121
Student: Yeah.12
Student: Sometimes.123
Student: Not always, not always.124
266
Student: I?I have a lightbulb and it gets really hot because when I125
when I put it inside a diff..because it?s a different kind of126
light bulb so it gets really hot and that?s not what usually127
lightbulbs do.  They just get not hot its just..128
Student: There?s different light bulbs for different purposes129
though.130
Student: I think?.  Like outside its its light its light outside but its131
still cold outside, so light doesn?t always get hot, get you132
heat.13
David: Yeah but that that is could be like the angle that the earth134
is at.  You could you never you could always think of135
that.136
Brian: Remember when we learned about137
Student: Yeah but it?s the light is not um solar138
Brian: The light. I mean the sun and the earth139
Student: I kind of agree with [?]140
Brian: I don?t think it?s the angle.  (Teacher: shh..) Remember141
when we learned about light.  I mean, the sun, the earth,142
and the planets, its like, what the an..what angle the earth143
is at.  And what angle the sun is at.14
Student: Well the sun doesn?t turn.145
Brian: What angle the earth is at.146
Student: And sometimes where the moon is.147
Student: We have the earth is rotating in the front.148
Student: Like rotating [?]?149
Student: No it's revolving150
Brian: Like right now we have winter and Asia, right now, or in151
China or (Student: California) over there over on the152
other side of the world, um they're having summer over153
there right now.154
Student: Like Australia15
Student: Yeah156
Students:[Mumbling.]157
Student: California?s not the other side of the world.158
Student: But they have different kinds of life over there at [?]  And159
they do different things and have different colors.160
Student: They can last longer than the other [?]161
Student:No they don?t162
Teacher: So does it have to do with the angle of the lens?163
Student: Yeah, yes.  I agree.164
Teacher: So when we set up our solar ovens do we have to put165
them at a certain angle in order to create heat?16
Student: Yes.167
267
Student: Remember when you were talking about light [?]? and168
you drew a diagram on the board and we had for our169
warm up?170
Student: Well maybe it would be different.  I think the angle171
would be..dot the angle would be the light and the boxes172
right here then the um the foil and right here so it would173
hit the foil and then come inside the um solar ovens.174
Student: Because if it's if the light is facing the other way then175
there wouldn't be, it can't just go around over the foil and176
then hit it and then go back in.17
Teacher: Frankie and then Kyle178
Frankie:[?]  I think it does have something to do with the angle179
because when we were doing that experiment with Miss180
Viglioti it um each angle got a different temperature like181
direct got the highest and then [?] and then the shadow.182
Student:[?]183
Kyle: It depends on how close on how close the heat is to the184
solar oven.185
Teacher: That's kind of similar to what Brian was saying, that it186
depends on how far the Earth is from the sun.  [?]  Ok, I187
notice that some of  you are getting a little antsy.18
What?moving around a little bit.  Um?let?s kind of189
back this up a little bit.  Let?s wrap it up with any final190
thoughts about what we talked about today.191
David: Um..I remember somebody saying that the sun and like a192
light bulb has like different kin..ah..heat like different193
temperatures of heat. But I don't remember who it was.194
So ahh..  But I think that since the sun is so far far away195
from us like the lamp would be like close to the solar196
oven then like they?re around the same temperatures.197
Student: I thought that198
Student: The sun and the in so? and the light, the sun and our19
light bulb would be at the same temperature?20
Student: No, not really.201
David: Like um the sun's really powerful and it hits us, right?202
But it's not on its full capacity because it, the light travels203
all the way over here so if the light bulb is like weaker204
than the sun and it's closer to the uhh solar oven then I205
think that they would be around the same area of206
temperature.207
268
Appendix H
Transcript 8
The following transcript is from a 5
th
 grade classroom.  In a discussion about
falling objects, the students were using the word ?gravity? to explain their reasoning.
Curious about what the students were imagining gravity to be, the teacher asked ?what?s
this word, gravity??
Teacher: What?s this word gravity?  That?s another big word.  We1
use that word a lot.  What does gravity mean?2
Ibrahim: Like, like if I throw this pencil up in the air, like it pulls it3
down with strength.4
Teacher: So gravity is a pull?5
Ibrahim: Yeah.6
Student: No7
Ibrahim: If we didn?t have gravity, then?8
Teacher: Gravity is a pull downward.9
Ibrahim: Yeah10
Teacher: Pulls things down. [Several students start dropping1
pencils] Anybody want to?12
Student: It speeds up.13
Teacher:? agree with that or challenge that?  Theodore.14
Theodore: Like um, gravity is like when you are like in space and15
um then like when you are trying to punch like um,16
somebody and your hand stops like that, because you go17
and go slow, first and second.18
Teacher: You go slow in space?19
Theodore: Yeah, you go slow in space.20
Teacher: Why do you go slow?21
Theodore: Because space like um, space has too much high gravity2
Teacher: Space has a high gravity.23
Theodore: Yeah.  Which which lets you, you walk slow, you punch24
slow, or you can do everything slow.25
Teacher: Um, anybody else? Julie.26
Julie: Remember when he said, the last thing I remember is that27
you said that if you throw a marker in the air, it had the28
possibility of going down or the possibility of going up?29
Only that it would go up a little while then it would come30
back down.31
Teacher: What does that have to do about gravity?32
269
Julie: Huh?3
Teacher: Huh?34
Julie: That, it could (??)35
Teacher: Does gravity making it go up, or does gravity make it36
come down?  What?s happening?  Gravity making it go37
down.  Okay.  Uh, Thimios.38
Thimios: I have two things to say.39
Teacher: Okay40
Thimios: First of all, space has a low type of gravity.41
Teacher: Oh, so you are disagreeing with Theodore?42
Thimios: Yeah, because, right there [written on the board], it says43
gravity pulls downward.  In space, it is like it just flies.4
And about like the circle thing, you can balance it.  But it45
is like, it concentrates on the middle. (??) balance.46
Teacher: So you are saying if it, if it a sphere on a flat surface, it is47
easy to make it move.48
Thimios: Yeah, because it?s concentrated.49
Teacher: Because the gravity is in the center, the gravity is on the50
middle.  But if it was on a curved surface, and it fit51
inside, then what would happen?52
Thimios:  You would have to like pick it up and throw it.53
Teacher: It would be harder to make it move and roll.  Okay54
Thimios: [??]5
Teacher: Theodore:56
Theodore: I disagree with Thimios.57
Students: [laughter]58
Theodore: I saw a movie, a movie where two people were talking59
about space, like um, like um, they look like, what?s the60
name again?61
Students: Astronauts62
Theodore: Yeah, astronauts.  And um, they said that they had a63
really high gravity.  Gravity, like um, like one day I64
watched a cartoon called Jackie Chan Adventures.65
[Students laugh].  Um, They showed when he was about6
to punch a man.  And then his hand moved so slow um67
Teacher: He was in space?68
Theodore: Yeah, he was in space.69
Teacher:  All right, we can talk about this.  Let?s talk about space70
and gravity.  We have two opinions. One, that space has a71
Ibrahim: low gravity72
Teacher: A low gravity, or a less gravity than Earth.  And the other73
one is that er space has a lot of gravity.  And Theodore74
says that space has a lot of gravity and that?s why75
everything moves so slow[ly].  Thimios says that space76
has a little bit of gravity and that?s why if you jump up,7
270
you would just keep going up and up and up.  Right?78
[Thimios shakes his head yes and raises his hand] Let me79
hear someone else before besides these two then I will80
come back to these two. Okay.  What do you guys think?81
Uh, Felix.82
Felix: Well, I think there is less gravity, be, if, if, you throw83
something in the air you know it?s going to come back84
down because gravity um, pushing it down.85
Teacher: Here on Earth?86
Felix: Yeah.87
Teacher: Uh, huh.8
Felix: And if we?re in space, there?s less gravity, we?d be flying89
up and jumping (??)90
Deena: I agree with Thimios just because well that that?s why if91
you threw a pen or anything in the air, it would fly down92
really fast because of the gravity.  It has a high gravity93
here.  And in er, in space, if you jump up, you would just94
like keep going up in the air because you have a low95
gravity.96
Santiago: Uh, I kind of agree with Theodore because like the sun it97
has like enough gravity to keep us from spinning around98
in orbit.9
Teacher: So gravity comes from the sun?10
Santiago: I think, I don?t know.101
Teacher: Well, why do you think gravity, lets, why do think102
gravity comes from the sun?103
Santiago: Because it, keeps other planets from rotating around it.104
Teacher: Okay, so because planets rotate around the sun, then105
gravity must be, the pull must come from the sun.106
Because it [the sun] stays still and the other planets are107
moving?  That?s your theory?108
Santiago: I don?t know.109
Teacher: Fine. Okay.  Lorenza10
Lorenza: I have to agree with all of them, because like11
Teacher: Even, both of them?12
Lorenza: Yeah.13
Teacher: Okay.14
Lorenza: When you walk, you walk slow, but you can talk slow,15
jump and just stay.16
Teacher: Here, on Earth?17
Lorenza: No, in space18
Teacher: Oh.  So, that supports what theory?  That there is a lot of19
gravity or a little bit of gravity?120
Lorenza: Like, like half.121
Teacher: Half? Oh, so not a little bit, not a lot but some.12
271
Lorenza: Yeah123
Teacher: Julie.124
Julie: I agree with, um, (??) if you go up and jump up?125
Teacher: That means a lot of gravity or a little bit of gravity?126
Julie: Um, a little bit of gravity.127
Teacher: Okay.128
Julie: Um, if you go up and (??) up and down, that (??) there is129
still(??)130
Ibrahim: I agree, I disagree with Theo because um, when you?re131
going in space and jump on the moon, you jump back up132
you stay um, um in space you don?t move anywhere, you13
don?t come down. But on Earth, if you try to jump, you134
come straight falling back down, so I think um, space has135
a lower gravity, and and Earth has a higher gravity.136
Teacher: Okay. Cristian.137
Cristian: Um, I agree with Theodore because, um, like, gravity138
makes you slow. Um and (??) in space walking like slow139
and like to the space shuttle, you have to walk slow, and140
like, it?s like (??) it feels like you need to be so strong, it141
feels like it is hard to walk or something.142
Teacher: So what does that mean?143
Cristian: That it has to like be strong to walk?14
Teacher: So you agree with Theodore.145
Cristian: Um, huh146
Teacher: Okay, Khawar.147
Khawar: I think in space there is a low center of gravity because148
like149
Teacher: Low center of gravity?150
Khawar: Well, like less gravity (??) In space when you go up, you151
stay up, but um in Earth, there is more gravity like152
because when you jump up, the gravity pulls you down153
fast, so I agree with him.154
Ibrahim: Um like I think um like um Santiago is right when like he15
said the sun like has it pull the earth and the planets156
together.  But like if you?re, um, um I think that he was157
right because how come the Earth is not moving away158
from the sun.159
Teacher: So because the Earth is not moving away from the sun,160
you say that the gravity is coming from the sun.161
Ibrahim: Um hum162
Teacher: Okay, ahh.  Who did I say was going to be next?163
Lorenza.164
Lorenza: Um, I was going to say that um, I kind of, um I still kind165
of agree with him, but um, but um (??) said, but um, but16
272
um that when we go up in the air (??) I think that we just167
will come down slowly.168
Teacher: You think that he will come down eventually.169
Lorenza: Yeah.170
Teacher: But, it just takes a longer time.  But does that mean more171
gravity in space or less gravity?172
 Lorenza: I am not really sure.173
Teacher: Don?t know, okay.  Thimios.174
Thimios: Can I do a demonstration?175
Teacher: Sure176
Thimios: If gravity is pulling down, right.17
Teacher: Um, hmmm178
Thimios: So if you were doing (??) punching going like that, it179
would just [he makes a slow motion downward punching180
movement] it would go to the floor.181
Teacher: Where?182
Thimios: In space because its, because gravity going downward.183
Teacher: Uh, huh184
Thimios: So it?s like that.  When you try to punch (??) and because185
you?re going so slow, it?s because when you go to space.186
It?s because your weight is divided by three.  So it makes187
you weaker, kind of.18
Teacher: So I would weigh one third less in space, or one third189
more?190
Thimios: One third less.191
Teacher: That?s a good diet.  Let?s go to space.  [laughter] Okay,192
Kevin.193
Kevin: I agree with Theodore, because um, when I was watching194
this cartoon, when the gravity went up, when they were195
playing, it was harder for them to to move faster.196
Teacher: Okay. So.   Because it was harder for them to move, you197
think more gravity.  That gravity makes it harder for you198
to move.  Okay, ahh, lets have Deena.19
Deena: I now agree with Theodore.20
Teacher: You changed your mind?201
Deena: Well, cause look it?s, it?s like, this has low gravity, (??)202
never mind, never mind203
Teacher: Go ahead204
Deena: No, never mind.205
Teacher: Okay, Kenny.206
Kenny: I agree with Theodore because um, um sometimes in207
space, um you weigh lesser than you would, so if it?s like208
um, it?s like as if you um were on a bungee string or209
something like that, and it?s like the moon or something.210
Teacher: What about the bungee string, moon?21
273
Kenny: It makes you like go, it?s like um it makes you jump212
higher, but213
Teacher: What?s it? What makes you jump higher?214
Kenny: The bungee thing or whatever.215
Teacher: You would jump higher in space or on earth?216
Kenny: Space.217
Teacher: Because--?218
Kenny: Because you going to weigh less.  And when you jump219
it?s like you?re going to have times three of what you20
would jump here.21
Teacher: Okay, um Felix.22
Felix: I agree with Theodore and Thimios, because there is just23
a little gravity in space.  And if you um, If you went up in24
the air, you uh would eventually come down because, um25
there?s a little gravity in space.26
Teacher: So just a tiny bit, but not as much as here?27
Felix: Yeah.28
Teacher: Uh, Santiago.29
Santiago: Uh (??) you said like that the sun or something like that230
the gravity pulled in a like a comet, or asteroid (??)231
Teacher: Okay232
Santiago: [??]23
Teacher: So what are you saying about where gravity comes from?234
Santiago: (??)235
Teacher: You still don?t know? But somewhere there?s gravity236
pulling some things in space closer to earth, like asteroids237
and comets and stuff.238
Santiago: Yeah, and like the moon239
Teacher: The moon240
Santiago: Yeah, it like (??) it controls like (??) the waves and so241
that makes like tidal waves, so that means the moon has242
gravity.243
Teacher: So the moon has gravity and that causes waves and tidal24
waves.245
Santiago: I think.246
Teacher: Okay, Julie?247
Julie: Um, I think um that in space, um, when you try to turn248
around in space, it takes a little long, and that?s what249
happens to the Earth when it tries to rotate, that;s why it250
takes twenty-four hours, it takes a day to rotate one whole251
turn and maybe um maybe there?s um that?s why (??) the252
earth has low gravity in the air.253
Teacher: A little gravity254
Student: Low gravity25
Teacher: Low gravity or high gravity256
274
Julie: Um, low gravity.257
Teacher: Theodore thinks there?s high gravity.258
Julie: Then I agree with Thimios.259
Teacher: You?re agreeing with Thimios.  Okay, ahh, Khawar.260
Khawar: You know what Kenny said, he said like he said that you261
could loose weight when you go to space and um that?s262
what he (used or knew).   And he said that he agreed with263
Theodore like when you use something, that Thimios (?)264
Teacher: All right, Thimios.265
Thimios: I just want to say something.  Cartoons are cartoons.26
They are not real life, okay people.267
Santiago: What do you mean they are not real life?268
Thimios: Gravity is the a pull downward.  (??) They?re saying269
gravity, they?re saying gravity is a pull upward!  It is a270
pull downwards.  Those comics are wrong.  They?re lying271
because there is not a lot of gravity.  So they are going up272
and space and they are not falling down.  Because there is273
no gravity there.274
Theodore: I disagree with Thimios.275
Students: [a lot of laughter and noise]276
Teacher: I can?t hear.  Go ahead Theodore.27
Theodore: I saw a movie, ?The Matrix? and how come they do that278
stuff and um279
Students: It?s a movie! (??) Shh!280
Teacher: Lorenza281
Lorenza: I tend to disagree with Theodore because they could be282
using a harness and pulling on wires.283
Teacher: You have seen them do stuff like that in a movie before284
with a harness?285
Lorenza: Yeah.286
Teacher: Deena.287
Deena: I kind of agree with both of them, it?s because like when I28
go in the water it?s like I can carry like when I go into the289
water, it?s like you loose a little weight cause like my290
little sister like her, she really can?t pick me up because I291
am really heavy and when I am in the water, she can pick292
me up.293
Teacher: So what does that say about gravity?294
Deena: I don?t know, it?s like.  I don?t know.295
Teacher: Are you connecting?  Does anyone know why she said296
that?  Rebecca.297
Rebecca:  Um, I think she saying this because, like if you are in the298
water, you are still staying down in the ground, but once29
you are in the water you feel like you are loosing gravity30
of yourself.301
275
Teacher: And what did you say?302
Rebecca: [??]303
Teacher: So being in water is like being in space?  Why?304
Santiago: Because it is more easier to pick up a rock in the water305
and in space.306
Teacher:  So, it is easy to pick up heavy things?307
Deontrae: No it?s easier to pick up heavy things in space.308
Students: (??) water makes it lighter, yeah.309
Teacher: So is being in water and being in space similar?310
Students: Yes. (??)  No.31
Teacher: Ah, hah one at a time, one at a time.  Julie.312
Julie: I think it is similar, um because um, water, you go into313
the water, you could um, you could kind of loose weight.314
You can push the water up.315
Teacher: Why do you say loose weight?  Explain that to me.316
Julie: Because if you like um you?re not, you?re not like on317
Earth.  On Earth you could just touch the water and pick318
it up like this in your hands, and then it will fall out319
though.  But when you?re in the water, you can just push320
the water off you, it?s like, um heavy.  It feels like you321
lost weight or something.32
Teacher: Okay, Santiago then Deena.323
Santiago: Like when you are in the swimming pool and like when324
you reach the bottom and then you can jump up high,325
from the bottom.  That? kind of like space.326
Teacher: Why is that like space?327
Santiago: Ah, You can jump up high.328
Teacher: Because you can jump high.  Ah. Deena.329
Deena: And it is kind of like what Thimios said.  Because it?s30
like.  That?s why it is really hard to walk on, that?s why is31
really hard to walk on water like on beaches. That?s why,32
it?s like sometimes it?s hard for me to walk on the water33
in beaches.34
Teacher: So, it?s hard for you to walk in the water like it?s hard for35
you to walk in space.36
Deena: Yeah, you have to, yeah, yeah.37
Teacher: Ibrahim.38
Ibrahim: I think water, um water, is the same as outer space39
because like when you are walking on the water, it?s like340
gravity is pulling your legs down.341
Deena: Yeah, it?s hard.342
Ibrahim: It?s like, when you?re walking like I went to um Ocean343
City, I went into the water, the water pulled me down.34
Teacher: Um hm.  Okay, pulled you down like what?345
276
Ibrahim: Like, like gravity is pulling me down.  And when I went346
to the sea, close to the sea, the water was pulling me347
down still, kind of.348
Teacher: Thimios.349
Thimios: The water, I think it has like a medium center of gravity,350
because in the water you gain weight.  You know,351
because it is harder to move and everything.352
Teacher: So some people, so you are disagreeing with the people353
who say you loose weight.  You say in water you are354
actually heavier?35
Thimios: Yeah, because you have to, because you have to like if356
you try to walk in the water, you walk like this [he takes357
steps with a long stride].358
Teacher: Like slow motion?359
Deena: Yeah, it?s hard.360
Teacher: Okay361
Student: And (??)362
Thimios:And, you know, it?s hard to pick up things in the water.363
Deena: Actually, it not.  Actually, it?s not.  It?s light.  It?s really364
light.365
Thimios: Well, if you tried to pick up like a big, big rock,36
Deena: It?s light.367
Deontrea: It?s light368
Students: (??)It?s real, real light.  You could pick it up easily, with369
your hands.370
Thimios: But then it wouldn?t weigh so much.  But in the water it371
gains weight.372
Deena: Actually, it doesn?t373
Kenny: No it doesn?t.  It looses weight.374
Students: (??) [Many students talk at once.]  I know.375
Thimios: Water has pressure. Water has pressure. Remember the376
Ranger Rick magazine we read?  It said (??)37
Student: Shh.378
Thimios: Water pressure makes it hard on you. So it?s like you379
have to like use all of your muscles to [many students380
start to talk]381
Teacher: So you are saying because of the water pressure, the382
water has pressure, which makes it hard for you to walk.383
Thimios: Yeah.384
Teacher: Okay, ah.  Felix.385
Felix: Ah, ah, I think that uh space and water, kind of similar386
because except space you would jump up in the air and if387
you?re down in the water sometimes, you would, um float38
back up if you are um, I don?t know.389
Student: Oooh!390
277
Lorenza: Um, I kind of agree and disagree with Thimios because391
like it?s like the rock (??) right, but it would be hard to392
pick it up because the rock, the water gives a lot of393
pressure on the rock.  So394
Teacher: What do you mean the rock looses weight.  Explain that?395
Lorenza: Like, it is kind of like it is lighter, below the water, like396
when you go to pick it up.  It won?t come up because the397
water gives a lot of pressure on it.  (???????)398
Teacher: Khawar.  Julie39
Julie: The reason why the rock in the water because Earth,40
when throw it on the ground, it will, um, land easier401
because of the gravity, but when you throw it on the402
water, it takes longer for the rock to go down to the403
surface.  Because how, how, how heavy the water is and404
how much pressure it is on it.  And because the rock is405
also heavy.  And that the weight of the water, it can just406
push it down. (??)  But on Earth the air is pushing it down407
and the air doesn?t weigh anything, so it will just push it408
down to the ground.409
Teacher: So the rock is pushing So you are saying that the rock410
moves slower in water not because of gravity, but41
because of the water pressure?  Is that what you are412
saying?413
Julie: Um hmm.414
Teacher: Okay, so you are saying that the air doesn?t have a lot of415
pressure so the rock falls quickly.  Santiago.416
Santiago: It?s not really like you are getting heavier, because all of417
the water around you is making you harder to move418
Teacher: So you are not gaining weight.419
Santiago: No420
Teacher: Uh, Ibrahim, Thimios and Theodore.421
Ibrahim: I agree with Lorenza like earlier what she said because42
she said um, like um, um like.  When you?re in space you423
fall down um like it takes a long time to fall down.  So424
like, if you drop something in the water, it?ll try, it floats425
up slowly.   So I think water, and um outer space is426
similar.427
Thimios: Can I do a demonstration?428
Students:Uhh!429
Thimios: Pretend you are a little green person with a little tiny430
body.  Everybody is saying that gravity is here or that431
Earth has a lot of gravity (??)  or gravity is coming down432
to you.  It would be harder to pick something up in the43
water because gravity is pulling you down here, like that.434
278
Teacher: So you are saying not only do you have to pick up the435
rock, but you have to fight gravity pulling the rock the436
down.437
Thimios: Or pulling the rock up438
Teacher: Gravity pulls the rock up?439
Thimios: No, gravity pulls you down but40
Deontrae: Gravity does pull the rock up.41
Thimios: That?s why the rock will be weighing more.   The people42
will be weighing more because, gravity is pulling down43
the rock.44
Teacher: Okay45
Thimios: That?s what everybody is saying (??)46
Teacher: Do you agree with that or disagree with that?47
Thimios: I agree that the gravity is pulling you down.  That?s the48
only thing I agree with these people.49
Teacher: Okay, Theodore.450
Felix: I?m confused.451
Teacher: Oh well, Thimios is saying that when you try to pick up a452
rock, gravity is pulling the rock down, and you?re trying453
to push it up.  That?s why it?s hard.  Because you have to454
fight against the weight of the rock and gravity.  Yes?45
Thimios is shaking his head yes.  Theodore.456
Theodore: Thimios said that the rock that the water makes457
something heavier. How come like when we put a beach458
ball in the pool, and so, and and then we push it down and459
Teacher:It goes under the water.460
Student: Shh461
Deena: Yeah!462
Theodore: It pops up really hard and um it may hit your face like463
pow!464
Teacher: So, it pops up with a great force?465
Felix: Yeah, it does!46
Thimios: It?s because it?s light.  It doesn?t weigh anything.467
Theodore: Basketball too!468
Deena: It has air inside.469
Thimios: A basketball only weighs four pounds. (??)470
Theodore: Four pounds is heavy!471
Felix: Ohh!472
Thimios: [??] only weighs five pounds.  It?s going to jump up and473
gravity is going to pull it down when it is in the air.474
279
Appendix I
Transcript 9
This transcript is from a second grade classroom.  The teacher has brought
magnets and is asking the students how do magnets ?work??  One question related to this
is concerning the ability of a paper clip that is touching a magnet to then pick up another
paper clip.  The transcript begins with a discussion regarding this.
Dalton: Well if it?s third (the paper clip) then, um, it would1
definitely pick the whole thing up.2
Teacher: Why?3
Dalton: Because um it?s um takes a lot of um it would take about4
two magnets to get put together to get- hold a rock up.5
Hold a rock.  And put two paper clips on? and he said6
he put it third and then he put the paper clip on the7
bottom and then picked it up and um um it can?t get very8
much magnetism through it because it?s just two little9
small paper clips.10
Teacher:  Okay, so you?re, are you saying there?s not enough power1
in one of these?12
Dalton: There?s enough, um, electric power in it but um it just13
couldn?t hold up the rock because it doesn?t take very14
much.15
Teacher: Okay.16
Dalton:  It doesn?t, ?cause, um, because um the power the magnet17
takes a little bit of power through each paper clip and um,18
it goes right down into it.  And um, the rock?s heavier19
than it so it would have to fall off.20
Teacher:  Okay so you?re saying that what Calvin described can?t21
happen because the rock is too heavy?2
Dalton: Yes.23
Teacher: But you also said that some of the power or whatever24
goes through the paper clips?is that true?25
Dalton:  Yeah but um if you?ve got a rock, it?s hard to put26
a rock on because some rocks aren?t made of metal.27
Teacher: Okay, Calvin do you want to??28
Calvin: Um? well, when I pulled it up I held it from the magnet29
and the rock so it wouldn?t fall down.30
Teacher: Oh, so you were holding.31
Calvin: The second time when I hold the rock.32
Teacher: Because the first time what happened?3
280
Calvin: Yeah, because the first time all I did was pick the magnet34
up and it went fweep!35
Teacher: Okay.  So when you did the experiment again you held36
the magnet here with the paper clip and then you held this37
rock and there were two paper clips hanging off the38
bottom of the rock and that worked?the paper clips hung39
off of the rock.  Okay, what do you think about that,40
Dalton?41
Dalton: Um it would only be possible if um you got the magnet42
and you stuck the rock on first.43
Teacher: Why?4
Dalton: Because the magnets are smaller than the rock and it, the45
rock takes more power into it so you can put three paper46
clips onto the bottom and pull it up without it dropping.47
Teacher:  Okay so does the rock pull something away from the48
magnet?49
Dalton: Kind of.50
Teacher: Kind of.  Okay.  Renee.51
Renee:  I think there is clay inside of it because clay does pick52
stuff?things, like paper and stuff.53
Teacher: Okay, so if there?s clay inside here, does that help it stick54
to things?  That helps the magnet stick?  Okay, does it5
stick to paper, like clay does?56
Renee: Ummmm.57
Teacher: Okay, so the clay that?s in here that you?re saying is in58
here- what does it help the magnet stick to?59
Renee:  Metal.60
Teacher:  Okay.  So is it?? Why does that work?  Because if I61
have clay and I can stick it to paper, but I can?t do that62
with this?63
Renee:  I mean if the clay was like [?] it could stick to paper64
[Teacher: Okay okay.] And something that?s metal then it65
can only stick to metal.6
Teacher: Okay.  Alright. Go ahead Taylor.67
Taylor:  Um I have a different one.  Yesterday when we were at68
the science Saturday we saw a reaction when this guy69
poured an um really cold water and then he put food70
coloring in it and he um, added? I forget what it was71
called, but when it put it in it got all bubbly?72
Teacher:  Very cool, very cool.  Kind of different from magnets,73
huh?  Okay.  Alright.  How about over here at this table?74
Do you guys have anything you want to add to this power75
idea?  Go ahead Evan.76
281
Evan:  I agree with Taylor about the little magnets inside the big7
magnet.  Because, um, a couple little magnets can stick to78
anything.79
Teacher:  Okay, so you?re ? you?re saying there are little magnets80
in here and if there?s a couple of them in here that?s going81
to make this stick?82
Evan: To most anything.83
Teacher: To mostly anything, okay.  So ? where is this power stuff84
coming from?  Go ahead Carla.  Oh I?m sorry is there85
something else you wanted to say?86
Dalton:  I don?t understand!87
Teacher:  So don?t I!?go ahead?8
Dalton:  Because Evan said um well?I was thinking in my head89
when he said that there?s magnets inside there?what are90
in the other?what are in the magnets that are inside the91
big magnet?92
Evan: Ohhh.93
Teacher: OOOH!  Uh oh!  [laughter]  [talking]  Let me ? let me94
make sure I?m understanding Dalton correctly?let?s put95
this back out on the floor here?hang on, hang on for a96
second here.  Evan says there are little magnets in here97
and you were agreeing with Taylor- right?  And Dalton?s98
question is: Okay, well if there are little magnets in here9
making this work, then what?s inside the little magnets?10
Okay?that?s what he wants to know [so do I!], that?s his101
question to you Evan.102
Evan:  Little pieces of metal.103
Teacher: Little pieces of metal?104
Dalton: Then what?s inside them?105
Student: Yeah- what?s inside the little pieces of metal?106
Taylor: Clay.107
Teacher:  Oh Taylor has an answer, she says clay.108
Student: Clay?!109
Student: And what?s inside of clay?  Play-doh!10
[Laughter.]11
Teacher:  Now wait a minute, wait a minute, we?ve got this magnet12
and we?re breaking it down into smaller and smaller13
pieces, but then how does that make it work?if that?s14
really what in here, we work our way back and way say15
okay: clay, bits of metal, little magnets, big magnet: how16
does that make this magnet work??  I don?t know!?  Ben?17
Ben: I disagree with Evan because last time (??) a piece fell off18
and there wasn?t any little pieces inside.19
Teacher:  Oooohh?. At the very end of class last time I was120
talking to some folks over here and we took two of these121
282
and they went?fwwp- they smashed into each other and12
a piece broke off and Ben is saying he saw that and he123
says there wasn?t a little magnet inside.  Hm.  What are124
you thinking Taylor?125
Taylor:  Because it was just blended in with that magnet126
because most of them are in the same color.127
Teacher:  Oh, so when this broke off?128
Taylor: There (?) the little pieces.129
Dalton:  Then how could they- then how could they be so close130
together that you couldn?t even see the lines in between131
them?132
Taylor: Because they were glued.13
Dalton: Well glue makes ?em fatter.134
Taylor: There was clay on top of em135
Student: Nuh uh clay?s ?136
Dalton:  Nuh uh clay?s not that black!137
Teacher:  Hang on?[Savannah: There?s food coloring.]138
Savannah?s ? with food coloring.  Okay.139
Dalton: Food coloring!140
Student: That would be [unclear: very weird?]141
Teacher: Hang on for a second?Savannah.142
Savannah:  You had um if there were magnets inside the ? those143
magnets it would be really fat!  It would be fatter because14
the little magnets are as thick as that.145
Teacher:  Okay so you?re saying this is too thin to have little146
magnets in it?if this really had little magnets in it it147
would be fatter?  This way?  Okay [nuh uh]148
okay?Calvin.149
Calvin: I disagree with Taylor and because well I think that150
there?s um.  That there?s um just the magnet the magnet151
is just there.  You know?  No other magnets.152
Teacher: So there?s there?s This is it?it?s a magnet, there?s153
nothing inside except ?magnet??the, whatever this is?154
Calvin: And the electricity?s in the magnet.15
Teacher:  And the electricity.  Alright- let?s get back.  Wait a156
minute, hang on for a second?hang on for a second.157
Remind me about the electricity.  I?m going to come back158
to that?yes?159
Student: I am confused with Taylor.160
Teacher: About what?  Be specific what are you confused about.161
Student: What she said before um Calvin was talking?162
Teacher: You mean about the clay inside?163
Student: Yes.164
Teacher: Okay?do you want to add on to that?165
Taylor:  Um?I don?t know what she?s talking about.16
283
Teacher:  Okay.  Hang on?let?s see if we can figure this out.  I?ve167
got hands over here going off too.  Emily?168
Emily:  I have a question [go for it] um um what is169
metal?  If that?s metal ? what?s inside it?170
Teacher: There?s a question: if this is metal then what?s inside.171
What?s inside?anybody have any ideas?  Ben?172
Ben:  The um what?s inside is like magnetite because173
magnetite made magnets so magnetite made magnets.174
Teacher:  Is this made of magnetite? [mm hmm]  What?s175
magnetite?176
Ben: A type of rock.17
Teacher:  So this is a rock?178
Ben: No.  It?s metal, because they made metal out of179
?.different kinds of magnetite.180
Teacher: Okay, so this is metal, and metal?s made out of181
magnetite?okay alright.  Who wants to add to that?182
Student: I agree with Ben because there?s rock inside it.183
Teacher: So there?s rock in here?  This is metal and there?s rock184
inside?  What do you think Emily?185
Emily: How do you know there?s rock inside?186
Teacher:  Renee?  How do you know?  How do you know if there?s187
rock inside?  [I have a question]  Katie?  Not sure?18
Dalton.189
Dalton:  Um?there can?t, there could be some metal on rocks190
because um we have um a park (?) right off of our lane191
it?s inside and I like to pick rocks up and throw ?em  and I192
saw some that had metal on the side of them.193
Teacher: Okay how does that relate to magnets?194
Dalton: Because magnet?it?s kind, because magnets are made195
out of metal and it had stuff kind of like metal.196
Teacher:  Okay so the rocks you found had something kind of like197
metal and that makes them kind of like magnets?198
Dalton: Yeah?it had the same kind that they use for magnets.19
Teacher:  Okay?Michaela?20
Michaela:  How do you know there?s [?] metal inside?201
Teacher: I dunno?that?s a question for you guys?how do you202
know there?s metal inside?  She ? Michaela asked a203
question, how do we know what?s in here?  Yeah?Emily204
how do we know there?s rocks in there.  Who?s got some205
ideas?  Carla?206
Carla: I think there?s metal in there because if there wasn?t207
metal in there it would it [?] metal around the outside and208
it probably wouldn?t pick up metal that well.209
Teacher: So it has to be all metal or it wouldn?t stick?210
Carla: Mostly metal.21
284
Teacher: Mostly metal.  Okay.212
Student: I know there?s clay in it because I went on this website213
about magnets and it said that there?s clay inside.214
Teacher: Hm?okay.  So, go ahead.  Taylor.215
Taylor:  I ? um agree with Ben because that?s a rock it?s just they216
shaped it into a rectangle.217
Teacher: Oh, okay!  so can I go out on the playground and make it218
do the same thing that this does?  [Students: No no no.]219
Dalton:  I?m confused.20
Student: That?s a special kind of rock.21
Teacher: Okay so I have to find this special kind of rock and that22
will work like this does.  How are you confused?23
Dalton:  Because um I?m confused by how do they make the rocks24
square?  They just get a machine and they push it flat and25
they push it sideways??26
Teacher:  Can I toss Calvin?s idea back out here for a second for us27
to discuss?28
Student:I don?t even know what it was?29
Teacher:I do.  He brought up electricity.  He said this is a magnet230
it?s only a magnet there?s nothing in it except electricity.231
Calvin: Electricity and the magnet.232
Teacher: Electricity and the magnet.  Alright, what?s electricity?23
Calvin: What I meant was whenever it attracts metal, we can?t234
see the electricity going to it?except we can feel it since235
it?s pulling.236
Teacher: So if I have two magnets or a paper clip and a magnet?237
Which one?238
Calvin: Either one.239
Teacher: Either one?I can feel something pull it.  Okay and240
you?re saying it is electricity or it?s like  electricity.241
Calvin: It is electricity that pulls them together.242
Teacher: It is electricity.  Okay, so then my question is: what is243
electricity?  Dalton?24
Dalton: [Pointing.] That?s electricity!  The light!245
Teacher:  That?s electricity?alright?is that what?s in here?246
What?s in here?  What?s making this work?247
Dalton: It?s wires!?[Students: Huh? What? Wires]248
Teacher:  Okay?hold on a second I think your classmates are249
asking you what do you mean by wires?  [yeah]250
Dalton: Um they have?you know how they make machine?cut251
a wire, it shocks you when you touch it?  And um I tried252
it with a light bulb in the bathroom.  [laughter]253
Teacher:  Okay?so you hooked a wire to a light bulb?254
Dalton: No, I touched it and it um shocked me.25
285
Teacher: Okay, what does that have to do with my magnet?  Are256
there wires in here?257
Dalton: It?s kind of like it?they used wires to put it in that? why258
um some of them have little holes left from it?I?ve seen259
em.260
Teacher: Okay- so to get stuff into the magnets? [what stuff?] I261
don?t know that?s what I?m trying to figure out.  Wait a262
minute?Dalton, finish your thought?263
Dalton: Um?it?s kind of like it takes the electricity from the264
wires and plugs it in and then there it?s a wire.  A big265
long wire.  And then they push a lever up and hit on and26
then um the electricity flows down from a big huge um?267
I?ve seen on tv?it has this big wall on the top and268
electricity comes down and it goes into the wire like this.269
And the magnet and then um it has um a certain kind of270
electricity.271
Teacher: Okay, so to make this magnet work somebody hooked it272
up to a wire pushed the on switch, stuff came down273
through the wire to the magnet [electricity] electricity274
did?and now the magnet works.275
Dalton: Mmm hmm.  It?s kind of like that?it?s magnetism.276
Teacher: It?s kind of like that?you keep telling me that?it?s kind27
of like that.278
Dalton: Well it?s not kind of like um?electricity.  It?s magnetism279
that flows down?280
Teacher:  Okay, do I have to plug this in to a wire?281
Dalton:  No?when you?re making it.282
Teacher: When I?m making it I have to plug it in. [yeah] and give283
it magnetism. [yeah] okay.  who else has some thoughts284
on that process?Savannah?285
Savannah:  I have two thoughts on that.286
Teacher:  Okay?go ahead.  Be nice.287
Savannah:  Make up your mind already!  Between it kind of has28
something?289
Dalton: That?s hard to do!  [Okay.]  It?s hard to do both at the290
same time!291
Teacher:  Okay so Savannah what?s your question.  Specifically292
what?s your question?what are you confused293
about?you?re saying make up your mind Dalton.294
Between what and what?295
Savannah:  Kind of and it is.  Or it isn?t.296
Teacher:  Okay?so you?re saying? is your question to Dalton: IS297
this electricity or is it kind of like electricity?  [Yeah.]298
Hang on?don?t answer yet?yes?29
286
Student:  I remember that one time in this classroom you did?we30
had two magnets and we put them in front of a car [yeah301
and what happened] and then one side it would push the302
car away and one side would pull it together.303
Teacher:  Oooh?so how?d that happen?304
Student: Electricity.305
Teacher:  Electricity?okay.  [?]  When you said the other side306
pushed it away?one side pulled it towards the other side307
pushed it away.  Okay?so what?s going on with that?308
Emily?309
Emily: Well I am kind of confused with what Dalton said.310
Teacher: Okay- can you specifically ask him?31
Emily:Make up your mind?it?s kind of the same what312
Savannah said, like, well, what what do you think in the313
magnet?because you?re saying two things.  Magnetism314
or electricity?  What?s in there?315
Dalton: I can?t figure it out?she?s asking me questions and I316
make um a good sentences up, so they? they complete!317
Then?that?s what Ms. McRae always tells us to do!  Use318
complete sentences!319
287
Appendix J
Transcript 10
??  what teacher what grade?
Teacher:  Did you and your partner talk about what is happening1
when solar energy heats water?  Or did you not get to that point?2
Student 1:  We didn?t get to that point.3
Teacher:  You didn?t?  So, you all need more time.  Brian?4
Brian:  Me and my partner, we thought that um what Brianna5
said?ahh?disagreed with her, that when?6
Teacher:  So you guys could have a debate back and forth?7
Brian:  But, when water would hit the top of it and spread to the8
bottom, send to put the thermometer at the bottom so it9
take the temperature at the bottom so that would have to10
be the temperature of the whole?of the water.1
Teacher:  Interesting.  Okay.  That?s interesting because so far what12
I?ve heard from the group is that the sun?s rays hit the13
water and travel down towards the bottom of the14
container.15
Student 2:  They spread out.16
Teacher:  I?m sorry, they spread.  I need to use your words.  They17
spread across, right?  That?s what you said.18
Student 2:  They spread all over.19
Teacher:  They spread all over.  So, does anyone have any other20
ideas, or could you clarify that for me?  What?s21
happening?  Anthony?2
Anthony:  Well, I had a new theory, um, when, the um,23
when?since the windows open, wind is coming in and24
it?s cold outside, so that comes in, it makes the room25
colder, so that that makes that room temperature.  Then,26
the sun is further away from the Earth now, so it won?t27
really heat as fast with the [inaudible]?28
Teacher:  Miguel?29
Miguel:  I was thinking that?[inaudible]?from the top of the30
window, it?s pointed up, so solar energy might come and31
go to?32
Teacher:  So, it might reflect off the window?  Back up?3
288
Miguel:  Yeah, and then?34
Teacher:  Chen?35
Chen:  I have lots of things to say?36
Teacher:  Talk loud.37
Chen:  I kinda agree with Miguel, but I actually think that it is38
not really air because if you?re, if the tall container is39
closer to the window, and solar energy might, it might40
have more solar energy than, um, the other container.41
And, I changed my mind and I disagree with him now,42
because the counter over there, it?s not so hot because the43
leaves make some of the shade over there, and the leaves4
are blowing, so some of it, like, solar energy will come45
only like a little bit at a time.  And?that?s it!46
Student 3:  The taller container is partly in the shadow.47
Teacher:   Yes, I am noticing that the sun is not as bright at the spot48
as when we first came in.49
Student 3:  So the, um?50
Teacher:  ?let?s think about it as being the same?what if it was51
just, it was not changing.  So, we?ll still take the52
temperature of those containers, but I don?t really have53
any other places in the room where there?s a constant54
source coming through the window.  So, let?s just5
suppose that the same amount of sunlight was hitting56
both containers the entire discussion that we?re having.  I57
want to get back, one more time, to what is happening58
when solar energy heats water.  Hunter?59
Hunter:  Well, what me and Brian talked about was when the solar60
energy heats the water, the water molecules are used to61
being like cold, and then when the sun heats it, they?re62
not, it?s not used to the heat, and then so the heat gets63
hotter.64
Teacher:  What do you all think about that?  Andy?65
Andy:  Well, I think I get Hunter and I think I agree with him.6
Because, like, um, the sun is heating both containers and67
they pick-up the same amount of heat and they measure68
[?] to be the same temperature.  And, I have something, I69
have, um, my Dad like makes coffee and heat puts water70
in, and puts it on the stove and since the fire comes up71
from the bottom it takes like two or three minutes to heat72
it up.  But, since there is like no fire underneath them, it73
will take a long time for it to absorb the sun, and get it all74
over the two liters.75
Teacher:  So, he?s saying that because it?s not a source as, I guess76
as hot as fire hitting the container right there, it?s going to7
289
take a lot longer for the heat, for the sun to, the water to78
absorb the heat?79
Andy:  Like, for instance, for example, if both thermometers said80
that the water was 73 degrees each in the beginning and it81
will take like one minute to go to 70-- or 75 degrees.  So,82
it?s going to take a long time to get to 100 or 90.83
Teacher:  And you don?t think that the size of the container or the84
shape of the container matters?85
Andy:  No.86
Teacher:  Do you think that they would both heat at the same rate?87
Andy:  Yeah.8
Teacher:  ?or they would both heat the same amount?  Okay.89
We?re gotten some new ideas about what is happening90
inside the container?to the water as the sun?s energy hits91
the container, or it is collected into the container.  Brian?92
Brian:  Um, I agree with Hunter, because since all the, it like93
stays out in the sun, and if it were hot, and more so,94
?[??]95
Teacher:  The water in those containers?  Interesting.  What do you96
all think about the molecules that Hunter?s saying the97
water molecules?what?s happening to those molecules?98
Hunter:  The water starts out cold, and then when the sunrays hit9
it, the water molecules, like, they?re?they?re used to10
being cold, and then the sun hits it, gets used to the heat101
and then the water gets hot.102
Chen:  They?re cold, they don?t really expand and when they?re103
hot, they expand.  When you boil water, it expands and104
there are bubbles on the top.  And, you will notice that105
you start out with maybe half a cup, and it expands you106
get?it expands.107
Teacher:  I noticed that last night when I was making pasta.  I had108
to turn the heat down because it was going to boil over.109
Sara?10
Sara:  Oh.  Well, what I think is happening to the molecules11
when you put it down the molecules that used to the12
water, like Jennifer said, and then, now that the sun?s not13
hitting them as much, now they are kinda like in the14
shade more.  It?s like, now they?re like going back to like15
room temperature cause the sun stopped heating them to16
make them hotter anymore.  The sun?s making, like17
colder.  Right now the shade?s like making it a little bit18
colder.  So, now I guess it?s not quite room temperature,19
but maybe a little less than it would have been if the sun120
had stayed there.121
290
Teacher:  I have one last question.  How does Hunter?s idea, or12
Hunter and Brian?s idea connect with Marianna?s idea?123
Cause we?Marianna?s idea was that the sun?the solar124
energy hits the container, hits the water, and spreads125
across?126
Marianna:  All over.127
Teacher:  ?all over, spreads all over.  As it travels down?  To the128
bottom?  Or not?129
Marianna:  Just spreads all over.130
Teacher:  As it just spreads all over where ever it can.  How do131
those two theories connect, or do they not connect, or132
what do you think?  Go ahead.13
Student 4:  Well, I think?Okay, this is mine, this is what I think.134
When a molecule gets like really hot and absorbs a lot, a135
lot of heat, and then it expands, some of the heat goes to136
the other molecule next to it, and then it goes on and on137
and on until the whole container?s full of heat.  And then138
they keep getting bigger and bigger and bigger.  And the139
container keeps getting hotter and hotter and hotter.140
Teacher:  So that would kinda then explain what Hunter?s?I think141
that kinda goes with Hunter?s?  Do you agree, Hunter?142
Do you feel comfortable with that?  Miguel?143
Miguel:  I agree with cause, um,  [inaudible]?14
Teacher:  Anthony?145
Anthony:  Well, I kinda agree with Marianna because, like, um, the146
molecules when they absorb the sun, it?s like a person147
eating a lot and then getting fat, and then it spreads to the148
next one like a virus.  And then it keeps going until the149
whole thing, until the whole thing?s hot.  But, when the150
sun?s blocking, when the sun gets blocked, it?s like an151
antidote to that virus, and certain, and the molecules get152
skinny again.   And it keeps passing down on and on and153
then it gets cooler again.154
Teacher:  Excellent.15
Student 5:  I kinda believe the same because, since like the water156
molecules are all over, the heat spreads towards the157
bottom, then it?s the same thing, and water molecules158
start to go up?[inaudible]159
Teacher:  Okay, I?m going to give you a piece of paper and I?m160
going to ask you to draw a diagram, with labels, so I can161
really understand it when I read it, of what you think is162
happening to those containers.  And then below it I want163
you to write what you think about which liter container164
will have the higher temperature.  They?re both the same165
amount of water inside.  Or, do you think it?s the same16
291
temperature?  Or do you think it would be the same?167
Even though, yes, our sun is kinda fading away.168
292
Appendix K
Young Students? Analogies
Introduction
The analogies from student scientific discourse presented in this dissertation have
been offered as evidence that analogies are based on schemas and are contradictions to an
expected schema?a story consistent with the choice of the base in generated analogies,
the multiple analogies and constructed analogies, and the same story that Lakoff (1987)
uses to understand categorization.  In the following sections, I present instances in which
young students compare astronauts in space to being in a swimming pool, magnets to
clay, and magnetism to electricity.  In each case, the students display confusion with their
own analogy.  In one case, the student is not able to relate her analogy to the case at
hand?that is to say, she states the analogical base but then cannot relate it back to the
target.  In another case, a student asserts a meaningful analogy but then confuses being
like with being an instance of.  And in the final analogy presented, the student explicitly
grapples with the question of ?it?s like or it is.?  I argue that when these analogies are
analyzed with respect to representational redescription, dual reference theory and
semantic field theory, a story consistent with categorization emerges.  Structure-mapping
and other theories that limit analogies to pairwise analyses of the target and base would
not predict and cannot account for the confusion young students show between
statements of class-inclusion and statements of superordinate-class-inclusion.
293
Space as a Swimming Pool
The following transcript is from a 5
th
 grade classroom.  In a discussion about
falling objects, the students were using the word ?gravity? to explain their reasoning.
Curious about what the students were imagining gravity to be, the teacher asked ?what do
you mean by this word, gravity??  In discussing this, some students claim that there is
more gravity is space?a claim that the teacher (and this researcher) finds surprising.  A
student explains that in space it is difficult to move.  Arguments against more gravity in
space include the astronauts? ability to jump high.  During this debate, the Deena presents
an analogy to reconcile the competing ideas but cannot explain how this analogy relates
back to gravity or the arguments that preceded:
Deena:  I kind of agree with both of them.  It?s because like when I go in
the water, it?s like I can carry like when I go into the water.  It?s
like you lose a little weight.  ?Cause like my little sister, like her,
she really can?t pick me up because I am really heavy and when I
am in the water she can pick me up.
Teacher:  So what does that say about gravity?
Deena:  I don?t know.  It?s like? [pause] I don?t know.
Teacher:  Are you connecting?  Does anyone know why she said that?
Rebecca:  Um, I think she?s saying this because, like if you are in the water,
you are still staying down on the ground, but once you are in the
water you feel like you are losing gravity of yourself.
Teacher:  So being in water is like being in space?  Why?
Santiago:  Like when you are in the swimming pool and like when you reach
the bottom and then you can jump up high, from the bottom.
That?s kind of like space.
Teacher:  Why is that like space?
Santiago:  Ah?you can jump high.
To understand this passage, in which Deena is able to draw a relevant and significant
analogy, but is unaware of her own reasoning, saying ?It?s like?I don?t know,? I turn to
Karmiloff-Smith?s theory of representational redescription.
294
In Beyond Modularity, Karmiloff-Smith develops what she terms a
?representational redescription? model of cognitive development.  In this model, she
identifies four levels of representational format.  These are Implicit, and Explicit 1, 2 and
3. (I, E1, E2 and E3)
At level I, representations are in the form of procedures for analyzing and
responding to stimuli in the external environment? Information embedded in
level-I representations is therefore not available to other operators in the cognitive
system? A procedure as a whole is available as data to other operators; however,
its component parts are not?
The E1 representations are reduced descriptions that lose many of the details of
the procedurally encoded information.   As a nice example of what I have in mind
here, consider the details of the grated image delivered to the perceptual system of
a person who sees a zebra.  A redescription of this into ?striped animal? (either
linguistic or image-like) has lost many of the perceptual details.  I would add the
that the redescription allows the cognitive system to understand the analogy
between an actual zebra and the road sign for a zebra crossing (a European
crosswalk with broad, regular, black and yellow stripes), although the zebra and
the road sign deliver very different inputs to the perceptual system?
The RR [representational redescription] model posits that only at levels beyond
E1 are conscious access and verbal report possible.  At level E2, it is
hypothesized, representations are available to conscious access but not to verbal
report (which is possible only at level E3)? we often draw diagrams of problems
we cannot verbalize?
At level E3, knowledge is recoded into a cross-system code.  This common
format is hypothesized to be close enough to natural language for easy translation
into statable, communicable form.
These levels have been empirically verified.  In one experiment described in Beyond
Modularity, children were shown playrooms of a boy doll with a single toy car and a girl
doll with three toy cars.  The children were asked to which doll the experimenter was
speaking if she said ?Lend me the car? or ?Lend me a car.?  By age 3, children have
achieved behavioral mastery at this task, correctly assigning ?the car? to the boy doll
(who has one car) and ?a car? to the girl doll (who has multiple cars).  However, around
295
age 5 and 6 these children start to make mistakes.  Karmiloff-Smith argues that
?representational redescription of each of these procedures into the more explicit E1
format makes it possible to link the common phonological form across the two
representations of form-function pairs? In comprehension they start to make mistakes as
to which of the two functions (numeral or indefinite reference [?one? or ?one of?]) is
intended.? (p 57)  That is to say, at the I-level students have a ?black box? algorithm for
arriving at an analogical map: they have a schema but aren?t aware of that schema, are
not aware of the instances from which that schema was derived, and are not able to pick
apart how a new item (in this case outer space) is judged to be a member of that schema.
Even more startling and provocative evidence for the implicit level comes from the
reports of the children as to the reasoning they report for making their choice:
the youngest subjects, even though they must have used  the contrast between the
articles to make their correct guess, explain this on the basis of real-world
knowledge, saying something along the lines of ?You must have been talking to
the boy, because boys like cars? Later in development, children explain their
correct guesses by referring to the contextual features?for example, ?You were
speaking to the boy, because he?s got one car.?  It is really rather late in
development, around age 8 or 9, that children make explicit reference to the
linguistic clue that all children must have in fact used. (p 59)
When the teacher asks ?Does anyone know why she said that?? she is, unwittingly
perhaps, adopting the representational redescription model of students? levels of
explicitness in their knowledge.  Deena is able to draw a correct and relevant analogy, but
is unaware of her own reasoning, saying ?It?s like? I don?t know.?  Rebecca, in hearing
the analogy, is able to abstract slightly?equating the feeling of being in water to ?losing
gravity.?  And Santiago relates this to space because in both places ?you can jump high.?
296
This raises an interesting question regarding my definition of analogy as a
conscious choice?a deliberate reclassification of the object, subject to a new cognitive
model (which may be interpreted as a distinct ?form of procedures for analyzing and
responding to stimuli in the external environment? (from Karmiloff-Smith, quoted
above)).  At what level of representational redescription are students able to make
conscious decisions regarding their response to stimuli and deliberately reclassify
objects?
Magnets as clay, magnets as electricity
The children in the above transcript are in the 5
th
 grade?placing them at an age
where Karmiloff-Smith observed linguistic abilities to be at the E2 and E3 levels.  In the
following transcript, I present data from a 2
nd
 grade classroom.  These students are
between the ages 7 and 8 and, according to Karmiloff-Smith, less able to reason explicitly
about concepts.  Here this is evidenced by the confusion the students have using the
analogies that they create.  In this transcript, the students are discussing what?s ?inside? a
magnet that makes it behave the way it does (lines 48 ? 56).
Renee:  I think there is clay inside of it because clay does pick
stuff?things, like paper and stuff.
Teacher: Okay, so if there?s clay inside here, does that help it stick to
things?  That helps the magnet stick?  Ok, does it stick to
paper, like clay does?
Renee: Ummmm.
Teacher:  Okay, so the clay that?s in here that you?re saying is in
here? what does it help the magnet stick to?
Renee:  Metal.
And later in the conversation (lines 92 ? 97):
Dalton:  Because Evan said um well?I was thinking in my head
when he said that there?s magnets inside there?what are in
297
the other?what are in the magnets that are inside the big
magnet?
Evan:  Little pieces of metal.
Teacher: Little pieces of metal?
Dalton: Then what?s inside them?
Student: Yeah- what?s inside the little pieces of metal?
Taylor: Clay.
Teacher:  Oh, Taylor has an answer, she says clay.
And following up on these comments (145 ? 151):
Student: I am confused with Taylor.
Teacher: About what?  Be specific what are you confused about.
Student: What she said before um Calvin was talking?
Teacher: You mean about the clay inside?
Student: Yes.
Teacher: Okay?do you want to add on to that?
Taylor:  Um?I don?t know what she?s talking about.
There is something reasonable in Renee?s statement?although neither she nor Taylor
can articulate it and other students may not see it.  There are a limited number of things
that can stick to, say, your refrigerator and not fall down.  Clay and magnets have crucial
differences, of course, but more intriguing here is that Renee explains her reasoning that
clay is inside magnets by stating: ?clay picks stuff up.?  Is she thinking that there is, quite
literally, standard modeling clay inside the magnet?  Or is it more plausible that she is
using clay as a prototypical member of the category of things that can ?pick stuff
up??but do so in a way that does not fit with our standard p-prim of carrying?  For while
she uses the term linguistically in the first sense, if we recognize her knowledge to be at a
less explicit level than language can effectively express, it is reasonable to believe that
she ?means? the second sense.  Taking into account that languages often name categories
after their prototypical or prevalent members (i.e. a Xerox machine for any copy
machine), I contend that Renee is creating an analogy between the magnet and clay,
298
assigning them to a category that she is terming ?clay? because clay is an everyday object
to a 2
nd
 grade student.  The students (including Renee and Taylor, proponents of the clay
?theory?) do not recognize that clay is serving as the name of a category and not the thing
itself and argue against the clay-theory by stating that ?clay?s not that black? and
suggesting perhaps the blackness is due to ?food coloring? (again, ?food coloring? may,
to a young student, be a fairly standard way of changing the color of objects and so is
used here as a way of signifying that color must be added?not necessarily food
coloring).
When the base of an analogy (termed the vehicle in the context of metaphor) is
used as both an exemplar and as an ad hoc name for a category, Glucksberg et al. (1997)
all this linguistic move ?dual reference.?  As an example of dual reference,
the phrase ?a responsibility is a shackle? can be used to refer to the concrete,
physical device that is made of metal, often has chains, can be locked around
someone?s arms and legs, and so forth, and it can also be used to refer to the
abstract, general category of constraining entities.  We refer to such abstract,
general concepts as attributive categories.  (Glucksberg et al 1997 p 334)
The authors claim that ?nouns can be used to make dual reference whenever a
superordinate category has not been lexicalized and a category exemplar is available that
is prototypical of that category.?  What I hope to have demonstrated with the above data
on ?clay? is that this dual reference is an ability that young students have, but, consistent
with findings from representational redescription do not have access to the ?duality? of
the reference: is a magnet a member of a superordinate category typified by clay or is it a
member of a category of things that are clay?
The question of ?being like? versus ?being? is addressed in the same magnets
conversation in a different context.  While Renee has identified clay as an appropriate
299
analogy because of the ability of both clay and magnets to pick things up, other students
are curious about the ability of a magnet to exert a force through things and one suggests
electricity as an analogy (lines 212 ? 214).
Teacher:  ?He said this is a magnet, it?s only a magnet, there?s nothing in it
except electricity.
Calvin: Electricity and the magnet.
After asked to explain, Dalton jumps in with a theory and the teacher summarizes his
comments (lines 251 ? 271):
Teacher: Okay, so to make this magnet work somebody hooked it up to a
wire, pushed the on switch, stuff came down through the wire to
the magnet?
Dalton: Electricity.
Teacher: Electricity did?and now the magnet works?
Dalton: Mmm hmm.  It?s kind of like that?it?s magnetism.
Teacher: ?It?s kind of like that??you keep telling me that??it?s kind of like
that.?
Dalton: Well it?s not kind of like um?electricity.  It?s magnetism that
flows down.
Teacher:  Okay, do I have to plug this in to a wire?
Dalton:  No?when you?re making it.
Teacher: When I?m making it I have to plug it in. [Dalton: Yeah.] and give it
magnetism. [Dalton: Yeah.]   Okay.  Who else has some thoughts
on that process?Savannah?
Savannah:  I have two thoughts on that.
Teacher:  Okay?go ahead.  Be nice.
Savannah:  Make up your mind already!  Between it kind of has something?
Dalton: That?s hard to do!  [Teacher:  Okay.]  It?s hard to do both at the
same time!
Teacher:  Okay so Savannah what?s your question.  Specifically what?s your
question?what are you confused about?you?re saying make up
your mind Dalton.  Between what and what?
Savannah:  Kind of and it is.  Or it isn?t.
The students recognize that there is a fundamental difference between the magnet being
electricity and it being like electricity, but Dalton is still unsure why he claims it might be
electricity in the magnet.  Were this an unreasonable assertion with no cognitive
300
?meaning? in his statement, it would be indecipherable to a scientist.  There is logic to his
statement.  Both are instances of the schema of action-at-a-distance?something rarely
seen in daily activity (and not shared by clay?at least not on visual distance scales).
And in this schema the property responsible for that action may have ?electricity? as a
prototype.  (Dalton, after suggesting electricity, is asked what electricity is and relates an
experience with being shocked.)  In fact, in a true scientific description of the
phenomena, electricity and magnetism are manifestations of the same electro-magnetic
force, and any motion produced by a magnet is due to the electrical manifestation of that
force.  Dalton is onto something?but his grasp is at the E-1 level of representational
redescription.  In essence, when Dalton states ?they used wires to put it [electricity] in.
That?s why, um, some of them [the magnets] have little holes left from it?I?ve seen
?em,? or when Renee claims that she ?went on this website about magnets and it said that
there?s clay inside? they are doing the same thing as the child who explained why the
experimenter must have been referring to the boy doll by saying: ?You must have been
talking to the boy, because boys like cars.?  Because these students don?t have access to
their own reasoning, they have to invent a plausible reason that is, in fact, less plausible
than the actual reason that they must have had, but not had access to.  This lack of access
to one?s own reasoning echoes diSessa?s findings (also reported in an chapter 5)
concerning a problem in which students are presented with bells made of the same
material, same length, same height, but varying widths.  Almost without exception
students predict (erroneously) that the thicker bells will have a lower pitch.  Though some
have cognitive access to the schema they are using to predict this and can refer to
301
particular instances from which this schema has been abstracted, not all do.  DiSessa
reports:
Although most subjects were ready with analogies?church bells compared with
jingle bells, xylophones, musical instruments of various sizes?I was struck that
some initially could not produce any example of the phenomenon they identified
to be at the root of the situation.  This, along with the rapidity and expressed
certainty of responses, heightened my confidence that a p-prim (or several) was at
stake rather than analogy.
I think these youngest students have trouble separating the schema from the instance (that
is clay from things-that-stick?a category typified by clay); but later (older students) have
trouble putting them back together (as with bigger-means-lower and the xylophone).
302
Glossary
Analogy: The claim of likeness where it is not routinely or automatically conceived.
Analogy includes similes and metaphors.  I define analogy to entail the choice of
an alternative cognitive model.
Category: Commonly considered ?a group or set of things, people or actions that are
classified together because of common characteristics.?  Known not to possess
rigorous propositional structure (i.e., they are not always grouped by common
characteristics), it has been suggested (Lakoff, 1987) that the boundaries between
categories are a result of cognitive models.
Cognitive model: ?Can be viewed as ?theories? of some subject matter? (Lakoff, 1987).
May be an ad hoc construction, a kind of ?story? of how the world works in a
particular situation.  Our cognitive models define our categories (for example, it
only makes sense to differentiate a bachelor from a non-bachelor when you have a
cognitive model in which people reach a marriageable age and enter monogamous
long-term relationships).
Metaphor: A figure of speech in which two things are compared, usually by saying one
thing is another, or by substituting a more descriptive word for the more common
or usual word that would be expected. Some examples of metaphors: the world's a
stage, he was a lion in battle, drowning in debt, and a sea of troubles.  As noted by
Glucksberg and Keysar (1990) one may note similarity by saying that copper is
like tin, but not the metaphorical construction copper is tin.  Metaphor, then, is
used as a comparison between two categorically related items when no commonly
accepted name for the category exists (for example, copper is metal).
Phenomenological Primitive: ?The intuitive equivalent of physics laws; they may
explain other phenomena, but are not themselves explained with the knowledge
system? (diSessa, 1993).  Examples include ?closer is stronger? (as an
explanation for why it is hotter closer to a fire), ?blocking? (as an explanation for
why a car is stopped when it hits a wall), and ?supporting? (as an explanation for
why a book is held up by a table).  These are basic stories that convey an
elemental sense of mechanism.
Prototype:  Defined operationally by Rosch, this is a category member that is easily
learned, easily identified, can be used to generalize about the entire category, and
comes to mind most quickly as a category exemplar.  Barsalou points out that
prototypes are a cognitive behavior and not cognitive structure.  Lakoff points to
the role of cognitive models as a possible cognitive mechanism for the prototype
structure of categories.
Schema: Short ?scripts? or stories that we have about the world and the way it works:
event schemas that are abstracted from our experience of certain events, image
schemas that provide structure for conceptualizations ? ?schemas of intermediate
abstractions [between mental images in abstract propositions] that are readily
imagined? (Palmer, 1996 p. 66) ?  and proposition schemas: abstractions that act
as models of thought and behavior and specify ?concepts and the relations which
hold among them.?  (Quinn, 1987)
303
Structure-mapping: ?The central idea is that an analogy is a mapping of knowledge
from one domain (the base) to another (the target) such that a system of relations
that holds among the base objects also holds among the target objects.  In
interpreting an analogy, people seek to put the objects of the base in one-to-one
correspondence with the objects of the target so as to obtain the maximal
structural match? Thus, an analogy is a way of aligning and focusing on
relational commonalities independently of the objects in which those relations are
embedded.? (Gentner and Jeziorski, 1993)
304
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Curriculum Vitae
Leslie Atkins
Department of Physics
Department of Education and Curriculum
University of Maryland
Education:
Ph.D, December 2004
Analogies as Categorization Phenomena: Studies from Scientific Discourse
University of Maryland
M.S., Physics, 2004
University of Washington
B.S., Physics, 1998
University of Virginia
Memberships:
Cognitive Science Society
International Society of the Learning Science
American Physical Society
American Association of Physics Teachers
Fellowships:
Graduate
NSF G/K-12 Fellowship Program
PRIME: Partnership for Research in Math-science-engineering Education
University of Washington, 2000-2001
Undergraduate
NSF Research Experiences for Undergraduates
University of Washington
Eotwash Group, Eric Adelberger
1996
NSF Research Experiences for Undergraduates
University of Virginia
Neutron Spin, Donal Day
1997
313
Teaching Experience:
University of Maryland
Physics 115, Fall 2004 (instructor of record)
Physics 115, Spring 2003 (co-instructor)
Physics 121 (TA)
The Governor?s School of North Carolina 2003, 2004
Physics
Self & Society
University Preparatory High School
10
th
 grade Environmental Chemistry, 2002
University of Washington
Lead Teaching Assistant, Physics 121, 122, 123. 1998, 2000
Physics 321 Lab TA.
Seattle Girls? School
Mentor
Garfield High School
Saturday-School tutor
Workshops:
Co-led:
Analysis of Student Inquiry in Group Conversations About Physical Science at NSTA 2003.
Building Capacity for Professional Learners.  Montgomery County Public Schools, 2003.
Workshop in physical science for in-service teachers.  King County Public Schools.  2001, 1999.
Summer Institute in Physics and Physical Science.  University of Washington, 1998.
Participant:
Doctoral Consortium.  International Conference on the Learning Sciences. 2004.
National Outdoor Leadership School, New Zealand.  2004.
Public Conversations Project: Inquiry as Intervention.  2003.
Preparing Future Faculty: Teaching and Learning in Higher Education. 2001.
Publications:
L.J. Atkins.  Phenomenological Primitives as Idealized Cognitive Models: A Basis for Graded Structure in
Scientific Categories.  Proceedings of the 2004 EDCI Symposium.
Atkins, L.J.  Student Generated Analogies in Science: Analogy as Categorization Phenomenon.
Proceedings ICLS 2004.
G.T. Seidler,  L.J. Atkins,  E.A. Behne,  U. Noomnarm,  S.A. Koehler,  R.R. Gustafson,
and W.T. McKean,   ?Applications of synchrotron x-ray microtomography to mesoscale materials.?
Advances  in Complex Systems.  Vol. 4, issue 4.
Atkins, L.J., E. Behne, G. Martinez, J. Rensberger, and G.T. Seidler.   X-ray microtomography of the
vascular canal network in permineralized Triceratops bone.  Advanced Photon Source User Activity
Report.
G.T. Seidler, E. Behne, L.J. Atkins, and A. Rendahl.  X-ray microtomography study of a model crumpled
membrane.   Advanced Photon Source User Activity Report
314
Presentations:
An Experimental Study of Bond-Orientational Order in Random Dense Packings of Spheres.
Atkins, L.J., Martinez, G., Seeley, L.H., and Seidler G.T.
APS March Meeting 2001
Functional vs. Structural Analogies: A Conceptual or Epistemological Basis?
Atkins, L.J., Hammer, D. and Hutchison, P.
127th AAPT National Meeting: Madison, WI August, 2003.
Phenomenological Primitives as Idealized Cognitive Models: A Basis for Graded Structure in Scientific
Categories.
Atkins, L.J.
Graduate Student and Teacher-Researcher Symposium, April 2004.
Student Generated Analogies in Science: Analogy as Categorization Phenomenon.
Atkins, L.J.
International Conference on the Learning Sciences, June 2004.
A graduate physics course in physics education research for physics education graduate students.
Scherr, R.E. and Atkins, L.J.
128th AAPT National Meeting, August 2004.