? 1982 
REVA RUTH BRYANT 
All Rights Reserved 
EFFECTS OF TEAM-ASSISTED INDIVIDUALIZATION 
ON THE ATTITUDES AND ACHIEVEMENT 
OF THIRD, FOURTH AND FIFTH GRADE 
STUDENTS OF MA THEM A TICS 
by 
Reva R. Bryant 
'' I 
Disserta tion submitted to the Faculty of the Graduate School 
of the University of Maryland in partial fulfillment 
of the requirements for the degree of 
Doctor of Education 
1981 
1 0plj I 
APPROVAL SHEET 
Title of Thesis: Effects of Team-Assisted Individualization on the 
Attitudes and Achievement of Third, Fourth and 
Fifth Grade Students of Mathematics 
Name of Candidate: Reva R. Bryant 
Doctorate of Education, 1981 
Thesis and Abstract Approved: 
Dr. Martin L. Johnson 
Associate Professor 
Early Childhood-Elementary Education 
Date Approved: 
ABSTRACT 
Title of Dissertation: 
Effects of Team-Assisted Individualization on the Attitudes and Achievement 
of Third, Fourth and Fifth Grade Students of Mathematics 
Reva R. Bryant, Doctor of Education, 1981 
Dissertation directed by: 
Dr. Martin L. Johnson 
Associate Professor of Early Childhood-Elementary Education 
The philosophy of individualized instruction has been embraced by many school 
systems in the United States. However, research has not consistently supported 
claims that this approach will increase academic achievement. Teachers have lodged 
multiple complaints related to demands and problems imposed upon them and their 
students as a result of implementing individualized programs. 
An attempt to alleviate the problems inherent in existing individualized programs 
has resulted in The Johns Hopkins University staff's development of Team-Assisted 
Individualization (T AI)--an approach based on the modification of a researched and 
widely used group-paced model of instruction with cooperative learning teams as 
one component. 
This eight-week study evaluated the effects of three treatments on the achievement 
and attitudes toward mathematics class of 504- students in grades three, four and 
five. TAI combined student team learning and individualized instruction. Rapid 
Progress Mathematics (RPM) students used the exact materials as TAI students, 
but omitted the team component. This treatment was included to determine whether 
any effects of the program were due to the combination of teams and individualized 
instruction or only to the materials and procedures. Control students were instructed 
with traditional materials and procedures. 
Results of the standardized mathematics test showed significant effects for 
treatment and grade. After adjustment for pretests, the treatment effects were 
in the order TAI > RPM > Control. Effects approaching significance were also found 
on the diagnostic test, but on this test the order of the treatments was TAI> Control > 
RPM. Results of the attitude scale clearly showed that TAI and RPM conditions created 
more positive attitudes than did the control classes, but there were no differences 
between TAI and RPM. 
Further research will be needed to assess and to adapt these methods for use 
over a longer time period and to clarify the relative contributions of the team component 
and the individualized instruction component of the TAI program. However, this 
study documents the effectiveness and practicality of combining team learning and 
individualized instruction. 
DEDICATION -
To my mother, Eva; 
my two sisters, Rose a nd Esther; 
and to my son, Michael--with love. 
11 
ACKNOWLEDGMENTS 
Special acknowledgment is directed to the following without whose assistance 
and cooperation this dissertation would never have been completed. My gratitude 
is expressed to: 
Dr. Martin L. Johnson of the University of Maryland who, as my advisor, was 
al ways supportive, accessible and a genius regarding his perception of where I needed 
guidance; 
Dr. Robert E. Slavin and Marshall Leavey of The Johns Hopkins University, 
two of the three authors of the TAI program who guided and advised me throughout 
the project and whose staff spent untold hours on the many project details that needed 
attention; 
Students, teachers and principals of the Howard County Public School System 
without whom there would have been no project; 
Dr. Maurice Kalin, Director of Program Evaluation, Research and Development 
for the Howard County Public Schools, who approved the project for use in the schools 
and who offered guidance and support throughout; 
Dr. John P. Madison, Dr. Frank Milhollan and Dr. Robert Wilson of the University 
of Maryland, additional committee persons whose advice and suggestions strengthened 
the dissertation; 
Kathy Lamm, a friend who did a masterful job of typing and proofing this document; 
and, 
To my many friends who provided continuous support and encouragement throughout 
the Doctoral program. 
- Reva R. Bryant 
iii 
TABLE OF CONTENTS 
CHAPTER ONE 
The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 
CHAPTER TWO 
Review of Relevant Literature . . 9 
The Johns Hopkins University Programs . 26 
Summary .. . 28 
CHAPTER THREE 
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 
CHAPTER FOUR 
Results of the Study . . . . . . . . . . . . . . . . . . . . . . . . . 46 
CHAPTER FIVE 
Conclusions and Discussion . 54 
APPENDICES . 64 
REFERENCE LIST 100 
BIBLIOGRAPHY . 108 
lV 
LIST OF TABLES 
1. Outline of Research Design . 4-3 
2. Means and Standard Deviations, Grade Three 
Achievement and Attitude Measures 4-6 
3. Means and Standard Deviations, Grade Four 
Achievement and Attitude Measures ....... . ..... 4-6 
4-. Means and Standard Deviations, Grade Five 
Achievement and Attitude Measures .... ..... . 4-7 
5. Means and Standard Deviations, Full Sample 
Achievement and Attitude Measures 4-7 
6. Individual Comparisons, Grade Three, RPM vs. Control 4-8 
7. Individual Comparisons, Grade Three, TAI vs. Control 4-8 
8. Individual Comparisons, Grade Three, TAI vs. RPM . 4-9 
9. Individual Comparisons, Grade Four, RPM vs. Control 4-9 
10. Individual Comparisons, Grade Four, TAI vs. Control 4-9 
11. Individual Comparisons, Grade Four, TAI vs. RPM 50 
12. Individual Comparisons, Grade Five, RPM vs. Control 50 
13. Individual Comparisons, Grade Five, TAI vs . Control 50 
14-. Individual Comparisons, Grade Five, TAI vs. RPM 51 
15. Analysis of Variance, TAI vs. RPM vs. Control 
Comparison of Performance on Pretests . . 51 
16. Analysis of Covariance, TAI vs. RPM vs. Control 
Posttest Comparisons on Achievement and Attitudes 52 
17. Analysis of Covariance, TAI vs. RPM 
Achievement and Attitudes . . . . ... .. . . . . 52 
18. Analysis of Covariance, TAI vs. Control 
Achievement and Attitudes . . . . . . .... .. . . .. . 52 
19. Analysis of Covariance, RPM vs. Control 
Achievement and Attitudes . . . . 53 
20. Reliability Estimates Test-Retest Correlations 53 
21. Test Correlations, CTBS and Diagnostic 53 
V 
CHAPTER ONE 
The Problem 
Introduction 
Never before have so many educational issues been in the forefront. Publics 
are questioning, accusing and suing. Budgets are being stripped. Students are 
staging walkouts. Drugs and crime are in the elementary schools. Vandalism is 
costing taxpayers millions of dollars each year. Family crises are increasing and 
have adversely affected students' emotional well-being. Child abuse is increasing. 
Each of these occurrences has an impact upon student learning. In addition to 
these major issues, pressure to meet the diverse needs of students from different 
cultures and to provide appropriate instruction for mainstreamed students as mandated 
in PL 94-142, as well as average, talented and gifted students, have added to the 
classroom teacher's responsibilities and taxed his or her energy level to the breaking 
point. In today's educational system it is not at all clear how these diverse issues 
can be addressed to the satisfaction of all parties. 
In a summary of six reports on mathematics education between 197 5 and 
1980, Suydam ( 1981) points out that teachers frequently do not differentiate instruction 
to meet individual needs. Of ten, the same techniques and the same materials 
are used with the entire class. Whole-class instruction need not preclude attention 
to differing needs, but most teachers make few accommodations to provide for 
individual differences. This is a sad commentary when one analyzes the broad 
range of needs that exists in every classroom. 
One method of providing for individual needs in the classroom is "ability 
grouping." Many different opinions exist about the value of this procedure. Research 
indicates that ability grouping can lead to the stereotyping of students which may 
lead to poor self-concepts (Begle, 197 5; Levenson, 1979; Combs, 1952; Rosenthal 
1 
2 
& Jacobson, 1968; Brassell, Petry & Brooks, 1980). Ability grouping may also lead 
to perpetuating segregation in the classroom along ethnic and socio-economic 
lines (Levenson, 1971; Findley & Bryan, 1975). A paper prepared by the U.S. Comm-
ission on Civil Rights (1976) states that the educational practice of ability grouping 
is the most common cause of classroom segregation. 
Borg (1965) found that self-concepts, feelings of belonging and a reduction 
of anti-social tendencies were favorable in heterogeneous settings. Goldberg, 
Passow and Justman (1966) found that students grouped heterogeneously in forty-
five New York City elementary schools had superior attainment to that of students 
in other grouping patterns and that the presence of gifted students had a positive 
effect on other students. 
Not unlike many other leading school systems, the Howard County Maryland 
Public School System has included in its philosophy the belief that students should 
receive instruction at their own level and rate of learning. Emphasis is placed 
on achievement, opportunities to increase computational skills, positive attitudes 
and acceleration. Awareness of the diversity of abilities in a single classroom 
brings one to the logical conclusion that Howard County teachers need access 
to programs of individualization which will assist with the implementation of the 
accepted philosophy. 
Although some teachers in Howard County have used available individualized 
mathematics programs, county test scores in elementary mathematics tend to 
support existing research which does not indicate increased academic achievement 
by students who have progressed through individualized programs. Teachers report 
that individualized instruction based on the IPI model is much more demanding 
on their time and is extremely difficult to manage. The checking component alone 
requires the assistance of aides or volunteers, both of which are often unavailable. 
3 
Teachers express concern that some students waste time on material they already 
know, and others move quickly through the units without clear understanding. 
They note that many students are bored by individualization and that working 
in isolation with infrequent direct-contact feedback may allow students to precede 
on work which is error laden, therefore, developing inappropriate habits. Working 
in isolation provides no opportunity for peer interaction which is crucial in the 
development of mathematics concepts and vocabulary as well as a clear understand-
ing of word problems. Waiting for checking and additional assignments leads to 
wasted time, frustration, boredom and potential discipline problems. If students 
are unable to receive appropriate feedback at the time it is needed, they may 
not be motivated to participate and may develop negative attitudes toward mathema-
tics (Aiken, 1961, 1970, 1976; Burbank, 1970; Callahan, 1971; Stanley & Campbell, 
1963; Dutton, 1956; Moore, 1973). 
Although many teachers support the concept of individualization, it is because 
of many of the above-cited reasons that individualization in mathematics seems 
to be diminishing in the schools as an instructional mode at a time when the need 
for individualization is accelerating. 
During the fall of 1980, a pilot study was conducted by The Johns Hopkins 
University in a Howard County fourth grade mathematics class. The newly-developed 
individualized program was designed to maximize learning opportunities for students 
with diverse ability levels through the inclusion of a management system which 
involved a cooperative learning team component. Students were allowed to perform 
many of the functions which created management problems in other individualized 
programs, while at the same time they received the motivation which was needed 
to alleviate some of the problems related to checking, monitoring, assigning, feed-
back and motivation. Students were grouped for instruction within the program 
4 
as well as expected to work independently. 
Pilot study results revealed growth in mathematics achievement. Results 
showed that the management system was effective and that it contributed to positive 
attitude changes in some students, as well as increased on-task behavior. Students 
of varying abilities and learning levels were able to work effectively within the 
cooperative learning teams. The success of the pilot would indicate that there 
is a high degree of probability that these results could be replicated in a larger 
setting. Therefore, it was desirable that this assumption be tested in a larger 
study. 
Statement of the Problem 
Most past attempts to individualize instruction were of a certain type; well 
conceived in theory but insufficient in practice. It is clear that an effective de-
livery system is mandatory--a system which effectively customizes a program 
to provide for individual differences among learners (Jeter, 1980). To this end 
a new program was developed at The Johns Hopkins University in 1980 with the 
major purpose of providing a structured management system component different 
from others which are available. The effects of this program on achievement 
and attitudes of students have been fully explored in a controlled classroom situa-
tion for six weeks. The need to determine if this individualized progam could 
be applied to a larger setting such as an entire elementary school was immediately 
evident. Hence, the purpose of this study was to compare attitudes and achievement 
over an eight-week period, of third, fourth and fifth grade students who were instructed 
in three different treatments: Team-Assisted Individualization (TAI) as developed 
by The Johns Hopkins University, Rapid Progress Mathematics (RPM) which 
includes only the materials of The Johns Hopkins University-developed TAI program 
5 
and the Regular Program (RP). Definitions of each treatment can be found at 
the end of this chapter. 
Significance of the Problem 
The Howard County Public School System is continuously looking for new 
and effective ways to improve instruction. This study has significance in that 
it directly relates to a specific need in this system. These arrangements and organ-
izational patterns have never been used in such a complete package. The pilot 
study leads one to predict that implications of the study could be far reaching. 
If this study shows that Team-Assisted Individualization (TAI) is an effective way 
to increase acceleration and achievement and at the same time provide appropriate 
instruction for the diverse needs of students in the classroom, it will provide an 
alternative to the current elementary school mathematics program that, until 
now, has not existed. 
Combining individualization and effective management components into 
TAI should provide solutions to problems which exist in current individualized 
programs. The specially designed materials will allow students of diverse abilities 
to receive instruction at their own level, and the treatment has been specifically 
structured to facilitate increased academic achievement through consistent support 
from teammates and the opportunity to progress at the student 's individual rate. 
Assignment to a team should provide motivation toward a common goal and should 
eliminate problems and frustration which consistently arise in classrooms where 
downtime is extensive. Students can check, assist and encourage others, are contin-
uously involved and are motivated by the team component to move quickly and 
accurately (DeVries & Slavin, 1976 & 197&). Therefore, it could be predicted that 
students who participate in a program with high motivation will out-perform and 
6 
will have better attitudes than students with little motivation, and it is logical 
that teachers will prefer to teach with a system which offers the components 
of TAI that are so necessary for successful management and implementation. 
In addition to assessing the impact of a form of cooperative learning in a 
regular classroom, this study will provide an extension to the growing literature 
on the use of cooperative learning strategies in educat ion . The data collected 
in this study will provide an opportunity to replicate findings of increased achieve-
ment and rate of learning. Because this specific form of cooperative intervention 
is new, this study will provide a stronger test of the concept of cooperation and 
individualization in the classroom . 
Definition of Treatments 
For more than ten years, instructional techniques involving learning teams 
have been researched at The Johns Hopkins University and, as a result, are widely 
used throughout the United States (DeVries & Slavin, 1976 & 1978). Although 
these techniques have been found to be an effective instructional strategy, the 
foundation is a group-paced model in which all students learn the same material 
at the same rate. The new program (TAI) is a merging of individualization and 
a management system which includes a team component. Complete descriptions 
of the different treatments are found in Chapter Three as well as the Appendices 
where actual materials are included. 
TAI: Team-Assisted Individualization is the individualized mathematics 
program designed by The John Hopkins University. High, average and 
low achieving students, as determined by a diagnostic test, are assigned 
to teams in which they proceed at their own rate to work at the appropriate 
place within individualized curr iculum materials which are organized 
into skills, units and blocks. Students contribute to team scores which 
are the result of the sum of the average number cor rect of all tests 
taken by all team members. Posted performance of teams shows students 
how their accomplishments compare with other teams. TAI will be 
7 
referred to as Treatment One (T 1) . 
RPM: Rapid Progress Mathematics is a varia tion of the TAI program which 
did not include the team component. Performance on the diagnostic 
test will determine the appropriate level at which students will begin 
instruction in the program. Procedures are identical to the TAI treat-
ment except that all aspects of the TAI team component have been 
omitted. RPM will be referred to as Treatment Two (T 2). 
RP: Regular Program teachers were instructed to proceed with their use 
of the current instructional strategies and mathematics materials with 
no changes in their regular program for the purpose of this study? 
RP will be referred to as Treatment Three (T 3). 
Definition of Terms 
Accuracy The sum of the average number correct of all tests taken 
Score: by all team members. 
Answer Sheets which contain the correct answers to the problems 
Sheets: which are included on the Skillsheets, Checkouts and Final Tests. 
Block: Collection of problems of equal difficulty which represents 
several skill areas. For example, a block might include several 
multiplication units, a numeration unit, a few fraction units 
and a word problem unit. 
Checkout: A final worksheet which consists of two parallel sets of ten 
items which relate to a particular unit and which must be completed 
prior to the final test. 
Cooperative The specific treatment which has been designed to implement 
Intervention: cooperative learning. 
Cooperative Any of several instructional strategies in which students 
Learning: work in small groups toward an identified common goal. 
CTBS: Comprehensive Tests of Basic Skills is the standardized test 
from which the Mathematics Computation scale was used to 
determine achievement by pre-and posttesting. 
Down time: Any amount of time during which a student is not exhibiting 
on-task behavior due to waiting for such things as instruction, 
checking or receiving assignments. 
Final Test: The evaluative test which is administered after instruction and 
assignments on a specific unit have been completed. 
H.: Hypothesis to be tested. 
l 
8 
Individual Sheet for students in RPM to record progress for a period 
Progress of one week. 
Record: 
Instruction An instructional worksheet which explains the skill to 
Sheet: be mastered and gives a step-by-step method for solving the 
problems. 
Math The student's individual record of units which have been 
Progress and which are to be completed. 
Form: 
Monitor: Rotating students whose responsibility is to locate and check 
the Final Test which is taken by a member from anothe r team. 
Pair: Two students who work together within a team. 
Progress The average number of units covered by each team member 
Score: times ten. 
Public Sheet on which the accomplishments of each team are 
Team Score posted. 
Sheet: 
Resource Any source of additional instructional help, such as reading, 
Help: diagnostic-prescriptive or resource teachers. 
Skillsheet: A worksheet which consists of twenty problems which introduces 
a subskill that leads to final mastery of the entire skill. 
Teacher A period of fifteen to twenty minutes during which the 
Review teacher instructs groups of students. 
Session: 
Teacher's Specific directions needed in order for teachers to implement 
Manual: TAI and RPM. 
Team: Four or five members comprised of high, average and 
low achievers as determined by the diagnostic test. 
Team Score: The sum of points earned on accuracy, progress rate and 
facts drills. 
Team The sheet on which each team member's scores are recorded 
Score for a period of one week. 
Sheet: 
Triad: Three students who work together within a team. 
Unit: A specific topic of study, such as addition, fractions and decimals 
which is arranged for study in a definite sequential hierarchy. 
Week: Four to seven days (see Chapter Three). 
CHAPTER TWO 
Review of Relevant Literature 
Research Related to Individualization 
Individualized instruction has no precise definition. In a broad sense it simply 
requires the adaptation of instructional strategies and materials to fit a student's 
specific needs and characteristics. Rousseau (1906), sometimes referred to as 
the father of modern pedagogy, wrote that teachers must study each individual 
carefully in order to learn what the child is, what his/her strengths are and where 
the child's potential lies. It is then the teacher's responsibility to create the proper 
environment for the child so that he/she can develop to the fullest extent of his/her 
abilities. Davidson (1971) writes that only three of Rousseau's educational demands 
have truly influenced education. They are: (1) the demand that children should 
have complete freedom of movement from the moment of birth; (2) that children 
should be taught to use their hands in a productive way; (3) that children should 
be educated through direct experience. 
In Educational Psychology Gates (1942) stresses the development of the total 
individual. Beginning with one's innate intelligence he elaborates on the rate of 
development in all areas--vocalizing, motor, emotional and social development, 
leadership adaptability, needs, ways of learning, degrees of creativity and mental 
health. Only by looking at each person as an individual and developing educational 
programs around that particular person can the schools be successful. 
In the first fifty years after the union of the states, schools differed very 
little from what they had been during the colonial period in New England. They 
remained one-teacher schools within crude buildings and with poor supplies. The 
instructional method was still on a one-to-one basis, and each pupil continued 
9 
10 
to obtain only fleeting periods of private help from the teacher (Reisner, 1930). 
The methods were generally characterized by drill, memorization and severe discipline, 
with the major educational aim being the indoctrination in the church doctrines 
and chatechism. Until the middle 1880's the teacher's time was almost fully used 
in hearing an individual's assigned recitations, with very little actual instruction 
(Knight, 194-0). Theoretically, the individual lessons should have helped the teacher 
meet the needs of individuals, but there were so many children waiting for a chance 
to recite that the amount of individual attention given to students was negligible. 
Individualized teacher-pupil recitation was not very popular because results were 
slow. Children may attend school for years and learn only a small amount of reading 
and writing. The communities did not wish to give more money for the support 
of schools, but when the monitorial school was introduced by Lancaster and Bell 
in England in 1797, people realized that education would be rather inexpensive 
by doing away with the individualized form of teaching. Citizens began to support 
public schools, and group instruction became popular. The monitorial method 
brought education to thousands of children who, had it not been for the Lancastrian 
schools, would never have had an opportunity to receive an education. Meeting 
the educational needs of a massive segment of the population led the public to 
the recognition that it was necessary to support public education. Great controversy 
over this led to the development of the graded school system as a means of improving 
the education of children (Thomas & Crescimbeni, 196 7). 
In 1888, Preston Search established a plan of individualized instruction in 
Pueblo, Colorado. Daily recitations were abolished, and students were permitted 
to advance at their individual rates of learning (Alberty & Alberty, 1962). Eight 
years later John Dewey challenged established practices of his time in an 
attempt to better society by liberating individuals and supporting the variations 
of children's abilities. His work at the University of Chicago Laboratory Schools 
r 
11 
eliminated arbitrary classification of grades and textbooks in an attempt to meet 
individual needs (Boyd, 1921). He stated that differences in capabilities, needs 
and preferences in individual children must be recognized, because one cannot 
assume that all minds work in the same way (Dewey, 1916). 
Frederick Burk at the training school of San Francisco State College seems 
to have been the first educator to systematize a plan for individualized instruction. 
In 1913 his staff developed self-instructional materials, and students were guided 
to advance through the assignments at their own rate while the teacher assisted 
with any difficulties and checked the work. After the student successfully passed 
a test on the material, a new assignment was given (Alberty & Alberty, 1962). 
Alfred Hall-Quest (1916), Professor of Education at the University of Virginia, 
pleaded for revision in the schools. He recognized that grouping students proved 
to be an immense time-saver over the method of individual instruction, but stated 
that grouping is inadequate and unjust to children. He requested quick revision 
of methods to help each individual reach his or her maximum potential. 
Carleton Washburne was also concerned about the fact that children learn 
at different rates. His answer to how this could be dealt with realistically within 
a class was the beginning of programmed instruction. His Winnetka Plan abolished 
grade failure and promotion, attempted to provide for the learner's individual 
differences, and at the same time, teach for specifics. Curriculum was divided 
into two parts. The first phase involved the use of individual work centers 
which were developed around the basic skills and knowledge which everyone had 
to master. The second phase focused on the child 's creativity, special abilities 
and interests (Bayles and Hood, 1966). 
Still in use today are some of the original features of Helen Parkhurst 's Dalton 
Plan which was put to use in the public schools of Dalton, Massachusetts, and in 
12 
1920, in the Dalton School of New York City. Students were conferenced, and 
e ach established a contract. Several teachers aided students as they progressed 
as opposed to a student's receiving the help of a single homeroom teacher. This 
plan was designed to help students of varying abilities to work unhampered by 
uniform grade expectations. This method of individualization caught the attention 
of professionals, because it was far superior to the old recitation method of indiv-
idualization of the nineteenth century (Brubacher, 1947). 
Henry Clinton Morrison's teaching method incorporated the steps of pretest, 
teach and test and focused upon the idea of mastery which, for him, meant only 
the achievement of a fundamental group of the material studied. His form of 
individualized instruction was similar to Washburne's in that both provided opportun-
ities for each student to work alone so that he or she could progress at his or her 
own rate. After Morrison, workbooks and worksheets became widely used in American 
education (Bayles and Hood, 1966). All of these plans called attention to the inad-
equacies of daily recitation and pointed out the need for providing instruction 
according to individual needs. 
Edward Thorndike found that in any classroom there are always children 
who are much poorer or much better than the average, even though the children 
may be close together in mental ability. He felt that the worst errors of 
teachers are for them to ignore the fact that these differences exist, and to create 
one set of fixed rules and habits for all children in the classroom, as well as to 
teach the class instead of the assortment of different individuals. He believed 
that variation in the amount, kind and quality of work and the amount of needed 
assistance vary for each child; that a competent teacher expects these differences 
and then teaches accordingly (Thorndike, 1913). Thorndike writes that the applica-
tion of all of the principles of education are affected by individual differences. 
13 
They affect the implementation of the aims of education as well as the adaptation 
of the curriculum to the talents and limitations of each student (Thorndike and 
Gates, 1931). 
By the mid-l 930's there was some degree of individualization through the 
offering of a variety of courses, and through grouping, but there was still strict 
adherence to subject matter. Progressive educators felt that individual needs 
of students could be met more effectively by providing life experiences. From 
this attitude came the Eight Year Study, one of the greatest efforts up to that 
time to prove this philosophy. 
Thirty-eight schools participated in the Eight Year Study from 1932 to 1940. 
Instruction included group work on common needs and individual activities on partic-
ular needs and interests. During this eight-year period certain colleges were persuaded 
to accept students without meeting the conventional requirements for entrance. 
At the end of the eight years the work of these students was compared to those 
with a traditional background. The 1942 published results indicated that success 
in a liberal arts college did not depend so exclusively as thought on the student's 
methods of study in the secondary school (Parker, 1963). 
The term individualized instruction can be ambiguous. Some interpret this 
simply to mean that a teacher should treat each student as an individual instead 
of a number in an assigned seat. Arthur Bestor (1955) writes that individualized 
instruction means much more. A teacher must be capable of leading each child 
through the continuous steps in his or her development, regardless of the child's 
achievement level. The teacher resembles a tutor except that the conditions are 
less ideal than those under which an actual tutor works. Instead of giving undivided 
attention to one individual or small group, the classroom teacher must parcel out 
his or her time to all the pupils who are at different stages of advancement, and 
l l~ 
some people argue that he or she is neither teaching a class nor offering individual 
instruction. 
Individualized differences became a primary topic in educational theory 
in the early years of this century. Until then there was a fixed curriculum. The 
elimination and the dropping out of less successful students provided the only method 
which was used to account for individual differences. 
Later use of ability tests determined which students should drop out, be 
placed in an undemanding "slow" classroom or guided into "high" classes which 
prepared students for higher education. The social theory behind this selection 
system was that each student should have the opportunity to procede as far in 
school as his or her abilities warrant which, in turn, would support the dropping 
out of less responsive students. 
Most schools tend to minimize the inconvenience of providing for individual 
differences so that the same curriculum can be taught. Rather than redesigning 
programs, most remedial instruction is an add-on to the regular program. Supple-
mentary instruction is provided with the intent of repairing the skill gaps so that 
the student can be placed back in the regularly paced program. This could be 
interpreted as an attempt to erase individual differences as opposed to meeting 
needs of different individuals. 
Research on child development by Piaget and others provides evidence that 
the quality and quantity of peer interactions in a child's school experience play 
a vital role. Through the interaction, children hear views of others, clarify their 
own and develop language and the ability to think logically. The lack of peer inter-
action in individualized programs creates many questions regarding the potential 
harm (Kam ii, 1971; Graham, 1977). One of the conclusions of the extensive second-
and fifth-grade observation studies of the Beginning Teacher Evaluation Study 
15 
conducted by the Far West Laboratory for Educational Research and Development 
was that peer interactions can affect cognitive development as well as classroom 
performance and achievement. The studies showed that a student's achievement 
level is correlated with the amount of time engaged in meaningful learning tasks 
and is related to the amount of time spent with teachers and peers in interactions 
about the specific academic content (Schneider, 1979). 
The burden that individualized instruction places upon the teacher is unreal-
istic, writes Bestor. The teacher must have strength in a large range of subjects 
and must be informed within the many levels of each subject. Bestor states that 
these demands are as great as those required of a professional clinical psychologist, 
are beyond the reach of most teachers, and that the success of any such program 
stands or falls on the ability of the individual teacher . In contrast, a classroom 
grouped homogeneously can be organized. Subject matte r can be planned system-
atically, and even a mediocre teacher cannot entirely destroy the effectiveness 
(Bestor, 1955). 
Management Through Individualization 
The most encouraging attempts to individualize instruction today are the 
ungraded or multigrade classrooms where children a re allowed to learn as much 
as fast as they can, or at least as much as the teacher can provide . Goodlad and 
Anderson (1959) advocated the development of the nongraded vertical pattern 
of school organization in order to accommodate differences among students. They 
recognized that the school structure must be designed to facilitate the continuous 
educational progress of each student . They also note that eliminating the stereo-
types of grade expectancies should influence students ' abilities to earn early ap-
proval of other children. Individual differences, instead of being a source of con-
16 
tention, would become a source of strength. Students of child development realize 
that the reality of growth defies the strict ordering of students' abilities and at-
tainments into a conventional graded structure. 
Individualized instruction has been categorized into several major types. 
Clearly specified behavioral objectives are a part of all individually diagnosed 
and prescribed programs. Materials and methods of implementation are coordinated 
with stated objectives. In self-directed programs, students and teachers coopera-
tively establish learning outcomes. Students have the freedom to select materials 
and the goal-reaching procedures. Personalized programs allow the student to 
choose his or her own objectives and then follow a prescribed program with special-
ized materials. The independent study approach allows the student to determine 
his or her own objectives, the topic of study and the means to attain the objectives 
(Pancella, 1973). Contracting as defined by Wilson and Gambrell (1973) is another 
variation of individualization. The student and teacher cooperatively agree upon 
specific learning objectives under the broad objective which is teacher-determined, 
and then negotiate a contract which takes into consideration the student 's learning 
pace, study method and evaluation. 
Driscoll (1980) writes that pace is the only individualized feature within 
individualized programs since students use the same materials in the same manner 
--through programmed texts or worksheets. Other objections to individualized 
instruction were published by the National Advisory Committee on Mathematical 
Education (NACOME, 1975). The report states that only computation is emphasized, 
that little, if any, peer interaction occurs and that little attention is given to prob-
lem solving and higher analytic skills. 
Many schools are attempting to implement individualized programs with 
materials that are not designed to provide clear directions for the student, nor 
17 
to provide immediate feedback--two of the most important components of an 
individualized program. Materials in use often require the teacher's teaching, 
providing feedback and giving directions. One of the benefits of individualized 
instruction is lost because students cannot easily move at their own pace. This 
occupies the teacher with the mundane management problems of the system instead 
of allowing the teacher to work with individual students. The built-in dependency 
on the teacher creates confused, frustrated students who must wait and, therefore, 
use instructional time in an off-task manner which decreases productivity (Kepler 
and Randall, 1977). 
Data from studies conducted by Soar and Soar (1972), Brophy and Evertsen 
(1976) and Good and Grouws (1977) seem to support the use of direct instruction 
when teaching basic skills. According to Rosenshine (1979), direct instruction 
is characterized by large group, teacher-directed instruction on specific goals 
with little choice of activity. These findings would create questions about the 
use of individualized instruction, but both concepts can be merged to form an 
e ffective instructional program. 
A study of the individualization movement which peaked in 1920 was published 
by Grittner (1971). He concluded that present individualization programs will 
decrease in use just as they did decades ago, and for the same reasons--cost, no 
achievement increases, excessive demands on teachers, excessive test-taking and 
isolation. Nix (1970), Osmundson (1972), Palow (1973) and Sutton (1976) agreed 
that teachers who use individualized programs are overworked. Additionally, Colvin 
(1973) found that teachers felt isolated in spite of inservice training which was 
designed to instruct teachers how to manage the programs but which, in fact, 
did not. Schoen (1976) writes that the results favoring traditional approaches 
far outweigh those which support individualized approaches and that the numerous 
18 
management problems which are a part of individualized programs are detrimental 
to learning. 
Individualization Within Content Areas 
One of the most popular attempts to individualize classroom reading and 
mathematics programs incorporates the use of kits such as those published by 
Scientific Research Associates. Students can be assigned to or select activities 
at an appropriate instructional level and are allowed to work at their own rate. 
The learning center is another attempt to individualize learning. This is 
a place where directions and materials are provided for children's individual or 
group use in an attempt to solve problems or answer questions. The center is designed 
to help the learner gain knowledge, maintain skills, develop attitudes, or all of 
these. 
Pancella (1973) clearly states in Harcourt Brace Javonovich's Concepts in 
Science that individualized instruction is not synonymous with independent study 
but is a program which provides for intrinsic motivation, increases the use of a 
variety of materials, increases opportunities for direct involvement, and allows 
each student to advance at his or her own pace. Concepts assumes that all students 
should master the same conceptual schemes, but that this should be allowed to 
take place in a variety of ways through different investigative processes. 
Many reports of individualized mathematics programs were reviewed by 
Schoen (1976) who concluded that student achievement was no better in self-paced 
programs than traditional programs in mathematics, that program effectiveness 
was not balanced by the extra expense, teacher effort and time and that self-paced 
programs fared worse than traditional programs when used in the upper elementary 
grades. 
Individually Guided Education (IGE) was developed at the Wisconsin Research 
19 
Development Center for Individualized Learning (1974-) and is a comprehensive 
system which is designed to help students learn at their own pace and instructional 
level. The two key components of IGE are identified by Klausmeier and others 
(1977) as instructional programming for the individual student and the administra-
tive-organizational arrangements that support the implementation. This innovative 
program involves the schools, community and teacher education programs, but 
leaves instructional decisions to the classroom teacher. Developing Mathematical 
Processes (DMP) is the mathematics component of IGE. The objective-based program 
allows for grouping and individualizing within an activity, within a topic and among 
topics. The philosophy of the program is based upon the beliefs that alternative 
approaches to the learning of a concept must be provided and that "no one under-
stands enough about instructional procedures or how children learn to routinize 
instruction" (Wisconsin Research & Development Center for Cognitive Learning, 
1974-, p. 66). 
A large scale evaluation of IGE has been designed to assess the program's 
impact. Results of the evaluation which is being conducted by the Wisconsin Research 
and Development Center for Individualized Learning are not complete, but preliminary 
results show that teacher satisfaction increases in relation to increases in the 
degree of implementation. It is common to find an increase in positive staff outcomes, 
but there has been no significant increase in student achievement (Romberg, 1976). 
Katzenmeyer and others (1976) found no differences in standard achievement measures 
in the pilot study involving IGE and non-IGE schools. 
One multi-media approach to individualization is embodied in Programmed 
Logic for Automatic Teaching Operations (PLATO), a computer-assisted delivery 
system which combines a responsive typewriter, films, slides, tapes and other 
audio visual aids with computer programs. This is not intended to be the total 
20 
curriculum, but an adjunct to the existing program. Traditional and PLATO classes 
were compared after data collection over a two-year period. Results showed that 
no differences occurred in achievement or attrition (Murphy and Appel, 1977). 
The teacher assigns an appropriate program to a student, after which the computer 
presents the material, monitors student progress and evaluates performance. 
The student determines the instructional pace and receives instant feedback for 
correct answers, as well as alternative information and instruction for incorrect 
responses (Jeter, 1980). 
In 1970 a consortium of eight Utah local educational agencies and the state 
education agency developed the Utah System Approach to Individualized Learning 
(U-SAIL) as their approach to the individualization of mathematics and reading 
for the elementary grades. Curricula were designed to assist in diagnostic/prescrip-
tive teaching, and interdependent components were designed to provide specific 
guidelines in planning, organizing, implementing and evaluating within a humane 
classroom environment. In mathematics, materials are coded to promote easy 
access, and use of manipulatives is stressed. After diagnosis, the student is assigned 
to groups according to identified need. The teacher presents to the group, monitors 
student progress, reteaches and conferences with each student as needed. Assignments 
vary among students, and immediate feedback is provided prior to reassignment 
or testing (Hales, 1978). Five evaluation studies on the U-SAIL system showed 
that students in the system showed significantly greater gains than students in 
non-U-SAIL programs. One of the studies showed that there was a relationship 
between achievement and the degree of implementation. High implementation 
classes significantly outperformed the low and middle implementation classes 
(Jeter, 1980). 
Individualized Mathematics System (IMS) is a complete mathematics system 
21 
for grades 1-6 which is developed on the hierarchial model. Each unit of instruction 
states behavioral objectives for teacher and student use. Instruction initially begins 
with seminars and leads to a student's working individually in each asssigned unit 
until mastery is demonstrated on the unit posttest. Monitoring within a unit is 
the responsibility of the student and is based upon recycling or branching as perform-
anc e on worksheet tests indicates. The teacher determines placement, mastery 
and initiates assignment to seminars. Pacing is dependent upon the students' indi-
vidual needs. The system requires considerable preparation as well as several 
aides, in addition to the classroom teacher (EPIE, 1974). 
The University of Pittsburg's Learning Research and Development Center 
directed an experimental project in K-6 classrooms which eventually became what 
is known as Individually Prescribed Instruction (IPO--a highly structured system 
for instruction in reading, science, spelling and mathematics. Individually prescribed 
learning activities allow students to work at their own pace and instructional levels 
after assignment according to placement tests. The system's foundation is a set 
of instructional objectives for each subject area sequenced into learning hierarchies 
and is grouped into instructional units. The teacher administers tests, diagnoses 
strengths and weaknesses, writes prescriptions, monitors student progress and 
instructs individual students. The scoring of tests is often performed by additional 
personnel (Bolvin, 1968). Duda (1970) analyzed the IPI program in an attempt 
to provide insight into its contribution to the field of individualization, and the 
Learning Research and Development Center for IPI conducted a comprehensive 
study of the program in 197 3-7 4 in a large number of second grade classrooms 
across the United States. Analysis of data collected over a period of three years 
showed that the degree of implementation and individualization had increased 
since 1963. Increases were noted in reading and mathematics placement scores 
22 
and teachers expressed greater confidence with the IPI system (Leinhardt, 1977). 
O'Daffer (1976), however, raises many concerns regarding the system. He questions 
whether or not the individual pacing is really what is meant by individualization. 
Students work basically in isolation and have few opportunities for interaction 
with peers. Concept development is minimal since there are few teaching aids, 
manipulatives or demonstrations. O'Daffer has observed the development of negative 
attitudes toward mathematics. Paraprofessionals were often instructing students 
while teachers were scheduling, prescribing, recording and testing. He concludes 
that careful consideration should be given to the true meaning of an individualized 
program and that programs should be built around true student needs. Oles (1973) 
found that the self-scoring component was enjoyed by students, but that 88% of 
the students cheated. 
The Comprehensive School Mathematics Program (CSMP), in its eighth year 
of the pilot study, is designed to overcome what the program developers identified 
as a typical problem in elementary school mathematics programs--the lack of 
appropriate verbal-language skills and abilities. Program components incorporate 
extensive use of the non-verbal languages of strings, arrows and minicomputer 
without the symbology and notation that are customarily used. Classroom management 
techniques allow for large and small group instruction and independent study. 
The individualization component is implemented through the use of worksheets 
and workbooks which increase in difficulty as the students progress through the 
series. Data collected in 1974, 1975 and 1976 favored the CSMP classes, but areas 
of concern include the cost factor and recruitment and training of teachers (Woodward, 
1980). 
Management Through Social Organization 
Over thirty years ago a small group of social scientists began to write about 
23 
the effectiveness of team competition within the classroom (Deutsch, 1949b; Sherif 
& Sherif, 1953; Coleman, 1959; Bronfenbrenner, 1970). Intrat eam cooperation 
and interteam competition seemed to promote peer support for academic success 
as well as more eagerness to assist peers. It provided teachers with an instructional 
strategy which allowed for a successful mode of dealing with the diversity 
in student achievement within the same class. 
Again, an upsurge in research on the attitudinal, social-interactive and academic 
effects of cooperative team learning in the classroom has occurred during the 
past decade. Teams are designed to promote peer interaction and to cooperatively 
pursue the study of an academic subject. Research on cooperative interventions 
shows that use of this strategy leads to an increase in students' mutual concern 
for peers as well as in the number of friends which were named (De Vries & Edwards, 
1973; Edwards & De Vries, 1974; Slavin, 1978c, 1979; Johnson, Johnson, Johnson 
& Anderson, 1976; Wheeler, 1977). This has resulted in areas of study such as 
social studies, mathematics and language arts. Edwards and DeVries (1974), Slavin 
(1980), Johnson et al (1976) and Wheeler and Ryan (1973) reported students' increased 
liking of school as a positive effect of cooperative structures. Additionally, Blaney 
et al (1977), and Slavin and Karweit (1979) reported that students learning coopera-
tively showed increased self-esteem. 
Johnson, Johnson, Johnson and Anderson (1976) hypothesized that structuring 
learning in different ways would lead to different patterns of interaction among 
students and would promote different learning outcomes. Their study used an 
individualized instruction approach where a student's goal achievement is unrelated 
to the goal achievement of the other students and a cooperative learning structure 
where all students are affected by the success of a student. The cooperative learning 
2~ 
group studied together, completed one assignment per group, assisted each other 
and received praise from the teacher on an individual basis. Results at the end 
of the 17-day study indicated that "cooperative interaction with peers, compared 
with studying individualistica11y, promotes altruistic behavior by children" (p. ~50). 
There was also some evidence to indicate that cooperative learning facilitates 
an intrinsic motivation to learn while the opposite may occur with individualized 
learning experiences. There was a positive correlation with cooperative learning 
and attitude, and higher daily achievement occurred with the cooperative learning 
group, but no differences existed between groups on an individual review test. 
However, when students in that setting and students in the individualized condition 
completed the review test individua11y, the cooperative group performed better. 
Artzt (1979) found that whole class instruction does not a11ow for enough 
time to work with individuals, too much time is used to answer questions which 
are relevant to only a few students, little student interaction occurs, certain stu-
dents tend to dominate discussions and others hesitate to ask questions. In an 
attempt to diminish these problems in her secondary mathematics classes, the 
homogeneous groups were divided into above average, average and below average 
tracks. Students subdivided themselves into teams of four to six members. Students 
worked cooperatively and submitted assignments from the team rather than individuaUy. 
New teams were formed after each unit test. Parents expressed satisfaction with 
the teaming component, discipline problems decreased, more work was completed, 
attitudes toward mathematics became more positive and grades improved. 
An extensive review of research by Johnson and Johnson (1979) found that 
student achievement was higher on problem solving tasks when in a cooperative 
learning situation as opposed to a competition learning situation. Michaels (1977) 
reviewed ten selected studies and found that in seven, competition promoted higher 
25 
achievement than both the individualistic or cooperative efforts. However, 
Davis, Laughlin and Komorita (1976) found that cooperation promoted higher achieve-
ment in concept attainment tasks than competition or individualistic efforts. 
Studies by Clifford (1971) as well as Scott and Cherrington (1974) found that compe-
tition resulted in higher achievement than individual efforts, although one study 
found no difference between the two efforts (Clifford, 1972). 
Educational critics (Holt, 1967; Johnson & Johnson, 1975; Postman & Weingartner, 
1969) have warned that consistent use of competition in education can have deleterious 
effe cts upon students. A field study reported by Sherif, Harvey, Hood, White and 
Sherif (1961) found that intergroup tensions can be alleviated by an organization 
which incorporates a mutual interdependence component as opposed to an intergroup 
competitive organizational structure. Blaney, Stephan, Rosenfield, Aronson and 
Sikes (1977) assessed the effect of interdependent learning groups, as opposed 
to traditional teacher-directed competitive groups, upon the attitudes and interper-
sonal liking of elementary school students. The study concurred with Gottheil 
(1955), Phillips (1956), and Phillips and D'Amico (1956) that positive interpersonal 
benefits result from cooperative interaction; that there was an increased liking 
for classes (Wheeler & Ryan, 1973) and that students manifested higher self-esteem 
(Lippitt & Lohman, 1965; Lippitt, Eiseman & Lippitt, 1969) in interdependent groups 
as opposed to students in traditional classes. This study did not measure academic 
performance, but a subsequent study suggests that academic performance of students 
in interdependent learning groups is as good as or is better than that of students 
in a traditional classroom setting (Lucher, Rosenfield, Aronson & Sikes, in press). 
Additionally, improvement in ethnic relations in the form of increases in cross-
thnic helping behavior has been the result of the use of cooperative techniques 
in the classroom (De Vries & Edwards, 1974; Weigel, Wiser & Cook, 1975). 
26 
A study of 45 first grade students by Johnson, Skon and Johnson (1980) showed 
that students solve problems best by working cooperatively. Groups comprised 
of high, middle and low achievers were assigned to one of three treatments--the 
cooperative group where students worked together on tasks, the competitive group 
whe re students competed against one another and those in the individual group 
where students worked independently. Students' tasks included solving verbal 
and spatial reasoning problems as well as categorizing and retrieving information. 
Cooperative group students scored significantly better on performance tests than 
did students in the other two treatments. 
The Johns Hopkins University Programs 
Aronson and associates (197 5, 1978) developed a teaching method, Jigsaw, 
whic h is designed to increase a student's sense of responsibility for his or her own 
learning and to foster peer cooperation and tutoring. Each student becomes an 
"expert" on one component of the instructional unit and then assumes the responsi-
bility for teac hing this to all members of the team to which he or she is assigned. 
Because team members have knowledge of only the assigned component, their 
unique information makes their contribution valuable to the success of the team. 
Task closure is contingent upon mutual cooperation among team members. As 
existing materials required rewriting by the teacher prior to implementation, Jigsaw 
II was developed to help alleviate this problem by assigning students to read all 
of the descriptive or narrative material which is to be studied (Slavin, 1978b). 
Teams-Games-Tournaments (TG T) is an imaginative classroom intervention 
which incorporates a cooperative group structure, active competition among individuals 
from different teams and games which are integrated into the curriculum (De Vries 
& Edwa rds, 1973). Results from t e n field experiments over a four-year period 
27 
show that TGT, when compared to traditional instruction, "has generally consistent 
effects on students' academic achievement, mutual concern, positive race relations, 
peer norms supporting academic achievement and positive attitudes toward school" 
(De Vries & Slavin, 1976, 1978, p. 31). 
A simplification of TGT, Student Teams-Achievement Divisions (STAD) replaces 
the TGT games with a quiz (Slavin 1977, 1978a, 1980a). Team cooperation is an 
integral part of the program, but it does not include individual competition. Games 
and tournaments are replaced by quizzes and rewards to students for special improve-
ment and effort on any given quiz rather than ability. Teachers are encouraged 
to a lternate the use of TGT and STAD in the classroom in order to maintain high 
student interest over a long period of time. 
One could conclude that involving students in an interdependent task will 
not ensure an improvement in achievement. Greater achievement gains were 
found where members in the cooperative group received rewards which were contingent 
upon total group performance. Deutsch (194-9) describes this as reward interdepend-
ence--a situation in which peer norms develop which encourage efforts that lead 
to group success and discourage behavior which could hinder the group success. 
Based on the findings on Aronson and associates (1975, 1978), DeVries and 
Edwards (1973), DeVries and Slavin 0976, 1978) and Slavin (1977, 1978a, 1980a) 
it appears that an intervention designed to facilitate a cooperative learning environ-
ment where students with diverse ability levels in mathematics can succeed and 
student attitudes and academic achievement will improve, would enhance the 
existing classroom curriculum. It is felt that such an intervention has not existed 
a nd that the issue of management of an individualized program has not been addressed 
adequately in the literature. 
28 
SUMMARY 
Research has focused on ways to individualize, but yet no suitable program 
has been found whereby students and teachers agree that it is motivating, manageable, 
flexible and tailored to meet the diverse ability needs of students. Individualized 
mathematics programs that were found suffered from various weaknesses which 
included limited feedback to students, no peer interaction, unrealistic planning 
demands on teachers, management difficulties, weak monitoring of student progress, 
built-in dependency on the teacher and the development of negative attitudes. 
The literature search has revealed that research on cooperative-learning 
techniques has shown that both internal and external validity are high and that 
the "positive effects of cooperative learning methods are well enough established 
to begin to evaluate extensions or adaptations of the methods to new uses, such 
as ( . .. ) individualization and mastery learning" (Slavin, 1980b, p. 110). 
The program proposed in this study was designed by The Johns Hopkins University 
staff in an attempt to overcome the weaknesses of existing individualized mathematics 
programs. Based upon the research it would be logical to expect that the use of 
Team-Assisted Individualization would improve the attitudes and achievement 
of students of mathematics, and this program, as described in detail in the next 
chapter, is the foundation of this dissertation. 
CHAPTER THREE 
Methodology 
Subjects 
The subjects for the study were 504- third, fourth and fifth grade students 
from six elementary schools in the Howard County Public School System in the 
State of Maryland. Male and female Black, Asian and Caucasian students from 
middle and upper middle socioeconomic communities comprised the sample, which 
also included students who received resource help in addition to the regular class-
room instruction. 
Selection of Subjects 
Howard County is a small geographic area in which there are 27 elementary 
schools. In accordance to the philosophy of this county's school system, schools 
participate in studies on a voluntary basis. For purposes of this study, within the 
set of eighteen volunteers, six schools were chosen to participate by utilizing a 
table of random numbers. It was determined by the researcher and the Director 
of Program Evaluation, Research and Development that these schools were represen-
tative of the total population as it relates to IQ, race, sex, socioeconomic status 
and achievement. 
Selection of Teachers 
The random selection of the six participating schools also provided for the 
selection of eighteen teachers who were representative of the third, fourth and 
fifth grade teachers of the Howard County Public Schools. The average teaching 
experience of the eighteen teachers was 8.6 years. Five teachers had earned a 
29 
30 
Bachelor's degree and thirteen had completed a Master's degree or equivalency. 
Teacher Inservice 
One week prior to implementation, two separate inservices of three hours 
each were conducted for the six Team-Assisted Individualization and the six Rapid 
Prog ress Mathematics teachers. Training included an introduction to the materials 
a nd teachers' manuals, established guidelines and timelines for the project and 
attempted to develop support system networks within each of the participating 
schools. Teachers were instructed to keep careful records of student and teacher 
a tte ndance. Students for whom test data were missing were excluded from the 
final a nalyses. 
Overvie w of Treatments 
Three treatments were utilized in this study. One, Team-Assisted Individuali-
zation (Slavin, Leavey & Madden, 1980), involved the use of cooperative task and 
reward structures in the classroom. The second, Rapid Progress Mathematics 
(Slavin, Leavey & Madden, 1981) was a variation of the Team-Assisted Individuali-
zation program in which only the materials were utilized. The third, Regular Program, 
was the curriculum which would have been used for instruction whether or not 
this study was in progress. 
Desc ription of Treatments 
Team-Assisted Individualization (T 1) 
Team. Students were assigned to four- to five-member teams which consisted 
of high, average and low achievers as determined by the diagnostic test. Boys, 
g irls and students of any ethnic groups in the classes were represented in the proportion 
in which the entire class was comprised. 
31 
Diagnostic test. The students we re pretested at the beginning of the project 
on mathe ma tics operations a nd were plac ed at the appropriate point in the individ-
ualized program ba sed on their performance on the diagnostic test (see Appendix, 
p. 95). Students we re assig ne d to skill c a tegories on which their performance 
was less tha n 80 per cent. 
Curriculum ma t e ria ls. For all opera tions skills, the students worked on individ-
ualized c urriculum mate ria ls that ha d the following subparts: 
1. An Instruction Sheet explaining the skill to be mastered and giving 
a step-by-s tep me thod of solving problems. 
2. Seve ral Skillsheets, each consisting of twenty problems. Each Skillsheet 
introduced a subskill that led to final mastery of the entire skill. 
3. A Chec kout which consisted of two parallel sets of ten items. 
4. A Final Test. 
5. Answer Sheets for Skillsheets, Checkouts, and Final Tests. 
Monitors. At the beginning of each class period, three students from different 
t eams we re appointed to serve as monitors for that period. Each received 15 points 
fo r the ir team (as though they had taken one test and received a perfect score) 
because he or she could not contribute to his or her team during the period in which 
he or she se rved as monitor. Monitors distributed, scored and filed appropriate 
test s . 
Te am study method. Following the diagnostic test, students were given 
a s ta rting place in the individualized mathematics units. They progressed through 
the ir units in their teams, following these steps: 
1. Students formed pairs or triads within their teams which were initially 
formed at random. Students located the unit on which they were working 
a nd brought it to the work area. The units consisted of the Instruction 
32 
Sheet, Skillsheets and Checkouts stapled together, and the Skillsheet 
Answers and Checkout Answers stapled together. 
2. In pairs, students exchanged Answer Sheets with their partners. In 
triads, they gave their Answer Sheets to the student on their left. 
3. Each student re ad his or her Instruction Sheet and asked teammates 
or the teacher for help, if necessary. 
~- When students had read the Instruction Sheet they began with the first 
Skillsheet in the unit. Students copied the problems from the stapled 
sheets onto their own paper prior to completing the tasks. 
5. Each student worked the first four problems on his or her own Skillsheet 
and then had his or her partner check the answers against the Answer 
Sheet. If all four were correct, the student progressed to the next 
Skillsheet. If any were wrong, the student completed the next four 
problems, and so on until he or she got one block of four problems correct. 
6. When a student successfully completed four in a row on the last Skillsheet, 
he or she took Checkout A, a ten-item quiz that resembled the last 
Skillsheet. On the Checkout, students worked alone until they finished. 
When the problems were completed, a teammate scored the Checkout. 
If the student got eight or more correct, the teammate signed the 
Checkout to indicate that the student was certified by the team to 
take the Final Test. If the student did not get eight correct, the teacher 
was called to explain any problems the student was having. The teacher 
had the prerogative of asking the student to work again on certain 
Skillsheet items or the student may have then taken Checkout B, a 
second ten-item test comparable in content and difficulty to Checkout 
A. Otherwise, students skipped Checkout B and went directly to the 
33 
Final Test. No student could take the Final Test until he or she had 
been passed by a teammate on a Checkout. 
7. When a student "checked out," he or she took the Checkout to a student 
monitor from a different team. Three students served as monitor at 
any given time. The monitors' responsibilities included finding the 
proper test and giving it to the student. The student moved his or 
her desk away from all other students to take the test. A student taking 
a test was not to be disturbed for any reason. After completion, the 
student located the monitor who scored the test. The student could 
check the scoring in the presence of the monitor and bring any discrep-
ancies to the attention of the teacher for resolution. The teacher 
had the option to check all Final Tests to keep track of student progress 
and to be sure that all scoring was accurate . When the tests were 
scored, they were given to the teacher to be entered on the student's 
Team Score Sheet, and the student checked the unit off on his or her 
own Math Progress Form. The student then wrote the unit and his 
or her score on the Team Score Sheet posted in the classroom . 
8. There is no "down time" in this model. Whenever a student 's partner 
left to take a Checkout or Test, to serve as a monitor or for any other 
purpose, the student joined another pair in his or her team to form 
a triad, or if the rest of the team was already a triad, the student formed 
a new pair with one of the students. 
Facts Sequence 
The system described above does not include addition, subtraction, multiplica-
tion or division facts. These facts were taught using a separate method. 
Preparation for the facts units occurred entirely out of class, as homework. 
34-
Students were able to take home facts practice sheets to study with parents, siblings 
or other students. Students worked first for accuracy, then for speed on each 
facts unit. 
Every Tuesday and Friday, all students took five-minute tests on the facts 
unit they were studying. The facts tests were comprised of ninety problems organized 
in three sections: A, Band C. The A section consisted of facts from 0-3; B contained 
4-6; and C contained 7-9 facts. When a student began a fact sequence, he or she 
had to first get 100% on Section A, then 100% on Section B, then 100% on Sections 
A and B taken together, then 100% on C, then 100% on B and C taken together, 
and finally 100% on A, Band C in five minutes, as indicated in the chart below: 
Facts Sequence 
1. A 
2. B 
3. A+ B 
4-. C 
5. B + C 
6. A+ B + C 
Five minutes should be more than sufficient for any one section, but as sections 
were added together, the task became more of a speed drill. A student stayed 
on each step of the sequence until he or she passed it and then proceeded to the 
next step. If a student passed step 6, he or she went to the next fact sequence 
(e .g., from multiplication to division). 
The fact tests were scored in class. Students exchanged tests with members 
of other teams after which the teacher read the answers, displayed them on an 
overhead projector or chart paper or distributed answer sheets. When a student 
passed a step in the sequence, he or she put a "15" by his or her name on the Team 
Score Sheet to indicate a perfect paper, gave the teacher the correct test and 
e ntered the date in the appropriate place on the Math Progress Form. 
If any students were having difficulty with their facts so that this interfered 
35 
with their progress through their skill units, they had the opportunity to use a 
grid until they passed the relevent facts sequence. If any students passed all of 
their facts sequences, they worked on their individualized units during the facts 
tests . The teacher provided homework for these students, as they were not studying 
facts at home. 
Teacher review session. As soon as students were working well in their teams 
on the individualized materials, the teacher began forming groups of one to ten 
students who were at about the same point in the curriculum and provided instruction 
in l 0-20 minute sessions. The purpose of these sessions was to discuss any points 
with which students were having trouble and to prepare students for upcoming 
units. During the time that the teacher was working with the small groups, other 
students were continuing on their own units. 
Curriculum organization. The curriculum was organized into nine skills: 
addition, subtraction, multiplication, division, fractions, decimals, numeration, 
ratio/per cent and word problems. The units in each skill area were arranged in 
a specific sequence in which each unit depended on the last unit's having been 
mastered. 
The units were also organized into blocks that included units from several 
skill areas but at about the same level of difficulty. Most blocks included at least 
one word problem unit. A block might have included, for example, several multipli-
c ation units, a few fraction units, a numeration unit and a word problem unit. 
Students could complete the units in a block in any order as long as they 
did the units in the same skill area in order. They were required to complete all 
the units in a block before they could proceed to the next block. 
Students maintained personal folders with all of their work and with a Math 
Progress Form on which they checked off units as they completed them. The teacher 
c hecked each student's Math Progress Form before making up groups for review 
36 
sessions, to see how students were doing and to prepare for anticipated difficult 
upcoming units. 
Record keeping. The Individual Progress Record was used to monitor student 
progress through the curriculum. An example of an Individual Progress Record 
appears as an Appendix. Each Individual Progress Record encompassed one week. 
The letter-number combination (such as M 13, D3) in the space for "unit," and the 
number correct on the fifteen-item test in the space for "score" were recorded. 
When a student passed a step in the facts sequence, this was indicated by putting, 
for example, MF2 to show that the student passed the second step in multiplication 
facts (the B section, 4's, 5's, and 6's facts). The facts units were designated AF 
(addition), SF (subtraction), MF (multiplication) or OF (division). Under the unit 
I 
designation, a "15" was recorded to indicate a perfect score (the 100% correct 
required to pass the unit). 
In addition to the Individual Progress Record, student progress was recorded 
on the Math Progress Forms. 
Team scores. At the end of each week, the teacher calculated team scores. 
Those scores were the sum of the average number correct of all tests taken by 
all team members (the Accuracy Score) and the average number of units covered 
by each team member times ten (the Progress Score). To calculate the Accuracy 
-<-
Score, the sum of the number correct on all Final Tests taken by all students in 
the team was divided by the number of tests taken and multiplied by ten. If a 
student attended class two days or less, his or her scores were not included in 
the Accuracy Score. To compute the Progress Score, the total number of tests 
taken by all students who were in the class three or more days were counted and 
divided by the number of students on the team who were present three or more 
days and multiplied by ten. For the purposes of scoring, a "week" was considered 
37 
four or more days. If a week had three or fewer days due to snow days, holidays, 
school assemblies etc., the week's scores were combined with those of the next 
week. 
The team score was equal to the sum of the Accuracy Score and the Progress 
Score. A sample Team Score Sheet appears in the Appendix. 
Team recognition. It was critical in Team-Assisted Individualization for 
tudents to feel that team success was important, and that any team could be 
successful. For this purpose, goals were set for team success that teams could 
achieve if they worked well together. Students could see how their accomplishments 
moved the team toward its goal as they posted their scores on the Public Team 
Score Sheet. The goals for different levels of team achievement and the awards 
for achieving them were as follows: 
Award Goal Reward 
GOODTEAM 35 Congratulations in class 
GREATTEAM 40 Certificate to each 
team member 
SUPER TEAM 50 Fancy certificate to 
each team member 
The GREATTEAM certificates were standard certificates for all students who 
made 40 or more team points. SUPERTEAM certificates were fancier and had 
spaces on which to put the team and team member's name. In addition to these 
rewards, the teacher was encouraged to give other special privileges to successful 
teams . It was also recommended that the team names be put on pieces of cardboard 
and then displayed on a bulletin board on which the team names were displayed 
under "GOODTEAM," "GREATTEAM" or "SUPERTEAM." It was very important 
that the teacher emphasize the importance of team success and the fact that 
38 
a ny team c ould a chieve it if they worked together. A sample SUPER TEAM certificate 
appears in the Appendix. 
If a short week e xisted because of vacations, assemblies, fire drills, etc. , 
t he teac her could lower the goals temporarily. If two weeks were combined, the 
teacher could raise the goals temporarily. 
Rapid Progress Mathematics (T ) 
2
Diagnostic test. TI1e students were pretested at the beginning of the project 
on mathematics operations and were placed at the appropriate point in the individual-
ized program based on their performance. 
Curric ulum materials. For all operations skills, the students worked on individ-
ualized c urriculum materials that had the following subparts: 
l. Instruction Sheet explaining the skill to be mastered and giving a step-
by-ste p method of solving problems. 
2. Several Skillsheets, each consisting of twenty problems. Each Skillsheet 
introduced a subskill that led to final mastery of the entire skill. 
3. A Checkout, whic h consisted of two parallel sets of ten items. 
4. A Final Test . 
5. Answer Sheets for Skillsheets, Checkouts and Final Tests. 
Monitors. At the beginning of each class period, three students were appointed 
t o se rve as monitors for that period. Their responsibilities were to distribute, 
score a nd file appropriate tests. 
Study method. Following the diagnostic test, students were given a starting 
place in the individualized mathematics units. They progressed through their units, 
fo llowing these steps. 
l . Students located the unit on which they were working and brought 
39 
it to their desks. The units consisted of the Instruction Sheet, Skillsheets 
and Checkouts stapled together, and the Skillsheet Answers and Checkout 
Answers stapled together. 
2. Each student read his or her Instruction Sheet and asked the teacher 
for help, if necessary. 
3. When they had read the Instruction Sheet, the students began with 
the first Skillsheet in the unit. Students copied the problems from 
the stapled sheets onto their own paper prior to completing the tasks. 
4. Each student worked the first four problems after any examples on 
his or her own Skillsheet and then checked the answers against the 
Answer Sheet. If all four were correct, the student proceeded to the 
next Skillsheet. If any were incorrect, the student tried the next four 
problems, and so on until he or she got one block of four problems correct. 
5. When a student successfully completed four in a row on the last Skillsheet, 
he or she took Checkout A, a ten-item quiz that resembled the last 
Skillsheet. On the Checkout, students worked until they we re fi nished. 
They scored their own Checkouts, using their Answer Sheets . If the 
student got eight or more correct, he or she signed the Checkout to 
indicate that he or she was ready to take the Final Test. If the student 
did not get eight correct, he or she called on the teacher to explain 
any problems. The teacher had the prerogative of asking the student 
to work again on certain Skillsheet items or the student may have then 
taken Checkout B, a second ten-item test comparable in content and 
difficulty to Checkout A. Otherwise, students skipped Checkout B 
and went directly to the Final Test. No student could take the Final 
Test until he or she had passed a Checkout. 
t+O 
6. When a student "checked out," he or she took the Checkout to a student 
monitor. Three students served as monitor at any given time. The 
monitors' responsibilities included finding the proper test and giving 
it to the student. The student moved his or her desk away from all 
other students to take the test. A student taking a test was not to 
be disturbed for any reason. After completion, the student located 
the monitor who scored the test. The student could check the scoring 
in the presence of the monitor and bring any discrepancies to the attention 
of the teacher for resolution. The teacher had the option to check 
all Final Tests to keep track of student progress and to be sure that 
all scoring was accurate. When the tests were scored, they were given 
to the teacher to be entered on the student's Math Progress Form and 
recorded by the teacher. 
Facts sequence. The facts were taught according to the procedures outlined 
on the preceding pages in the TAI guidelines. 
Teacher review sessions. As soon as students were working well on the individ-
ua lized materials, the teacher began forming groups for instruction as described 
on the preceding pages in the TAI guidelines. 
Curriculum organization. The curriculum is organized identically to the 
TAI c urriculum as previously described. 
Record keeping. Record keeping procedures were identical to those in the 
TAI treatment. 
Regular Program (T ) 
3
The curriculum is that which would have been used for instruction whether 
or not this study had occurred, but was restricted to the same nine units of study 
41 
which were included in T 1 and T 2 (numeration, addition, subtraction, multiplication, 
division, fractions, decimals, word problems, ratio/per cent). 
Test Instruments 
The standardized Comprehensive Tests of Basic Skills Mathematics Computation 
(1 973) subscale was administered as pre- and posttests to assess achievement. 
The forty-eight item test was forty-five minutes in length. 
A sixteen-item questionnaire was given as a pre- and post measure to assess 
students' attitudes toward mathematics and feelings of competence in mathematics. 
The scales were "Liking of Mathematics" and "Feelings of Competence in Mathematics" 
and were presented in a Likert-type format where students ' agreement or disagree-
ment with each statement was indicated by circling one of the following responses: 
YES! yes no NO! This was adapted from scales used previously by Slavin (1978c). 
A diagnostic test was administered to determine appropriate placement 
in the program. The test was designed as a part of the Team-Assisted Individuali-
zation package and consisted of sixty-four items of varying levels of difficulty 
which were representative of the nine skills included in the materials (Slavin, Leavey 
& Madden, 1980 & 1981). 
Treatment Schedule 
The timeline for implementation of the study was as follows: 
March 16 a.m. inservice for T 1 teachers 
p.m. inservice for T 2 teachers 
March 17 administered diagnostic pretest 
March 18 administered achievement pretest 
42 
March 19 administered attitude survey pretest 
March 20 administered make-up tests 
March 23 project began 
May 22 project ended 
May 26 administered diagnostic posttest 
May 27 administered achievement posttest 
May 28 administered attitude survey posttest 
May 29 administered make-up tests 
Outline of Research Design 
A Pretest-Posttest Control Group Design outlined in Stanley and Campbell 
(1963) was used to assess the effects of the use of specifically designed individual-
ized mathematics materials and the accompanying management system. Utilization 
of table of random numbers determined assignment of schools to treatment groups 
T , T 2 or T 3. Within each school, one teacher in grades three, four and five, or 1
a total of three teachers was assigned by the Principal to participate in the study? 
This study made it possible to assess the effects of using Team-Assisted 
Individualization in a classroom setting which was comprised of students with 
varying levels of ability. Use of analysis of covariance with pretests and grade 
level as covariates reduced the possibility that any differences found were due 
to pre-existing differences. 
Threats to the generalizability of the findings to the regular classroom were 
minimized by specific features of the experimental design . First, regular classroom 
teachers who were encumbered by all of the usual problems of teaching full time, 
implemented the treatments. This would be true in most situations in which any 
findings would be applied. Second, the use of the treatment with materials only 
(T ) also demanded a restructuring of the classroom and a new curriculum. Treat-
2
ment T 3 groups proceeded according to what they would have done if they had 
. I 
not participated in the study. The design can be described as follows: 
RO T 0 
1 
RO T2 0 
RO T 0 
3 
Table 1 provides a complete outline of the research design for this study. Only 
TABLE l 
Outline of Research Design 
Teacher Grade Level School Treatment 
I 3 A Tl 
II 4 A Tl 
III 5 A Tl 
IV 3 B T2 
V 4 B T2 
VI 5 B T2 
VII 3 C T3 
VIII 4 C T3 
IX 5 C T3 
X 3 D Tl 
XI 4 D Tl 
XII 5 D Tl 
XIII 3 E T2 
XIV 4 E T2 
xv 5 E T2 
XVI 3 F T3 
XVII 4 F T3 
XVIII 5 F T3 
one of the three treatments was assigned to a given school. Different treatments 
in progress concurrently within the open-spaced structures would have confounded ' I 
t he treatments. It was also the intent of the researcher to build a support system 
among teachers in a given building who were participating in identical programs. 
Hypotheses to be Tested 
Providing appropriate instruction for students of high, average and low abilities 
within the same classsroom has geen a goal for many individualized mathematics 
programs. However, available programs have not provided solutions to many of 
the problems which are previously discussed. Thus, this study was designed to 
answer the following question: Will the use of Team-Assisted Individualization 
result in a difference in third, fourth and fifth graders' achievement and attitudes? 
With the aforestated question in mind, the researcher formulated the following 
hypotheses: 
H 1: There will be a significant difference among treatments on the 
mean scores of the achievement pre- and posttests in favor of the 
T 1 classes. 
H2: There will be a significant difference among treatments on the 
mean scores of the attitude pre- and posttests in favor of the T 1 
classes. 
H3: There will be a significant difference among treatments on the 
mean scores of the achievement pre- and posttests in favor of the 
T 2 classes over the T classes. 3 
H4: There will be a significant difference among treatments on the 
mean scores of the attitude pre- and posttests in favor of the T 2 
classes over the T 3 classes. 
Analysis of covariance was used to determine statistical significance which was 
predicted to be at the p = <.05 level on all comparisons. It was determined that 
45 
grade level and pre-test scores on both the CTBS-math and diagnostic tests would 
be used as covariates. It was agreed that if significance were found, pairwise ( I 
comparisons would be made using the procedures as diagrammed below: 
Data Analysis 
The data analysis was comprised of four basic parts. The first set of analyses 
examined the effect of TAI on students' attitudes toward mathematics. The second 
set of analyses dealt with the effects of RPM on students' attitudes towards math-
emetics . The third set analyzed data to determine the effect of TAI on students' 
achievement. A fourth set of analyses dealt with the effect of RPM on students' 
ach ievement. Data analyses are outlined as follows: 
Analysis I: Attitudes of T and T students at all grade levels. 
1 2 
Analysis II: Attitudes of T and T students at all grade levels. 
1 3 
Analysis III: Attitudes of T and T students at all grade levels. 
2 3 
Analysis IV: Achievement of T and T students at all grade levels using 
1 2 
pretest achievement on the CTBS-math and grade level as covariates. 
Analysis V: Achievement of T and T students at all grade levels using 
1 3 
pretest achievement on the CTBS-math and grade level as covariates. 
Analysis VI: Achievement of T and T students at all grade levels using 
2 3 
pretest achievement on the CTBS-math and grade level as covariates. 
Analysis VII: Achievement of T and T students at all grade levels using 
1 2 
pre test achievement on the diagnostic test and grade level as covariates. 
Analysis VIII: Achievement of T 1 and T students at all grade levels using 3 
pretest achievement on the diagnostic test and grade level as covariates. 
Analysis IX: Achievement of T and T students at all grade levels using 
2 3 
pre test achievement on the diagnostic test and grade level as covariates. 
CHAPTER FOUR 
Results of the Study 
( . 
The data were analyzed by means of analysis of covariance, with the CTBS-
TABLE 2 
Means and Standard Deviations, Grade Three 
Achievement and Attitude Measures 
Instrument -X TAI SD -X RPM SD X Control SD 
CTBS - Math Pre 19.84 6.23 19.73 7.94 17.06 6.88 
Post 24.33 7.87 22.65 8.65 19.43 9.19 
Adjusted 23.58 22.01 21.26 
Diagnostic Pre 23.02 6.44 25.35 9.04 21.19 8.86 
Math Test Post 29.65 6.92 26.65 9.13 24.57 9.72 
Adjusted 29.95 24.99 26.40 
Liking of Pre 21.55 6.44 25.07 5.98 23.05 5.33 
Math Class Post 25.24 6.73 26.72 5.34 20.89 6.74 
Adjusted 26.20 25.72 21.02 
Self-Concept Pre 23.32 3.69 24.31 5.62 24.50 4.27 
in Math Post 25.62 4.11 25.00 4.78 25.34 4.76 
Adjusted 26.02 24.83 25.06 
TABLE 3 
Means and Standard Deviations, Grade Four 
Achievement and Attitude Measures 
Instrument -X TAI SD X- RPM SD X Control SD 
CTBS - Math Pre 33.02 7.82 30.29 11.74 26.39 8.95 
Post 34.46 7.19 36.00 10.39 28.73 10.49 
Adjusted 31.91 35.62 31.45 
Diagnostic Pre 38.17 6.82 39.39 13.73 32.69 9.02 
Math Test Post 39.77 7.98 45.04 14.50 37.33 11.27 
Adjusted 38.25 42.32 41.23 
Liking of Pre 26.45 5.59 24.83 5.24 23.00 5.72 
Math Class Post 25.77 5.70 25.86 4.86 23.27 5.54 
Adjusted 24.94 25.79 24.07 
Self-Concept Pre 25.70 4.36 24.52 4.39 24.06 4.24 
in Math Post 25.52 5.06 25.05 4.04 24.02 4.92 
Adjusted 24.80 25.21 24.53 
46 
4-7 ' 
math and diagnostic pretests and grade level as the covariates. The experimental factor ? I 
was treatment: T 1 vs. T 3 and T 2 vs. T . Three comparisons were computed: T vs. 3 1 
TABLE 4 
Means and Standard Deviations, Grade Five 
Achievement and Attitude Measures 
Instrument -X TAI SD -X RPM SD X Control SD 
CTBS - Math Pre 36.59 7.29 35.29 8.97 38.11 6.4-3 
Post 39.57 6.15 36.15 9.23 39.11 7.34-
Adjusted 39.69 37.27 38.05 
Diagnostic Pre 4-5. 94- 6.12 4-3.4-7 9.83 4-5.54- 7.86 
Math Test Post 51.33 6.35 4-6.62 12.36 50.93 8.4-1 
Adjusted 50.4-1 4-8.02 50.38 
Liking of Pre 25.37 5.59 25.11 3.99 23.51 4-.4-3 
Math Class Post 24-.37 6.08 24-.02 5.4-5 21.54- 5.15 
Adjusted 23.84- 23.66 22.23 
Se lf-Concept Pre 25.67 3.96 23.91 4-.54- 24-. 97 4-.06 
in Math Post 26.22 3.53 24-.89 4-. 4-2 24-.14- 4-.55 
Adjusted 25.68 25.4-8 24-.05 
TABLE 5 
Means and Standard Deviations, Full Sample 
Achievement and Attitude Measures 
Instrument X- TAI SD -X RPM SD X Control SD 
CTBS - Math Pre 30.18 10.08 28.51 11.59 29.25 11.27 
Post 33.12 9.4-3 31.4-5 11.31 31.02 11.86 
Adjusted 32.4-0 32.18 30.95 
Diagnostic Pre 35.83 11.54- 36.11 13.4-0 35.05 13.05 
Math Test Post 4-0.4-0 11.39 39.35 15.10 39.68 15.10 
Adjusted 4-0.31 38.96 4-0.14-
Liking of Pre 24-.37 6.23 25.02 5.09 23.23 5.07 
Math Class Post 25.09 6.19 25.51 5.35 21.93 5.75 
Adjusted 24-. 94- 25.01 22.56 
e lf-Concept Pre 24-.87 4-.13 24-.23 4-.89 24-.56 4-.16 
in Math Post 25.80 4-.23 24-.97 4-.4-2 24-. 4-0 4-.72 
Adjusted 25.57 25 .14- 24-.4-4-
) 
48 :,( 
T , T vs. T and T 2 vs. T 3. Means, adjusted means and standard deviations by 
''i , I 
2 1 3 
grade level are shown on Tables 2, 3 and 4 and by treatment in Table 5. 
It was hypothesized that there would be significant differences on the mean 
scores of the achievement and attitude posttests (controlling for pretests and 
grade level) in favor of TAI over all groups and that there would be significant 
differences on the mean scores of the achievement and attitude posttests in favor 
of RPM over Control. 
Grade three comparisons show in Tables 6 and 7 that a statistically significant 
effe ct (p < .001) was found in favor of RPM and TAI students over Control students 
in Liking of Math. TAI students showed a statistically significant higher performance 
on the diagnostic test than Control students (p < .003), as shown in Table 7, and 
were significantly higher than RPM students on the diagnostic test (P<,001), as 
shown in Table 8. 
TABLE 6 
Individual Comparisons 
Grade Three, RPM vs. Control 
Instrument F df p < Direction 
CTBS - Math .342 1,83 N.S. 
Diagnostic 2.44 1,94 N.S. 
Liking of Math 19.113 1,89 .001 RPM > C 
Self-Concept .088 1,89 N.S. 
TABLE 7 
Individual Comparisons 
Grade Three, TAI vs . Control 
Instrument F df p < Direction 
CTBS - Math 2.789 1,75 N.S. 
Diagnostic 9.23 1,88 .003 TAl >C 
Liking of Math 20 .598 1,86 .001 TAl >C 
Self-Concept 2.039 1,85 N.S. 
49 ) :,.: 
TABLE 8 
Individual Comparisons ) i. . 
Grade Three, TAI vs. RPM ) 
:,( 
Instrument F df p < Direction 
CTBS - Math 1.856 1,91 N.S. 
Diagnostic 22.7 56 1,101 .001 TAI > RPM 
Lik ing of Math .008 1,102 N.S. 
Self-Concept 2.133 1,101 N.S. 
Comparisons of grade four students show (Table 9) that RPM students scored 
at a statistically significant level over Control students on the CTBS - math test 
(p <.003) and Liking of Math scale (p < .001). Table 10 shows a statistically signif-
icant difference in the performance of Control students over TAI students (p<.004) 
on the diagnostic test and, in Table 11, RPM students over TAI students on the 
CTBS - math test (p< .005) and diagnostic test (p. < .001). 
TABLE 9 
Individual Comparisons 
Grade Four, RPM vs. Control 
Instrum ent F df p < Direction 
CTBS - Math 9.083 1,88 .003 RPM > C 
Diagnostic ?6 41 1,94 N.S . 
Liking of Math 14.142 1,88 .001 RPM > C 
Self-Concept . 952 1,88 N.S . 
TABLE 10 
Individual Comparisons 
Grade Four, TAI vs. Control 
Instrument F df P < Direction 
CTBS - Math . 060 1,92 N.S . 
Diagnostic 8.763 1,96 .004 C >TAI 
Liking of Math .525 1,90 N.S. 
Self-Concept . 038 1,90 N.S . 
.. 
50 
TABLE 11 
Individual Comparisons 
Grade Four, TAI vs. RPM 
Instrument F df P < Direction 
CTBS - Math 8.273 1,85 .005 RPM > TAI 
Diagnostic 12.039 1,91 .001 RPM > TAI 
Liking of Math .816 1,83 N.S. 
elf-Concept .302 1,83 N.S. 
Table 12 shows that there was a statistically significant difference in the 
pe rformance of the grade five Control students over RPM students (p < .031) on 
t he diagnostic test and of RPM students over Control students on the self-concept 
cale (p < .032). TAI students also showed a statistically significant difference 
on the self-concept sc ale over Control students (p < .008) as shown in Table 13. 
Table 14 shows that there was a statistically significant performance on both the 
s tanda rdized (p <.014) and diagnostic tests (p < .050) by TAI students over RPM 
students. 
TABLE 12 
Individual Comparisons 
Grade Five, RPM vs. Control 
Instrument F df p < Direction 
CTBS - Math .301 1,116 N.S. 
Diagnostic 4.759 1,124 .031 C > RPM 
Liking of Math 3.182 1,118 N.S. 
Self-Concept 4.724 1,117 .032 RPM >C 
TABLE13 
Individual Comparisons 
Grade Five, TAI vs. Control 
Instrument F df p < Direction 
CTBS - Math 2.244 1,110 N.S. 
Diagnostic . 006 1,115 N.S . 
Liking of Math 3.445 1,116 N.S. 
Self- Concept 7.294 1,114 .008 TAl >C 
51 
TABLE 14 
Individual Comparisons ' I 
Grade Five, TAI vs. RPM 
Instrument F df P < Direction 
CTBS - Math 6.266 1,101 .014 TAI > RPM 
Diagnostic 3.914 1,108 .050 TAI > RPM 
Liking of Ma th .036 1,103 N.S. 
Self-Concept .143 1,102 N.S. 
As shown in Table 15, pretest comparisons of academic performance between 
classes in the different treatments show no significant differences (p > .10). However, 
the RPM groups entered the study with a better liking of math than did other groups. 
TABLE 15 
Analysis of Variance, TAI vs. RPM vs. Control, Full Sample 
Comparison of Performance on Pretests 
Instrument F df p < Direction 
CTBS - Math .826 2,431 N.S. 
Diagnostic .299 2,466 N.S. 
Liking of Math 4.14 2,448 .02 RPM >TAbC 
Self-Concept .783 2,445 N.S. 
The results of the overall comparison of TAI vs. RPM vs. Control on the 
standardized mathematics test (CTBS) showed that effects for treatment were 
not statistically significant. After adjustment for pretests and grade level by 
using the analysis of covariance the adjusted posttest scores were in the order 
TAI >RPM >Control. Statistically significant effects (p<.001) were found for the 
Liking of Math measure (RPM > TAI > Control) and for Self-Concept (p<.017; TAI> 
RPM > Control). 
The comparison of TAI vs. RPM (Table 17) showed that TAI students scored 
ignificantly higher than RPM students on the diagnostic test (p < .052), but there 
52 
TABLE 16 
Analysis of Covariance, TAI vs. RPM vs. Control, Full Sample ( I 
Posttest Comparisons on Achievement and Attitudes 
Instrument F df p < Direction 
CTBS - Math 2.74 2,429 N.S. 
Diagnostic 2.60 2,464 N.S. 
Liking of Math 11.60 2,446 .001 RPM >TAI >C 
Self-Concept 4.11 2,443 .017 TAI >RPM > C 
were no other differences. However, as is shown in Table 18, TAI students scored 
significantly higher than Control students on the CTBS-math test (p < .021), the 
Liking of Math scale (p < .001) and the Self-Concept scale (p < .008). 
TABLE 17 
Analysis of Covariance, TAI vs. RPM, Full Sample 
Achievement and Attitudes 
Instrument F df p < Direction 
CTBS - Math .305 1,282 N.S. 
Diagnostic 3.79 1,305 .052 TAI > RPM 
Liking of Math .015 1,293 N.S. 
Self-Concept 1.27 1,291 N.S. 
TABLE 18 
Analysis of Covariance, TAI vs. Control, Full Sample 
Achievement and Attitudes 
Instrument F df p < Direction 
C TBS - Math 5.36 1,282 .021 TAI> C 
Diagnostic .194 1,304 N.S. 
Liking of Math 16.26 1,297 .001 TAI >C 
Self-Concept 7.22 1,294 .008 TAI >C 
Al though RPM students scored higher than Control students on the CTBS-
math test and lower than Control students on the diagnostic test (p < .052), only 
the latter of these findings was statistically significant. RPM students scored 
53 
'( . \ 
significantly higher than Control on the Liking of Math scale (p < .001), and marginally 
higher on the Self-concept scale (p < .07 5). 
TABLE 19 
Analysis of Covariance, RPM vs. Control, Full Sample 
Achievement and Attitudes 
Instrument F df P < Direction 
CTBS - Math 2.87 1,292 N.S. 
Diagnostic 3.82 1,317 .052 C > RPM 
Liki ng of Math 19.37 1,300 .001 RPM > C 
Self-Concept 3.19 1,299 N.S. 
Test-retest correlations of pre- and posttest instruments are shown in Table 
20 , and the correlations of the standardized (CTBS) mathematics test and the 
diagnostic test are reflected in Table 21. 
TABLE 20 
Reliability Estimates 
Test-Retest Correlations 
Instrument r 
CTBS - Math .86 
Diagnostic .91 
Liking of Math .52 
Self-Concept .63 
TABLE 21 
Test Correlations 
CTBS and Diagnostic 
Test r 
CTBS and Diagnostic Pretests .81 
CTBS and Diagnostic Posttests .86 
) 
CHAPTER FIVE '>. 
Conclusions and Discussion 
I 
-I. I 
Four hypotheses were formulated in the proposal which served as the foundation 
for this study. It was hypothesized that: 
l. There will be a significant difference among treatments on the mean 
scores of the achievement pre-and posttests in favor of the TAI classes. 
2. There will be a significant difference among treatments on the mean 
scores of the attitude pre- and posttests in favor of the TAI classes. 
3. There will be a significant difference between treatments on the mean 
scores of the achievement pre- and posttests in favor of the RPM classes 
over the control classes. 
4. There will be a significant difference between treatments on the mean 
scores of the attitude pre- and posttests in favor of the RPM classes 
over the control classes. 
Conclusions 
Full sample analyses (Table 5) show that TAI students gained more than RPM 
and Control students on both the CTBS-math and diagnostic tests, but that the 
gain was not statistically significant (Table 16). Tables 17 and 18 show that TAI 
students gained at a statistically significant level over RPM on the diagnostic 
test and over Control on the CTBS-math test, but hypothesis one was not confirmed 
by the results of the study. 
Confirmation was not received on hypothesis two. RPM students showed 
a statistically significant gain (p < .001) over TAI and Control students on the Liking 
of Math scale, but TAI students gained a statistically significant amount (p < .017) 
over RPM and Control students on the Self-Concept scale (Tables 5 and 16). No 
54 
) 
55 )( 
statistical significance was found between TAI and RPM on both attitude scales 
) 
(Table 17), but a statistically significant gain was found in favor of TAI over Control )( I 
.?:  
students on both the Liking of Math scale (p < .00 l) and the Self-Concept scale 
(p < .008, Table 18). 
Hypothesis three cannot be confirmed. Results in Table 19 indicate that 
there was no significant difference between the RPM and Control groups on the 
CTBS-math test and that the Control group outperformed the RPM group on the 
diagnostic test (p <.052). 
Hypothesis four was not fully confirmed. Although RPM students gained 
at a statistically significant level over Control students on the Liking of Math 
cale (p <.001), no significant difference could be determined on the Self-Concept 
scale (Table 19). 
On the CTBS -mathematics test, both Team-Assisted Individualization and 
Rapid Progress Mathematics groups gained more than the Control group, while 
on the diagnostic test the Team-Assisted Individualization groups gained more 
than Rapid Progress Mathematics groups. This can be interpreted to mean that 
the results of the CTBS comparisons support the effectiveness of the individualized 
instructional program itself, while the results of the diagnostic test support the 
importance of the team component as an addition to the individualized program. 
Because the results from the CTBS -mathematics and diagnostic tests are different, 
firm conclusions are difficult to draw; but the fact that the Team-Assisted Individual-
ization classes gained the most overall on both measures supports the conclusion 
that the TAI treatment is effective in increasing student achievement. The unclear 
factor is whether this is due to the use of the team/individualized instruction combi-
nation or just to the program of individualization. 
Attitude effects were more distinct than achievement effects. They clearly 
56 
showed that Team-Assisted Individualization and Rapid Progress Mathematics 
tr eatments increased student attitudes toward mathematics and toward their own 
mathematical abilities more than the Control condition. Because no differences 
we re found between TAI and RPM on attitude measures, it must be assumed that 
the attitude effects are due to the use of the individualized mathematics program, 
not spe cifically to the combination of this program with the team component. 
This is contrary to Hypothesis 2 in that it was expected that the team component 
wou ld add to student attitudes. This finding, however, does correspond to impressions 
which resulted from classroom visits where it was observed that students very 
much enjoyed both the Team-Assisted Individualization and Rapid Progress Mathematics 
treatments. 
Discussion 
Overall, the results of the study support the usefulness of Team-Assisted 
Individualization for increasing student mathematics achievement and student 
a ttitudes toward mathematics. While there were major differences in the total 
sam ple, some differenc es could be determined in favor of TAI at specific grade 
levels (Tables 7, 8, 13 and 14 ). 
It was hypothesized that TAI would outperform both RPM and Control groups 
in t e rms of achievement. Perhaps the treatment was too short to effect such 
a change, or there may have been other factors since there is no way to control 
the actual procedures of implementation that teachers actually use. 
There is a possibility that other mathematics learning was occurring that 
wa s not measured by the tests which were used in the study. Perhaps different 
findings would have occurred if another standardized test had been used, since 
some of the items on the CTBS-math test were unrelated to the units in the TAI 
program. This is a possible explanation as to why little significant difference 
57 
was found on the TAI groups' CTBS-math test performance over other groups. 
Further studies should be concerned about the appropriateness of the achievement 
measure . 
In an attempt to develop some insight into the feelings of the teachers who 
pa rticipated in the program, a short questionnnaire was developed and administered 
to the TAI and RPM teachers. Team-Assisted Individualization and Rapid Progress 
Mathematics teachers reported very positive attitudes toward the program on 
the pa rt of both TAI and RPM, but neither received a stronger preference over 
the other. Of the six teachers involved in each treatment, five in each indicated 
that they would definitely use their method next year, and one teacher in each 
treatment predicted that she might use the same method. Four teachers in each 
treatment thought that their classes learned more than in their usual program, 
one t eacher in each treatment felt that their classes learned about the same as 
in their usual program and one teacher in each treatment felt that their classes 
learned less. Two TAI teachers and one RPM teacher felt that their method was 
the best they had used to teach mathematics, three in each treatment liked it 
be tter than their usual methods, one in each treatment liked it the same as their 
usual methods, and one RPM teacher liked RPM less than her usual method. 
A side note involves remarks from some teachers in the study. Comments 
included statements that certain students who seemed to have no desire to learn 
the basic facts did indeed learn them; that some students who were major discipline 
problems were no longer problems; that students who did not like mathematics 
and who were not performing assigned tasks won awards; that parents seemed 
e nthused and supportive of the programs; and that both high and low achievers 
could work successfully in the same classroom. 
In summary, there are some ambiguities in the results, but overall the study 
58 
showed that Team-Assisted Individualization, the first attempt to combine coopera-
t ive learning methods with individualization, had significantly positive effects 
on both achievement and attitudes. Rapid Progress Mathematics had positive 
effec ts on student attitudes, but its effects on achievement were unclear. Class-
room observations and a teacher survey gave the impression that both TAI and 
RPM are enjoyable methods of learning and teaching mathematics. 
Contribution 
Use of Team-Assisted Individualization provides teachers with a mathematics 
program which supports the Howard County Maryland Public School System's philosophy 
t hat students should receive instruction at their own level and rate of learning 
with e mphasis on achievement, opportunities to increase computational skills, 
positive attitudes and acceleration. It also provides for the diverse abilities of 
students who are assigned to a single classroom. 
The problems related to other individualized programs such as requiring 
the assistanc e of aides or volunteers, demands on the teacher, massive "checking" 
responsibilities, student wasting of time on known material or moving quickly 
through units without clear understanding, bored students, working in isolation, 
no peer interaction, infrequent feedback, proceeding on work which is error laden, 
waiting for checking and additional assignments, frustration and discipline problems 
a re minimized through the use of Team-Assisted Individualization. It has now 
bee n shown that TAI provides an alternative to the current elementary school 
mathematics program which has never before existed which addresses most of 
the concerns related to individualized instruction expressed in the literature, and 
that these issues have been resolved. 
Bestor (1955) wrote that the burden that individualized instruction places 
on the teacher is unrealistic and that the demands are beyond the reach of most 
59 
t eachers. Kepler and Randall (1977) wrote that materials in many individualized 
programs are not designed to provide clear directions to students, don't provide 
immediate feedback, create management problems for teachers and allow students 
to wait. Nix (1970), Osmundson (1972), Palow (1973) and Sutton (1976) agreed 
that the use of individualized programs overworked teachers. Schoen (1976) felt 
t hat the management problems which are part of individualized programs are detrimental 
t o learning. The succ ess of Team-Assisted Individualization shows that these need 
no longer be concerns. 
This study supports the research related to cooperative team learning which 
shows that positive attitudinal and ac ademic effects result from their use and 
that it provides teachers with an instructional strategy which allows for a successful 
way of meeting the diverse needs of students within the same class (Coleman, 
1959; DeVries & Edwards, 1973; Slavin, 1978c, 1979; Johnson, Johnson, Johnson 
& Anderson, 1976; Wheeler, 1977). Additional research is supported by Team-Assisted 
Individualization which showed that there was an increase in the liking of mathematics 
a nd self-concept (Gottheil, 1955; Phillips, 1956; Phillips & D'Amico, 1956; DeVries 
& Slavin, 1976, 1978). 
Limitations 
Length of the study could be a contributing factor to many of the unclear 
results. It would be de sirable to extend a similar study over a period of a school 
year as further described in this chapter. 
For students who had been receiving instruction in a structured setting, it 
is possible that a period of adjustment to the independent learning component 
with the a c companying responsibilities delayed an immediate and total participation 
in the assigned treatment and that, over a longer period of time, this would have 
been eliminated. 
60 
It is probable that the confusing performance patterns between grade levels 
we re partially due to differences between teachers rather than the differences 
in tre atm e nts. When dea ling with such a small number of teachers, it is very likely 
tha t this did affect results. This is further supported by analyzing results of the 
t eacher-a ttitude survey. Negative and positive teacher attitudes are known to 
im pac t upon learning. An interesting study would result if teachers were randomly 
select e d from the total population as opposed to the random selection of schools 
a nd the utilization of the teachers who happen to be assigned therein. 
The ultimate study would include the random selection of students throughout 
the county and the placement of these students in a particular treatment, but 
this is an impossibility in a public school system. 
There was a general impression on the part of the researcher that the attitude 
0? the building principal had a strong influence on the success of the program. 
One principal was so enthusiastic that new groups were formed and teachers were 
reass igned to accommodate the study. In a few others, the principal was involved 
on a daily basis, offered continuous support and learned details of the program. 
In two schools the principal had minimal participation. It might be interesting 
to study the effects of principal involvement on the success of an innovation such 
as TAI. 
Many times the key to successful instruction is controlled by a teacher's 
management and organizational techniques rather than the knowledge of content. 
Li mitations of individualized programs have been discussed previously in detail. 
Team-Assisted Individualization offers a solution to the multiple problems named, 
as well as provides a unique and workable management system which should make 
a pos itive impact in any classroom. Teaching ability is as diverse as the academic 
needs and abilities of students. TAI provides a successful alternative to the teaching 
61 
of mathematics which, hopefully, will be used as widely throughout the country 
a s are STAD, TGT and Jigsaw. 
Implicat ions for Howard County 
It would be desirable to expand this study in the project schools. There is 
a strong feeling on the part of the researcher that, if the program were expanded 
over the period of a school year, results would show more consistent improvement 
in all four areas tested (CTBS - math, diagnostic, Liking of Math and self-concept). 
The intensive use of the Team-Assisted Individualization and Rapid Progress Mathematics 
programs over a period of eight weeks as was done in this study should be modified 
so that the program is in use for 2 to 3 weeks at a time. At the end of each of 
these periods, instruction should be directed to such units as measurement, geometry 
and other units not included in the material, and toward units with more emphasis 
on application of skills which lend themselves to activity-oriented and/or teacher-
directed lessons. 
Funds should be budgeted for the purpose of purchasing one set of materials 
fo r each of the nonproject schools and accompanying teacher inservice. Volunteer 
teachers within each school should have the opportunity to use the program. 
The elementary supervisors should receive inservice on the program as well 
so each can serve as a resource person who is easily accessible to teachers in their 
assigned areas; however, the primary responsibility should be assumed by the elementary 
supervisor who is responsible for the mathematics curriculum for the county. 
Inservice should include both the Team-Assisted Individualization and Rapid 
Progress Mathematics procedures so that teachers can opt to implement the program 
in the manner that is most appropriate for them. If results of the one-year study 
provide statistically significant results in favor of either TAI or RPM, the elementary 
supervisors should be responsible for conveying the results to the Director of Curriculum, 
principals and teachers who are using the programs. 
62 
It would be desirable to monitor the progress of the project students to determine 
if they score higher in mathematics on the required standardized tests which are 
administered to students each year, as well as to whether a greater percentage 
of these students enter classes in higher mathematics than do control students. 
A similar new study should be made at the middle school to determine if 
Team-Assisted Individualization will provide a successful alternative at that level. 
"Research on cooperative learning techniques represents an unusual event 
in the history of educational research. The techniques arose out of social-psychological 
t heo ry; they have been evaluated in numerous field experiments that were generally 
high in both internal and external validity; and they are in use in hundreds of classrooms 
across the country and in Israel" (Slavin, 1980b, p. 110). The addition of Team-
Assisted Individualization to cooperative learning techniques in the classroom 
should provide another successful alternative to those which presently exist . 
Additional Research 
As in any program of research there is a need for further investigation of 
limitations, interactions and extensions of findings, some of which might include 
t he following as related to Team-Assisted Individualization: 
1. What are optimum uses of TAI materials? 
2. What effects on attitude and achievement would result if the TAI program 
were integrated with other units and extended over the period of one 
school year? 
3. What effects does selection of the teacher have on the successful implementation 
of the TAI program? 
4. Can additional units be developed and successfully incorporated into 
the TAI program? 
63 
5. With what types of students is TAI most effective? 
6. To what degree does student training for working in TAI affect the 
success of student achievement and attitude? 
7. How can TAI be adapted for use with students who have special needs? 
Appendices 
Mathe matics Attitude Questionnaire . 65 
List of Units by Skill Area . 67 
Mathematics Progress Form 71 
Team Score Sheet (TAI) . . 78 
Individual Progress Sheet (RPM) . 79 
Facts Progress Form . 80 
Facts Sheets . . . . 81 
Sample Instruction Sheet 85 
Sample Skillsheet . . . 86 
Sample Skillsheet Answer Sheet 89 
Sample Checkout . . . . . . 92 
Sample Checkout Answer Sheet 93 
Superteam Certificate (TAI) 94-
Diagnostic Test . . . . . . 95 
64-
=--------- -~~---- ..,.-c;..,.. 
65 
ame __________________  
Teacher- - ------------- -
Mathematics Attitude Questionnaire 
How do you feel about your math class? Please tell us how you honestly 
fee l. This is not a test, and no one in your school will know what you write down. 
DIRECTIONS: 
1. Read each sentence carefully. 
2. Think about how well the sentence describes the class you are in now. 
3. C ircle one (and only one) of the four words across from the sentence. 
Circle YES! if you think the statement is definitely true for you. 
Circle yes if you think the statement is mostly true for you. 
Circle no if you think the statement is mostly not true for you. 
Circle NO! if you think the statement is definitely not true for you. 
EXAMPLES: 
1. I like spinach. YES! yes no NO! 
Circle the letter that tells how you feel about spinach. 
--If you really love spinach, circle YES! 
--If you kind of like it, circle yes. 
--If you do not like it, circle no. 
--If you hate it, circle NO! 
2. I do not like to roller skate. YES! yes no NO! 
--If you do like to roller skate very much, you would circle NO!, 
becausethe sentence is definitely not true for you . 
66 
l. I like this math class very much. YES! yes no NO! 
2. I am good at math in this class. YES! yes no NO! 
3. This math class is always a lot of fun. YES! yes no NO! 
4. I can do almost all the math problems YES! yes no NO! 
in this class. 
5. This math class is sometimes boring. YES! yes no NO! 
6 . I'm proud of my math work in this class. YES! yes no NO! 
7. This math class is the best part of YES! yes no NO! 
my school day. 
8. I have trouble learning new things in math. YES! yes no NO! 
9 . I look forward to math class every day. YES! yes no NO! 
l 0 . I'm doing the best work I can do in math. YES! yes no NO! 
11. I wish I didn't have to go to this math class. YES! yes no NO! 
12. I can do math as well as anyone else in YES! yes no NO! 
this class. 
13. I like learning math in this class. YES! yes no NO! 
14. I'm a success in math. YES! yes no NO! 
15. I like my math class this year better than YES! yes no NO! 
last year. 
16. I worry a lot when I have to take a math YES! yes no NO! 
test. 
67 
LIST OF UNITS BY SKILL AREA 
umeration 
N l Numeration - Tens 
N2 Numeration - Hundreds 
N3 Numeration - Working with Ones, Tens and Hundreds 
1 4 Numeration - Thousands 
N5 Numeration - Greater than, Less than 
6 Numeration - Ten Thousands 
N7 Numeration - Hundred Thousands 
8 Numeration - Place Value 
N9 Numeration - Millions and Billions 
10 Numeration - Approximation 
Addition 
AF l Understanding Addition 
Al Adding Three Numbers 
A2 Adding One Digit and Two Digits 
A3 Adding Two Digits and Two Digits 
A4 Adding Three or Four Single Digit Addends 
A5 Adding Three Digits and Three Digits 
A6 Adding One or Two Digits to Three Digits 
A7 Introduction to Addition with Renaming 
A8 Addition: Renaming with Tens and Hundreds 
A9 Renaming Twice in Addition 
A l O Renaming When Necessary 
A 11 Column Addition 
Al2 Adding Four or Five Digits: No Renaming 
13 Renaming More than Twice 
Al4 Adding Money 
Subtraction 
Sl Subtracting Two or Three Digits Minus One Digit: No Renaming 
S2 Subtracting Two or Three Digits Minus Two Digits: No Renaming 
3 Renaming Tens as Ones 
4 Renaming Tens as Ones for Subtraction 
S5 Renaming Hundreds as Tens 
S6 Renaming Hundreds as Tens for Subtraction 
S7 Renaming Twice in Subtraction 
S8 Zero in Subtraction I 
S9 Zero in Subtraction II 
S l O Renaming Thousands in Subtraction 
11 Subtracting Money 
l. 
68 
.\11 ul tiplication 
M l Introduction to Multiplication 
M2 Multiplying Multiples of Ten by One Digit 
13 Multiplying Two Digits by One Digit: No Renaming 
4 Multiplying Two Digits by One Digit with Renaming 
iV15 Multiplying Multiples of 100 by One Digit 
M6 Multiplying Three Digits by One Digit with Carrying 
M 7 Multiplying Three Digits by One Digit: Zero in the Tens ' or Ones' Place 
18 Multiplying by Multiples of Ten: No Carrying 
~9 Multiplying by Multiples of Ten with Carrying 
Ml O Multiplying Two Digits by Two Digits: The Long Way 
M 11 Multiplying Two Digits by Two Digits: The Short Way: No Carrying 
M 12 Multiplying Two Digits by Two Digits: Carryi ng Once 
1\ 13 Multiplying Two Digits by Two Digits: Carrying Twice 
M 14 Multiplying Three Digits by Three Digits 
M 15 Multiplying with Money 
Division 
D l Introduction to Division 
02 Labeling in Division 
03 Two Digits -c-One Digit = Two Digits 
D4 Two Digits"'" One Digit = One or Two Digits 
D5 Division with a Remainder 
06 Checking an Estimate: Too Much 
D7 Checking an Estimate: Too Little and Too Much 
08 Two Digits "'" One Digit = One Digit 
D9 Two Digits -;- One Digit = Two Digits: With Renam ing, No Remainde r 
0 10 Two Digits 7 One Digit = Two Digits: With Renam ing and Remainder 
I 11 Two Digits"'" One Digit = Two Digits: Zero in t he Ones ' Place 
O 12 Mixed Practice: Two Digits"'" One Digit 
D 13 Three Digits "'" One Digit = Three Digits 
D 14 Three Digits "'" One Digit = Two or Three Digits 
D l 5 Three Digits "'"One Digit = Three Digits : Zero in the Tens ' Place 
D 16 Mixed Practice : Three Digits "'" One Digit 
D 17 Four Digits "'" One Digit = Three or Four Digits 
018 Two or Three Digits"'" Multiples of Ten= One Digit : No Renaming 
D 19 Three Digits -c-Multiples of Ten = Two Digits: Est imating with Renaming 
D20 Three Digits ;- Multiples of Ten = Two Digits: With Renaming and Remainder 
D2 l Three Digits "'"Multiples of Ten = One or Two Digits 
022 Three Digits -;- Two Digits = One Digit: Learning to Estimate 
D23 Three Digits 7 Two Digits = One Digit: Est imat e Too High or Too Low 
D24 Three Digits 7 Two Digits = Two Digits: Learning to Estimate 
D25 Three Digits 7 Two Digits = Two Digits: Estimate Too High 
D26 Three Digits "'" Two Digits = Two Digits: Estimate Too Low 
027 Three Digits"'" Two Digits = Two Digits : Mixed Practice 
D28 Fou r Digits"'" Two Digits = Two or Three Digits 
69 
Common Fractions 
fl Identifying 1/2, 1/3, 1/4, 1/5, and 1/8 
F2 Numerators Greater Than One 
F3 Frac tions and Whole Numbers 
F4 Com paring Fractions I 
F 5 Comparing Fractions II 
F6 Addition of Fractions I 
F7 Subtraction of Fractions I 
F8 Equivalent Fractions 
F9 Factors 
F 10 Greatest Common Factor 
F 11 Simplest Name 
1:;- 12 Addition of Fractions II: Simplest Name for the Sum 
F 13 Subtraction of Fractions II: Simplest Name for the Difference 
F 14 Mixed Numerals 
F 15 Addition with Mixed Numerals: Like Denominators 
F 16 Least Common Multiple 
F 17 Least Common Denominator 
F 18 Addition with Unlike Denominators 
F 19 Subtraction with Unlike Denominators 
F20 Dividing to Find Mixed Numerals 
F 2 l Addition of Fractions III: Improper Fractions in the Sum 
F22 Subtraction of Fractions III: Mixed Numerals in the Difference 
F 23 Multiplying Fractions I 
F24 Multiplying Fractions II: Fractions with Whole Numbers 
F25 Multiplying Fractions III: Fractions and Mixed Numerals 
F 26 Multiplying Fractions IV: Mixed Numerals/Mixed Numerals and Whole 
Numbers 
F27 Multiplying Fractions V: Reciprocals 
F28 Dividing Fractions I: Dividing a Fraction by a Fraction 
F29 Dividing Fractions II: Fractions and Whole Numbers 
F30 Dividing Fractions III: Fractions and Mixed Numerals 
F 31 Addition of Fractions IV: Mixed Numerals with Unlike Denominators 
F32 Subtraction of Fractions IV: Mixed Numerals with Unlike Denominators 
Decimal Frac tions 
Dee l Decimal Tenths 
Dec2 Decimal Hundredths 
Dec3 Decimal Thousandths 
Dec4 Deci mal: Place Value 
Oec5 Comparing Decimals 
Dec6 Adding Decimals 
Dec? Subtracting Decimals 
Dec8 Multiplying by 10, 100, 1000 
Dec 9 Multiplying by .1, .01, .001 
D c lO Multiplying with Decimals 
Dee l 1 Dividing by Tenths 
Dec l2 Dividing by Hundredths 
Dec l3 Dividing Whole Numbers with a Decimal Divisor 
Dec l 4 Writing Decimal Fractions for Common Fractions 
70 
\V o rd Problems 
w l word Problems: Addition or Subtraction 
\V 2 Word Problems: Multiplication 
\V 3 Word Problems: Addition or Multiplication 
W4 Word Problems: Addition, Subtraction or Multiplication 
W5  Word Problems: Division 
\V6 Word Problems: Multiplication or Division 
W7 Word Problems: Addition, Subtraction, Multiplication or Division 
Ratio and Per Cent 
R l Ratios: An Introduction 
R2 Equivalent Ratios 
R3 Ratios: Time, Rate and Distance 
R4 Ratios and Money 
P 1 Introduction to Per Cent 
P2 Decimals and Fractions for Per Cent 
PJ Per Cents for Decimals and Fractions 
P4 Using Per Cent in Multiplication: Fraction Method 
P5 Using Per Cent in Multiplication: Decimal Method 
71 
Team Assisted Individualization 
Ma th Progress Form 
You may decide to start with a ny skill (addition, multiplication, and so 
on) you wish within a block . You must comple t e all of the units on that skill 
before moving to another skill. You must complete all of the skills within a block 
before start ing a new block. 
Oat Test Date Test 
Comp . Score BLOCK l Comp. Score BLOCK 2 
N l Numeration - Tens N2 Numeration: 
Hundreds 
A Intro-Understanding 
Addition A5 Adding Three 
Digits and Three 
Al Adding Three Numbers Digits 
A2 Adding One Digit and A6 Adding One or 
Two Digits Two Digits to Three 
Digits 
A3 Adding Two Digits and 
Two Digits S Intro-Understanding 
Subtraction 
A4 Adding Three or Four 
Single Digit Addends Sl Subtracting Two 
or Three Digits Minus 
One Digit-No Renaming 
S2 Subtracting Two 
or Three Digits Minus 
Two Digits - No Renaming 
BLOCK 3 BLOCK 4 
N3 Numeration-Working S3 Renaming Tens 
with Ones & Hundreds as Ones as Ones 
A7 Introduc tion to Addition S4 Renaming Tens 
with Renaming as Ones for Subtraction 
A8 Addition-Renaming in S5 Renaming Hundreds 
Tens a nd Hundreds as Tens 
A9 Renaming Twice in S6 Renaming Hundreds 
Addition as Tens For Subtraction 
W l Word Problems: AlO Renaming When 
Addition or Subtraction Necessary 
A 11 Column Addition 
72 
You may decide to start with any skill (addition, multiplication, and so on) you 
ish within a block. You must complete all of the units on that skill before moving 
to another skill. You must complete all of the skills within a block before starting a 
new block . 
Date Test Date Test 
Comp. Score BLOCK 5 Comp. Score BLOCK 6 
57 Renaming Twice in Al2 Adding Four 
Subtraction or Five Digits- No 
Renaming 
S8 Zero in Subtraction I 
Al3 Renaming More 
59 Zero in Subtraction II Than Twice 
N4 Numeration - Thousands Al4 Adding Money 
N5 Numeration - Greater SlO Renaming Thousands 
Than Less Than in Subtraction 
N6 Numeration - Ten Thousands S 11 Subtracting Money 
M 1 Introduction to 
Multiplication 
BLOCK 7 BLOCK 8 
M2 Multiplying Multiples M7 Multiplying Three 
of ten by one digit Digits by One Digit: 
Zero in the Tens' 
M 3 Multiplying Two Digits by or Ones ' Place 
One Digit (No Renaming) 
M8 Multiplying by 
M4 Multiplying Two Digits by Multiples of Ten: 
One Digit with Renam ing No Carrying 
M5 Multip lying Mulitiples of M9 Multiplying by 
100 by One Digit Multiples of Ten With 
Carrying 
M6 Multiplying Three Digits 
by One Digit with Carryi ng W3 Addition or Multipli-
cation 
W2 Word Problems: Multipl i-
cation D 1 Introduction to 
Division 
D2 Labeling in Division 
73 
You may decide to sta r t with a ny skill (a ddition, multiplication, and so on) you 
1ish within a block. You must complete all of the unit s on that skill before moving 
t o anothe r skill. You must complete a ll of t he skills within a block before starting a 
e w block . 
Da t e Test 
Comp . Sc ore BLOCK 9 Date Test 
Comp. Score BLOCK 10 
Ml O Multiplying Two Digits 
by Two Digits : The Short Fl Identifying l /2, 
Way 1/3, 1/4, 1/5 and 
1/8 
M 11 Multiplying Two Digits 
by Two Digits : The Long F2 Nu merations 
Way - No Car r ying Greater Than One 
D3 Two Digits -(?One Digit == W5 Word Problems: 
Two Digits Division 
D4 Two Digits -( One Digit == D6 Checking an Estimate : 
One or Two Digits Too Much 
D5 Division with a Remainde r D7 Checking an Estimate: 
Too Little & Too 
W 4 Add ition, Subtraction or Much 
M ul tiplica tion 
D8 Two Digits .;- One 
Digit == One Digit 
N7 Hundred Thousands 
BLOCK 11 
BLOCK 12 
F3 Fractions a nd Whole Numbe rs 
N9 Millions and Billions 
F 4 Compar ing Fractions I 
D9 Two Digits .;- One 
F5 Compa ring Frac tions II Digit == Two Digits 
with Renaming, no 
Ml 2 Mul t iplyi ng Two Digit s by Remainder 
Two Digits Carrying Once 
DlO Two Digits-=One 
M 13 Mul tiplying Two Digits by Digit == Two Digits 
Two Digit s Carrying Twice with Renaming and 
Remainder 
N8 Numeration: Place Value 
F6 Addition of Fractions I 
F7 Subtrac tion of 
Fractions I 
F8 Equivalent Fractions 
74 
You may decide to sta r t with any ski ll (addition, multiplica t ion, and so on) you 
jsh wi t hin a block . You must complete all of the units on that skill before moving 
to a nother skill. You must complete all of t he skills within a block before starting a 
ne w block . 
o ci t e Test Date Test 
-c omp . Score BLOCK 13 Comp. Score BLOCK lt+ 
Mlt+ Multiplying Three Digits 
by Three Digits Dl l Two Digit s -,-
One Digit = Two Digits: 
M Multiplying with Money Zero In the Ones' 15 Place 
F9 Factors 
D12 Mixed Practice 
- Two Digits -,- One 
F 10 Greatest Common Factor Digit 
F 11 Simplest Name 
Fl2 Addition of Fractions 
II: Simplest Name 
W6 Word Problems: Multipli-
cation o r Division for the Sum 
Fl3 Subtraction of 
Fractions II: Simplest 
BLOCK 15 Name for the Difference 
Flt+ Mixed Numerals 
W7 Wor d Problems: Addition, 
Subtraction, Multiplication 
or Division Fl5 Addition with 
Mixed Numerals: 
Like De nominators 
NlO Numeration : Approximation 
D13 Three Digits-,-One Digit= 
Three Digits BLOCK 16 
Dec l Decimal Tenths 
D 14 Three Digits -,- One Digit = 
Two or Three Digits 
Dec 2 Decimal Hundredths 
D l 5 Three Digits -c One Digit= Dec 3 Decimal Thousandths 
Three Digits Zero in t he Te ns ' 
Place 
Dec t+ Decimals: 
Place Value 
Dl6 Mixed Practice : Three 
Digi t s-,- One Digit 
Dl7 Four Digits-One 
Digit = Three or Four 
Digits 
75 
You may decide to start with any skill (addition, multiplication, and so on) you 
wish within a block. You must complete all of the units on that skill before moving 
to another skill. You must complete all of the skills within a block before starting a 
new block. 
Date Test Date Test 
Comp. Score BLOCK 17 Comp. Score BLOCK 18 
Dec 5 Comparing Decimals D22 Three Digits c 
Two Digits = One 
Dec 6 Adding Decimals Digit: Learning to 
Estimate 
Dec 7 Subtracting Decimals 
D23 Three Digits 
Dl8 Two or Three Digits Two Digits = One 
Multiples of Ten = One Digit: Digit: Estimate 
No Renaming Too High or Too Low 
D 19 Three Digits 7 Multiples D24 Three Digits 7 
of Ten = Two Digits Two Digits = Two 
Estimating with Renaming Digits: Learning 
to Estimate 
D20 Three Digits 7 Multiples 
of Ten = Two Digits: With D25 Three Digits 7 
Renaming and Remainder Two Digits: Estimate 
Too High 
D21 Three Digits 7 Multiples 
of Ten = One or Two Digits D26 Three Digits 7 
Two Digits = Two 
Digits: Estimate 
Too Low 
BLOCK 19 
D27 Three Digits 
F 16 Least Common Multiple Two Digits = Two 
Digits: Mixed 
F 17 Least Common Denominator Practice 
? 
F 18 Addition with Unlike 
Denominators 
F 19 Subtraction with Unlike 
Denominators 
F20 Dividing to Find Mixed 
Numerals 
D28 Four Digits -;-Two Digits= 
Two or Three Digits 
76 
You may decide to start with any skill (addition, multiplication, and so on) you 
w ish within a block. You must complete all of the units on that skill before moving 
to another skill. You must complete all of the skills within a block before starting a 
new block. 
Date Test Date Test 
Comp. Score BLOCK 20 Comp. Score BLOCK 21 
?-----------------.-
F21 Addition of Fractions III: F28 Dividing Fractions I: 
Improper Frac tions in the Dividing a Fraction 
Sum by a Fraction 
F22 Subtraction of Fractions F29 Dividing Fractions 
III: Mixed Numerals in The II: Fractions and 
Difference Whole Numbers 
F23 Multiplying Fractions I Dec8 Multiplying 
by 10, 100, 1000 
F24 Multiplying Fractions II: 
Fractions with Whole Numbers Dec9 Multiplying 
by .1 , .01, .00 1 
F25 Multiplying Fractions 
III: Fractions and Mixed Declo Multiplying 
Numerals with Decimals 
F26 Multiplying Fractions IV: Deel l Dividing by 
Mixed Numerals With Mixed Tenths 
Numerals and Whole Numbers 
Dec 12 Dividing by 
F27 Multiplying Fractions V: Hundredths 
Reciprocals 
BLOCK 22 
F30 Dividing Fractions III: 
Fractions and Mixed Numerals 
F31 Addition of Fractions III: 
Fractions and Mixed Numerals 
F32 Subtraction of Fractions 
IV: Mixed Numerals with 
Unlike Denominators 
Decl3 Dividing Whole Numbers 
with a Decimal Divider 
Decl4 Writing Decimal Fractions 
for Common Fractions 
77 
You may decide to start with any skill (addition, multiplication, and so on) you 
wish within a block. You must complete all of the units on that skill before moving 
to another skill. You must complete all of the skills within a block before starting a 
new block. 
Date Test 
Comp. Score BLOCK 23 
Pl Introduction to 
Per Cent 
P2 Decimals and 
Fractions for Per 
Cent 
P3 Per Cents for 
Decimals and Fractions 
Pl/- Using Per Cent 
in Multiplication: 
Fraction Method 
P5 Using Per Cent in 
Multiplication: Decimal 
Method 
Rl Ratios: An Introduction 
R2 Equivalent Ratios 
R3 Ratios: Time, Rate and 
Distance 
R4- Ratios and Money 
TEAM SCORE SHEET 
Team FANTASTIC FouB Week Beginning MARCH 2 .3 End i ng MARCH 2 7 
Student 1 2 3 4 5 6 8 9 10 
ANDY Unit I :![)f):/ 
BAKER Score I / 1-/- 15 
CINDY Uni t ?5>9:\ ????: _?.it>\ 
DOREN 
Sco r e 15 
ELAINE Un it I <,2,-\@t 
FOR.D S c o r e 
GEORGE Unit tjj)~'.gi/lP~~!~l~~~i@ 
HALL S core 
Unit 
Score 
Total Numbe r Total Number Accuracy No . . Units No . Studerit s Progress Accuracy Team 
Correct of Tests Taken ? Score Comoleted X 1 0 on Team [ Score Scar,~ Score 
/ 7 / ,2 = / ,'f I ,20 7 + = ~ +/j 'f 1= 
- [ - (12. }(. /0) - ~ - s ~ 
~=1r1m1-m~-~~~a-~~-,a.a,aa~~~~~=~cm~I 
G-REATTEAM ! 
79 
Week Begin ning __M___6-R-C1\- )__ "3 En d i ng ,M ~1-J- 2..7 
Individual Progress Record 
S tudent 1 2 3 4 5 6 7 8 9 10 
AN D'? Unit v~ "D7 Ph" 
Bq J(~ Score 11-f ,5 1!:) 
LJ Np'1 Unit s -J S.8' $~ MP3 
D OREN Score 15 13 I .5 15 
{NE Unit P l.- F"7 
17-P Score 15 (5 
fl);'& Unit Der I 'Dec. 2- C>a 3 
fel,L Score I s t :2._ (5 
IS Unit M 12.. fa\ I;, DF I jV\ /if 
o,,.Jf/5, Score I c; I J..f ,c;- I .3 
k IM Unit All A12- Sfc 57 S8 M~t W3 
J,.... f;.wl > Score / s /Lf I c:; l c; ,~ I <; /3 
M I 1-(G' Unit D~ D~ 3 T? 4- pc:; 
/V E-l5,cYJ Score I ~ ,s I c:; I c; 
D C. TA VIA Unit h:,,.. 5 Tuc, l,, 
'Pot'? Score It+ I c; 
Q Ui ll)N Unit Fl ~ I=- 14/ i=(- 1., D/.3 I> l'f ()J 7 Yns E Score Is I 4- ' I~ I , 1'1-
s ARA Unit NS $ '1 5/D !lF~ s I( 
1 D<KE<.. Score 15 I~ l'-1- I C:: I J./-
uf !...Sv'-A Unit l J,q- q 
VAAJtJ Score I Ci 
w I J-1- Unit rT. ) 1-""t Dl.., n P C-? D ;2..h 
'f.Al}JR2 Score I -t., ,~ I'< 1 "f 
ot..;:y,.JPA Unit ,:.. . q f,-'/D n F I ().)Lt 
ZA-V< Score I t; I c; ,<: 13 
Unit 
Score 
Unit 
Score 
Unit 
Score 
80 
Team Assisted Individualization 
Facts Progress Form 
Addition 
A 
B 
A+B 
C 
B+C 
A+B+C 
Subtraction 
A 
B 
A+B 
C 
B+C 
A+B+C 
M ul tiplica tion 
-----A 
-----B 
----- A+B 
-----c 
-----B+C 
-----A+B+C 
Division 
-----A 
-----B 
-----A+B 
-----c 
-----B+C 
-----A+B+C 
\la n1 e 81 Work A 
Addition Facts 
Date only B 
these A& 
Nu ni ber  Correct 
problems --- C 
B & I 
A B & 
A ,
7 2 7 1 6 6 3 4 2 9 
+3 +2 + 1 +3 +O +2 +3 +2 +l +2 
6 8 0 9 6 8 7 8 4 8 
+O +3 +2 +l +3 +l +2 +3 +l +O 
3 + 2 = 5 + 2 = 4 + 3 = 5 + 0 = 2 + 3 ~: 
9 + 3 = 0+0 = 1 + 2 = 5 + 3 = 8 + 2 = 
B 
1 4 7 6 2 6 8 2 3 5 
1 5 +4 +6 +5 +6 +4 +6 +4 +5 +6 
8 1 6 4 0 9 5 5 3 0 
+4 +6 +6 +5 +4 +5 +4 +5 +6 +5 
7 + 5 = 7+4= 8 + 5 = 0+6 = 1 + 4 = 
3+4= 9 + 6 = 4+6= 9+4= 2+5 = 
C 
5 7 3 1 2 2 4 9 7 6 
+7 +8 +9 +7 +8 +9 +7 +9 +7 +9 
2 1 9 6 4 4 5 8 5 0 
+7 +9 +8 +7 +8 +9 +8 +7 +9 +8 
1 + 8 = 0 + 7 = 8 + 9 = 8 + 8 = 0 + 9 = 
7 + 9 = 6+8= 3 + 8 = 3 + 7 = 9 + 7 = 
Subtraction Facts A 
B s2 
Narne ------- Work A and B 
only C 
Date- ------- these Band C 
Number Correct- --- problems -- A, Band C 
A 8 8 5 1 4 3 7 2 7 8 
- 2 - 0 - 2 - 1 - 3 - 2 - 3 - 1 - 2 - 1 
6 6 10 9 7 12 9 10 11 0 
- 2 - 3 - 3 -2 -1 . -3 -1 -2 -3 -0 
2 - 2 = 4 - 1 = 6 - 0 = 8 - 3 = 5 - 1 = 
3 - 3 = 11 - 2 = 5 - 3 = 4 - 2 = 9 - 3 = 
B 
6 8 12 12 13 7 7 11 9 11 
- 4 - 6 -4 - 6 -4 - 6 - 5 - 6 -4 - 5 
9 7 13 5 4 15 10 10 10 12 
- 5 - 4 - 6 -5 -4 -6 -5 -6 -4 -5 
9 - 6 = 11 - 4 = 13 - 5 = 6 - 6 = 8 - 4 = 
5 - 4 = 8 - 5 = 14 - 6 = 14 - 5 = 6 - 5 = 
C 
9 11 8 12 13 9 11 11 17 15 
- 8 - 9 - 7 -9 - 7 :- 9 - 8 - 7 - 9 - 8 
14 12 10 16 8 13 9 15 16 15 ? 
- 7 - 8 - 7 -9 -8 -9 -7 -9 -8 -7 
18 - 9 = 16 - 7 = 10 - 8 = 12 - 7 = 14 - 8 = 
7 - 7 = 13 - 8 = 17 - 8 = 10 - .9 = 14 ~ 9 = 
Multiplication Fpcts A 
B 83 
,nie Work A and B 
only C 
1te these Band C 
Jm be r Correct problems- A, Band C 
9 4 0 1 8 2 3 3 5 6 
X 0 X 3 X 2 X 1 X 3 X 2 X 3 X 1 x3 x2 
3 0 4 7 7 9 2 9 0 5 
X 2 X 3 xO X 3 X 2 x 1 x3 X 2 X 1 X 2 
1 X 3 = 0 XO = 8x2= 6 X 1 = 6x0 = 
1 X 2 = 9 X 3 = 4 X 2 = 6x3= 5 X1  = 
1 2 4 4 6 2 0 0 6 7 
x5 X 6 x 4 X 6 X 5 x4 x5 X 6 x4 x5 
1 3 7 3 5 5 9 5 9 2 
X 4 x5 X 6 x4 X 6 x4 X 6 X 5 x4 X 5 
7 X 4 = 9 X 5 = 6x6 = 8 X 6 = 8x4 = 
8 X 5 = 3 X 6 = Ox4 = 4x5 = 1 X 6 = 
5 1 4 9 8 0 6 2 3 3 
X 7 x8 x9 X 7 X 9 X 8 x9 X 7 x9 X 8 
8 1 9 4 8 7 7 2 6 3 
X 8 X 7 X 9 X 8 X 7 X 9 X 7 X 9 x8 X 7 
6 X 7 = 5 X 8 = 7x8 = 5 X 9 = 1 X 9 = 
0 X 9 = 4x7 = 2x8 = Oxl = 9 X 8 = 
Division Facts 
__ A 84 
,am e- -------- Work -- 8 
late only __ A and B --------- these -- C 
1u mber Correct- ---- problems -- 8 and C 
__ A, Band C 
3 ) 12 1 ) 0 2)2 1 ) 6 3 ) 21 2) 10 3 ) 3 1 ) 3 
2 ) 6 3)6 2) 16 1 ) 5 2 ) 8 3 ) 27 2 ) 4 ?- 1 ) 8 
2)0 1 ) 2 3 ) 0 1 ) 1 3 ) 9 1 ) 7 3 ) 15 2 ) 12 
14 :- 2 = 24 :- 3 = 4: 1 = 18.;. 2 = 18 :- 3 = 
> 
) 
5 ) 15 6 ) 36 5)0 4) 16 6) 54 5 ) 10 6)18 - 4)12 
4)4 5) 40 4) 24 6 ) 12 5) 30 4)0 6 ) 42 4 ) 28 
5)5 6) 24 4) 32 6)0 5) 20 6) 30 4 ) 36 5 ) 35 
20 7 4 = 25 :- 5 = 6 -:- 6 = 48 -:- 6 = 8 :=- 4 = 45 :- 5 = 
9)9 7)0 9) 63 8) 32 7) 28 8 ) 8 9) 18 7 ) 42 
8 ) 56 9 ) 45 7 ) 63 9) 36 8) 48 7) 21 9) 54 8 ) 72 
7 ) 35 ,. 8 ) 16 9) 72 7 ) 7 8) 0 9 ) 0 7) 49 8) 64 
24 7 8 = 14 +7  = 27 ~ 9 = 40 :- 8 = 56 :- 7 = 81 -: 9 = 
? 
85 
l NSTRlJCTION Sr-lE ET: Aodit\9 Ohe Digit ~"'d Tv1() Di~;t-.s 
You wiU be ab!e to do problems lit(e this: 
l,'t 
+- 3 <Di 
~rs+epJ 
v,H-\~N 'JOv.. ad~ Q e>vte di~it t\l..\WlerC\l +o a +wo oti~it YlwMerctl 
~1~-sT o.c(c{ +he ones'plCtce 
'-:4 \) 
... "' 3 .?: !,  
w~1-re. '' 1 '? in the\~{" here in +~e ones' pla~c.e. 
"'lf ; 
-t 3' 
(2 ncl s+e.0 
. THfN Qctv( +he +enr~P I" c.e . 
r~?lL 
r f' 
+ ..: i3 
7 
,_ 
--=----==---=:-~ 
Ai ?S ki! lsl1eet: .Ad-. di ng One DJ gi t an dliw. o ?oi g&,.6t s 
Step I: Add the ones. 
1) 28 $ 7/./ SJ 65 II) 80 5) q3 
ti !.~ +2 !.._8 +3 
CJ.. 
7) 55 'J) 4 I 'I) f 2 J(j 54 / 
!_!:I + I +-6- f 4 ' . 
1... ..., .- ? 
l?J 23 13) 50 ,~J t1 6 ,s s?,  , 
Ll -e- 5- + 2 .,,. 5 
TR? I C 
16) 63 11) eae  1s) 91 1qJ q2 ,c5J 60 ti + ?...6i u u 
I ' 
1
~sheet: Adding One Digit and Two D;9its 
. Step 2: Now, add the tens. 
1) LIi 2) 33 g)q2 Lt) 52 
+ g t 3 + 5 
=?;i q 6 Ll 7 q 
6) 66 6J 13 1) 8ll i}IO 
,. 2 / 4G? 6 + ti t- q 
- q 
ca i q 
. - . aszLJ 
8 q 
q) 2q ,cv LI 7 II} 56 lfl q2 -
t- 5 +-.  2. Ll t9 6 
' q q . q - ~ g 
13) 71 15} 60 16) 25 
. i- i -+" ""'  t.1l. 
i -8 . ?.'/ 
17) 3 l IS) 52 19) ll 6 2o)q5 
+ 6 Ll + 2 + q ? 
1 6 ?g q 
.,.. 
,_ 
' 
-A2 , . - . ss 
Ski I/sheet: Adding One Digit and Two Dig ifs 
Step 3: Add and write the answer. 
0 26  2) 35 3) 86  q) 11 
Ll -1~ 1 ~$ !~ 
$> 62 6l 51 
+ q + 5- 
q} 15 to> 82 . II) 99 12) 25 
.,.f. .. ~ :;:) 
+ ' ... l.f Ll 
l3J 75 15) 53 . IQ 61 
y ' u -t- I 
lg) I LI l'l1 91 20) 32 
+....  ,., .Ll .-+- -7 
l\ J ---= - == . . 
.__ 89 
.__ ? Ski/lsheet Answers: Addin!J Ont. DijitQnd Two Di,"5 
----- ? 'I  
Step I: Add the ones. ? ? 
,) 28 2) 74 ~ 65 ll) so 5) q3 
+ I t ?3 + 2 + 8 + 3 
....... ? . .. :i.u:u ~ - ? 
c1._ ; 7 .7 'a _{a_ 
.i 
Q ?a'-1 ? 1) 55 i) I q) 12: 
-'/ 
e 16) + ..... . ll . I 54 - ::..-8 , . .,  + 6 - . ::_I} ~ -u' ._(p ;Cf - .2?- ? :3 -~ 
. 
-
.. .. 
1,1) 12. 1'9 23 ' , 13)! -5
. . 
A0 . 1q) J.16 I~ 5 ? 81 
~+ ~., Ll . + 2. + 5 41111M CWWW.  ..rMO 
7 - --?:C... f :.  : .5..  . , :3 L(Q~ 
16) 63 11) 22 IS) 9 I 20) 60 
-1-'- 6... . + 2 L'1 Ll? 
~q_: 5 5 ~ 
. . 
., 
Al 90 
~ _ki/lsh~t A_~ers:AJd inj Ont 1);3it and T"'o D;,;ts 
Step 2: Now, add the tens. 
I) LI I 2) 33 g) q 2 Q) 52 
..: ... g t.3 +5 .g. 7 
-~'i 36 . 97 ~q -
5) 66 6)73 7) 8ll 9) 10 
+ 2 ~t =6  +- 0J.f  ;- q ~.,. 
.. -~.ca .. . t1q :~9 {q 
q) 2LI Id. .. ll7 
Ill 56 (e) 92 -
Ll - __.2..  Ll ?'r 6- ,2.q ;:l
; ... ,lq  .. . :5'1 , .9... 8 
---
13) 77 Ill) 95, 15) 60 (6) 25 
i" I t- I. . . ?..fl 
..............I d L! 
:78 ???15g ~8 29 
11) 91 I'd) 52 1q) ll6 20Jq5 
+ 6 .!_~ + 2 + ~ . m?? -
31 5.6 48 qq 
. . . 
p 
/1l Ski_ll~_heet . A~swers:AcJJ,-,,jOne. Oi.3itQt1df~o D;j;-t.5 
Step 3: Add and wrHe the answer. . ? 
I) 26 2) 85 3) 86 lJ) 71 
Ll + I + g i9 7 .., l!lilQtll =-LL sn - mm 
28 30 ?)1 73 
I 
$:) 62 &} 51 7) q2 8) _qg 
+ - .,. 2 ...., , wL,I.  .+..,._ ..,,.5....  + 4F'i>ii:n:r:S><8 
L/ij -9?
6  1 / 
(o~ 5fo 9 
q) 15 to) 82. . II) 99 12) 25 
+ g .+  --' -a- ,.q.,.,  Ll ~
I '8 i3 37 29 ? . 
19l -75 114) LI I 15) 53 IQ 61 
+ I t- 8 .t.. . ,_ _I .. . 
c:s .. w Ll 
70 49 5<B C:,2 
It} I '4 fq) 81 20) S2 
+.._  q t 6 . +IA --  39.
7.  
18 ?. ? 'tl7  ? I 
. . 
-~ -
; . 
112 92 
?ui1@@C:@: ~DUg Adding One Digit and Two Digi+s 
A: Add B: Add 
I) 32 6) 72 I) .85 6) 6LI 
Ll -8? 7 . ;.-....I.  + 5 
2) 61 7) I I 2) 56 7) 91 
t .. -_L,I,  + 6-- !--2 ?i-- -7 
8) 55 
g 3) 30 S) Lf5 t - Ll -i? g -? ?-
' 
LI) 43 ? 78 q) 12 
Ll . + I + 6 
10) 67 10} l\ 6 
Ll :::....3 
;; ~2 . .. . 93 - - . 
? ?inl@@~@llilU b\fnl@W@[i'?~. Addin.9 One Oi9if and Two Digits 
Checkout Answers A : Checkout Answers B: 
1Jo 27 'ua 8 6 
~o 65 ~o 58 
~o gq ~o 3q 
@.}o L/5 @Jo 7C/ 
@o q7? ?a 2. 6 
?a 7q ?a 6q 
77a -17 7lo q 8 
?o 58 ~)o '-19 
. 
@o 29 ?a 18 
lJ@o 6q lJ@o '-I q 
--
94 ~J 
i ' 
- ? 
~ 
-J.-
v-\. 
u.l 
~ -
\ 
-~~) -.~  
~ 
........ 
.... .~....  
'.!; 
? 
?as 
95 
Name- ---------- Period 
Teacher ----------- Date 
Diagnostic Test 
1. 6 8. 89 15. 52 
(Al) + 7 (S2) - 37 (M3) X 3 
2. 12 9. 461 16. 39 
(A2) + 4 ( S4 ) - 138 (M4) X 7 
3. 38 10. 517 17. 300 
(A3) + 16 ( SS ) - 388 (MS) X 4 
4. 608 11. 706 18. 437 
(AS) + 427 (SS) - 188 (M6 ) X 5 
5. 14 12. 730 19. 307 
(S l ) _ 8 ( S9) - 578 (M7) X 8 
20. 12 
6. 29 13. 5 (MS) X 30 
(Sl) - 3 ( Ml ) X 2 
7. 35 14. 20 
(S3) - 7 (M2) X 4 
96 
21. 27 28. 8 ) 473 
(MlO) X 43 (Dl3) 
29. 30) 690 
22. 607 (D18) 
(Ml 4 ) X 127 
30. 70) 374 
(D19) 
23. 2 ) 14 31. 33 ) 177 
(Dl) 
(D22) 
24. 3 ) 96 
(D3) 
32. 27 ) 892 
(D 24) 
25. 6 ) 27 
(D6) 
33. 14 ) 813 
26. 6) 72 (D24) 
(D9) 
27. 5 ) 71 
(DlO) 34. Write a fraction for this picture. 
(Fl ) 
97 
35. Ci rel e the I ar ger fraction. 
(F4) 
2 2 
3 7 
36. 1 4 
(F6) 6 + 6 = 
37. 
(F7) -
7 - 4 = 
8 8 
38. 1 D 
(F8) = 3 9 
39. What is the greatest common factor of 12 and 16? 
(F9) 
40. 
(F15) 3-32 - 1-1 = 3 
41. Add and simplify if necessary. 
(Fl6) 
98 
42. Subtract and simplify if necessary. 48. Add and simplify if 
(Fl 9) 8 (F31) necessary. -
9 7 _}_ 
3 4 --
4 1 
+ 2 14 
43. Write ~ as a mixed numeral. --
(F20) 
49. Write as a decimal: 
(Dec 1) 
44. 3 7 3 -7: 
(F23) -x - = 5 8 100 
50. Write as a decimal: 
(Dec2) 
45. Multiply and simplify if necessary. 3 
(F26) 1 
2 3 1000 7-x 6- = 
3 5 
51. Circle the smaller 
(Dec4) decimal . 
. 2 ' .18 
46. -1 -..  -3 (F28) = 3 8 52. 13. 7 + 6. 95 = 
(Dec5) 
47. Divide and simplify if necessary. 
(F29) 
3-1 _. 2 53. 41. 3 - 6.11 
= 
2 . = (Dec6) 
99 
54. 7. 25 
(De c 7) X ? 7 
55. 2.5) 35 
(De clO) 
56. Write : as a decimal fraction. __ 
(Decl 2) 
57. Write. 05 as a percent. --
(Dl) 
58. Write 60% as a fraction. Simplify if necessary. 
(P3 ) -----
59. What is 3% of 126? ---
(PS ) 
60. 4 is 10% of what number?- --
(P6) 
61. f of fruits in a grocery bag are apples. What percent of fruits are 
(Rl) 
apples? ---
62. Fill in the missing numerator of the second ratio: 
(R2) 
3 D 
= 
8 120 
63. How long wil I it take if you travel 14 miles per hour for 98 miles?_ __  
(R3) 
64. If you buy 6 pieces of candy for a total of 3 cents, how much will 24 pieces 
CR4) of candy cost? ____  
REFERENCE LIST 
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of Educational Psychology, 1961, 52, 19-24. 
Aiken, L.R., Jr. Attitude towards mathematics. Review of Educational Research, 
1970, 40, 551-596. 
Aiken, L. R., Jr. Update on attitudes and other affective variables in learning 
mathematics. Review of Educational Research, 1976, 46, 293-311. 
Alberty, H. B., & Alberty, E. J. Reorganizing the High School Curriculum (3rd 
ed.). New York: Macmillan, 1962. 
Aronson, E., Blaney, N., Sikes, J., Stephen, G., & Snapp, M. The jigsaw route 
to learning and liking. Psychology Today, 1975, ~' 43-50. 
Aronson, E., Stephen, C., Sikes, J., Blaney, N., & Snapp, M. The Jigsaw Classroom. 
Beverly Hills, California: Sage Publications, 1978. 
Artzt, A. Student teams in mathematics class. Mathematics Teacher, 1979, 
72(7), 505-508. 
Bayles, E. E., & Hood, B. L. Growth of American Educational Thought and Practice. 
New York: Harper and Row, 1966. 
Begle, E. G. Ability Grouping for Mathematics Instruction: A Review of the 
Empirical Literature. Stanford University, California: Stanford Mathematics 
Education Study Group, 197 5. (ERIC Document Reproduction Service No. 
ED 116 938) 
Bestor, A. The Restoration of Learning. New York: Alfred A. Knopf, Inc., 1955. 
Blaney, N. T., Stephen, C., Rosenfield, D., Aronson, E., & Sikes, J. Interdependence 
in the classroom: a field study. Journal of Educational Psychology, 1977, 
69(2), 121-128. 
Bolvin, J. O., & Glaser, R. Developmental aspects of individually prescribed 
instruction. Audiovisual Instruction, October, 1968, 828-831. 
Borg, W. R. Ability grouping in the public schools: a field study. Journal of 
Experimental Education, 1965, 34, 1-97. 
Boyd, W. The History of Western Education. London: A & C Black, 1921. 
Brassell, A., Petry, S., & Brooks, D. Ability grouping, mathematics achievement, 
and pupil attitudes toward mathematics. Journal for Research in Mathematics 
Education, 1980, 1J..(l), 22-28. 
Bronfenbrenner, U. Two Worlds of Childhood. New York: Russell Sage Foundation, 
1970. 
100 
101 
Brophy, J . , & Eve rston, C. Learning From Teaching: A Developmental Prospective . 
Boston, Mass.: Allyn and Bacon, 1976. 
Brubacher, J. S. A History of the Problems of Education Ost ed.). New York: 
McGraw-Hill Book Company, Inc., 1947. 
Burbank, I. K. Relationships between parental attitude toward mathematics 
and student attitude toward mathematics and student achievement in mathematics 
(Doctoral dissertation, Utah State University, 1968). Dissertation Abstracts 
International, 1970, 30, 3359A-3360A. 
Callahan, W. J. Adolescent attitudes toward mathematics. Mathematics Teacher, 
1971, 64, 751-755. 
Clifford, W. J. Motivational effects of competition and goal-setting in reward 
and nonreward conditions. Journal of Experimental Education, 1971, 39, 11-
16. 
Clifford, M. Effects of competition as a motivational technique in the classrrom. 
American Educational Research Journal, 1972, 2_, 123-137. 
Coleman, J. S. Academic achievement and the structure of competition. Harvard 
Educational Review, 1959, 29, 339-351. 
Colvin, D. Evaluation of CPMP (Grades 6-12 Continuous Progress Mathematics 
Prora_m) 1972-1973. Final Report of ESEA (Elementary and Secondary Education 
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085 400) 
Combs, A. W. Intelligence from a perceptual point of view. Journal of Abnormal 
and Social Psychology, 1952, 49, 562-573. 
Comprehensive Tests of Basic Skills. McGraw-Hill, 1973. 
Davidson, T. Rousseau and Education Accordin to Nature. New York: Charles 
Scribner's Sons, 1971. Reprinted from 1898 edition. 
Davis, J., Laughlin, P., & Komorita, S. The social psychology of small groups: 
cooperative and mixed-motive interaction. In M. Rosenweig & L. Porter (Eds.), 
Annual Review of Psychology. Palo Alto, California: Annual Reviews, 1976. 
Deutsch, M. An experimental study of the effects of cooperation and competition 
upon group process. Human Relations, 1949, ~' 199-231. 
Deutsch, M. A theory of cooperation and competition. Human Relations, 1949, 
~' 129-152. 
De Vries, D. L., & Edwards, K. J. Learning games and student teams: their 
efforts on classroom process. American Educational Research Journal, 197 3, 
.!Q, 307-318. 
DeVries, D. L., & Edwards, K. J. Student teams and learning games: their effec ts 
on cross-race and cross-sex interaction. Journal of Educational Psychology, 
1974, 66, 741-749. 
102 
DeVries, D. L., & Slavin, R. E. Teams-Games-Tournament: A Final Report on 
the Research. Baltimore: Center for Social Organization of Schools, The 
Johns Hopkins University, 1976. 
De Vries, D. L., & Slavin, R. E. Teams-games-tournaments (TGT): review of 
ten classroom experiments. Journal of Research and Development in Education, 
1978, .!1_(1), 29-38. 
Dewey, John. Democracy and Education. New York: The Macmillian Company, 
1916. 
Driscoll, M. J. Research Within Reach: Elementary School Mathematics. St. 
Louis, Missouri. CEMREL, Inc., 1980. 
Duda, M. J. A critical analysis of individually prescribed instruction. 
Educational Technology, 1970, .!.Q, 47-52. 
Dutton, W. H. Attitudes of junior high school pupils toward arithmetic. School 
Review, 1956, 64, 18-22. 
Edwards, K. J., & DeVries, D. L. The Effects of Teams-Games-Tournament and 
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