A CASE STUCY OF THE EFFECT OF AN
iNNOVATWE APPROACH N A COLLEGE GENERAL
' EDUCATION MATHEMATlCS COURSE

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
ROSE ANN SCHWARTZFISHER

1973

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ABSTRACT

A CASE STUDY OF THE EFFECT OF AN INNOVATIVE APPROACH
IN A COLLEGE GENERAL EDUCATION
MATHEMATICS COURSE

BY

Rose Ann Schwartzfisher

The population of the study consisted of college

students enrolled in a general education mathematics course.

The purpose of this study was to record and describe

the behavior of the subjects taking this course. Specifi-
cally, the purposes were concerned with four problems:

1. To determine if the method used in the study
was effective in the students' learning of
computational skills;

2. To determine if the method used in the study
was effective in the students' learning of
the historical problems of mathematics;

3. To determine the effect of the method used in
the study on the students' attitude toward
mathematics.

Data were collected for the analysis of the study

from results of the mathematics placement test} placement

Rose Ann Schwartzfisher

post-test, historical problems in mathematics test, problem
solving test, and an attitude questionnaire.

The Friedman two-way analysis of variance test using
the mathematics placement pre-test and post-test investi-
gated whether the size of the score on computational and
basic skills depended on the time during the semester when
the test was given. The null hypothesis of no differential
effect was rejected at the .001 level.

The Wilcoxon matched-pairs signed-ranks test using
the historical problems in mathematics test investigated
whether students after taking this course were better able
to answer the questions on historical problems in mathe-
matics. The null hypothesis of no difference was rejected
at the .001 level of significance.

The Wilcoxon matched-pairs signed-ranks statistical
test was used to investigate whether students were better
able to solve problems after taking this course. The null
hypothesis of no difference was rejected at the .001 level
of significance.

No statistical test was used to investigate students'
attitude towards mathematics. The responses to the ques-
tionnaires were summarized using key words or phrases. It
was concluded that this course improved the attitude of

students toward mathematics.

A CASE STUDY OF THE EFFECT OF AN INNOVATIVE APPROACH
IN A COLLEGE GENERAL EDUCATION

MATHEMATICS COURSE

BY

Rose Ann Schwartzfisher

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

College of Education

1973

ACKNOWLEDGMENTS

The writer is indebted to many, particularly:
my committee chairman, Dr. William Fitzgerald.
Without his help this study would not have been
completed.
my guidance committee for their advice and
direction.

the subjects of this study.

ii

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . .

LIST OF TABLES O O O O O O O O O 0

LIST OF

Chapter

I.

II.

III.

IV.

APPENDICES O O O O O O O 0

INTRODUCTION 0 O O O O O O 0

Purpose . . . . .
Importance of the Study
Limitations . . . .
Assumption . . . .
Overview of Procedure
Succeeding Chapters .

BACKGROUND OF THE PROBLEM AND REVIEW
LITERATURE O O C O O O O O 0

Introduction . . . . . . .
Review of the Literature . . .
Summary . . . . . . . . .

METHOD OF THE INVESTIGATION . . .

Introduction . . . . . . .
Background . . . . . . . .
The Population and Sample . . .
Method of Instruction . . . .
Instruments Used in the Study . .
Testing Sequence . . . . . .
Summary . . . . . . . . .

RESULTS AND DISCUSSIONS . . . .

Introduction . . . . . . .

Case Studies for Computation and Basic

OF

THE

Skills

Case Studies for Historical Problems in

Mathematics . . . . . . .

iii

Page

ii

vi

\lmmU'll-‘H l-‘

11
24

26

26
27
28
29
44
45
46

47

47
48

49

Chapter Page
Case Studies for Problem Solving . . . . . 52

Case Studies for Attitude Toward Mathematics . 64

Summary . . . . . . . . . . . . . 69

V. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS . . 70
smary O O O O O C O O 0
Conclusions . . . . . . . . . . . . 71
Recommendations . . . . . . . . . . 77

SELECTED BIBLIOGRAPHY . . . . . . .

APPENDIX 0 O O O O O O O O O O O O O O O 86

iv

LIST OF TABLES

Table Page
3.1 Highest Level of Mathematics Background . . . 28
3.2 Years Since Last Math Course . . . . . . . 29
3.3 Class Distribution of Sample . . . . . . . 29
4.1 Computational and Basic Skills Results . . . 49
4.2 Scores of Fifty-four Students on Computational

and Basic Skills Under Three Conditions

Of Time 0 O O O O O O O O O O O O 50

4.3 Ranks of Fifty-four Students on Computational
and Basic Skills Under Three Conditions

Of Time 0 O O O O C O O I O C O O 51
4.4 Results of Historical Problems in Mathematics . 52
4.5 Scores of Historical Problems in Mathematics . 53

4.6 Scores for Problem Solving Ability and
Procedures Used . . . . . . . . . . 56

LIST OF APPENDICES

Appendix Page
A. Instruments Used . . . . . . . . . . . 87
B. Procedures . . . . . . . . . . . . . 96

C. Materials Used . . . . . . . . . . . . 118

vi

CHAPTER I

INTRODUCTION

Purpose

This research reports the effects of an innovative
approach to the teaching of a general education mathematics
course. The methods used included laboratory experiments,
demonstrations, overhead projector, programmed materials,
problem sessions, tutorial sessions, and films; as well
as lectures and textbook.

The purpose of this study was to record and describe
the behavior of the subjects taking this course. Specifi-
cally, the purposes were concerned with four problems:

(1) To determine if these methods were effective in the
students' learning of computational skills; (2) To deter-
mine if these methods were effective in the students'
learning of the historical problems of mathematics; (3) To
determine the effect of these methods on the students'
problem solving ability; and (4) To determine the effect
of these methods on the students' attitude toward mathe-

matics.

Importance of the Study

 

The instruction of general education mathematics
courses in college deserved attention for several reasons.

1

Students beginning such a course have a great variety of
backgrounds. These courses are taken by students who may
have had four years of high school mathematics, very little
mathematics taken several years ago, or by those who have
forgotten everything or failed to understand it in the
first place. Concerning the great variability of back-
grounds among college students, McConnell and Heist, using
American College Examination (ACE) scores, wrote:
Most college instructors are probably well aware of
difference in student ability, but it is doubtful that
most of them realize the degree of diversity that
exists or adapt their teaching procedures accordingly
(10).

Experience in the instruction of these mathematics
courses indicated that these students did not respond well
to lecture classes. Ryan wrote:

Students with educational deficiencies, underachievers,
do not do well in classes in which information is
presented primarily or solely through lecture. They
do better when methods are used which provide oppor-
tunity for immediate feedback, when material is
presented in Optimum sequence and when students are
active rather than passive (70).

Students in these classes often have attitudes that
are not conducive to learning mathematics. Stein has used
the phrase "captured student" to convey the mood of:

. . . failure, fear, frustration and perhaps hatred
that such a student frequently brings to these
mathematics classes (52).

The dropout rate for general education mathematics

courses using the lecture-textbook method have been high.

Fall semester of 1972 showed that the dropout rate at

Saginaw Valley College was 45 per cent in the morning
section and 50 per cent in the afternoon section.

General education mathematics courses often have
poor attendance. Even though attendance records are
usually not kept, the faculty members who have taught the
course over the past four years have noted absenteeism to
be high. This is not a problem peculiar to Saginaw Valley
College. Arendsen noted that at Michigan State University:

Of 143 students registered for the class during the
winter quarter of 1971 a check revealed that less
than 40 students attended each of the last two large
lectures which were held each Thursday (57).

Another reason for conducting the research was an
attempt to improve the instruction and the level of
achievement in general education courses. Examination
of fall 1972 post-tests indicate that less than one per
cent would be eligible to enroll in the college algebra
course. This has been the only class at Saginaw Valley
College where students were able to eliminate their
deficiencies in mathematics to enable them to enroll in
college algebra.

Jones and Arendsen commented on the need for further
research and changes in a remedial algebra course. Jones
commented:

Approximately fifteen years of lecture method has
left the student in a position requiring him to take
a remedial algebra class in college. We then offer
the student another lecture course which is signifi-
cantly different from courses in his prior experience

only in the ratio of number of students to lecturer.
The students have been trained to view mathematics

as a spectator sport in which the number of spectators

is ever increasing as the opportunity to perform is
ever decreasing.

A major change of method is due in the teaching of
Mathematics 082 and such change should be in the
direction of individualized instruction. Such methods
as audio-visual-tutorial learning, programmed learning
and computer assisted learning should be considered
for immediate adoption to relieve the plight of the
students in Mathematics 082 (67).

Arendsen recommended:

The use of a mathematics lecture-laboratory situation
incorporating a lecture with laboratory experience such
as audio-tutorial techniques, programmed texts, slide
rules, probability materials, desk calculators, and

computers should be investigated at the college level
(57).

The Committee on the Undergraduate Program in
Mathematics (CUPM) of the Mathematical Association of
America has directed its attention to the problem of con-
struction of a general mathematics program and it noted:

Many of the students in basic mathematics courses have
seen this subject matter in elementary and high school
without apparent success in learning it there. It is
often the case that a second exposure to essentially
the same material, similarly organized, is no more
successful even though an attempt is sometimes.made

to present the subject matter in a more 'modern'
manner.

. . . it is our belief that the type of student

described can also be greatly helped by reform in the
mathematics curriculum.

The student must be actively involved throughout the
course and should be encouraged to formulate problems
of their own based on their experiences. Full
advantage should be taken of the playful impulses of
the human mind; interesting tricks and seemingly

magic ways of solving problems are to be exploited
(20).

Limitations

 

This study limited its considerations to an instruc-
tional approach that directly influenced the achievement
gains in a general education mathematics course in the
specific areas of computation and basic skills, historical
problems of mathematics, problem solving and attitude.

This study made no attempt to evaluate the follow-
ing: (1) the superiority of this method over the lecture-
textbook method but simply to report the results; (2) the
superiority of one test over another in the area of compu-
tational and basic skills but to use the instrument that
has been part of the evaluating system at Saginaw Valley
College; (3) the advisability of smaller or larger classes
but to use the intact groups as they registered at Saginaw
Valley College; and (4) the times of day but to accept the
scheduling as determined by previous scheduling.

The justification for these limitations was that
the study was primarily concerned with using the normal
settings at Saginaw Valley College to determine the feasi-
bility of this method as a viable approach to the teaching
of the general education mathematics course.

This study was also limited by the length of time
of the study and the investigator's ability to implement

the goals of the instructional methods.

Assumption

 

It was assumed that no student had an opportunity
to practice the questions on any examinations.
The integrity of all tests was maintained between

the dates at which the examinations were given.

Overview of Procedure

In order to determine the effectiveness of this
method on computational and basic skills, a pretest was
given. During the semester it was given again and at the
end of the semester a similar test served as a post-test.
The scores were compared for differences over time.

In order to determine the effectiveness of this
method on problem solving, a random sample of 10 students
were selected. These students were pretested for their
ability and procedures used to determine whether two five
digit numbers were prime numbers. Each was given 10 minutes
and was allowed to use any of the materials in the room
which included pencil, paper, blackboard, chalk, electric
calculator, textbooks, slide rule, and any of the laboratory
materials available during the semester. They were allowed
to ask questions of the investigator which required a yes
or no answer. The investigator could decline to answer
if the question was too leading. At the end of the semester
the 10 students were post-tested using two different five
digit numbers. The students were observed as to the

techniques that they used as well as the solution. The

students were also asked to determine the path of a shadow
as a person moves away from the light source in a straight
line, when a person moves in a circle around the light
source, and the path in general for any path of the person.

To investigate the effect on the students' knowledge
of some historical problems of mathematics, questions were
asked on quizzes given during the semester to determine their
knowledge of these historical problems of mathematics. A
test was given at the end of the semester to test their
mastery learning at the 75 per cent level.

To investigate the students' attitude toward mathe-
matics a questionnaire was given to the students at the end

of the semester.

Succeeding Chapters

 

Chapter I has given the background and purpose of
this study, limitations of the study, assumptions made in
the study, and an overview of procedures.

In Chapter II the background of the problem and a
review of related studies is provided.

Chapter III contains a detailed description of
the study.

Chapter IV consists of the results and discussion.

Chapter V consists of the summary, conclusions and

recommendations.

CHAPTER II

BACKGROUND OF THE PROBLEM AND REVIEW

OF THE LITERATURE

Introduction

 

According to Goodlad, the current curriculum reform
dates back to the years following World War II; however, he
states it is usually linked more recently with the Russian
satellite of 1957. Goodlad wrote:

This spectacular event set off blasts of charges and
counter-charges regarding the effectiveness of our
schools and stimulated curriculum revision, notably
in mathematics and the physical sciences (38).

Goodlad continues by analyzing the strengths and
weaknesses of the mathematics programs that were part of
the revised curricula. He contended that, although there
was great uniformity among the various projects and programs,
objectives were "vague," "not stated," or had a "mystical
quality." He felt this was particularly true regarding
structure and concept-attainment.

Each of the major math programs that came into
existence as a result of curriculum reform are well known
and each group has published a wealth of materials. It
might be fruitful to examine certain aspects of these

programs. In 1963 The National Council of Teachers of

Mathematics published a bulletin, An Analysis of New

 

8

Mathematics Programs, (24) in which an overall analysis
was made of eight of the better known programs. The follow-
ing criteria were used in this evaluation:

Social Applications
Placement

Structure
Vocabulary

Methods

Concepts vs. Skills
Proof

Evaluation.

A resume followed each of these topics, usually in the form
of questions. The members of the committee noted that
mathematics is in a state of flux and they left it to the
reader to decide his own position on each of the above
topics. Even though the committee did not take a more
definitive position, they did consider each of the above
topics crucial to a mathematics program.

In a comparative study of the eight programs in-
cluded in the bulletin, Hall noted that:

. . . whereas each program was unique nevertheless
commonalities were evident. Among the latter that
appeared most frequently were: concept oriented;
stress upon understanding, thinking, and reasoning;
search for patterns, ideas and principles. The
language of mathematics was considered important and
thus a precise vocabulary essential. In some programs
new content was introduced; others introduced topics
earlier grade-wise than is dictated by tradition. A
few stressed the discovery method, in-service training
for teachers, development of aids, films and materials,
etc. This is by no means a thorough presentation of
what may be considered a 'new mathematics' program;
however, it would serve as a guide as to what is
deemed important by those who have developed such
programs. It should be noted that all eight programs
were designed for the typical child or the more able
learner (65).

 

 

 

 

 

 

10

The School Mathematics Study Group recognized that
their programs were designed for the more able students
when they reported:

It should be noted that much of the success of the
so-called new mathematics has been demonstrated with
middle-class children (25).

Conference members from the joint meeting of the
U. S. Office of Education and the National Council of
Teachers of Mathematics also recognized this neglect:

Our intent here is to consider the mathematical need

and proper instruction in mathematics for that category
of youth referred to by Dr. Conant as 'social dynamite'--
those who possess no skill, who are unemployable and
unschooled. . . . Our range of interest will include
mathematics for those students who are potential
dropouts, as well as for those who remain in school,

but who, for one reason or another, exhibit a pattern

of low achievement in mathematics (26).

Reasons for updating the math curriculum appear in
much of the literature and include the following taken from

Frontiers in Mathematics Education. These appear in summary

form:

1. Mathematics must grow and change to meet the demands
of a changing world.

2. Other fields of knowledge are making increased use
of mathematics.

3. Inability to predict the skills needed in the future.

4. Experiments indicate the value of earlier intro-
duction of selected concepts and skills.

5. New development in child psychology and a greater
understanding of the learning process require
elimination of rote memorization and meaningless
drill.

6. Large numbers of our population have been inade-
quately prepared in mathematics.

11

7. Greater realization that the development of persons
proficient in mathematics begins in the elementary
school.

8. Schools must produce more highly trained technical
people and also informed and literate citizens in
mathematics to enable them to understand their
technological world (21).

These same reasons can be advanced for up-dating and
improving the instruction of mathematics for the students in

general education courses today.

Review of the Literature
This study examined an innovative approach to the
teaching of a general education mathematics course. The
related literature was reviewed in the five areas: (1) labo-
ratory instruction; (2) programmed instruction; (3) indi-
vidualized instruction; (4) remedial instruction; and

(5) discovery learning.

Laboratory Instruction

 

Laboratory instruction in this study referred to a
teaching technique which utilizes activity by the student
with materials other than blackboard, paper for writing,
or library reference materials.

The idea of a mathematics laboratory has been with

us for a long time. In the May, 1970 issue of The Arithmetic

 

Teacher Kristina Leeb-Lundberg (46) describes the original
kindergartens as they developed under the influence of
Froebel in Germany in the early 1800's. The kindergarten

she described includes grids on the tables and chalkboards,

12

studies of shapes, sand for measuring volume, early versions
of attribute blocks, multibase arithmetic blocks, pattern
blocks, linkages, and even lattice boards which could be

used as geoboards are today.

One of the more influential proponents of mathematics
laboratories was E. H. Moore. In his presidential address
before the American Mathematical Society in 1902, he dis-
cussed the state of abstract and applied mathematics, then
went on to discuss the state of the teaching of mathematics.
In discussing elementary mathematics he said:

Would it not be possible for the children in the grades
to be trained in power of observation and experi-
mentation and reflection and deduction so that always
their mathematics would be directly connected with
matters of thoroughly concrete character? The response
is immediate that this is being done today in the
kindergartens and the better elementary schools. I
understand that serious difficulties arise with children
of nine to twelve years of age, who are no longer
contented with the simple, concrete method of earlier
years and who, nevertheless, are unable to appreciate
the more abstract methods of the later years. These
difficulties, some say, are to be met by allowing the
mathematics to enter only implicitly in connection with
the other subjects of the curriculum. But rather the
materials and methods of the mathematics should be
enriched and vitalized. In particular, a grade

teacher must make wiser use of the foundations
furnished by the kindergarten (11).

In 1927, Austin proposed a laboratory approach to

high school geometry. He stated:

The keynote of the laboratory idea is discovery by
means of experimentation. Pupils should be permitted
to observe the laws of geometry operating in concrete

form before they are required to do logical thinking
(29).

13

In 1954 the NCTM published a yearbook entitled
Emerging Practices in Mathematics Education. One of the
major sections of this yearbook was entitled "Laboratory
Teaching in Mathematics." In the first section, we find
the statement:

Laboratory techniques have long been used in public
schools in such areas as science, dramatics, home
economics, and shop. Teachers have long been urged
to use laboratory techniques in the teaching of
mathematics. Enough teachers are doing that, so
that we may well consider laboratory teaching as
one of the emerging practices in teaching mathe-
matics (3).

Thus, it appears that at times through the years,
there have been collections of voices asking for a more
intuitive and less deductive approach to the teaching of
mathematics at all levels.

Current writers generally attribute the forces
behind the development of mathematics laboratories to the
work of Piaget, Bruner, Gattegno, etc. (63).

The literature of mathematics education at the

present time has frequent articles discussing mathematics

laboratories. Two issues of The Arithmetic Teacher (Oct.,

 

1968 and Jan., 1970) have had this topic as their central
theme.

Kieren (43) provides the most extensive review of
the research literature dealing with laboratory learning

during the period 1964-1968 in the Review of Educational

 

Research, October 1969. Most of the research activity has

been aimed at determining the effectiveness of the use of

14

specific materials on the achievement of children. The
use of the cuisenaire rods has been studied in several
studies with mixed results. Callahan, Crowder, Hollis,
Locow, and Nasca all report that the use of the rods
promoted more learning by children while Brownell, Fedon,
and Haynes report either mixed results or no significant
differences. Passy reports a negative effect from the
rods.

Davis (43), in his comprehensive report of the
Madison Project stated that the goals of the project were
to prepare classroom experiences developed in a specific
manner, to train teachers in the use of the experiences,
and to test the lessons. The methodology of evaluation
makes it difficult to report the results in statistical
terms. Davis (1967) reported an interview study by a
clinical psychologist, H. Barrett, done in connection with
the Madison Project. After a year of conducting interviews,
Barrett concluded that students in grades 6 and 7 like
physical activity and courses involving it and do not like
doing sophisticated mental tasks. Barrett's data formed the
basis for the Madison Project decision to use physical
materials and laboratories at grade K—9 and also in college
courses for prospective mathematics teachers.

In an unpublished manual, About Mathematics Labo-

 

ratories, Fitzgerald writes:

There is at this point no empirical evidence which
will convince the unconvinced that mathematics

15

laboratories are the best way to accomplish these
aims. Many have found through experience that the
school as a system makes the implimentation of a
laboratory a difficult task to accomplish.

This writer, however, has never seen a teacher who,
after shifting from a teacher-dominated, total-class
approach to an individualized, activity-oriented
approach, has chosen to shift back again (63).

Programed Instruction

Programed instruction is based on the assumption
that by splitting the instructions into a very large number
of minimal steps, it becomes possible for every student to
be able to put together whatever is necessary to arrive at
the competencies required.

Regarding programed learning and instruction, Fey

wrote:

Programed instruction plays an important research
role as a means of simulating classroom instructional
methods under rigorously controlled, repeatable condi-
tions. However, programed instruction was originally
conceived as a replacement for traditional classroom
teaching--as a tool that could offer pacing and feed-
back on an individual basis. Since mathematics has
been a popular topic for developers of programed
material, there has been a great deal of research

on the effectiveness of this new instructional
medium. The most frequently debated and tested
question was about the relative merits of programed
and teacher-directed instruction. Results of media
comparison studies have failed to establish the
superiority of programed or conventional instruction.
In fact, the evidence is sharply contradictory. 2011
(1969) reviewed research in programed instruction and
found that in 13 comparative studies differences
favored programed in 3 cases, traditional methods

in 3 cases, and neither method in 7 cases (35).

The one great stumbling block to the use of programed
instructions is that it is extremely boring. According to

Dienes,

16

An adult who is motivated to learn something in order
to improve himself in his job can discipline himself
and say, 'I must learn it,‘ and go through a thousand-
page text to pass an examination. But he does this to
earn the privilege of higher pay. For children, on
the whole, the reward has to be much more immediate
than that (4). -

Another objection to programed instructions is that
in our normal environment, we do not learn in minimal steps.
Dienes suggests that:

Some kind of nonlinear programming would seem necessary,
which would combine learning small items with inter-
action with the environment and, consequently, with

the added depth that such interaction can produce in

the way of motivation. Computerized instruction may
here provide a partial solution. Any program of
mathematical activities should be one in which children
can engage in exploratory activities in relation to
their environment (4).

Therefore, it seemed reasonable that research in
programed instruction move toward defining the interactive
effects among various types of programing, subject matter,
and toward determining how programed materials can be most
effectively used in conjunction with standard teaching
procedures. For example, Morgan (68) and Callister (59)
examined stress, anxiety, and achievement interactions in
programed and conventional classes. In 1966 Wiebe (71)
tried to find the most effective combination of programed
and teacher-directed instruction with low—achieving students.
According to Fey:

It seems fair to conclude that, far from supplanting
teacher classroom instruction, programing will probably
prove to be an important instructional device to be

effectively integrated into a total instructional
plan (35).

17

In this study it was the author's intent to inte-
grate programed instruction into the arithmetic phase of the

course 0

Remedial Instruction

 

In the past decade many programs have been insti-
tuted to alleviate problems associated with the academically
deprived student. These bear various labels indicating
objectives such as: “remedial," "preventive," "enriching,"
and "supportive." Two important conferences were held
which stressed the need for more emphasis and research in
this area of mathematics. One was held in 1965 and was a
joint effort by the U. S. Office of Education and the
National Council of Teachers of Mathematics. The other
was sponsored by the School Mathematics Study Group (SMSG)
in 1964. The following exerpts were taken from the reports
of these two conferences:

From the very beginning SMSG recognizes perfectly well
that we were doing something for only part of the
school population. We have made a remarkable amount
of progress, but we are far enough along to realize
that the rest of the school population, the students
who are not doing well in mathematics, must be given
attention (25).
At all levels, however, the emphasis and attention
have been directed toward the above-average mathe-
matics achiever (26).

Not only is it evident that a large segment of the

student body has been by-passed by curriculum changes in

mathematics but it is now realized that our modern

l8

technological society demands that ways and means be

explored to reach these students. Stein believes that

colleges:

and

have a responsibility to meet the captured student
not halfway, but exactly where he is: and to help
him go as far as he is able in the direction that

he chooses or is compelled to follow (52).

Others have made similar comments. Stubblefield
Whitman of Florida A. & M. state:

In most mathematics departments, freshmen general
education mathematics courses affect more college
students than all of the other mathematics offerings.
Yet, the general education courses are of little inter-
est to these departments, . . . . In such courses,
many developing colleges are finding 'rough diamonds'--
able but reluctant or handicapped students--who are
potential 'late bloomers.‘ They are finding that
effective teaching can release mental fetters by
breaking away from excessive formalism and stereotyped
thinking. Furthermore, they find that as students
experience intellectual success they raise their
self-expectations and travel new routes to higher
achievement. A most effective way of improving
student experiences in freshman courses is through
programs which involve the instructors in creating
fresh approaches to their teaching. They are setting
up active learning situations by using intrinsically
interesting materials, by providing students with
attainable goals, and by allowing flexibility in
course content (53).

Recently, Melvin Kranzberg wrote:

We must remember that approximately one-third of
American college-age youngsters are now in college,
and their aversion to required science courses would
seem to manifest a disregard or even disrespect for
science. Students now are concerned with the quality
of life; and they wish to participate more actively
in society. Above all, they are motivated by humane
and social considerations. Science education has

not responded satisfactorily to changing motivations
(45).

19

There are colleges and universities that are trying
to alleviate the problems that plague these students. The
City University of New York implemented an open admissions
policy in September 1970 and it is the University's
objective to provide a college sequence of courses which,
regardless of a student's previous academic achievement,
will give him the opportunity to attain the background
necessary to choose and pursue a college course of study
(37).

Staten Island Community College has implemented
the open admissions policy mentioned above. Ablon reports
on the program constraints, the program structure, the
students enrolled, the placement of students, the program
content and the results of their program for the fall of
1971. He states:

1. The dropout rate was 45% during the first semester
of the program, but as a result of changes, has

fallen to about 25% and seems to be holding there.

2. Of those who do complete the program and go on to
a pre-calculus course 81% pass.

We believe that there are four ingredients any open
admissions program must have to be effective.

1. The capacity to accept every student at his own
level and work from there to wherever he wants to

go.

2. The ability to continue working with a student
until he reaches or changes his goals.

3. The cooperation of an administration eager to go
out on a limb and spend the money required to
make the program work. -

20

4. Teachers who honestly believe in the right of
everyone to try to develop to his capacity regard-
less of his past experiences.

Mix these ingredients into any reasonably well thought-
out program, hope for a liberal sprinkling of Hawthorne
Effect, and do it (27).

Jones wrote on the approach used to solve the
problem at Lansing Community College, Lansing, Michigan.

The student may enroll in one of three remedial courses
with initial placement based upon entrance examination
scores. The courses are Basic Arithmetic, Beginning
Algebra, and Intermediate Algebra. The student has the
choice of a traditional lecture course or a programed
instructional course in both Basic Arithmetic and
Beginning Algebra. Intermediate Algebra is offered

in a choice of traditional lecture or audio-tutorial
mode of instruction. All of the courses are offered

on the basis of a 10 week term but the 10 week time
periods are used only for fee assessment and record
keeping. The key idea is the time flexibility within
the 3 courses. . . . Student reaction to these pro-
cedures is positive and the administrative problems
which accompany the increased flexibility have been
solved (67).

Individualized Instruction
The need to individualize has been a concern of
educators for many years. Wood and Haefner (19), Foster,
Kaufman and Fitderald (22), and White (72) expressed a
need to use individualized instruction.
The Comprehensive School Mathematics Program has
been concerned with individualizing instruction since 1966.
Kaufman and Haag said:
It has become clear to many people who think about
mathematics education that the traditional process
of fitting all children in a class to one instructional

program must be replaced by a process by which programs
are designed to fit the individual children (23).

21

Arendsen noted:

Fortunately, efforts have been made to individualized
instruction at all levels of education and for the
large number of students attending today's schools and
colleges. Commercial companies are producing methods
of instruction designed to individualize teaching.

The Westinghouse Learning Corporation has produced
individualized learning materials which are in use

in the elementary grades at the McCulloch School in
Jackson, Michigan. In this system each child has a
yearly instructional plan (which may be changed by

the teacher) designed for him by computer. The child
proceeds through the system on his own, receiving help
from the teacher when needed. A visit to the school
showed that the system was well received. Students

in all classes were busy learning, and it was reported
that there were very few discipline problems. A child
working on his studies has no time to cause disturb-
ances (57).

The need to individualize on the college level has
been receiving more than lip service in the last five years.
Michigan State University has developed Structural Learning
and Training Environments (SLATES) in response to the need
for individualized instruction on the campus.

Ohio State University has implemented a program
called CRIMEL (Curriculum Revision and Instruction in
Mathematics at the Elementary Level). In 1971 Riner
reported on the initial development of the program which
was designed to individualize mathematics instruction. In
1972 Waits reported on the modified CRIMEL experiment
stating that:

The basic spirit of the CRIMEL program at Ohio State
is to allow the individual student to learn the

mathematics he needs at a pace consistent with his
background, interests, goals, and ability (48).

22

As a result of modifications to the experiment,

Waits reported:

The overall attitude of the student in the CRIMEL
program was excellent. There was a great improvement
in attitude and morale in the 3 credit hour sections
over the first implementation. As the result of the
retesting aspect of our program, many students felt
that the Mathematics Department was actually trying
to help them, not flunk them out! (55).

Because of the variety of backgrounds of the students
in the general education course at Saginaw Valley College,

the need to individualize was apparent.

Discovery Learning

In this form of learning, the student actively
engages in the process of forming mathematical ideas for
himself, but does not necessarily use any physical devices

in learning mathematics.

The late Max Beberman, of the University of Illinois
Committee on School Mathematics, believed that the use of

discovery leads to greater understanding by actively

involving the student:

A second major principle which has guided us in
developing the UICSM program is that the student
will come to understand mathematics if he plays an
active part in developing mathematical ideas and
procedures. To us this means that after we have
selected a body of subject matter to be learned we
must design both exposition and exercises in such

a way that the student will discover principles and
rules (7).

Bruner hypothesizes four benefits of discovery

learning: increased intellectual potency, intrinsic

23

motivation, the learning of the heuristics of discovery,
and enhanced use of memory, and gives his View of discovery
teaching:

Discovery teaching generally involves not so much
the process of leading students to discover what is
'out there,‘ but, rather, their discovering what is
in their own heads. It involves encouraging them to
say, Let me stop and think about that; Let me use my
head; Let me have some vicarious trial-and-error.
There is a vast amount more in most heads than we are
usually aware of, or that we are willing to try to
use. You have got to convince students . . . of the
fact that there are implicity models in their heads
which are useful (32).

Other distinguished educators and researchers
support the use of discovery learning: Allen (28), Hartung
(42), Davis (43), Hawkins (43), and even Ausubel, who has
been a strong critic of discovery learning, admits:

Learning by discovery has its proper place among the
repertoire of accepted pedagogic techniques available
to teachers. For certain designated purposes and for
certain carefully specified learning situations, its
rationale is clear and defensible (30).

Hence, there is a great deal of support for, and
interest in, a type of teaching or learning the essence
of which lies in inductive generalization.

Kieren reported in the Review of Educational Research:

There was no lack of published written work defending

or attacking the effectiveness of discovery methods

of instruction. As Cronback pointed out there are a
multitude of different approaches that have been applied
under the rubric of 'discovery.' Thus, the theoretical
discussion and the research in discovery learning
suffered from lack of precision and lack of communi-
cation caused by the variations in connotations of
'discovery.' As part of a conference report (edited

by Shulman and Keislar, 1966), Davis attributed several

24

positive effects to a discovery learning experience,
especially in a classroom group. He suggested that
the child is rewarded for his efforts by the pleasure
of his own discovery and his ability to bring closure
to a problem setting himself, that the child is
motivated by the opportunity to display his ideas

in competition with others, and that the student
learns to act independently in checking the veracity
of his ideas. Bruner (1966) concluded that discovery
settings present opportunities for students to draw
for themselves relationships between things they

know and the learning task at hand. In addition,
discovery methods build problem-solving skill by
providing experience in pushing ideas to their
logical limit and in effectively forming concise
hypotheses. Bruner, like Davis, suggested that
thinking acts are reinforced by the discovery accom-
plished and that a reflective attitude is developed
in students.

Ausubel, in a counter-argument to those of Bruner and
Davis, stressed that discovery processes are less
efficient than didactic or expository processes and
that discovery is not needed for meaningful acquisition
of knowledge. Bittinger (1968) cited literature sug-
gesting that discovery attitudes can be sustained via
didactic teaching. He further noted Cronbach's
hypothesis that discovery methods may not work well
for anxious, dependent students. Becker and Maclead
(1967), in reviewing literature relating discovery and
transfer, found no conclusive research evidence that
discovery methods foster transfer. This find sug-
gests that discovery methods might not facilitate the
acquisition of superior problem-solving capabilities.
Nonetheless, Becker and Maclead found that the liter-
ature indicates a conflict between methodologies that
sponsor maximum understanding and those that sponsor
maximum motivation to continue learning.

The research published during 1964-1968 on discovery

methodologies in mathematics instruction did little
to resolve the theoretical conflict outlined above (43).

Summary

Chapter II was concerned with reviewing the liter-
ature in each of the following five areas: laboratory
instruction, programed instruction, individualized instruc-

tion, remedial instruction and discovery learning.

25

Some conclusions from the literature which are
important to the study are:

1. Typically, present college instruction of
general education mathematics classes leaves much to be
desired.

2. Individualizing instruction is a goal worthy
of more attention.

3. Laboratory instruction has been with us a long
time. Enough teachers are using laboratory techniques so
that we may consider it as one of the emerging practices
in teaching mathematics.

4. Programmed instruction will not supplant teacher
classroom instruction but will probably prove to be an
important instructional device to be integrated into a
total instructional plan.

5. Discovery learning has a great deal of support
and interest.

6. Remedial instruction in mathematics is beginning

to receive the attention that it needs.

CHAPTER III

METHOD OF THE INVESTIGATION

Introduction

 

The purpose of this study was to record and describe
the behavior of the subjects taking this course. Specifi-
cally, the purposes were concerned with four problems:

(1) To determine if this method was effective in the students'
learning of computational skills; (2) To determine if this
method was effective in the students' learning of the
historical problems of mathematics; (3) To determine the
effect of this method on the students' problem solving
ability; and (4) To determine the effect of this method

on the students' attitude toward mathematics.

This chapter includes: (1) background; (2) the
population and sample; (3) the method of instruction;

(4) the instruments used in the study; and (5) the testing
sequence.

This course was designed to implement the goals of
the Committee on the Undergraduate Program in Mathematics

(CUPM), outlined in A Course in Basic Mathematics for

 

Colleges.

26

27

Background

 

Saginaw Valley College is Michigan's newest four
year state supported institution of higher learning. It is
comprised of the Colleges of Liberal Arts, Business Adminis-
tration, Education, Fine Arts, and the Basic College. Thus
far in its development, main emphasis has been on the
liberal arts. The student body is predominantly drawn
from the tri-county area of Saginaw, Bay, and Midland.

The mathematics department at Saginaw Valley College
offers a general education mathematics course, an integrated
algebra and trigonometry sequence, and a calculus sequence
as freshman level mathematics courses. Students who enroll
in the first course of the algebra or calculus sequence are
required to take a 35 minute, 25 response, multiple-choice
placement examination (Appendix A.l). If a student scores
17 or more correct responses, the student is permitted to
take the first course in the college algebra and trigonometry
sequence. (If a student wishes more advanced placement,
further testing is required.) If the student scores 15 or
16, the student is considered to be marginally prepared for
college algebra and a review is recommended. If a student
scores below 15, the student is required to improve his or
her math background either by taking the general education
mathematics course or by self study. Exceptions to the
foregoing are often made on the basis of personal interview.

Any student at Saginaw Valley College may select the general

28

education mathematics course to fulfill a natural science
Basic College requirement. (Mathematics majors are
restricted to fulfilling their natural science requirement
outside of their major. This restriction is often not
followed since many students fulfill their Basic College

requirements before selecting a major.)

The Population and Sample

The population in this study is the set of all
students who have enrolled, are presently enrolled, or will
enroll in the general education mathematics course at
Saginaw Valley College.

The sample was comprised of two intact sections of
general education mathematics offered winter semester of
1973 at Saginaw Valley College. The author served as the
instructor for both sections. Personal data collected on
each student were used to describe the sample in detail.
There were 30 males and 24 females in the study who
completed the course. Table 3.1 indicates the mathematics
background of the students who completed the course for
the two sections.

TABLE 3.l.--Highest Level of Mathematics Background (In
Per Cent to the Nearest Tenth).

 

High School Mathematics

General Math 1 Year 2 Years 3 Years 4 Years College

 

9.3 24.1 27.7 13.0 18.5 7.4

 

29

The mathematics background was quite varied. With 7.4% of
the students having had college algebra and 9.3% of the
students having had only general math in high school, the
difficulty one would have in trying to lecture to this
diverse group of students appeared obvious. Table 3.2
indicates the years since the last math course was taken
by the students. The time since the last mathematics
course ranged from one semester to 15 years. Table 3.3
indicates the class distribution of the sample. The infor-
mation in these tables pointed to the wide variety of
students in the class. The motivations, goals, interests
and recall of these students were probably quite different.

TABLE 3.2.--Years Since Last Math Course (Per Cent to
Nearest Tenth).

 

0-1 2—3 4-5 More than 5 Years

 

22.2 40.7 24.1 13.0

 

TABLE 3.3.--C1ass Distribution of Sample (Per Cent to
Nearest Tenth).

Freshmen Sophomores Juniors Seniors

 

74.1 18.5 3.7 3.7

 

Method of Instruction

 

The concept approach to teaching, which stressed

the ideas and basic understandings that underlie our number

30

system, was used. In practice the method was accompanied
by many activities and a utilization of tangible aids and/or
models that were intended to promote learning and make the
experience an enjoyable one. This methodology stressed

main ideas, generalizations, structure, understanding, and
relationships in a mathematical environment. A procedure
was instituted that permitted flexibility, large and small
group activities, utilization of laboratory materials,
frequent change in pace, numerous concrete aids and models,
and involvement of tutors.

A routine was established to give the student a
framework within which they could operate and yet enjoy a
great degree of freedom.

The text used during Winter Semester for the
arithmetic phase (first five weeks) of the course was
Computational Arithmetic by Charlene Pappin. It was
published by Glencoe Press in 1972. Along with this book,
prepared materials were handed out and laboratory materials
were introduced. For the remaining 10 weeks of the course
no textbook was used. Materials had been prepared and
were handed out in class. Laboratory materials were also
used during this phase.

The class met four days per week and each class
period was 50 minutes long. At the beginning of the hour
the day's work would be explained. This explanation was

limited to 5-10 minutes. Occasional unannounced quizzes

31

were given at this time. When a particular assignment was
finished by all (or almost all) of the students, at the

end of the class or the next day, the students with the

aid of the instructor would summarize what they had learned.
It can be seen that there were no lectures as we tradi-
tionally envision them in a math class.

The classroom became a laboratory in which students
could construct, manipulate, measure and explore mathe-
matical ideas, and participate in demonstrations, problem
solving, and discussions. Every effort was made to make
the class enjoyable for the students and to give them
experiences in success. There were homework assignments
which related to the material covered in the classroom.

In an attempt to make mathematics more relevant
and desirable to the students, much effort was put into
the development of everyday mathematics problems.

A lesson plan follows for a typical class session.
While no two days were exactly alike, the general format

was adhered to.

Typical Lesson Plan

 

Beginning of the hour:

1. Pass out answer sheet to homework. Have students
check homework and answer any questions. Possible
quiz.

2. Pass out classroom activity for the day.

3. Introduce (or explain) the activity.

32

10 to 15 minutes past the hour:

4. Have the students work in groups of four or
five to perform the activity and draw con-
clusions. During this time circulate among
the groups to give guidance and help where
needed.

10 to 15 minutes before the end of the class time:

5. Determine the progress the students have made.
If it requires another class period to complete,
allow the students to continue working. Other-
wise, use these 10 to 15 minutes to check the
results and summarize.

The basic arithmetic operations with whole numbers,
integers, rational numbers and real numbers from a compu-
tational standpoint were carefully covered in Charlene
Pappin's Computational Mathematics text which was the
required text for the course. The text is designed for
individualized instruction of mathematics. The students
were able to pre-test their competency in each of the
eleven areas covered in the text to determine if it was
necessary for them to work through the material.

In order to disguise the drill of the basic arith-
metic operations on whole numbers, number bases were
introduced. This also allowed for the opportunity to
introduce exponential notation, place value and the
importance of the number zero. Cuisenaire rods were used
to represent the numbers in the different bases. The

students were also led to discover the algorithm for

converting from base 10 to a smaller base. For example,

33

2310 is represented in base 10 by:

 

 

 

 

 

 

 

 

2 tens 3 units

and in base four by:

 

 

 

 

 

 

 

 

1 sixteen 1 four 3 units

(Since there is no 16 in the set of rods, it was necessary

to create a sixteen by taping two eights together.)

1 0

2 + 3x4 .

10 1134 = 1x4 + 1x4

Thus, 23

or 2310 = 1x42 + 1x4l + 3x40.

Dividing both sides by four we have:

 

 

 

23 1x42 1x41 ®x40
10 //,-J
————-= + x +
4 4 ,,«“’4 4
l/f’
/.
4) 23 ////
10 .m’”
2.9. //
(3)"
and as a result 5 = 1x4l + 1x40. Dividing this equation on

both sides by four, we have:

 

And as a result 1 = 1x40. Dividing this equation on both

sides by four we have:

1 (:)x40

 

. Thus, 23 = 113
4 4 10 4

IO

em

After converting several numbers from base 10 to a smaller
base, the students were able to conclude that by successive
divisions of the quotients of the base 10 number by the
radix of the smaller base, it was possible to obtain the
smaller base number by using the remainders from the
successive divisions.

Also, the advancing algorithm for addition in any
number base was developed using odometers.

"Guess My Rule" (Appendix C 3.1) was used to intro-
duce inductive reasoning, to practice basic operations of
whole numbers, to encourage students to work together, to
build a foundation for functions and graphing in two
dimensions, to discover finite differences and to form

conjectures.

35

For example: The subjects played "Guess My Rule"
on numerous occasions until they discovered a systematic
method to determine the value of m in the equation,

y = mx + b. While playing the game, a table was made and
the points plotted on a graph. The value of m was called
the first difference and a column for the first differences
was made. Having had algebra before, several students
suggested substituting a pair of values for x and y and
solving for b. This is certainly a valid method for
obtaining b and so they were asked if they could discover
another method for obtaining the value of b. They noted
that the y entry in the table when x=0 was always the same
value as b. With this information they were able to deter-
mine the rule for data when the first difference is a
constant. After discovering the method to determine the
rule, the students were asked to determine the relationship
between the values of m and b and the graphs that they
drew.

Having completed this discovery, the students then
performed several experiments whose data could be described
by linear equations.

Realizing that all rules are not linear, the
students used the rule y = x2 + 2x + 1 to make a table of
ordered pairs. The entries in the first difference column

were not alike.

36

The students were also asked to plot the ordered
pairs and noted that the graph was not linear. The author
suggested adding a second difference column. The students
noted that these entries were alike. The students tried
several rules until they were satisfied that this would
be the case for a second degree equation. The students
were asked to study their rules and tables of data to
determine the relationship that existed between their data
and their rules. They were asked to consider their rules
to be of the form y = ax2 + bx + c and to find a, b, and
c from the data. They were asked the following questions:

1. What is the significance of the y entry if
x=0?

2. Is there any connection between the second
difference column and a?

3. Is there any connection between the first
difference column and a and b?

In this way they were led to discover that the y
entry if x=0 is the value of c, the constant in the second
difference column divided by two is the value of a, and
the sum of a and b is equal to the first entry in the first
difference column when x=0.

Having made this discovery, the students used physi-
cal materials whose data could be described by a second
degree equation.

9 Deductive reasoning was introduced by considering a
cube with lines drawn on it that divide each face into nine

squares. The following questions were asked:

37
1. How many small cubes would be formed if the
large cube were cut apart?
2. How many cuts would be needed to do this?

3. If the pieces were rearranged between cuts,
could the job be done with less than six cuts?

Many students have a very poor comprehension of
three-dimensional geometric figures and cannot satisfactorily
picture them when looking at two-dimensional representations.
They were asked to build a model with their Cuisenaire rods.

The chain of logic should be:

1. One of the cubes to be formed by the cuts is
the one in the center of the big cube.

2. This center cube has six faces.

3. Since each of its faces lies in a different
plane, only one face can be cut at a time.

4. Therefore, six cuts are necessary to form this
cube.

Another type of problem that was used to introduce
deductive reasoning was:

Choose a number.

Triple it.

Add four.

Subtract your original number.

Divide by two.

Subtract two.

What do you conclude about the result of this

number trick?

Initially, students chose specific numbers and
inductively arrived at a conclusion. The students were
then asked to use a box for their chosen number and dots

for the constant added. Since the box represented any

number, but the same number throughout the trick, the

38

students were given the opportunity to write some simple
proofs for some number tricks.

Negative numbers were introduced through the use
of the number line. The operations with negative numbers
were motivated by use of the "Postman Stories“ found in
Robert Davis‘ text Discovery in Mathematics.

The conversion from the English measurement system
to the metric system and conversely were used to demonstrate
the use of ratio, proportion, per cent, measurement,
accuracy and precision.

After the students were familiar with the metric
system, the English measurement system and the history behind
their development, it seemed that the World Calendar would
be a point of interest. Price indexes were used to demon-
strate a possible need for the World Calendar and the
current interest in the development of the World Calendar.
Price indexes also served as a practical application of
basic arithmetic operations on the real numbers.

A Source Book of Mathematical Applications compiled
by a committee of the NCTM served as a source of relevant
problems for daily life.

The geoboards and geo—board activity sheets were
used to investigate, intuitively, the geometric properties
of plane figures. The figures were formed by stretching
rubber bands around the nails on a geoboard. The geoboards

used have nails arranged in a rectangular array, with the

39

nails one unit apart vertically and horizontally. The

unit of area is then the area of a square, one unit on a
side (Appendix C.4). The following concepts were developed
on the geoboards: area of squares, rectangles, parallelo-
grams, trapezoids, pentagons, hexagons, octogons, triangles;
similar figures, congruent figures, perimeters, right
triangles and the Pythagorean Theorem, square roots,

linear equations, slope of a line, concave and convex
figures, symmetry, reflections and rotations, midpoint

of a line segment and Pick's Theorem.

[The factor game (Appendix C.3.2) was used to practice
the concept of prime and composite numbers. The Sieve of
Erasthosthenes was used to discover a method to determine
whether a number was prime or composite. The students were
able to determine that it was necessary to cross off
multiples of all primes up to and including the square
root of n. All numbers that were not crossed off were
prime numbers.

After discovering the above the students were then
able to discover that it was only necessary to divide by
the primes less than or equal to /H to determine whether n
was prime or composite.

The Euclidean Algorithm was developed by the use of

Cuisenaire rods (see Laboratory Manual for Elementary

 

Mathematics by Fitzgerald et al.) to find the greatest

 

common divisor.

4O

Graphing in two dimensions was introduced while
playing "Guess My Rule." Appendix B.7 gives two interesting
examples of plotting exercises. The Madison Project "shoe
boxes" were used to collect data. The students represented
their data by a table, plotted the data on a graph and
found a general rule to describe it. The students were
given ample practice in plotting points from coordinates
in preparation for sketching the graphs of functions that
were covered later in the semester. To discover the rela-
tionship between the circumference of a circle and its
diameter, the following example was used: A set of six
circular objects of different diameters was distributed.
The students were asked to cut a piece of calculator tape
of length equivalent to the distance around the object.
The subjects were then asked to mark the diameter of the
object on the abscissa of a set of Cartesian coordinates,
and at this point on the abscissa attached the calculator
tape in the ordinate direction. In this manner the class
was led to discover that the circumference of a circle is
a linear function of its diameter.

Data from class quizzes and tests were used to
make histograms, calculate averages, medians, and modes.

The numbers from one to 100 were written on a roll
of adding machine tape and stretched across the front of
the room. The story of how Gauss solved this problem in

his head was told to the class. The numbers in this problem

41

form what is called a number sequence. Number sequences

 

and their properties were derived from practical appli-
cations such as the relationship between the length of a
pendulum and time in seconds, the relationship between the
number of generations back and the number of ancestors in
the generations, the piano keyboard, chain letters, markings
on a ruler, finding the relationship between the number of
folds of a piece of paper and the number of thicknesses, a
number trick with digital roots, and a number guessing

game using base two or binary number sequences.

After the students completed the handouts on
sequences, they were expected to be able to identify and use
simple arithmetic sequences, geometric sequences and to
recognize the smaller square and cube numbers. They were
also expected to appreciate the wide variety of topics in
which these sets of numbers appear.

Because exponential notation had been introduced
using number bases, the concept of scientific notation was
an easy concept to learn. It was introduced as a means of
conveniently expressing large numbers and then applied to
the work with decimal logarithms.

Logarithms were introduced by pairing a geometric
sequence with an arithmetic sequence to form an exponential

function. For example:

42
logarithms
0 1 2 3 4 5 6 7 8 9 10
numbers
1 ' 2 4 8 16 32 64 128 256 512 1,024
or logarithms
0 l 2 3 4 5 6 7 8 9 10
numbers
1 3 9 27 81 243 729 2,187 6,561 19,683 59,049
The students were able to recognize that the numbers from
the first set were powers of two and the numbers from the
second set were powers of three. Consequently, we called
the logarithms base two logarithms and base three logarithms,
respectively. The students were asked to multiple two
numbers, say, 4 x 8 = 32 and they noticed that the log of
32 was the sum of the log four and the log eight. Likewise,
they were asked to divide two numbers, say 81/3 = 27.2 The
students noticed that the log of the quotient was the log
of the numerator minus the log of the denominator. After
working multiplication and division problems with base two
logs and base three logs, base ten logs were introduced.
At first, the work with decimal logarithms was limited to
the logarithms of the first ten positive integers so that
emphasis could be placed on some simple theory and the
discovery of the relationship between the scientific
notation form of a number and its decimal logarithm. A
brief introduction to the slide rule was given along with

some practical applications of logarithmic scales.

43

The concept of area as (length) x (width) was used

as an example of the product:

x2 + x + 2x + 2

x2+3x+2

(x+l) (x+2)

 

 

 

 

 

 

before using the distributive axion or the rule for the
product of the means and the product of the extremes. When
considering problems of the form: (x-l) (x+2) or (x-l)

(x-2), it was difficult for the students to draw areas for
these, so they were asked to fold paper until the idea became
easy for them.

The concept of volume as (length) x (width) x (height)
was used as an example of the product: (x-2)(x-2)(x-2).
Dissectable cubes were available so that the students could
handle the pieces as they found the product.

The formulas for,

(xia)(xia) = x2 i 2x + a2

(x+a)(x—a) = x2 - a2

(x+a)(x-b) = x2 + ax - bx - ab

(xia)(xia)(xia) = (xia)3 = x3 i 3ax2 + 3a2x i a3

were discovered using the area and volume concept.

44

The students were asked if they could recognize a

pattern from the following information that they had dis—

covered.
(x + a)0 = 1
(x+a)l=x+a
(x + a)2 = x2 + 2ax + a2

3

(x + a)3 = x + 3ax2 2 3

+ Ba x + a

Hint: Consider only the coefficients.

Since some of the students had algebra before, they were
able to recognize it as Pascal's triangle and the coeffi-
cients as the binomial coefficients. A coin tossing
experiment showed a very elementary application of Pascal's
triangle.

During the development of any concept, some
historical facts were included either by the instructor
or in three or four minute reports from the students. The
variety of applications was intended to present the uni-
versality of mathematics, to make clear to the students
that they were expected to think, and equally as important
was the idea that doing mathematics can be an enjoyable

and rewarding experience.

Instruments Used in the Sipgy

 

As mentioned in the background, the mathematics
department employs an instrument which is used for placement

of students. It is this instrument which was used in the

45

testing of computational and basic skills. For attitude
changes, problem solving, and historical problems in
mathematics, the investigator devised her own tests or
questionnaires. The historical problems in mathematics
test was patterned after a test written by Avital. The
items on the attitude questionnaire were those recommended
by the National Council of Teachers of Mathematics (12).
The problem solving experiment was patterned after “The
Playground Problem" which is being used by the USMES
(Unified Science and Mathematics for Elementary Schools)
project. However, "The Playground Problem" is more open
ended and the question is not essentially mathematical
where the prime number problem and shadow problem have a

specific mathematical problem setting.

Testing Sequence

 

After a brief introduction to the course, the first
day of class was used to pretest for computational and
basic skills. At this time the department's placement
test was given. During the next two days, but not during
class, the 10 randomly selected students were interviewed
for their problem solving ability and procedures used to
solve the problems. During the semester but before we
discussed a specific historical problem, the problem was
included on a quiz for pretesting purposes. There was no

pretesting for attitude.

46

At midsemester time the placement test for compu-
tational and basic skills was administered again and was
designated as mid-test. The last two days of class were
used for follow-up testing designated the post-test. During
this time the attitude questionnaire and the historical
problems in mathematics were given. A test similar to
the placement test was given as a post-test for computational
and basic skills. The 10 randomly selected students were
post-interviewed for their problem-solving ability and
procedures used but not during class. Testing, with the

exception of problem solving, was done on a group basis.

Summary

In this chapter, the investigation of an innovative
approach to teaching general education mathematics courses
was described. The instruments used included the depart—
ment's placement test, a historical problems in mathematics
test, an open-ended attitude questionnaire and a test to
determine problem solving ability.

A detailed description of the sample was given
including the class distribution, years since taking a math
course and mathematics background.

The method of instruction and materials-and activi-
ties used in the instruction were discussed. A typical
lesson plan and a summary of teaching techniques used in

this class were included.

CHAPTER IV

RESULTS AND DISCUSSIONS

Introduction

 

Data were collected for the analysis of the study
from results of the mathematics placement test, placement
post-test, historical problems in mathematics test, problem
solving test, and an attitude questionnaire.

Thirty students registered for the A. M. section
and thirty-four for the P. M. section. Ten students were
randomly selected from the 64 students to be pre-tested
and post-tested for their problem solving ability and
procedures used. All of the students participated in the
testing of historical problems in mathematics, computa-
tional and basic skills, and attitude.

Before considering the analysis of the data, mention
should be made of the data which, for various reasons,
could not be collected. At midsemester time there were no
withdrawals. However; it is a college policy that during
the eleventh week of class, all students who are doing
marginal work (C_ or lower) are to be notified so that they
may withdraw. Six students were notified and they withdrew
from the course. Two students, for personal reasons, received

incompletes, and two students took jobs during the last four

47

48

weeks of the semester and withdrew from college. By the
end of the semester eight students had withdrawn and two

students had incompletes, leaving a total of 54 students

in the sample.

Case Studies for Computation
and Basic Skills

 

The department placement test served as the pre—test
and mid-test for the computational and basic skills. A test
similar to the placement test was used as the post-test. By
a similar test is meant that the questions were essentially
the same but the numbers, symbols, or names used were
changed. Also, the position of the correct response was
changed in the possible choice of answers. For example,
one of the questions on the pre-test was:

Out of a class of 75 students 3 received the grade

of A. The per cent of the class receiving the

grade of A was:

(a) 3% (b) .03% (c) 2.5% (d) 4% (e) none of these.

For the post-test the corresponding question was:

Out of a group of 125 people 5 had cancer. The
per cent of the group having cancer was:

(a) 2.5% (b) 4% (c) 3% (d) .03% (e) none of these.
Table 4.1 gives a summary of the results for the three tests.

According to Siegel (16) the Friedman two-way analysis
of variance by ranks for k matched samples provides a method
for testing whether k matched samples have been drawn from

the same population (Table 4.2).

49

TABLE 4.1.--Computational and Basic Skills Results.

 

 

 

 

 

Pre-test Mid-test Post-test
Class Average Median Average Median Average Median
A.M. 7.1 6 12.9 13 17.3 18
P.M. 8.2 7 12.4 12 19.8 21
Combined 7.7 6 12.6 12 18.6 19

 

H : the different times of the test have no differ-
ential effect.

the different times of the test have a differ-

 

l ential effect. That is, the size of the scores
depends on the time during the semester when
it was given.
Decision:
2 12 2 2 2
xr = [(61.5) +(100.5) +(l62) ]—3(54)(3+l)
(54)(3)(3+l)
2

95.097x2 = 13.816 and the decision is to reject H0
at .001 level of significance. The conclusion is that the
scores on the computational and basic skills depended on
the time during the semester when it was given.

Case Studies for Historical Problems
in Mathematics

 

 

Questions on quizzes given during the semester
served as pretest scores for the historical problems in
mathematics. The investigator's test for historical
problems in mathematics patterned after the test by Avital
served as the post-test instrument (Appendix A.2) Table 4.4

gives a summary of the results.

50

TABLE 4.2.--Scores of Fifty-four Students on Computational
and Basic Skills Under Three Conditions of Time.

 

 

Subject Pre Mid Post Subject Pre Mid Post
1 16 16 23 28 9 8 l4
2 3 7 13 29 9 13 18
3 12 10 23 30 9 14 19
4 6 6 17 31 4 10 17
5 15 22 23 32 11 9 22
6 12 18 19 33 19 23 25
7 14 IO 15 34 4 10 17
8 10 21 22 35 8 10 21
9 6 11 14 36 3 5 ll

10 7 6 14 37 5 14 20
ll 9 10 13 38 3 20
12 5 8 19 39 2 8 l4
l3 7 13 21 4O 19 16 24
14 21 24 25 41 17 21 25
15 4 9 14 42 18 24 25
16 16 18 24 43 6 13 21
17 4 9 44 10 20 23
18 ll 45 10 14 22
19 13 18 46 7 18 23
20 11 15 18 47 10 16 23
21 12 17 22 48 15 22
22 10 ll 49 14 22
23 10 50 13 17 23
24 3 18 51 6 9 17
25 12 14 21 52 6 7 15
26 15 20 25 53 8 8 16
27 7 ll 24 54 4 5 l4

 

 

51

TABLE 4.3.--Ranks of Fifty-four Students on Computational

and Basic Skills Under Three Conditions of Time.

Mid

Subject

Post

Pre

28

29

30

31
32

33
~34

35

36

37
38

39

4O
41
42
43

44
45
46

47
48

49

50

51
52

53

54

 

162

100.5

61.5

 

 

Mid

 

Post

Pre

Subject

 

1.5

1.5

1.5

1.5

10

11
12

13

14

15

16

17

18

19

20

21
22

23

24

25

26

27

 

52

TABLE 4.4.--Results of Historical Problems in Mathematics.

 

 

 

Pre-test Post-test
Class Average Median Range Average Median Range
A.M. .4 0 0-2 17 18 12-20
P.M. .7 0 0-4 17.3 17 10-19
Combined .6 0 0-4 17.2 17 10—20

 

The relative magnitude as well as the direction of
the differences for each pair is known for the data collected
for the historical problems in mathematics. Therefore, the
Wilcoxon matched-pairs signed-ranks test is the appropriate
test to use to analyze these data (16) (see Table 4.5).

HO: there is no difference between the pre-test
scores and the post-test scores for the
historical problems in mathematics.

H1: the post-test scores are greater than the pre-
test scores on the test for historical problems
in mathematics. That is, students after taking
the course were better able to answer the
questions on historical problems in mathematics.

Decision: 2 = -6.4 < -3.1. Therefore, H0 is
rejected at the d = .001 level of significance.
That is, students after taking the course were
better able to answer the questions on
historical problems in mathematics.

Case Studies for Problem Solving

 

On the second day of class the 10 randomly selected
subjects were individually interviewed for their ability
and procedures used to determine whether two five digit

numbers were prime. The author served as the interviewer.

53

TABLE 4.5.--Scores of Historical Problems in Mathematics.

 

Rank of Rank with less

 

Subject Pre-test Post-test d d frequent sign
1 1 16 15 21.5
2 O 13 13 5.5
3 0 18 18 41.5
4 0 19 19 51.0
5 0 l7 17 30.0
6 0 16 16 21.5
7 0 18 18 41.5
8 0 18 18 41.5
9 0 16 16 21.5

10 0 14 14 88.0
11 0 12 12 3.0
12 0 18 18 41.5
13 1 20 19 51.0
14 2 16 14 8.0
15 0 18 18 41.5
16 1 l7 16 21.5
17 0 17 17 30.0
18 1 13 12 3.0
19 0 20 20 54.0
20 2 19 18 41.5
21 0 18 18 41.5
22 0 18 18 41.5
23 0 19 19 51.0
24 0 14 14 8.0
25 l 16 15 21.5
26 1 14 13 8.0
27 2 19 17 51.0
28 O 12 12 3.0
29 0 l8 18 41.5

54

TABLE 4.5.--Continued.

 

 

Rank of Rank with less

 

 

 

 

Subject Pre-test Post-test d d frequent sign

30 O 18 18 41.5

31 0 l6 16 21.5

32 0 15 15 13.5

33 2 17 15 13.5

34 0 16 16 21.5

35 0 18 18 41.5

36 0 17 17 30.0

37 0 19 19 51.0

38 0 l6 16 21.5

39 O 10 10 1.0

40 3 18 15 13.5

41 2 18 16 21.5

42 4 19 15 13.5

43 0 l7 17 30.0

44 2 17 15 13.5

45 2 19 17 30.0

46 O 17 17 30.0

47 O 18 18 41.5

48 0 16 16 21.5

49 O 18 18 41.5

50 2 19 17 30.0

51 0 19 19 51.0

52 0 15 15 13.5

53 O 17 17 30.0

54 O 18 18 41.5

T = O
54 X 55 54 X 55 X 109
Mean = ——————— = 742.5 and S. D. = = 116.5
4 24
0 - 742.5

2 = = -6.4

55

Each subject was allowed 10 minutes and was allowed to use
any of the materials in the room which included pencil,
paper, blackboard and chalk,welectric calculator, textbook,
slide rule and any of the laboratory materials available
during the semester. They were allowed to ask questions

of the investigator which required a "yes" or "no" answer.
The investigator could decline to answer if the questions
were too leading. At the end of the semester, the subjects
were interviewed for their ability and procedures used to
determine whether two different five digit numbers were
prime.

The problem was read to the subjects and each was
given a copy of the questions. The investigator ascertained
that each subject knew the meaning of "prime number." The
subjects were scored as follows:

0 - if the subject guessed the answer

0 - if the subject calculated at random

1 - if the subject divided with primes only

1 - if the subject divided with primes sequentially

l - if the subject divised an algorithm

1 - if the subject used any of the materials besides
pencil, paper, blackboard and chalk

l - if the subject determined the correct answer.

The maximum points available were five (see Table 4.6).

HO: there is no difference between the pre-test
scores and the post-test scores for the
Problem Solving Experiment.

56

TABLE 4.6.--Scores for Problem Solving Ability and Procedures

Used.

 

Rank of Rank with less

 

Subject Pre-test Post-test d d frequent sign

P l 0 l 1 1.5

P 2 0 l 1 1.5

P 3 l 3 2 3

P 4 0 0 0 deleted

P 5 0 4 4 8

P 6 O 4 4 8

P 7 l 4 3 5

P 8 0 3 3 5

P 9 0 4 4 8

P10 1 4 3 5

T = 0

 

the post-test scores are greater than the pre-
test scores for the Problem Solving Experiment.
That is, students after taking the class, were
better able to solve the problem than they
could before the class.

Decision: There is a large number of ties and the

exact null distribution is not applicable
unless ties are considered. Therefore,

29 = 512 and (512)(.001) = .512 would allow
one point in the region of rejection. There-
fore, with T = 0, H0 is rejected in favor of
H1 concluding that students were better able
to solve the problem after taking the class.

In addition to determining whether two numbers were

prime, the subjects were asked to solve a problem that had

not been discussed during the class nor had they been pre-

tested for their procedures used and ability to solve the

problem.

They were asked whether your shadow gets shorter

57

or longer as you walk away from a lamp post; if someone
disagreed with you, how would you convince that person that
you were right? There were three other questions relating
to the shadow:

1. What would the path of the shadow look like if
the person walked in a straight line?

2. What would the path of the shadow look like if
the person walked in a circle?

3. What would the path of the shadow look like in
general, that is, for any path that the person
walks?

The method for collecting the data for this part

of the study involved the use of a tape recorder, note taking
by the investigator and the students' notes. The author
found the tape recorder a great aid in refreshing her memory

for students' comments.

Subject Pl

 

Subject Pl answered the question by writing that
the shadow gets longer and then it gets shorter. He used
a flashlight and pencil to demonstrate what would happen.
Then he changed his answer to longer.

In response to the path of a shadow when a person
walks in a straight line, he immediately responded, "I don't
know." He used a flashlight and pencil to determine that

the path was a straight line.

58

For the path of a shadow when a person walks in a
circle he responded, "I don't know that either." Then he
moved the pencil in a circle under the flashlight and con-
cluded that the path was "almost a circle but a little more
like an ellipse."

In response to the path of a shadow in general, he
also didn't know this answer. When he used the flashlight

and pencil, he still was unable to draw any conclusions.

Subject P2

 

Subject P2 immediately responded that the shadow
got longer. She used a flashlight and pencil to show what
the shadow would look like. Her response to the path of a
shadow when a person walks in a straight line was, "I don't
know." Then she drew a straight line on a piece of paper
and followed the straight line with the pencil and concluded
that the shadow followed the straight line. For the path
of a shadow when a person walks in a circle, she responded,
"I don't know." She drew a circle and followed the circle
with the pencil and concluded that the path was a circle.
In response to the path of a shadow in general, she responded,
"It follows the path the person is walking." Then she
"verified" her statement with the flashlight and pencil
and appeared satisfied that her answer was correct as she

indicated that she was finished.

59

Subject P3

 

Subject P3 responded, "The shadow gets longer, but
that must be wrong as that is so obvious." He used the desk
lamp and pencil to verify his statement. In response to the
path of a shadow when a person walks in a straight line, he
used the lamp and pencil and concluded that it was a straight
line. For the path of a shadow when a person walks in a
circle, he drew a circle and followed it with his pencil.

He concluded that the path was "almost a circle, but a
little tilted." In response to the path of a shadow in
general, he immediately responded, "The same path but I
think it really depends on something." He used the lamp
and pencil trying various paths but couldn't decide that
it depended on anything and so he wrote his answer as "the

same path."

Subject P4

 

Subject P4 answered the question by writing that
"the shadow gets longer as long as you were in the rays of
the light." First he tried to use the overhead light and
a ruler but he couldn't get a shadow that was sharp enough
to demonstrate what would happen. He then used the lamp
and a pencil to demonstrate that the shadow gets longer.
In response to the path of a shadow when a person walks in
a straight line, he drew a straight line and followed it
with a pencil concluding that the path was a straight line.

In response to the path of a shadow when a person walks in

60

a circle, he responded, "I don't know." He drew a circle
on a piece of paper and followed the circle with a pencil.
He moved the paper to three different places under the
light and said that the path was "the same as the person"
and wrote that the "path was a circle." In response to
the path of a shadow in general, he immediately responded,
"It is the same as the person is moving." He then used

the lamp and pencil to demonstrate his statement.

Subject P5

 

Subject P5 answered the question by writing that
the shadow gets shorter. Then in trying to convince someone
else, she used the desk lamp and pencil. After examining
the shadow she changed her answer to longer. In response
to the path of a shadow when a person walks in a straight
line, she immediately responded, "I don't know." She drew
a straight line on paper from one corner to the other and
followed the line with her pencil. She concluded that the
shadow followed the line. For the path of a shadow when a
person walks in a circle, she said that she didn't know
that either. She drew a circle and followed it with her
pencil under the lamp. She moved the circle farther away
from the lamp for five consecutive tries. She concluded
that the path was a circle but the diameter increased as
one gets farther from the lamp. In response to the path

of the shadow in general she responded, "The path is the

61

same as the path of the person." Then she proceeded to

use the lamp and pencil to demonstrate her conclusion.

Subject P6

 

Subject P6 answered that the shadow gets longer.
He drew similar triangles to demonstrate that the shadow
gets longer. In response to the path of a shadow when a
person walks in a straight line, he used the desk lamp and
a ruler to demonstrate that the path would be a straight
line. In response to the path of a shadow when a person
walks in a circle, he tried to demonstrate what would
happen using the lamp and a ruler. Because the ruler was
awkward, he used a pencil as the person and concluded that
the path was elliptic. In response to the path of the
shadow in general, he used the desk lamp and pencil to
check the path and concluded, "It is about the same as
the person but will be farther away if the person is farther

away from the lamp."

Subject P7

 

Subject P7 answered the question by responding,
"The shadow gets shorter." He used the desk lamp and a
pencil to demonstrate what would happen and commented that
the shadow got longer. Then he used the flashlight instead
of the desk lamp and responded, "It still gets longer." He

changed his answer to longer. He used the desk lamp and

pencil to determine the path of the shadow when a person

62

walks in a straight line and concluded that the path was
a straight line. Similarly for the circle, he concluded
that the path was a circle. In response to the path of
the shadow in general, he used the desk lamp and pencil
and concluded that the path was "like the path of the

person."

Subject P8

 

Subject P8 answered the question by writing that
the shadow gets longer. He used similar triangles to demon-
strate what would happen to the length of the shadow. In
response to the path of a shadow when a person walks in a
straight line, he answered, "It's a straight line." He
used the desk lamp and a pencil to determine the path of
a shadow when a person walks in a circle, and concluded
that it was "elliptic and not quite circular." In response
to the path of a shadow in general, he used the desk lamp
and pencil to determine what would happen. He wrote, "the

same path but the period isn't the same as the person's."

Subject P9

 

Subject P9 wrote, "the shadow gets longer until out
of realm of light." She used similar triangles to demon-
strate that the shadow gets longer. She used the desk lamp
and a pencil to demonstrate that the path was a straight
line when a person walks in a straight line. She moved

the pencil in circles and concluded that the "path is a

63

circle with larger diameters as you walk away from the
light." In response to the path of the shadow in general,
she used the desk lamp and pencil and concluded that it
was “the same path but with different amplitudes“ and she

explained what she meant by different amplitudes.

Subject P10

 

Subject P10 said that he thought that "the shadow
gets longer." He used the desk lamp and ruler to demon-
strate that it would get longer. Because the ruler was
too long, he used a pencil as the person to demonstrate
that the path would be a straight line if the person walked
in a straight line. Similarly, he demonstrated for the
circle and concluded that the path was a circle. To the
question of the path of the shadow in general, he responded,
"the path he walks." He used the desk lamp and pencil to

demonstrate this.

Summary

All of the students concluded that the path of the
shadow gets longer when a person walks away from a lamp
post. Likewise, they were able to conclude that the path
of a shadow is a straight line if they walk in a straight
line. If a person walks in a circle, all of the subjects
concluded that it was a circle, circular or elliptic. Nine

of the students were able to conclude that the path of the

shadow in general would be like the path the person walked.

64

One student was unable to arrive at a conclusion in this
instance.

So, after taking this class one can conclude that
all but one of the students were able to solve the shadow
problems. This does not conclude, however, that they did
not have these skills before taking the class.

Case Studies for Attitude
Toward MathematiCs

 

Because the concept of attitude is difficult to
quantify and because the questionnaire was open-ended, the
investigator decided to use the technique of key words or
phrases to summarize the data collected. The key words and
phrases were selected from the students' comments. The
per cent responses were calculated on the basis of one
response per subject. The subjects were asked to write
whatever came to their minds but not to evaluate the
instructor. (They were told that they would be given a
subsequent chance to evaluate the instructor.) They had
no previous knowledge that they were going to be asked to
fill out the questionnaire. They were not informed that
this form was an attitude questionnaire. This was not
communicated to the subjects because it was felt that to
have done so might have biased their answers.

Also, it was intended that the first three questions
would be answered in a general nature and the last three

questions would be answered with respect to this course.

65

It soon became apparent when reading the questionnaires that

most of the subjects answered all of the questions in re-

lation to this course.

to fill out the questionnaire.
the subjects had
asked to fill it

were spontaneous

It took no longer than seven minutes for any subject

accurate measure of their attitudes (5).

1.

How do you feel about mathematics?

Keyiwords or phrases

Like it; enjoy it; at ease with it; love it;
very good; like it very much; very inter-
esting; like it and have a deeper apprecia-

tion of it; not afraid of it anymore;
it and it strengthened my use of it;

it and it was fun; like it and it was
interesting . . . . . . . . .

Worthwhile; necessary; essential; useful;

challenge; exact and consistent;
foundation of cosmos . . . . . .

Ok; don't mind it . . . . . . .

Don't like it; don't really care for it
What do you like best about it?

Key words or phrases

 

Everything we did; classroom procedure
Math materials we use . . . . . .

Everyday math; applications; usefulness

Variety of problems; easy to understand;

learned new things; friends; history
Arithmetic; number problems . . . .

Algebra; equations . . . . . . .

From this and the fact that
no previous knowledge that they would be
out, one can conclude that their responses

and hence they would tend to be a more

% responses

 

. 51.9
. 29.6
C 12.9
C 5.6

% responses

 

. 7.4
. 29.6
. 9.2
. 9.2
. 9.2
. 14.9

it
Comments such as these are of a general

4.

66

Gives satisfaction; challenge; thinking
involved; logic; concreteness; reasoning
involved . . . . . . . . . . .

What do you like least about it?

Key words or phrases

 

Nothing 0 O O C O O O O O O O C

When I don't understand; and when others do

Too much to remember; too wide; memorizing;
time consuming . . . . . . . . .

Just about everything; material covered .

. 20.4

% resppnses

 

Logarithms; algebra; equations; story problems;

history . . . . . . . . . . .

Class too large; noisy; grading on handouts;
tests; homework; hard for me; when not clear

enough . . . . . . . . . . .
Not knowing all I would like to know . .
Blank O O O O I O O O O O O O

*
Calculus; theory; proofs; trig. . . . .

applicable to this course.

0 14.9
. 13.0
. 11.1
. 3.7
. 11.1
. 18.5
. 3.7
. 3.7
. 20.4

nature and are not

Has this course influenced your feelings toward

mathematics?

Keypwords or phrases

 

Positive: very much so; positivelyl; enjoyed

it; for the better; better understand;
decided to major in math; fun; know its
worth it; not afraid of it anymore; very

successful course; strengthened my use of;

strengthened my confidence; different

approach. (These comments were preceded
by "yes" and then an explanation why it
improved.) . . . . . . . . . .

Other: needed, necessary . . . . . .

No change: already liked, still do . .
still don't like . . . . . . . .

Blank: . . . . . . . . . . . .

% responses

 

. 61.1
. 18.5
. 13.0
. 5.6
. 1.9

67

What do you like about this course?

 

 

Key words or phrases % responses
New ways and techniques to solve problems . . 11.1
The math materials . . . . . . . . . . 5.6
Applications; explorations . . . . . . . 7.4
Wide variety so never got boring . . . . . 9.2
Good review; learned something; something really
needed; understood for first time . . . . 24.0
Very interesting . . . . . . . . . . . 5.6
Eliminated fear of math . . . . . . . . 3.7
Classroom procedures; no lectures; relaxing
atmosphere; fun . . . . . . . . . . 14.9

No textbook . . . . . . . . . . . . 5.6
Taught well . . . . . . . . . . . . 11.1
Everything . . . . . . . . . . . . 1.9

What would you do to change this course to make it
more interesting?

 

 

Key words or phrases % responses
Don't know; blank . . . . . . . . . . 9.2
No change needed; leave as is; interesting enough 24.0
Have reference book; textbook . . . . . . 5.6
Eliminate arithmetic text (redbook) . . . . 3.7

More history . . . . . . . . . . . . 1.85
No history . . . . . . . . . . . . . 1.85

More classroom materials; more classroom

More slowly; clearer expectations . . . . .

Longer than 15 weeks . . . . . . . . .
Meet only 3 times a week . . . . . . . .

participation . . . . . . . . . . . 18.5
Have field trips; more to daily life . . . . 9.2
Fewer topics . . . . . . . . . . . . 9.2
MCIG topics 0 o o o o o o o o o o o 1.85
More handouts . . . . . . . . . . . . 1.85
Fewer handouts . . . . . . . . . . . 1.85
More general; delve deeper . . . . . . . 3.7

3 7
l 8
1 8

mm

68

Comments Made by Students
During Semester

 

"I'm a clockwatcher. Every few minutes I usually

say, 'Oh, so many minutes left.‘ But not in this class.
I'm really enjoying it."

"The time goes so fast, the hour is gone by without

even noticing where it went."

"I'm trying to get my wife to take this course."

"After this I couldn't take another math class of

the other type."

"I prefer working in class, at least I can get help."

"This is an interesting course, I'm glad I'm

taking it."

"Boy, I need this!"

"I learn the most by working in class. Then what I

don't fully understand, I can work out at home."

"I enjoy it. I find it relaxes me and in high

school I dreaded going to math classes."

"I really like the relaxed atmosphere."

"If Harold can get it and understand it, then this

is a good way to learn."

learned.

mat .

to know

for me."

"I'm signing up for more math."

"I've got a job and I'm really using this stuff we

"I've got a job and now I could use more of this

"I really never knew before how important it was
math."

"I've changed my major to math."
"I want you to know that I really enjoyed the class."

"I didn't do well because there was too much noise

69

"I usually tune the instructor out after 5 minutes,
so class is a waste of time. But not in this class."

Summary

The analysis of the study has been presented in
this chapter. The statistics consisted of four parts. The
Friedman two-way analysis of variance test using the mathe-
matics placement test investigated whether the size of the
score on computational and basic skills depended on the
time during the semester when the test was given. The null
hypothesis of no differential effect was rejected at the
.001 level.

The Wilcoxon matched-pairs signed-ranks statistical
test using the historical problems in mathematics test
investigated whether students after taking this course were
better able to answer the questions on historical problems
in mathematics. The null hypothesis of no difference was
rejected at the .001 level of significance.

The Wilcoxon matched—pairs signed-ranks statistical
test was used to investigate whether students were better
able to solve problems after taking this course. The null
hypothesis of no difference was rejected at the .001 level
of significance.

No statistical test was used to investigate students
attitude toward mathematics. The responses to the question-
naire were summarized using key words or phrases. Also, a

list of student comments were presented.

CHAPTER V

SUMMARY, CONCLUSIONS,AND

RECOMMENDATIONS

Summary

The purpose of this study was to record and describe
the behavior of the subjects taking this course. Specifi-
cally, the purposes were concerned with four problems:

1. To determine if this method was effective in the

students' learning of computational skills;

2. To determine if this method was effective in the
students' learning of the historical problems of
mathematics;

3. To determine the effect of this method on the
students' problem solving ability; and

4. To determine the effect of this method on the
students' attitude toward mathematics.

The study was conducted during the Winter semester
of the 1973 Academic Year within the Department of Mathe-
matics, Saginaw Valley College.

Classroom materials and evaluation instruments were
developed and prepared during the Fall Semester of 1972.

Chapters I, II, III, and IV of this thesis were
written to provide the reader an understanding of the

problem, its background, its analysis and its resolution.
70

71

Conclusions

 

Purpose One

 

The writer concludes that this class had a signifi-
cant effect on the subjects' ability to answer questions on
the computational and basic skills test. This effect was
uncovered using the Friedman two-way analysis of variance
by ranks test. The Friedman test determined that the rank
totals differed significantly at the .001 level of signifi-
cance. That is, the size of the score depended on the time
during the semester when the test was given.

It is interesting to note that, in comparison to
Fall Semester, 1972, where less than 1% were eligible to
register for college algebra, 67% of these students had
17 or better making them eligible to register for college

algebra.

Purpose Two

 

The writer concludes that this class had a signifi-
cant effect on the subjects' ability to answer the questions
on historical problems in mathematics. This effect was
uncovered using Wilcoxon matched-pairs signed-ranks test
and the magnitude of the effect is to be found in Table 4.5.
It is more interesting to note that the average on the pre-
tests was .5 while on the post—test it was 17. If one is
interested in mastery learning, 81.5% of the class scored

80% or better on the post-test. It was the intent that

72

75% of the class would score 75% or better on the post-test.
So, in the area of historical problems in mathematics one

might conclude that 80/80 would be a reasonable expectation.

Purpose Three

 

The Wilcoxon matched-pairs signed-ranks test un-
covered significant results concluding that students after
taking this class, were better able to solve problems than
they could before the class. The magnitude of the effect

is to be found in Table 4.6.

Purpose Four

 

The effect of this class on students' attitude is
summarized in Chapter IV (pp. 65-67). It should be pointed
out that there was no entry attitudinal data on these
students. However, given the content of the statements
on the questionnaires, the writer concludes that the methods
used in this class improved the attitude of the subjects
toward mathematics. Sixty-one percent of the students
responded that their feelings improved while only 5.6%
responded that they still don't like mathematics. No
student responded that he disliked math more now than he
did before.

The fact that no student responded that he disliked
math more now than he did before is relevant when comparing
with the Fall Semester of 1972 which used the lecture-

textbook method. Thirty-three percent of those students

73

wrote that they "hate it, now," "detest it, now," "dislike
it, more," "absolutely hate it!," or "it discouraged me."

The following comments are of a more general nature:

Many days it was necessary to remind the students
that the class was over.

Students frequently asked to take the geoboards,
tower puzzles, Cuisenaire rods, etc. home with them over
week-ends to "play with" or to show friends and family.

After students learned to share their knowledge,
they also brought their own calculators to share with others
as there weren't enough for students to have their own.

Several of the students took the math materials to
their place of employment to show their fellow employees
how to get mathematical solutions. Many of the students
had played with these before and never knew that there was
math involved.

Fourteen of these students have registered for
Math 001 this spring half semester. (Math 001 is a refresher
that has been added this spring for those students who would
like more math but don't feel ready for college algebra.)

Three students have told me that they were changing
their major to math as a result of taking this course.

Only one student commented on the noise and there
certainly was more of this than in a "traditional" class.

These observations in addition to the comments

and questionnaires led the writer to conclude that the

74

attitude toward mathematics had improved for many of the
students.

These methods have been used in the same course by
the author and another faculty member at Saginaw Valley
College during the subsequent semester after the data for
this theses were collected. The other faculty member had
taught the course for four years with unsatisfactory results.
He was extremely impressed with the outcome after using this
method. It was the first time that he enjoyed teaching the
course.

Using the lecture-textbook method the wide variety
of students in a class such as this (Tables 3.1, 3.2, 3.3)
has created a difficult problem. It appeared to the writer
that this problem was basically eliminated because of the
interaction of the students. The students who understood
the material very readily assisted the students who were
having more difficulty understanding it. In fact, one
might conjecture that the heterogeneity of the group is
a contributing factor to the success of a course using the
method of this thesis.

The author recognizes that this paragraph is of a
subjective nature and necessarily so as she is giving her
opinion of the students' behavior. It took about four
weeks for the students to adjust to this approach in terms
of classroom behavior. The right and need to communicate

Openly with their fellow students led to unnecessary

75

"horseplay" which was not conducive to learning. Also,
during this time students would often ask, "What's the
answer?" without trying to do the work themselves. It
appeared to be a difficult transition for some of the stu-
dents to "discover" the answers themselves. However, after
this initial adjustment period, the author felt that this
was the approach to use in a general education mathematics
class with such a wide variety of backgrounds.

A comment might be in order regarding the use of
tutors. Tutors were available eighteen hours per week to
assist the students in this class (see Appendix B.4). How-
ever, very few students took advantage of this opportunity.

The author discussed this with the students at
various times during the course. In this atmosphere stu-
dents felt that they had become acquainted with members of
the class. It was easier to get help from their classmates
than from the tutors because their classmates understood
the difficulties that they were having better than the
tutors. From this limited experience, it would seem that
the use of tutors is not an important consideration in the
success of a course such as this.

With regard to the problem solving experiment, one
student did not improve his score on the prime number
portion (p. 59). In checking the attendance records, he

was absent for two weeks during which time prime numbers

76

were covered. The student was present for most of the
remaining course and performed well on the shadow portion
of the experiment.

For the shadow portion of the experiment, the
subjects either used a desk lamp or flashlight for the
lamp post and a pencil for the person to determine the
path of a shadow when a person walks in a straight line,
in a circle, or in any path. A more analytic approach
could have been employed. Namely, that the rays of the
lamp can be said to form a plane. The rays of the lamp
on the head of the person walking in a straight line form
a curve that intersects the plane in a straight line.
Similarly, the rays of the lamp on the head of a person
walking in a circle form a curve that intersects the plane
in a circle. The rays of the lamp on the head of a person
walking in any path forms a curve that intersects the plane
in a path similar to the one that the person is walking.
This is essentially what the students were observing but
none expressed their solution in this form. There had
been discussion of the intersection of curves with planes
in this course but no transfer had taken place in writing
the solution to this problem. This could indicate that the
geometry taught had not been sufficiently related to the

real world.

The cost of the materials used in this course which

were not consumable was about $1,000.00. The movies were

77

rented at a cost of $6.00 each. Those designated by an
asterisk are movies that the author recommends (see
Appendix C.l).

Because this is a case study of the effects of an
innovative approach on general education mathematics stu-
dents, the author presents the following recommendations

for further study.

Recommendations

 

1. Further individualization which would allow
students to learn the mathematics he needs at a pace con-
sistent with his background, interests, goals, and ability
be investigated in a general education mathematics course.

2. More extensive use of physical math materials
be used in the general education mathematics course.

3. Schedule course for two hour blocks of time so
that students can finish the task that they have started.
Determine whether the scheduling improves computational and
basic skills or attitude.

4. .A research experiment should be carried out
testing the purposes of this study with a control group

using the lecture-textbook method.

SELECTED BIBLIOGRAPHY

78

10.

11.

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Clark, John R., ed. Emerging Practices in Mathematics
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Dienes, Zoltan. "Some Relections on Learning Mathe-
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Engelhart, Max D. Methods of Educational Research.
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Kline, Morris. Mathematics in Western Culture. New
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McConnell, T. R., and Heist, Paul. "The Diverse
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Moore, Eliakem H. "The Foundations of Mathematics."
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National Council of Teachers of Mathematics.
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NCTM Thirty-third Yearbook. A History of Mathematics
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National Society for the Study of Education. Indi-
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Polya, G. How to Solve It. Garden City, N.Y.:
Doubleday and Co., Anchor Books, 1957.

 

Siegel, Sidney. Nonparametric Statistics for the
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Skemp, Richard R. The Psychology of Learning Mathe-
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Penguin Books Ltd., 1971.

 

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New York: Harcourt Brace Jovanovich, Inc.,
1972.

Wood, Ben D., and Haefner, R. Measuring and Guiding
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1948.

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1971.

 

The Department of Public Instruction. Michigan:
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No. NDEA-310, 1961.

 

Foster, G.; Kaufman, B.; and Fitzgerald, Wm. A First
Step Towards the Implementation of the
Cambridge Mathematics Curriculum in a K-12
Ungraded School. Cooperative Research Project

 

 

 

No. 5-405. Tallahassee, Florida: Florida State

University, 1966.

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Kaufman, B., and Haag, V. The Comprehensive School
Mathematics Program--An Experiment in Content-
Orientation and Individualization of Instruction.
Carbondale, Illinois, Central Midwestern
Regional Educational Laboratory, Inc., 1971.

 

National Council of Teachers of Mathematics. Ap
Analysis of New Mathematics Programs. Washington,
D.C.I 1963.

School Mathematics Study Group. Conference on Mathe-
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Leland Stanford Junior University, April, 1964.

 

U. S. Department of Health, Education, and Welfare,
Office of Education. The Low Achiever in
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1965.

 

 

C. Periodicals

 

Ablon, L. J. "A Modular Approach to Preparatory Mathe-
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Vol. 79, No. 10 (Dec., 1972), 1126-1131.

 

 

Allen, Frank B. "Teaching for Generalization in
Geometry." The Mathematics Teacher, Vol. 43
(Oct., 1950). 245-251.

 

Austin, C. A. "The Laboratory Method in Teaching
Geometry." The Mathematics Teacher, XX (1927),
286-294.

 

Ausubel, David P. "Learning by Discovery: Rationale
and Mystique." Bulletin of the National
Association of Secondary School Principals,
Vol. 45 (December, 1961), 18-58.

 

 

Becker, J. P. "Some Notes on the First International
Congress on Mathematical Education." The

American Mathematical Monthly, Vol. 77, No. 3
(19701, 299-302.

 

 

Bruner, Jerome S. "The Act of Discovery." Harvard
Educational Review, Vol. 31 (Winter, 1961),
21-32.

Davis, Robert B. "The Madison Project's Approach to
a Theory of Instruction." Journal of Research
in Science Teaching, Vol. 2 (1964), 146-162.

 

 

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45.

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Fehr, H. F. "Reorientation in Mathematics Education."
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Fey, James. "Classroom Teaching of Mathematics."
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No. 4 (October, 1969), 535-551.

Freudenthal, H. "Why to Teach Mathematics so as to
be Useful." Educational Studies in Mathematics,
1968, pp. 3-8.

Gavurin, Lester L. "Open Admissions--Mathematical
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Goodlad, John I. "Changing Curriculum of America's
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Harvey, J. G., and Pownall, M. V., eds. "Report of
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Heimer, Ralph T. "Conditions of Learning in Mathe-
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APPENDICES

86

APPENDIX A

INSTRUMENTS USED

PLACEMENT EXAMINATION

HISTORICAL PROBLEMS IN MATHEMATICS TEST
PROBLEM SOLVING TESTS

ATTITUDE QUESTIONNAIRE

MATH BACKGROUND

87

Appendix A.l
MATHEMAT ICS PLACEMENT EXAMINATION

Inclusion of the Mathematics Placement Examination
itself is impossible since that could terminate the exami-
nation's usefulness.

A review of the examination shows that questions
covering the following material appear on the teSt:

1. Percent, proportion, scientific notation.

2. Operations with signed numbers, exponents
(integral and rational), and rational algebraic
expressions.

3. Solution of linear and quadratic equations.

4. Factoring second degree equations.

5. Simplification (including multiplication and
division) of algebraic expressions of one and
two terms, complex fractions.

6. Verbal problems.

Students with a "good" background in first year high
school algebra should have little trouble with the test.
There is no geometry on the examination. The test will
entrap the student who would make such errors as:

(x"1 + fl).1 = x + Y or (x2)3 = x5.

88

89

Appendix A.2

HISTORICAL PROBLEMS IN MATHEMATICS

Name:

Date: April 13, 1973

 

Instructions: Give short and precise answers, using only the

space allotted.

Make your answers as clear and readable as
possible.

If you don't know an answer to a question
write "don't know."

Some people say that the introduction of a special
symbol for the number zero is one of the greatest
inventions of humanity. EXplain:

 

 

 

 

a. What is an "algorithm?"

 

 

 

b. What does one use the Euclidean Algorithm for?

 

 

 

a. What is a theorem?

 

 

b. What is the Fundamental Theorem of Arithmetic?

 

 

There are problems in mathematics which can easily be
formulated and which nevertheless haven't been solved
yet, even though mathematicians have tried very hard
to solve them. List at least one such problem. (Give
as many details as you know).

90

 

 

 

 

a. In what way has there been established a basic
connection between algebra and geometry?

 

 

 

 

b. When was this connection originated and by whom?

 

 

What modern studies has the discovery of the conic
sections made possible?

 

 

 

a. Who introduced place value in their number system?

 

b. What base did they use?

 

c. Give 2 examples where that base is used today.

 

 

a. What is inductive reasoning?

 

 

 

b. Give an example.

 

 

0. Give a limitation of inductive reasoning.

 

 

 

a. What is deductive reasoning?

 

 

 

10.

91

b. Give an example.

 

 

c. Give the outstanding advantage of deductive reasoning.

 

 

d. Give the outstanding disadvantage of deductive
reasoning.

 

 

From the soil of Egypt the science of Geometry--explain.

 

 

 

92

Appendix A.3

PROBLEM SOLVING

The zip code in East Lansing is 48823. Someone in East
Lansing asked if this is prime. What would your answer
be? How would you determine whether it is prime?

A zip code in this area is 48713. Is this zip code prime?

93

Problem Solving, Continued.

1.

a. Is the number 48827 prime?

b. Is the number 48719 prime?

Students were discussing whether your shadow gets
shorter or longer as you walk away from a lamp post.
What would your answer be? If someone disagreed with
you, how would you convince that person that you were
right?

What would the path of the shadow look like if the person
walked in a straight line?

What would the path of the shadow look like if the person
walked in a circle?

What would the path of the shadow look like in general,
that is, for any path?

94

Appendix A.4

QUESTIONNAIRE

How do you feel about mathematics?

 

 

 

What do you like best about it?

 

 

 

What do you like least about it?

 

 

 

Has this course influenced your feelings toward
mathematics?

 

 

What do you like about this course?‘

 

 

What would you do to change this course to make it more
interesting?

 

 

95

Appendix A.5

MATH BACKGROUND

Name

 

Class Standing (circle one) Fr. So. Jr.

Major

Sr.

 

Math courses since 8th grade

 

 

 

 

How long has it been since your last math class?

 

APPENDIX B

PROCEDURES

SEMESTER PLANS

SAMPLE DAILY PLANS
TOPICS COVERED

TUTORING SCHEDULE
REFERENCE TESTS

SAMPLE ASSIGNMENT SHEET
SAMPLE HANDOUTS

SAMPLE QUIZZES

96

JANUARY

Monday

a"

8 Digital
roots
Axioms under
x and +

Arithmetic
sequences

Fibonacci,
sequences

Inductive
reasoning

15

22 Prime
factors
Composite
nos.
H.C.F.
L.C.D.
Euclidean alg.
Computations
with
rational
numbers

29 Denominate
numbers
and
arithmetic
of
measurement

Appendix B.l

Wednesday Thursday
3 Pretest 35 4 Numbers
minutes Number line
Brief Exponents
Introduction Bases
Decimal System
10 Axioms under 11 Graphing
- and ? Linear
Division equations
algorithm Histograms,
Order of mean,
operations median, and
mode
17 Geometric 18 Geometric
sequences and
arithmetic
Power
sequences
sequences
24 Fund. Thm. of 25 Mult. and
Arithmetic div. of
Decimal fractions fractions
Place value Repeating,

Scientific
notation
Addition and

subtraction

of fractions

31 Metric system
Errors in
approximate

numbers.

97

terminating,
non-

12

19

26

terminating

decimals

Friday

Negative
naturals

Order

Absolute value

Inequalities

Rational
operations
on negative
integers

Graphing

35 min. test
on Chapters
l-3 and
handouts

Rational
numbers

Ratio,
proportion
percent

FEBRUARY

Monday

5 Word problems

Squaring both
sides of an
equation

12 Graphing
circle
ellipse

l9 Washington's
Birthday

26 Quadratic
equation

Polynomials

14

21

28

98

Wednesday Thursday
1 Square root
Irrational
numbers
Square root
algorithm
Equality
Arithmetic 8 Graphing
of expon-
ents

Square roots
and
exponents

Graphing
hyperbola

15 Logarithms

Interpopula-
tion and
extrapola—
tion

Logarithms 22 Quadratics
graphing

problems

Mid-test
35 min.

Friday

2 Solution of
elementary
equations

Arithmetic of
exponents

9 Graphing
linear and
quadratic
equations

16 Logarithms

23 Quadratics
graphing
problems

MARCH

Monday

5 Pentagons
Parallelogram
Trapezoids
Similar figures

Congruent
figures

12 Factoring
sum and
difference
of 2 cubes

19 Algebraic
fractions
sums and
differences

26 Exponents
and
radicals

l4 Factoring

21 Algebraic

28 Exponents

99

Wednesday Thursday Friday
1 Intuitive 2 Geometry
geometry Pythagorean
Area, squares .theorem
rectangles, Di:tanc:
triangles ormu a
Perimeters
7 Parallel 8 Factoring 9 Factoring
lines squares of difference
. Bi m'al f 2
Perpendicular no 1 S O
. squares
lines

15 Systems of 16 Systems of

linear linear
equations equations
graphically

22 Algebraic 23 Algebraic

fractions fractions- fractions-
multiplica- division
tion

29 Exponents 30 Test 35 min.
and

radicals

and
radicals

100

APRIL
Monday Wednesday Thursday Friday
2 4 5 Pascal's 6 Sum and dif-
triangle ferences
d
Binomial cube
theorem
9 11 Review 12 Attitude l3 Computational
question- and basic
naire skills
. . -t t
Historical pOSt es
problems in
mathematics
post-test

16 Last day of
class

Returned
Historical
problems in
math

Reviewed
post-test
results

Day and date

Classroom

Materials

Homework

Objectives

101
Appendix B.2
SAMPLE DAILY PLANS

Wednesday January 3, 1973

Introduce course
Departmental pre-test 25 min.

Pretest

Computational Arithmetic (abbreviated C. A.
from now on)

C. A. pp. 1-9 explanations

C. A. pp. 15-16 1-42 exercises

1. To learn to identify counting numbers
(natural numbers) whole numbers, even
whole numbers, odd whole numbers and
integers

2. Use the number line.

 

Day and date

Classroom

Materials

Homework

Objectives

Evaluation

Thursday January 4, 1973

Introduce number line, numeration system
demonstrator, and odometers.

Number line, numeration system demonstrator
odometers, handout

C. A. pp. 9-14 explanations
C. A. pp. 16-17 43-62 exercises

1. To learn to use exponential notation

.2. To learn to read and write numbers in

the decimal system

3. To discover advancing algorithm for
addition in any base

4. To learn the concept of place value and
the number zero.

Quiz tomorrow

 

Day and Date

Classroom

Materials

Homework

Objectives

Evaluation

102

Friday January 5, 1973

Quiz, "Guess My Rule" If students don't
volunteer, use random number generator and
introduce the concept of "equally likely."

Quiz, handout on checks, Cuisenaire rods

C. A. pp. 19-27 explanations
C. A. pp. 46-47 1-34 exercises

1. Continue advancing algorithm discussion
for addition and to practice addition
of whole numbers

2. Checks--application of paraphrasing in
decimal system

3. Encourage students to work together--
group interaction

4. Build foundation for function, graphing,
inductive reasoning and to form conjectures.

Quiz on Monday

 

Day and Date

Classroom

Materials

Homework

Objectives

Monday 1973

January 8,

Quiz, make histogram from data from last quiz,
Guess My Rule

Quiz, graph paper, rulers, handout, Cuisenaire
Rods

C. A. pp. 27-45 explanations

C. A. pp. 47-48 35-66 exercises
l. Inductive reasoning

2. Data collection

3. Practice addition and multiplication of
whole numbers

4. Introduce negative numbers through two
dimensional graphs

5. Group interaction

 

103

Appendix B.3

TOPICS COVERED

Following is a list of major topics listed for the

course during the time this study was in progress.

1.

6.
7.
8.

9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.

Basic Arithmetic Operations with whole numbers, integers,
rational numbers and real numbers; prime factorization

Ratio, Proportion, Percent, Measurement
Equality and order axioms
Properties of real numbers

Sums, differences, products and quotients of signed
numbers

Absolute value
Order of operations

Polynomials--sums, differences, products, quotients,
and factoring (linear and quadratic)

Signs of algebraic fractions

Reduction of Algebraic fractions

Sums, products and quotients of algebraic fractions
Integral, rational, zero and negative exponents
Radicals

Solution of first and second degree equations
Application Problems

Systems of Linear Equations

Graphing

Problem solving

Inductive and deductive reasoning

Historical Problems in Mathematics

Metric system

Geometry, Pythagorean Theorem, distance formula, etc.
Logarithms

Histograms, mean, median, mode

Binomial theorem

Sequences

104

Appendix B.4

TUTORING SCHEDULE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Day
Time Monday Tuesday Wednesday Thursday Friday
8-9
9-10 Mike B.
Rm. 217
10-11 Joyce N. Don H. Don H.
Rm. 218 Rm. 310 Rm. 217
11-12 Joyce V. Joyce V.
Rm. 218 Rm. 310
12-1 Sue R. Rick D. Sue R. Sue R. Rick D.
Rm. 235 Rm. 233 Rm. 217 Rm. 218 Rm. 217
1-2 Matt K. Matt K. Matt K. Matt K.
Rm. 218 Rm. 238 Rm. 218 Rm. 217
2-3 Tim H.
Rm. 235
3-4 Jerry S. Don H.
Rm. 218 Rm. 217
4-5
5-6
6-7
7-8
8-9
9-10
Tutors' math background
Don H.--161 to 262, 313, CS 147
Joyce V.--161 to 262, 313, 372, CS 147
Sue R.--CS 147, 157, 331, 332
Joyce N.--l6l to 262, 370, CS 147, 157, 331, 332
Jerry S.--l6l to 262, CS 147, 157, 291, 292, 331, 332
Rick D.--16l, 162
Matt K.--121, 122, 161, 162, CS 147, 157, 331, 332
Tim Hazzard—-16l to 262, 370, CS 157, 310, 331
Mike B.--121, 122, 161 to 262, 420, CS 147

Reference Texts

 

1.

10.

ll.

12.
13.

14.

15.

16.

Accent on Algebra

Computational
Mathematics

Discovery in
Mathematics

Essential Mathematics

Explorations in
Mathematics

Geo-board Activity
Sheets

Graph Gallery

Laboratory Manual
for Elementary
Mathematics

Laboratory Manual

Mathematics--A
Developmental
Approach

Mathematics--A
Human Endeavor

Men of Mathematics

Number Sentence
Games

Palatable Plotting

A Source Book of
Mathematical
Applications

Using the
Cuisenaire Rods

105

Appendix B.5
Author

Pat Boyle

Charlene Pappin

Robert B. Davis

Doris Stockton

Robert B. Davis

Cech
Tate

Joseph P.
Joseph B.

Pat Boyle

Fitzgerald et a1.

Allyn J.
Washington

Allyn J.
Washington

Harold R. Jacobs

E. T. Bell

Dale Seymour
Margaret Holler
Nancy Collins

Pat Boyle

Compiled by
A Committee of
the NCTM

Jessica Davidson

Publisher

 

Creative Publi-

cations, Inc.,
1971

Glencoe Press,
1972

Addison-Wesley
Publishing Co.,
1967

Scotts,
and Co.,

Foresman
1972

Addison-Wesley
Publishing Co.,
1967

Ideal School Supply
Co., 1971

Creative
cations,
1971

Prindle,
Schmidt,
1969

James E.
Assoc.,

Publi-
Inc.,

Weber &
Inc.,

Freel &
Inc., 1972

James E. Freel &
Assoc., Inc.,
1972

W. H. Freeman &
Co., 1970

Creative Publi—
cations, Inc.,
1971

Creative Publi-
cations, Inc.,
1971

Bureau of Publi-
cations Teachers
College Columbia,
Univ., 1942

Cuisenaire Co.
America, Inc.,

of
1969

106

Appendix B.6

SAMPLE ASSIGNMENT SHEET

Assignments:

1/15/73

1/17/73

1/18/73

1/19/73

1/22/73

1/24/73

1/25/73

1/26/73

1/29/73

1/31/73

2/1/73

2/2/73

2/5/73

Handout--to be handed in
Arithmetic Sequences
Fibonacci Sequences

Handout-~to be handed in
Geometric Sequences
Sequences of Squares

Review for 35 min.

test on 1/19/73

Test in class 35 min.

P 101 - 114
Exercises P

P 115 - 142
Exercises P

P 151 - 172
Exercises P

P 172 - 182
Exercises P

P 189 - 212
Exercises P

P 217 - 233
Exercises P

P 233 - 256
Exercises P

P 261 - 268
Exercises P

P 281 - 301
Exercises P

P 309 - 328
Exercises P

143 - 145
145 - 149
183 - 184
186 - 187
213 - 216
257 - 259
259 - 260
Skip rest
279 - 280
302 - 307
329 - 333

(1 — 32)
(33 - 100)
(1 - 58)
(59 - 91)
(l - 71)
(l - 40)
(41 - 75)
of Chapter
(1 - 48)
(1 - 40)

(l — 52)

107

Appendix.B.7.

Application of Paraphrasing Numbers

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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108

Appendix B.7
Sample Handout - Used to derive the advancing algorithm.
Handout 2c
Use odometers:

Add: 21 List steps:
43

___ a.

b.
C.
d.
e.

f.

Possible Algorithm:

 

 

 

109

Sample Handout

77
78
75
76
75
82
83
90
76
87
68
71
76
49
71
87
80
93
74
80
67
69
82
90
82
79
83
89

Scores for C1ass-—A. M. Section
Make a histogram using the data below which are the
scores for this class on the last test.
Calculate the mean, median and mode.
Plot the mean, median and mode on the graph.

Be sure to label the axis.

'Longhude

Longitude and Latitude is a very real example of ordered pairs and graphing.
is angular distance, measured in degrees, East or West from the prime
meridian (Greenwich, England) while latitude is the distance North or South from

the equator.
gitude and

Here is a graphing exercise that may prove interesting if you are able to cor-

Each point on the surface of the earth can be designated by its ion-
a one-to-one correspondence between points (locations) and
ordered pairs (longitude, latitude).

latitude,

110

Longitude and Latitude

rectly plot each location accurately and then connect these points in order.

Nd—h—l—lu—l—h—b—Ia—t—I
959908979199)pr

99°N9’WPPN.‘

( 90.0w,
( 88.0W,
( 86.0w,
( 85.5w,
( 84.5w,
( 83.5w,
( 83.9w,
( 82.0w,
( 81.0w,
( 80.5W,
( 80.8W,
( 81 .8W,
( 82.0w,
( 81 .0W,
( 75.5W,
( 75.5w,
( 75.0W,
( 73.0W,
( 71.0W,
( 71.1W,

30.0N )
30.2N )
30.0N )
29.5N )
29.8N l
28.0N )
27.0N l
25.0N )
25.0N )
25.7N )
27.3N l
29.0N )
30.5N )
31.7N )
35.0N )
37.0N )
38.5N )
41.0N )
41.5N )
42.5N )

21 .
22.
23.
24.
25.
26.
27.
28.
29.
30.
31 .
32.
33.
34.
35.
36.
37.
38.
39.
40.

( 70.0w,
( 67.0w,
( 68.0w,
( 68.0w,
( 69.0w,
( 70.0w,
( 70.0w,
( 72.0w,
I 75.0w,
( 76.0w,
( 76.3W,
( 79.0w,
( 79.3w,
( 82.0w,
( 83.0w,
( 84.0w,
( 82.8w,
( 82.6w,
( 83.0w,
( 83.5w,

44.0N )
44.5N )
46.0N l
48.0N )
48.0N )
46.5N )
45.5N l

45.0N )—

45.0N )
44.0N )
43.2N )
43.0N l
42.0N )
41.5N )
41.3N l
41.6N )
42.0N )
43.3N )
44.2N )
45.2N )

41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51 .
52.
53.
54.
55.
56.
57.
58.
59.

( 84.3w,
( 85.0w,
( 86.0w,
( 86.0w,
( 87.0w,
( 87.6w,
( 88.0W,
( 87.0w,
( 86.0w,

( 84.0w,

( 85.0w,
( 85.0w,
( 87.0w,
( 88.0W,
( 90.5w,
( 91.0w,
( 90.8w,
( 92.2w,
( 90.0w,

45.0N )
45.3N )
44.0N )
42.2N )
41.7N )
41.7N )
43.0N l
45.0N )
45.5N )
45.8N )
46.0N )
46.4N )
46.0N l
47.0N )
46.0N )
46.2N )
46.8N )
46.5N )
47.8N )

an all-- |

l00

 

Hm

 

 

 

 

 

 

111

 

 

 

 

 

 

 

EAST

 

 

 

 

N92

08

 

 

30

 

 

 

 

 

 

 

 

 

 

112

Plot the following points on a single set of coordinate axes, then connect the

points with segments between consecutive points, in the order indicated by the
scheme at the bottom, to form five closed figures. Place the coordinate axes in the
center of your graph paper in order to accommodate the completed figure.

1.

2.

10.

11.

12.

13.

14.

15.

16.

17.

(15, 0)
(14, 5.5)
(12, 9)
(12, 0)
(11.75, 3.25)
(10, 6.6)
(9,12)
(9,8)
(6.6,10)
(5.5, 14)
(325,115)
(15,121
(1.5, 0)
(0,15)
(—1.5, 12)
(—1.5, 0)

(-3.25, 11.5 )

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

(-5.5, 14)
(-6.6', 10)
(-9, 12)
(—9, 8)

(-10, 6.6)
(—11.75, 3.25)
(—12, 9)
(—12, 0)
(-14, 5.5)
(—15, 0)
(—14, —5.5)
(—12, —9)
(—11.75, —3.25)
(—10.5, —5.5)
(—9, —12)
(—9, —8)

(—6.6, -—10)

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

(-5.5, —14)
(-3.25, -11.75)
(0, -151
(-1.5, —12)
(—1.5, -3.5)
(3.25, -11.75)
(5.5, —14)
(6.6', —10)
(9,. —12)

(9 ,—8)

(10.5, —5.5)

( 11.75, —3.25 )
( 12, —9)

(14, — 5.5)
(1.5, —12)

( 1.5, —3.5 )

1-2-3—7—10—14—18—20—24—26—27—
28—29—32-35—37—41—43—47—48-1
4—5—6—8—9-11—12—13—45-46—4
16—15—17—19—21—22-23—25—30—31-16
33—34—36—38-39—33

113

 

31.

32.

33.

34.

35.

114

Appendix B.8

SAMPLE QUIZ

Quiz 7

Each number in this sequence 5,11,17,23,29, . . . is a
prime number. What are the next 5 numbers in the
sequence? Are they all prime numbers?

a.
b.
c.
d.

35, 41, 47, 53, 59 yes
35, 42, 50, 59, 69 no
35, 41, 47, 53, 59 no
35, 42, 50, 59, 69 yes

What is the 100th term of the sequence 2,5,8,1l,14 . . .?

a.
b.
c.
d.

The

d.

302
299
300
takes too long and there is no simple way to find out

keyboard of a piano forms a Fibonacci sequence:

statement isn't true, its the guitar.

13 keys form a chromatic scale, of these 8 are the
white keys, 5 are black, of the 5 black keys 3 are
grouped and 2 are grouped; the 1 corresponds to 1
chromatic scale.

1,2,3,5,8,13 is represented on all musical
instruments and therefore must be on a piano.
because Fibonacci designed the keyboard.

"Chain letters" is an example of:

a.
b.
c.
d.

arithmetic sequence
Fibonacci sequence
a sequence of squares
geometric sequence

Who were the first to introduce place value in their
number system and what base did they use?

Greeks; base 12
Romans; base 5
Babylonians; base 60
Descartes; base 10

115

Sample Quiz

36.

37.

38.

39.

4o.

 

Quiz 8
The first people to use place value were . The
base they used was . Two examples where that

base is used today are

 

a. Greeks; 2; counting; slide rules

b. Romans: 100; Roman numerals; books

c. Middle ages; 10; counting; praying and rosary
d. Babylonians; 60; degrees; time

Fractions are called rational numbers because

a. people who understand fractions are more "rational"
than those who understand whole numbers only.

b. because fractions aren't as difficult to understand
as "irrational" numbers.

0. a fraction, the ratio of 2 integers, and "rational"
is a derivative of the word ratio.

d. none of these.

 

Deductive reasoning means that:

a. one reasons from the specific to the general.
b. one reasons from the general to the specific.
c. one must always use mathematics.

d. one is playing mathematical games.

Inductive reasoning means that:

a. one must always use mathematics.

b. one reasons from the general to the specific.
0. one is playing mathematical games.

d. one reasons from the specific to the general.

When we play "Guess My Rule" we are using:

a. deductive reasoning
b. inductive reasoning
c. neither (a) or (b)
d. both (a) and (b)

116

Sample Quiz

Quiz

Determine the rule for the following two problems.

waH
\IU'IUJl-J

 

Y = , and when X = 100 Y

 

 

 

17
30
37

 

Y = and when X = 40 then Y =

 

 

117

Sample Quiz

Quiz

 

 

Simplify:
1. (-4X2Y3)(2XY)(XlOY-l) =
2. (v2w5)(vw'l)4 =

2

3

3. (—125 r3s9) =

 

4. \3/-64X3Y6 =

 

 

 

Evaluate:
1
5. 2-92 =
Z
6. 273 =
1 _Z
8 l6 0

 

 

APPENDIX C

MATERIALS USED

LIST OF MOVIES

LIST OF MATERIALS
SAMPLE GAMES

C.3.l GUESS MY RULE
C.3.2 FACTOR GAME

C.3.3 CONTIGUOUS BOARDS

MADISON PROJECT BOXES

118

Appendix C.1

LIST OF MOVIES

Associative Properties *

Commutative Properties *

Distributive Properties *

Equations and Graphs

Ordered Pairs & Cartesian Product
Unsolved Problems/Two Dimensions Film I
Unsolved Problems/Three Dimensions Film II
Let Us Teach Guessing *

Mathematical Induction

Mr. Simplex Saves the Aspidistra *
Topology

Challenging Conjectures

What is Mathematics and How Do We Teach It:

Nim and Other Oriented Graph Games

More Solutions of Linear Equations
Equations and Graphs of the Parabola
Hyperbola, Ellipse and Circle
Progressions, Sequences and Series *
Infinite Series and the Binomial Expansion
Equations with Unknowns in the Exponents
Using Logarithms to Solve Equations
One—to-one Correspondence *

Counting *

Sets: Union and Intersection *

Man and the Computers *

Infinity *

119

'k

120

Appendix C.2

LI ST OF MATERIALS

Demonstration Slide Rule

Dandelin's Cone, Conic Sections

Coaxial Cylinders

Sectioned Cylinder

Binomial Cube

Hemisphere with Spherical Sector

Three Dimensional Coordinate System Model
Styrofoam Solids Set

Metric Chart

Liter Block--dissectib1e

Liter Case, plastic

Liter Blocks

Metric Liquid Measure Set

Radian and circle demonstrator

Chalkboard graph chart--polar and rectangular
Chalkboard graph chart--log and semi—log
Coordinate system transparencies

Ball Abacus

Binary Counter

Numeration system demonstrator

Madison Project--independent exploration materials
Multi-Base arithmetic blocks and work cards
Cubic inch blocks

Meter sticks

Geoboards and geoboard activity sheets, activity card set
Slide rules--c1assroom set

Tape measures--automatic rewind

Cuisenaire Rods

Wang Calculators Rulers
Odometers Scissors

Number line Graph paper

121
Appendix C.3.1
GUESS MY RULE

The students were presented with a set of ordered
pairs from which they had to determine some rule or function
that associates the first member of the ordered pair to the
second. The set of ordered pairs was obtained in this
fashion: some membercfifthe class presented a first number
for an ordered pair. The leader, in this case the author,
responded with the appropriate second number which was
obtained by mentally applying the rule to the given first
number. After each response the class was allowed to guess
what rule of correspondence was being used. The author
limited the functions to linear, quadratic and some simple
exponential functions.

A variation of this procedure was to give each
student a 3 x 5 card upon which he could write some function
of his own choosing. These cards were collected, shuffled,
and redistributed. The class was divided into groups of
about five students and "Guess My Rule" was played with
one of the group being the leader and using the rule that

he selected from the cards.

122
Appendix C.3.2
THE FACTOR GAME

The game is played as follows:
Red player places a red mark on a number (n).

Blue player places a blue mark on each uncovered proper
factor of n.

Blue now selects an uncovered number (m) to cover with
a blue mark.

Red player covers each uncovered factor of m with a red
mark.

Play continues until no number remains which has any
uncovered factors.

If a player places a mark on a number which had no
uncovered factors, the other gets to lead in the next
two plays.

The game is over when there are no more legal moves.
The score is the sum of the numbers covered.
If one player doesn't cover all of the factors available

to him on a turn, there are two different rules which
can be used:

1. The other player can cover the missing factors.

2. The missing factors remain uncovered.

The rule should be determined before the game begins.

123

The Factor Game

 

3

4

5

6

 

q

10

12

 

)3

14

I5

lb

17

18

 

I9

20

2|

22

25

24

 

25

26

27

28

29

so

 

3|

 

32

 

33

 

34

 

35

 

36

 

 

 

 

lO

 

l4

l6

I7

18

19

20

 

24

26

27

28

29

30

 

.34

36

37

38

39

4O

 

44

46

47

46

49

So

 

54

56

57

59

59

60

 

64

66

67

68

69

7o

 

74

76

777?

79

90

 

8‘1

86

87

8?

89

90

 

 

 

 

 

,94

 

 

_96

 

 

97

98

 

99

 

Hm

 

 

124

Appendix C.3.3

CONTIGUOUS BOARDS
Rules for Contiguous Playing Boards:

Each table has 2 cubes (and 3 dice) with 4 through 9
on each cube.

To play board 1, write the possible sums and differences
of the numbers 4 through 9 in the squares with a 0 in
the square on top. (Each player has a board.)

Determine who will toss the cubes first and then rotate
turns (in a clockwise direction).

After you have tossed the cube, the tosser may either
add or subtract the two numbers (for the first few tosses
it usually doesn't make any difference which you do).
Then cross off the sum or difference on your contiguous
board. The first person who gets the numbers crossed

off wins.

Note: For board 1, time will not be a factor for most
of you. However, from board 2 on it probably
will be. So, determine the time that you are
allotting before you start board 2, 3 and 4.

For board 2 you may add, subtract, or multiply.

For board 3 you may add, subtract, multiply or divide.

For board 4 you will use 3 cubes and you may add,
subtract, multiply or divide.

Note: For board 3 and 4, allow only those divisions
which do not have a remainder.

125

Appendix C.4

 

 

Three students were examining the contents of this box one
day. Marilyn said, "This is a puzzle I've seen before. The
object of the puzzle is to transfer the discs from the center
peg to either of the other two pegs, ending with the discs
arranged in the same order as at the start (smaller discs on
top of larger discs).

 

 

 

g j

There are only two rules in moving the discs: (1) only one
disc may be moved at a time, and (2) a larger disc may
never be placed on top of a smaller disc. "Frank and Mark
said they would like to try it.

CAN YOU DO IT STARTING WITH 5 DISCS?

 

Tower Puzzle instruction card no. 1

 

 

CAN YOU DO THE PUZZLE STARTING WITH ONLY 3 DISCS?

To transfer 3 discs, Frank said it took him 11 moves. Mark
did it in 7 moves, and Marilyn said it took her 10 moves.
They each tried again. Mark said "This is the shortest way
to do the puzzle with three discs; it should take 7 moves."
DO YOU AGREE?

IS THERE A WAY TO TELL WHETHER YOU HAVE THE MINIMUM
(SMALLEST) NUMBER OF MOVES OR NOT?

CAN YOU DO THE PUZZLE STARTING WITH 2 DISCS? 4 DISCS?
6 DISCS?

 

Tower Puzzle instruction card no. 2

 

 

126

 

 

Frank asked, "Is there any relation between the number of
discs and the minimum (smallest) number of moves needed
to transfer the piles?" Mark suggested they make a table
to keep track of the numbers. "Let the number of discs
be the number and let the minimum number of moves to
transfer all the discs be the A number,“ he added.
Marilyn said, "This is something like the game Guessing
Functions, when you put a number in and get a number out
and figure out a rule that works."

Number of discs-+C) A*- bfixfinmmlnumber of moves needed to
-ff_-- transfer the discs
2
3 7
4
5

 

CAN YOU COMPLETE THE TABLE ABOVE? FILL IN THE NUMBER ON
THE PAPER INCLUDED IN THE BOX.

 

Tower Puzzle instruction card no. 3

 

 

CAN YOU FIGURE OUT ONE RULE THAT WORKS FOR ALL THE PAIRS
OF NUMBERS IN THE TABLE?

WRITE YOUR RULE (USING A ANDD) , THEN CHECK IT BY TRYING
VARIOUS PAIRS OF NUMBERS FROM THE TABLE.

 

Tower Puzzle instruction card no. 4

 

 

127

 

 

"I think the graph of these pairs of numbers will lie along
a straight line," said Mark.

DO YOU AGREE? CAN YOU MAKE A GRAPH OF THE PAIRS OF NUMBERS

IN THE TABLE?

A f

 

 

 

 

+0

 

 

 

 

 

 

 

MAKE YOUR GRAPH ON THE GRAPH PAPER INCLUDED IN THE BOX.

 

Tower Puzzle instruction card no. 5

 

 

IF YOU STARTED WITH lOO DISCS AND IT TOOK ONE SECOND FOR
EACH MOVE, HOW LONG WOULD IT TAKE TO TRANSFER THE PILE?

 

 

 

Tower Puzzle instruction card no. 6

128

 

 

Brian and David were examining the contents of this box one
day. Brian said, "I've seen this game before. The object
of the game is to interchange the red and white pegs.

You must move the pegs according to the following rules:

 

white + + red

i) The white pegs (tees) must move only to the right:
the red pegs (tees) must move only to the left.
ii) You can only move one peg at a time.
iii) You can move a peg into an adjacent hole.
iv) You can jump, but only a single peg of the opposite
color (you can't jump two pegs).

See if you can do it.

CAN YOU INTERCHANGE THE FOUR PEGS USING THE RULES ABOVE?

 

Peg Game instruction card no. 1

 

 

David said that if he started with two pegs on each side of
the center hole, he needed 8 moves to interchange them.

DO YOU AGREE?

Brian suggested that a table be made to keep track of the
number of pegs on each Side (the number of pairs of pegs)
and the corresponding number of moves. "Let the number of
pairs of pegs be tne number and let the number of moves to
interchange them be the number," he suggested further.

number of pairs of pegs +(J A + number of moves

 

 

mmwaI-I

CAN YOU FILL OUT THE REST OF THE TABLE (ON THE PAPER INCLUDED
IN THE BOX)?

 

Peg Game instruction card no. 2

 

 

129

 

 

David said, "This is like the game, Guessing Functions; you
give me a number (D) , I use a rule on that number and get a
number out (A)." CAN YOU FIND A RULE FOR THE PAIRS OF
NUMBERS IN THE TABLE?

WRITE YOUR RULE (USINGIJ AND A)' THEN CHECK IT BY TRYING
VARIOUS PAIRS OF NUMBERS FROM THE TABLE.

 

Peg Game instruction card no. 3

 

 

David tried to graph the pairs of numbers from Brian's table.
CAN YOU MAKE A GRAPH OF THE PAIRS OF NUMBERS IN THE TABLE?

MAKE YOUR GRAPH ON THE GRAPH PAPER INCLUDED IN THE BOX.

A

A

 

 

 

 

 

 

+0

 

 

 

 

 

 

 

 

 

 

 

Brian said, "I bet the points lie on a straight line!"
DO YOU AGREE?

 

Peg Game instruction card no. 4

 

 

If you took a white rod and used one side of it like a rubber
stamp, what is the least number of times you could use it and
cover the whole surface of a yellow rod?

Don said he got 17. DO YOU AGREE?
Marilyn said she got 22. WHO IS RIGHT?

 

Centimeter Blocks instruction card no. 1

 

 

 

130

 

 

Marilyn, Don and Jerry made this table with the various rods.
(They called the "number of times stamped" the surface area).

 

Color of rod Surface Area

 

 

White 6
Red

Light Green

Purple

Yellow 22

CAN YOU COMPLETE THIS TABLE?

 

Centimeter Blocks instruction card no. 2

 

 

Martha and Frank made their table a different way. Martha
decided that since it took 5 white rods to make the same
length as a yellow rod, she would call that one "5" and make
her table like this:

Number of
White Rods + D A + Surface Area

 

 

White 1 6
Red 2

Light Green

Purple

Yellow 5 22

CAN YOU FINISH THIS?

 

Centimeter Blocks instruction card no. 3

 

 

Ida Mae said that the table on card 3 reminded her of the
game of "Guessing Functions." The "number of white rods"
would be thetJ number, and the "surface area" would be the
number. Ida Mae said that she could even figure out a
rule which would tell here the "EJnumber" if She knows the
Ainumber.
CAN YOU FIND THE RULE?

 

Centimeter Blocks instruction card no. 4

 

 

 

131

 

I (IAN YOU GRAPH THE PAIRS OF NUMBERS IN THE TABLE ON CARD 3?

 

Centimeter Blocks instruction card no. 5.

 

USING THE GRAPH CAN YOU FIND THE SURFACE AREA FOR THE BLUE ROD?
THE ORANGE ROD?

USING YOUR RULE CAN YOU FIND THE SURFACE AREA FOR THE BLUE ROD?
THE ORANGE ROD?

 

 

 

Centimeter Blocks instruction card no. 6

 

How many different shapes and sizes can you make by stretching
a rubber band around some of the nails on this geoboard?

 

Geoboard instruction card no. 1

 

Let's try something. Stretch a rubber band around four nails,
like this:

 

 

(the black dots are supposed to be the heads of nails)

If this is said to have an area of "one square," can you make
a shape that you think would have an area of "2 squares"?
"3 squares"? "1 1/2 squares"?

 

 

 

Geoboard instruction card no. 2

132

 

 

Don says that this shape has an area of 2 1/2 squares.
DO YOU AGREE?

That this shape has an area of 2 1/2 squares.
IS HE RIGHT?

 

 

Geoboard instruction card no. 3

 

 

Frank says that this shape has an area of 1 1/2 squares.
What do you think?

Frank's friend Martha says Frank is wrong. ‘She says the
area is 2 squares. WHO IS RIGHT?

 

Geoboard instruction card no. 4

 

 

CAN YOU FIGURE OUT THE AREAS WITHIN THESE SHAPES?

 

a)..@.. 13).“... c).....
d) : . . : e) .

 

 

Geixnrd1hmxrmfifioncxud1xx 5

 

 

 

133

 

 

CAN you make up some interesting shapes and find the areas
within them?

Can a friend of yours make up a shape that is too hard for
you to find the area of?

Can you make up one that he can't find the area of?

 

Geoboard inStruction card no. 6

 

 

Have you ever played the game "Guessing Functions?" If you
have you might like to try this. Suppose you were told a
certain number of nails, and you were told that you must
touch all the nails with a rubber band. CAN YOU PREDICT
WHAT THE LARGEST POSSIBLE AREA WITHIN THE FIGURE WILL BE?
(Nails within the figure are illegal).

 

Pat says that if you tell her "3 nails," the area will be
1/2. DO YOU AGREE? .

Bill says that if you tell him "5 nails,“ the area will be
1 1/2. IS HE RIGHT?

Louise says that if you tell her "6 nails," the area will
be 2 1/2. WHAT DO YOU THINK?

 

Geoboard instruction card no. 7

 

 

134

 

 

Gary and Bob made this table for their answers.

 

 

Nails Area
4 1
5 1 1/2 ‘
- 6 2

For example, if you were told "6 nails"

This is This is This is
illegal! legal! legal!

DO YOU THINK THEY ARE RIGHT?

CAN YOU FILL IN ANY MORE PAIRS OF NUMBERS?
CAN YOU GRAPH YOUR PAIRS OF NUMBERS?

CAN YOU GUESS THE RULE?

 

Figure A4.8 Geoboard instruction card no. 8

 

 

WHAT HAPPENS IF YOU NOW PLAY WITH DIFFERENT RULES AND ALLOW
ONE NAIL WITHIN THE FIGURE, NOT TOUCHED BY THE RUBBER BAND?

DOES IT MAKE ANY DIFFERENCE HOW MANY NAILS THERE ARE WITHIN
A FIGURE NOT TOUCHED BY THE RUBBER BAND?

CAN YOU WRITE A NEW RULE TO FIT THIS IDEA?

 

Figure A4.9 Geoboard instruction card no. 9

 

 

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