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A COMPARATIVE STUDY OF THE TRADITIONAL CALCULUSCOURSE VS. THE CALCULUS & MATHEMATICA COURSE

BY

KYUNGMEE PARK

B.S., Seoul National University, 1987

M.S., University of Illinois, 1990

THESIS

Submitted in partial fulfillment of the requirements

for the degree of Doctor of Philosophy in Education

in the Graduate College of the

University of Illinois at Urbana-Champaign, 1993

Urbana, Illinois

Q^310TcL^3p

UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

THE GRADUATE COLLEGE

SEPTEMBER 1992

WE HEREBY RECOMMEND THAT THE THESIS BY

KYUNGMEE PARK

FYTTTT.FD ^ COMPARATIVE STUDY OF THE TRADITIONAL CALCULUS

COURSE VS. THE CALCULUS & MATHEMATICA COURSE

BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

t Required for doctor’s degree but not for master’s.

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SUBMITTED BY KYUNGMEE PARK AS ONE OF THE

REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

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A COMPARATIVE STUDY OF THE TRADITIONAL CALCULUSCOURSE VS. THE CALCULUS & MATHEMATICA COURSE

Kyungmee Park, Ph.D

College of Education

University of Illinois at Urbana-Champaign, 1993

K. J. Travers, Advisor

The C&M (Calculus & Mathematica) course is the computer

laboratory calculus course at the University of Illinois. The purpose

of this study was to evaluate the C&M course, and to compare the

mathematics achievements and attitudes of the C&M group with

those of the traditional group. As a methodology, the investigator

combined quantitative and qualitative methods. The instruments of

the quantitative research were the achievement test and the attitude

survey. The statistical analysis method of the quantitative data was

ANCOVA (analysis of covariance). For the qualitative research, the

investigator observed the computer laboratory and the classroom,

interviewed the students, asked them to complete the questionnaire,

and analyzed their solution files, concept maps, and the class

materials.

The result of the achievement test was that the C&M group,

without seriously losing computational proficiency, was much better

at conceptual understanding than was the traditional group. To focus

more on students' conceptual understanding, a new instrument, the

concept map, was used. The investigator designed two analysis

methods to compare the students’ concept maps, and the results from

both methods were favorable to the C&M group. The attitude survey

results indicated that the C&M group's disposition toward

mathematics and computer was far more positive than that of the

traditional group

The advantages of the C&M course were the students'

exploration through calculations and plottings, visualization of ideas

by Mathematica graphics, and the students’ collaborative activities in

the laboratory. On the other hand, not all of the results were positive.

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The students' high dependency on Mathematica, the black-box

syndrome (students' blind execution of commands without

understanding underlying concepts and procedures), and the time-

consuming quality of the course were all possible drawbacks.

However, the benefits of the C&M course were enough to offset those

negatives. For this reason, the investigator is optimistic about the

potential of the C&M course as an alternative approach to current

calculus courses.

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To my family

ACKNOWLEDGEMENTS

I would like to thank my family for the love and support

during my graduate career. I am also grateful to my academic

advisor. Dr. Travers, for his generous guidance and everlasting

encouragement. Finally, I would like to express my deepest

appreciation to Dr. Glidden, Dr. Hamisch, and Dr. Porta for their

willingness to discuss my work in detail and the numerous

suggestions.

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TABLE OF CONTENTS

CHAPTER PAGE

I INTRODUCTION 1

Statement of the Problem 2

Purpose of the Study 3

Background of Significance of Study 3

Quantitative Research Hypotheses 4

Qualitative Research Questions 5

Limitations and Assumptions 5

n LITERATURE REVIEW 7

A Rationale for Change 7

Evolution in the Teaching Calculus 9

Computers in Calculus 10

Use of Programming Languages and Software

Packages 10

Use of Computer Algebra Systems 13

On-going Calculus Reform Projects 16

Use of Programming Languages and Software

Packages 16

Use of Computer Algebra Systems 17

Mathematica 19

Concept Maps 20

Novak's Concept Map 21

Theoretical Background 22

Educational Applications 23

m EXPERIMENTAL DESIGN AND PROCEDURES 25

Sample 25

Course Methodology 25

Environment 25

Quantitative Research 26

Experimental Variables 26

Instruments 27

Statistical Analysis 31

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TABLE OF CONTENTS (CONTINUED)

CHAPTER PAGE

Qualitative Research 32

Instruments 33

Data Analysis Procedure 38

Pilot Study 40

IV RESULTS IN THE CONTEXT OF MATHEMATICS LEARNINGCharacteristics of the Samples 43

Achievement Test 43

Results of Testing the Hypotheses 44

Further Analysis 46

Conclusions on the Achievement Test 52

Concept Maps 52

Analysis Method 52

Examples 54

Concept Map Data 60

Statistical Findings 61

Non-Statistical Findings 64

Misconceptions 65

Students' Evaluation 65

Interview 66

Conclusions on the Interview 72

Observation 73

Details of the Observation 73

Suggestions from the Observation 79

Solution Files 79

Findings from the Solusion Files 79

Exams 84

Exam 1 84

Exam 2 85

Exam 3 86

Final Exam 87

Courseware 89

Description 89

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Review of the Lessons 89

Students' Difficulties in the Lessons 93

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Attitude Survey 99

Results of Testing the Hypotheses 100

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Questionnaire Ill

Observation 119

Details of the Observation 119

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Observation in the Classroom 131

Conclusions on the Observation 133

Solution Files 136

Findings From the Solution Files 136

Courseware 140

Characteristics 140

Evaluation 143

Drop-out Pattern 145

VI SUMMARY OF THE RESULTS 147

Question 1 147

Question 2 151

Question 3 153

Question 4 155

Question 5 158

Vn CONCLUSIONS AND RECOMMENDATIONS 161

Conclusions 161

Suggestions 165

Limitations 165

Recommendations 166

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REFERENCES 168

APPENDDC

A Pre-Achievement Test 174

B Post-Achievement Test 178

C Attitude Survey 183

D Concept Map Sheet 185

E Item-total Statistics 189

F Give it a Try Problems 191

G Exams 200

H Observation Note 206

VITA 207

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CHAPTER I

INTRODUCTION

The quality of calculus instruction is a barometer of reform in

mathematics education. This is the case because preparation for

calculus has been the focus for the organization of secondary school

mathematics. Although many other mathematics courses are as

important and practical, calculus currently is in the unique position

of being the gateway from secondary school to college mathematics.

Furthermore, since the language of calculus has spread to all

scientific fields, successful mastery of calculus is essential to

satisfactory performance in science. Unfortunately, there are huge

gaps between calculus as presently taught in schools and colleges and

the way calculus is used in everyday application. As the National

Research Council (1989) reported, "many of those who do finish

[calculus] learn little beyond a series of memorized techniques now

more commonly performed by computers" (p. 52).

Calculus may be regarded as nothing more or less than a course

in how to use the tools of differentiation, integration, and

approximation to make precise measurements (Brown, Porta, & Uhl,

1991). However, most calculus courses available today have ignored

this basic fact, focused on laborious paper-and-pencil calculations on

tricky problems, and as a result have little purpose other than to

train students to pass tests.

The teaching of calculus is the natural vehicle for introducing

applications of science and engineering, and those applications give

the proper shape to calculus. Without applications, a calculus course

is in danger of resembling

a guided tour through a carpentry shop, with instruction on how to use

each tool, but giving no sense of how to use them to build a thing of

beauty and utility, . . . , or a language class where grammar and syntax

are taught systematically, but where there is little conversation,

composition or reading of literature (Lax, 1986, p. 69).

There has been widespread agreement that there should be a

revolutionary improvement of calculus instruction. In 1986, the

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calculus reform conference at Tulane University agreed that "the

syllabus should be leaner, contain fewer topics, and that it should

have more conceptual depth, numerically and geometrically"

(Douglas, 1986, p. v).

Those who saw the need for considerable revision in calculus

instruction often cited the need to incorporate new technology as a

motivating force. There is a sense of agreement emerging that

students can benefit from the introduction of technological tools into

the curriculum. The availability of technology with both numerical

and symbolic capabilities removes the necessity of covering many of

the techniques and drills that now form a large part of calculus.

Reducing the mechanics of hand calculation to technology makes

calculus instruction more applicable to real-world problems and

focuses more on fundamental ideas. Furthermore, students' interest

in calculus can be stimulated, since access to technology enables

them to explore a variety of examples and to solve realistic applied

problems of considerable complexity. In short, technology has the

potential to be used to heighten the understanding of, and insight

into, the concepts of calculus, to improve the pedagogy of the calculus

course, and to influence the choice of topics to be taught.

Statement of the Problem

The department of mathematics at the University of Illinois has

offered two different freshman calculus courses entitled Calculus and

Analytic Geometry II: the C&M (Calculus and Mathematica) course

and the traditional course. The C&M course was designed by two

mathematics professors, Horacio Porta and Jerry Uhl. They chose

Mathematica, one of the most powerful and easy-to-use computer

algebra systems, for writing the calculus courseware. Their goal is to

change the delivery of calculus from lectures and printed texts to a.

laboratory course through an electronic text . Significantly, there is no

textbook for this course; instead, a sequence of electronic notebooks

is used.

Each notebook opens with "basics" problems introducing many

of the new ideas, followed by "tutorial" problems in techniques and

application. Both problem sets provide full solution to support

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students' learning. The notebook closes with a section called "give it a

try", which contains problems for the students to solve. Students can

make use of the standard word processor and calculating software

with graphic capabilities to build their own notebooks to solve these

problems, which are electronically submitted for comments and

grading.

The instructional focus is on the computer laboratory and the

electronic notebook, with half an hour a week for lecture and another

half hour for organized discussion, exams, and so forth. Basically, the

C&M course is a laboratory course with minimum lectures, while the

traditional course teaches calculus in a traditional setting with no

computer.

The C&M course was first opened in the spring semester of

1989, and now is in its fourth year. Even though several informal

evaluations for the C&M course as well as comparisons with the

traditional course have been made, none of these has been in depth.

Purpose of the Study

The purpose of this study is to compare the mathematics

achievements and attitudes of the students in the C&M course with

those of the students in the traditional course, and to evaluate the

C&M course from an educational perspective. The investigator used

qualitative and the quantitative research methods to understand

better the outcomes of the C&M course. The quantitative research

focused on the comparison of the outcomes of the two groups, while

the qualitative research emphasized the evaluation of the C&Mcourse. The quantitative research used the result of an experiment

which involves the testing of well-defined hypotheses, whereas the

qualitative research was based on the data collected in an open-

ended way which allowed the investigator to concentrate on the

contextual aspect of the C&M course.

Background and Significance of Study

There is widespread consensus that calculus courses at present

tend to be excessively technique- and skill-oriented, with correspon-

ding agreement that calculus should be a more concept- and

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application-oriented subject. Much of the current reform of calculus

instruction is centered around the use of computers. As computers

have become cheaper, smaller, user-friendly, and also more efficient,

many people have started to explore their possible applications for

the calculus reform.

"Laboratory calculus course" is not a new term; doing calculus

in laboratories has become one of the major trends in calculus

reform. This term has been heard much more frequently since the

advent of the personal computer and the ready availability of

numerical, graphics, and symbolic software suitable for use in

calculus classes. Nonetheless, many professors and instructors still

are reluctant to adopt the laboratory calculus approach. One reason is

that computer lab environments are not available, but the more

important reason is that there have not been many evaluations of

laboratory courses that can be used as criteria.

In order to document the perceived validity of laboratory

calculus courses, this study has attempted to provide objective,

legitimate, and authentic evaluation of the C&M course. This

evaluation may eventually catalyze the current calculus reform.

Quantitative Research Hypotheses

This study examined the effect of exposure to Mathematica

environment on the scores of achievement test and attitude survey.

A comparison between two groups was examined, using the following

null hypotheses:

HI - There is no statistically significant difference in the scores

on the achievement test between the C&M group and the traditional

group.

HI (A) - There is no statistically significant difference in the

conceptual understanding scores between the C&M group and the

traditional group.

H1(B) - There is no statistically significant difference in the

computational proficiency scores between the C&M group and the

traditional group.

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H2 - There is no statistically significant difference in the scores

on the attitude survey between the C&M group and the traditional

group.

H2(A) - There is no statistically significant difference in

attitude toward mathematics scores between the C&M group and the

traditional group.

H2(B) - There is no statistically significant difference in

attitude toward computers scores between the C&M group and the

traditional group.

Qualitative Research Questions

The qualitative research addressed the following five

questions:

1. What is the role of Mathematica in the development of

concepts?

2. How can Mathematica be used in the development of

calculational skills?

3. In what specific way does Mathematica facilitate the

students' exploration and discovery^ learning?

4. In the C&M course, what different cognitive procedures

occur in the learning process?

5. How does the C&M course provide a cooperative learning

environment?

Analytic reflection on observations, questionnaires, interviews,

concept maps, solution files, and class materials provided important

insight for answering those questions. Data of each type were

categorized, interesting patterns were described, and unusual

student understandings were noted.

Limitations and Assumptions

1. This study was limited to the students enrolled in MATH 132

at the University of Illinois during the spring semester, 1992.

2. The experimental groups were probably representative of a

typical freshman calculus class at a large-size state university.

^ learning in which the principal content of what is to be learned is not

presented but should be discovered by the learner.

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3. The instruments utilized in the quantitative research were

reliable and valid. Support for reliability and validity of the achieve-

ment test and the attitude survey are presented in chapter III.

4, The students involved in this study honestly responded to

the attitude survey and the questionnaire.

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CHAPTER II

LITERATURE REVIEW

A Rationale for Change

Calculus has held a special place in the college mathematics

curriculum. It has been an introductory mathematics course and it

has served a variety of audiences. Calculus courses have served as an

introduction to "what mathematics is all about" for liberal arts

students, as an introduction to the "language of science" for science

and engineering students, and as an introduction to "fundamental

mathematical notions" for those who would go on to be mathematics

majors. In short, calculus has been the foundation for college

mathematics.

Now, however, there are large cracks in that foundation. There

appears to be general dissatisfaction with calculus teaching, both

among students and college professors. This dissatisfaction does not

arise simply from the availability of alternative first year courses

such as discrete mathematics; rather it comes from the perception

that calculus courses, as currently taught, do not meet the needs they

should satisfy.

What follows are frequently voiced complaints about the

teaching of calculus, and eventually the reasons to change the

methods of calculus instruction:

1. Calculus texts that were widely used one or two decades ago

did not have as many pages as modern texts which commonly run

over 1000 pages. More and more fields require their majors to

complete calculus, and each wants to include some applications and0

some special emphases related to their special needs. New contents

have been continuously added, but rarely has anything been

removed.

2. The current practice in calculus teaching and texts is

perceived to drive students away from scientific and mathematical

careers; it tends to be a barrier to students. Many of them lose

interest, fail, and drop out. A recent study showed that among

300,000 students who begin a mainstream calculus course annually,

only 140,000 finish the course with a grade of D or higher (Roberts,

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1991). What we should do is to find ways to encourage and not

discourage students, to keep them in the "pipeline" (White, 1987).

3. As presently taught, calculus, is essentially irrelevant for the

nearly half of the college students who do not go on to use

mathematical tools in their careers; these students who were

introduced to mathematical thinking via calculus are "ill-served" by

current versions of the course.

4. The current calculus courses are superficial and permit

"mimicry" (Davis, 1986). Typical calculus courses do not develop in-

depth understanding. They fail to prepare science students for

applications of mathematics to their fields, and fail to convey to

mathematics students a sense of mathematics and mathematical

thinking.

5. Too much of the time in current calculus courses is spent

carrying out routine algorithmic manipulations, which students will

not long remember. This is done at the expense of both conceptual

understanding of calculus and an appreciation of mathematical

processes. In other words, today, the major emphasis in calculus

instruction is placed on imparting specific mathematical facts and

algorithms, rather than on understanding and developing an

inquisitive attitude, analytic abilities, and problem solving skills.

6. Students have a tendency to view calculus as the

memorization of formula and believe that "to do mathematics" is "to

compute" because most of their calculus class is devoted to

algorithmic computations.

7. The applied problems are artificial and degenerate in one or

two ways. They are often slight variations of an example worked out

in the text; thus students need only to substitute new numbers into

the prescribed recipe.

8. The current hour-long examinations evaluate only a subset

of students' skills. The tests should be changed to open-ended exams,

take-home exams, or oral exams with essay questions or standard

questions in non-standard format.

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The availability of computers with both numerical and

symbolic capabilities removes the necessity for covering many of the

techniques and for many of the drills that now form a large part of

the calculus courses. The use of computers can free both instructor

and students from computational tedium, thus allowing them to focus

on conceptual rather than computational matters. Moreover, many

topics of calculus can be dramatically illustrated with the help of

computers. In conclusion, calculus instruction should make use of

computers, which are increasingly available and have the potential

for significantly improving the teaching of calculus.

Evolution in the Teaching of Calculus

The fact that computers are tools to change the way all

mathematics, and especially calculus, is being taught, has been

pointed out over the years by various mathematicians and educators.

It might be helpful to recall briefly a few publications and meetings

that have taken place since 1980 that have contributed to the

incorporation of mathematics and computers (Hodgson, 1987).

1980--Mindstorms: Children. Computers and Powerful Ideas.

by Seymour Papert. In this book, the computer was presented as an

"object-to-think-with."

1984—

Several sessions of ICME (International Congress on

Mathematical Education)-5 were devoted to the teaching of calculus

and to the effects of symbolic manipulation systems on the

mathematics curriculum.

NCTM 1984 Yearbook, Computers and Education .

1985—

A symposium in Strasbourg on the topic: The Influence

of Computers and Informatics on Mathematics and its Teaching

(Proceedings published by Cambridge University Press, 1986).

1986—

-The Tulane conference: To Develop Alternative Curricula

and Teaching Methods for Calculus at the College Level (Proceedings,

Toward a Lean and Lively Calculus, published by MAA, 1986).

1987—

The Washington colloquium: Calculus for a New Century

(Proceedings, Calculus for a New Century: A Pump. Not a Filter.

Published by MAA).

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1988—The programs of both the AMSXMAA annual meeting

and the ICME-6 (Budapest) contain many activities related to

computers and the teaching of calculus.1990—

The MAA report Priming the Calculus Pump:

Innovations and Resources. The purpose of this report was to

disseminate and promote a change in calculus instruction.1991—

The MAA report The Laboratory Approach to Teaching

Calculus. In this report, several calculus reform projects were

explained and evaluated.

Moving Beyond Mvths: Revitalizing Undergraduate Mathematics, by

National Research Council.

Computers in Calculus

During the past two decades, there has been considerable

research that investigates the advantages and disadvantages of using

computers in teaching introductory calculus courses. Held (1984)

stated that "an ideal first testing ground for the notion of using the

computer to replace by-hand manipulation as a tool in concept

development is a course in introductory calculus" (p. 3).

Various approaches have been suggested and tried. Although

no very clear preference has been indicated among them, two

approaches do appear to be in favor. One of the most popular

approaches has been to devise a computer calculus course in which

the basics of programming are taught and the students learn to write

programs involving topics from calculus, or to utilize calculus

software packages. The other approach has been to design the

calculus course using computer algebra systems (CAS) such as

Macsyma, muMath, Maple, Derive and Mathematica, which can

handle almost any calculation and operation without programming.

Most of the main tools of the early studies were student

programming and software packages while those of the recent

studies have been CAS.

Use of Programming Languages and Software Packages

Most of the main tools of the studies conducted before 1980

were programming languages and software packages.

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Achievement and Attitude

One of the first formal projects designed to integrate computers

into college calculus courses came from the Center for Research in

College Instruction of Science and Mathematics (CRICISAM) in 1969.

The CRICISAM developed an experimental textbook - The CRICISAM

Computer-Oriented Calculus, which was tested in 50 institutions.

According to the overall survey, 40% of the instructors found that the

CRICISAM approach was more effective than the traditional

approach; 54% reported that they couldn't find any difference

between the two approaches, and the remaining 6% insisted that

they had better success with the traditional approach (The Center for

Research in College Instruction of Science and Mathematics, 1971).

Rice (1973) at Georgia State University conducted an

experiment that compared three different methods of teaching in

freshman calculus. The three methods, traditional instruction,

computer-assisted instruction (CAI), and programmed packet

(written material paralleling the computer programs), were used in

the teaching of limits, derivatives, and integrals of functions. The

result showed no signifrcant difference among the three teaching

methods for any of the concepts. But it suggested that CAI produced

slightly higher scores and the students' positive attitudes toward the

computer were enhanced.

Strawn (1974) investigated the effects of computer-assisted

instruction on learning six topics in calculus: geometric properties of

function, max-min problems, related rate problems, integration by

parts, integration of rational functions, and evaluation of improper

integrals. He found that the students in the computer group were

superior to the traditionally taught group only in their performance

of related rate problems. The students' performance did not differ

significantly in the other five topics. Similar results were reported by

Basil (1974) in his investigation of differences between students who

used BASIC programming language and those who used a calculator.

He concluded that writing computer programs does not have overall

advantages in the learning of elementary calculus. However, in these

two studies, favorable attitudes toward calculus and computers were

found.

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In 1989, Hamm (1989) conducted an experiment to explore the

association between a computer-oriented instruction and a non-

computer-oriented instruction at Brookhaven College in Texas. The

students in the computer-oriented group were taught with the use of

microcomputer calculus software for in-class presentation and

homework assignment while those in the other group learned in a

traditional setting with no microcomputer intervention. He

administered an MAA calculus readiness test in a pre-test and an

experimenter-developed achievement examination in a post-test for

measuring the students' achievement in three topics: functions and

limits, differentiation, and antidifferentiation. He concluded that the

microcomputer in introductory calculus instruction does not

significantly affect either student achievement or student attitude

toward mathematics.

Conceptual Understanding and Computational Proficiency

Bell (1970) at Cornell University, conducted a study

attempting to determine the effectiveness of a computer-oriented

calculus course. The control group studied with a calculus manual

written by Bell, while the experimental group used a similar manual

that included six computer-oriented problem sets. He concluded that

a computer-oriented calculus approach was an effective method of

promoting conceptual understanding and increasing students'

interest without weakening the ability to apply the techniques of

calculus.

The work of LeCuyer (1977) and Daughdrill (1978) focused on

the aspects of computational achievement. LeCuyer performed an

experiment comparing sections of a college mathematics survey

course in which the students in one section learned programming

language (APL) within the context of the course, and in the other

section the students learned the same topics without using the

computer. Daughdrill conducted similar research with the BASIC

language. Both of them found no significant differences in

computational skills between classes taught with computer

programming and classes in the traditional manner. These results

diminished the apprehension that the use of computers to perform

computation might lower students' computational achievement.

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Lesh (1987) conducted an experiment with two groups: a

utilities group and a control group. The students in the utilities group

used to a computer software package that computed what the

student requested and plotted the corresponding lines for each step.

The control group was given a computer package that told the

students which computations to perform, and which also graphed the

correct lines for the final solution. The result was that the utilities

group showed higher score on the computation part of the final

examination than did the computation group. This suggests that

students do not have to perform computation repeatedly to maintain

computational proficiency, and additional learning tools (computational

and graphic facilities) lead to a significant difference in conceptual

performance.

Limitations

The above studies indicate that the incorporation of computer

programming language and software package into calculus

instruction did not always improve students' achievement and

attitude. One of the reasons may be that the calculus curriculum did

not change; in other words, it was difficult to fit computer

programming and software packages into the traditional calculus

curriculum. Another probable reason was that students had major

difficulties in computer programming and they had to wait a long

time to get numerical and graphic results. These unsuccessful

experiences seemed to require a different method and to expedite

the use of computer algebra systems in calculus instruction.

Use of Computer Algebra Systems

Most of the recent studies conducted after 1980 have used

computer algebra systems as a tool.

Achievement and Attitude

Heid (1984) at the University of Maryland explored the

potential of computer-based concept-oriented calculus course with

muMath as a replacement for the conventional calculus course. Data

were collected on experimental groups, who used the computer as

the primary executors of basic skills, as well as on a traditionally

taught comparison group. Heid collected class transcripts, interview

results, photocopies of student assignments, quizzes, exams, and field

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notes for open-ended analysis. She found that the experimental class

students had a remarkably deep and broad understanding of calculus

concepts and performed almost as well on routine skills. She

concluded that the computer as a tool appeared to play an important

role in concept development and in encouraging a variety of

mathematical explorations beyond those of a traditional curriculum.

In a non-experimental study at Colby College, Hosack and Lane

(1985) used Macsyma in an introductory calculus course and

investigated its potential for improving calculus instruction.

Macsyma is an interactive computer algebra system that allows the

user to define an expression and apply and manipulate an operation

without programming. Macsyma was used for solving homework

problems, serving as an "answer book" for problems that students

themselves generated, exploring concepts, and solving examination

questions. Students felt comfortable with the user-friendly

characteristics of Macsyma and were excited by its various

capabilities. Hosack and Lane recommended the use of CAS, which

has the potential for significantly improving calculus instruction.

Another exploratory study was undertaken by Hawker (1986)

to determine the association among achievement, attitude, and drop-

out rate when skills used in a business calculus course were replaced

by one of CAS, muMath. Although no significant differences were

observed between the treatment and control group in achievement

and attitude, students in the treatment group did as well as, or

slightly better than, control group students on conceptual problems

without decline on mechanical problems.

Freese, Lounesto, and Stegenga (1986) experimented with

introducing computer symbolic mathematics into the first year

calculus classes at the University of Hawaii. In the experiment, a

modiHcation of the muMath software package, that included a

powerful integration package and a facility to graph functions, was

used. The students' response to that course measured by a written

questionnaire was favorable. They liked the ability to mix symbolic

mathematics with graphing. Particularly, the students commented

that they would like to have the program illustrate how it does each

procedure.

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Conceptual Understanding and Computational Proficiency

Computers can perform computational procedures more easily

and more quickly than can students who have spent several years

practicing these procedures by hand. Accordingly, the use of

computers allows the student to spend less time on developing

computational skills, as computational skills for their own sake are

no longer a desired goal. Then, what is the impact of de-emphasis on

computational skills on students' understanding of other

mathematical content? Attempts have been made to examine the

effects of using computers on computational proficiency and

conceptual achievement.

In her landmark study. Held (1984) suggested that by the use

of graphic and symbolic manipulation computer programs (muMath),

it might be possible to teach concepts before skills, without loss of

understanding of content in calculus. She found that the students

with computers showed better understanding of concepts and

performed as well on routine skills as did students who had

practiced the skills with paper-and-pencil. She asserted that the void

created by a de-emphasis on skills can be filled by an increased

emphasis on the development of mathematical concepts. She

concluded that the computer was a useful tool in a concept-oriented

introductory calculus course and stated:

Hand calculation is time consuming, and at its end students have often lost

sight of their initial goals as well as the interrelationships among the

mathematical concepts involved. The computer can be used to provide the

results of algorithm executions, not only saving time usually spent on

hand execution of these procedures but also giving students quicker and

easier access to exemplars of a concept. A wider range of exemplars can

be used in instruction, and students might be less distracted than when

they must produce the exemplars by hand (Held, 1988, p. 3).

Limitations

Although those innovations seem to provide the driving force

for the incorporation of computers in calculus courses, not very much

progress has been made toward exploring the potential pedagogical

application of computer algebra systems to the fullest extent.

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1 6

On-going Calculus Reform Projects

After the Tulane conference, "Toward a Lean and Lively

Calculus", in 1986 and the Washington symposium, "Calculus for a

New Century", in 1987, many projects started to get underway to

reshape the teaching of calculus, using powerful symbolic

computation software and the display of graphics. Among them, six

promising projects were selected. A theme running through the

following projects is the power of the computer to improve the

"versatility" of students' thought processes in calculus.

Use of Programming Languages and Software Packages

The calculus reform at Dartmouth College spans four courses;

two single variable calculus courses, one multivariable calculus

course, and one differential equations course. The focus of all four of

the calculus courses has been on the underlying geometry of the

subject, and the development of the students' ability to interpret the

analytic information geometrically. The format of the course consists

of a lecture supplemented by 5-10 minutes of computer demonstration.

The computer is routinely used in the classroom, and students are

expected to use the computer for doing their homework. The

predominant tool is the language True BASIC, supplemented by

occasional use of commercial software. Because Dartmouth College is

running an experimental section in conjunction with a traditional

offering of each course, an assessment of the benetits was obtained.

The students who had taken the new calculus course performed as

well as did the students in the traditional section on a conventional

test of skills (Baumgartner & Shermanske, 1990).

The calculus project at the University of Michigan (Dearborn)

schedules an 80-minute, once-a-week computer laboratory with IBMPC in addition to three fifty-minute lectures per week. In the

computer sessions, the instructor introduces on the blackboard the

topic of the day at the beginning of the class. Then the students are

given a handout with computer experiments and exercises designed

to explore the topic in detail. The main softwares used are MicroCalc

(by H. Flanders) and Exploring Calculus (by J. B. Fraleigh and L. I.

Pakula). The goal of the project is to make a calculus course suitable

for department-wide adoption, working within the standard syllabus.

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involving students more actively in the learning process, encouraging

more interaction among students, engaging students in large

problems, and providing opportunities for students to use mathematics

coherently. Even though no comparison of test scores is available, the

responses to the questionnaire showed that many of the students

considered the computer sections to be beneficial. (Hoft & James,

1990).

The calculus reform at Purdue University is more a reform of

pedagogy than of content. The centerpiece of the project philosophy

is that students should construct their own understanding of each

mathematical concept. Hence, the primary role of teaching is not to

lecture, explain, or attempt to transfer mathematical knowledge, but

to create situations for students that will foster their mental

development. Under this belief, new curricula were developed, which

mixed a theoretical development with concrete applications and

which emphasized ideas as opposed to techniques of calculation. The

mathematical programming language, ISETL (instructional set

language), was adopted as the main tool because it involves students

in a level of programming that is essential to the theory of learning.

The interfaces have been created that make it possible for students

to utilize Maple, in conjunction with ISETL. The students in the

computer course took departmental common Hnal exams during the

first two years of the project. Although the students in the computer

courses spent less class time on mechanics than the students in the

regular courses did, the former averaged 2% to 6% higher than the

latter on the final exams. (Schwingendorf & Dubinsky, 1990).

Use of Computer Algebra Systems

Brock University in Canada introduced the use of Maple into a

calculus course. The pedagogical goals of the course are: increasing

the students' confidence and enjoyment when doing mathematics;

providing more time in lectures for the explanation of calculus

concepts and mathematical language by relegating to Maple numeric,

algebraic, and graphic manipulations; and presenting more

exploratory situations, "what if" situations, thereby rekindling the

spirit of discovery (Muller, 1991). The comparison with the

traditional course was done using three criteria; failure rates.

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withdrawal rates, and average grade. The results based on these

criteria were favorable to the course with a Maple lab. In 1988, the

average grade for all students who volunteered for laboratories was

eight percent higher than the corresponding average for those who

did not have the opportunity to take laboratories.

Project CALC (Calculus As a Laboratory Course) at Duke

University developed a three-semester calculus program based on a

laboratory model. In CALC, lecturing is limited to brief introductions

to new topics and responses to requests for more information. Each

class splits into two lab groups; each group has a scheduled two-hour

lab every week. Each lab team, which consists of two students,

submits a written report almost every week. After receiving

comments from the instructor, the team revises and resubmits the

report. The key features of the CALC are: real-world problems,

hands-on activities, discovery learning, the writing and revision of

reports, group work, and the effective use of available tools. The

principle software packages used in the labs are Derive (for symbolic

and graphic computation), MathCAD student edition (for numerical

and graphical computation and for discovery experiments), and EXP(for technical word processing). Comparisons of the CALC with the

conventional courses in the areas of basic skills, concepts, non-

routine problems solving, attitudes, and writing are in process.

According to the students evaluation of the course at the end of the

semester, the CALC received high numerical ratings. The consensus of

the students was that they had worked hard and that it was a good

course (Smith & Moore, 1990).

Since the early 1970's the University of Iowa has offered

optimal computing laboratories (BASIC calculus labs) that run

concurrently with calculus I and II and linear algebra. These labs are

based on simple programs (locally written) in BASIC, which are

largely numerical rather than symbolic or graphical. In 1990, a

network of NeXT computers was installed for an "accelerated

calculus" course. The NeXT lab is dedicated to accelerated calculus,

whereas the BASIC labs share public computing clusters with all

students. The main tool of the NeXT lab is Mathematica Software

with a combination of prepared "Notebooks" and student

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programming. The laboratory is used in two directions; weekly

"electronic homework", and "team project" regarding scientific

application in depth. The accelerated calculus course develops

materials that present calculus as the language of science. This course

is the result of "calculus reform" which includes up-to-date scientific

computing as part of the revision. (Stroyan, 1991).

Mathematica

Mathematica is a computer algebra system which was

developed in 1988 by Stephen Wolfram, a professor at the

University of Illinois. Mathematica is readily available to novice

computer students with a few minutes introduction, starting with

how to turn on the computer. Thus, almost no class time needs to be

spent on the mechanics of using the computer system. Mathematica

can be used in four different ways. (Wolfram, 1988).

First, Mathematica can be used as a calculator . However,

Mathematica can do far more than a traditional calculator can; it can

handle symbolic and algebraic operations, and graphic represen-

tations as well as numerical operations. Users can make their own

definitions and write programs in Mathematica not only with

numerics but also with symbolic and graphic expressions.

Second, Mathematica is a programming language . We can write

programs in Mathematical much as we do in a language like C. Also

Mathematica is an interpreter, since we can run programs as soon as

we have typed them in.

Third, Mathematica is a language for representing mathemati-

cal knowledge . We can take any information from textbooks and

enter it directly into Mathematica. Fundamental to much of

Mathematica is the notion of "transformation rules," which specify

how Mathematica should treat expressions with different forms.

Fourth, as well as being a calculator and a language,

Mathematica can serve as an environment for computing .

Mathematica provides an environment in which to set up, run and

document calculations and programs. For example, one can create

"notebooks," which consist of ordinary text, mixed with graphics and

live Mathematica input.

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Brown, Porta, and Uhl (1990 p. 43) noted that "In two years,

Mathematica has already revolutionized the desk top of mathemati-

cians and scientists all over the world. Pencils and writing paper

have been assigned a new role as more and more scientists begin to

rely on Mathematica as their calculating companion of choice.”

However, the greatest impact of Mathematica has been in the

classroom. Hassen (1989) stated:

We are convinced that Mathematica will have an overwhelming impact

on doing, learning, and teaching mathematics. It is our belief that

Mathematica will pave the way for revolutionizing the teaching of

mathematics; we are already witnessing this. Several experimental

Afar/temarie a -based courses in calculus and precalculus are already

offered in some universities in the U.S. (p. 20).

Mathematica can also be applied and used not only in

universities, but also in secondary schools. Mathematica can be an

invaluable aid to visualization and exploration of mathematical

knowledge. The use of Mathematica forces students to think about

the problem and its solution on a completely different level, which

can lead to a better understanding. Furthermore, the flexibility and

functionality of Mathematica enable the students to enhance their

mathematical thinking, thus promoting guessing, Hnding appropriate

procedures and exact solutions, and verifying those solutions.

Concept Maps

As we have seen, over the past twenty years computer

hardware and software have developed to the point where they can

perform nearly all of the symbolic manipulation that once formed

the core of traditional calculus courses. As a consequence, calculus

instruction now can focus more on the concepts of calculus than on

its computations. If calculus courses adopt this focus, how can we

evaluate student mastery? We can not use the instruments that we

used for traditional courses because they focus on computations. One

promising technique to assess student conceptual understanding is

concept maps.

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Novak’s Concept Map

Concepts and theii relationships shape the students' knowledge

of mathematics. One of the goals in mathematics education is to build

powerful conceptual structures, but conceptual development is very

difficult to assess and measure.

There have been several representation forms for knowledge

structure and learning tasks: cycle diagrams, flow charts, organization

charts, predictability trees, and semantic networks (Johnson,

Pittelman, & Heimlich, 1988). However, none of these forms except

semantic networks, thoroughly reflects the theory of learning and

theory of knowledge. Concept mapping developed by Novak is

another representation form, and it has more promise than other

representation schemes for both education and research.

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Figure 1. Simplitied Model for Concept Maps

Concept maps are two-dimensional graphic representations of

concepts, propositions, and their relationships. They are graphic

organizers which represent content diversity, superordinate-

subordinate relationships, and interrelationships among subordinate

concepts. Concept mapping is a process by which students explain

their understandings of a content area by hierarchically organizing

their ideas, making associations among the concepts, and indicating

the interrelationships (Novak, 1977).

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The hierarchical relationships of the concepts and propositions

represent the degree of specificity. The higher on the hierarchical

model, the more general the concepts. Moving down on the model

leads to more specific and detailed concepts. The relative positions of

concepts and propositions in the map determine the superordinate

and subordinate relationships among them.

Researchers have been troubled by the fact that any variety of

paper-and-pencil test can not validly measure students' knowledge,

especially their conceptual understanding. It is reasonable to

hypothesize that a concept mapping procedure is potentially a better

assessment of students' knowledge structure and the extent of their

conceptual understanding resulting from learning activities.

Theoretical Background

The idea behind concept mapping is based on Ausubel's

cognitive learning theory. From the perspective of Ausubelian

psychology, concept development involves hierarchical organization,

progressive differentiation among concepts, and integrative

reconciliation of concepts (Novak & Godwin, 1985).

One of the fundamental principles of cognitive psychology is

that cognitive structure has a hierarchical organization. Ausubel's

learning theory starts from this hierarchical organization : more

inclusive, more general concepts and propositions superordinate to

less inclusive, more specific concepts and propositions. In order to

construct a hierarchical concept map, students should think what is

the most inclusive, less inclusive, and least inclusive concepts in any

given subject matter. This requires active cognitive thinking.

Concepts in cognitive structure proceed to progressive

differentiation, where greater inclusiveness and greater specificity of

regularities are recognized and more propositional linkages with

other related concepts are discerned. Ausubel's principle of

progressive differentiation supports the idea that meaningful

learning is a continuous process where new concepts acquire greater

meaning as new relationships are constructed. So concepts are never

completely learned but are always being learned, modified, and

made more explicit and more inclusive as they become progressively

differentiated.

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Integrative reconciliation occurs when two or more concepts

are recognized as relatable in a new propositional meaning and/or

when conflicting meaning of concepts are resolved. Meaningful

learning is enhanced when the learner recognizes new relationships

between related sets of concepts or propositions. Concept maps that

show valid cross links between sets of concepts can suggest students'

integrative reconciliation concept. Novel integrative reconciliations

are the major product of creative minds. Therefore, the extent of

cross links constructed by students might represent their creative

minds.

Educational Applications

As educators begin to realize that conceptual understanding is

more important than isolated computational skills, they take more

interest in the concept mapping procedure. Novak and Gowin (1985)

explored various uses of concept mapping; as an evaluation tool, as a

curriculum guide tool, as a research tool, and as an instructional tool.

The concept mapping procedure can be used for an evaluation

tool because it externalizes student's cognitive structure. The student

constructs a map of the relationships between the concepts taught by

the lesson. Then the teacher analyzes the map to determine the

degree of understanding both for individual students and for the

entire class. The weakness of traditional tests is the insensitivity to

the conceptual understanding and the structure of the knowledge. In

this sense, concept mapping as an evaluation tool can be considered

as an alternative to remedy the deficiency of traditional tests.

As a curriculum guide , concept maps, which show relationships

not between places but between ideas, are somewhat analogous to

road maps (Novak, Gowin, & Johansen, 1983). Just as we do with road

maps, we can construct a global concept map showing the major

ideas to be considered in a semester or a year, then move to specific

concept maps showing a three- or four-week segment, and finally

draw a detailed concept map for one or two days of instruction.

These three levels of maps—global, more specific, and detailed—are

helpful for both students and teachers by providing information

about where we are, where we have been, and where we are going.

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Furthermore, concept maps can help the determination of the

sequence of topics and the organization of the content in each topic.

Concept maps are a valuable research tool, for they provide a

method to understand and measure the knowledge structure and

cognitive growth (Hamisch, 1991). They are thought to be a valid

and reliable approach for assessing and predicting learning

achievement. In assessing cognitive growth, researchers have

compared the novices' concept maps with those of experts in terms

of amount of organization, depth of structure, and cognitive

sophistication. For instance, Novak and Musonda (1991) found that

there were many common points in the conceptual sophistication of

elementary school students, high school students, and college

students. Although the best high school or college students were

better than the best elementary school students, the latter

constructed more valid and sophisticated concept maps than poor

high school or college students.

As an instructional tool , concept maps can be used by teachers

to explicitly show students the knowledge of concepts and their

relationships. By seeing these relationships, the contents of the

subject become meaningful to the student who may experience

difficulty in understanding. By making a cognitive connection early,

the student is less likely to experience conceptual problems in the

future (Bartels, 1991). Moreover, concept maps are useful to help

students to acquire and to recall coordinated ideas and meanings. At

the same time, the teacher can see the records of the students'

cognitive structures by reviewing their concept maps. Then s/he is

able to determine what relationships are missing or incorrect and

adjust his/her instruction to make changes in the students' cognitive

structures.

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CHAPTER III

EXPERIMENTAL DESIGN AND PROCEDURES

25

Sample

An experimental study was conducted with the students

enrolled in MATH 132 (second semester freshman calculus course)

during the spring semester of 1992. The experimental groups for the

quantitative research consisted of two sections of the C&M course

and two sections of the traditional course. The numbers of students

in the C&M group and the traditional group were 26 and 42

respectively. Only one section (10 A.M. class) of the C&M course was

the target group for the qualitative research.

Table 1

Course Schedules

Group Schedule Number of Students

C&M 10 A.M. 12

11 A.M. 14

Traditional 11 A.M. 22

3 P.M. 20

Course Methodology

Basically, the C&M course was taught as a laboratory course

with lectures being held to a minimum. The class met at 10 A.M. on

every Monday, Wednesday, and Friday. The students learned

calculus by working with computer lessons called "Mathematica

Notebooks" installed in computers. Thus, the lab became the

classroom. In the meantime, the students and the instructor met in

the classroom whenever classroom discussion was necessary, roughly

once per week. The students' grades were determined by the

performance on computer assignments and the scores on handwritten

tests.

Environment

The laboratory consisted of 30 Macintoshes, each with 4

Megabytes of RAM, running Mathematica off a hard disk. The

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26

computers were connected, using AppleShare, and two Macs were

used for distribution of lessons and collection of solution tiles—the

C&M students were expected to transmit regularly the solution files

(assignments) to their instructor.

In the lab, the computers were arranged in the formation

shown in Figure 4, screens facing toward the center. The instructor's

position was at the center of the lab looking over the students'

shoulders. Hence, the instructor could monitor the work being done

on each terminal and quickly provide aid for a student in trouble.

This allowed the instructor to contact students at crucial points and

thus minimize distractions and frustrations that students might

experience.

^tranceJTassignment

board

lab

manager

desk

printer

lab assistant

desk

Figure 2. Arrangement of the Lab

Quantitative Research

E?^pgrimgn tal YariablgsThe independent (treatment) variable was teaching/learning

methods, in other words, the exposure of students to Mathematica

environment. The two dependent variables included students'

achievement in calculus, and their attitudes toward mathematics and

computers.

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eidthty (>jio*.oit»il) Mkiô^tiUS

ol ^^h9U lo 9fXii??,0«T»

iwftcyfcru t»«wilt<_.tl faldtitty )cyxi etfT

bin «ain.r’%', bu'^'W «i5b«fhv *iiTii-.ttfit ,tcli5^'T oi )<»«r

Irfl tu

Instruments

The achievement test and the attitude survey were utilized to

measure the dependent variables in the study.

Achievement Test

Pre-achievement test. The items in the pre-achievement test

were designed to measure the mastery of prerequisite content for

introductory calculus. The investigator determined that selected

items in the achievement test for grade 12 used in SIMS (Second

International Mathematics Study) were appropriate for this purpose.

Therefore 16 items were selected from the SIMS (Travers, 1981)

achievement test and the format of those items was slightly revised

for college students (Appendix A). The investigator did not estimate

the reliability of the pre-achievement test because it had already

been estimated and reported by SIMS researchers.

Of the sixteen items, eight were designed for measuring

computational proficiency, and the other eight were used for

evaluating conceptual understanding. All the items in the pre-

achievement test were show-all-work type and scored with a

uniform grading scale because the difficulty levels of the items were

homogeneous. Table 2 provides the information about the cognitive

level, the content category, and the detailed topic of each test item.

Post-achievement test. The post-achievement test contained

sixteen items designed to estimate students' overall understanding of

calculus. Although the investigator wanted to include more post-test

items, sixteen items were the maximum due to the tight course

schedule and limited time allowed for the test. Like the pre-

achievement test, each set of eight items was designed either for ,

measuring conceptual understanding or computational proficiency,

and all the problems in the post-achievement test were show-all-

work type questions. The post-achievement test had unequal weight

because several items consisted of subproblems, and the difficulty

levels were not homogeneous. Table 3 provides the information

about the post-achievement test.

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rt«'. )’»i k') bn* la^coa & ,10

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o»u»o5 ’4h 01 «t»b t^rsw ny..vU

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jol “oilVfs aSuad) tiigb Vi ioa

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,idoi adt KiahtvO’q ^ îdoT .iax^’Wi'^lomo/t vra visv

irM lHarfPi.*;>iii:»^ odi

28

Table 2

Classification of the Pre-achievement Test Items

Item Cognitive

Level

Content

Cateeorv

Detailed

Tonic

1 CU Functions Definition of a function

2 CP Functions Zeros of the function

3 CP Trigonometry Simple trigonometric equation

4 CU Exponents & Solving an exponential equation

5 CP logarithms Solving a logarithmic equation

6 CP Limits &

continuity

Limits associated with finding the

derivative of f2

7 CU Limits &

continuity

Concept of continuity

8 CU Differentiation Relationship between an even function

and the derivative at 0

9 CP Differentiation Finding of actual derivatives

10 CU Differentiation Ability to find dy/dx when x and y are

presented as a third variable

11 CU Differentiation Identification of graphs

12 CP Differentiation Meaning of the derivative of a distance

function

13 CU Integration Integral as the area under a curve

14 CP Integration Relationship between the integral and

its antiderivative of f(x)

15 CP Integration Definite integral

16 CU Integration Area enclosed between two intersecting

curves in the plane

Note . CP means computational proficiency and CU means conceptual

understanding.

5

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29

Table 3

Classification of the Post-achievement Test Items

Item Cognitive

Level

Weight Content

Categorv

Detailed

Tonic

1 CU 5 Integration Meaning of the Fundamental

Theorem of Calculus

2 CP 6 Integration Definite integration

3 CP 5 Integration Setting up an integral formula

and its calculation

4 CP 5 Integration Finding of an integral

5 CP 6 Integration Integration by parts

6 CU 5 Integration Relation between integration by

parts and the product rule

7 CU 5 Series &

approximation

Application of geometric series and its

derivative to find an expansion

8 CU 5 Series & Expansion of e*^ in power of x

9 CP 5 approximation Taylor's formula

10 CP 5 Series & Convergence interval

11 CP 5 approximation L'Hdpital's rule

12 CU 5 Differentiation Relation between the sign of a derivative,

and rising and falling of the graph

13 CP 6 Differentiation Finding of actual derivatives

14 CU 6 Differentiation Identifying the graph of a function and

that of its derivative

15 CU 6 Differentiation Sum and product of two functions when

the signs of their derivatives are given

16 CU 6 Differentiation Identifying the model of a differential

equation and its graph

Grading procedure. To obtain reliability of experimental results,

a dual grading system was adopted: the grading was done and

inspected by both the investigator and one of the teaching assistants.

Before grading both tests, the investigator consulted with the

teaching assistant to establish some general guidelines of how to

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30

distribute the points. To be able to use the same criteria, a very

detailed list was made, specifying the points for correct start-up, and

partial credit assigned to each error type on each problem. In order

to score objectively, both graders rated problem 1 for all students

before going on to problem 2, and continued in that manner. In most

cases, the two scores for each student were the same, but if not, a

compromise was assigned. The two graders attempted to grade the

pre- and the post-achievement test as impartially as possible.

Attitude Survey

Students' attitudes toward mathematics and computers were

measured at the beginning and at the end of the semester. The

attitude survey (Appendix C) was adapted from the attitude

instrument developed by Sandman (1973) and from the attitude

questionnaire used in the SIMS.

The attitude survey was based on (I) attitudes toward

mathematics which had four dimensions: (a) mathematics as a

process, (b) mathematics and affect, (c) cooperative learning, and

(d) value to society; and (2) attitudes toward computers.

1. Attitudes toward mathematics.

a. Mathematics as a process (items 1, 5, 10, 18, and 23)—

The items in this category were designed to measure how

students view mathematics as a discipline. To have a positive

view of mathematics as a process is to view mathematics as a

field where speculations and heuristics rather than just rules

are important, where the body of knowledge is growing rather

than fixed, and where opportunity for creative people exists.

b. Mathematics and affect (items 2, 4, 9, 12, 14, 22, and

24)—The questions in this category were divided into four

subcategories; competence (items 2 and 22), enjoyment (item

9), motivation (items 12 and 14), and anxiety (items 4 and 24).

The items were designed to determine students' personal views

of themselves as learners of mathematics. These items

measured the extent to which students feel confident in their

mathematical ability, enjoy studying mathematics, want to

make achievement in mathematics, and are not anxious about

mathematics.

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c. Value to society (items 8, 17, and 20)—Three items

measured students' view of the usefulness and importance of

mathematics to society. A positive view is one in which

mathematics is seen as useful in everyday life and important in

preparing for an occupation.

d. Cooperative learning (items 3, 7, 15, and 19)—Four

questions were addressed for diagnosing the students' feelings

about cooperative learning.

2. Attitudes toward computers (items 6, 11, 13, 16, and 21)—

Five items in this category assessed students' attitudes toward

computers.

The attitude survey used a 5-point Likert scale with responses

ranging from strongly agree to strongly disagree. Because nine items

(4, 7, 10, 11, 14, 15, 16, 18, and 24) were negatively worded,

responses were recoded on a 5 -point scale, with 1 representing the

most negative view and 5 the most positive. The scoring of the

attitude survey is found in Table 4. For each student, a mathematics

attitude score (maximum total score 95 points) and a computer

attitude score (maximum total score 25 points) were computed and

added.

Table 4

Values of Response in Attitude Survey

Response Positive Item Negative Item

Strongly Agree 5 1

Agree 4 2

Undecided 3 3

Disagree 2 4

Strongly Disagree 1 5

Statistical Analysis

This study used the analysis of covariance (ANCOVA) as an

analysis method. ANCOVA is a statistical analysis method that

combines the analysis of variance with regression analysis (Kirk,

1982). ANCOVA is particularly useful in an experimental study in

which the subjects are not randomly assigned to groups.

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The experimental groups in this study—the C&M group and the

traditional group—were intact groups, not randomized groups. Thus,

they might have inherent differences in attitude and achievement,

and these extraneous variables are capable of causing evaluation

bias. Therefore the analysis method needs to reflect the initial

differences and remove them. For these reasons, the investigator

used ANCOVA as a data analysis method in this study.

The ANCOVA procedure involves measuring one or more

covariates as well as the dependent variables. The covariates

represent a pre-existing variation that has not been controlled in the

experiment, one that is believed to affect the dependent variable.

Through ANCOVA, the dependent variable can be adjusted so as to

reduce the effects of the uncontrolled source of variation represented

by the covariates. The potential advantage is the reduction in

experimental error, thus increasing power and the reduction in bias

caused by differences among experimental units.

ANCOVA rests on the same assumptions as the analysis of

variance plus three additional assumptions regarding the regression

part of the covariance analysis. ANCOVA also assumes (Stevens,

1990, p. 166)

1. a linear relationship between the dependent variable and the covariate

2. homogeneity of the regression slopes for covariate

3. measuring the covariate without error

Qualitative Research

The data from the achievement test and attitude survey are

not sufficient for the explanation of students' understanding of the#

calculus concepts. This research needs a more holistic view of the

C&M course than that which can be provided by quantitative data.

The purpose of qualitative research is to understand the processes

which ultimately determine success or failure. “Data gathered in an

open-ended fashion can be a source of well-grounded, rich

descriptions and explanation of processes” (Miles & Huberman, 1984,

p. 15) as they occur in the C&M course.

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Instruments

The qualitative methodology used in this study was based on a

field-method philosophy of open-ended data gathering and analysis.

The investigator gathered the data in the form of observation notes,

questionnaire responses, interviews, solution files, class materials,

and concept maps.

Table 5

Qualitative Data Collection

Source Pilot Studv Main Studv

Observation 5 weeks 16 weeks

Nov. 11 - Dec. 14 (1991) Jan. 16 - May. 6 (1992)

Mon, Wed (9:00-10:00) Mon, Wed, Fri (10:00-11:00)

Questionnaire 12 students

Interview 12 students

Solution 3 solution files 12 solution files of

Files each lesson

Class review of courseware review of courseware

Material and quizzes and exams

Concept

Map24 concept maps

Observation Notes

The investigator engaged in extensive and detailed

observations of the C&M course throughout one semester.

Observation notes reflected the frequency of the instructor's

statements, dyadic contact (instructor-students, lab assistant-

student), and students' interactions, as well as the descriptions of

students' activities and their difficulties in manipulating the

computers or understanding the content of the lessons.

To record the details of observation systematically, the

observation note was used. The purpose of the observation system

was to trace the patterns and the changes of activities in the lab. The

frequency of each activity was noted and recorded in the class on the

days prior to the homework due date and to the exam date (13

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34

times). The coded observations fell into one of the following

categories:

1. Instructor's statements—instructor's general announcement

concerning (a) the schedule of the discussions, exams, and

assignment due date and (b) the lesson content.

2. Dyadic contact—private interaction, initiated by the students,

the instructor, or the assistants.

3. Students' interaction—interaction among students in order to

share ideas to understand the lessons or to complete the

assignments.

Sometimes it was not clear who initiated the interaction, and

how many students were involved in the interaction. The

investigator concedes that part of the observation was subjectively

coded.

Questionnaire

Most of the questions centered on the students' learning

procedure and their evaluations of the C&M course. The goal of the

questionnaire was to investigate the following eight issues.

1. Motive of taking the C&M course was:

a. Why did you choose this course and what did you

expect to learn in this course?

2. Background knowledge in computers was:

a. Did you have any computer experience before this

course? What kind of computer experience? Is that experience

helpful for this course?

3. Experiences with the C&M course were:

a. How many hours per week have you usually spent in

the lab?

b. Have you ever used the computer and Mathematica to

explore mathematics beyond the requirements of course

assignment? If so, what for?

4. Experiences with Mathematica were:

a. How long did it take for you to become comfortable

with Mathematica and what particular problems did you have

with it?

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3 5

b. Do you think that your experience with Mathematica is

helpful in learning other computer programming languages like

Pascal or BASIC? Do you think Mathematica is totally different

from other programming languages?

5. Perceptions of how particular aspects of the course affected

the students' learning and attitude were:

a. Could you give me a concrete example in which

learning with Mathematica is particularly helpful? Be as

specific as possible.

b. Are you more or less confident in mathematics than

you were before taking this course? What aspects of the C&Mcourse made you more (less) confident?

c. Do you agree that lots of plotting, calculating, and

exploring by computers give a good perspective of underlying

principles? If so, which Mathematica command or tool was the

most helpful?

6. Comparison with the traditional calculus course were:

a. Some people say the C&M course and the traditional

course present different views on what calculus is. Do you

agree or disagree? Could you give me your reasons for

agreement or disagreement?

b. Do you think you might have less ability in hand

calculation than the traditional calculus course students? If so,

have you developed anything to replace hand calculation

ability?

7. Evaluation of the C&M course were:

a. In your opinion, what is the strongest point of this

course? What is the weakest point of this course?

b. Do you like the format of the C&M lesson: basic-

tutorial-give it a try-literacy sheet? Do you think there is some

redundancy in the four steps?

c. Are you going to take other C&M courses again like

MATH 242 or 245? Would you recommend this course to your

friends? Why or why not?

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36

8. Suggestions for improving the course was:

a. Do you have any suggestions for on-screen lessons,

assignments, classroom sessions, classroom exams, literacy

sheets, style of learning, instructor, or assistants?

Every student was encouraged to write openly, freely, and

specifically about his/her experience, opinion, and evaluation of the

course. The investigator hoped that the questionnaire, with its open-

endedness, provided the space for students' voices and concerns.

Interview

The investigator informally interviewed the students

concerning their understanding of important concepts in calculus.

The interview questions were loosely structured around the

following six questions:

1. Does the slope of the tangent line to the graph of the function

f at point (x, f(x)) mean the derivative f '(x)?

2. Is the integral of a function over a closed interval a number

or a function?

3. How do you solve maximum and minimum problems?

4. What is the Fundamental Theorem of Calculus and how does

it establish a connection between integration and differentiation?

5. How do you determine the convergence interval of a power

series when computers are available? How about when they are not

available?

6. What is L’Hopital's rule, what is Taylor's formula, and how

does L'Hopital's rule come from Taylor’s formula?

The first and the second questions were asked both the C&Mgroup and the traditional group in order to compare their responses.

The other four questions were addressed only to the C&M group.

Solution Files

The Students' solution files contained the full solutions of

assigned problems in "give it a try" and "literacy sheet." Each of the

student solution files sent to the instructor was printed out before

being returned to the student. As written documents, the student

solution files were studied, and particular notes were made on

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indicated the range of the his/her understanding when incorporated

with the interview and concept map data.

Class Materials

The approach of the C&M course was different from that of the

traditional one in various aspects. To understand the C&M course and

find the interesting patterns of the students' learning, the

investigator reviewed the class material including the courseware

and the exams.

Concept Maps

A concept map is a schematic device that represents a set of

concepts embedded in a framework of a proposition. In the concept

map, the most general and inclusive concept is listed at the top of the

map and successively more specific concepts are subsumed below. Aconcept map can demonstrate visually how the concepts are

hierarchically related and how concepts on the same level are

horizontally related. Constructing a concept map requires the

students to externalize their thinking by mapping out their

conceptual structure of a subject showing those relationships.

At the end of the semester, two examples of the concept maps

(Appendix D) were given to the students. Then, the investigator

briefly explained how to construct a concept map and asked the

students to make their own concept maps.

The scoring criteria used in the study follow:

1. Propositions— If the relationship between two concepts

indicated by the connecting line and linking words was meaningful, 2

points were scored.

2. Hierarchy—The map shows valid hierarchy when each

subordinate concept is more specific and less general than the

concept drawn above it. For each valid level of the hierarchy, 5

points were scored.

3. Cross links—The valid cross link means meaningful

connection between one segment of the concept hierarchy and

another segment. Five points were scored for each cross link that is

both valid and significant. For creative cross links, extra points are

given.

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38

4. More concepts--The students who included meaningful

concepts which were not given, deserved to gain 3 points.

5. Misconceptions-'Three levels of misconceptions were

considered. Five points were deducted for an occurrence of a major

misconception. Successively, three points or one point were

subtracted according to the extent of the misconception.

The total score was computed by combining scores on the five

criteria.

LEVEL 1

LEVEL 2

LEVEL 3

LEVEL 4

Scoring for the above model is

Propositions (if valid)

Hierarchy (if valid)

Cross links (if valid and significant)

More concepts (the concepts with *)

Misconceptions

Total

11 * 2 = 22

4 * 5 = 20

2 * 5 = 10

3*3= 9

0_

61 points

Figure 3. Scoring Model

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3 9

Data Analysis Procedure

The large number of data sources used in this study made it

important not only to reduce the data efficiently but also to establish

a clear method to display them. Qualitative data analysis used in this

study consisted of concurrent flows of activity: data reduction, data

display, and conclusion drawing/verification (Miles & Huberman,

1984).

1. Data Reduction—Data reduction refers to the process of

selecting, focusing, simplifying, abstracting, and transforming the raw

data in written-up field notes. The data was reduced and

transformed through selection, discarding, paraphrasing,

summarizing, and being subsumed into a larger pattern.

2. Data Display—Data display is an organized assembly of

information that permits conclusion drawing. Better displays can

facilitate valid qualitative analysis. Among several types of display

(matrices, graphs, charts, and networks), the investigator used a

chart as a data display tool.

3. Conclusion Drawing/Verification--From the beginning of data

collection, the investigator began to note regularities, patterns, and

explanations. At the same time, the meaning emerging from the data

was tested for its plausibility.

Figure 4 . Components of Data Analysis: Interactive Model

s;

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40

Data reduction, data display, and conclusion drawing/verifica-

tion were the three streams of qualitative data analysis, and these

three types of analysis activity formed an interactive, cyclical

process (Figure 4).

Pilot Study

The pilot study was done in the fall semester of 1991. The

investigator administered the post-achievement test and attitude

survey to estimate their reliabilities. One section of the traditional

course and one section of the C&M course participated in the pilot

study. The reliability was estimated by using Cronbach’s alpha

coefficient formula. The formula (Mehrens & Lehmann, 1984) is

a = ^^[1-K-1

Where K = number of items in the test

S = standard deviation of the set of test scores

Si = standard deviation of a single item i

This general formula, rather than the Kuder-Richardson

formula, was thought to be appropriate since the items in the post-

achievement test were not scored dichotomously, nor did they have a

uniform scale.

The traditional course had already covered all the content for

the post-achievement test. However, the C&M course had not dealt

with the content corresponding to questions 7, 8, and 9 at that time.

Thus, all the questions in the post-achievement test were administered

in the traditional course and these three items were removed for the

C&M course. Accordingly, separate SPSSX procedures were run.

The reliability estimates of the post-achievement test for the

traditional course and the C&M course were .826 and .813

respectively. Tables 6 and 7 provide more detailed information about

the post-achievement test.

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4 1

Table 6

Mean. SD. and Reliability Estimate for the Post-achievement Test

(Q1 - 013)

Course Traditional C&MClassification total CU CP total CU CP

Number of Items 13 6 7 13 6 7

Means 2.12 1.68 2.51 3.04 3.86 2.34

Standard Deviation 3.87 3.16 4.48 4.07 3.70 4.38

Reliability .825 .650 .746 .813 .739 .719

Table 7

Mean. SD. and Reliabilitv Estimate for the Post-achievement Test

(01 - 016)

Course Traditional

Classification total CU CP

Number of Items 16 8 8

Means 1.99 1.71 2.27

Standard Deviation 3.78 3.48 4.08

Reliabilitv .826 .651 .714

According to the result of the item-total correlation analysis

(Appendix E), the question with negative item-total correlation was

replaced with a more dependable one.

Table 8

Means. Standard Deviations. and ReliabiliU' Estimates for the

Attitude Survey

Test Number Means Standard Reliability

of Items Deviation Estimate

Attitude 20 3.66 0.99 .879

The reliability of the attitude survey was estimated also by

using Cronbach's alpha coefficient formula. The reliability estimate of

the attitude test was .879 (Table 8). The result of the item-total

correlation analysis was provided in Appendix E.

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After the pilot study, the committee members and the

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learning. Since the attitude toward collaborative work was

considered an important variable in the explanation of the nature of

laboratory study, we included four items diagnosing attitude.

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CHAPTER IV

RESULTS IN THE CONTEXT OF MATHEMATICS LEARNING

4 3

Characteristics of the Samples

The characteristics of the subjects within groups were noted

because the comparison by the achievement test used an intact

group instead of a randomized one.

Table 9

Mean and SD of Age. Gender, and College Mathematics Courses Taken

Group N Age

Mean 5D

Male Female Mathematics Course

Mean SD

C&M 26 20 1.9 21 (81%) 5 (19%) 1 0.5

Traditional 42 19 1,1 33 (79%) 9 (21%) 1 0.4

Table 10

Distribution of Maiors

Group Liberal Arts Engineering Business Agriculture

and Science Administration

C&M 6 (23%) 19 (73%) 1 (4%) 0 (0%)

Traditional 8 (19%) 24 (57%) 8 (19%) 2 (5%)

No difference between the two groups was found in the

percentage of each gender, the mean of age and number of

mathematics courses taken at the college level. However, there was

substantial difference in the distribution of majors between the two

groups: 73% and 4% of the C&M students and 57% and 19% of the

traditional students majored in engineering and business

administration respectively.

Achievement Test

Before the instruction began, the C&M and the traditional

groups were administered the pre-achievement test (Appendix A) in

order to measure the students' prior knowledge of calculus. The pre-

achievement scores were used as covariates for the statistical

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44

analysis. At the end of the instruction, the students in both groups

took the post-achievement test (Appendix B). In an effort to test the

students both on conceptual understanding and on computational

proficiency, eight items of the pre- and post- achievement tests

addressed the concepts and applications of calculus, and another

eight items concentrated almost altogether on calculations.

Before using ANCOVA (analysis of covariance), the following

two checks were made to determine whether ANCOVA was

appropriate: first, the investigator examined the linear relationship

between the dependent variables and the covariates by checking the

correlation coefficients (Table 11); second, the interaction between

the covariates and the treatment variable was inspected to test

whether the regression slopes for the covariates were homogeneous

(Table 12).

Table 11

Correlations Between the Dependent Variables and the Covariates

Dependent Variable Covariate Correlation Coefficient

post-test scores pre-test scores 0.80

post-CU scores pre-CU scores 0.60

post-CP scores pre-CP scores

Note . CU means conceptual understanding and CP meanscomputational proficiency.

Table 12

Interactions between the Covariates and the Treatment Variable

Source F p

post-test scores * treatment 0.66 0.42

post-CU scores * treatment 0.03 0.87

DOSt-CP scores * treatment Q.Q4 0.85

In sum, ANCOVA was a legitimate analysis method in this

study because the two conditions were satisfied.

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4 5

Results of Testing the Hypotheses

The null hypotheses addressed in chapter I were:

HI --There is no statistically significant difference in the scores

on the achievement test between the C&M group and the traditional

group.

H1(A)--There is no statistically significant difference in the

conceptual understanding scores between the C&M group and the

traditional group.

H1(B)--There is no statistically significant difference in the

computational proficiency scores between the C&M group and the

traditional group.

Using ANCOVA, the investigator tested the post-achievement

scores at the .05 level of significance. The calculated F-value 4.49

(Table 13) was larger than the critical F-value for the appropriate

degree of freedom (Fqs, i, 65 - 3.99); thus, HI was rejected. This

result indicates that the exposure to Mathematica environment

played a significant role in determining students' achievement

scores.

Table 13

Summary Table of ANCOVA for the Achievement Scores (Total!

Source SS Df MS F D

Covariate 3222.3 1 3222.3 119.78 .001

Treatment 120.7 1 120.7 4.49 .038*

Within 1748.6 65 26.9

Tables 14 and 15 are the results of the ANCOVA of the

pertaining to null hypotheses H1(A) and H1(B). When ANCOVA for

the conceptual understanding scores and the computational

proficiency scores between the two groups were performed, the

calculated F-values were 16.40 (Table 14) and 0.93 (Table 15),

respectively. The F-value of the conceptual understanding scores was

significantly larger than the critical F-value (3.99). Thus, H1(A) was

rejected, indicating that there was a significant difference in the

conceptual understanding scores between the two groups at the .05

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46

level of significance. On the contrary, the F-value of the

computational proficiency scores was smaller than the critical F-

value. Hence, H1(B) was not rejected, indicating that there was no

significant difference in the computational proficiency scores

between the two groups at the .05 level of significance.

Table 14

Summary Table of ANCOVA for the Conceptual Understanding Scores

Source SS Df MS F p

Covariate 642.0 1 642.0 39.00 .001

Treatment 270.2 1 270.2 16.40 *oqWithin 1070.9 65 16.5

Table 15

Summary Table of ANCOVA for the Computational Proficiency Scores

Source SS Df MS F p

Covariate 857.3 1 857.3 41.24 .001

Treatment 19.3 1 19.3 0.93 .339

Within 1351.2 65 2Q.g

Further Analysis

The C&M group outperformed the traditional group on ten out

of sixteen questions on the post-achievement test: seven out of eight

conceptual understanding problems; and three out of eight

computational proficiency problems. Among the ten questions, the

three items (6, 8, and 14) of the conceptual understanding showed

the significant p values for a t-test. Summaries of relevant statistics

are shown below.

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Table 16

Mean of the Post-achievement Scores

C&M Traditional

Mean Adjusted Mean Mean Adjusted Mean

Total 58.4 57.9 54.7 55.1

CU 29.5 29.2 24.9 25.1

CP 28.9 28.8 29.8 29.9

Note. The adjusted mean is the mean calculated from ANCOVAprocedure that considers the pre-existing differences reflected in the

pre-achievement scores.

Table 17

Mean and SD of the Post-achievement Scores

Category Item Weight C&MMean foercentaeel SP

Traditional

Mean (oercentaee) SP

P

(t-test)

CU 1 5 3.7 (74%) 1.8 3.2 (64%) 1.6 0.26

CU 6 5 3.7 (74%) 1.1 2.8 (56%) 1.6 0.02*

CU 7 5 2.9 (58%) 1.9 3.4 (68%) 2.1 0.32

CU 8 5 4.0 (80%) 1.8 2.8 (56%) 1.9 0.01*

CU 12 5 3.5 (70%) 1.6 3.1 (62%) 1.6 0.40

CU 14 6 4.3 (72%) 2.2 3.0 (50%) 2.0 0.01*

CU 15 6 3.7 (62%) 1.3 3.5 (58%) 1.6 0.51

CU 16 6 3.7 (62%) 1.3 3.1 (52%) 1.2 0.05

CP 2 6 4.0 (67%) 1.6 4.6 (77%) 1.5 0.11

CP 3 5 3.2 (64%) 1.5 3.6 (72%) 1.2 0.28

CP 4 5 3.5 (70%) 2.4 2.7 (54%) 2.5 0.24

CP 5 6 3.9 (65%) 1.6 3.8 (63%) 1.5 0.80

CP 9 5 2.5 (50%) 1.7 3.0 (60%) 1.6 0.29

CP 10 5 3.7 (74%) 2.0 3.6 (72%) 1.7 0.72

CP 11 5 3.8 (76%) 1.6 4.2 (84%) 1.4 0.33

CP _L2 6 4.3 (72%) 1.4 4.4 (73%) 1.7 0.69

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48

A detailed interpretation of the results in two categories

follows.

Conceptual Understanding

Generally, the C&M group was better able to solve most of the

conceptually oriented questions. In particular, there were large

differences in the scores on questions 8, 14 and 16. These three

questions required the translation of mathematical ideas in symbolic

or algebraic form to an equivalent representation in graphic form,

and vice versa. The comparative superiority of the C&M group on

these three questions indicates that computers could improve

students' abilities to translate formulas to graphs, and vice versa, by

providing quick plots.

To solve question 14, students should be able to identify the

graph of a function and that of its derivative. During the semester,

the C&M students worked with computer plottings until they

developed a feeling for the relationship between the up-and-down of

a function, and the sign of its derivative. Undoubtedly, the C&Mstudents did benefit from their visualization experiences with

Mathematica.

In the answers to question 16^, an interesting phenomenon

was found. Among the students who did not receive full credit on

this question, four students in the C&M group correctly matched

graphs but mismatched formulas, whereas five students in the

traditional group could identify only formulas. This sharp contrast

indicates the different foci of the two courses, and the corresponding

different abilities of the two groups. During the semester, neither

group learned differential equations, which is a topic relevant to

question 16. Moreover, neither the C&M courseware nor the

traditional calculus texts dealt with problems similar to question 16.

Certainly then, the students' responses to question 16 were not

influenced by extraneous factors but only by their own abilities.

Question 1 asked the students to relate the integral and the

derivative using the Fundamental Theorem of Calculus, and question

^ Choose both the mathematical model and the graph from the lists that best fit

the described function f(t). . . .

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4 9

6 addressed the relation between the integration-by-parts formula

and the Product Rule of differentiation. Three weeks before the post-

achievement test, these concepts were given to one section from each

group, and the students in those sections were asked to construct the

concept map.

Table 18

Mean of the Scores on Questions 1 and 6

Question 1 Question 6

C&M Standard C&M Standard

Participated 3.8 3.4 3.8 3.1

Not Participated 3.6 U 3.4 2J

The two sections which participated in the concept map test received

higher scores on these two questions than did the other two sections

without concept mapping experience.

Question 7 required students to find the expansion of —

-

(1 - x)^

by using the fact that the derivative of -7-^—- is^—- . The

(1 - X)(1 .

traditional group did better on this question than the C&M group did.

One of the possible explanations is that partial credit was given to

the students who solved this problem by using Taylor's formula

instead of the required derivative property. Several students in the

traditional group, but no students in the C&M group got partial credit

with that approach. Therefore, these partial credits could have

increased the mean of the traditional group.

On the other hand, the C&M group was better able to sketch the

graph of a function when its derivative was given (question 12), to

justify their answers by providing examples or counter-examples

(question 15), and to explain the relation between the graphs of e^^

and n increased (question 8). The C&Mgroup's superior performance on these non-routine and applied

problems might be an evidence of their deep understanding of the

corresponding concepts in each question.

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Computational Proficiency

The traditional group performed better on five out of eight

computational proficiency questions. Among the five questions, two

(questions 2 and 13) were the questions which required no more

than the direct application of learned techniques (to compute

derivatives and integrals). Thus, there is no doubt that the traditional

group was better at calculations with simple techniques.

In a sense, question 43 was a different version of question 1,

which asked to establish a connection between integration and

differentiation, using the Fundamental Theorem of Calculus. To solve

question 4, students should be able to understand thoroughly the

Fundamental Theorem of Calculus, and apply this theorem to the

given integral. Certainly, the C&M group's higher score on this

question was related to the higher score on question 1. Fewer

students in the C&M group than in the traditional group considered

this question as the extension of the previous calculation questions

and tried to solve by several integration techniques.

Question 3"^ was a relatively easy problem which involved two

steps: setting up the formula, and doing the integration. However,

this question entailed one tricky part, "(9 + t) thousands." The

answers that used (9 + t) instead of 1000 (9 + t) lost 2 points for this

mistake. In the test, the traditional group paid more attention to

"thousand", and reflected it in the formula. As a result, the traditional

group received higher scores on this question. The possible reason is

that the traditional group had more chance to encounter problems

with tricky parts, and was more trained to handle those problems.

The C&M group showed more ability to solve integration

problems related to the integration-by-parts formula (question 5).

Among the several integration techniques, the C&M courseware

focused on two techniques: integration-by-parts and integration-by-

3 Calculate F '(t) given that F(t) = I— dx

jo(k^x^)

Suppose that the birth rate in a certain country t years after 1970 was (9 + t)

thousands of births per year. Set up and evaluate an appropriate integral to

compute the total number of births that occurred between 1970 and 1990.

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5 1

substitution. Hence, the more concentration on integration-by-parts

might be one of the reasons for the C&M group's superiority on this

question.

The traditional group performed better on question 9 than the

C&M group did. This question is a typical hand calculation problem

which frequently appears in exercises and tests in the traditional

course. Thus, the traditional group might have had a greater amount

of experience with the problems just like this one, while the C&Mgroup had not had much experience. Based on this fact, the relatively

low mean scores of the C&M group is a natural result.

The further investigation of students' answers to question 9

revealed a somewhat noticeable pattern. The most usual way of

solving this problem (the expansion of e** cos[x]) is applying Taylor's

formula both to e‘* and to cos[x], multiplying two expansions, and

then simplifying the result. This can also be solved by applying

Taylor's formula to e-* cos[x]. The first method involves less

calculation than the second one; thus, calculus texts almost always

present the first method and most of the students in the two groups

took that one. Meanwhile, several students in the C&M group but no

student in the traditional group tried the second method. This

implies that the students in the traditional group seemed to store the

most convenient solution method in their memory and to apply the

proper solution procedure when they confronted problems, whereas

the students in the C&M group seemed to be able to figure out the

solution procedure after they were faced with problems. Apparently,

this difference originates in the traditional group's more experience

and the C&M group's less experience in hand calculation exercises.

L'Hopital's rule is the key to solve question 11. The traditional

group received the second highest scores on this item. This confirms

that the traditional group is good at problems related to rules and

techniques.

Question 10 (to find the convergence interval of the given

series) was one of the three computational proficiency items on

which the C&M group outperformed the traditional group. This

outperforming is partly explained by the fact that a large part of the

C&M courseware was covered by visual plottings related to

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facilitate their understanding of the three techniques of finding

convergence interval.

Conclusions on the Achievement Test

The results of the achievement test suggest that the C&Mgroup, without seriously losing computational proficiency, was much

better at conceptual understanding than was the traditional group. In

particular, the concept-oriented items related to graphs (questions 8

and 14) and the meaning of the theorems (questions 1 and 6) yielded

the largest difference in performance. On the computation-oriented

items, the C&M group was slightly inferior to the traditional group.

But the loss of the computational proficiency was trivial enough to

accept the null hypothesis. This may be an evidence that learning

with Mathematica does not always weaken students' calculational

abilities.

Concept Maps

One section of the C&M group and one section of the traditional

group participated in the evaluation by concept map. First, the

concept mapping procedure was explained to the students with easy

examples from elementary and high school mathematics. After that,

concept map sheets (Appendix D) were handed out and assigned as

homework. Because of the differences in the content that the two

groups learned, slightly different concept lists were given. All the

three lists A, B, and C (Appendix D) were provided to the C&M group,

and only the first two lists A and B were given to the traditional

group. Nine students in the C&M group and 15 in the traditional

group handed in the completed concept maps.

Analysis Method

The purpose of the concept mapping procedure was not to

compare the scores, but to recognize the different patterns of

students' mathematical understanding. Nevertheless, the

standardized scores for the students' concept maps were necessary to

make more legitimate comparisons. Thus, the students' concept maps

were analyzed by two methods.

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5 3

The first method was to grade the students' concept maps

according to five criteria: propositions, hierarchy, cross links, more

concepts, and misconceptions. Two points were given to meaningful

propositions, and 5 points were scored each for valid hierarchy and

for significant cross links. The students who included more

meaningful concepts which were not given deserved to gain 3 points.

According to the extent of the misconceptions, 1, 3, or 5 points were

deducted.

The second method was to examine the congruence coefficients

between the teacher's concept map and the students' concept maps

using the software MicroQAP, a program for the computation of the

generalized measure of association between the two data matrices

(Costanzo et al., 1983; Hubert et al., 1981 and 1985).

The basis for the statistics calculated in MicroQAP is the

comparison of two data matrices, A and B, organized as a square

matrix. In the analysis of concept maps, A is the matrix constructed

from the teacher's concept map and B is the matrix from each

student's concept map. The following example demonstrates the

method of creating the matrix.

Figure 5. Simple Example of the Concept Map

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1 2 3 4 5 $ 7

1 0 1 1 0 0 0 0

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3 0 0 0 0 0 0 1

4 0 0 0 0 0 0 0

5 0 0 0 0 0 1 0

6 0 0 0 0 0 0 1

2__JL_ 0 _Q__Q__Q__L__Q

Figure 6. Matrix Created from the Concept Map in Figure 5

Examples

Figures 7, 8, 9 and 10, 11 are the dichotomous examples

indicating the differences in the complexity of the high and low

scoring maps. The three concept maps (Figure 7, 8, 9), even though

not perfect in the concept listing and connecting lines, demonstrated

an attempt to use more concepts and to list relevant interrelation-

ships between concepts, while the two concept maps (Figure 10, 11)

showed a relatively simplistic view of the calculus concepts without

making any cross links.

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Figure 7. High Scoring Map Demonstrating the Understanding of

Interrelationship Between the Concepts

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Scoring Propositions 8 * 2 = 16

Hierarchy 6 * 5 = 30

Cross links 1*5=5More concepts 4 * 3 = 12

Misconceptions Q

Total 63


Figure 8. High Scoring Map Demonstrating the Large Number of

Relevant Concepts Identified

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Total 57


Figure 9. High Scoring Map demonstrating the Thorough

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Total 47


Fi gure 11. Low Scoring map Demonstrating the Relatively Simplistic

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60

Concept Map Data

Table 19 presents the students' scores from the first analysis

method and their congruence coefficients from the second analysis

method

Table 19

Students' Scores and Congruence Coefficients

Student Concept Map (A)

Score Coefficient

Concept Map (B)

Score Coefficient

Concept Map (C)

Score Coefficient

C&M (1) 77 .83 63 .55 51 .65

C&M (2) 61 .48 42 .41 44 .67

C&M (3) 72 .68 53 .70 57 .83

C&M (4) 77 .67 41 .56 42 .49

C&M (5) 71 .72 63 .52 51 .74

C&M (6) 73 .56 45 .70 37 .57

C&M (7) 50 .63 55 .64 42 .68

C&M (8) 87 .88 61 .64 57 .56

C&M (9) 60 .50 44 .52 42 .47

Trad (1) 79 .64 40 .15

Trad (2) 72 .53 43 .18

Trad (3) 72 .53 47 .63

Trad (4) 72 .58 55 .44

Trad (5) 46 .40 34 .26

Trad (6) 62 .68 39 .48

Trad (7) 44 .31 33 .12

Trad (8) 66 .51 43 .41

Trad (9) 74 .56 41 .49

Trad (10) 55 .48 36 .50

Trad (11) 100 .73 50 .70

Trad (12) 73 .51 24 .22

Trad (13) 67 .43 41 .40

Trad (14) 71 .67 44 .41

Trad (15) ZL_ ^0 _^1 22

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6 1

Statistical Findings

What follows are the major findings from the statistical

analysis of the concept maps.

First, the total scores of the C&M group were generally higher

than those of the traditional group. In particular, there was a big

difference in the scores on the concept map (B), but a relatively small

difference in the scores on the concept map (A). This might have

originated from the fact that a large part of the C&M curriculum was

covered by the topics of series and approximation related to the

concept map (B), whereas the traditional course handled these topics

only for a short period.

Table 20

Mean and SD of Concept Map Scores

Concept Map (A)

Criterion C&M Traditional

Mean SD Mean SD

Propositions 29.8 0.7 27.2 2.8

Hierarchy 24.4 3.9 27.7 5.9

Cross links 10.0 5.0 5.0 3.8

More Concepts 7.0 4.7 10.8 10.6

Misconceptions -1.4 2.2 -2.3 3.3

Total 69.8 11.1

Criterion C&MConcept Map (B)

Traditional

Concept Map (C)

C&MMean SD Mean SD Mean SD

Propositions 15.6 0.9 15.2 1.0 22.0 0

Hierarchy 25.0 3.5 23.0 5.9 21.7 2.5

Cross links 3.3 2.5 1.3 2.3 6.7 7.1

More Concepts 9.3 4.4 4.0 4.6 0.7 1.3

Misconceptions -1.3 3.0 -2.8 2.3 -4.0 1.9

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6 2

Second, the C&M students' concept maps showed stronger

congruence with the teacher's concept map than the traditional group

did.

Table 21

Mean and SD of Congruence Coefficients Between the Teacher's

Concept map and the Students' Concept Maps

Concept Map (A) Concept Map (B) Concept Map (C)

Mean SD Mean SD Mean SD

C&MTraditional

0.66

0.53

0.14

0.11

0.58

0.37

0.10

0.18

0.63 0.12

Third, there was a certain correlation between the scores from

the first analysis method and the congruence coefficients from the

second analysis method. But the correlations were not strong (0.52 to

0.67). The possible reason is the fact that the first analysis method

gives the credit for more concepts and cross links, whereas the

second analysis method does not reflect those two criteria (Table 22).

For instance. Figure 8 is one of the examples of high scoring

maps and Figure 10 is that of low scoring maps based on the first

method. However the congruence coefficient of the former is .55

while that of the latter is .63. The former gained 5 and 12 points for

cross links and more concepts, and the latter received only 3 points

for more concepts. This difference resulted in the superiority of the

former in the total scores. But, in the aspect of similarity to the

teacher's map, the latter is preferable to the former.

Table 22

Correlations Between the Total Scores and the Congruence

Coefficients

Concept Map (A) Concept Map (B) Concept Map (C)

Correlation 0.67 0.^1 0x52

Fourth, cross links was the area where a distinctive difference

was found. For example, more students in the C&M group connected

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6 3

the derivative and the integral, or differentiation and integration by

the Fundamental Theorem of Calculus. Another example is the

relation between the power series and the geometric series; five out

of nine students in the C&M group and four out of fifteen students in

the traditional group linked the two series and stated that the

geometric series was a special case of the power series. Also the

interpretation of integration-by-parts as the Product Rule of

differentiation, and the explanation of integration-by-substitution as

the Chain Rule of differentiation could be one of the cross link

examples with favorable scores for the C&M group.

Fifth, there was strong correlation between the concept map

scores and the post-achievement scores. The high correlation

coefficient 0.82 (Table 23) implies that the majority of the top scores

on the concept maps were made by high achieving students, and the

majority of the bottom scores were from the low achieving group in

the achievement test. Furthermore, the correlation coefficient

between the concept map scores and the conceptual understanding

scores (0.71) was larger than that between the concept map scores

and the computational proficiency scores (0.59).

Table 23

Correlations between the Concept Map Scores and the Achievement

Scores

Concept Map Scores

Post-achievement Scores 0.82

Post-CU Scores 0.71

Post-CP Scores 052

Sixth, the cross link scores and the total scores showed strong

positive correlation (0.73) although the scores on the other four

criteria also had positive correlation with the total scores (Table 24).

This means that the students who did well in linking concepts in

different branches generally received high total scores. In fact, the

heavy weight (5 points) given to valid cross links partly contributed

the strong correlation between the cross link scores and the total

scores.

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Correlations Between the Total Score and the Scores on the Five Criteria

Propositions Hierarchy Cross links More concepts Misconceptions

Total 0.26 0.54 QJ2 0.40

Non-statistical Findings

From a further analysis of the concept maps, the following

three points were noticed. First, most of the students in both groups

tried to include more concepts beyond the given lists. However, there

were substantial differences in the concepts chosen by the two

groups. Most of the concepts selected by the C&M group were

related to graphs (increasing function, decreasing function, maximum

& minimum, concavity), visual interpretation (cohabitation and

barriers), and visual examples (kissing parabola). In contrast, most of

the concepts chosen by the traditional group were the various terms

(arithmetic series and alternative series), applications (area, volume,

density, and weight), and techniques (integral test, root test,

comparison test, and alternating series test). This disparity could be

illustrated by the different texts and approaches of the two courses.

The C&M course did not handle as many techniques and terms as the

traditional textbooks did; instead it visually dealt with a limited

number of topics rather profoundly.

Second, most of the C&M students started the second concept

map with "expansion", whereas roughly one third of the traditional

students placed "Taylor’s formula" at the top of the map. This

difference might also be explained by the different emphases of the

two courses.

Third, the logical connectives used to make the propositions

tended to be equally simple on all the maps regardless of the

students’ scores. Very simple vocabulary—"found by", "allows

calculation", "some have", "equivalent to", "look for", "have an inverse

relationship expressed by"—instead of mathematical jargon was used

to explain how the concepts were related.

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65

Misconceptions

Following common misconceptions were found in the students'

concept maps. First, almost all the students who could not recognize

that the Fundamental Theorem of Calculus was connected both to

differentiation and to integration placed the theorem in the

integration branch; only one student put the theorem in the

differentiation branch. Undoubtedly, this misconception originated in

the fact that the Fundamental Theorem of Calculus was introduced in

the integration chapter.

Second, several students in the traditional group

misunderstood the term, "convergence principle"^. They seemed to

consider the convergence principle as the general principle of the

series having convergence intervals; thus, they placed this term in

the level between the series (the power series or the geometric

series) and the techniques of finding convergence intervals.

Third, many students did not connect the geometric series and

the convergence interval. They seemed to have the misconception

that only the power series has the convergence interval.

Students' Evaluation

Twenty-one students among the twenty-four who participated

in the concept mapping procedure filled out the evaluation form that

asked for their feelings about developing concept maps. The overall

responses of the students were favorable. Apparently, the students

considered that concept mapping procedure was worthwhile and

useful. Also they felt that the concept mapping was strongly related

to thinking and ideas, rather than to memorizing and symbols.

However, they seemed to have a hard time to complete the concept

maps because their reactions were close to being difficult,

challenging, and diffident.

5 Given a power series

a[0] + a[l] X + a[2] + a[3] x^ + . . . + a[k] x'^ + . . .

If for some number x = R, the (infinite) list of individual terms

{a[0], a[l] R, a[2] R2, a[3] R3 a[k] R^ . . . }

stays bounded, then power series

a[0] + a[l] X + a[2] x^ + a[3] x^ + . . . + a[k] x*' + . . .

converges on any interval [-r, r] as long as 0 < r < IRI.

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".Ipjqiib Mul>rr|a)ni îlJ

Ifipohibini Ofl) ni tJn>buU lenii‘^5R ^

f i t»iIT diqiiiuTq *'d) l»oo}£t»fcifloJUTi

jtiii io olqibiiiha yf' ,zm tirjotfnq ad; âbi-côa

n| nw! .'**• * foi /«d

aiiia to9^ 5 taa/moa )oii bii) btifn*^

no'iaaoftoaîui adJ /ii*»i c»i^ 5: \!StfTT lars^'m. c^*}' A

r- •;!«»' i^vnoj adj ju&d j;D y<«n>'^titff5

:mi9iik^,d * stoai^S

.'! Ip-. ... i,*rratvi ?ti'f %ixomft innbu'U a«o ^ffihawT -

iM' ijrif'^ ailif jiir. bÛ'i atffbaaoTq ^niqqnfri Kpoooa aJjf <ii

iqojfio'i r.a*q<)J5v“ab Jirodft RgiuiaaV ‘t-fjî i£»l baii'-f

.'.uauxi^j I'/U .^'MiaiiiqqA .aidr.TO’/i;^ ipaw^rjaaaijtc

viii q ^n'^qam iqaaooa miti ;;^bian»o

bcjuiar V 4iTi‘;':n^tr i^d.no- aiJ) j«i1j oÂ J ^^»^^*

0.4u‘d</iyt intfi :îa..Wi.Twn b) njsrti tdillin .arabi bniiv^nciliiLdl o*

' b HaP k oJ bswa* •ojU

,UtiaOut> :g«iad Oit »wv/ ii»di a»a#aa<5 «q«in

Jrtat)i35rb oofi 4*iii|ri!^fsib

'

-.i'

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. . .• <?e fjiJ# i- . > > *t l?St| * W (IJ* i^ (OjA

la fsli /»dai1id*) .55 - 9n^s toI

1 . M {i> / . < i«*q e^/\j| Idviroii .r>t1i .6i*?''JOp

. t .. y4 '/ (Td *ji ujjt t J in» * f0]« I

,

JJti > t 1 0 M |iK>? cf t ,1-J ljivi:»}0' n‘i

i. V. ?

¥ Ljb.<

66

Table 25

Mean and SD of the Students' Responses.

Classification Mean SD

easy (1.0) -- difficult (5.0) 4.1 0.9

waste of time (1.0) — worthwhile (5.0) 3.9 0.7

memorizing (1.0) -- thinking (5.0) 4.5 0.6

useless (1.0) -- useful (5.0) 3.8 0.9

trivial (1.0) — challenging (5.0) 3.7 1.0

diffident (1.0) -- confident (5.0) 2.2 0.7

one way (1.0) -- many different ways (5.0) 2.8 0.9

symbols (1.01 - ideas (5.0) 4.4 0.7

On the other hand, the students generally expressed neutral

opinions about whether the concept maps were constructed in one

way or in many different ways, even though they were told that

their concept maps might not be similar to the concept maps

proposed by others. This implies that the students were so

accustomed to uniqueness of the correct answer that they could not

accept the idiosyncrasy of concept maps.

To sum up, most of the students conceded that the concept

mapping procedure was valuable. But due to the lack of experience

thinking about the calculus concepts in-depth, they had difficulty in

developing concept maps.

Interview

Among the six questions, the first and the second questions

were asked of both the C&M group and the traditional group for

comparison. The other four questions were addressed only to the

C&M group.

First, the investigator showed the following graph to the

students in the C&M course and to those in the traditional course,

and asked "Does the slope of the tangent line to f(x) at point (a. f(al)

represent the derivative of the function f at the same point?

t )il

r V '

’ " n ilM-

1'

:^m>y^ '2L/^bfjj2 aJj iQ. Q2 hfiif flffaMV *1

e-p-l

r

V?

iiZ

0 0 f

A u > :•

( (t «.C

f

.0

6 .?

.xiL

i-*

(ru) - <ai|

(Oir? ^[rdwdiirt>w -- ^0 1) >* 5l/«w 5;

(0?) - (OJ) arrtsfK)«i3me

\0.ty ial9za — Î0. 1^

' U0.^> ii»l|!*i40i*»b « (Jj) -U/hi

lO.t) (fvll) 3n^tJn>ib

*t:8‘ * icJTi'liib -• (O.i) vi'^ ^nd'

- ubuJL..i:it5tiLrLiO.

3

4

iltli!9ii ir>ef *>111*5 lUtjyB^ »?di <<0

9<20 ii? j oiirfu; <qixK ^mi Jf/erAt «noiniqo

<C'1J ilfol V4»d» • tti to Y»W

U ..71 :*'*-«> -»itj oj uianii v5 mo id;>^7i wfsn:

m sm',7 "Siif l*dl wilqmt *mi ,iti>tilo xd ln?oc|ojq^>.*

.

tofl b'u'.:> lawcrt* icft-rtoo «»dr Viî«*3ii5tipii4d > 0 ;

b 2bx>i>oo V IW>C2 ,qt» nrnii oT~ - 9H.

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id ff&Jtffobiu ail luod*

r.''^’T--V

itti-onF.-ii;!' to*>^ ^ ba4 wnl ^jiu .ifi 'UeMfji xj« ©d) jjaocsA^

ait qôxj i U;t3 ;ra bifja quat^ Mibj died to esw

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.,

^ ' ' ‘

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** 1;4*:cilibi-’l •n) «d dtOdi.oi ibo*. 9dt af alfiVÛlJf

'"k '‘'»-

J

y = f(x)

67

Fi gure 12. Graph Used in the First Interview Question

Most of the C&M students answered affirmatively, while the

responses of the traditional students were half positive and half

negative.

The reason for the different responses might be explained in

the following way: the traditional students were first introduced to

the derivative via the slope of the tangent line, and then taught the

more formal epsilon-delta definition of the derivative. They

identified the derivative as a slope of the tangent line rather than by

the complicated epsilon-delta definition. Typical examples of the

tangent line in the texts look like Figure 13.

Figure 13. Examples of the Tangent Line

Thus, the traditional students were not familiar with the graph in

Figure 12 as a tangent line. All these facts probably caused the

students to reply that the slope of the line did not represent the

derivative in the graph in Figure 12.

On the other hand, the C&M students did not learn the

derivative as a slope of the tangent line, nor were they taught the

epsilon-delta definition. Instead, they recognized the derivative as an

(ij*'

X

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*.t:LiFf adt of-bstU dqioD ^_mai3' t..-

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r; 5fl|im <0î9tlib wl* lol .

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>(i*. 1»* noitl’^'iy^ i*#ino> • loia

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68

instantaneous growth rate. This simple introduction might have

helped those students to answer correctly.

Second, the C&M and the traditional students were asked

whether the integral of a function over a closed interval is a number

or a function . Most of the C&M students could answer correctly. Here

are two responses given:

r"The symbol f(x)dx means the area between f(x) and the x-axis

between x = a and x = b. It is a number, not a function."

"The integral of a function over a closed interval gives us a numberwhich denotes the area underneath the curve of f(x).

In sharp contrast, fewer students in the traditional course gave

the right answer. The probable reason is that the real difficulty in

hand calculation of a definite integral lies in finding an

antiderivative. Thus the traditional students are likely to lose the

central idea, and answer that the integral of a function over a closed

interval is a function. However, the C&M students usually can get the

result of integration quickly and easily by Mathematica commands.

Third, the C&M students were asked how to solve maximum

and minimum problems . Generally, they were successful. Most of

them precisely described their own plans for approaching maximum

and minimum problems. The responses of four students are as

follow:

1. "First, find the quantity to be maximized or minimized. Compute the

derivative f *(x) and tlnd the points of f '(x) 0. Then, evaluate f at

each point of f \x) = 0 and the two end points. The Hnal step is to

answer the question posed in the problem."

2. "Compute the values of f at all points f *(x) = 0 and at the end points.

The largest of those values is the maximum value of f and the smallest

of those values is the minimum value of f.”

Therefore, they are apt to consider f(x)dx as a number.

3.

"Look for zeros of the derivative. Then these points will give you the

highest or the lowest point."

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.»i il »jiBiitr Util *«tS ’*tli ti a>»r» aididovj

oa m wiJ Uiii la^. «n»,i3ali *5 if bnsi

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f

4. "If f '(x) is positive to the left and negative to the right, then the point

indicates the maximum value of f. When the conditions are reversed,

the point indicates the minimum value."

The first two explained the complete procedure of solving

maximum and minimum problems with absolute extrema, while the

second two addressed only the part of the procedure concerning

relative extrema. A common fact was that the students did not use

the term "critical point." Instead they used "the point of f \x) = 0" or

"f '(x) is positive to the left and negative to the right." Furthermore,

about one third of the students used the expression similar to that in

the fourth answer above: f '(x) is positive to the left and negative to

the right, which is based on visual image. The students worked on a

variety of maximum and minimum application problems—greatest

and least, shortest and tallest, fastest and slowest, largest and

smallest, and best and worst—with the corresponding plots. This

plotting experiences might be the reason that many students visually

interpreted maximum and minimum problems.

Fourth, the investigator asked the students: what is the

Fundamental Theorem of Calculus and how does it establish a

connection between integration and differentiation? Almost all the

students could accurately verbalize the connection between the

integral and the derivative in their own words even when they failed

to recall the whole theorem correctly and used erroneous notations.

One of the interesting facts was that three students could not

remember the formula of the theorem, but they were able to

reconstruct the appropriate formula from the consideration of the

relation between the integral and the derivative. Among the students

who easily recalled the theorem, no one stated the condition that f(x)

is continuous on a l x l b, most likely because continuity was not

even mentioned in the courseware.

On the other hand, the students’ explanations of the connection

between integration and differentiation varied, but carried the same

basic idea:

"The Fundamental Theorem of Calculus says that differentiation andintegration are inverse processes."

Sill r?.t ^ W £^ '^9Pq Uy " >m *iw#iit55? «!’ *'ufW ,, : - ditb.? t iimi/MOL 9ft3 t^v^»br.i

ir.ttsifiifn 9«t: ^lilsli^ni!i' inf£#qft

^ 5l^? ifnoo Uol^q7.o owl JttQ

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® i • 7 .diawair^b^ rrf ^&oftfr*'n oava

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4

70

"The main idea of the theorem is that differentiation and integration

undo each other."

"The Fundamental Theorem of Calculus implies that if we first integrate

the function f and then differentiate, the result is the function f. Sodifferentiation cancels the effect of integration."

"Integrals are antiderivative."

"The two are opposites. You can go from the derivative to the integral

and back."

Another notable fact was the method of proof the students

remembered. Only two students could reproduce the proof given in

the courseware. The others simply said that you could prove the

theorem by executing Integratelf 1x1. (x. a. bl1 and flb1 - flal with

specific function f, and by comparing them. This may indicate that

the students prefer a verification to a theoretical proof.

Fifth, the students were asked how to determine the

convergence interval of the power series . Most of their first reactions

focused on the use of plots. Many said something like: "get the

several expansions of the series up to high degrees, plot them

together, and watch the cohabitation interval." They gave priority to

the intuitive visual approach. The investigator asked again how to

determine the convergence interval when computer plots were not

available. All the students mentioned the singularity method, the

ratio test, and the convergence principle. However, fewer than half of

the students indicated that the singularity method required explicit

form for the function that originated the expansion. Here is how one

student who did know this condition explained it:

"The ratio test and the convergence principle can be used any time. It is

usually easier to use the singularity method when f(x), the representa-

tion of the power series, is known."

The investigator was surprised that the initial replies of all the

students were centered on the visual plots, and by their pertinent

explanations of "cohabitation" intervals as convergence intervals. But

their responses to the subsequent question were slightly

disappointing. Although they could remember the three methods,

more than half of them could not appropriately explain the details of

the methods and when each method is applied.

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7 1

The last questions were ’’What is L'Hopital’s rule, what is

Taylor’s formula, and how does L’Kopital’s rule come from Tavlor’s

formula?" Actually, these are calculational skill-oriented, not

concept-oriented questions. However, most of the students could

remember both the rule and the formula, and successfully connected

L’Hopital’s rule and Taylor’s formula.

For L’Hopital’s rule, all twelve students explained it either in

symbolic form or in sentence form:

"Under suitable conditions, the limit of the ratio of the function f(x) andg(x) is equal to the limit of the ratio of their derivatives."

In particular, they verbalized "under suitable conditions" or "0/0"

instead of "indeterminate form."

For Taylor’s formula, three students did not give the factorial

symbol in the denominator, which is the most common error:

fw = f(0) + X + xU . . . +

.

They were asked again, and two out of the three repeated the

following procedure and finally completed the correct formula:

f(x) = a(0) + a(l) X + a(2) x^ + a(3) x^ + . . . + a(n) x“ + . . .

f(0) = a(0), a(0) = f(0)

f '(x) = a(l) + 2 a(2) x + 3 a(3) x^ + . . . + n a(n) x“-^ + . . .

f ’(0) = a(l), a(l) = f(0)/l!

f "(x) = 2 a(2) + 3 2 a(3) x + . . . + n (n-1) a(n) x“*2 + . . .

f "(0) = 2 a(2), a(2) = f'(0)/2!

As to deriving L’Hopital’s rule from Taylor’s formula, half of the

students could successfully answer. Their answers were either in a

sentence:"replace the numerator and the denominator by the early parts of their

series based on Taylor's formula, and then divide."

or with symbols:

f(0)x+^^x2 + O[xl’

1 im - = 2

gxo)x + s::mx2-.o[xf

f’(0)

g’(0)

The students’ good retention of L’Hopital’s rule and Taylor’s

formula can be explained in the following way. The C&M courseware

provides the simplest form of L’Hopital’s rule:

G3

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'.fniol 'boi

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itOTT^j horn Oil) ii aU) ni io^ 1«rx%e

. *jt .(ju^,..i ^ . .- V K i'/ODj •» {*)^ y

•r*U- j

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id' ‘tndi -n?7-ô ufo owi ban iiaiii fadr*

'‘.*l2Jimol")ârmo ad» iijiatqfliot, bdi stiibaoo'q ^tâlvyofloV

... t- *i (>v« (€)i *' * * * (^)4“ * "

-;,f ^ »1L Kt 4 ^ ^ *^X £ 4 r (£> ? 4 (J)tt -• (jt>* X

^ iiv^rn »» (tM • (0)|^Jl ,

.

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artJ lo s'lniiqbF'J jnlrfijsb :>) li^ -

p ni ^raio ww iL^vtiftk jJodf fsvpiiuP »>baia

^ /*

bn® w^siadssw ’?dJ ' <Alrpat‘ 3

".(X.M.'^. hO!» y«Uf f^-my »Vb*T tw*«ef ^ ^

-'i.l,*| -fe-'.'.-'- , . 6i<0, .

5». .:sl(;il«>*>(»..'ifriv JO'

' n,a«-V Lm f _zl—;—^

>

il

jll0nrfe4i>a^'y , /kW JtîwoUo^ <^ja tti ad -lao tiurorrol^

^ lilt m nist<\ iMamU idf fvoiq

72

Suppose f(x) and g(x) are functions with f(0) = g(0) = 0, g’(0) = 0, then

*->0g(x) g'(0)

instead of

If Jim f(x) = 0 = Jim g(x) or Jim f(x) = «» = Jim g(x),

then lim = lim^

.

Also, Taylor's formula was reduced to the polynomial of f(x) at zero

without remainder:

f(x) = f(0) +1! 2! n!

instead of the polynomial of f(x) at the arbitrary point a with

remainder:

f(x) = tk = o k

n-*4)

(n + 1)!

(X - a)"** for some z between a and b

Conclusions on the Interview

First, the C&M students more clearly understood the nature of

the derivative and the integral than the traditional students did.

Probably, the simple definition of the derivative as an instantaneous

growth rate and the direct introduction to the definite integral

induced their correct understanding.

Second, when the C&M students spoke about the concepts or

the procedures asked, the wording was often clearly their own. Also

they preferred easy expressions such as "the point of f '(x) = 0" or

"f '(x) is positive to the left and negative to the right" instead of the

formal mathematical term "critical point."

Third, the C&M students could reconstruct the formula when

they had a deep understanding of the basic idea behind the formula.

The students who forgot the Fundamental Theorem of Calculus could

figure out the formula because they firmly grasped the connection

between integration and differentiation. A similar case is Taylor's

formula. The students who recalled an erroneous formula, repeated

the derivation procedure of Taylor's formula and subsequently

corrected their wrong formulae.

fSO

r: ^

•j*l* U - ‘#>4 /B* ^ €r.oi>a«u1 rt <3r‘>!| as- <j4Bt^

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***“ ^‘ JL^ (nd » lLLL«jl gpdif

.m \t lu »A^‘i‘on'{loq 3rf> oi bo:>ob9i ir*r gijamoi

iob(iM;<nndt \ IfrOsftlit;

z ^ vo)i2 . ,im^(fl IV l| '

TL,

iS

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. i. ‘•'tf‘i»* to\ %) -1^— ’ k)£^2i1(1 44 , i ^

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:. gr ' '

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ni»li4 )9i»«ii ' ; ti *>rxi<Vt]C>b Mii V» oorli^xOab o(<|ntu s»(ft <vWttdof|^‘*

^ 01 (^botflxysias iîib v/iJ |s4#t

^ ^nlbiJ.JljniCiic » :

iO ' TU<Xl6 <0<apiJ*l • ^'‘V^^W

(mlA rrwo ibrti ctdllto %Tib»ov 'V'tJ J «dt

10 “O t«‘(x)f^ V io io?aq a‘‘‘r liatjn icftyt^ ‘•t y^3 i{>

joii »di s>vii*%*4ii Imj j 4)«* *fli 03 svliltoq ^4 (x) *ij

*.|#|i:iq fwljrfe"* '-âf fiiobf.'tiaifl*CT;

tdmtmi^y jiil i»iiC5b*^g lÂD

wciminot edi r?Hi d ••fO lo igaibiTfeuvdfc^y^ q^‘ 7 <di(»

Mm'6:> roiJt'iUO mjîpadT Un iniuao'^ sUJ Jojtol odw eirdabuf* ^(TT

d;II \Mir o¥xrjt^>'^d fffurmo's. aiO »o»

' tiîrbii noni/iîiata1l«t biw aoiJaijiaJm rw^wiad

iw»»rf^s£jai ^Jo<!iio1 i*»02r/iorit^ oj o*iw ^Wi^hiiÊ j^<lT Msinhol^

4'«

iwi# îts&i^ iwlxs»y Jo arHiti»w*i qbi*»v!*«vb

73

Observation

Details of the Observation

The First to Fourth Weeks

On the first day, the instructor started his explanation with

very basic procedures: how to turn on the computer, how to start

work on a lesson, and how to do the homework. The first assignment

was four problems of "give it a try" and nine questions of "literacy

sheet" in Lesson 4.01: Empirical Approximations. The investigator

frequently stopped by the lab except during the scheduled

observation and found that many of the students stayed in the lab

late and struggled to complete the first assignment. Predictably, they

were nervous about working with computers and about the writing

component of the assignment.

In the C&M course, the choice and the order of lessons

depended on the instructor's decision. The first lesson in the class

observed was in Empirical Approximations, the first part of the

fourth section (Series and Approximation). Other instructors usually

started with Numbers and Algebra, the first part of the first section

(Starting Out). Both approaches seemed to have sound reasons. The

former approach includes many plots that demonstrate powerful

graphic capabilities, while the rather easy latter approach can reduce

the students’ difficulties at the beginning and encourage them. As the

instructor expected, the students appeared to be attracted by the fact

that they visualized the meaning of interpolation by producing their

own graphics.

Generally, it is believed that students are more capable of

understanding concepts in geometric forms than in symbolic forms.

During the third week, the students learned two approximations: one

by interpolating polynomials and the other by least square

polynomials. First, the instructor outlined the methods of two

approximations and explained the main difference between the two:

approximations by interpolating polynomials using running

polynomials through data lists, and approximations by least square

polynomials using running polynomials near data lists. After that, the

students had a chance to plot and compare two approximations. This

experience was apparently helpful for the students who could not

4k

-’tfl

.

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Ji ‘rt adT ab o) v^-od tfia ,m.'^«.ol ^ i«*

.c^s tr nohw. bna ’’/Ti £ »> >o Hiri^d(Mq jutil

. »; *^rit3viii «dT .»nabaAniioT</qA L .J0> ft?

bt: 9»l) gniiob Jq<wx^ tt#J ^di ^6 b^sîqptf^;î3a^sp^

fiftl %dl ni b^yiJk i)ab!>ux4 aal lu ytttfrt i.di -uuoi bn£ noliivi^^do^

ycurtl JrJ) «/l> 8>n>lqmot> oi Jb^înUi tft£ M%t

9m lood* ruiA 4'. a»n/fJO'J dil# gfllilO^ iiOOnsfi

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.di lo TiA'j ?n 7>, alii _.v ' . mu^qqA tauhrima ac .i£w bt> rpiido

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edi hfitKt o' di'"»a jj»tLif;J8)

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dOtttiOT ftCd d I'vJlil 74ti3 1diil£l alidw oldo«Tg

«/ dÂxiiOr: 1 bo* “ainr*.4g0d oil! i£ gabino?Hit5 »dx

odi \,d.

^ usi#t::'qqi airtolwlk

f'diit • nr -.-td poiufc^wini^lo ^niiM ain^ # yodi ii:îiJ^

.drtsqc ; orom n« «in^‘boiV Udl bi'r9ilod ii ’>

^ c^dt Ermc^ <ll aKjoOti?C*o ^ntbodl^f. bnn

c**?^,*^ ^fnibajn atii fcniiil vttHdQ

wopt W*bo 14J» toi tf/naouyjoq t*^^^46qînr yd

04f^ io eb05fbi«t ^ âiJriio iorî^dtsi ^Jasi'J 4fl£}w>rylo<|

t'dl ^fti4-^wiiMJ ftilj<(i orii fit,<aitlqxi? bnw vjioi*£fltJ<*»‘Tqq£

# jjd'QOJi gabo Hjsi'mor^lv ; v*'at4.<»»c|;idi04 yd «cwjitife'?or^*

>iii I yd RiK)M&mlyo:\qq6 bna ,tlaU - fi^tfvuD elutmontl^^

.li4i uiM mil fiifib wq xUnmOiTytoq iwin.rin anhu iIau )»1yloq

fidi -#Ôl3»-r4£Cît;js trwKTmoo boJi H>lq ul « Ud »:>4

irjif oii# clin^wqqi .W'w »«sh

traditional explanation. Even the students who had already

comprehended seemed to reinforce their understanding by the

provided plots.

Polynomial Curve Polynomial Curve

A similar case was noticed in the learning procedure of

geometric series. The instructor explained the basics of the geometric

series (—— = l+ x-i-x2 + ... + x"+ . . . ) in the classroom discussion1 - X

session, and then the students started to work on the C&M lesson on

computers in the lab. The idea of approximation was primarily

illustrated by the plotting process. The screen subsequently showed

the plots of T 9 , 1 - x^ + x^ - x^, 1 - x^ + x"^ - x^ + x^ and^1 + x^

1 - x^ + x"^ - x^ + x^ - x^^ + x^^, and led the students to compare

them. By watching and comparing the plots, the students could figure

out that 1 - x^ + x'^ ... + (-l)"x^^ converged to^

n increased.

Some of the students were not able to understand the main concept

until they saw the plots. This observation led the investigator to

confirm that the students' direct plotting experiences are more

effective than the explanation with blackboard and chalk in the

classroom. Furthermore, those plotting experiences appeared to act

as cognitive bridges to the abstract understanding of principles of

geometric series.

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- « ••' ^‘

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:

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,v,'

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ÔCT) iSlIJ *!-7p^n^W. lasiib 'fciilflbCUi UilJ

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rjj o^ 1>i$i5l|9qxa

. . i^ iifqbphq vr gittbW^Wftbqji/ qxl»; Pi ifv()iaM‘V5

7 5

The Fifth to Eighth Weeks

The students approached the C&M lesson according to their

study habits. Roughly three different patterns of learning the lessons

were found. In the first pattern, the students did not follow all the

problems in "basics” and "tutorial." They skipped the rather easy

problems and moved to the next problems which were challenging to

them. In the second pattern, the students proceeded through the

lesson step by step, even though they did not have to work all the

problems in that lesson. They might have been comfortable to learn

old ideas in new ways. In the third pattern, the students directly

started from the "give it a try" problems assigned as homework. It

was inevitable for those students to go back and forth frequently

between "basics" and '"tutorial", both of which gave clues for solving

homework problems.

The investigator expected that it would take more time to

complete an assignment when an individual started from the point of

"give it a try" than when s/he started from the beginning. However,

the amount of time required to finish an assignment depended on

one individual's understanding level of the calculus concept, not of

one of the three study patterns. Every student had his/her own

learning style and could be categorized into one of the three.

Moreover, the student's learning style appeared to be distinctively

set with no crossover in any case observed.

One of the important concepts the students learned during this

period was L'Hopital's rule. By using the command Series[/tf/ic/io/i],

the students learned the proof and the various applications of

L'Hopital's rule. Unlike the formal and strict proof in the traditional

texts, the proof of L'Hopital's rule provided by the C&M courseware

was simple and clear.

In[l]:= numerator = Series[f[x], {x, 0 ,2}] /. f[0] -> 0

Out[l]:= f[0] X + 02LJ^ + 0[xf2

In[2]:= denominator = Series[g[x], {x, 0, 2}] /. g[01 -> 0

Out[2]:= g’[0] X + +0[xf2

J

^«4ii uf fw«8ol MAO ^ ft»rt;w€/rq9»^ j

waoi4*l V> tfi'/ig'^lifc aânrls *4id*;d

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i-ifirj b')s|ifii ’Uiwlxi hiu '«f/4d” ni î-iiardon

oj 'i tirfw tiu^^WcTii ,|:x^4*t :>/U or ’>na fin»W«?jq(

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.ia ion, fefo no«idl

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i»»iJi cmiibci^ !»di nTaaitî biim o«U p\

if iiic; iTOf' tt bonîsaa fi ^ iroi^ otmU*

xVic9ifp^t iiri.ri haa ia#'f OS Unabtji^ wdj lot arw

;4irvio4 To1 «oiii ’'vfj d^vr id* narA^yJ

*l;Ori»lDOj ii'»a'W_:'fTlOff

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tNlo nif\«id bid fotilima \’Tr''3 o^vfll ddMc

>*1; '.»ni> omt b^vQoaaiip ad blcoo bm /Xyw

tiavii;>n sjlt oJ tauaqqa sifjft iMjnaaf i1n!>bafa oiir .iivoef^M

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i. 4»iic,iJs .:iHqx{ ' wU biui Voiq arf) •fftsbjilt «!|"

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4fu uîooo OabiY^:?^ ofur ytfiiiqdH.lîo bxnq ^ .li^J

s -< -tof* A uti « r«>

-'; Cs. T '0

<- f«'j \ Iv'i trf' .twISliaHiV: » mk*Joic»c^ll^

o » r fij’s ».\CVuQ•Si '

F«'

7 6

ou,[3i = noi ^ (Xm . f[oi g"ioi) X + o(xi^g'[01 2g'[01 2 g'[0f

Sending x to 0 gives^

*^0g[x] g' [0]

One student (P.K.) said that he had already learned L'Hopital's

rule in high school and used it to solve limit problems. He added that

he was surprised because the proof is so easy. Most of the students

seemed to understand the proof of L'Hopital's rule without difficulty.

The investigator realized that students are not always afraid to learn

the proof of theorems. If a rather easy proof with appropriate

difficulty level is given, students are probably willing to accept it.

The Ninth to Twelfth Weeks

The later part of chapter IV (Approximation and Series) deals

with somewhat complicated topics which were not familiar to

students. However, as the students started the new topic in chapter

III (Integration), they were refreshed. They liked the simple

definition of integration; the C&M courseware does not follow the

typical order of introducing the integral: Riemann sum, indefinite

integral, and definite integral. Also, they seemed to feel comfortable

with easy explanations, which were mostly done by visualization. For

example, in Lesson 3.01, three basic properties of the integral^ were

visually explained and then confirmed by choosing specific values or

functions instead of strict and formal proof.

The investigator had a chance to talk to a student who had

dropped the same course with a traditional approach during the

previous semester. He (D.S.) said that he had dropped the course

because he had been intimidated by the intricate procedure of

introducing the integral and by the laborious proofs of properties

which were intuitively obvious. He added that "there is no reason to

prove a property unless there is some doubt about the result, and I

rb fC ,b ,b

fix] dx = f[x] dx + f[x] dx K fix] dx = Ka C a

f[x] dx

f[x] dx = f[t] dt

V

cy

IT>0 ^ ^

Wi I 101^•'

5« , iO -*V VN.' ,

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id> >0 J«oM Of «l VwTtq ^rti aavn^-xJ aiJ

iL^^/iJiw iolm )0j TfJ b(t|U£»9b^

irn^si ji b^fnlt m emabK ' isits ijifl'

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^ b»<teirt>Jrt '/ Mil .1^ jll

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*> -

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4

-1

77

have never doubted the properties given in the text. Most of the

proof is nothing but a bunch of equivalent statements and they don’t

do anything for me." From these statements, the investigator could

conclude that he was satisfied with the brief introduction of the

integral and the intuitive explanation of the basic properties

provided in the C&M courseware.

On the other hand, the formula of integration-by-parts was

introduced as a simple consequence of the Product Rule for

derivatives. The first example in the lesson showed this procedure

with a specific function and value. After that, the general formula

Juv' = uv

-

Ju'v was induced in the same way. Based on the

observation, the students easily understood the formula and strategy

of choosing u and v. This indicates that students generally have less

difficulty when the concrete examples are provided before the

general case.

The Thirteenth to Sixteenth Weeks

The lab atmosphere was such that as soon as one student

figured out something, it was shared with nearby students and then

with others throughout the entire lab. It was true that this

disseminated was usually a method of how to rather than a complete

understanding . Nevertheless, whenever a student explained to

someone else what s/he had understood, the act of explanation

appeared to contribute to his/her own understanding as well as to

the understanding of the others. For example, in problem G.7

.

2lJ in

Lesson 3.07 which requires finding the center of gravity, one student

(M.V.) explained to the next:

"You know that at each distance x there is a corresponding mass. First,

add all of these values of mass times x, X 5 e * dx. To find the center of

^ An infinitely long straight rod of variable density is laid out on the non-

negative x-axis. If its density is 5 e * pounds per foot x units to the right of 0,

then what is the total weight of the rod?

What number b > 0 would you set so that the weight of the rod to the right of b

is in your opinion negligible?

Where is the center of gravity?

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j 0</6 cwi« t'«i 5 1o ?:»iiud U ltf<! aouHr-^ |o<S>TCf

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’ ad «idr obuJ^»ooD

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.Me vw antt'oo adi n\ babivoiq Û¥f clo«.. u 1 -dio ^ p|?V

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» dJtW

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I*

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.V* *

.

yaa «U8 ii/», ftlumÔl aif tKK>i,'-abnu -Utm^ jjbato ndl .iiOU»vTjl<fe -

woi 'î’*'i5lXPa "int'Otfl® i*d< lijS^^fbin stifl bjti ‘ Ho

y»oHod bsibWiyiq *» ta;o>fic4 5 rntocao 7*^

jdJUisaiatfi ^’• .ô«> vau .. iJid a*>iP» oiorfqwiritlB OnJ MiT ^ 7

if-j ! bui y<fta»l di^v# bourtt e&'^ ii^$iLMi»frto:< jo ;

.>ifj md’ 2UU akî L ,dil luodjuoif^jf ;nri<Uo A)hff

aiilSK’ifi-'uJ nu'lj 'o. jRi ylUoRu a^p ,hm^tPimd>

on Sa. ic'qxa ioabo'r b‘

^ adUjLûlqxa Ho boofmbfto bfH ’>«V. Mioaixijii'

0) *,ii volbKî‘’at>ei> /r^/o *?u\2 id ^ 9>ud»uiTw'> ot*£*»i8aqq^

i.i ^,$S,0 ol ,9lqmi.X9 aib |<^ri«iboa5t-rwiw ac!*.

' C 6trt> i*no .yti.ia^ »o isiuar tdJ l.dtsfi*' wlnrw ûrdw TO.r* ao« ^‘jjtw Oft> oJ w ydkiqia t y,H)

i-njil. <wv

3niik)oinrv.{»7 4 rta&’ gm

} fb bcil ©T » 1 . Gtttfl to »TiM )n> Ht 11151'

:fer'

. :t\

îr,v'

CJ

#..

«»dii

,;> • iijusl .tf ^farat MtUiaav to bot id^Lu.:

.0 Jt' VxJV Wf *tao^ *-a ^ VI yilivttb 1 ! tJ J»‘X ovtiljp^<''. » 10 fdii^v* ImlOf Od' ft le 4t HOm* s

<) »«!$*> 0^ bfa id. lo odi omU 1 7?* of)y utoow 0 < d »*'j»t*

" V. toi4tii4iaiMaoo *> at

gravity, you need to divide this value by the equation 5 e * dx. So the

Jo

equation for the center of gravity is X 5 e * dx / 5 e * dx."

His explanation seemed to clarify his own understanding as well as to

provide new information to the fellow student. The formula

I.

x5 e'^^dx / ISe'^^dx spread to the next student, and on to the next

0 Jo

student, and finally to almost the entire lab.

The application problem about parametric planets in Lesson

2.08 (G.5., Appendix F) appeared again in Lesson 3.08 (G.8.). The

former lesson taught this problem with parametric equations and

their plots, while the main tool to use to solve this problem in the

latter lesson was integration-by-substitution. In other words, the

students were faced with the same question twice and explored it

from two the different aspects. Revisiting this tough problem and

considering it from various angles required hard work of the

students. But this experience seemed to provide the students a more

global view of the problem and the understanding of the

interrelationship between the two lessons.

Providing frequent feedback in mathematics courses--

especially for example in an introductory calculus course--seems to

be essential for correcting students' mathematical misconceptions. In

the absence of such coaching, students are likely to develop a

misunderstanding that is hard to undo, and that may become an

obstacle to further learning. The students' misunderstanding arose

during the learning of integration-by-substitution, illustrating the

importance of feedback. There might have been several reasons that

caused the trouble in the substitution procedure:

b Mb]f '[u[x]] u'[x] dx =

I

f '[u] du. Among them, the primary reason

A[a]

might have been the failure to understand that:

I

jiil oft ’t>

IS

OJ 1.

- ,t

1

**

! «C*Kt;o-: ootiv -irti itvlytii o» b^s« uo^f .'(liv'Erji

. — • • i7fc

ii .

*‘

\ ', ‘'ll <^3S

*

i.^

r.

* ^ ^*

tâ^v >r ;>r (b^liUrtobtiu nvtfO '(Ihi.ij n5 'itf' £lH

.'. u'Tnî ja:buM woH-jli rJi ol onr-Tô^nt won oSwiq

H} oj no l>«u^ HH-sa Vdt <^1 U \ a

‘ (ta^ t'Uma aiii MOmifi r*t li iabuj*

Ch>rr J rtl i^*URlq oro^msi^q luOflA mjldoiq oohdoi^Qqc

ouT (.K Oi «0.l r/ 'V. I (*T xihuaqqA V.t.O) 30^,^

txftir ?nofc*H/^ jhJ^rjKif.q r?T'‘** ghU lliîf^ ftOlusljj r-into^

mU aJ '•i")ftfC'Ui oi ifit n[ U : aîm di'iklw .''jiolq ôdi

-'.•<tui fil ’tmiv'ii; *ûv (rl TSlI^

1 bar ^|^‘-» * nmtijjjp wiict • '•f bajiH f5ije»bui?

bT* mtloHil <* v< !jnijini )*.;! » tu^wStib •im oWj imy\

' rti* ''> yK* v CHiTit L:Mi4>prji • »J07S®^ ‘rtinl h '^ftfUfbhcPb^^

iMt^ >: at’ui}' "bi'/’Tic OJ

ill* ’iv> ^n;brEHJ2pbtio n1> bn« inaivloiq Mfi lo

* ? .airm<*?f3! o>?* aril irâwio^

'•-eoiiuo^, rfbftiH>ooi >tjattoaa1 0

‘r "iit

I

01 '^n§32-- ifiiiitio ?uii/‘''iB'' x^c.*ibul>o*iftf >'*•' alqiT^fiX^ iCt

lo .rriorii

lft:»{i<uii«4-^m 'yjnabu. * 3udO!?Txo:» lol ^‘

i nqr^vSb fsi /Mil J^nabiii* 4;nfB lb Jl:n^B 3 Ifilil Jlîb<lfi^?rlobuu<:irn

'j' >'.<;i wTiith'^t *jo3bti,V5 '^'il > ^ntmxoJ lauiibl oi alosj- cfO

,, -fij îiUiinBuU' noi^bidadc. rohfiâtn 1o ôimaar jitiiob ^

'|ri|il|||j| / Lciovot- i/fld idgi</\ «)||t‘‘rjci<j{rni

'4^S4i

jS’

•q%ubou^\iJ «obviiu?dair 5jy al a^djott

j . i<n«i

fiS'i

^ -

Mull i^xtilJB'ioA

/A nj .- . L

xli t*]’v f(xji/1\aii,di tf^nafeflu <H riuUtoS "iiii av«^’ id|im

S' X'"

In the integral f '[u] du, the symbol u is treated as a variable, and

u[a]

u[a] and u[b] are treated as numbers;

In the integral

/•b

f '[u[x]] u'[x] dx, u[x] is a function, x is a variable, and a

ja

and b are numbers.

Several students confused the above facts and made major enors in

the substitution procedure. The instructor immediately corrected

their misunderstanding, and seemingly, their confusion disappeared

after the instructor's appropriate feedback. This would indicate that

the instructor’s immediate and frequent feedback in the form of

conversations in the lab, comments on assignments, and meetings

with individual students by appointments are effective devices to

remedy student misunderstanding.

Su ggestions from the Observation

The main purpose of the C&M courseware is teaching calculus

not Mathematica. The complete understanding of the Mathematica

commands used in the lessons might not be really necessary because

Mathematica is just a medium and a tool for conveying the ideas of

calculus. However, the investigator wants to note that the students

need more understanding of the Mathematica commands. Sometimes

the students copied the command lines in the previous part and

blindly used them. That problem may disappear if more time is

allotted for the explanation of the Mathematica commands, or the

brief glossary of the commands is attached in the printed

courseware^.

Solution Files

Findings from the Solution Files

The students’ solution files were a fertile ground for indicating

the depth of their mathematical understanding because of the non-

routine and concept-oriented nature of the assignment problems. In

many cases, the types of understanding shown on students’ solution

^ The students can find a concise explanation of the commands by typing

?? command. That inquiry was not utilized during the observation.

1^(t ti t) if. t6<Jmxi c^rfl u/f? fuV \ ] hmûAi al

j*a' l*V*1

>l>drauci ftv bvisjn »i« 'dji? hnt [tli

#4

MfiSSm* •* » . Ji ,rfj UVa l. ^l=r j eri»

’• ars* d b<Jt"

i»3 i4fn tifesm Msfli »vodA oHi b3«rV»oa

^00 'îîsitvîTtm r»ffT .cmibTJixrttj ffobu3md«i«

'.»>>* roi?a1fto5 ^loHl baa . . fUblrt<‘?'Tdbn:oafiij

•t t» flj-oitv 1 t .uow ir<rr\:drJtdl>?iT Mr i

'R30iq<t* >di

1o n.iol Ddt at )o»M#pir>i brui Maib.^fn,rnt adi

>ifi bi‘^5 *»r?miig(-ve rto e{f*'»rt.*r.oo ,dfi d/li ai iOot)itiavai»vT

M

13 3‘^d bi 9Ct»q^frq (Ui^

i.' UBw »A*^Vi *i((j Vi aniboAJ' âlqincw^ ^ ***^'* ™ g^ .-~

.)^vj«3*wi <T3i.*>:4»->a<T vi )ou Jci§io Faf>âJ ^fli f'

'o -Hti ^mv:i>tifio tol looi « fc>JT^ muibam « tz?it 4l , ^ >hsir»» |wadi .irfi ^)oii ol'«Jn4w u>Jfi|b29yax ads

lainiJafiiOt? Ckal\siw%<Vu^Vk odJ V? j^flibfiSU*i5blti? aidftt bôn ^'

Zi fii»q adj di b/iri^mca adi ^44*»bo3? ^d*

ti amlf ;y%pm )i ia.7 :qaît) yErn rr.sidotq laifV ./rtatliyboan yIbnHd

adl to ^4bo3;jnrtrndr5 ivVUdmii^^juH ô coiscflalqja =adJ ir' Uoirofli: s.^

boiriit'i aiJ r<i baibfUfi t! cbii««wr-Au Hr?' rmk Wwuo^’ife

•K -wmf - (îI±jss2iMQ^Lm ,aigQ tagAbm ;

'

- _

w $ maw. «^!iT ijoauloa aaoabtfii aiff-

•iô« adJ s«oi'-»ad âibaEJmbad ibi^ rftqab adl

»? RfrialdoTtj fno*t»r adi aw\tin batnaiTO'iq^^RO'^ bdE ^fliJuot

anhuik>i» 'e»«‘ji3Mi «o nwpiM! aaibuiî'jabnu aaciy’ ^*a*®3* yncrn

Q '' a ,

“ '

,

~j^'^ 1ijftttfVi yd itf>orc?>»oa oiit • floilxttiifcjJta twifl hs9

K«D/ttys^t(jki H*l) ijc Miilii ^jwi iiiw Mftr ti

80

files reflected a thoroughness and analytic ability far beyond that

expected. The students more frequently proved their ability to solve

problems rather than just to demonstrate learned techniques.

Beyond the Lesson

With the aid of the numeric and graphic capabilities of

computers, the students seemed able to see the global pictures of

what they learned. Relegating computation and plotting to computers

freed the students to think about what was going on, and even to

anticipate the content of the following lesson.

The purpose of problem G.3.b.^ in Lesson 4.02 (Approximation

by Expansions) is to understand that the cohabitation interval grows

as we use more of the expansion. In addition, most of the students

noticed the existence of the convergence interval (even though they

did not use the exact terminology) as well as the point of the

problem. Even the student who received the lowest score in the

assignment reported the convergence interval in his own term

"barriers":

The lines will never break through the "barriers" at 2 and -2. No matter

how large the powers of the approximation, they will never compensatefor the fact that they are approximations and not the actual function

that controls the plot.

The ability of the students to bring together a variety of

techniques (numerical, symbolic, graphical) to attack problems

seemed to deepen their understanding of the mathematics involved.

High Dependency on Computers

Mathematica is the main tool for most of the "give it a try"

problems. But part of the "give it a try" problems basically requires

the hand calculations, and in these problems, Mathematica is Just an

auxiliary tool for checking the results. One of the undesirable aspects

of learning with computers is that students have a tendency to rely

on computers too much.

^ Plot 1/(4 + x^) and the sum of the first terms of its expansion through the x''

(x’°, x’*^) terms on the same axes for -2 < x < 2. Describe what you see.

tft/0 }>u»^ iii:5HMl?lK)»ii4 B

3/Ioa ot vMtJds b^YMq Yltiia^tKnV int*wnf> 24noU*ie ort’F

.

;aifyt£ul3ik< J>suTWi «J*viSaOfndl> * J?4^t YCâidoiq

' 3#dqt4|i -immuii 5(i|il6

1o iciytaiq UfJolg adj a>« >' boraa9> •srti 4*iaiuq/noo

. m gfl/llo(

f>ox- noijfftuqiboa xpdl iMw

r/i if'tvs bnti ,di.* i«’»y tudw UH>v*fc 3lf^r <J ^losbujj^ 9At baayl

ootstÎ gniwoliOit «iJ ’io

iir>4iMi?urnqqA) ^ u^;->J m <^.d.^ O {iio'doiq \o ^«ocn«q

Urijjqt i>»iii£«d6'io> ndl Mrti bnaiaiabny ol *1 (inot^#qj<a yd• •• V

fjAst>iri« aril lo liooi j50ir*UAii ttl .rn/Uwiqxsi atrl^ 1: snom aw

a. 'rvi'odi ofiYSi Uvjoiai ^l5Aa)JmAua *j»U lo/-^';ciat4Us baaiiQfi

<afb Ki tnru<4 ad4 t* i’^w is (y||0lb4>mv»' 9W t>ib

,d? Ai arts ortV/ :>dl .maldaKi;

.rniaj owe* iirt tn iRravJni aon(r|ji' vftoaârt*

."naiTUSd”

Tt

A

.if»Mi oK t arfi Hj<i4\df iâf* lllvf taoit ftfl

5--r>>v>qi«.u3 1?V4*; liu s’ .nuiwiwfji^viqq* S« mw<ic| orli w^rt

Uii» • ;» ’>0- fiffP AlWihj(mUo>»qtl* •»dl Ittlfl ;ai»*S tol

' .p fq U) d<hiiiferj

To y_iaii4i,Y s >9#iji303 jnii'^ oi aiii 1:0 y^Silids ^rtT**-

wi;4c.rq rt'^wir. (bphlqvij .ailodfRyv .l»^>mouiJi) - ’pittdsal

boYloYfif ?4Uama(li^ a*li Jo ^nibnuriabAo li&iti nâab fcamsaa

'\ii « n avijt" orti laom *^J1 loos main arii si

iTijCiiai *Û*oo^4 cfttîdoTiq "<(TJ i ii avij|‘‘ sitJ lo StiK} tu8 -Jifaaick-iq

fts 5Sii|, fi âi<i5tdf>i

S5

&S

.-.v A

"~si'

i:r>'>• 1-,

fll

“^1 '4l (' •/fawiv* lo 4iav VU 0<»> fA?

' n ?/rinw^ i t ^ ' *\<>i ®d4' >*o *«r»a451 . -* .

'»

•E'M-

E

S'>'

1

# hja M

'E-

8 1

Problem G.l.io in Lesson 4.06 is a typical example. For full

credit, students should find convergence intervals of the given power

series using the ratio test or the basic convergence principle, and

then illustrate the result with appropriate computer plots. But two

students applied neither the ratio test nor the basic convergence

principle. Instead, they visually found the convergence interval by

plotting the sums of the power series up to the nth degree and n+1

degree and then by choosing the bifurcation point of the two sums.

In that case, the students could get the correct convergence interval

when the end points were integer, but otherwise they could not

determine the precise convergence interval. For example, two

students guessed the convergence interval [-3, 3] whereas the exact

answer was [f-^, iT^]. These students missed the point of the

problem because they were excessively dependent on plottings by

Mathematica.

Another example is problem G.4. in the same lesson which

requires the students to determine whether the given infinite sums

are convergent, and to find what they converge to. For the infinite

sum Xn=0

(-If log[2f

n!students are expected to answer that this is the

series for e** with x replaced with log[2]; thus this series converges

to e'^og[2] = 1/2. After that, they can use computers to check the

answer by looking at a large partial sum;

In[l]:= Sum[(-l)''n (Log[2]''n)/n!, {n, 0, 100)]//N

Out[l]:= 0.5

However, only three students approached the problem in this way

and answered correctly. The rest of the students could not recognize

that the infinite sum had the form of the series for e-’^ with x

replacement; instead, by choosing large n, they directly used

Mathematica to find what the infinite sum converged to. In fact this

is an easy problem once students recognize the series e-’^. However,

the students who clung to Mathematica as a primary calculation

executor failed to figure out the correct solution process.

Find convergence intervals of the following power series: Illustrate with

appropriate plots. . .

Ujy^ >5'i >\> t o moWdvr

rrri? s»iî lo tlfivif^ttil i>'>«»jgioi^woo fertH i)!t>6d? ’ ' uuH .ifb'^nv

• Jiitjioffilq oixfid scft vj iâi oh®”f »<ii *^'1^

ov'* .liolq 5)j!h<yjTfn ii&itr'y- «rt. îint?,*jlfi

V;f(tCj^l^)'/n03' Îj 70*1 . lflfl9t)WJ<8

^ *jmaiiii 3^>fi*»’^ti>yiw)0 UnifO^ vidi ,tig

N-c l-r 'i*.-^^b rfio o:b w qu i» <tn qni:iA<^

m;i>» o /*f *5dJ JOioq n'oJlBaii;>id t«d» <it n&ifj h***

I'nyipjnl ^Mrt-j:.»ovfioo Td) lag Huoa daabim srtj ujIj «1

’ ,.f> biiKin '(^J aJî vôdJo tud wmO>q bii^ *ifi «adw

»#),

iqr*»<. io'"i 9*iftagiavcoo ^trâiq odt 'innnaTob'

iv;iA diii i£5Tqdvy tt ,f't gjg;>gi

a/ti idtoq m:’ b‘ n» unab’tK awiff S ttawwmi

’ '.jun.'tio^q '<Q tfr'^Ln -^’•>6 pad diaivô^

daiMW orn-jt o^yî* -r^l ai X^y w^hor^ ** ftlqiiifjx^, .rarfmiiA ^

sen / 'JltJlltfli A&’ ** ' ’ ^ÛT71*ital> At i3A®tH.dt aHt iiJ-ltWpAl

^tiniVri K'^ -01 4f <iw,

t>A® ,j«cg[Wâno 2114

^*T -

lU *'-i.< Hill j3Wv:hi* j^s S'»3aQA'> i it îiwbip''? 4~ ’ '^ mill*

•a ’«,.( AS^

#flgv,¥’io3 fAhoe i-idj H»di :(l]goi lUi'w booBlq^^ x mb|f 10I

,.•*'0 Jli-'.t j iil .;alt>qfiioa oil' ufc-"

’J .*'iii vo-tA Xit - <W

,tl ,;0 .hiHnîliOcU

mm ^ ''"=

"i«f1 uf maldc<iq .‘.U b'ldc^kOicnja ooirl? *1.1^^,

,3,> :in«?oM

10 >T bUHE>7 tj0sh:,4i »(U %o 1?^ o(fT

i dji>' * 5> t fiiTO} orii

chb i jsi fft <>i ôg'b^ftu'j rnn?- «4tu3iif* aib Surw bniJ o; ;^v'iU5s®'iA*feJi

,it>vyffOh .’“a «auj? uifr â5‘7#rt8 aoffo mdj-^rnq *1

frafriti^ Jr Ur v*oUi>i^ÂlbH oiiw JWiahtti? u<0

v.'OOOî; flOltttio« IDOTIOO (Ub llHi 3til®n OJ«, .

• ^

,tl

dU'*' !TKiJ‘;3» W'W&q ftfliwi>tJo\ 34l tu llnyiStH}, a4MJt’gl3^1*<15 bft*''

1u - '* "(CKrfq aithijat<iq»

82

Partial Credit

The objective of problem solving often held by students is

obtaining a correct answer. But if a student cannot produce a

complete answer, the second objective is to write down a reasonable

procedure in order to get partial credit. When the first objective was

impossible to obtain, the C&M students tried to achieve the second

objective. Thus, it was rare for the students to give up at the start,

and, consequently, they produced an "all or nothing" answer.

Problem G.3.a^i. in Lesson 3.07 was a good example of earning

partial credit. The range of the student scores in this problem was

1.5 to 3 points. The sample answer which received full credit was:

Since this integral does not have a standard antiderivative, I can find an

approximation by using the fact that

. ooI-

100

= +

1 1 100100

Mathematica can compute the decimal answer for

which is:

e'*/V 1 + x'^ dx

In[l]:= NIntegrate[E''(-x)/Sqrt[l+x''4], {x, 1, 100}]out[l]:= 0.127414

However, this answer is still off by ; ’^/V 1 + x‘* dx which is the error.

1100

Since V 1 + x^ > 1^

In[2]:= esterror

Out[2]:= E-ioo-E-b

e'*/V 1 + x'^ dx <

Jioo

= Integrate[E''(-x),

e'* dx

Jioo

{X, 100, b}]

In[3]:= N[esterror, 10]

Out[3]:= 3.720075976 lO'^

-

1

2.718281828*’

Since the second term will go to zero as b->o®, the remaining term is the

important one for the value of the error.

This means that 0 < e'*/V 1 + x"^ dx < e * dx < 3.720075976 lO'"*^.

Jioo Jioo

^ ^ Come up with a reasonably accurate estimate of e ’'/V 1 + x'* dx.

Discuss the accuracy of your estimate.

1

Mar? iwiM j.tfvlcHJ mîdtnq Hj ovlrj''jdd ofTT -\

i v .^“^4 jnabajii li loQ .Tov^îui i^.<jMflkido

i JKff

t

ff! r flkl'^TW OJ !)VtJ3ii>|T5o t>nOOs>^ atll jtwV/rflm 5K'*JQcno^

:#•-» cvfl>3i^^ nt:if^ ^jibfiTO tag «•) ni OTmOctoih

ioo .^ aÛ .^5»ibc ot taiti ^lU c>: aidjaao- nh

:.fi »<» q« avig o) uoTbL'iii ^l^ TO^ w si «»'n‘ .av»Wido

"gniHiorr io tU” hb b^>ubo4; ,\(Un^f^sFm7 Dajt

^ u .tv t(, booq fl ?Aw tt).£ iiC'itiaJ nr .‘»S C»0 .

*

ivvi aiiii ui frioM ^nabjtte ojHj lo a^njn adT 4^tw“it>vîr»sq

trw itbjj'j Miil UvbDyi ri->idv^/ ra^riM* afqmii* aiH { CV '

.-

nn lirul lA^ 1 .9VCwv*ral>ii/»« ' if<6<iti;: Ov«rf «3^<>^

.

I ^

afrfi m> 5dt |AI*W ^ u rj»mU£rt<?y^_

(

i.^

:i-

tb N • M ‘ 3 1 «... le »m um^^l^. jdJ sn jfiTOi i:«a

t*{

^ V,-- ..

(,bS' i ,ft <.vv>v.

^ ^ ym.r(

j

..m*' *di tJ livWw tb'^TTV',* V 1 x>:^

ad .. .S

r.h * 9I

2 rb^Jf+ lVV ’j

f tV

'1(4 ,0^t ,a) ,u*;''.dlatfct:*a;al = t4iitaSw *.\M^'

•''1.

- 1««

'.L.-e ,

:• * ,fV-• • — -, X , ,

..‘S-

-r

M «1J» fftTitJifiL'-.i viii! ,^>*-4 wi CV315 oi «><{ wjai ^00'.#<»a •'tj :

js *.1 »4t Iq t»aUt 4tU igi -jqft jctTiSr^^^ ‘

-J' -. .

' « '

.r*»7t 3tW, r > *'a j vT7V\ ”

'aI*

^

.. «f . . . . ,

'V-

SJ

> 0 Itlio jfniiOd li;dTsm »--

>91

1

!,, \ it,, C^- .

'xm- ^ 0S-4’'-

«

...f'

*\

'

.

^ I f

'î'rll\* io * rtil-t- qu vnoO*

iitmiur Tvov i<> vatirvqa adi •W’>?r4 •

83

100

So

j

e'*/V 1 + dx estimates

decimal places.

e**/V 1 + x^ dx to at least 43 accurate

The lowest scored answer (1.5 points) for this question was:

First, let's check with the standard method:

In[l]:= NIntegrate[E'^(-x)/Sqrt[l+x''4], {x, 1, t}]

Out[l]:= Integrate[ , } ^E Sqrt[l + x^]

Since that produced nothing helpful, we can now use the NIntegrate

command to get an approximation of the value:

In[2]:= NIntegrate[E'^(-x)/Sqrt[l+x'^4], {x, 1, 100}]Out[2]:= 0.121414

The error for this approximation can be assumed as follows: the estimate

for 1 to 100 is significantly greater than the extremely small value of

the integration from 100 to infinity.

Let's see how accurate this statement is:

In[3]:= esterror = NIntegrate[E^(-x)/Sqrt[l+x''4], {x, 100, 1000}]Out[3]:= 3.64782 lO'^*

Compare to the value we confirmed above, any further integration from

above 100 would be useless. We are saying that the approximate

integration from 1 to 100 plus the integration from 100 to infinity (or

1000 here) would be equal to the integration from 1 to infinity. Since

the integration from 100 to infinity is so extremely small and

insignificant, it can be ignored in our answer and we can assume that

the approximation is accurate to the decimal places we have displayed.

The second student made an effort to complete this problem even

though he could not come up with the appropriate formula for the

problem: e‘*/Vl + ^ e’* dx

With the aid of a calculating and graphic tool, each student

developed his/her idea whether it led to the correct answer or not.

The fact that the students received at least partial credit because

they did not hand in "all or nothing" answer can be partly explained

by the availability of computers. Having a powerful tool of

calculation and plotting made the students go through to the end

confidently without quitting.

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Exams

Exam 1

The range of questions in exam 1 (Appendix G) was the first

half of the "series and approximation" section. The exam consisted of

twelve questions, in which four questions (1 to 4) were relatively

easy computational problems, seven questions (5 to 11) were

designed to test more advanced computational ability and conceptual

understanding, and the remaining question (12) was a complicated

problem for measuring conceptual understanding with some

creativity.

The investigator expected that the students would do well in

questions 1 to 4 and would have difficulties with question 5 to 11.

However, there was not much difference in the average scores of 1 to

11. The fact that the students received approximately the same

scores in the first category problems (1 to 4) and the second category

problems (5 to 11) could be interpreted to mean that the students

were better in conceptual understanding than in computations. The

students sometimes misperformed the computations in the relatively

easy problems and showed high understanding in the rather difficult

problems.

One of the noticeable facts was that most of the students tried

to figure out their own solution plans for question 12:

Question:

If you suspected that the data from an experiment was coming from an

exponential function y = a e* or a power function y = a x**, explain howyou would decide between the two and how you would computereasonable values of a and b after you had decided.

Answer 1:

log y = log [a e*l log y = log [a x**]

log y = log a + X log y = log a + b log x

To find out if the data is coming from an exponential function, I wouldplot L=log[y] vs. X. If the data resembles a straight line, it is from an

exponential function. To find out if the data is from a power function, I

would plot logly] vs. log[x] and again look for a linear relationship. Foreither type of function, the y-intercept is In a.

Answer 2:

If I suspected the data was coming from an exponential function, then y= a e*, log[y] = log[a] + x. Since log[y] is a line function of x, I would plot

the data in the form (x, log[y]). If a line could then be drawn through or

nearly through the points, I would suspect an exponential function. 1

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85

would then compute a reasonable value of a by using the formula log[y]

= log[a] + X.

If I suspected a power function, y = a x**, log [y] = log[a] + b log[x], then I

would plot the data in the form (log[x], log[y]). If a line could be drawnthrough or nearly through the points, I would assume a power function

relationship and determine a and b by using log y = log[a] + b log[x] and

plugging in values for x and y from the data.

In fact, this is a tough problem which requires in-depth

understanding of the concepts in exponential function, power

function, and approximation. Compared to question 12, the students

made little effort to solve other questions. They did better in non-

routine, applied problems than in routine, typical problems. This

phenomenon was the strong point and at the same time the weak

point of the C&M students.

Exam 2

Exam 2, which was composed of five questions (Appendix G),

covered the latter half of the series and approximation section and

the part of the integration section.

Question 1 consisted of two sub-questions which asked the

students to find convergence intervals. All three methods—the

singularity method, the ratio test, and the convergence principle—

could be applied to the first sub-question, and the ratio test and the

convergence principle were appropriate for the second one. In fact,

the easiest method was the singularity method for the first sub-

question and the ratio test for the second one. One noticeable fact

was that the students absolutely preferred the ratio test. The three

students who used the singularity method, even checked their

answers by applying the ratio test in addition. On tlie other hand,

most of the students tended to apply the same method to the next

question that they answered; only two students mixed the three

methods in the two sub questions.

To summarize, the students’ criterion for choosing the method

depended on individual preference rather than on the appropriate-

ness of the method for each question. Especially, they showed a

strong tendency to use the ratio test. Moreover, once the students

chose one method, they continued to apply the same method to the

subsequent questions.

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86

Question 2 asked the students to evaluate the three definite

integrals. The strategies of calculating integrals were integration-by-

parts for the first integral, and simple substitution for the second the

the third ones. The students did well on the first integration, but

some of them did not perform well on the second and the third ones

because they tried to apply the integration-by-parts formula. This

indicates that the students tend to use the most recent method they

learned (integration-by-parts) rather than the old method

(substitution).

Question 3 required the students to find the power series in

power of (x - 1) that represents the function f(x) = ^. Most

x^ - 2x + 5

of the students successfully substituted t = x- l,t+l=x, and

obtained g(0 = — — . However, in the process of expanding g(t) in

t^ + 4

power of t, some of the students applied Taylor's formula instead of

the geometric expansion, which is the easiest and the most

convenient way of expanding in this case. Here, the students also

showed the inclination to adopt the latest method (Taylor's formula)

they had acquired.

To sum up, the students' overall scores were high even though

the questions required a lot of hand calculation, which were usually

done by computers. One of the most remarkable findings from the

second exam was that the students tended to take the most recent

method they had learned when several methods were available.

Furthermore, they were inclined to apply the same method to the

subsequent question regardless of its different nature.

Exam 3

The investigator examined the question types of the

examinations of MATH 132 for the past few years and found that

almost 90% of the questions were asking students to:

solve, find, calculate, determine, sketch, graph, evaluate, and what is?

Most questions asked for straightforward calculations or posed

template problems that were taught over and over again in the

course and that were in the textbook. Only 10% of the questions had

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87

high-order challenges, and most of those were template word

problems that fit a standard pattern. Some examinations had

different patterns, but the mainstream examinations had the

proportion of 90% of calculation and 10% of thought.

However, the question types of the C&M course were little

different. In exam 3 (Appendix G), there were two of the "explain"

type and two "why" type questions among a total of six questions. To

answer those questions, a thorough understanding of the following

basic properties of integrals was required:

If f[x] < g[x] for all X with a < x < b, then g(x)dx;

f(x)dx =I

f(x)dx -I- I f(x)dx for any number c with a < c < b;

jf(x)dx

=J

f(x)dx+J

^Kf(x)dx = K^Kf(x)dx = K| f(x)dx for any number K;

If f[x] is increasing on [a, b], then f[a](b-a) <[ f(x)dx< f[b](b-a).

Almost all the students correctly explained the above four properties

in their own words and appropriate visual aides.

It is a common belief that "tests drive the curriculum"; tests

give students the "bottom line" of what they are expected to know,

and that bottom line determines what they will study and learn. In

this context, the concept-oriented questions in exam 3 might provide

the students with a motive to focus on conceptual understanding.

Final Exam

Several sections of MATH 132 at 10:00 o'clock took the

common final exam (Appendix G). A slightly revised form of that

final exam was given to the C&M students. Twelve students of the

C&M group and 31 students of the traditional group participated in

the test. What follows are the mean and SD of the two groups on the

common items (items 1-4) and those on the similar items (items 5

and 6).

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88

Table 26

Mean and SD of the Scores on the Common Final Exam

Item C&MMean SD

Traditional

Mean SD

1 7.3 3.2 7.0 2.5

2 7.7 3.7 7.6 2.4

3 7.2 3.7 7.5 3.0

4 7.4 3.6 7.2 3.0

5 9.3 1.7 8.4 1.9

6 8.8 2.1 8.8 1.4

Items I to 6 47.7 11.0 46.5 7.7

Items I to 4 29.5 8.5 29.4 6.1

Tables 27 and 28 provide the ANOVA (analysis of variance) for

the 6 items and 4 items

Table 27

Summary Table of ANOVA for the 6 items

Source SS Df MS F D

Treatment 10.8 1 10.8 0.14 0.707

Within 3102.3 41 75.7

Table 28

Summary Table of ANOVA for the 4 items

Source SS Df MS F p

Treatment 0.2 1 0.2 0.00 0.951

Within 1914.1 41 46.7

The null hypothesis addressed is that there is no significant

difference in achievement scores between the students in the C&Mgroup and the traditional group. The calculated F-values were

significantly less than the critical F-value (Fqs 41= 4.08). Therefore,

the hypothesis was not rejected at the .05 level of significance. Even

though the difference in the achievement scores between the two

groups was not significant, the fact that the C&M group outperformed

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the traditional group in the traditional course-oriented test was an

encouraging result. The C&M students' performance on the common

final exam seemed to support the claim that learning with

Mathematica does not impair students' calculational abilities.

Courseware

Description

The C&M calculus courses have used the electronic text called

Mathematica Notebook. The lessons are contained in a number (about

thirty) of separate Mathematica Notebooks that the students work

on.

Basically, the format of the C&M calculus course is that of

problems and solutions. The Mathematica Notebook consists of an

introductory "guide” and four styles of problems: "basics", "tutorial",

"give it a try", and "literacy sheet." Every lesson starts with a brief

"guide" which announces the new ideas of the lesson. The "guide" can

be compared to an introduction to a chapter in a printed textbook.

Then come the "basics" problems which introduce many of the new

ideas on the subject matter. The "basics" is followed by "tutorial"

problems, which present the techniques and applications. Each

"basics" and "tutorial" problem provides full solution and detailed

explanations which students can easily understand. These solutions

and explanations are a basis for the students to use in the following

section, "give it a try", which includes problems for the students

themselves to do. Most of the problems in "give it a try" are similar

to the "basics" and "tutorial" problems, but some of them are not.

Some are routine, some are exploratory and challenge, and the others

require ingenious intuition. The last section, "literacy sheet" is a li^t

of mathematical facts and concepts that students are supposed to

know, and the questions that students should be able to deal with

away from the computer (Davis, Porta, & Uhl, 1990).

Review of the Lessons

The C&M courseware consists of four sections: Starting Out

(Lesson 1.01-1.03), Differential Calculus (Lesson 2.01-2.10), Integral

Calculus (Lesson 3.01-3.08), and Series and Approximations (Lesson

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4.01-4.08). What follows are the descriptions and interpretations of

distinct aspects of each lesson.

Lesson 2.02: The Chain Rule Unlocks the Secrets of Logarithms

The Chain Rule is introduced through several exploration

examples designed to lead students to find the rule for themselves.

Then this rule is reinforced in the following way:

Recall that s[t] grows s’[t] times as fast as t Accordingly f[g[x]] growsf '[g[x]] times as fast as g[x], and g[x] grows g'[x] time as fast as x. As a

result flg[x]]] grows f '[g[x]] g'[x] times as fast as x.

This intuitive explanation of the Chain Rule in the C&Mcourseware is simple and clear, while the proof of the rule in the

traditional texts, which uses the property of derivative,

differentiability, difference quotient, and product law, is somewhat

complicated.

Lesson 2.03: Powers. Products, and the Trigonometric Functions

Traditional calculus books prove the Product Rule, D[f(x) g(x)] =

f(x) g'(x) -I- f(x) g(x), using the sum and product laws for limits, the

definitions of f(x) and g'(x), and the fact that lim f(x + h) = f(x).h—>0

However, such an abstract approach regarding continuity is no longer

used after the proof of the Product Rule. For this reason, the C&Mcourseware does not include abstract continuity, but uses a different

point of departure instead. The Product Rule is obtained from the

Power Rule, f(x) D[log(x) f(x)] = f(x), which is the Chain Rule applied

to logarithmic differentiation. In short, the two approaches use

different prerequisite rules for the Product Rule.

Lesson 2.04: The Race Track Principle

The Mean Value Theorem, which establishes the connection

between the rising or falling graphs and the sign of the derivative, is

the principal theoretical tool of differential calculus. The C&Mcourseware and the traditional calculus texts have a different

approach to the Mean Value Theorem.

The traditional calculus texts first give a preliminary result

called Rolle's Theorem, which expedites the proof of the Mean Value

Theorem. On the contrary, the C&M courseware first introduces the

Race Track Principle, if f(a) = g(a) and f(x) 1 g'(x) for x 2 a, then

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f(x) 1 g(x) for X 2 a, an active form of the Mean Value Theorem. When

the Mean Value Theorem is introduced via the Race Track Principle,

students can understand the former as a rather easy consequence of

the latter.

Lesson 2.07: Tangent Lines

Most traditional calculus texts introduce the tangent line in the

very first chapter. They formally present the definition of tangent

line and explain the other concepts of calculus on the basis of the

understanding of tangent line. In contrast, in the C&M courseware,

the definition of tangent line is placed after the explanation of

differentiation, derivatives, and the Chain Rule. Although the

students have not been consciously aware of the concept of tangent

line, they have an intuitive feeling for an idea of tangent line before

formal definition. This kind of explanation seems to make the

students comfortable with that concept and facilitate their

understanding.

The explanation of tangent line is followed by Newton's

method, which is an important tool of numerical mathematics.

Compared to traditional calculus texts, Newton's method is much

more emphasized in the C&M courseware. In the traditional calculus

texts, most differential equations are analytically handled by several

special methods. In that case, students are apt to lose sight of the

general idea that every differential equation has a solution, and that

the solution is uniquely determined by its initial data. Numerical

methods including Newton's method are universal and give students

an intuitive grasp of existence and uniqueness of solutions (Lax,

1986).

On the other hand, the students can easily determine the initial

point for Newton's method, when they practice Newton's method by

using Mathematica because the screen shows the coordinates of the

point wherever the cursor is placed.

Lesson 3.01: Area and the Integral

Traditional calculus courses emphasize both the integral as a

measurement of area, and area in terms of the integral. Also they

follow the typical order in introducing integral: Riemann sums,

indefinite integral, and definite integral. However, in the C&M

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courseware, integral is defined in terms of area, and not vice versa.

This eliminates the need for Riemann sums and makes the

Fundamental Theorem of Calculus more accessible.

Lesson 4.01: Empirical Approximations

One of the most distinguishable differences between the C&Mcourse and the traditional course appears in the teaching of series

and approximations. Traditional calculus books have only a few

chapters dealing with series and approximations. On the contrary, a

relatively large part of the C&M courseware is devoted to the topics

of series and approximations (interpolation, approximation by

expansions, error estimates, power series, and so on), and those ideas

permeate the entire C&M courseware. Also the C&M course directly

approaches infinite series and power series by means of their

application to approximation, while the traditional course studies the

series as mathematical objects first and then applies to

approximation afterwards.

Lesson 4.04: Convergence: The Explanation of Our Observation

Lesson 4.04 presents the method of determining rQ, the radius

of convergence of X ; if the power series X represents a

n=0 n=0

function f(x), then find the distance ro to the nearest complex

singularity of f(x) and its derivatives. The restriction that the power

series X with a prescribed point 0, represents a function f(x) is

n=0

expanded in two directions in the later lessons. First, Lesson 4.06

deals with the case that we do not know f(x). In that case, the basic

convergence principle or the ratio test is applied to find R i ro.

Second, Lesson 4.08 handles the generalized case that the prescribedoe

point is any real number k, i.e., X an(x-k)“.

n=0

The first treatment of the determining the convergence

interval in Lesson 4.04 only deals with the special case while the

next treatments in Lessons 4.06 and 4.08 cover the more general

case. The vehicle for this kind of approach is a spiral curriculum, in

which topics are taken up again and again, and the later treatment

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93

being less intuitive and more formalized than the previous one.

Simplifying the topic--determination of convergence intervals--in

the early stage does not mean the distortion of the basic idea. Rather

it would mean presenting partial or incomplete structures in the way

that later learning would serve to fill out or complete the structures.

In the light of this, the spiral approach promotes the intuitive

understanding of the interrelationships among the topics.

Lesson 4.06: Power Series

L'Kopital's rule provides an important algebraic manipulation

technique for solving the limit of an indeterminate form. In most of

the traditional calculus texts, L’Hopital's rule is proven by applying

Cauchy's Mean Value Theorem^^

However, the C&M courseware takes a different approach to

L’Hopital's rule. The main tool of the proof is the command

Series[/w/ic/io/i]. Actually, Taylor's formula is the basis for the

calculating Ser\es[function]

.

Thus, the underlying principle of

proving L'Hopital's rule in the C&M courseware is Taylor's formula,

whereas the underlying principle in the traditional texts is Cauchy's

Mean Value Theorem.

Students' Difficulties in the Lessons

Lesson 4.01: Empirical Approximations

The purpose of problem G.3. (Appendix F) is to show the

drawback in interpolating polynomials. Using the data given, the

students were asked to find the interpolating polynomial fitting the

data, and to predict the values not included in the original data

points. The problem was designed so that the predicted values from

the interpolating polynomial were not acceptable. Thus, the students

were required to be aware that interpolating polynomials were not

always a good predictor even though they yielded a reasonable

description of the given data.

Some of the students caught the point of the problem and

backed up their opinion with plots. However, some of the students

could not do this problem well. Apparently, they were accustomed to

^ ^ Suppose that the functions f and g are continuous on the closed interval [a,

b] and differentiable on (a, b) then there exists a number c in (a, b) such that

[f(b) - f(a)l g’(c) = [g(b) - g(a)] f(c)

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mathematics having correctness and preciseness; thus they did not

dare to suggest that the interpolating polynomial failed to predict the

values. To show the powerful as well as the unreliable aspects of

interpolating polynomials is necessary in order to make students

able to handle the approximation of the realistic data appropriately.

Lesson 4.02: Approximations by Expansions: Quotients of Polynomials

Problem G.5. consists of eight sub-questions which require the

students to find the expansions of 1/(1 - x^), 1/(1 + x^), 1/(1 - x^),

1/(1 - x/2), 1/(2 - x), l/(a - x), x^/(a - x^), and 1/(1 - x)^ from the

given expansion 1/(1 - x) = l+ x + x^ + ... + x*^ + .... Almost all

students did well on the first four questions because they simply

needed to substitute x by x^, -x^, x^, and x/2. A few students had

difficulty in sub-questions 5, 6, and 7 which included one more step of

changing —^— to — (—

l

—). The main point of the last question is

a - x^ ^I

x^a

that the series can be calculated by differentiating the geometric

series term by term, D[Normal[Series[l/(l-x), {x, 0, 9}]], x].

Instead of this, most of the students approached the last question by

Normal[Series[l/(l-x)^, {x, 0, 8}]]. They certainly knew

D[l/(1 - x)] = 1/(1 - x)^, but they failed to connect this property to the

question.

Lesson 4.03: Approximations by Expansions: Integration

The purpose of problem G.lO.b. (Appendix F) is to make

students aware of the fact that approximation by interpolating

polynomials does not always accord with approximation by

expansion. Although most of the students noticed the difference

between two the approximations based on the result they got from

computers, they had difficulty figuring out the reason; the slight

difference between the coefficients of the interpolating polynomial

and the partial expansion is caused by the fact that the interpolating

polynomial fits exactly at the interpolating points while the partial

expansion does not.

Lesson 4,06: Power Series

Problem G.7.C. aims to derive the tangent line function from

Taylor's formula:

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95

f[x] = a[0] + a[l] X + a[2] x^ + a(3] x^ + . . + a[n] x“ + . .

.

According to Taylor's formula, a[0] = f[0], a[l] = f '[0].

Therefore, g[x] = a[0] + a[l] x.

g[x] is the tangent line function of the graph of f[x] at the point {0, f[0]}.

Almost all the students recognized the relationship between Taylor's

formula and the tangent line, but most of them did not mention the

specific point {0, f[0]}. Also, in problem G. a., two students confused

the interval [-r, r] for r < R and the interval [-R, R] = {x; -R i x i R}.

Seemingly, these two mistakes were originated from the students'

lack of attention not from a lack of knowledge.

Lesson 4.08: Expansions in Power of (x - a’l

Problem G.5. asks the value of lim f[x]/(l + cos[x]) when f[TT] = 0> K

and f '[Tf] = 5. The students could get the solution by applying

L'Hopital's rule or the expansion in power of (x - Tf). Although the

application of L'Hopital's rule is much easier than the expansion in

power of (x - Tf), most of the students chose the second method.

In hand calculation, if the method with simple computation and

that with complicated computation are both available, students

generally prefer the simple method. They tend to avoid intricate

calculations whenever possible. In contrast, when computers are

available, students' criterion for determining the solving procedure is

not the number of calculations involved because they do not have to

be concerned with the calculational procedure. Rather they are

concerned about whether the solving procedure is fundamentally

sound. According to this criterion, the expansion method is the best

choice for the above question even though it requires several steps

of calculation. The students who used the expansion method followed

these command lines and then reported that the limit did not exist.

In[lJ:= Numerator = Series[f[x], {x. Pi, 2}] /. {f[Pi]->0, r[Pi]->5)In[2]:= Denominator = Series[(l + Cos[xl}, {x, Pi, 5}]

In[3J:= Numerator/Denominator

Oul[31: + HPi] + 0[-Pi + I]-Pi + X

Problem G.2.a. requires the students to account that: neither x

nor Log[x] has an expansion in powers of x; but if a > 0, then

functions have expansions in power of (x - a). Every student tried to

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96

explain in his/her own way. Most of the responses provided a

pertinent explanation such as:

"Neither of these two function has an expansion in powers of x because

each has a complex singularity at x = 0. These functions can be expanded

in powers of (x - a) where a > 0 because the complex singularity is now a

distance of a from x = a. Therefore, both functions have power series

representations to the power of (x - a) that converge on [a - r, a + r] for

r < a."

But several students missed the point of the question and answered:

"You can't take the square root of or the log of a negative value. If youlook at the plots of Sqrt[x] and Log[x] you will see that they are

undefined."

The first student caught exactly the idea of the question, and

thoughtfully wrote down the answer. But, the second student could

not relate this question to what he had learned in the lesson, and,

consequently, produced an irrelevant answer.

Lesson 3.07: Integration bv Parts

The best way of estimating the size of an improper integralb

f(x)dx by hand is to integrate to a finite limit b, I f(x)dx, and

f I.then let that finite limit approach, b-><» . However, most of the

students did not follow the intermediate process (finding the finite

limit); instead they directly used <» as an upper end. For example:

IX e*^* dx = [

'^-

Idx =....= ——— - 1. (e‘^“ - e®)

Jo 3 0 jo 3 3e3- 9

In Mathematical infinity {^o) can be taken as an integration range,

Integrate[f[x], {x, a, Infinity}]. Hence, the direct use of as an

upper end might be the reflection of the students' habits in using

Mathematica.

As expected, many students suffered the difficulty of choosing

u and V in the integration by parts formula, Juv' = uv-Ju' v. Even

though there is a general guideline of determining u and v, students

learned the integration-by-parts technique by trial and error. The

usual problem solving experience is: students choose u and v, and

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follow the procedure of integration by parts; if the calculation

becomes complicated, they reverse u and v, and try again. However,

one thing unexpected was found in the students' solution files; the

students who made the wrong choice between u and v sometimes did

not alter their choice even when the formula became long and

student explained that:

"The first thing we do is to set our u and v prime. We do this in a way so

that we can integrate v and differentiate u easily, and so that we can fit

them into our integration-by-parts formula. We will set our u equal to

Sin[Pi x] because we know the derivative is equal to Pi Cos[Pi x]. We nowset the V prime equal to the remainder of our function, which is x.

When we integrate this we get x^/2."

He went on with his calculation and finally arrived at the correct

answer Pi'^ even though he did not take the short cut by choosing

u = X and v' = sin[x]. He seemed to be very confident in his choice

between u and v because he continued to the end without hesitation.

The fact that the C&M students are very confident in their decisions

and calculational steps can be one of the positive aspects of using

computer in a learning procedure.

Lesson 2.08: Parameters

At the beginning of the lesson, the students seemed to confuse

parametric equations and polar equations. But this confusion cleared

up quickly when they progressed through several problems and

recognized the difference, except that, they had difficulty in the

problem G.l.b.i. which is:

What is the lowest point on the curve specified parametrically by

X = x[t] = c«'* + t

y = y[t] = (t^ + 4t - 16) Log[t]

Most of the students approached this problem in the following

two steps:

i) first, find the point t such as y'[t] = 0In[lJ:^ X[tl = E*(t78) + t;

In[2J:= y[t] = (t'^2 + 4t - 16) Logft];

In[3]:= SoIve[t'^2 + 4 t - 16 == 0, t]//N

Out[3J:= {{t -> 2.47214}, {t -> -6.47214}}

complicated. For example, in the integration

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Since Log[t] is undefined for t < 0, the curve is undefined for t i 0.

Therefore, the lowest point is between t = 0 and t = 2.47214.

At this point dy/dx = 0.

In[4]:= PIot[y'[t], {t, 0.001, 2.475}, PlotRange -> {-1, 1}]

Out[4]

The value of t is close to 1.7:

In[5]:= FindRoot[y’[t] == 0, {t, 1.7}]

Out[5]:= {l-> 1.68211}

ii) then, find the lowest point

The lowest point on the curve is:

In[6J:= low = {x[1.68211], y[1.68211]}

Out[6J:= {2.91611, -3.35019)

However, several students did not follow the above two steps.

Instead, they solved this problem by the convenient shortcut; they

directly got the lowest point of the curve with Mathematical

s

ownfunction^3

Mathematica gives the x, y coordinates of the point when we place the

cursor on the target point and press the command key.

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CHAPTER VRESULTS IN THE CONTEXT OF AFFECT

Attitude Survey

The questionnaire data provided rich information about the

students' perceptions and attitudes as seen at a single time, while the

data from the attitude survey, which was administered at the

beginning and the end of the semester, indicated how students'

perceptions and attitudes had changed during the course. The

attitude survey had two major parts: attitudes toward computers,

and attitudes toward mathematics, which had four sub-areas:

mathematics as a process, mathematics and affect, value to society,

and cooperative learning.

Before applying ANCOVA (analysis of covariance), the

investigator checked two assumptions of ANCOVA: first, whether

there was a linear relationship between the dependent variables and

the covariates (Table 29); second, whether the regression slopes for

the covariates were homogeneous, that is whether there was no

significant interaction between the covariates and the treatment

variable (Table 30).

Table 29

Correlations Between the Dependent Variables and the Covariates

Dependent Variable Covariate Correlation Coefficient

post-test scores pre-test scores 0.71

post-AM scores pre-AM scores 0.72

post-AC scores prg-A.C Sggrgs (in

Note . AM means attitudes toward mathematics and AC means

attitudes toward computers.

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Table 30

Interactions Between the Covariates and the Treatment Variable

Source F p

post-test scores * treatment 2.36 0.13

post-AM scores * treatment 3.04 0.09

post-AC scores * treatment 0.23 0.63

Since the two assumptions were satisfied, ANCOVA was an

appropriate statistical analysis method in this study.

Results of Testing the Hypotheses

The null hypotheses addressed in chapter I were:

H2—There is no statistically significant difference in the scores

on the attitude survey between the C&M group and the traditional

group.

H2(A)"There is no statistically significant difference in

attitudes toward mathematics scores between the C&M group and

the traditional group.

H2(B)—There is no statistically significant difference in

attitudes toward computer scores between the C&M group and the

traditional group.

Using an analysis of covariance, with the pre-attitude scores

being used as the covariates, the post-attitude scores were tested at

the .05 level of significance. The calculated F-value was 19.2 (Table

31), which was larger than the critical F-value (Fqs, i, 65 = 3.99).

Therefore, H2 was rejected, indicating that there was significant

difference in student attitudes between the two groups at the .05

level of significance.

Table 31

Summary Table of ANCOVA for the Attitude Scores (Total)

Source SS Df MS E C_Covariate 1143.7 1 1143.7 69.1 .001

Treatment 317.4 1 317.4 19.2 .001*

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Table 32

Summary Tabic of ANCOVA for the Scores on Attitudes Toward

Mathematics

101

Source SS Pf MS F p

Coyariate 1076.3 1 1076.3 76.60 .001

Treatment 129.0 1 129.0 9.18 .004*

Within 913.3 65 14.1

Table 33

Summary Table of ANCOVA for the Scores on Attitudes Toward

Computers

Source SS Df MS F p

Coyariate 164.1 1 164.1 68.5 .0001

Treatment 49.0 1 49.0 20.5 .0001*

Within 155.6 65 2.4

According to the results of the ANCOVA data pertaining to

attitudes toward mathematics (Table 32) and attitudes toward

computers (Table 33), the hypotheses H2(A) and H2(B) were also

rejected.

Further Analysis

In addition to the hypothesis test, the attitude suryey data

were analyzed. On the whole, the students in both groups had fairly

positiye attitudes. One of the reasons might be the fact that 74% of

the C&M group and 59% of the traditional group were from

engineering, and those students generally had fayorable attitudes

toward mathematics and computers. Moreoyer, most of the non-

engineering students also had positiye attitudes because they had

enough motiyation to take the course yoluntarily.

From the data in the next page (Table 34), it is concluded that

the attitudes of the C&M group shifted more strongly than did those

of the traditional group toward the highest rating, although the

attitudes of both groups became more positiye.

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Table 34

Mean and SD of the Attitude Scores (Total)

C&M Traditional

Pre-survey Mean 86.3 84.2

SD 5.8 6.0

Post-survey Mean 91.8 85.8

SD 4J2

Attitudes Toward Computers

Five out of the twenty-four questions referred to attitudes

toward computers. As expected, the C&M students showed a

remarkable change in attitudes about the use of computers in

learning mathematics. For every question, the t-test was done to

check whether there is statistically significant difference in the post

means of the two groups. The summaries of the responses to these

questions appear below.

Table 35

Mean and SD of the Scores on Attitudes Toward Computers Items

Mean SDC&M 20.21 (18.7)2 1.93 (2.7)“*

Traditional 17.4 (17.1) 2.4 (2.71

Note . 1 and 3 are the data from the post-survey, and 2 and 4 are the

data from the pre-survey.

6. Everyone should learn something about computers.

Mean S£ E

C&M 4.2 (4.2) 0.8 (0.7) 0.722

TrajiUgnal 4.3 (4.n 0.8 (0.9)

11. Using a computer makes learning mathematics more mechanical

and boring.

Mean SD o

C&M 4.0 (3.6) 1.0 (0.9) 0.014*

Traditional 3.4 (3.4) 0.9 (0.7)

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Table 35 (Continued)

13. Using a computer can help you learn many different

mathematical topics.

Mean SD p

C&MTraditional

3.6 (3.7)

3.3 (3.2)

1.0 (0.9)

0.9 ro.9i

0.192

16. If you use a computer, you don't have to learn to compute.

Mean SD p

C&MTraditional

4.0 (3.8)

3.6 (3.5)

0.9 (1.0)

0.9 n.oi

0.034*

21. Solving word problems is more fun if you use a computer.

Mean SD p

C&MTraditional

4.3 (3.5)

2.9 (2.9)

0.7 (0.9)

. M.ILQ1

0.001*

On all five questions, the responses of the C&M group in the

pre-survey were more positive than were those of the traditional

group. This initial difference can be explained by the fact that the

C&M group were already aware of the different approach of the

course before registering. Thus, the members of the C&M group were

the students who initially had a positive opinion about computer use

in mathematics learning.

The significant differences in the scores occurred on items 11,

16, and 21. These differences implied the C&M students' more

favorable disposition to the computer and their enjoyment of the

C&M course. However, the C&M students' opinion on item 13 was

negatively changed compare to the scores on the pre-survey. They

seemed to believe that they did not learn more topics by using

computers (even though they could come up with various examples

within each topic ).

Item 6 addressed the fundamental question about the use of

computers. The end-of-semester ratings of the two groups on this

item were close to each other (4.2 and 4.3). It means that the

From the responses to the questionnaire

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traditional students equally agreed to the basic fact that everyone

should learn something about computers. Yet, they were not sure

whether or not learning mathematics with computers is more fun.

Cooperative Learning

The greatest difference in attitude changes occurred in items

which addressed cooperative learning.

Table 36

Mean and SD of the Scores on Cooperative Learning Items

Mean SDC&M 16.2 (14.5) 2.0 (2.0)

Traditional 14.7 H4.n 3.1 (2A)

3. I like to solve problem by working with others.

Mean SD p

C&M 3.8 (3.5) 0.7 (0.9) 0.061

Traditional 3.4 (3.3) 1.1 (1.2^

7. Solving problems with others lowers self-confidence.

Mean SD pGfeM 4.1 (3.8) 0.6 (0.8) 0.415

Iiaditignal 3.9 (3.7) 0.8 (0.9^

15. I prefer to study mathematics by myself.

Mean SD pC&M 3.9 (3.4) 0.7 (1.0) 0.073

Traditional 3.5 (3.3) 1.0 (1,0)

19. When I do mathematics with other students, I realize I am not

the only one who can't understand.

Mean SD pC&M 4.3 (3.8) 0.6 (0.7) 0.014*

Traditional 3.9 (3.8) 0.8 (0.8)

Possibly, the reason for greater movement toward a positive

reaction was that the C&M group had had a favorable cooperative

learning experience during the semester. In fact, considerable

t ^ 81

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105

interaction among the students and the camaraderie of working in

natural groups were observed in the C&M lab.

On the other hand, the correlation coefficient of the scores on

the achievement test and the scores on the cooperative learning

items were examined (Table 37).

Table 37

Cooperative Learning Scores

C&M Traditional

Correlation Coefficient -0.05 -0.51

In the C&M group, the correlation coefficient was near zero.

This means that the C&M students, regardless of their achievement

levels, liked to converse and discuss problems with their classmates

and benefited from the collaborative learning activities in the lab. In

contrast, the negative correlation coefficient of correlation in the

traditional group implies that the high-achievement-level students in

this group usually preferred individual learning although the low-

achievement-level students perceived the peer sharing as beneficial.

Mathematics as a Process

Five items addressed students’ view of the nature of

mathematics. A summary of the means and differences appears

below.

Table 38

Mean and SD of the Scores on Mathematics as a Process Items

Mean SDC&M 18.5 (18.0) 2.5 (1.9)

Traditional 17.9 (17.61 2.7 (2.6)


1. New discoveries in mathematics are constantly being made.

^5 The iterated principal factor analysis of the four cooperative learning items

resulted in a high communality estimate (2.30). Thus, the sum of those items

can be a legitimate measure of attitudes toward cooperative learning.

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1. New discoveries in mathematics are constantly being made.

Mean SD p

C&MTraditional

2.9 (3.0)

3.0 (3.0)

1.1 (0.7)

1.1 (0.9)

0.681

5. Most mathematics problems can be solved in different ways

Mean SD p

C&MTraditional

4.1 (3.8)

4.0 (3.8)

0.6 (0.5)

0.7 (0.8)

0.500

10. Most of the learning of mathematics involves memorizing.

Mean SD p

C&MTraditional

4.0 (3.6)

3.3 (3.2)

0.8 (0.8)

1.1 (1.0)

0.011*

18. There is

problems.

little place for originality in solving mathematics

Mean SD p

C&M 3.8 (3.5) 0.8 (0.8) 0.297

Traditign^l 3.5 (3.4) 0.9 (l.Q)

23. Mathematics helps one to think logically.

Mean SD p

C&M 3.7 (4.0) 0.9 (1.0) 0.233

Traditional 4.0 (4.1) 0.8 (0.8)

Many students were undecided at the beginning and at the end

as to whether new discoveries in mathematics are constantly being

made (item 1). This uncertainty can be explained by the fact that the

main topics of calculus (differentiation and integration) were

developed during the seventeenth and eighteenth centuries, and

were the only ones appearing in the textbook used. Therefore, the

students did not have the opportunity to encounter recently

developed topics.

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At the end of the semester, more students in the C&M group

than in the traditional group agreed that most mathematics problems

can be solved in different ways (item 5), and that mathematics could

be learned without memorizing (item 10). On these two items,

changes in students' perceptions reflected their experiences during

the semester in a predictable manner. The C&M group had had

plenty of plotting experience and seemed to consider visualization as

a different way of solving problems. Also, the more positive reaction

of the C&M group and the more negative reaction of the traditional

group to item 10 indirectly indicates the amount of memorization

involved in their different learning procedures. The traditional

students were required to memorize many formulas, whereas the

C&M students usually figured out the formulas by themselves by

working with computers.

On the other hand, the students in both groups did not feel

strongly that mathematics helps one to think logically (item 23). The

direction of the changes suggests that the calculus curriculum in both

groups did not help students to perceive that mathematics build

logical thinking.

Value to Society

The attitude survey included three items on the students' view

of the usefulness and importance of mathematics to society.

Summaries of the students' responses follow.

Table 39

Mean and SD of the Scores on Value to Society Items

Mean SDC&M 10.7 (10.2) 1.4 (1.2)

Traditional 10.5,(10.2) 1.4 (1.5)

8. I want to work at a job which requires mathematics.

Mean SD p

C&M 3.5 (3.2) 1.0 (0.8) 0.255

Traditional 3.2 (3.2) 0.9 (0.9)

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108


17. Mathematics is helpful in understanding today's world.

Mean SS p

C&M 3.4 (3.3) 0.9 (0.8) 0.284

Tr^qitiQnftl 3A OJ) 0.7 (0.7)

20. It is important to know mathematics in order to get a good job.

Mean p

C&M 3.9 (3.7) 0.8 (0.7) 0.396

Tragitipnal (3 .6) 0.8 (0.9)

There were not many differences in the responses to the items

in this category. The two groups initially rated the three items

similarly (within .2 of each other), and rated these items similarly at

the end (within .3 of each other) again.

The data of item 8 indicate that the C&M group felt more

strongly than did the traditional group that they wanted to have a

job which required mathematics. This trend was predictable from the

differences in students' major areas of study. The C&M group was

more heavily populated with engineering majors than was the

traditional group; thus the former reacted more favorably to this

item than did the latter. In addition, the explanation for the

generally low ratings (3.2-3.5) in this item could lie in the fact that

the members of the two groups were freshman who did not yet have

a concrete plan for the future.

The overall performance of the two groups on item 17 was also

relatively low. Presumably, the students in the two groups were not

convinced that mathematics is helpful in understanding today's

world.

Mathematics and Affect

Seven items questioned students on their competence,

enjoyment, motivation, and anxiety in learning mathematics.

Summaries of responses to these items appear below.

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Table 40

Mean and SD of the Scores on Mathematics and Affect Items

109

Mean SDC&M 26.4 (24.9) 2.9 (3.3)

Traditional 25.4 (25.4) 2.6 (2.6)

2. I usually understand what we are talking about in mathematics

class.

Mean SD b

C&M 3.4 (3.6) 0.9 (0.8) 0.586

Traditional 3.5 (3.4) 0.9 (0.71

4. When I cannot figure out a problem, I feel as though I am lost in a

maze and cannot find my way out.

Mean SD E

C&M 3.7 (3.2) 0.9 (1.0) 0.098

Traditional 3.3 (3 .2) 1.0 (U)9. Mathematics is something which I enjoy very much.

Mean SD pC&M 3.9 (3.7) 1.1 (0.8) 0.233

Traditional 3.6 (3.7) 0.8 (0.7)

12. I have a real desire to learn mathematics.

Mean SD EC&M 3.9 (3.5) 0.6 (0.6) 0.289

Traditional 3.7 (3.6) 0.9 (0.8)

14. The only reason I’m taking this course is because I have to.

Mean SD pC&M 3.7 (3.8) 0.9 (0.8) 0.732

Traditional 3.8 (3.8) 0.9 (1.01

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Mean SD p

C&MTraditional

3.8 (3.5)

3.7 (3.8)

0.7 (0.9)

0.9 (0.8)

0.469

24. Mathematics is harder for me than for most students.

Mean SD p

C&MTraditional

3.8 (3.6)

3.8 (3.6)

0.8 (0.9)

0.7 (0.6)

0.654

The C&M group showed a greater change in perceptions

did the traditional group on items 4, 9, 12, and 24. Their more

favorable reaction to item 9 could be directly reflective of the C&Mstudents' enjoyment in the course during the semester. At the same

time, the cooperative learning experiences of the C&M students

might affect their reaction to item 24 by reducing their worry that

mathematics is harder for them than for most students. Likewise, the

C&M group’s more positive reaction to item 4 can be easily explained

in light of the learning environment of the lab; help from lab

assistants was always available and collaborative problem solving

was common. Finally, the students' enjoyment and satisfaction with

the C&M course seemed to motivate further learning of mathematics

(item 12).

On the items asking students' competence (2, 22), the C&Mgroup showed contradictory reactions: increased competence based

on the responses to item 22, and decreased competence indicated by

the ratings on item 2. These inconsistent reactions to the two similar

questions might originate in the fact that the C&M students restricted

the term "mathematics class" on item 2 to classroom discussion

sessions. In the classroom meeting, the instructor usually previewed

the contents of the lesson students were supposed to work on. Thus,

the C&M students might have had difficulty understanding these

brief previews. Consequently, they answered that their

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understanding in mathematics class (classroom discussion session)

was not so good.

In both groups, there was almost no difference between the

responses to item 14 at the beginning and those at the end. In fact,

the statement of item 14--the only reason they are taking this course

is because they have to—was a kind of fixed fact that had nothing to

do with students' experiences during the semester.

Conclusions on the Attitude Survey

The overall response of the C&M group was more positive than

that of the traditional group. In particular, there were substantial

differences in the items of attitudes toward computers, cooperative

learning, and mathematics as a process: 3 items of attitudes toward

computer; 2 item of cooperative learning; and 1 item of mathematics

as a process all showed a significant t-test result at the .05 level.

However, the C&M students did not totally change their

perceptions and attitudes about mathematics and computers as a

result of their involvement in the course. The semester's experience

was just a small part of their entire mathematics experience. Thus,

they seemed to change their perceptions and attitudes gradually and

thoughtfully, not suddenly. Also the effect of the C&M course on

perceptions and attitudes undoubtedly interacted with other

variables such as learner characteristics.

Questionnaire

All the twelve students responded to the questionnaire. What

follows is the brief description of their replies.

1. Why did you choose this course and what did you expect to

learn in this course?

There were two important motives in choosing the C&M course.

Half of the students decided to take the course because their friends

spoke highly of, and recommended, this course; the other half chose

this course because they just thought it would be more interesting to

learn calculus with the aid of a computer. The students' expectation

of what they would learn in the C&M course was little different from

what the investigator anticipated. Part of the students thought that

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anticipated that the C&M course was just a regular course plus a

computer.

2. Did you have any computer experience before this course?

What kind of computer experience? Is that experience helpful for

this course?

Most of the eight engineering students answered that they had

had a substantial amount of computer programming experience,

especially, three of them had taken CS (computer science) 121 The

four non-engineering students had minimal exposure to computers

such as word processing experience. Thus, all twelve students

entered the course with at least a little computer experience. The

students added that their experience made them more comfortable

with computer manipulation, but it was not particularly helpful for

the course. In fact, the amount of their previous computer experience

had nothing to do with their success in the course in terms of the

scores on assignments and exams.

3. How long did it take for you to become comfortable with

Mathematica, and what particular problems did you have with it?

It took the students from 2 weeks to 4 weeks to become

comfortable with Mathematica. The students reported that they did

not have much difficulty from the beginning. The main problems

were with the commands, especially how to use those commands,

and the various options on the computer itself. One of the students

complained that:

"I couldn’t really get into the separate enter keys of all the fonts. It is as

confusing as a desktop publishing program."

4. How many hours per week have you usually spent in the '

lab?

There were a relatively large variation in the answers to this

question. The range of hours per week the students spent in the lab

was 6 to 12. As expected, the students who spent more time in the

lab tended to receive high scores; in other words, the time they

invested and the scores they got were parallel. In many regular

The course title is "Introduction to Computer Science." This course covers

PASCAL language.

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113

There were a relatively large variation in the answers to this

question. The range of hours per week the students spent in the lab

was 6 to 12. As expected, the students who spent more time in the

lab tended to receive high scores; in other words, the time they

invested and the scores they got were parallel. In many regular

courses, it is possible for bright students to perform well on exams

and assignments with little effort and time. But in the C«&;M course,

high scores almost always mean much effort and time.

5. Could you give me a concrete example in which learning with

Mathematica is particularly helpful? Be as specific as possible.

Most of the answers to this question were centered on the

advantage of visual presentations. One of the students replied:

"The visualization of functions being studied is extremely helpful. It

makes you understand what you are doing and why rather than just

how to get an answer. You don't have to waste a lot of paper and time

doing calculations which do nothing to aid in your understanding of the

topic itself."

About the specific topics in which learning with Mathematica is

particularly helpful, five students’ opinions were as follows:

"Studying power series would be difficult and boring without the

abilities to make quick plots."

"I think that Mathematica really gives me a good insight into

convergence intervals. The graphing capabilities and the ability to use

a large number of terms make the concept a lot easier to understand

than just formula."

"Visualizations are most beneficial in the learning of the trapezoidal

rule."

"Parametric equations. I saw the way the regular 132 students learn it

and I didn't like it all. Then I tried C&M and found it really wasn't that

bad after all."

"Expansion. It was really helpful to see on a graph what happens whenyou take 5 terms or 10 terms."

6. Do you agree that lots of plotting, calculating, and exploring

give a good perspective of the underlying principles? If so, which

Mathematica command or tool was the most helpful?

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114

All the students agreed that plotting and calculating activities

provide a good perspective of the underlying principles. One of the

students pointed out:

"By looking at many similar problems within a short time, it is easier to

understand exactly what is happening."

As the most helpful Mathematica conunand or tool, the students

listed Plot, Series, Expand, Integrate and ListPlot. Several

students applauded the command Show because it is possible to

overlay a number of plots and compare them.

7. Are you more or less confident in mathematics than you

were before taking this course? What aspects of the Mathematica

calculus course made you more (less) confident?

Nine students answered that they were much more confident

in mathematics than before. Their reasons for being more confident

included:

"When I am behind in a particular lesson, the computer lesson allows

me to spend more time on and review the lesson until I feel confident."

"Before I took this course, I just blindly followed steps to solve a problemwithout real understanding of why or where the steps came from. But,

now I can figure out the steps and understand the underlying conceptsbefore they are applied to problems."

"The overall structure of the course and the way it is outlined really

gave me conHdence."

In contrast, one student said that his confidence had not

changed and two students replied that they were less confident than

before. The reason for being about the same level of confidence was:

"... the lack of hand calculating made me less confident But I feel I

understand the concepts much better raising my confidence."

The two students who negatively answered complained that:

"Mathematica makes me lazy, I can't even remember really well how to

do simple integrals."

"Last semester I could visualize everything myself and do complexfunctions in my head. Now I can't."

1

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115

8. Have you ever used the computer and Mathematica to

explore mathematics beyond the requirements of the course

assignments? If so, what for?

The answers to this question were a little disappointing. Only

two students had used Mathematica for exploring the materials being

taught in other courses: one for PHYCS 106 and other for MATH 225.

However, most of the students had voluntarily used Mathematica

within the C&M course to check their answers by looking at

Mathematica plots. They commonly said that they wanted to explore

several things with Mathematica but they did not have enough time.

9. Do you think that your experience with Mathematica is

helpful in learning other computer programming languages like

Pascal or Basic? Do you think Mathematica is totally different from

other programming languages?

Three students simply answered that they were not sure

because they did not know any other programming languages. Six

students stated that there were probably enough similarities in all

programming languages that the learning of one would benefit in the

learning of others; but their knowledge of Mathematica was not

sufficient to give them an advantage in learning other languages. The

other three said that their experience with Mathematica might be

helpful in learning other computer programming languages. One of

them added that:

"My knowledge of recursion formulas from computer science made it

easier to understand iteration formulas for integration. So myexperience with Mathematica might also give me some benefit in

learning other programming languages.”

10. Some people say the Mathematica course and the

traditional course present different views on what calculus is. Do you

agree or disagree? Could you give me your reasons for agreement or

disagreement?

Most of the students agreed with the statement. As the reason

for agreement:

"I think that Mathematica presents a more real-world and useful view of

what calculus is rather than just how to solve specific problems which

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116

are quickly forgotten and easily misapplied. I feel like I am really

learning with Mathematica and not just memorizing certain things."

"I believe that the Mathematica course approaches calculus from a

calculational perspective instead of a theoretical perspective.”

"Yes, they do present different views. The traditional calculus stresses

the ability to use formulas, where the Mathematica course stresses the

concepts behind the formulas."

"The regular course is more "hands on” work and Mathematica does a lot

of the work for you."

11. Do you think you might have less ability in hand calculation

than do the traditional calculus course students? If so, have you

developed anything to replace the hand calculation ability?

Seven students conceded that they might have less ability in

hand calculation than do the traditional calculus students. However,

they believed that they might have a better understanding of the

concepts overall or how to approach new problems that they were

faced with. The representative response of the seven students was:" I probably do have less ability in hand calculation, but I believe I

learned Just as much. I think that using Mathematica helps you learn

more on how a problem solution develops - there's more thinking

involved."

The responses of the other five students were similar to this

student’s response:

"I don't feel that I would have less ability in hand calculation than dothe regular calculus course students. I might not be fast in calculating

problems by hand, but the literacy sheets make sure that I know how to

calculate problems by hand."

12. In your opinion, what's the strongest point of this course?

What's the weakest point of this course?

As the strengths of the C&M course, the students listed:

"It allows you to go at your own pace."

"Its strongest point is that there is a lot of help available. Since youwork in the lab on homework assignments, there is always help right

there."

"The explanations in Basics and Tutorial are very helpful and generally

quite understandable."

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"The C&M course gave me the feeling that I was really doing something,

not just plugging and chugging away."

"... repetitious lessons develop skills."

"The strongest point is that you get to see why things happen by

changing the equation slightly and seeing what happens."

"This course provides more common applications."

The responses to the weakest point of the C&M course were

centered on the fact that the course requires a lot of time in the lab.

An interesting pattern was noticed in the students' responses. The

students with substantial computer experience talked more about

the hard work required in the course than did the students with less

computer experience. Possibly this indicates that the students with

previous exposure to computers expected more of a free ride in the

course because of their experience.

On the other hand, three students talked about the lack of hand

work. They pointed out:

"The weakest point is not enough hand work or classroom lecture."

"Weak - lack of examples of how to do things without the computer."

"... lack of written out work."

Also two students who had Macintoshes at home complained

about the high price of Mathematica software. They said that if the

student version of Mathematica did not cost so much ($175), they

would buy it and do the assignments at home.

13. Do you like the format of the C&M lesson: Basics-Tutorial-0

Give it a try-Literacy sheet? Do you think there is some redundancy

in the four steps?

The students generally liked the format of the C&M lesson.

Their opinions were:

"Even though there is some redundancy in the four steps, each step is

beneficial."

"Yes, I do like the format because of how the problem progress step by

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118

The students generally liked the format of the C&M lesson.

Their opinions were:

"Even though there is some redimdancy in the four steps, each step is

beneficial."

"Yes, I do like the format because of how the problem progress step bystep."

"There is a significant amount of redundancy, but it would be difficult to

do many of the Give-it-a-try problems without the examples in Tutorial.

Also, this redundancy serves to reinforce the most important skills."

"I think that Basics and Tutorial could be combined but otherwise it is

OK."

"Sometimes, Tutorial is too weak to back up Give-it-a-try."

14. Are you going to take other Mathematica courses again like

MATH 242 or 245? Would you recommend this course to your

friends? Why or why not?

Seven out of twelve responded that they would take the C&Mversion of MATH 242, and ten out of twelve answered that they

would recommend the C&M course to their friends. Some of the

students added that they would recommend this course with a

warning that it would take a lot of time and effort each week. One of

the students who replied positively to both questions mentioned

that:

"Several of the department in the College of Engineering use

Mathematica. So I would recommend this course to my friends,

especially those in engineering. This course makes it possible to use

calculus to solve "real" problems which are very helpful in the Held of

engineering.

"

15. Do you have any suggestions for on-screen lessons,

assignments, classroom sessions, classroom exams, literacy sheets,

style of learning, instructor, or lab assistants?

Several representative suggestions were:

"One aspect of the computer assignments and literacy sheets which I

don't like is the fact that the first one or two problems are difficult and

require much work, while the following problems tend to be simple. I

find it discouraging when I need to struggle too much with the first

problem. I also realize that part of the reason why the following

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119

problems seem easier is because they build upon the skills developed in

the first problem. Perhaps it might be better if the first few problems

gradually lead the student into the new material, instead of assuming

that the student absorbed all the material presented in Basics and

Tutorial."

"I would suggest that, for the first day of a new lesson, the instructor gothrough the Basics section with the class together, and maybe add to the

statements and explanations so that we aren't missing entire ideas or

concepts which may be very important and even vital to completing the

assignment. Then we can easily understand what to look for and discuss

in the Give-it-a-try problems."

"My main problem in the beginning was the fact that I was really

unclear on what an answer to a problem entailed. There needs to be

more emphasis in the beginning of the semester on what is expected

from the student in terms of homework answers."

"This class is only 3 hrs but the assignments take up over half of mytime in comparison to my other classwork. I think that this class should

be worth more credit hours."

"Try to have an instructor present on top of the lab assistants."

Observation

Details of thg Observation

The First to Fourth Weeks

During the first week, most activities were focused on

becoming comfortable with the computer setting. As the investigator

expected, the beginning activities seemed not to be smooth because

the students needed to learn both calculus and Mathematica

simultaneously. For most of the students, this meant a little difficulty

and frustration at the start.

However, roughly more than half of the students already had

experience in dealing with Macintosh computers and those did not

have much trouble. Also the students with no Macintosh experience

were not seriously hampered by the manipulation procedures,

because they had also had a certain amount of computer experience

with other types of computers. Since the Macintosh computers and

the C&M lessons were fairly user-friendly, it was possible for the

students to use common sense in determining appropriate

procedures.

The small problems that the students encountered was a

message at the beginning of each lesson. Whenever the students

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120

open the lesson, the screen shows the message, "The document is

locked. You will not be able to save any change." The purpose of this

message is just to warn students that the lessons can not be changed

under the same file name, but can be saved as a different file name.

At first, this message appeared to perplex the students. One student

(E.S.) said that when she first saw the message, she thought she had

made a fatal error, but she got used to the message.

During the first two weeks, interactions among students did not

occur frequently. The students worked individually rather than

cooperatively. They just asked the instructor or lab assistants about

the technical problems in manipulation. Apparently, the students

found the first two weeks of the course stressful. For this reason,

their perceptions and success with the course seemed to be critical.

From the third week, the students' frustration began to clear

up. Most students seemed to be comfortable with manipulating

procedures. It was surprising that it took only a couple of weeks for

the majority of the students to be at ease with the technical details.

Being familiar with the computer procedures made the students able

to concentrate all their effort on learning the materials. Probably the

reason for this situation was that the C&M electronic text, which

showed Mathematica instruction in context, eliminated the need for a

separate unit on Mathematica programming.

Another noticeable fact during the third and the fourth weeks

was that the students started to interact with their classmates. But

the interaction was initially limited, in that most of the inter-student

talk dealt only with comparing the results on the computer screen.

Even though these activities could not be categorized as real

cooperation in its strictest meaning, the fact that they began such

interaction was a significant change.

The Fifth to Eighth Weeks

During this period, the investigator paid much attention to the

"black-box syndrome" (Kenelly & Eslinger, 1988)—students blindly

use the computer courseware without understanding underlying

concepts and procedures. Sometimes the students appeared to just

execute the Mathematica commands in "basics" and "tutorial" by

superficial mouse clicking, then watch the computational and graphic

i ^T“ lid) KWOJk Olb .’I|6)li34 -Vtl n:sq^^

•I'ii Ur : «»:. ^-?uaf5 till aw»« oi %idji od Jr?a UiV' -«aV

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121

results. They appeared to merely copy the command lines in "basics"

and "tutorial", and then substitute the numbers or formulas in "give

it a try". This behavior increased especially when the content of the

lesson was somewhat difficult. For example. Lesson 4.03

(Approximation by Expansions: Integration) and Lesson 4.04

(Convergence: The Explanation of Our Observations) were handled

during the fifth and sixth weeks, and the black-box syndrome was

recognized more frequently in rather difficult Lesson 4.04.

The Mathematica output in the form of plotting can be reduced,

magnified, or shifted by dragging the outside box. In the fifth week,

one student (A.B.) realized this function of Mathematica by chance

and seemed to be very curious about it. He experimented with the

reducing and magnifying of several plottings, and then conveyed

what he experienced to the next student. The next student also tried

that function with the same curiosity. From this scene, the

investigator noticed that the students were inquisitive about the

novel function of Mathematica.

Editing (cutting, copying, or pasting), and changing the face of

letters (plain, underlining, or italic) can be done both by pulling

down the menu to the desired one and releasing, and by typing

special combinations of the keys. These methods in the C&M context

are exactly the same as those in Microsoft Word. At the start, most of

the students used the method of pulling down and releasing, but the

number of students using the convenient key combinations gradually

increased. In the eighth week, almost all the students used the key

combination method.

The Ninth to Twelfth Weeks

As the semester reached the middle phase, the students

abandoned the strange feeling about the lab, and their relationship

became intimate. The students seemed to think of the computer lab

as a comfortable place to work in the company of their classmates;

they appreciated the support and encouragement of the instructor

and the lab assistants.

The observation was done three times a week, and the

corresponding three different pairs of lab assistants helped students.

Actually, among the four assistants (one male and three female)

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involved, one female assistant was teaming up with the other three

assistants. When the students had a question, they raised their hands

and waited until the available assistant helped them. Thus, there

should be no difference in the frequency of contacts between the

assistants and the students because every two lab assistants had an

equal chance of interaction. However, according to the records of the

frequency of the lab assistants' contacts with the students, the

students apparently preferred one particular lab assistant.

Table 41

Frequency of the Lab Assistants’ Contacts with the Students

Date Monday Wednesday Friday

Assistants S.M. MJ. S.M. J.W. S.M. J.G.

Gender female female female female female male

Frequency Ranking 2_ 4 _J 1 2

It was hard to figure out the reason for this preference because

the lab assistants seemed to have about the same kindness and

willingness to help. In their knowledge of mathematics and

computers, they were also at the same level. The investigator

continued to focus on their attitudes toward students and finally

found the clue. At a glance, there was no noticeable difference, but

there were subtle differences between the lab assistant with more

frequent contact and those with less frequent contact. The most

popular lab assistant (J.W.) among the four often used expressions

like "I also had the same difficulty when I took this course", "That's

right", "So far so good" to encourage the students. Also she would ,

stay with one student longer than did other assistants; she did not

simply point out the wrong ideas or commands, but instead elicited

ideas from the students to guide them to correct ideas. Such delicate

differences produced a big difference in the frequency of the

contacts. Presumably, this also made a big difference in the impact

the lab assistants had on the students.

In the meantime, the more frequent contact with the male

assistant (J.G.) cannot be explained by the same logic, because the

•ait t *Uo difw q#* 3iQifn*A; '*9^ s*Um*'*) .bjî/lovw

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123

other female assistant (S.M.) kept a more favorable attitude toward

the students and sympathized with their difficulty. Presumably, the

students had the stereotype of male superiority in mathematics. To

sum up, for students, lab assistants' gender and attitudes toward

them seemed to be the criteria which determined to whom they

asked for help.

On the other hand, the investigator realized that students often

preferred to use the command they learned first. For example. The

commands Integrate//N and NIntegrate produce exactly the

same result. A student (K.K.) continuously used Integrate//N

although NIntegrate was more convenient. The probable reason is

that in the courseware, Integrate[function, {x, min, max}]//N

was introduced first, and then NIntegrate[function, {x, min,

max}] appeared with the simulation of the trapezoidal rule and the

area under the curve.

The distribution of the students' majors was eight from

engineering, three from liberal arts and science (one, psychology;

one, biology; and one, mathematics) and one from business

administration. The students certainly showed strong interest in the

problems which were relevant to their majors. Accordingly, they did

a good job with the problems in their fields. For example, the student

from business administration quickly finished problem G.8. in Lesson

3.07 (Appendix F), the application problem of finance. The eight

engineering students seemed to confidently complete problem G.6. in

Lesson 3.07 (Appendix F), the application problem dealing with error

propagation via iteration.

In the courseware, most of the application problems in the

fields of science, engineering, and business can be understood with

common sense, but some of them require a certain amount of

background knowledge. When the students confronted the problems

requiring specific knowledge, they sought help from the expert

students majoring in those fields. In this context, the class

encompassing students with mixed majors offers the advantage of

peer tutoring.

On the other hand, no C&M course can cover all the content and

problems in the courseware; parts must be skipped. Hence, it might

41

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124

also be desirable to open the course for students in specially related

fields such as electrical engineering and computer science. In that

case, the curriculum could be adjusted to be field-oriented. The

emphases of calculus in engineering, business, mathematics, and non-

mathematical science are not identical. Several sections with

different focuses within a course might offer a more profound basis

for students' further study because such sections could deal with in-

depth concepts required for each major.

During the final two weeks, the students studied Lesson 2.08

(Parameters) and Lesson 3.08 (Substitution). In Lesson 2.08, there

were various types of parametric curves including spiral, hyperbolic

spiral, cycloid, ellipse, hyperbola, folium of Descartes, cardioid,

limagon, three-leaved rose, four-leaved rose, and lemniscate.

Although those curves were introduced as a tool to explain the

properties of parametric equations, plotting the curves also provided

a secondary benefit to the students. As finals got closer, the students

felt the pressure of final exams and were a little depressed.

Consequently, during this period, the fascinating curves could be a

refreshing distraction for the students who really enjoyed plotting

them.

Observation Notes

Frequencies

The behaviors of the students and the instructor, and their

interactions appeared to be most active on the days prior to the

homework due date and the exam date. For this reason, the

observation note was filled out (Appendix H) on the days

immediately preceding each homework due date and the exam date.

The investigator's major concern was whether the elapse of time

affected the distribution of activities across three categories and

corresponding subcategories.

First, the frequencies of the instructor's statements (Figure 16)

and the contacts between the students and the assistants (Figure 17)

were almost constant.

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125

Figure 16. Frequency of the Instructor’s Statement

Note . The thicker line represents the frequency of schedule

announcements and the thinner line represents the frequency of

comments on lessons.

frequency

Figure 17. Frequency of the Contacts Between the Students and the

Assistants

Note . The thicker line represents the frequency of student-initiated

contacts and the thinner line represents the frequency of assistant-

initiated contacts.

Second, in the category of dyadic contact between the students

and the instructor, there also was not much difference (Figure 18).

The total frequency of two sub-categories (student-initiated and

instructor-initiated) of the student-instructor contacts was almost

steady, even though there was alternation in each sub-category. Less

student-initiated contact means more instructor-initiated contact

because the frequencies of the two sub-categories were

complementary. This implies that the instructor could contact a

limited number of students during the lab hour.

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126

Figure 18 . Frequency of the Contacts Between the Students and the

Instructor

Note . The thicker line represents the frequency of student-initiated

contacts and the thinner line represents frequency of instructor-

initiated contacts.

Third, there was significant change in the category of students'

interaction (Figure 19). The students' interaction increased by

degrees both in pairs and in groups. Although the general trend of

interaction was growing, there were some fluctuations according to

the difficulty levels of the assignments. Tough homework

(Assignment #6) naturally provoked more interaction and

cooperation among the students, while simple homework

(Assignment #7) was usually done individually.

frequency

Figure 19 . Frequency of Students' Interaction

Note . The thicker line represents the frequency of interaction

between two students and the thinner line represents frequency of

interaction among more than two students.

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Except the change of frequencies, there were a couple of

noticeable facts. One was that the purpose of the dyadic contact was

gradually shifted from questions about computer manipulations to

questions about the content of the lessons. The other was that on the

day before the exams, not much interaction among the students or

with the assistants was recorded. Instead, the frequency of the

dyadic contact with the instructor increased. This apparently

indicates that as far as the written exam is concerned, the students

prefer to contact the instructor.

Manipulation Difficulties

From the beginning, the students had only a little trouble in

computer manipulations and were able to overcome those difficulties

within a couple of weeks. Among the minor difficulties, the following

six appeared to be the most frequent.

First, the students made syntax errors such as starting the

command with a small letter, or using a parenthesis instead of a

bracket. Particularly, the students with previous computer

programing experience seemed to have more confusion because they

had been accustomed to different syntaxes.

Second, one of the most frequent errors the students made was

to skip the preliminary steps and execute only the crucial command

lines. As a consequence, they got strange results. In order to avoid

this, they should follow the lesson step by step and execute all the

commands to get the desired results. The students often presumed

that unexecuted commands were already executed because they

could see the commands which were already installed. The function

of the semi-colon might have compounded this confusion. When thp

students used a semi-colon at the end of the command line, the

screen did not show the result even though the command was

executed. They could only differentiate whether the command was

executed or not by checking input numbers. However, it did not take

much time for the students to overcome this error by getting

accustomed to following and executing every command.

Third, the other common error the students made was not

clearing the variables by the command Clear[van‘a^/£s]. The

instructor told the students several times to clear all the variables

&

.21

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128

previously used. But the students repeatedly forgot to clear the

variables, and this caused the strange results. Sometimes students

could get the desired results without clearing the variables, and

probably these experiences created the misunderstanding that the

variables did not have to be cleared every time. Although this error

lasted longer than others, this was also transient like the previous

ones; after a month, the students did not make this mistake.

Fourth, besides the syntax errors described above, one last

major source of student difficulty was in the determination of

appropriate scaling along the x-axis and the y-axis. This difficulty

had been expected because the students tried to choose a scale that

illustrated prominent features of a graph without understanding

where those features occur. When the students confronted unsatis-

factory graphic results in early lessons, they tended to conclude that

they had made errors in the input of the function and not in the

scaling. After the students had had some practice with the graphs of

the function, they became adept at reacting appropriately by

deciding to change the scales. Thus, the students appeared

reasonably able to overcome the difficulty in scaling. If this difficulty

did remain unsolved, the reason was that the students neither

analyzed the algebra properly nor thought carefully about how the

change of scale affected the shape of a graph.

Fifth, when the students used the Macintosh SE-30, they were

inconvenienced by the small screen; the letters on the far right did

not appear, nor did the reducing and magnifying box. In that case,

the students had to move the entire lesson on the screen to the left

to find the reducing and magnifying box. However, this inconvenience

did not cause much trouble in the scheduled observation. The

students could almost always secure the Macintosh II or IIcX with

large screens because there were not many students (usually 12

students in the observation section and 3-4 students in other

sections).

Sixth, almost every student asked the lab assistants, the

instructor, or their classmates how to change the input mode to the

text mode, and how to type the superscript and subscript. Even

though those questions are trivial, they are absolutely necessary.

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129

Because the text mode, the superscript, and subscript were used

almost all the time, the students never inquired twice once they

were familiar with those techniques.

Interactions

One of the most noticeable features of the observation notes

was the students' interaction. Early fears before observation were

that the computer would teach the student who worked and listened

in quiet isolation; no talking would be needed once the student

started the investigation through computer lesson. Contrary to these

fears, the use of computers tended to increase rather than decrease

the students' interactions with their classmates. In the lab,

considerable mutual support and camaraderie developed among the

students. The students seemed to feel free to ask their peers for help

and enjoy the camaraderie of working in natural groups. The

students clustered together, sought advice from peer experts, and

showed off their products. Unlike written sheets of paper, writing

and graphics on a computer screen were looked upon as public

information.

At the beginning, the students consulted the instructor or the

lab assistants for guidance on problem solution, for corroboration of

the result, or for interpretation of alien output. As the semester

progressed, the instructor and the assistants were no longer the only

source of authority. By the middle of the semester, the students

seemed equally comfortable in enlisting the help of their colleagues

when they confronted difficulties, although the instructor and the

assistants were their first resort for checking answers or errors.

Apparently, they recognized the strength of their colleagues through

the informal group structure that arose during the semester and they

actively sought opportunities for consultation with their classmates

about assignments. By the way, the students showed two types of

accommodation for cooperation; some students worked only with

specific classmates; and the others floated back and forth among

several students.

Researchers suggest that a wide range of students in cooperative

learning situations have significantly higher achievement gain in

mathematics. The results of the collaborative projects are very

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130

positive (Adams & Hamm, 1990): students feel more satisfied and

perceive their work as less difficult; students take responsibility for

their partner's understanding as well as for their own; and

cooperative learning adds to students' and teachers' enjoyment of

learning and improves the educational experiences of both.

Experiencing the "synergic" results of group work is rewarding and

personally empowering. Making use of collaborative learning

strategies benefits almost everyone. Preventing gifted students from

becoming bored also can help not-so-academically-talented students

learn. By collaborating with other students, the gifted student can

help others move foreword while preparing for the leadership roles

that are so important to his/her future.

The investigator could confirm those advantages by observing

the lab. The students frequently conversed and discussed topics with

classmates, and their conversations and discussions seemed to be a

primary source for the understanding of calculus. If the C&M and the

traditional course are compared to music, the former sounds like a

choir performance while the latter sounds more like a solo

performance. Peer tutoring or coaching was found frequently, and

peer sharing in a supportive small group was common. The students

got help from peer experts as well as from the instructor and the lab

assistants.

The exceptional students in the class seemed to gain more

understanding by working with a wide range of peers. Certainly the

less capable students benefited by talking and discussing with the

high level students to develop ideas and solve problems. Above all,

the cooperation seemed to be most beneficial to the average

students. The probable reason is that the average students can

improve their mathematical understanding if a little aid is provided.

Actually, the student in the middle ranking, D.B., made much

progress. The investigator witnessed his working with other students

frequently and later found that his performance on assignments and

tests was much improved.

Observation in the Classroom

The students and the instructor regularly met in the classroom.

The activities in the classroom varied according to the progress of the

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131

laboratory work, but the main activities were lectures, discussions

and exams.

In lectures, the instructor, in the traditional manner,

summarized the content of the previous computer lesson that the

students had completed, and briefly previewed the content of the

next computer lesson that they would work on. The summary of the

concepts in the previous lesson seemed to make the students

consolidate the concepts they had learned, and the array of the

crucial concepts in the following lesson seemed to make them

organize the the concepts they would learn. From the change of the

students' facial expressions, the investigator noticed that the

instructor's explanation provided them the global portrait of the

concepts which had been already acquired and would be acquired

through computers.

In discussions, the instructor or the students put forward some

problems or topics, and the instructor diagnosed the students'

current knowledge, and then, based on the diagnosis, explained the

ideas mainly by dialogue. Sometimes, the students asked questions

about difficult problems in assignments, and talked about their ideas.

Even though there were some individual differences in participation,

every student tried to take part in the discussion.

Two different instructors were involved in the pilot study and

the main study. According to the style of the instructor, there were

some differences in the activities and atmosphere of the classroom.

The professor involved in the pilot study strongly emphasized

learning with computers. He usually used the computer and the

overhead projector in the classroom to demonstrate and explain

concepts. There were more discussions than lectures. On the other

hand, the other professor, who taught during the main study, focused

both on learning with computers and hand calculation. He used the

blackboard and chalk rather than the computer and the overhead

projector for explanation and demonstration. In addition, most of the

classroom sessions were executed in the form of lectures.

Accordingly, the atmosphere of the classrooms were quite different;

the atmosphere of the class during the pilot study was lively and

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liberal, while that during the main study was rather calm and

organized.

Based on the responses to the questionnaire, classroom

observation, and informal talks with the students, the investigator

found that several students worried that they might have less

computational skills than the students in the traditional course.

Attending the classroom discussion gave the students a feeling of

relief because they felt that the classroom meeting compensated for

the weakness of computer instruction. The students apparently

preferred the computer lessons, but they seemed to still have a

stereotyped idea that mathematics problems should be solved by

hand. The students appeared to enjoy the learning in the computer

lab, and at the same time they appeared to be mentally satisfied by

attending the classroom discussion and solving "literacy sheet"

problems by hand.

Conclusions on the Observation

Exploratory View

Greeno (1990) contrasted two different roles—didactic and

exploratory—for computers in education. From the didactic

viewpoint, computers are a tool for presenting instruction in a

systematic and individualized way. The main goal of this approach is

to provide for effective learning with minimum error on the part of

the students. From the exploratory viewpoint, instruction is treated

much less systematically. The computer system affords the

environments in which students can investigate through interactions

and experimentation.

The didactic view is based on a theory that considers cognition

as a system of knowledge structure and procedures, and learning as

acquisition of cognitive structures and procedures. On the other hand,

the exploratory view reflects a theory that considers cognition as

activity and learning as a strengthening of capabilities for situated

activities.

After observing the calculus lab for one semester, the

investigator realized that the viewpoint of the C&M course was close

to the exploratory perspective. The course placed the greatest value

on students inquiring and investigating ideas, with the instructor

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133

playing the role of a guide and a coach. The learning environment of

the C&M course provided a situation in which students were engaged

in different activities, often working together, and calling on the

instructor to help when they needed it. This was far from the

didactic-structural approach, which placed the greatest value on

teaching through heavily-structured, "canned" lectures with students

carefully paying attention to the instructor or working quietly and

individually.

Instructor’s Role

The interactive nature of the C&M course spontaneously

changed the role of the instructor to one of coach or resource person.

This, in turn, shifted the instructor's focus from teaching to helping

students learn how to learn. Helping and working with the students

in the lab seemed to provide the instructor a better insight into the

level of student understanding than that obtained through a

traditional class or test. The instructor was fairly well acquainted

with the details of the individual students' ability, interest, and

progress. Part of the reason was that the class size was small, but the

main reason was that the laboratory setting forced the instructor to

interact with the students more openly.

The semester of the main study was the second time the

instructor had taught the C&M course, but he had had plenty of

experience teaching regular calculus courses. The instructor always

tried to give individual attention to every student. When he

contacted the students, he listened to their questions or ideas

carefully, indicated the points of agreement, and then corrected their

thought without criticism. He usually gave positive reinforcement to

the students and praised them to promote their self-confidence;

moreover, his words never sounded like mere lip-service. His

language and behavior seemed to provide an emotionally supportive

lab and classroom environment for his students.

Lab Assistants' Role

Twenty-one lab assistants and one technical assistant worked

in the lab which was opened 78 hours per week. The lab assistants

were chosen from among undergraduate students who had

performed well in the C&M course. Their essential role was to help

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134

students with technical details and to explain mathematical ideas to

them.

Each pair of lab assistants was assigned for scheduled

observation. They knew each other fairly well and the members of

all three pairs worked well with their partner. They paid careful

attention to students' performances in the lab and were eager to help

them. They seemed to be very enthusiastic about their role in the

lab.

The quality of the lab assistance was generally good. There

appeared to be no serious problem in the lab assistants' attitudes

toward the students. However, the lab assistants sometimes just

pointed out the wrong commands or ideas, and directed the correct

information to the students. It was recommended by the investigator

that they be trained to use questioning techniques to induce the

students' reflective thought. The positive or negative words of the

lab assistants had influence on the students' conftdence. Thus it

would be better if the lab assistants were disciplined to use tips to

promote the students' self-confidence. To be more helpful lab

assistant, it is necessary for the lab assistants to have a broader

educational perspective.

The lab assistants were undergraduate students one or two

years older than the C&M students. As lab assistants, undergraduate

students were not always inferior to graduate students, because the

knowledge of mathematics and that of mathematics education were

in a different dimension. Even though undergraduate assistants

might have less mathematical knowledge than did graduate

assistants, the former did not necessarily have less knowledge of

mathematics education. In some aspects, the undergraduate

assistants might understand the difficulties of the students more

easily than did graduate assistants because they had about the same

level of mathematical understanding. Moreover, the students

considered an undergraduate lab assistant as one of their colleagues

and felt free to ask for help. If only graduate assistants were hired,

the students might hesitate to seek help because they were afraid of

asking a stupid or trivial question.

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135

In the C&M course, the role of the lab assistants could be

compared to that of a conu’ade or a teammate, while that of the

instructor might be likened to a mental coach who facilitates his

students' inspiration. Sometimes this classification was meaningless—

the lab assistants played the instructor's role, and vice versa. To

conclude, the lab assistants were another crucial contribution to the

C&M course.

Black-box Syndrome

There has been no consensus as to whether the black-box

syndrome is beneficial or harmful. Students do not have to know

every underlying concept and detailed procedure. A certain amount

of "automatized process" or "simplification" is necessary for students

to grasp the global picture of calculus. In light of this, the black-box

syndrome has a positive effect. However, computers should not be

used to simply calculate and plot what students used to do by hand.

Students should think about the process of why specific commands

are chosen and how computers work, and they should spend time

thinking about the meaning of numerical and graphic results. Based

on these facts, the black-box syndrome is a somewhat undesirable

phenomenon.

On the other hand, the black-box syndrome is not an exclusive

phenomenon of learning with computers. Learning without

computers also includes the black-box syndrome to a certain degree;

the typical example is the blind use of the Chain Rule. Students do

not always think about underlying concepts and procedures. Every

student has his/her own "automatization" or "default" during

calculation, and such a simplified process is necessary for efficient

learning.

In sum, like the black-box syndrome in traditional learning,

that in the computer laboratory approach can be considered as a

necessary evil which has both benefits and drawbacks; thus, the

investigator would call it "gray-box syndrome." Learning calculus

with computers involves the negative aspect of blind execution

without understanding as well as the positive aspect of reducing the

thought process.

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136

Solution Files

The assignment problems in "give it a try" and "literacy sheet"

required the students to analyze problem situations rather than to

follow routine procedures. The questions asked the students to

demonstrate mathematical ideas in several different representations

—numerical or algebraic results, tables, and graphs, and to explain

the details of each step.

Findings from the Solution Files

Students* Writing

Students in the C&M course frequently were asked to write

explanation of what they were doing and what was happening in

their answers. The answers required in traditional calculus courses,

mostly composed of symbols, are rigid, brief, and do not give

students much opportunity to talk about their own thinking. In

contrast, an important component of the C&M course was students*

writing in detail about their mathematical thinking and the

explanation of their answers. These activities seemed to facilitate

students* mathematical understanding and force them to reflect what

they were doing.

What follows are the examples of students' writings in early

assignments.

Question (Lesson 4.02, G.3.e.);

Write a few words on how you think the plot of the sum of the

beginning terms of the expansion of a given function is related to the

plot of the given function.

Answer:The beginning terms of the expansion of a function are the mostimportant for the shape of the plot especially close to 0. The effect of

using more of the terms of the expansion increases the cohabitation of

the plots for intervals of x.

The beginning terms of the expansion are the least precise ones. Theyset the guidelines by which the rest of the expansion follows. Because of

this, they can be used to approximate, but only in a very small andlimited interval around 0. The more terms you use, the more you are

defining the plot, and the more accurate plot you will have. It's like fine

tuning a television, the more you goof with the dial, the more youdiscover that no resolution is perfect, and that there is always room for

improvement.

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Question (Lesson 4.03, G.l.b.ix.):

Describe what seems to be happening.

Answer:As more terms of the expansion of Sin[x] are used, the range of x-values

for which the expansion gives a good approximation of Sin[x] increases

in both the positive and negative directions.

In early assignments, the students wrote explanations of the

problems when clearly required to do so as in the above questions.

But in the usual questions, most of the students did not sufficiently

explain why they were doing each step and what they found. They

just briefly showed the results by picking up the appropriate

commands and executing them. The instructor continuously asked

the students to include their own explanations, observations, and

conclusions. As the instructor requested, the students began to write

more thorough explanations of each question. But the next problem

was that the use of terms in writing was often imprecise, and the

expressions were sometimes vague. For example.

You could find a more accurate approximation of Pi by expanding

ArcTan[x] infinitely [instead of "to a larger number of terms"].

When we see more factors [instead of "terms of the expansion of Sin[x]"]

the graph gets better [instead of "of the expansion is a better

approximation to the graph of Sin[x]"] . . .

As one increase the range of x value one must also increase the

magnitude of the power [instead of "number of terms"] used in the

approximation. . .

The polynomials we create are good for limited regions of the graph

[instead of "intervals"] around the origin.

... the limit is 10/(-Tf + Tf) + 0 + 0 + ... = undefined [instead of "limit does

not exist"]

From the traditional viewpoint, this writing style is not

acceptable because the students used their own language rather than

mathematically correct language. However, if we consider the fact

that calculus students do not always have to mimic the precisely

defined terms and subtle expressions used by pure mathematicians,

this writing style should not be discouraged.

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yir *511 *11(1? lofl hU) •xP V^^Jao/n ,ejioa«3Dp tcupo arff lit («fi

•aiTr Unuol boi q^9 ank* ai>w /itefiira^

oiRmoT^Qu 3(0 qo I'thb^ Uiinisi wti jati|

!>3Ju yUjocrttinoo idhioiuni ftiH m^rfi it i/ii/a«a bnn glntwnnsoo• ^ rv^

t^s jiOoDcviaaifci vaoiiiKiuicÂ) • tfi .\i;iirt*bi <jJ msrf^bW^

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»,ist ^»o.‘ "sbiBotK? •!’'w 1i .nv"v«f‘irt

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9d icTO tdiiod: alvli anii'r/i 'fldl

138

After the students overcame two difficulties in writing—the

lack of explanations, and the ambiguous terms and expressions, their

writing was gradually improved.

According to Small (1991), there are two levels in writing: a

"rough draft" aids a student in organizing thought and formulating

problem approaches while a "polished report" requires the student to

describe a logical reasoning process to make connections to interpret

results and to make conjectures. The C&M students sometimes

outperformed and reached the latter level answer in the questions

that require the former level answer. For example, in the problem

for explaining error propagation via iteration, most of the students

made the conclusion at the level of "polished report." Here are two

samples of their writing.

The original error builds upon itself as the iteration progresses. Small

errors at the beginning result in larger errors at the end. For this

reason, small errors at the beginning are definitely something to worryabout.

As the iteration progresses, the effect of the original error is magnified.

The larger the n value becomes the larger the error becomes. For small

n values, the error is small. For large n values, the error is large. Small

errors are nothing to worry about in the beginning of the iteration

process. However, they make a significant difference towards the end.

Discrepancy Between the Scores on "Give it a Try" and "Literacy Sheet"

Generally, the ability to solve problems by hand is a stepping-

stone to successful computer work, and the understanding acquired

through computers strengthens the ability to calculate by hand. Thus,

the ability to solve problems by hand and computer work reciprocally

help each other.

For most of the students, the scores on "give it a try" were

comparable to those on "literacy sheet". However, some of the

students showed inconsistency between the scores on the computer

assignment (give it a try) and the hand-written assignment (literacy

sheet). For instance, J.G. received 23.5 points (ranking him tenth out

of twelve students) in the "give it a try" but marked 17 points

(ranking him first out of twelve students) in the "literacy sheet" in

Assignment 2.

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lof jtftr^j laf/f aiiiiiw'tj b»?ido»r 0.1 ,j^’'fwlsni to=i

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139

The most probable reason for this discrepancy is the students'

innate strength in one of the two; some students are better in

computer work than in hand computation, and some are not. Based

on observation, the other possible reason is that the students with

higher scores on "literacy sheet" and lower scores on "give it a try"

have a tendency to spend less time in the lab. Assuming that a

student grasps the basic concepts of the lesson, s/he is expected to

invest much time in showing a complete understanding of the

computer assignment but not necessarily of the hand-written

assignment. J.G.'s discrepancy is better explained by the second

reason. He appeared to complete computer assignments faster than

others. He certainly understood the lesson (as we confirmed in his

high "literacy sheet" score), but he neither paid careful attention nor

spent enough time on the "give it a try" assignment, which required

much attention and time.

Intimate Relation Between the Students and the Instructor

In their solution file, the students usually wrote their personal

experiences. For example:. . . Sure, 1 am interested in learning more, but please slow down. I amlosing my grip on this math. The past three days (and the past few months)I have spent every possible moment in this lab. I am going to go nut . . .

From this kind of message, the instructor realized the students'

difficulties or excitement that otherwise could not be expected. This

realization was reflected in the discussion session and the revision of

the courseware.

According to the students' messages, the instructor also sent

back to them the response messages as well as the grades on their

assignments. For instance:

Presentation 10/10 Mathematics 10/10

Comments: Your work is very good. We are slowing down: don’t panic.


Comments: Your mathematics is fair. Your answers are rather obscure.

Your presentation needs work


Comments: This is certainly much better work than you did on the last

lesson. You have the ability to do even better work when you apply

yourself.

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Comments: Your work has been excellent lately - keep it up!

Moreover, the communications between the students and the

instructor seemed to make their relationship intimate and informal.

Courseware

Ch aracteristics

After reviewing the C&M courseware, the investigator found

that this courseware was written with four related objectives in

view: applications, readability, flexibility, and concreteness.

Applications

The power of calculus lies in that it provides precise answers to

realistic problems. The C&M course places emphasis on concrete and

realistic examples and problems that serve both to demonstrate the

applicability of calculus in the investigation of scientitic questions

and to promote the development of calculus theories.

Various types of realistic problems are contained in the C&Mcourseware. The scientific problems found in the C&M courseware

include:radioactive decay, nuclear reactors, predator-prey models, spread ofinfection, pressure altimeters, pollution elimination, reflecting

mirrors, highway construction with splines, programming a robot to

use a router to cut given shapes, safe trajectories for airplane landing,

fluid flow, projectile motions, chemical reactions, spirals and cams, andsafe drug doses.

The economic problems in the C&M course include:

continuous compounding, financial planning for college expenses,

national debt projections, inflation studies, car loans, annuities

mortgages, actuarial calculations, and the ripple effect of spending.

All of these problems are introduced and supported by real data, so

students can see that they are dealing with important problems of

the real world.

There are big differences in the complexity of problems as well

as the problem types. The problems in traditional calculus courses

are usually limited by artificial constraints. For example, all

coefficients are small integers, all polynomials have integer roots, or

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t/ih'y nli It tiWAioor •ntî-' hithq ‘ibklU^t'^ls zwohif^

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141

all complicated formulas are reduced to a simple formula. These

kinds of problems are unavoidable because, if not, the required hand

calculations are beyond the scope of the students' abilities. With the

calculation and plotting power of Mathematical the C&M course deals

with real data and problems not deformed by the students* weakness

in manipulative skills.

Readability

Many of the traditional mathematics textbooks are full of

carefully and precisely defined words that are used by pure

mathematicians in their most subtle and difficult analysis. In some

cases, difficulties in learning mathematics originate in language

difficulties. The language of C&M courseware is relatively clear and

simple so that the students can easily read it. Also the C&M course

maintains an informal writing style which makes the students

comfortable and makes the material less intimidating. For example:Hot ziggety! (p. 50).

How sweet it is! Numerical and graphical prospecting pays off (p. S3).

Nifty! It sure looks as if g[x] doubles on intervals of len^ 2.1. Can yousee why? (p. 110).

You can't handle this differential equation now, but stick around for a

while because we'll get to it later (p. 114).

Flexibility

With regard to the selection, sequence, and treatment of

mathematical topics, the C&M approach is flexible. For example, if the

instructor does not like a particular topic or method, then s/he can

replace it. Similarly, students can do the same type of editing

whenever they like. Students can make their own summary or core

notes by cutting, pasting, and adding from the C&M courseware. They

can use the powerful color ability of the computers for highlighting.

At the same time, flexibility can be applied to increase the

quantity and variety of problems. One of the common beliefs in

mathematics education is that the more problems the students work,

the better they understand the concepts involved. Assuming this, the

C&M course is an effective tool for instruction. The C&M courseware

can be considered as a kind of "problem bank." Students can change

the data of problems and repeatedly practice similar problems until

they are confident in using the solution techniques. Instead of a

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,

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single printed problem, each problem can be expanded indefinitely

by reexecution with different parameters, ranges, functions.

Concreteness

Most of the traditional calculus books start by considering an

abstract concept and then moving to definite and concrete examples

of the concept. Like the several recent calculus reform projects, the

C&M courseware takes an alternative approach—starting from

concrete examples and exhibiting the process of abstraction to isolate

the concept, and finally the subsequent extension and generalization

of that concept.

The C&M courseware uses calculations and plotting as dynamic

devices to explore and discover concepts such as the meaning of

derivatives, exponential growth, the Mean Value Theorem, the

Fundamental Theorem of Calculus, and the convergence of series.

Based on their observations by plotting and calculations, the C&Mstudents can announce correct results and explain them in their own

words. As a consequence, they are quite convinced and fully

committed to the validity of what they have found (Brown, Porta, &Uhl, 1990).

The C&M courseware neither presents nor asks for formal

proofs. The reason for this is that formal proofs seem unrelated to

the successes of calculus learning, and clear intuitive arguments are

quite adequate. Therefore, the C&M courseware includes intuitive

proofs instead.

Evaluation

The investigator tried to evaluate the C&M courseware

according to the evaluative criteria suggested by Bangert-Drowns,

and Kozma (1989) for instructional software:

I. Operational Criteria

A. User-friendliness

B. Speed of execution

C. Treatment of operational errors

II. Instructional criteria

A. Presence and quality of pre instructional introduction

B. Degree of learner control

C. Frequency and variety of practice exercises

D. Use of a variety of symbols

E. Motivational quality

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143

User-friendliness is perhaps the most important of all the

operational criteria because it pervades the operation of the entire

software. Students can see Mathematica commands and routines in

context and pick them up very quickly. They are able to operate

Mathematica with only minimal training and minimal dependence on

manuals. After a couple of weeks, syntax is not much of a problem at

the level needed for successful performance. They can figure out

commands “intuitively”; that is, commands which are designed to

meet the expectations of the students require little effort.

Delayed computer responses have a detrimental effect on

motivation, creating a frustrating and boring learning environment.

However, Mathematica is one of the fastest computer algebra

systems. Students can promptly see the results of calculation and

plotting that they want. Thus, the C&M courseware minimizes the

amount of time and attention required to learn calculus and

maximizes the amount of time and attention directed to calculus

content.

Operational errors are mistakes that students make while they

are using software, such as pressing the wrong keys, typing incorrect

commands, or using incomplete or redundant brackets. Mathematica

(on Macintosh or Next computer) is very flexible regarding the

treatment of such errors. Students can easily cancel any commands,

undo mistakes, or edit entries. The computers generate some error

beeps when students do unexpected activities, place the cursor on

the incorrect part, or show each unit of the bracket by highlighting to

confirm the correct command line.

The C&M courseware presents the "guide" at the beginning of

each lesson that states instructional objectives or provides pre

instructional introduction to prepare students for the content that

they will be studying. One thing that might be suggested is to include

more content relevant to previous and subsequent lessons. This

simple strategy could stimulate cognitive processes and schemata

that would foster efficient integration of new information with

knowledge already in long-term memory.

As to control, the C&M courseware appears to be flawless.

Students can determine their learning pace and sequence, select any

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144

lesson, and stop it at any time. Such controls are expected to facilitate

learning by increasing a student's sense of competence and by

encouraging initiative and inquiry. However, it is important to keep

in mind that too many options might be overwhelming for some

students because they do not always make optimal self-instructional

decisions. Courseware should strike a balance between directed

purpose and learner freedom.

The C&M courseware has two steps of problem sets: "give it a

try", and "literacy sheet". Practice examples and exercises are

frequent and range from questions that require simple labels to

others which require complex inferences, predictions, calculations,

and plotting.

The variety of symbols used to present information seems to

have profound effects on students' learning. Since Mathematica has

superb graphic capabilities, most of its presentations are coordinated

with plottings. Through this medium, students have rich

opportunities to receive, transform, and encode what they have

learned. To fully make use of the multi-representation capability of

Mathematica, including more problems which can be solved by

various methods (numeric, approximation, and plotting) is

recommended.

The motivational quality of the C&M courseware appears to be

outstanding. Challenging real-world problems are the device

employed to motivate students, and the use of attractive (even

three-dimensional) graphics, colors, and brightness provokes

students' imagination. However, the investigator wants to suggest the

addition of one or two game-like problems in each lesson. An

instructional courseware with too many such problems may distract

students from attending to important instructional elements, but a

limited number of game-like problems with occasional challenges

and surprises is a good device to stimulate students' learning.

Drop-out Pattern

Table 42 presents the number of students who started (initial),

added, dropped, added and then dropped, and finished (final) the

course.

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145

Table 42

Number of Students in Each Group

Group Initial Add Droo Add —> Drop Final Net % Change

C&M (1) 11 3 3 2 13 +18%

C&M (2) 12 6 3 1 15 +25%

Trad (1) 36 1 10 0 27 -25%

Trad (2) 24 3 6 0 21 - 1 3%

In Figure 20, the solid lines represent the numbers of students

in the two C&M sections and the dotted lines represent those in the

two traditional sections. The most noticeable fact is that the number

of students in the C&M group increased while that in the traditional

group decreased. Through use of this simple interpretation, two

distinctive phenomena were found.

Figure 20 . Number of Students in the C&M and the Traditional Group

First, three students added and then dropped the C&M course

whereas no student joined and then withdrew from the traditional

course. The interesting point here is that all three students stayed in

the course less than two weeks. This indicates that the students who

joined the course later expected a free ride, but they were

disappointed by the required hard work. Presumably, the students'

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146

initial difficulties with computers expedited their dropping the C&Mcourse.

Second, the adding and dropping pattern of the two traditional

sections was routine, a few adding and then consecutively dropping.

Two traditional sections had exactly the same pattern even though

there were some minor differences in the numbers of students; in

the first group, one student added and ten students subsequently

dropped, and in the second group, three students added and then six

students dropped. On the other hand, the two C&M groups, which had

some fluctuation in the number of students, did not show any

specific adding and dropping pattern. Apparently, the novel

experience with computers caused a change of the students’ opinion

of the course, and raised the variations in the number of enrolled

students.

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rki 'TG^>b«)2

CHAPTER VI

SUMMARY OF THE RESULTS

This chapter provides the answers to the research questions

generated in chapter I. The results here are based on the

quantitative data obtained from the achievement and attitude

survey, and the qualitative data based on the interview,

questionnaire, observation, solution files, and concept maps.

Question 1

What is the role of Mathematica in the development of concepts?

The visualization which used the full capability of the

Mathematica provided the students with more sound conceptual

understanding. Most of the topics in the C&M courseware were

coordinated with Mathematica plottings. Thus, the C&M students

could visualize the mathematical meaning that they were supposed

to know. For example, visualization by Mathematica was a powerful

tool for learning the derivative by the difference quotient, the

trapezoidal rule, and the convergence interval.

First, the students learned the approximation of the derivative

by the difference quotient by comparing the plots of {cos[x+.0001] -

cos[x] }/.0001 and -sin[x], and determining the relationship between

the two plots.

Plot both (cos[x + .0001] - cos[x])/.0001 and -sin[x] on the same axes.

In[lJ:= f[x_l = Cos[x];

In[2]:= plot = Plot[f[x+.0001] - f[x])/.0001, {x, 0, 2 Pi}];

ln[3J:= D[f[x], x]

Out[3]:= -Sin[x]m

In[4]:= minussineplot = Plot[-Sin[x], {x, 0, 2 Pi}];

In[5J:= Showiplot, minussineplot]

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148

It looks like a great approximation since we see only one function.

In[6]:= difference = ((Cos[x+.0001] - Cos[x])/.0001) - (-Sin[x]));

In[7]:= PIot[difference, {x, 0, 2 Pi}]

The difference quotient f[x] and f[x] run within of each other the

whole way.

If the derivative of cos[x] by the difference quotient is taught

by the theoretical approach, it should be placed after the Mean Value

Theorem because it involves careful use of the theorem. With the aid

of Mathematical the C&M students learned this concept very early

(Lesson 2.01). By using Mathematica to answer the questions, the

students could form an idea of what a derivative is.

Second, the students visualized the trapezoidal rule by

producing their own graphics in which the plot of polygonal

approximations fits the plot of the integrand as the interval goes to 0.

By working on the following consecutive graphics, the students

naturally acquired the trapezoidal rule.

In[l]:= fixj = Sin[x];

In[2]:= functionplot = Plot[f[x], {x, 0, 1}];

In[3]:= jump = 1/3;

points = Table[{x, f[x]}, {x, 0, 1, Jump}];

In[4]:= traps[x_]:=Graphics[{GrayLevel[.7S], PoIygon[{{x, 0},

{x, f[x]}, {x+jump, f[x+jump]}, {x+Jump, 0}}]}];

In[5}:= brokenlineplot = Graphics[Line[points]];

In[6]:= Show[trap[0 jump], trap[l jump], trap[2 jump],

functionplot, brokenlineplot,

Ticks->{0, jump, 2 jump, 3 jump}]

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149

3 3

In[7]:= jump = 1/6;

points = Table[{x, f[x]}, {x, 0, 1, Jump}];

In[8]:= brokenlineplot = Graphics[Line[points]];

In[9]:= Show[trap[0 jump], trap[l jump], trap[2 jump],

trap[3 jump], trap[4 jump], trap[5 jump],

functionplot, brokenlineplot, Ticks->{0, jump,

2 jump, 3 jump, 4 jump, 5 jump, 6 jump}]

6 3 2 3 6

The illustrations show that, for large n's, the broken line segments are

hard to distinguish from the original curve. Consequently, for large n’s.

the area under the straight-sided figures should be very close to the

exact

1

measurement. 0

sin[7c x] dx .

The third example is the visualization of convergence intervals.

The students learned several techniques for determining

convergence intervals. In each case, the results were corroborated by

plotting, and these corroborated results were apparently more

effective than the results gained by the ratio test or the basic

convergence principle. For instance, in the power series (1 - x -i- x^/2

- x^/3 + - x^/5 -1-. . . -h (-1)" x"/n), one student demonstrated his

understanding of the convergence interval in the following way:

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150

Using the Basic Convergence Principle:

For any positive number x = R, the list of terms is {1, -R, RV2, -R^/4,

RV4. -RV5. . . . (-l)"R"/n, . . .}

If R > 1, the terms in the power series will become progressively larger.

However, if R < 1, the terms get smaller and smaller. This can be

illustrated graphically:

In[l]:= R = .9;

In[2]:= ListPlot[Table[{n, ((-!)“ R")/n}, {n, 1, 50}],

PlotStyle->PointSize[0.01]]

On the other hand, if R > 1, the terms get large:

In[3]:= R = 1.1;

In[4]:= ListPIot[TabIe[{n, ((-!)“ R“)/n>, {n, 1, 50}],

PlotStyle->PointSize[0.01]]

Therefore, the power series converges on every interval [-r, r],

provided 0 < r < 1. This convergence can be illustrated with a plot of the

power series:

In[5]:= sum9 = 1 + Sum[((-1)“ R“)/n, {n, 1, 9}];

In[6]:= sumlO = 1 + Sum[((-1)" R")/n, {n, 1, 10}];

In[7]:= Plot[{sum9, sumlO}, {x, -1, 1}]

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151

As seen in the above examples, the C&M courseware placed

heavy emphasis on student-produced visualization that set up and

illustrated underlying principles. Those three examples show how

Mathematica plottings can isolate and explain concepts more

satisfactorily than long tedious paragraphs.

Question 2

How can Mathematica be used in the development of

calculational skills?

It is quite natural to assume that Mathematica would degrade

students' calculational skills because Mathematica, not students,

perform most of the calculations. The results of the quantitative

research--the slightly lower scores on computational proficiency

itemS”Support this assumption. However, students' experience with

Mathematica might reinforce their ability to handle algebraic

manipulative skills in the following three aspects.

First, students were required to change the usual expressions

to the appropriate Mathematica input forms and to translate the

results from Mathematica to the suitable expressions. For example,

when students were asked to plot the broken line that joins the four

points on the curve {0, f[0]}, {1/3, f[l/3]}, {2/3, f[2/3]}, {1, f[l]),

where f[x] = sin[7l x] for 0 < x < 1, they should have figured out the

following steps:

In[l]:=f[x_\ = Sin[Pi x]

/n/2/;= jump = 1/3

In[3J:= points = Table[ {x, flx)}» {x, 0, 1, Jump}]

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On the other hand, when students got the result,

In[4]:= Series [1/(1 - x), {x, 0, 10}]

Out[4]: = 1 + X + + x"* + + X* + + x^® + 0[x]^ ^

they should have been able to interpret that the output is the just

approximation of 1/(1 - x) to x^^ (here 0[x]^^ stands for the

remainder after the x^® term.) These processes appeared to have the

unexpected side effect of improving students* algebraic skills.

Second, the students relegated most of the calculations to

Mathematica and they themselves only examined the results by

Mathematica. The time for checking the results of several problems

by Mathematica equals the time for calculating one problem by hand.

Apparently, plenty of checking experiences provide a certain sense

of choosing appropriate calculational procedures. For instance,

maximum and minimum problems usually require several steps with

complicated calculations. If students solve maximum and minimum

problems by hand, they are apt to be confined to following each

calculational step which requires much time; consequently, they lose

the global thought about the problem. Moreover, the values and

coefficients of the problem should be simplified when using hand

calculation. But, with the aid of Mathematical original data from the

real world can be used without distortion. Lesson 2.06 (greatest and

least, shortest and tallest, fastest and slowest, best and worst)

includes various application problems of maximum and minimum

with real data. Based on observation, most of the students could look

through "tutorial” problems within an hour. After that, they gained a

sense of choosing proper steps for solving maximum and minimum

problems.

Third, if the students with weak calculational skills are forced

to calculate for themselves, they might not even try to attempt to

solve the problems. However, when Mathematica is available,

students do not have to be diffident and can think more about the

calculational procedures because computers do not "mess-up" on

calculations. In this case, Mathematica play the minor role of

psychological motivator in the development of the students'

calculational skills.

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153

In summary, even though Mathematica do not play a direct

role in the development of calculational skills, they have some

second hand effects on strengthening algebraic skills.

In what specific wav does Mathematica facilitate the students'

exploration and discovery learning?

The C&M students acquired most of the rules and formulae by

using Mathematica to explore through calculation. The Chain Rule and

the integration-by-parts formula are representative examples.

What follows is the introduction to the Chain Rule in the C&Mcourseware:

Let's check out the derivative of the composition of two function;

Here is the derivative of sin[x^]

;

Out[l]:= 2x Cos[x2]

This is interesting because the derivative of sin[x] is cos[x] and the

derivative of x^ is 2x. It seems that the derivative of sin[x2] is

manufactured from the derivative of sin[x] and the derivative of x^.

Here is the derivative of (x2+sin[x])*:

Out[2]:= 8 (2 X + Cos[x]) (x^ + Sin[x])^

This is interesting because the derivative of x* is 8x^ and the derivative

of x2 + sin[x] is 2x + cos[x]. It seems that the derivative of (x^ + sin[x])* is

manufactured from the derivative of x*. the derivative of sin[x] and the

derivative of x^.

On the other, hand, the integration-by-parts formula was

explained in the following way: converting the Product Rule of

differentiation into instrument for calculation of integrals:

Question 3

/n^7y.= D[Sin[x''2], x]

In[2]:= DKx''! + Sin[x])''8, x]

In[l]:= D[x E^'x, x]

Out[l]:= E* + E* X

This tells us that xe* + e* = D[xe*, x]

So, xe* = D[xe*, x] - e*

As a result,

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Out[2]:= 1 + 4E^

Check:

In[3]:= Integrate[x E^x, {x, 0, 5}]

Out[3]:= 1 + 4E^

Now let's reflect on what we did. Our success in calculating xe*dx b y. 0

the method we used above was based on our ability to replace

fS

xe*dx b y. 0

two easily calculated integrals.

You are entitled to protest that what we did was just an isolated trick. The

response is that unless this idea is systematically developed, then it will

remain just an isolated trick. The next problem develops the idea in full.

The strategy of teaching the Chain Rule and the integration-by-

parts formula was guided discovery learning. The C&M students

were not simpW told the rule, D[f[g[x]]] = f '[g[x]] g'[x] or the formula,

Juv' = uv

-

j

u'v. Instead, the C&M lessons guided the students

through the examples leading up to a conclusion and let them find

the actual rules. When the students could not Hgure out the rules

with given examples, they were allowed to try more examples and

test ideas until they discovered the rules on their own. Without

Mathematical it might be impossible for the students to formulate

the rules for themselves.

Right after the exploration examples, the courseware provided

the application problems of the Chain Rule and the integration-by

parts-formula. Based on observation, most of the students correctly

answered within a short time. From this fact, the investigator could

confirm that if students are encouraged to discover rules for

themselves, the rules are internalized better than if they are simply

accepted in the form handed down by the instructor. Learning the

Chain Rule and the integration-by-parts formula by discovery

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155

seemed to make the rules more readily available for problem solving

and for understanding related mathematical topics.

Question 4

In the C&M course, what different cognitive procedures occur

during the learning process?

The cognitive learning procedure of the C&M group could be

featured by the bottom-up process. The learning of the relationship

between the up-and-down of a function and the sign of its derivative

(Lesson 2.02) was one of the representative examples of the bottom-

up process.

Bottom-up process

(1) The f '[x] curve represents the instantaneous growth rate.

When f ’[x] is negative, the f[x] curve is going down.

When f '[x] is positive, the f[x] curve is going up.

The solid line represents the f(x) and the dotted line represents the f '(*).

Figure 21 . Instantaneous Growth Rate

If students learn the rules (1) first and then study the plots (2)

in terms of the rules (1), this would be deductive or top-down

process , in the sense that a knowledge representation structure has

been retrieved from memory and is now guiding the process in

solving a problem. On the other hand, making a decision on (1) from

(2) is an inductive or bottom-up process because students learn

mathematical principles by exploring through examples and

discovering the rule for themselves.

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156

The learning procedure of the C&M students was closer to the

latter, bottom-up process. Instead of providing the relationship

between the up-and-down of a function and the sign of its

derivative, the students developed a feeling for the subjects and

worked with examples until they had a conclusive idea. Accordingly,

the direction of the cognitive process was from specific to general.

Experimenting with a variety of examples on the same topic seemed

to induce certain analytic thought patterns in the students, and to

lead them to the general rules through their own intuitive

processing. By channelling the students’ thought processes in this

way, they could often easily be led to the formulation of ideas which

would have been much more difficult to arrive at through their

normal non-analytic thought processes.

If Mathematica were not involved in the learning environment,

the inductive or bottom-up process might be almost impossible. Even

when the inductive procedure was possible, the effectiveness of

learning might not be satisfactory. When we assume the superiority

of the bottom-up process (even though there has been no consensus

on which process is more desirable and effective), one of the primary

advantages of learning with computers is to facilitate the inductive

learning process.

Analysis Based on Cognitive Psychology

According to Schoenfeld (1987), the acquisition of (1) in Figure

21 involves two stages: the declarative stage—students encode a set

of cues; the knowledge compilation stage--the cues are combined into

students’ generalized knowledge (Figure 22).

The bottom-up process can be interpreted based on Bruner's

three modes of representation and the spiral model, and Piaget’s

developmental cognitive model. The visual examples (2) in Figure 21

could be equivalent to Bruner’s iconic representation or Piaget’s

concrete operations, and the rules (1) in Figure 21 could be

equivalent to Bruner’s symbolic representation or Piaget’s formal

operations. According to Bruner, iconic and symbolic modes of

representations are related developmentally, and each mode

depends on the preceding one. On the basis of Piaget’s theory,

students’ mental development becomes increasingly sophisticated as

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they work through from concrete operations to formal operations. In

short, it might be said that the Bruner and Piaget theories support

the bottom-up process.

157

generalized knowledge

(rules)

compilation

a set of cues

(examples)

Figure 22 . Bottom-up or Inductive Process

Figure 23 . Information Flow

In the aspect of cognitive psychology, the inductive approach

seems to have the following two potential advantages. First, the

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158

knowledge from the bottom-up process can be more likely to be

stored in long-term memory than that from the top-down process.

While working through concrete examples by Mathematica, students

have already incorporated several isolated facts in short-term

memory. Thus the extension to long-term memory occurs more

easily. Figure 23 is the knowledge acquisition procedure based on the

memory architecture model.

Second, the bottom-up process is more able to facilitate

conceptual understanding. By comparing the visual examples,

students can invent and construct the framework, the relationship


derivative. A stable cognitive structure can be formed by the process

of figuring out the general rules.

Question 5

How does the C&M course provide a cooperative learning

environment?

Learning with computers is often viewed as an isolated,

individualistic matter; a student sits alone with the computer and

struggles to understand the subject and solve the assigned problems.

This process can often be lonely and frustrating. However, the

learning process of the C&M students was not individualistic. As the

semester progressed, there was a gradual evolution in students'

cooperation. No group structure was artificially imposed; nonetheless

the number of students engaged in group work increased.

Recent research on cooperative learning has indicated that

cooperation is considerably more effective than interpersonal

competition or individualistic efforts in promoting achievement and

productivity (Johnson & Johnson, 1989). The cognitive or affective

benefits accrued from engaging the students in cooperative learning

are increased skills or conceptual understanding, and improved

attitudes or motivation. Furthermore, systematic and frequent use of

cooperative learning procedures has a profound positive impact upon

the lab climate; the lab becomes a community of learners, actively

working together in small groups to enhance each student's

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159

mathematical knowledge, proficiency, and enjoyment (Davidson,

1990 ).

In this study, the primary source of the big differences in the

scores on the post-achievement and attitude survey was the

different teaching and learning methods. The next most important

reason could be credited to students’ cooperation because the

collaborative activities in the laboratory apparently helped most of

the students improve their ability to articulate mathematical ideas.

In the following three aspects, the C&M course fostered natural

opportunities for the development of collaborative learning.

First, the C&M courseware offered many opportunities for

creative thinking, for exploring open-ended situations, for non-

routine problems that merit discussion. The students could often

collaboratively handle challenging situations that were well beyond

the capabilities of individuals. They could learn lessons and solve

problems by talking, listening, explaining, and thinking with others

as well as by themselves.

As the students became comfortable with collaborative

computer work, they increasingly took advantage of opportunities

for discussions of such key components as multiple representations,

selection of appropriate mathematical procedures to solve problems,

and interpretation of problem results. In particular, challenging

assignments induced repeated opportunities for peer tutoring; the

more frequent interaction occurred when students learned the

difficult lessons.

Second, the physical setting of the C&M lab was one of the

contributors to the students' collaboration. It provided supportive

conditions for students' group work: room for several students to

work at a computer terminal; chairs with coasters to facilitate

students switching position at a computer keyboard; and wide aisles

to allow the instructor and the lab assistants to easily move

throughout the lab.

Third, the open and liberal atmosphere of the C&M lab seemed

to facilitate natural cooperation among the students. Again, as a

consequence of this collaboration, the lab atmosphere tended to be

more relaxed and informal, help was readily available, questions

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were freely asked and answered, and even the shy students found it

easy to be involved. Furthermore, students-instructor, student-

student, and students-assistant relations tended to be closer and

more pleasant than those in a traditional approach.

To conclude, the C&M lab was a natural learning vehicle for

cooperative group v/ork. The nature of the C&M courseware, and the

physical setting and atmosphere of the C&M lab, both contributed to

an environment in which cooperative learning thrived. Computers

were used to amplify the cooperative learning, and the collaborative

learning facilitated the learning with computers.

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161

CHAPTER VnCONCLUSIONS AND RECOMMENDATIONS

Conclusions

The purpose of this study was to evaluate the Calculus &Mathematica course—the laboratory calculus course at the University

of Illinois. The evaluation was done in the following three aspects:

the students' performance in the course, their activities in the lab,

and the C&M course itself.

First, in regard to the students' performance, the C&M students,

without seriously losing computational proficiency, were much better

at conceptual understanding than were the traditional students. The

ANCOVA result was caused us to reject the null hypothesis pertaining

to conceptual understanding, but not reject the null hypothesis

pertaining to computational proficiency. This indicates that there was

significant difference in the conceptual understanding scores, but no

significant difference in the computational proHciency scores. In

addition to ANCOVA, the scores on each item were tested by a t-test;

especially, the concept-oriented items related to graphs and the

meaning of theorems resulted in the signiHcant p values for a t-test.

The overall attitudes of the C&M students were more positive

than those of the traditional students. From ANCOVA, the null

hypothesis for the students' attitudes were also rejected, indicating

that there was significant difference in attitudes between the C&Mstudents and the standard students at the .05 level of significance.

Like the achievement items, the t-test was done for each item of the

attitude survey; especially, the items regarding attitudes toward

computers, cooperative learning, and mathematics as a process

yielded a significant t-test result.

To focus more on students' conceptual understanding, a new

instrument—the concept map—was used. The comparison of

students' concept maps was done by two methods; the total score

based on the five criteria (propositions, hierarchy, cross links, more

concepts, and misconceptions) and the correlation coefficient

between the teacher's concept map and the student's concept map.

The results from both methods were also favorable to the C&M

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162

students. Among the five criteria of the first method, distinctive

difference was found in cross links; in other words, the C&M students

were much better at linking concepts in different branches.

Second, regarding the students' activities in the lab, one of the

most remarkable characteristics was their exploration through

calculations and plottings. In the traditional calculus courses, the

instructor announces the mathematical theory and then reinforces it

with examples and exercises, and students recite the theory and

solve problems illustrating the theory. However, the learning pattern

of the C&M students was dramatically different. The experimentation

by redoing, reformulating, rethinking, adapting, and making changes

led the students to discover the basic concepts and principles: the

Chain Rule, the Trapezoidal Rule, the Fundamental Theorem of

Calculus, the convergence interval of the series, and the relationship


derivative. Furthermore, those experiences appeared to act as

cognitive bridges to the abstract understanding of each concept or

principle.

The students' responses to the questionnaire indicated that

they had a feel for "doing" mathematics instead of "watching"

mathematics and were enthusiastic about their experiences with

Mathematica. Several students stated that they were satisried with

the brief introduction and the intuitive explanation provided in the

C&M courseware. They added that they were not comfortable with

the style of traditional texts: the intricate procedure of introduction

and the laborious proofs which were intuitively obvious.

Most of the C&M students were confident and daring in

contrast to the typically diffident students from the traditional

course. A pleasant side effect of the lab was the rapport which was

established among the students. A lot of learning occurred when the

students converged around a computer, worked together, and shared

and developed ideas. In fact, this cooperation was one of the keys to

the success of the C&M course.

In the C&M course, the medium is Mathematica and the

message is mathematics (calculus). Based on observation,

Mathematica syntax was not hard for the students to learn. In the

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163

lessons the students could see Mathematica commands and

Mathematica routines in context and pick them up very quickly

because the C&M courseware was fairly user-friendly. After a couple

of weeks, Mathematica syntax was not much of a problem at the

level needed for successful performance.

Third, regarding the evaluation of the course, the C&M course

allowed the students to spend less time on computations and better

direct themselves to conceptual understanding. In such a situation,

there was an increase in the students' conceptual achievement

without a serious decrease in computational achievement. Moreover,

such an approach produced an improvement in the students'

attitudes toward mathematics by relieving them of the tedium and

source of errors involved with computations, and an improvement in

attitudes toward the computer by making the students more

comfortable with it and by introducing them to some of its

capabilities.

The calculating and plotting capabilities of Mathematica helped

the students discover and test results of calculus in much the same

way that a physics or chemistry student uses the laboratory to

discover and test scientific laws. Those capabilities provided the

opportunities for the students to consider the more open-ended

questions and to encounter the more realistic problems than those

found in traditional calculus texts. In a word, the use of Mathematica

strengthens the course by helping the students learn a more lively

calculus.

The C&M course changed the delivery of calculus from lectures

and texts to a laboratory course through an electronic interactive

text. There have been a lot of projects in which the existence of the

computer lab is the impetus for a reshaping of the calculus course

and a rethinking of the goals of calculus instruction. The C&M course

is one of those projects. However, the approach of some of those

projects and that of the C&M course are different. The former sets up

the ideas of calculus traditionally and then implements those ideas

using computers, while in the latter, computers are used both for

setting up the ideas and for implementing those ideas.

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’ ^ 3ttl?is9»i3iqnt* w1 bsit r^di qa 5, 'ttsif'

165

On the whole, the advantages and benefits of the C&M course:

visualization of ideas, discovery learning by exploration, and

cooperative learning were enough to offset its drawbacks: the

students' high dependency on Mathematical the black-box syndrome,

and the time-consuming quality of the course. By virtue of this, the

investigator is optimistic about the potential of the C&M course as an

alternative approach to current calculus courses.

Suggestions

It was noticed that the background knowledge of the students

was not homogeneous. Even though ten out of the twelve C&Mstudents had taken MATH 120 (Calculus and Analytic Geometry I)

during the fall semester of 1991, they entered the course with

varying levels of preparation. Some needed more review on

precalculus and some did not. This was noticed in the observation

and confirmed by the large standard deviation (5.8) of the pre-

achievement test results. Accordingly, the C&M courseware should

provide common background knowledge of precalculus to fill in the

gaps between the students' knowledge and the content of the lesson.

The given review of precalculus is too cursory to be of much help.

Undoubtedly Lessons 1.01 to 1.03 are a kind of warming-up section,

but the goals of those lessons are focused on the aspect of the

language of Mathematica, not on precalculus concepts. The C&Mcourseware may appeal more to students if it includes additional

content in precalculus review to give weaker students a better

chance to catch up.

Limitations

The following limitations of this study were noted.

First, each experimental group was an intact group; in other

words, the samples were simply available samples, not random ones.

Although no significant difference between the C&M group and the

standard group in the variables of age, gender, and the number of

college mathematics courses taken, the differences in the distribution

of majors showed that the two groups might not be equivalent.

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-1

The main goal of the calculus reform projects, including that of

the C<&M course, has been the reshaping of both the content and the

pedagogy of calculus instruction. The investigator's opinion is that

the reshaping of the content in the C&M course was almost flawless.

The C&M course was not a traditional course plus Mathematica; new

courseware and the corresponding curriculum were created; the

content and the approach to each lesson were quite different.

According to these changes, the pedagogy of the course was also

shifted from passive learning to learning by doing—students' active

involvement in the learning process. However, the reform of the

pedagogy seemed not to be sufficient. What the investigator wants to

suggest is in-depth reform of the pedagogy. For instance,

constructivism might be applied to the C&M courseware. The display

of each mathematical concept could be broken down by the

developmental steps based on cognitive psychology. There might

exist the most suitable pedagogy for the laboratory calculus course,

and the thorough reflection of that pedagogy could make the C&Mcourse more attractive.

On the other hand, as well as the positive aspects, some

unfavorable qualities of the C&M course should be mentioned. First,

the students who depended too much on Mathematica used it as a

primary calculating and plotting executor, even to solve the hand

calculation problems. Second, the students' blind execution of

commands without understanding underlying concepts and

procedures was an instance of black-box syndrome. Third, the lab

atmosphere was such that as soon as one student figured out

something, it was shared with nearby students. But, this

dissemination was usually a method of how to rather than a

complete understanding. Fourth, the C&M course required more

commitment and time than did the traditional course. Some students

complained that the traditional course was easier and required much

less time. Others noted that they were not doing the same thing that

they saw in the traditional course and they believed that the

traditional course should be changed. This feature of the C&M course

could be one of its strong points.

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166

However, the investigator believed that this difference was not a

factor in the results.

Second, the quite favorable results of both the quantitative and

the qualitative research could partly have originated in the

Hawthorne effect. In other words, the better performance of the C&Mstudents could be caused not only by the computer treatment but

also by the knowledge that they were in a study and had been

singled out for special attention.

Third, a self-selection process involved both the students and

the instructors. Before enrolling in the C&M course, the students had

already known that the teaching method of the course was different

from that of the usual courses. Thus, the members of the C&M group

might have been more energetic and adventurous than those of the

standard group. In the same way, the instructors who volunteered

for the C&M course might have been more committed. The

instructors worked hard, the students saw and appreciated that and

studied harder as a consequence. All these factors could have

produced the C&M students' outperforming the standard ones.

The discussed limitations and concerns all need to be

considered before generalizing about the results of this study to

other population.

Recommendations

One of the purposes of an exploratory study is to open the way

for further research. The following Ust contains the recommendations

for further research in this area.

First, the main study throughout one semester might enable

one to reveal the short-term effect of the C&M course, but could not

assess the long-term effect such as Will there be more mathematics

majors? Are engineers and scientists better prepared? Can the

students retain the concepts of calculus longer? Thus, longitudinal

studies eventually need to be undertaken to determine the long-

term effect of using computers to replace doing mechanical skills by

hand.

Second, in this study, the numbers of male and female students

— 14 female and 54 male—were too small to draw a legitimate

# I

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167

conclusion about the relationship between the gender and the use of

computers. Hence, the gender-computer issue deserves further

study.

Third, further research needs to be done to ascertain the

interactive effect of the computer treatment with the students'

achievement level.

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^

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,

io ^;r4>i^ .<9Sî) .H .j-u'»flto;i J> .M

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I

> . fi' ,d2»«f

iy:~K>'.jnxZ (YM ,acAdJin2

4^C oM >i4vr?i5‘flPiiâb<i^^J! ioa<iw/,«H3 ,yjit»v4alf

w»j«5i •> b ^ t*^9yi>;;)0»^p adi V« cbolf A' .M ^

,os^yyM iiM»J jK*^h&i$ib irioip(><I)^ oi ^ ^1 xc .ijwfciuinaat

k> *{0^4) /I #t«afitW^

jp ,xâii A ;ao*U*.yi3.iP^

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.t-tS> .’J m >iibk»b t.W »Gd.*'ixj»tA ifaiiJi.^ ‘Ckf • ,j»o<^mfUliW

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.A<^€CS: 1

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lilf T t*J ytewfc fTOlSTuI^/^ dA (^2?J) .M ,1>4>H

HHpqii: > tlfi rt» i^Ki^:*flO*) b/t»» >llb£a idl |. ^’

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171

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^ vr*f.qcnoD 5taiii»40u^'

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^noli^nîrtb lir!d)oon'» oqoofroo.'J •

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172

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Ducharme). Tallahassee: Author.

A t

-1

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The National Research Council (1989). Everybody counts: A report to

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Company, Inc.

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APPENDIX APre-Achievement Test

174

1. A function is said to be additive if f(x4-y) = f(x)-i-f(y). Give an

example of an additive function.

2. Find the points on the x and y axes where the curve y =

intersects.

(2x-H)

(x^+2x+3

)

3. Find the x between 2TT and 4‘lt where sin(x) = 12

4. If you know that 10^=4, then what is

5. If you know that logjj2 = 12

then what is logjj32?

i

!

'j.j; '?.> )

'=T

* ‘ »- k

;.i HjvJv) ^ ii SiVUiJjbfi ^ *!•

.nobrnul ovr)ibb^ 34(^*xs

’si?

.:%

z^L. y av-.ij <5iti Xpiaio^ siH *>«<•

r>^f:-^r'

•=L '-. -r' .<. . ,

,»\Ai

i * (Aiah TVJJ^'V nS i: ^ tvi?^ ^

X •

‘

4 H

r(4*- >0l »i <‘r.-li ii;dj ^oroi v0{ U .

IV ^ '-'VK. C im

a

Ilf T

1

•\,5^C»l *itfO t

175

6.

Find the value of limh-^^~'u

~^n

7.

Find the value of^i^x-4l(x^-4x+3

)

X- 1

8.

If f is an even function (i.e., f(-x) = f(x)), then what is f '(0)?

9.

Find the derivative of (3x +1)^.

10. When x = 2cos(t)d X

and y = sin(t), find.

in terms of t.

dy

f- M

'I

‘

*

r = lui) nob-sul •» r U .H

176

11.

Sketch the graph of a function f(x) such that f '(0) > 0, f '(1) < 0,

and f "(x) is always negative?

12.

The velocity of a body moving in a straight line t seconds after

starting from rest is (4t^-12t^) meters per second. How many

seconds after starting does its acceleration become zero?

13.

The graph of the function f is shown for 0 1 x 1 10. What c makes

I

f(x)dx

0

as large as possible?

y

14.

You know that 3f ’(x) = x^ - 5, and f(2) = 1. What is f(0) equal to?

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tl

p'-

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.3 'f f n., ,;f

I « boî.C ‘x - (x)' )c fwL* v'Jfli i*t>y

177

1 2x15. Find

'o(2x2+1)'

dx

16. Find the area between the curve y = 2x^ - 6x^ + 3 and the line

y = 2x - 3.

APPENDIX B

Post-Achievement Test

178

1.

How does the Fundamental Theorem of Calculus establish a

connection between the integral and the derivative?

2.

Calculate:

•3

t)

dt

b.

f2n

J-ln

sin ^ dx2

3.

Suppose that the birth rate in a certain country t years after 1970

was (9 + t) thousands of births per year. Set up and evaluate an

appropriate integral to compute the total number of births that

occurred between 1970 and 1990.

Sinxj_d^

(k^ + x“)

4. Calculate F '(t) given that F(t) =

0

I V^ , ji 4t^ r

H xiovia.m

JwT

<. uitu^tAl 1o fiOMOoifl Ifil03ictuli)»u^ Ort) i

't' *i3Avi'idb $CU b:lf 9lft lOW.^Cid nOU03kl/IOD

« *

'JTÎL/Tis^ r.rvi > (Unm iuxrc^ & ni Ji«i tiJnd '•ûoqqii^ .£"

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J

iK<?l hfui QVî iiiSv/taif b. uMcoO

:

r-E

I,5 Qsv;-g {ty H .

-• ‘J) *. ';ts^ i/j .=»

179

5. Calculate by integration by parts:

rc

a.

j

X log(x)dx

b. e‘^*dx

6. What is the integration by parts formula and how is it related to

the product rule of differentiation (i.e., (fg)' = fg+fg') ?

7. The derivative of^ ^

is^ ^ 2 •

Using the expansion of(1-xV

find the expansion of(1-xy

in powers of x.

1

1-x ’

8. Consider the graphs of f(x) = e’^ and g(x) =

1 + + — +2n

2! 3! n!

graph of f(x) and g(x) as n increases.

Describe the relation between the

f ,1 ji t. od ( ifoyrr? >(d floilttgisn* s**!! tl lifiW .d

? * W) ,.;?.|) «.(> ilii ?4Vt j

!'^'Xi 1

* Ifo »io*? s?ix^ od) afifzTT 7“'“)i.t 7 7 i») a'/tiftvrwt) sw<T^r ->

4'^-

.A *0 rtf - 'o rtiÛiUOX^ btujI,

X;,.^V. -:

•• !>i^ 4 k

%

<S

m-It

^ £,.ii

s

"K

tfi

i

'ti; D^Âr1©i:* lir.i|^|r*5i JsdhisÛ

X*J* t>ni: ’’. =- (je}\ % idqiwrs i»b "foD^ ,H

^ n.c» >5 e ;

..^. JC!,, (£:

AiNOi3lO«r a tfc (;cV^ ^»4U ^ aq*Tj<

E .

ar

180

9.

Use Taylor's formula to obtain the expansion in powers of e ’^ cos x

through the x'^ term.

10.

Give convergence intervals for the power series

1 + 2 X + 4 x2 + 8 x3 + 16 x"* + 32 x5 + . . . + 2° x° + . . .

11.

Find lim -SiO-JL

x->0 X + x^

12.

Sketch the graph of a function f(x) such that f '(x) is negative for

all X with 1 1 X i 3 and f '(x) is positive for 3 i x i 5. Which x in [1,

5] make f(x) the smallest?

13.

Differentiate the following functions with respect to x.

1 - 3x + 2x^cl. O

(1 + x)^

^0 •'tt \o il»woq 4ii 3ih rtUid6 OJ 4iIurmo1 e«U

tmtiJ ^Jrt» t{* joxiii

itftfTsa iswoq sri* w>^ fbjnrWf ^*viO ,01,

. , r 4- ... 4- ^ 1, CC H- ^y. o! 4-' a f! +:^si .'**•

I

trj

ti UV V l&dl (JS>% coHpfiffI « V' ilqfiT^ d^SsiZ St

ilîW t ir JL : f. vO »i=00 1 ^ i y 1 i \»v^

^n^fUrr>.^ îh'

3^

oi f|l,W:* ^00bî»t ' i y.*0J^jsrt J

181

b. cos X + 2x^)^

14.

Six functions are plotted below. Three of them are derivatives of

the other three. Match the plot of each derivative with its

conesponding function.

a. b. c.

15.

Suppose f(x) and g(x) arc two functions with f '(x) > 0 and g' (x) >

0 for all X values. This means both f(x) and g(x) go up as x

changes from left to right. Does it mean that the product f(x) g(x)

also goes up as x changes from left to right? What happens to the

sum f(x) + g(x)? To illustrate your answer, choose appropriate

functions.

16.

For each of (i) (ii) (iii), choose both the mathematical model and

the graph from the lists below that best fit the function f(t).

f/u4r io .^1^^ b >miq vim RdOi)o^ii/t ,?u2 >1

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182

i. When a drug is injected into the body of a patient, the drug amount

in the body decreases at a rate proportional to the time elapsed.

Define f(t) as the amount of the drug left in the body t hours after

the drug was administered.

ii. The temperature of a turkey changes at a rate proportional to the

difference between the current temperature of a turkey and the

temperature of an oven. Define f(t) as the temperature of the

turkey t hours after baking. (Assume that the temperature of an

oven is constant and higher than the temperature of a turkey).

iii. A chain letter was designed to be sent to members of a large

nationwide organization. The orgarnization's membership has

remained stable during the last year. The rate at which new people

received the letter is jointly proportional to the number who have

already received the letter and the number of members who have

not yet received the letter. Define f(t) as the total number of

members who had received the letter within t months after the

beginning of 1990.

Models

a. If y' = k(r-y) where y = f(t), then f(t) = r+ce’^^

b. If y' = k(r+y) where y = f(t), then f(t) = -r+ce^^

c. If y’ = ky where y = f(t), then f(t) = ce^^

d. If y' = ky(r-y) where y = f(t), then f(t) = r/(l+ce‘^*’^^

e. If y' = ky(r+y) where y = f(t), then f(t) = -r/(l+ce'*^*'^)

Graphs

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183

APPENDIX CAttitude Survey

Circle the choice that best describes your feelings.

SA = strongly agree

A = agree

U = undecided

D = disagree

SD = strongly disagree

1. New discoveries in mathematics are

constantly being made.

SA A U D SD

2. I usually understand what we are talking

about in mathematics class.

SA A u D SD

3. I like to solve problem by working with

others.

SA A u D SD

4. When I cannot figure out a problem,

I feel as though I am lost in a maze and

cannot find my way out.

SA A u D SD

5. Most mathematics problems can be

solved in different ways.

SA A u D SD

6. Everyone should learn something about

computers.

SA A u D SD

7. Solving problems with others lowers

self-confidence.

SA A u D SD

8. I want to work at a job which requires

mathematics.

SA A u D SD

9. Mathematics is something which I enjoy

very much.

SA A u D SD

10. Most of the learning of mathematics

involves memorizing.

SA A u D SD

11. Using a computermakes learning

mathematics more mechanical and boring.

SA A u D SD

12. I have a real desire to learn mathematics. SA A u D SD

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184

SA = strongly agree

A = agree

U = undecided

S = disagree

SD = strongly disagree

13. Using a computer can help you learn

many different mathematical topics.

SA A U D SD

14. The only reason I'm taking this course

is because I have to.

SA A U D SD

15. I prefer to study mathematics by myself. SA A U D SD

16. If you use a computer, you don't have to

learn to compute.

SA A U D SD

17. Mathematics is helpful in understanding

today's world.

SA A U D SD

18. There is little place for originality in

solving mathematics problems.

SA A U D SD

19. When I do mathematics with other

students, I realize I am not the only one

who can't understand.

SA A U D SD

20. It is important to know mathematics

in order to get a good job.

SA A U D SD

21. Solving word problems is more fun

if you use a computer.

SA A U D SD

22. I am good at working mathematics

problems.

SA A U D SD

23. Mathematics helps one to think logically. SA A U D SD

24. Mathematics is harder for me than

for most students.

SA A U D SD

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185

APPENDIX DConcept Map Sheet

Concept maps are two-dimensional graphic representations of

concepts and their relationships. They are graphic organizers which

represent superordinate-subordinate relationships, and interrela-

tionships among subordinate concepts. Following are examples of

concept maps from elementary and high school mathematics.

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186

Note: It is easy to add "area, parallelogram, trapezoid, rhombus,

polygon, ....

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187

Construct your own concept map with the concept list below.

Tips: Place the most inclusive concept at the top and show

successively less inclusive concepts at lower positions on a

hierarchy. Then, specify the appropriate linking words which

indicate the relationships between concepts.

Concept maps are idiosyncratic. Your concept map may not be

similar to the concept map proposed by others; but they both

may be correct and valuable.

Including more concepts which are not given, and making cross

links between the concepts in other branches deserves extra

credit.

Concept List (A)

Calculus Differentiation Integration

Derivative Integral

Instantaneous growth rate Tangent line

Fundamental theorem of calculus

Chain rule Product rule

Integration by parts Integration by substitution

Distance Velocity Acceleration

Concept List (B)

Expansion Power series Geometric series

Convergence interval Ratio test Convergence principle

Taylor's formula L'Hopital's rule

Concept List TOApproximation of functions Empirical approximation

Interpolating polynomial Least square polynomial

Data fit by a linear function Data fit by a power function

Data fit by a exponential function

Log-log paper Semi-log paper

Running polynomials through data list

Running polynomials near data list

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188

After completing the concept map;

Circle the choices which closely describe your feeling about

developing concept maps.

easy X X X X X difficult

waste of time X X X X X worthwhile

memorizing X X X X X thinking

useless X X X X X useful

trivial X X X X X challenging

diffident X X X X X confident

one way X X X X X many different ways

symbols X X X X X ideas

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APPENDIX E

Item-total Statistics for the Post-achievement Test and

the Attitude Survey

189

Item-total Statistics for the Post-achievement Test

Item Item-total Correlation

Standard C&M Standard

Alpha If Item

Standard C&MDeleted

Standard

Qi .344 .539 .363 .826 .798 .823

Q2 .383 .450 .379 .828 .816 .826

Q3 .800 .460 .811 .788 .801 .793

Q4 .445 .545 .367 .815 .792 .820

Q5 .660 .340 .598 .806 .809 .809

Q6 .268 .498 .214 .825 .797 .827

Q7 .358 .821

Q8 .366 .822

Q9 -.010 .834

QIO .472 .491 .495 .813 .797 .812

Qll .563 .330 .583 .806 .810 .807

Q12 .499 .245 .504 .811 .815 .812

Q13 .693 .558 .742 .799 .801 .801

Q14 .232 .481 .250 .827 .799 .825

Q15 .165 .672 .238 .831 .796 .826

016 .783 .687 _J22 JM .778 .797

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Item-total Statistics for the Attitude Survey

Item Item-total Correlation Alpha If Item Deleted

Qi .327 .878

Q2 .653 .867

Q3 .672 .865

Q4 .423 .875

Q5 .107 .881

Q6 .622 .868

Q7 .706 .864

Q8 .710 .866

Q9 .245 .880

QIO .676 .867

Qll .342 .877

Q12 .731 .864

Q13 .102 .884

Q14 .347 .877

Q15 .420 .875

Q16 .342 .877

Q17 .447 .874

Q18 .474 .874

Q19 .551 .871

020 Sin

191

APPENDIX F

Give-it-a-Try Problems

Lesson 4.01

G.3)

G.3.a)

Often in the science laboratory, experiments are done and

measurements are made and then a polynomial is fit through the

data points.

The following hypothetical data points {x, y} might measure

X = grams of nutrient added to one gallon of water and

y = grams of algae alive in the gallon of water one week later,

data points = {{0.2, 0.9}, {0.4, 1.8}, {0.6, 4.6}, {0.8, 6.8},

{1.0, 9.0}, {1.2, 4.6}, {1.4, 1.9}, {1.6, 1.2}, {1.8, 0.7}}

An interpolating polynomial is fit through the data points.

Clear[x]

yinterp = InterpoIatingPolynomial[datapoints, x]

Then the interpolating polynomial is used to predict outcomes not

included in the original data points. For instance if 0.75 grams of the

nutrient are added to one gallon of water, then the interpolating

polynomial predicts:

yinterp/. x->. 75

grams of algae at the end of the week.

This seems in harmony with the data.

But if 1.75 or 0.25 grams of the nutrient are added to one gallon of

water, then the interpolating polynomial predicts:

{yinterp/.x->1.75, yinterp/.x->.25}

grams of algae at the end of the week. Are these acceptable?

If not, then does a fourth or fifth degree least squares polynomial

"through" these data points yield more sensible predictions than the

interpolating polynomial?

Back up your opinions with plots.

G.3.b)

Extrapolation is the art of trying to use given data to try to predict

what will happen outside the range of the given data.

Here is a hypothetical situation:

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192

A small company in Spruce Pine, North Carolina produces prefabricated

dog houses. Beginning production in 1983, they have no trouble

producing as many dog houses as they can sell and they make a nice

profit on each house sold. Their sales record for the years with 1983 as

year 0 is:

sales = {{1, 1950}, {2, 3826}, {3, 5555}, {4, 7071}, {5, 8314},

{6, 9238}, {7, 9807}}

The production manager notices that the yearly increases in sales are

tapering off and passes an interpolating polynomial through these data

points to try to predict sales in future years.

Find the interpolating polynomial through the sales figures and plot

it along with the sales data on the same axes for 1 i t i 13. Does this

interpolating polynomial yield a reasonable description of the flow of

the data for the first seven years?

Is this interpolating polynomial a reasonably good predictor of future

sales?

Lesson 4.03

G.lO.b)

Take nine equally spaced points x^, X2, . . . , X9 in [-1, 1] starting with

Xj = -1 and stopping with X9 = 1.

Pass the interpolating polynomial through the list of points

{{xi,Sin[xi]}, {x2,Sin[x2]}, . . .

,

{X9, Sin[x9]} }:

jump = 2/8

Clear[x]

points = Table[{x, N[Sin[x]]}, {x, -1, 1, Jump}]

yinterp = InterpolatingPolynomial[points, x]

Comparing this interpolating polynomial with the expansion in

powers of x:

N[Normal[Series[Sin[x], {x, 0, 7}]]]

Notice anything suspicious? If so, then what?

What happens if you use a higher degree interpolating polynomial

arising from more than nine equally spaced points?

What happens with functions other than Sin[x]?

Do you think that approximation by interpolating polynomials and

approximation by expansions are unrelated ideas?

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Lesson 3.07

G.6) Error propagation via iteration: against us and

Recall that we used integration by parts to prepare

values of e* dx for various values of n's.

We started out by setting Int[n] = e’^ dx and

for use

a table of the

then we integrated

by parts to learn Int[n] = e - n Int[n-1].

We used this iteration by entering the exact value of

IntfO] x^ e’^ dx = e’^ dx = e - 1 .

Clear[x, int, n]

lnt[0] = E - 1

Then we typed the iteration formula:

Int[n_] := Int[n] = Expand[E - n Int[n • 1]]

Then we made a table of exact values:

ColumnForm[TabIe[{"Int"[n], Int[n]}, {n, 0, 15}]]

Now let's see what happens if we do the same but instead of entering

the exact value lnt[0] = e - 1, we enter a rounded off decimal

approximation of e - 1:

N[E - 1]

Clear[x, lint, n]

IInt[0] = 1.71828

Type the iteration formula:

IInt[n_] := IInt[n] = Expand[E - n IInt[n - 1]]

Make a table of values:

ColumnForin[Table[{"Int" [n], N[IInt[n], 20]}, {n, 0, 15}]]

Now let's make a table comparing the exact value and these values:

ColumnForm[Table[{"Int'' [n], N[Int[n], 20], N[IInt[n], 20]},

{n, 0, 15})]

For small n's, the exact values Int[n] are very close to the

approximate values IInt[n], but for larger n's there are dramatic

discrepancies.

Let's try to see what's going on:

Suppose exactO stands for the exact value of ln[0]:

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approximate value (exactO + errorO) for lnt[0]:

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G.6.a.i)

If the errorO is .0001, then how far off the correct values are the

values of ln[0], Int[6], Int[14], and Int[15]?

If the errorO is .00000001, then how far off the correct values are

the values of ln[0], Int[6], Int[14], and Int[15]?

G.6.a.ii)

Describe the effect of the original error as the iteration progresses.

Do small errors at the beginning result in larger or smaller errors at

the end? Are small errors at the beginning anything to worry about?

Now it's time to turn the tables.

We are going to tackle the problem of finding a very accurate

Mathematica's integrator will not touch this one:

Integrate[x'^Pi E^'x, {x, 0, 1}]

We can try for a numerical approximation:

NIntegrate[x'^Pi E'^x, {x, 0, 1}]

But what if we want better accuracy?

Although iteration worked against us when we used approximate

value, Ihere is a way to make it work for us. The idea is simple: Wejust iterate backward instead of forward in the hopes that as we

iterate backward the error will shrink.

approximate value

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Try this.

G.6.b.i)

Put I nt[k] =IJo

dx

How is Int[15] related to I x” e* dx ?f-Jo

G.6.b.u)

Use integration by parts to obtain a formula for Int[k] in terms of

Int[k-1] for k = 0, 1, 2, . . 15.

G.6.b.iii)

Let exactO stand for the exact value of lnt[0] x** e* dx and report

on the error introduced to the calculation of Int[15] x’* e^ dx by

incorporating an error into the value of lnt[0]: lnt[0] = exactO +

errorO and iterating.

G.6.b.iv)

Why can you say at a glance that 0 < lnt[0]

If you cannot see this at a glance, then look at the following plot to

see how the areas line up:

Plot[{0, x'^tlS + Pi) E'^x, E), {x, 0, 1}, AxesLabel->{”x", ”y”}»

PlotRange->All]

G.6.b.v)

If you use the value 0 = lnt[0] = exactO + errorO, then why is

lerrorOl < e < 3?

G.b.b.vi)

If you use the value 0 = lnt[0] = exactO + errorO, then how many

accurate decimals of Int[15] . f'..Jo

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G.6.b.vii)

Give a more accurate estimate of the true value of x” e* dx than

the estimate given by:

Clear[x]

NIntegrate[x''Pi E'^x, {x, 0, 1}]

G.8) Yuppie calculations

G.8.a.i)

You put invest A[0] dollars with and investment outfit with the

stipulations that:

—-> Payment will be made to you or your heirs at a rate p[t] dollars

per year t years from now.

—> If A[t] dollars is the amount in the fund t years from today, then

the these A[t] dollars are to accrue interest compounded continuously

at a rate of 100 r percent where the rate r is held constant.

—> lim A[t] = 0.t->00

(This arrangement is sometimes called a perpetual annuity or perpetuity.)

Explain why

A’[t] = -p[t] + r A[t]

Then explain why

e-*-^ A’[t] - re-'-t A[t] = -p[t] e-'-K

G.8.a.ii)

Look at:

CIear[A, t, r, p]

D[EM-r t) Alt], t]

Then integrate both sides of

e-f^ A’[t] - re-^^ A[t] = -p[t] e'^^

from 0 to oo to explain why the original investment A[0] is given by

A[0] = e"'’^p[t]d t

(Some alert persons may note that the original investment A[0] is a Laplace

transform of the payoff rate p[tj.)

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G.8.b)

The formula above explains what the financial people call present

value. They say that a profit-making scheme that will pay profits at

a rate of p[t] dollars per year t years from now has a present value of

where 100 r percent is the projected interest rate over the future.

So the present value is nothing but the amount of initial investment

A[0] that it would take to get payments at the rate of p[t] dollars per

year t years from now in the perpetual annuity set-up studied in

part a) above.

You have a profit-making scheme that is projected to pay profits at a

rate p[t] = (100000 -i- t) e'^/^ dollars per year t years from now.

Assuming a projected interest rate of 6 percent, what is the present

value of your scheme? Still assuming a projected interest rate of 6

percent, how much would you have to plunk down for a perpetual

annuity that would pay you at the same rate?

What is the projected total take on this scheme?

How many years would it take for this scheme to play out in the

sense that the future take will be next-to-nothing?

Lesson 2.08

G.5) Parametric planets

This problem was adapted from an article by Donald Saari in the American

Mathematical Monthly, February, 1990.

For most of our history, there was a vigorous debate about whether

the Sun or the Earth is the center of the solar system. Many persons

wonder why it is important. Many others wonder why astronomers

of antiquity had so much trouble predicting the paths of the planets.

The truth is that their view that the Earth was the center of the solar

system made the job of charting the motion of the other planets very

difficult.

To see why, let’s study a simplified version of the Sun-Earth-Mars

system.

Here are some simplified data:

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198

Both the Earth and Mars move on orbits that are nearly circular and

both orbits are in the same plane.

One astronomical unit is the distance from the Earth to the Sun. Mars

is about 1.52 times as far from the Sun as is the Earth.

Setting the Sun at the origin, measuring distance in astronomical

units and measuring time t in Earth years with the Earth and Mars in

alignment on the x-axis when t = 0, we can give a pleasing plot of the

motion during the first none months of the first year.

Clear[t]

{xearth[t_], yearth[t_]} = {Cos[2 Pi t], Sin[2 Pi t]>

{xmars[t_], ymars[t_]} = {1.52 Cos[2 Pi t/2], 1.52 Sin[2 Pi t/2]}

orbits = ParametricPlot[({xearth[t], yearth[t]}, {xmars[t],

ymars[t]}}, (t, 0, 9/12}, AspectRatio->Automatic,

DisplayFunction->Identity]

sun = Graphics [RGBColor[l, 1, 0], PointSize[0.1], Point[(0,0}]]

Show[orbits, sun, DisplayFunction->$DisplayFunction];

Here is a plot of the Mars’s orbit as charted from Earth during the

first 2 Earth years:

marsorbit = ParameticPlot[{xmars[t], ymars[t]} - (xearth[t],

yearth[t]>, (t, 0, 2}, AspectRatio->Automatic,

DisplayFunction->Identity]

earth = Graphics[RGBCoIor[0, 0, 1], PointSize[0.1],

Point[{0,0}]]

Show[marsorbits, earth, DisplayFunction->$DisplayFunction];

From Earth, it appears that Mars is doing a little dance.

G.5.a)

Explain the presence of the number 2 in the denominators inside the

parametric equations

xmars[t] = 1.52 Cos[2 Pi t/2]

ymars[t] = 1.52 Sin[2 Pi t/2]

G.5.b)

Plot the orbit of Mars as charted from Earth for the first 2 Earth

years.

Discuss how an old time theorist would have a lot of trouble

explaining the result.

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G.5.C)

Jupiter sits 5.20 times as far from the Sun as is the Earth. It also

moves in a nearly circular orbit in the plane of the Earth's orbit.

Jupiter takes about 12 Earth years to complete one trip around the

sun.

Setting the Earth at the origin, measuring distance in astronomical

units and measuring time t in Earth years with the Earth and Jupiter

in alignment on the x-axis when t = 0, give a plot of the motion of

Jupiter as observed from the Earth during the first five Earth years.

G.5.d)

Give representative plots depicting the motion of Jupiter as charted

from Mars.

G.5.e)

Give representative plots depicting the motion of Earth as charted

from Jupiter.

Lesson 3.08

G.8) Earth and Mars

The same as Question G.5. in Lesson 2.08

G.8.a)

Explain the presence of the number 2 in the denominators inside the

parametric equations

xmars[t] = 1.52 Cos[2 Pi t/2]

ymars[t] = 1.52 Sin [2 Pi t/2]

G.8.b)

Measure in astronomical units the length of one complete orbit of the

Earth around the Sun.

Measure in astronomical units the length of one complete orbit of

Mars around the Sun.

G.8.C)

Measure in astronomical units of the length of one complete orbit of

Mars around the Sun as charted from Earth.

Is your answer the same as it was in part b) above?

If they are the same then try to explain why.

If they are different then try to explain why.

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200

APPENDIX GExams

The First Exam

Write the first three non-zero terms and the nth term of the power

series in powers of x for the following function: (No explanations

reqyiirgd)

1)

f(x) = —1—+ x3

2)

g(x) = Cos [x3]

3)

h(x) = x^ e‘*

4)

k(x) = Sinh [2x]

Each of the following functions has a power series in powers of x that

converges on any interval [-r, r] provided that r < R. Find the largest

value for R in each case and give a brief reason for your answer.

5)

f(x) =8 - x3

6)

g(x) = Arctan [3x]

X

x^ + 2x + 27) h(x) =

V -

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luoy 10^ bna iiai?3 m Jl

— it

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Ic?1 OU5 - (4>a <d

c-'k>“x

For each of the following power series in powers of x, identify the

function represented by the power series: Explain your answer

briefly .

201

8) 1 - x^ + — ^ + . . . + — + . .

.

2! 3! n!

9)l+2x + 3x^ + 4x^ + ...n x“’^ + . . .

10. Find the coefficients a, b, c of the interpolating quadratic

polynomial p(x) = ax2 + bx + c that fits the data: {(1, 2), (0, 3),

(- 1, 6)}

11. If you needed a polynomial of degree 5 that would fit the

function f(x) = Sin [2x] for small yalues of x . which polynomial

would you choose? Explain briefly .

12. If you suspected that the data from an experiment was coming

from an exponential function y = a e’^ or a power function y = a x^>,

explain how you would decide between the two and how you

would compute reasonable yalues of a and b after you had

decided.

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202

The Second Exam1.

For each of the following power series in x, find the largest value

of R that you can with the property that the series converges on

[-r, r] for r < R. Justify your answer .

a) 3 X - 9 + 27 x3 - + (-If-' 3“ x“ +

b) 1+-A?— +—si— + + sli +(2)(2)

( 3 )(22

)(n+l)(2“)

2.

Evaluate the following integrals;

fa) X Inx dx

Jo

b) cos^(2x) sin(2x) dx

r*

c)

V9 + x^

dx

3.

a) Find the power series in powers of (x - 1), including the nth

term, that represents the function f(x) = 1

x^ - 2x + 5

b) Find the largest convergence interval that you can for the power

series in part a).

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203

4. Use power series to evaluate the limit

,in,cos(2x)^

sin(3x^)

(Note that the numerator and denominator functions both approach

0 as X approaches 0.)

5. Given a function f(x) and a point x = a, how do you find the

equation of the "kissing parabola" k(x) to f(x) at x = a? Apply the

procedure to find k(x) for f(x) = x"^ + e'’^ at the point x = -1. (Do not

simplify k(x).)

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204

The Third Exam1.

If f[x] i g[x] for all x's with a 1 x 1 b, then why it is automatic that

f[x] S[x] dx2.

Explain the formula

jf[x] dx

^ Jf[x] dx

+j

f[x] dx for any number c

with a < c < b.

f

.b -b

3.

Explain the formula|

Kf[x] dx^KI f[x] dx for any number K

(this means K 1 0, K=0, or k 2 0).

4.

Suppose f[x] is increasing on [a, b]. Why is guaranteed that.b

f[a] (b - a) ^ If[x] dx ^ f[b] (b - a) ?

I.

5.

a) Use the rectangular graph of the polar equation r = 2 + 3 cos(t)

0 1 1 1 2ir to sketch carefully the polar graph of the this equation.

0

b) Find the slope of the polar graph of r = t at the point where t = TT /2.

6.

a) Find the rectangular equation of the ellipse centered at (3, 2) with one

focus at (6, 2) and eccentricity e = 3/4.

b) Find the foci of the hyperbola.

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205

The Final Exam

1. Evaluate the improper integral xe’^* dx or show that it

diverges.

2. Use Taylor’s Theorem to find the partial expansion in powers of

(x - 1) up to the (x - 1)^ terms for the function f(x) = x^^^.

3. Find the interval of convergence of Xn-i n 4

4. Find the sum of X onn = 1

^

5. Evaluate the following integrals.

(a) - (c) are the C&M version and (a)' - (c)' are the standard version,

(a)I

e®^ * cosx dx

(b)

(c)

1

i

y'I

X sinx dx

dx

(a)'I2 x^ e^^dx

1(b)

'

(c)

(9x2 + 25)3/2

dx

. f x3 - x2 - X -h

j X^ + x2

^ dx

6. (a) and (b) are the C&M version and (a)' is the standard version.

(a) Sketch the polar graph of r = 1 + 2 cos t by using the rectangular

graph of this equation.

(b) Find the values of t in 0 1 1 1 2Tf for which the polar graph passes

through the origin.

(a)' Find the area of the smaller loop of the curve r = 1 + 2 cos t.

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APPENDIX HObservation Note

206

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VITA

Kyungmee Park was bom October 15, 1965 in Seoul, Korea. She

attended Seoul National University, receiving a Bachelor of Science

degree in Mathematics Education in 1987. After graduation, she

taught mathematics in Kum-Ok Women's High School and Dae>Young

High School in Seoul. She enrolled at the University of Illinois in 1989

and received a Master of Science degree in Mathematics in 1990. She

was supported by a fellowship from Korean Government during her

graduate studies in the University of Illinois. She is a member of the

Phi Kappa Phi honor society.

Any

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