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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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THIS IS TO CERTIFY THAT THE FORMAT AND QUALITY OF PRESENTATION OF THE THESIS
SUBMITTED BY KYUNGMEE PARK AS ONE OF THE
REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
ARE ACCEPTABLE TO THE COLLEGE OF EDUCATION.
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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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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
V RESULTS IN THE CONTEXT OF AFFECT 99
Attitude Survey 99
Results of Testing the Hypotheses 100
Further Analysis 101
Conclusions on the Attitude Survey 110
Questionnaire Ill
Observation 119
Details of the Observation 119
Observation Notes 124
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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CHAPTER PAGE
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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20
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.
most general,
inclusive concept
subordinate,
intennediaiy
concepts
most specific,
least inclusive
concepts
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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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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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
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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
investigator decided to include the test items regarding cooperative
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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50
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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>7, 7->6, and 6->7. Thus, the corresponding matrix is
1 2 3 4 5 $ 7
1 0 1 1 0 0 0 0
2 0 0 0 1 1 0 0
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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Nfisconccptions QTotal 87
Congruence Coefficient .88
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
Congruence Coefficient .55
Figure 8. High Scoring Map Demonstrating the Large Number of
Relevant Concepts Identified
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Hierarchy 4 * 5 = 20
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More concepts 0
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Total 57
Congruence Coefficient .83
Figure 9. High Scoring Map demonstrating the Thorough
Understanding of the Given Concepts
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Total 44
Congruence Coefficient .31
Fi gure 10. Low Scoring Map with the Lack of Interrelationship
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Total 47
Congruence Coefficient .63
Fi gure 11. Low Scoring map Demonstrating the Relatively Simplistic
View with Few Relevant Concepts
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
Iota] 9.1 40.7 JA. 4LQ_ 7.2
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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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Table 24
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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•ura|f>>f ,<lt»ht>i ?y:um»;U bOA niiyg •Hla«s^u>
root J9^ Izj^^tni) n90fJ’f(f jc b«ti J)d|bw fonj
jSilO'^ ^irw^zih tlrf i' (»40l J!:'-i»JH THJfchiiqf^^
owl dfli k) B.>?a»soi«3q4^ fcn6 «|r?j iff'JifiYtib <piii
8<#t zjt iniA #aii/ (nd-»| t^‘t£ui loa bib srwoo IvdAD pdl
birliH'fU i ilifw ylUnrlv II ?jt.3)ia{ jtib »:4ofl4tjrs4 Uftohi?i»m
^^lt<tl^o^ol<i mttiei ^ojqot 1o ladiRiiD
fri9!3tW^7 f/9at!^04i OjHj b5»it0^i al09bf}Jl ?/i :^>. 5fil ki iZOfTT *fcliQOa3
i«o©isb«Tl >> biMi / »(Jiiawo4ia5^sarwilw^/’«oi^ftB(v^ lUtw qim-^
ijfft .q«m jKi)p,!h> ^--lU .Sii ’‘fiti'Oino'i b«o|Jiq tm^Lm '
jfO k. «M4ufqm3 bam^iqud ^ oidit J^rit \
PJ -i ..Mtioo4 owl
-1 3 <J^ - '> %dt *b>*4T_^
»l» Ifj tBolbt^i |.i|irT 9ili Hn no Jilqmin x^»po «d i.v
bft> niqmbt .‘c^5V vWic^ia “*«oWli
ttA y*ffui*’ j*K>V«iobt" 1,
'
^'^ iiuoliiyb/pd** ,’*5i0riKjJi<ol£O .
.,r-'»
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.
m rr-j
ojiiiiiiiiimiM
aHi tn Uu»(?l Tww '"itTMjqoofj^ri«?m noifi^oo
%UT»^o>n iO‘< binoo m’Im* eiUdhtiir J»i^. Jgpmh .-‘.'R Jqs:»ii03^
r»!
. rfj<^ 5nt^dorK»:> -tjw Jo i&rti
:t< fii m<ru>aHi t>(i) bnu^tfi m>ijnT^^iqi oJ ntHisitAatlib
nisriootU art» doo xfjjto irohung^^toi "JfT'
i^*4Jsr':^rK» noi.«f|- . to9¥tn ?i<(i .^fboJiioabnU .daftuvj
! »i **:>#• lo rm osdT Ifc^i.-rionbriun adj )*»dl lOitl ^dl
i^
".Ipjqiib Mul>rr|a)ni ^ilJ
Ifipohibini Ofl) ni tJn>buU lenii‘^5R ^
f i t»iIT diqiiiuTq *'d) l»oo}£t»fcifloJUTi
jtiii io olqibiiiha yf' ,zm tirjotfnq ad; ^abi-c^oa
n| nw! .'**• * foi /«d
aiiia to9^ 5<U ».• iswoa '^tU) «ii** noerwj'iid bvt'; aitj
oQ^jsjnvr?*, j *: '».j»rtO 1 rtt^ b»i;n
Mr i^d> taa/moa )oii bii) btifn*^
no'iaaoftoa^iui 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. b^U'i atffbaaoTq ^niqqnfri Kpoooa aJjf <ii
iqojfio'i r.a*q<)J5v“ab Jirodft RgiuiaaV ‘t-fj^i 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 o^A J ^^»^^*
0.4u‘d</iyt intfi :^ia..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
'
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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
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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 — ^I0. 1^
' U0.^> ii»l|!*i40i*»b « (Jj) -U/hi
lO.t) (fvll) 3n^tJn>ib
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- 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 Vi^i«*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 q^oxj i U;t3 ;ra bifja quat^ Mibj died to esw
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.,
^ ' ' ‘
t 03 dcist^ ^iuwcito> o<U'bowo/!»
** 1;4*:cilibi-’l •n) «d dtOdi.oi ibo*. 9dt af alfiV^UlJf
'"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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r; 5fl|im <0^i9tlib wl* lol .
m bft^abotjiu iti • kflotttbiii *idi^ <8«r ipniwxriSCil ^di
ju^oid Mil' iSiJft ,^-nl irtlirgrij idi \0> d^ltiia csiU^iJv :«ll
>(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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\ ’4vt>o^rTi 3 cii t i(9boii >u>d
ttc ifit qryw , Xiiiollif'itJ. mil UstM flU^J
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9 iftwtAjp bkroo uii^^ a/t^ Vi hoM JBQiiiQ
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• 0Oi>mb;4 • wo TcnJmtfa t lA il M * t j •• jR ^ i— /•' j ^.*1 I
TOlljDItni f ..^ tVf’zJl I'i<ruJ4ij tolot> $ « Vf» loxj::tlni T*
(rjl ^ 3YJvj *ii? 3i» ^I'jaiufc iliUv - *<[j^ '
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.»i il »jiBiitr Util *«tS ’*tli ti a>»r» aididovj
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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
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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:
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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.
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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
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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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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
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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
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iL^^/iJiw iolm )0j TfJ b(t|U£»9b^
irn^si ji b^fnlt m emabK ' isits ijifl'
a!ohq<yspqi rtiiw ]cotq hjm^ mUtn m ,t/rtJViGif{ lit loo^if 9fl|
tf )r|^9:iii oJ jcifiJfw tfkf«*dOiq hval ^WaiTUb
i,*;orr?;C tMus ;iOiitailX'>'KjfyAy VI 1& mq Tsjfil "'fl
uJ Jon <aiqoi i»eijf3.nq(ri»>^ unjW9(tt*JZ dii*\
laiqsrirj rri irqoJ v#s>u tub i»rss*i la .v-9vaWt.B -»i:«buj(a
^ b»<teirt>Jrt '/ Mil .1^ jll
v/cMm) ton 4.50b di^:^r!»o3 ©li*' ir?ottri«;aiiti lo
»uwf iticT^'iStti *41 1^ i5bi^ Isoiqv
2lJiJio*Trtoa iWii l>omf>3r^. ,fsT5©Jaj boii ,(lK’i^attti
tO'H .iiaj iilutf<iv (Cbisom dqSw jf?noO«a»lqxd {iSa r&iw
9 i:>w otlj to ©ain^qtit^ ’jitjtd 6vidi ,l0.t dosit^i rt* ^Iqmiixa
JO 'j\yo^n |.i|sf>od3 <<l bofini^oi.' tiailJ t>n«
Joo^ Umot ta% J3hr>i %bn*i cdw, lOobjtrtA oJ i^'b» m a,X3niid3 9, bad ;' 'iv.
©dJ $n.iob Jli;i05qt:;« t4(ini»4TiJa^^ « mKv a^inoo aib b^qqo'tb
r o‘b bed ad i»i« bki C^.U) aH v*K>ivf<tq
r^iuLooo^ i4? xd t»,^KijitmUn} ftbad Ixrf aH «»?rjii5ad
1o aHj \i U(t£ isn^i^m ytx»»iii
'lO^h-’i Vi\ 41 XT^^C babbx .JEiicdvdo oi9>j© ,»oi**V
*> -
h
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?
itti U.I rS U7vt;» »t^imqor; aiil V'
j 0</6 cwi« t'«i 5 1o ?:»iiud U ltf<! aouHr-^ |o<S>TCf
V jjiKdvnr jd'* iwrtflJ inoi^ 'to’^ ^<iAi >• norl:;ii£x?tliti l»i>J i U cf^' -v
’ ad «idr obuJ^»ooD
4?4nx|OTq orjJM^ adi Ho floiicjisiqia i/biatnl aib ban bna^^wi
.Me vw antt'oo adi n\ babivoiq ^U¥f clo«.. u 1 -dio ^ p|?V
^ 'li? Tii« ic donaarw<»o^ ^fqrnifl a t«' bk'&b<nifti
a^ub^uQiq ^ vt) ^xowon. ttof al 9drof i^wue^a .'niTi criT ayfan-'Iiob
A»::i^it3l »Hi ,iivU iai>A aui% # t)«fi8 noi»:>f»Dl,
» dJtW
arti <10 a.Tin? a;i: «i WV*%^v’ o 1 v« a VaT
I*
^ .,
.V* *
.
yaa «U8 ii/», ftlum^Ol aif tKK>i,'-abnu -Utm^ jjbato ndl .iiOU»vTjl<fe -
woi '^i’*'i5lXPa "int'Otfl® i*d< lijS^^fbin stifl bjti ‘ Ho
y»oHod bsibWiyiq *» ta;o>fic4 5 rntocao 7*^
jdJUisaiatfi ^’• .^o«> 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 ak^i 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‘
^ adUjL^ulqxa Ho boofmbfto bfH ’>«V. Mioaixijii'
0) *,ii volbK^i‘’at>ei> /r^/o *?u\2 id ^ 9>ud»uiTw'> ot*£*»i8aqq^
i.i ^,$S,0 ol ,9lqmi.X9 aib |<^ri«iboa5t-rwiw ac!*.
' C 6trt> i*no .yti.ia^ »o isiuar tdJ l.dtsfi*' wlnrw ^urdw TO.r* ao« ^‘jjtw Oft> oJ w ydkiqia t y,H)
i-njil. <wv
3niik)oinrv.{»7 4 rta&’ gm
} fb bcil ©T » 1 . Gtttfl to »TiM )n> Ht 11151'
:fer'
. :t\
^ir,v'
CJ
#..
«»dii
,;> • iijusl .tf ^farat MtUiaav to bot id^Lu.:
.0 Jt' VxJV Wf *tao^ *-a ^ VI yilivttb 1 ! tJ J»‘X ovtiljp^<''. » 10 fdii^v* ImlOf Od' ft le 4t HOm* s
<) »«!$*> 0^ bfa id. lo odi omU 1 7?* of)y utoow 0 < d »*'j»t*
" V. toi4tii4ia<i fmii«t<)o »uo\f d *11
trw>iMaoo *> 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.
* ^ ^*
t^a^v >r ;>r (b^liUrtobtiu nvtfO '(Ihi.ij n5 'itf' £lH
.'. u'Tn^i ja:buM woH-jli rJi ol onr-T^o^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 lli^if^ ftOlusljj r-into^
mU aJ '•i")ftfC'Ui oi ifit n[ U : a^im di'iklw .''jiolq ^odi
-'.•<tui fil ’tmiv'ii; *^uv (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 ir^awio^
'•-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<b
lislil b‘UB ;»/b*l8J Oi b;ui4 !> Ifilil Jl^ib<lfi^?rlobuu<:irn
'j' >'.<;i wTiith'^t *jo3bti,V5 '^'il > ^ntmxoJ lauiibl oi alosj- cfO
,, -fij ^iiUiinBuU' noi^bidadc. rohfi^atn 1o ^oimaar 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 \ ] hm^uAi 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 '^i^isitv^iTtm 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)itiava<p ;»
oi -»vbJ9tbs arue ?ini:fn^i^»oqcjft yd ^
5fTibn$4«i^biujf kfi *^5*^MJ*, g .
,i’’uol«o grti loityt 2i !>i»vT
M
13 3‘^d bi 9Ct»q^frq (Ui^
i.' UBw »A*^Vi *i((j Vi aniboAJ' ^alqincw^ ^ ***^'* ™ g^ .-~
.)^vj«3*wi <T3i.*>:4»->a<T vi )ou Jci§io Faf>^aJ ^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 b^on ^'
Zi fii»q adj di b/iri^mca adi ^44*»bo3? ^d*
ti amlf ;y%pm )i ia.7 :qa^it) yErn rr.sidotq laifV ./rtatliyboan yIbnHd
adl to ^4bo3;jnrtrndr5 ivVUdmii^^juH ^o coiscflalqja =adJ ir' Uoirofli: s.^
boiriit'i aiJ r<i baibfUfi t! cbii««wr-Au Hr?' rmk Wwuo^’ife
•K -wmf - (^iI±jss2iMQ^Lm ,aigQ tagAbm ;
'
- _
w $ maw. «^!iT ijoauloa aaoabtfii aiff-
•i^o« adJ s«oi'-»ad ^aibaEJmbad ibi^ rftqab adl
»? RfrialdoTtj fno*t»r adi aw\tin batnaiTO'iq^^RO'^ bdE ^fliJuot
anhuik>i» 'e»«‘ji3Mi «o nwpiM! aaibui^i'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 YC^aidoiq
' 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
ootst^I 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' vftoa^art*
."naiTUSd”
Tt
A
.if»Mi oK t arfi Hj<i4\df i^af* 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 n^aab fcamsaa
'\ii « n avijt" orti laom *^J1 loos main arii si
iTijCiiai *^U*oo^4 cftt^idoTiq "<(TJ i ii avij|‘‘ sitJ lo StiK} tu8 -Jifaaick-iq
fts 5Sii|, fi ^ai<i5tdf>i<i a#^uU nj bfti^ adl
ai« */ri4Jbau art! lu ?.iO a<U
oj ^:;r<aljnaj ir'^Yvert xm'ibuU iflrti e^^iTii^ciimoo rtii /.’ ^aiiirut^, lo
... , rt04im,
^4 gg3 '
.«vt>
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»i^i lo tlfivif^ttil i>'>«»jgioi^woo fertH i)!t>6d? ’ ' uuH .ifb'^nv
• Jiitjioffilq oixfid scft vj i^ai oh®”f »<ii *^'1^
ov'* .liolq 5)j!h<yjTfn ii&itr'y- «rt. ^iint?,*jlfi
V;f(tCj^l^)'/n03' ^Ij 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 aJ^i v^odJo tud wmO>q bii^ *ifi «adw
»#),
iqr*»<. io'"i 9*iftagiavcoo ^tr^aiq 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 diaiv^o^
daiMW orn-jt o^y^i* -r^l ai X^y w^hor^ ** ftlqiiifjx^, .rarfmiiA ^
sen / 'JltJlltfli A&’ ** ' ’ ^^UT71*ital> At i3A®tH.dt aHt iiJ-ltWpAl
^tiniVri K'^ -01 4f <iw,
t>A® ,j«cg[W^ano 2114
^*T -
lU *'-i.< Hill j3Wv:hi* j^s S'»3aQA'> i it ^iiwbip''? 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 .hiHn^iliOcU
mm ^ ''"=
"i«f1 uf maldc<iq .‘.U b'ldc^kOicnja ooirl? *1.1^^,
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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
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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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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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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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m’-pi law ’Sn ^ tnoalaJ iii diar^aii ijgjfl
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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)
CfttJ ncrtj tiol aiom ^Mlfi a/ittwJni
f»i» ’^'i “>0 noiunlnmoi»f>- ,»i4oi ^iJ.sn'r/tilq ii?,-
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{j- ' *fU .y 4:510 '>uij4 afalqmoDni io ^iiiuntrt^ tta^ui biuov?. Ji
. »: !< 5*'J ,o mo lirl 01 5>fT«>e blwo'^ . 151^1 WfU
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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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fl.r5ijr.‘i oi lobu*^ al ^ihnwo<3o «i fti«imou’^o<i ^iflbBfocpeJnr
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/ ^tiJ|lh' -iil) jjo <9ife
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io ? •,
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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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97
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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98
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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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
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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)
Table 38 (Continued)
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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Table 38 (Continued)
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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Table 39 (Continued)
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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Table 40 (Continued)
22. 1 am good at working mathematics problems.
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."
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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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jaiV»»rlpV:^ >i±’ ii> fsiSg MC »lf S .ilsWviq^
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
^ 4^iit «ui $i vhixM^ tuapi um:>i4vM;r.':*.Mr|q ^ tfjS a. ti ^ t4|JU3 Jl i }*<h.J^ .£fiii»ldOni< i^!.n
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"
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oib x<6 l/n^qmnii ^ai»w
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Uvi lo pd;:o|&?w
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m ^ 41 ^na# -iwmmm) rjojtca e:a«biiie
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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
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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*'*) .bj^i/lovw
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„= b<»4i tiiiwat ^ bujin tiu «J7f>iTjM l>#*iMUnao
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wo-.w lo^pcn) te'sl !*ih»< r>iOd;
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isr; V^ ftd'i- «i ooLrislBb \ld a -li-^yliBrvi eaaii^TOO^-'
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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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n<>ba^iju I i«i rftf ^uibidm joi .qu muz
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\uT .aIqo»'** .tuii b<&«rU9l x**'^ bajoamna «4; jscu <i<i
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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
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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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t;v v,m iir^:- ,di tfttao o«H odw j i/irtfi^I
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lariio a:/i aO ,5«mw9? «xiOiriffett3t/b «wm 'V ffl
ftb?in x>dw ,wisa^£>w t5iSio '«iU ,band
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l^&av'^ biii ^^bh* .K^ioqcKoa wb n*^ M*d9
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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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ttaitt ;*Ai5 It *;ni(K( pth t^tMaCl/tl ^^ILiA^ky
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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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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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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.
Presentation 5/10 Mathematics 6/10
Comments: Your mathematics is fair. Your answers are rather obscure.
Your presentation needs work
Presentation 9/10 Mathematics 8/10
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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Presentation 10/10 Mathematics 10/10
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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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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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,
5
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Consequently xe*dx is given by:
Jo
In[2J:= ((X E'^x/a->5) - (x E'^x/.x->0)) - (E'^S - E*0)
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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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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the bottom-up process.
157
generalized knowledge
(rules)
compilation
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(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
between the up-and-down of a function and the sign of its
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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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
between the up-and-down of a function and the sign of its
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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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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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
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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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.*; . 'i ^ • .‘L- 4 ’''
The National Research Council (1989). Everybody counts: A report to
the nation on the future of mathematics education . Washington,
D.C.: The National Research Council.
Travers, K. J. (1981). Second international mathematics study
detailed report for the United States . Champaign, IL: Stipes
Publishing Company.
Tucker, T. W. (1990). Priming the calculus Pump: Innovations and
resources . Washington, DC: The Mathematical Association of
America.
White, R. M. (1987). Calculus of reality. In L. A. Steen (Ed.), Calculus
for a new century (pp. 6-9). Washington, DC: The Mathematical
Association of America.
Wolfram, S. (1988). Mathematica: A system for doing mathematics bv
computer . Redwood City, CA: Addison-Wesley Publishing
Company, Inc.
i i'
- *“• mP
f-] «r> r« A iloiisedH i^'»obf>i adT"-
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. >o'Y'f»03!bl>A -v;n h'fi>wl>©5i 4^^%Qli/d
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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
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?
8bt » -
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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
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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
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AiNOi3lO«r a tfc (;cV^ ^»4U ^ aq*Tj<
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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)^
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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).
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if 'I lii j*i<? woia^ Mliii ^ moil 4 adi. V
^.
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
f g h
rr
/
-r
J
1
i
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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
m
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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
t
ms.
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" ^re U "
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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.
tiln
mu’ t -
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tlJ
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1
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1
1
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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
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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1V
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]:
Itti:
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CIear[x, Int, n]
lnt[0] = exactO
In terms of exactO the exact values of lnt[0], Int[l], . . and Int[15]
are:
Int[n_] := Int[n] = Expand[£ - n Int[n - 1]]
ColumnForm[Table[{"Int''[n], N[Int[n], 20]}, {n, 0, 15)]]
Now watch what happens when we build in an error = errorO into an
approximate value (exactO + errorO) for lnt[0]:
Clear[x, Int, n]
lnt[0] = exactO + errorO
Int[n_] := Int[n] = Expand[E - n Int[n - 1]]
Then the corresponding values of lnt[0], Int[l], . . and Int[15] are:
CoiumnForm[Table[{"Int"[n], N[Int[n], 20]}, {n, 0, 15}]]
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
e* dx are guaranteed provided no
other calculational errors are made?
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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 -
M3
0 /IGKB^^IA
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luoy 10^ bna iiai?3 m Jl
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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.
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