My Fifty Years of CalculusMy Fifty Years of Calculus Abstract At the end of the fall 2015 semester,...
Transcript of My Fifty Years of CalculusMy Fifty Years of Calculus Abstract At the end of the fall 2015 semester,...
Paper ID #17610
My Fifty Years of Calculus
Dr. Shirley B. Pomeranz, The University of Tulsa
Shirley Pomeranz Associate Professor Department of Mathematics The University of Tulsa
Research and Teaching Interests: Boundary Element Method and Finite Element Method, NumericalMethods, Engineering Applications of Mathematics, Applications of Mathematica, Women in Mathemat-ics
Dr. Peyton James Cook Ph.D., The University of Tulsa
Department of Mathematics
c©American Society for Engineering Education, 2017
My Fifty Years of Calculus
Abstract
At the end of the fall 2015 semester, our Department of Mathematics at The University of Tulsa
relocated to newly renovated offices, and I had the task of emptying my office drawers and
cabinets after twenty-eight years in the same office. I found all of my calculus notebooks that I
had saved from the late 1960s, when I was an undergraduate and took my first calculus courses.
After more than thirty years of teaching calculus, and in observance of my fiftieth anniversary of
having taken my first calculus course, I would like to share some of my experiences in learning
and teaching calculus. As an undergraduate at Barnard College, I took a sequence of calculus
courses at Columbia University that was intended for physics and engineering majors. I now
teach in a mathematics department that is within the College of Engineering and Natural
Sciences at The University of Tulsa, so my observations are relevant with respect to calculus for
engineering students.
Much has stayed the same, but the use of technology, student demographics, student
academic/social support, the curriculum, and the way calculus is taught are some things that have
changed, comparing my calculus experiences from 1967 to those of my students in 2016. Not all
the changes appear to be for the better, and there are tradeoffs. The discussion focuses primarily
on anecdotal examples, although some statistical data are included.
1. Introduction
There are studies on the teaching of calculus at the university level that give detailed histories of
the pedagogical changes over the years. Some researchers who carried out such studies are
Friedler, whose work includes studying the connection between teaching and calculus
enrollments and who also examined the texts that were used at various times [4]; Rosenstein,
who noted the calculus texts at use in different eras to discover some ways in which the teaching
of calculus has changed [7]; Tucker, whose research describes the history of the mathematics
major and collegiate mathematics in the United States [10]; and Zuccheri and Zudini, the authors
of a text presenting a global view of an European, Brazilian, and American calculus history [11].
The presentation in this paper is from a very much reduced perspective and is based on the
author’s experiences. At the end of the fall 2015 semester I was happy (at first) when I
discovered that I had saved all of my calculus notebooks from the late 1960s, when I was an
undergraduate and took my first calculus courses. I was less happy when I also discovered some
graded homework and tests with comments such as “This work is not good”, which I suppose is
a gentle was of saying “this is nonsense, tear it up and start again” (a comment which I did
receive on some graded graduate school homework later in my academic career). There were
Columbia University graded calculus tests from my four-semester calculus sequence; 1B, 2B,
3B, and 4B; and some material with questions involving metric spaces and compact sets (which I
cannot even imagine covering in the calculus courses that I currently teach, even though these
are rigorous courses for science, engineering, and mathematics majors, and most of the students
are excellent).
In the late 1960s, Columbia University had three distinct calculus sequences: Calculus Sequence
A, supposedly the most computational and easiest; Calculus Sequence B, more theoretical and
harder (primarily for engineers and physics majors); and Calculus Sequence C, for the most
interested and talented students. As a physics major, I was in the calculus sequence B.
In spite of (or maybe because of) the comments on my mathematics work, I eventually obtained
my Ph.D. in mathematics. After a total of over thirty years of teaching calculus, and in
observance of my fiftieth year anniversary of having taken my first calculus course, I (referred to
as “the author” in the following material) would like to share some personal experiences and
some general observations in learning and teaching calculus.
2. Disclaimer
This paper compares, in some measure, what calculus was like for some students at Columbia
University fifty years ago with what calculus is like for some students at The University of Tulsa
at the present time. The author’s intent is to use Columbia University and The University of
Tulsa as representatives of part of the calculus scene in academia, noting some weaknesses and
some strengths. In no way is this paper a complete exposition but is, instead, selective of certain
facets of the learning and the teaching of calculus. It is also to be noted that the two universities,
Columbia and The University of Tulsa, are quite different institutions with their quite different
student bodies. I shall attempt to place my observations in a more general framework.
3. Calculus Course Descriptions
3.1 Then and Now
As of spring 2017, both universities use the same calculus textbook by James Stewart [9], and
the university website academic bulletin descriptions of the calculus sequences of courses appear
similar. One difference is that Columbia University (http://www.math.columbia.edu/)
(http://www.math.columbia.edu/programs-math/undergraduate-program/calculus-classes/) uses
four semesters to cover the calculus sequence, whereas The University of Tulsa uses three
semesters. This means that Columbia University students have one more course to fit into their
schedule. Alternatively, with just the three-semester calculus sequence, the author has never
managed to finish covering all the topics in the syllabus for multivariable calculus.
The description of calculus courses from the Columbia University Bulletin 1969, Columbia
University, Department of Mathematics is given in Appendix I.
The description of calculus courses in The University of Tulsa Department of Mathematics,
extracted from the current online Undergraduate Bulletin, (http://utulsa.catalog.acalog.com/) is
given in Appendix II.
3.2 The University of Tulsa Department of Mathematics, 2016-2017
As of 2017, The University of Tulsa Department of Mathematics uses ALEKS
(https://www.aleks.com/), for mathematics course placement. ALEKS (Assessment and
LEarning in Knowledge Spaces) is a web-based tool that can be used to assess math proficiency
and skill level. All (or nearly all of) incoming freshmen and transfer students take the ALEKS
placement assessment online. It tests for courses below the level of calculus and for calculus-
readiness. Transfer credits (including AP and IB credits) are used to determine readiness for
courses above Calculus I. Some students, especially international students, who may not have
credits but who are ready for higher-level courses, will take proficiency exams for course credit.
The ASEE Mathematics Division has sponsored many conference sessions with
presentations/papers related to the use ALEKS. For example, a couple of topics are: predicting
student success in calculus; and benefits of a tutorial mathematics program for engineering
students enrolled in precalculus (https://www.asee.org/papers-and-publications/papers).
The Mathematics Department Chair at The University of Tulsa has recently given some
guidelines for faculty members to assist in student advising. His suggestions for ensuring proper
mathematics course placement of students include that, in addition to the use of ALEKS, faculty
who are student advisors should review the actual high school mathematics courses taken by
students. It may be that evidence of a very strong high school precalculus background (including
trigonometry) is a better indicator of student academic success that having taken a high school
calculus course. Incoming students without strong high school mathematics educations or strong
mathematics aptitudes need to be advised accordingly.
4. Some Anecdotal Student Experiences and Perceptions: Then and Now
4.1 Columbia University Academic Year 1968-1969: Student Assessment
The author’s experiences as a physics major at Barnard College (1967-1971) (as remembered
fifty years later) are generally consistent with the following material copied directly from The
Columbia-Barnard Course Guide, April 1969 [3]. This publication contains the descriptions of
the 1968-1969 departmental programs and student course evaluations and opinions (based on
faculty and student questionnaires). The following responses to calculus course experiences are
from students at Columbia University, Barnard College, and The Columbia University School of
Engineering for the 1968-1969 academic year. This course assessment material is copied
verbatim without regard for tactful phrasing. The author feels that the comments are typical for
what many mathematics department student experiences were at that time, regardless of the
particular institution. No negative interpretations are directed at Columbia University. Note that
the Columbia University Mathematics Department and courses have changed greatly since the
following comments were made. However, since that is not part of the author’s experiences,
those changes are not part of this paper.
“For many students calculus, not Contemporary Civilization or Humanities is the most
memorable course at Columbia; the memory is not pleasant. Cognizant of the fact that nearly half
the students in the College take introductory calculus, the mathematics department has set up three
tracks to suit their varying needs. With the exception of the sophisticated C-sequence, these
offerings are rarely adequate, often horrendous. The problems of calculus are two-fold: misplaced
students in improperly oriented courses, and well-meaning but incompetent instructors.
The mathematics department prides itself on offering a calculus course tailored to the
background of every freshman. For those who have had little or no calculus in high school, the
department offers Calculus IA, emphasizing practical math, and Calculus IB, stressing theoretical
formulations. For those with a good background in differential calculus, and the desire for a
theoretical orientation, Calculus IIB is recommended. Calculus IIIA and section two of Calculus IIIB
are for freshmen with strong preparation in both integral and differential mathematics. The most
dedicated and experienced can take Calculus IC. Their courses are impressive on paper but
ineffectual in practice.
The A-sequence is designed to teach applied mathematics; it suffers from students taking the
course description too literally. Freshmen, particularly engineers in Calculus A have virtually no
interest in mathematics per se; they view calculus as a tool for use in the physical or social
sciences and desire only to learn how to use it. Yet the A-sequence employs a superficial
discussion of theory to teach practical methods; too often this approach is distasteful, confusing,
and non-instructive.
In direct contrast the B-sequence, which teaches theoretical math, suffers from the
department’s literal interpretation of the course descriptions – Calculus B frequently neglects the
application of calculus to numerical problems. Many students come out of the sequence knowing
three proofs of the mean value theorem but lacking the ability of solve a rate problem.
Only the C-sequence succeeds, but it is of such limited appeal that no more than twenty-five
people take it.
Clearly there must be a redefinition of the orientation of the A and B sequences. Calculus A
must be more practical than it is now, more relevant to problems in the physical and social
sciences. While cutting away all theory may render the sequence intellectually barren, it will enable
students to appreciate and master with greater facility physics, economics, and other disciplines
making use of the techniques of differentiation and integration. The math department must realize
that it is toward this end that most students in the A-sequence take calculus.
The B-sequence, too, ought to be more practical. If somewhat fewer topics were covered,
problem solving techniques could be added without sacrifice of rigor and theoretical sophistication.
The orientation might then place the sequence somewhere between the present A and B tracks.
Both tenured faculty and Joseph Fels Ritt instructors teach calculus. All suffer from the same
problem; they are mathematicians whose interest and love is for abstract, sophisticated topics.
Consequently they are unable to communicate with their students and are frustrated by the
knowledge that what they think elegant most students find insipid. Perhaps a solution might be to
hire instructors who lack mathematical brilliance and sophistication, but are steeped in calculus and
can teach it. Unfortunately this may be easier said than done.
Until the necessary changes are made, calculus at Columbia will continue to evoke epithets
bordering on the obscene. Calculus at Columbia shows the students how well-taught high school
mathematics really was.”
The Columbia Daily Spectator [6], the student newspaper [in Volume III, Number 4,
December 8, 1969], featured an article by mathematician, Professor Serge Lang, Don't Blame Us
if You Flunk Math. In this article many of the (still) current issues in teaching calculus to students
in general, and to engineers in particular are raised. A question is posed: “What kind of people
should be taught what kinds of math at Columbia”. This question (in its general form) is still
debated today in academic mathematics.
It is also of interest to see articles from that period discussing the pros and cons of the
mathematics department teaching of differential equations courses to engineering students versus
the engineering department teaching of this course.
There is an article from September 18, 1968 mentioning the total number of students and the
number of women students enrolled as engineering majors – “184 students enrolled in the class
of 1972 Columbia University School of Engineering and Applied Science, including six girls”.
That the number of women enrolled is miniscule comes as no surprise. But what caught the
author’s attention is that the phrasing was the number of girls enrolled as engineering majors.
4.2 The University of Tulsa Academic Years 2000-2016: The Author’s Calculus I, II, and III Student
Course Evaluation Comments
There are student course evaluation comments suggesting changes in how the author teaches
calculus. In Calculus II, Mathematica (http://www.wolfram.com) was introduced and used for
some homework problems and projects. In some semesters, Interdisciplinary Lively Application
Projects (ILAPs) (http://www.maa.org/press/maa-reviews/interdisciplinary-lively-application-
projects-ilaps) were used; that is relevant to the reference below to “projects”. The use of ILAPs
did not work as intended for the author’s calculus classes. Instead of making the material more
relevant for students, the projects appeared to antagonize most of the students. The author feels
that this is because much faculty time and effort is needed to make these projects appropriate for
students, and the instructors involved did not have the time required. This also resulted in the
calculus courses inclusion of too much material for the semester (in the author’s opinion). Some
student course evaluation comments follow:
Explain what the goal is of what we're doing with each equation.
Have less conceptual teaching, more working problems.
I believe that a more in depth explanation of the reasoning behind some of the formulas
we learned would help make the information clearer.
Some of the tests were difficult to do without calculators, and the length of the tests were
a bit long. The pace of the course went very fast.
The course is overwhelming. I am trying to stay caught up. Quite frankly, I do not know
of anyone who has completed even close to all of the suggested homework problems, and
as a math major, I wish I felt more confident about being able to remember these
concepts long-term.
I like to work with my professor when developing a proof or learning a subject, and this
course did not have enough interaction for me.
The biggest difficulty that I faced in this course was being forced to learn Mathematica
so quickly. The time would have been better spent, in my opinion, working on the
solvable problems that would help me learn the material better and prepare for the exams.
I will say that when I did get Mathematica code correct I really did like the program.
Explain Mathematica better, but that probably would have taken too much class time.
The projects were handed out with very little instruction.
5. Some Analyses for the Author’s Calculus Courses, 2000-2016
The author has performed some data analysis to determine what trends, if any, may apply to
various aspects of her calculus courses. The following graphics indicate the data and results.
Figures 1, 2, and 3, respectively, display overviews of the total Calculus I, Differential Calculus,
Math 2014 enrollments; Calculus II, Integral Calculus, Math 2024 enrollments; and Calculus III,
Multivariable Calculus, Math 2073 enrollments, by semester, from spring (S) 2000 through fall
(F) 2016. It can be observed that enrollments are larger for the traditionally “on-sequence”
courses of Calculus I and Calculus III during the fall semesters and for Calculus II in the spring
semesters. Also noted is a trend of increasing enrollments. There is a surge in the Calculus I
enrollments that appears in the fall 2013 and 2014 semesters and ripples through the subsequent
corresponding Calculus II and Calculus III enrollments. This is probably due to the surge at that
time in students selecting computer science as their majors.
Figure 1. Bar Plot for The University of Tulsa Total Calculus I Student Enrollment Data:
Calculus I, Math 2014 enrollments (by semester, from spring 2000-fall 2016)
The sizes of the Calculus I and Calculus II classes taught by the author are given in Figures 4 and
5, respectively. Her Calculus III class sizes have averaged about 50 students each. The class sizes
in the calculus sequence generally have been increasing in recent years. This is due to an
increase in the number of students in the College of Engineering and Natural Sciences requiring
calculus courses, without corresponding funding to hire new mathematics faculty members.
Further, when new faculty have been hired, the emphasis is on research productivity (although
Spring
Fall
good teaching ability is also required). Therefore, new hires do not significantly affect the class
sizes of the calculus courses.
Figure 2. Bar Plot for The University of Tulsa Total Calculus II Student Enrollment Data:
Calculus II, Math 2024 enrollments (by semester from spring 2000-fall 2016)
Figure 3. Bar Plot for The University of Tulsa Total Calculus III Student Enrollment Data:
Calculus III, Math 2073 enrollments (by semester, from spring 2000-fall 2016)
The University of Tulsa has been experiencing financial difficulties for the past couple of years,
partially due to the deterioration of endowment investments in the oil market. In particular this
affects the ability to hire new faculty members, resulting in larger calculus class sizes. The goal
is to keep enrollment in calculus courses under 45, but this is not always feasible.
In any fall or spring semester, the author generally teaches one or two sections of either Calculus
I, II, or III, depending upon what is assigned by the department chair. The author’s grade
distributions for Calculus I, II, and III for academic years 2000 through 2016 were fairly
consistent within a course and across courses, except for two fall 2016 sections of Calculus III.
Spring
Fall
Spring
Fall
The author is aware of her higher grading for her two sections of Calculus III taught during the
fall 2016 semester. Possible reasons are that (1) the author had large classes and used multiple-
choice tests. Students did have to show all their work, but the multiple choice aspect gave
students a way to check or partially check their work; and (2) the students were exceptionally
well prepared (more so than in the past).
Figure 4. Bar Plot for Author’s Calculus I Class Sizes
Figure 5. Bar Plot for Calculus II Author’s Class Sizes
The failure rates for students in the author’s calculus I classes have ranged from 3% to 28%, with
a general decreasing trend. The class grade-point average (GPA) ranged from 2.4 to 3.4,
remaining roughly at 3.0 in recent years.
The failure rates in the author’s calculus II classes ranged from 2% to 20%, with no trend noted.
The class GPA ranged from 2.5 to 3.3, and no trend was observed.
For the author’s calculus III classes, the failure rates ranged from 0% to 13%, with a general
decrease for recent years. The class GPAs ranged from 2.8 to 3.5, increasing over the years, with
the high value of 3.5 attained in the fall 2016 semester (with possible reasons as described
previously in this section).
For both Calculus I and Calculus III, there were generally higher GPAs coupled with lower
failure rates in the fall semesters than for the spring semesters. This could be explained by the
fact that the fall semester is the “on-sequence” semester for Calculus I and Calculus III.
However, this trend was not noted for the author’s calculus II classes. Related statistical analysis
is given in Appendix III.
At The University of Tulsa, it is generally the decision of each individual instructor whether/how
to use technology for students’ work and for teaching purposes. For example, some instructors
use WebAssign (http://www.webassign.com/) as a tool for students to use in doing homework
and find this tool very helpful. Other instructors feel that students can too easily use a trial-and-
error approach, inputting different answers to a given question/problem, until the correct answer
is obtained. This approach defeats the purpose of WebAssign, which is to promote a better
understanding of solution processes.
The author has tried the use of applied projects (ILAPs), more in-class student participation (e.g.,
students working on the board), in-class small group problem-solving, and the use of overhead
projections of lecture material. These did not work well for the author, possibly because they are
not consistent with her own natural style of teaching. The result is that the author currently
teaches her calculus students in the traditional lecture format.
The University of Tulsa instructors of Calculus I and Calculus II have teaching assistant (TA)
support. The instructors and TAs meet periodically to share teaching information. This ensures
transmission of teaching information from older, more experienced faculty members to new
faculty members and that new faculty members teaching these courses are introduced to
successful practices for teaching calculus. However, most new mathematics faculty members are
experienced, successful teachers and use their own personal teaching styles.
6. Student Community
For mathematics majors at The University of Tulsa, the Department of Mathematics has a
Mathematics Commons Room in which mathematics students may work and meet informally,
hopefully enhancing a sense of community. Literature on mathematics opportunities and some
mathematics journals are made available. There is also the traditional MATH Lab in which
advanced undergraduates can assist calculus students in a “drop-in” setting. Columbia University
offers similar amenities to its students.
The commons room is a relatively new feature, only in use for a couple of semesters, so data on
its impact or effectiveness is not yet available. However, during the past year the author has
noted a stronger sense of community among students. The author has observed more groups of
students discussing and working collaboratively on homework, etc. (in compliance with
academic integrity policies).
During the fall 2016 final exam period, The University of Tulsa College of Engineering and
Natural Sciences provided a complimentary buffet table for students. The author found this to be
a nice touch.
7. Contemporary National Perspective on Calculus
The Mathematics Association of America (MAA) report, Insights and Recommendations from
the MAA National Study of Calculus [1] is a comprehensive study. This report contains over
one-hundred and forty pages of detailed analysis and provides insights into what pedagogy
works well and what can be problematic, what changes can create significant improvements and
which are less effective.
This MAA report include these comments:
The traditional Calculus I course was designed for engineers and physical scientists. It is
not clear that such a course will meet the needs for the future.
Men and women are taking calculus for very different reasons, with men predominantly
heading into engineering, computing, or the physical sciences and women going largely
into the life sciences or teaching.
For example, Klingbeil, Mercer, Rattan, Raymer, & Reynolds [5] described a calculus
course that was designed for engineering students by including applications related to
engineering, and showed it retained more students in the major.
The MAA report also states:
“One of the clearest lessons learned from this study is that there are no simple solutions. We did,
however, identify seven practices that were common among the colleges and universities chosen
for the case study visits. The last seven chapters of this volume each revolve around one of these
practices, which we summarize here as recommended best practices; 1. Attention to the
effectiveness of placement procedures. 2. Proactive student support services, including the
fostering of student academic and social integration. 3. Construction of challenging and engaging
courses. 4. Use of student-centered pedagogies and active-learning strategies. 5. Coordination of
instruction, including the building of communities of practice. 6. Effective training of graduate
teaching assistants. 7. Regular use of local data to guide curricular and structural modifications.”
8. Changes for the Better and Changes for the Worse
During the author’s senior year in high school, calculus was offered to select students at her high
school for the first time. The author’s parents decided that there was no need to begin calculus
prior to college, and so she did not take calculus until her freshman year in college. This is quite
different for contemporary students, many of whom enter college, and place out of some of the
calculus sequence courses. There is concern that for too many students this placing-out of
college calculus courses is not appropriate. Students may think that they understand calculus
when they really don’t.
The author’s calculus notes, taken while a student in her 1967-1969 calculus courses, show that
epsilon-delta material was included in first semester calculus and specialized trigonometric
substitutions and other integration techniques were included in second semester calculus. These
topics are no longer included in many calculus courses. The specialized integration techniques
are now implemented by calculators and computers. Stunningly, the third semester topics
included inner-product spaces, normed vector spaces, metric spaces, compactness, completeness,
and uniform convergence, currently topics in more advanced mathematics courses. The fourth
semester course was basically linear algebra.
A textbook used by the author is Calculus on Manifolds [8]. This is an advanced calculus book
that the author (of this paper) now considers as rather theoretical for the multivariable calculus
course she took in the fall 1969 semester.
One of the Columbia University calculus professors gave each student in the class an oral
calculus exam. Each student met individually with this professor for about an hour and was
asked to solve problems. There was also a mathematics graduate student teaching assistant (TA)
present. The author remembers being asked questions by the professor and not understanding
what was being asked. The author then looked at the graduate TA, who rephrased the question in
a way that the author could understand. TAs were important then, as they are now, in getting to
know a student on a more personal level. Also, it is unusual for undergraduates in calculus to be
given oral exams. The author has never done this and is not aware of any of her colleagues at
The University of Tulsa having done so. One factor that preclude this is large class sizes.
Calculators, computers, and convenient internet access were/are some of the factors driving
changes in the calculus syllabus. Student demographics continue to change. There is more
attention paid by academic institutions to non-traditional students: part-time, single-parent,
working, older, women and minority, and international students, etc. Institutions have developed
academic/social support structures and services to help “level the playing-field” for such
students. For example, at The University of Tulsa there is the Center for Student Academic
Support (CSAS) https://utulsa.edu/campus-life/student-academic-support/ CSAS offers a wide
range of services to academically assist students.
Some improvements in (and outside of) the calculus classroom include:
Academic institutions’ sensitivity to student diversity.
Pedagogy for students with different learning styles.
Technology used both for enhancing teaching and as a tool in problem solving (e.g.,
WebAssign (developed in 1997) and Mathematica (initial release date of 1998)).
Some current issues in (and outside of) the calculus classroom include:
Students’ increased attitude of (unwarranted) “entitlement” (observed as early as 2010:
http://www.chronicle.com/article/Students-Should-Check-Their/126890) [2].
Students being over-extended with respect to discretionary heavier course loads and
extra-curricular activities.
Students’ inappropriate use of technology in the classroom (e.g., cell phone use).
9. Conclusions
Even though the courses were offered at different times and different places, i.e., at Columbia
University versus The University of Tulsa, it is the author’s opinion that what she has observed
at these two times and places may be representative of the evolution of the teaching of calculus
at many academic institutions during the past fifty years. The 2017 website descriptions of the
calculus courses and the corresponding sample syllabi for Columbia University and The
University of Tulsa are remarkably similar, and the same textbook is now used at both
institutions. This may indicate that this text is a useful text, oriented toward applications that
students will use in their science, engineering, and mathematics, etc., fields, and that these
calculus courses are now focused more on what their student clientele needs than they were in
the recent past.
The author notes that the smaller calculus class sizes she experienced as a student were helpful.
Decreasing class sizes might be difficult to achieve, considering contemporary funding issues,
but having well-prepared TAs is important now, as it was then. If possible, separate the class
sections of students who have seen calculus before from those students who have no prior
calculus experience. Rewarding faculty for excellence in teaching, similarly to how research
productivity is rewarded would be constructive. Faculty members who advise students must
ensure that incoming students have a strong pre-calculus background and that students without
strong high school mathematics backgrounds or strong mathematics aptitudes are advised
accordingly.
It appears to the author that the teaching of calculus has trended from “instructor-centered” to
“student-centered”, and that this is generally for the better. The author hopes that developments
in teaching calculus proceed in this direction.
Acknowledgments
The author thanks reviewers for their advice.
Bibliography
[1] David Bressoud, Vilma Mesa, Chris Rasmussen, Editors, Insights and Recommendations
from the MAA National Study of College Calculus (2010-2014), Mathematical Association of
America (MAA) Press, 2014, http://bit.ly/25cxNqs.
[2] Elayne Clift, From Students, a Misplaced Sense of Entitlement, Chronicle of Higher
Education, March 27, 2011, http://www.chronicle.com/article/Students-Should-Check-
Their/126890.
[3] Columbia-Barnard Course Guide, April 1969, Number 7, published by The Ted Kremer
Society.
[4] Louis M. Friedler, Calculus in the US: 1940-2004,
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.522.3340. [5] Nathan Klingbeil, Richard Mercer, Kuldip Rattan, Michael Raymer, and David Reynolds,
Rethinking Engineering Mathematics Education: A Model for Increased Retention, Motivation
and Success in Engineering, 2004 ASEE Annual Conference Proceedings, Salt Lake City, Utah.
[6] Serge Lang, Columbia Daily Spectator, Volume III, Number 4, December 8, 1969, Don't
Blame Us if You Flunk Math, Columbia Spectator Digital Archive,
(http://spectatorarchive.library.columbia.edu/), University Archives, Rare Book & Manuscript
Library, Columbia University in the City of New York.
[7] George M. Rosenstein, One Hundred and Fifty Years of Teaching Calculus,
http://apcentral.collegeboard.com/apc/members/courses/teachers_corner/19515.html.
[8] Michael Spivak, Calculus on Manifolds, The Benjamin/Cummings Publishing Company,
Menlo Park, CA, 1965.
[9] James Stewart, Calculus - Early Transcendentals, Edition 8E, Cengage Learning, 2016,
ISBN-10: 1285741552.
[10] Alan Tucker, The History of the Undergraduate Program in Mathematics in the United
States, http://www.ams.sunysb.edu/~tucker/, MAA History article.
[11] Luciana Zuccheri and Verena Zudini, History of Teaching Calculus, Springer New York,
2013.
Appendix I The following is the description of calculus courses from the Columbia University Bulletin
1969-1970: Mathematics Courses for Freshmen The systematic study of mathematics begins with one of the following two-term sequences: Mathematics C1101-C1102 (referred to below as Calculus IA, IIA), Mathematics C1103-C1104 (Calculus IB, IIB), Mathematics C1104-C1203 (Calculus IIB, IIIB), Mathematics C1107-C1108 (Calculus IC, IIC),
Mathematics C1201-C1202 (Calculus IIIA, IVA), Section 2 of Mathematics C1203-C1204 (Calculus IIIB, IVB). The A-sequence Calculus is a standard course devoted to the differential and integral calculus; it is intended for students who need calculus primarily for its application. The B-sequence Calculus is devoted to the same topics as A, but is also intended to develop a theoretical understanding of the mathematical concepts. The C-sequence Calculus covers much of the same material as A and B but the approach is on a mature level and the course demands creative imagination and unusual ability to think abstractly. In Calculus IA-IIA, the student has the option to attend a large lecture class held by Mr. Eilenberg or one of the smaller classes taught by selected graduate students and staff, without supplementary recitations. In most of the courses, the lectures are supplemented by recitation periods which meet once a week in small groups. Assignments to recitation sections are usually made at the first lecture. Mathematics C1101x-C1102y. Calculus IA, IIA 3 pts Prerequisite: trigonometry Functions. Derivatives, especially of rational and radical functions. Applications (motion, curve tracing.) Integrals. Integration by parts and substitution. Fundamental theorem of calculus. Applications (area, volume, length, work, energy.) Elementary transcendental functions. Applications (radioactive decay, vibrations.) Vector-valued functions. Applications (motion of a mass point). Functions of several variables. Partial derivatives. Mean value theorem. Taylor’s theorem for one and several variables. Mathematics C1103x-C1104y. Calculus IB, IIB 3 pts Prerequisite and topics: see statement above in “Courses for Freshmen” Mathematics C1107x-C1108y. Calculus IC, IIC 3 pts Prerequisite and topics: see statement above in “Courses for Freshmen” Mathematics C1201x-C1202y. Calculus IIIA, IVA 3 pts Analytic geometry using vector language. Multiple integrals. Line integrals. Green’s theorem. Techniques of integration. Infinite series. Power series. Application to geometry and physics. Mathematics C1203x-C1204y. Calculus IIIB, IVB 3 pts The same topics as in Mathematics C1201-C1202, with greater emphasis on the understanding of the mathematical concepts and logical structure.
Additional course descriptions from the Barnard College Announcement for 1970-1971: Mathematics C1107x-C1108y. Calculus IC, IIC 3 pts The same material as Course IA and IIA. The terminology and style are thoroughly modern. Intended for students who have facility with discussions on an abstract level, or who appear likely to develop such
facility early. Admission is by examination, given by the Columbia Mathematics Department during Freshman Week. Mathematics C1207x-C1208y. Calculus IIIC, IVC 3 pts The material of Calculus IIIA, IVA, plus additional topics. Terminology and style are thoroughly modern.
Appendix II
The following is the description of calculus courses from the current online The University of
Tulsa Undergraduate Bulletin, (http://utulsa.edu.catalog.acalog.com/):
MATH 1103 Basic Calculus [3 credits]
(3 hours) Calculus for students of business and the social and life sciences. Logarithms.
Exponential functions. Introduction to differential and integral calculus. Prerequisite: MATH
1093.
MATH 2014 Calculus I [4 credits]
(4 hours) Theory and application of the differential calculus of polynomial, exponential,
logarithmic and trigonometric functions. Graphical, numerical and analytical solutions to applied
problems involving derivatives. Introduction to the integral. Prerequisites: MATH 1163 (Pre-
calculus Mathematics) or equivalent, and passing score on the University mathematics placement
examination.
MATH 2024 Calculus II [4 credits]
(4 hours) Definite and indefinite integrals of functions of a single variable. Improper integrals.
Infinite series. Introduction to differential equations. Emphasis on applications of calculus and
problem solving using technology in addition to symbolic methods. Prerequisite: MATH 2014.
MATH 2073 Calculus III [3 credits]
(3 hours) Vector geometry, algebra and calculus. Partial and directional derivatives. Double and
triple integrals. Vector fields. Line and surface integrals. Theorems of Green, Stokes and Gauss.
Prerequisite: MATH 2024.
Appendix III: Calculus I, Calculus II, and Calculus III DWF Rates
We recognize that the data analyzed were not obtained from an actual random sampling process.
However, the data obtained is a possible result from sampling large populations of students with
characteristics similar to those attending The University of Tulsa and taught by the author.
We conducted location tests (Kruskal-Wallis and K-Sample T). Specifically, the null hypothesis
is that the two populations from which the data came have the same location (median or mean).
The Kruskal-Wallis test does not assume that data come from a normal distribution nor does it
assume equality of population variances. The K-Sample T test does assume that data come from
a normal distribution and equality of population variances.
Differential Calculus, Calculus I, Math 2014 DWF Rates:
Figure A1. Calculus I DWF Rate Data
The Calculus I data in the Figure A1 graph was used to further analyze the DWF rates. Both tests
have a very small p-value leading us to reject the null hypothesis that both populations sampled
have the same location (median or mean). We can say that if the two populations have the same
location, we obtained unusual data. Consequently, we reject the null hypothesis that the two
populations have the same location fall versus spring for the Calculus I DWF rate data.
Integral Calculus, Calculus II, Math 2024 DWF Rates:
Figure A2. Calculus II DWF Rate Data
F
F
F F
FF
F
SSSS
S
S
S
S
2002 2004 2006 2008 2010 2012 2014Year
0.05
0.10
0.15
0.20
0.25
0.30
RateCalculus I DFW Rate
F Fall
S Spring
Statistic P- Value
Kruskal-Wallis 10.5188 0.0000289043
K- Sample T 44.4231 0.0000155164
F
F
F
F
FF
F
F
F
F
FF
SSS
S
S
S
S
S
S
S
S
2008 2010 2012 2014Year
0.05
0.10
0.15
0.20
0.25
0.30
0.35
RateCalculus II DFW Rate
F Fall
S Spring
Statistic P- Value
Kruskal-Wallis 0.242784 0.633338
K- Sample T 0.390027 0.539012
The Figure A2 graph suggests the possibility that the spring and fall sections might have
different medians or means. We tested the null hypothesis that both populations sampled have
the same location (median or mean). The Kruskal-Wallis and K-sample T tests both have large
p-values. If the null hypothesis is true, we obtained typical data. We lack evidence that the two
populations from which the data came have different measures of location for fall versus spring.
Multivariable Calculus, Calculus III, Math 2073, DWF Rates:
Figure A3. Calculus III DWF Rate Data
The Figure A3 graph suggests the possibility that the spring and fall sections might have
different medians or means. We tested the null hypothesis that both populations sampled have
the same location (median or mean). The Kruskal-Wallis and K-sample T tests both have small
p-values. If our null hypothesis is true, we obtained unusual data. Consequently, we have
evidence that the locations (median or mean) of the two populations differ fall versus spring.
F
FF
F
S
S
2014 2015 2016Year
0.05
0.1
0.15
0.2
RateCalculus III DFW Rate
F Fall
S Spring
Statistic P- Value
Kruskal-Wallis 3.42857 0.0417947
K- Sample T 39.164 0.00332539