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, Numerical Methods, 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.

Spring Fall 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 good teaching ability is also required). Therefore, new hires do not significantly affect the class sizes of the calculus courses.

Spring Fall 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)

Spring Fall 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. 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:

Calculus I DFW Rate Rate S 0.30 S S S 0.25 S S 0.20 F F F F F 0.15 F F 0.10

0.05 F Fall Year S Spring 2002 2004 2006 2008 2010 2012 2014 Statistic P- Value Kruskal- Wallis 10.5188 0.0000289043 K- Sample T 44.4231 0.0000155164 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: Calculus II DFW Rate Rate

0.35 S F S 0.30 S F 0.25 F S

0.20 FS F S 0.15 S F F S F S S S 0.10 F F 0.05 Year 2008 2010 2012 2014 Statistic P- Value Kruskal- Wallis 0.242784 0.633338 K- Sample T 0.390027 0.539012 Figure A2. Calculus II DWF Rate Data 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:

Calculus III DFW Rate Rate 0.2 S

0.15 S

0.1

F 0.05 F F Fall F F Year S Spring 2014 2015 2016 Statistic P- Value Kruskal- Wallis 3.42857 0.0417947 K- Sample T 39.164 0.00332539 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.