My Fifty Years of Calculus

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.

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