insightsandrecommendations fromthemaa collegenationalstudyof calculus EDITORS DAVID BRESSOUD VILMA MESA CHRIS RASMUSSEN Insights and Recommendations from the MAA National Study of College Calculus Edited by David Bressoud,1 Vilma Mesa,2 Chris Rasmussen3 1 Mathematics, Statistics, and Computer Science Department, Macalester College, 1600 Grand Ave., Saint Paul, MN 55105-1899, [email protected]. 2 School of Education, University of Michigan, 610 East University, Ann Arbor, MI, 48109-1259, [email protected] 3 Department of Mathematics and Statistics, 5500 Campanile Drive, San Diego State University, San Diego, CA 92182-7720, [email protected] Editors’ Note: This work was supported by NSF grant DRL 0910240. The opinions expressed in this volume do not necessarily reflect those of the National Science Foundation. Table of Contents Preface v David Bressoud, Macalester College; Vilma Mesa, University of Michigan, Ann Arbor; Chris Rasmussen, San Diego State University Chapter 1 The Calculus Students 1 David Bressoud, Macalester College Chapter 2 The Impact of Instructor and Institutional Factors on Students’ Attitudes 17 Gerhard Sonnert, Harvard University; Philip Sadler, Harvard University Chapter 3 The Institutional Context 31 Natalie E. Selinski, Leibniz-Universität Hannover, Germany; Hayley Milbourne, San Diego State University Chapter 4 The Calculus I Curriculum 45 Helen E. Burn, Highline College; Vilma Mesa, University of Michigan, Ann Arbor Chapter 5 Placement and Student Performance in Calculus I 59 Eric Hsu, San Francisco State University; David Bressoud, Macalester College Chapter 6 Academic and Social Supports 69 Estrella Johnson, Virginia Tech; Kady Hanson, San Diego State University Chapter 7 Good Teaching of Calculus I 83 Vilma Mesa, University of Michigan, Ann Arbor; Helen Burn, Highline College; Nina White, University of Michigan, Ann Arbor Chapter 8 Beyond Good Teaching: The Benefits and Challenges of Implementing Ambitious Teaching 93 Sean Larsen, Portland State University; Erin Glover, Oregon State University; Kate Melhuish, Portland State University Chapter 9 Calculus Coordination at PhD-granting Universities: More than Just Using the Same Syllabus, Textbook, and Final Exam 107 Chris Rasmussen, San Diego State University; Jessica Ellis, Colorado State University, Fort Collins Chapter 10 Three Models of Graduate Student Teaching Preparation and Development 117 Jessica Ellis, Colorado State University, Fort Collins Chapter 11 How Departments Use Local Data to Inform and Refine Program Improvements 123 Dov Zazkis, Arizona State University; Gina Nuñez, San Diego State University Appendix A Surveys and Case Studies Methods 133 Jessica Ellis, Colorado State University, Fort Collins Appendix B Survey Questions and Codebook 139 Gerhard Sonnert, Harvard University; Jessica Ellis, Colorado State University, Fort Collins Preface David Bressoud, Macalester College Vilma Mesa, University of Michigan, Ann Arbor Chris Rasmussen, San Diego State University Calculus occupies a unique position as gatekeeper to the disciplines in science, technology, engineering, and mathematics (STEM). At least one term of calculus is required for almost all STEM majors. For too many students, this requirement is either an insurmountable obstacle or—more subtly—a great discourager from the pursuit of fields that build upon the insights of mathematics. Over 35 years ago, Robert White, then President of the National Academy of Engineering, declared that the time had come to turn calculus from a filter to a pump. Lynn Steen, former President of the MAA, echoed this sentiment in the National Research Council report Calculus for a New Century: A Pump not a Filter (1988). This was the document that heralded the start of the many efforts that collectively became known as Calculus Reform. Many decades later, we seem to have made little progress. Calculus is still a filter, but until 2010 we knew very little about who takes it, how it is taught, or what makes for effective calculus programs that promote rather than inhibit students’ continuation into successful careers in science and engineering. Existing knowledge on the effects of class size, placement procedures, use of technology, or pedagogical approaches was either not specific to calculus or of a very local nature. This knowledge is more critical now than ever before because the landscape of college calculus has changed. With the explosive growth of Advanced Placement and other calculus courses in high school, roughly three-quarters of all students who eventually study calculus take their first calculus course in high school,1 and almost half of all students who go directly from high school into a four-year college matriculate with Calculus on their high school transcript. In our large public universities, the primary engines for the production of scientists and engineers, the percentage of students taking Calculus I having already earned a 3 or higher on an AP Calculus exam exceeds 25%. Many of our most mathematically talented students place directly into Calculus II. At the same time, our colleges and universities find themselves in the vise created by the dramatic growth in the number of incoming students hoping to pursue careers in engineering or science pressed against the drastic budget cuts that have forced departments to reduce the number of full-time faculty and to teach calculus in ever larger classes. In addition, departments have been slow to address the changing demographics of the students who will need to go into STEM fields. Success rates in calculus for women, students from underrepresented minorities, economically disadvantaged students, and first generation college students have always been disappointing. The loss of these students is a luxury our nation cannot afford. This report, a summary of selected findings from the Mathematical Association of America’s (MAA’s) study of Characteristics of Successful Programs in College Calculus (NSF, DRL 0910240), constitutes two preliminary steps toward addressing these issues. First, it establishes a base of knowledge of who takes Calculus I and why, what their preparation has been, what they experience in the classroom, and how this affects their confidence, enjoyment of mathematics, and intention to persist in the study of mathematics. Second, it identifies institutional practices that contribute to the retention of STEM students. This is the first nationwide investigation of college-level Calculus I in the United States to combine both large-scale survey data and in-depth case study analysis. 1 This estimate is based on the following calculations: In 2014, 400,000 US students took an AP Calculus exam. According to the National Center for Education Statistics (NCES, 2012), these represent 53% of all high school calculus students, implying that roughly 750,000 students take calculus in high school each year. From the Conference Board of the Mathematical Sciences (CBMS) data (Blair et al 2013) and our survey data, approximately 500,000 students enroll in Calculus I in college each year, with about half of them repeating the calculus course they took in high school. Therefore, about 250,000 students each year see calculus for the first time as a college Calculus I course. vi PREFACE Why this Report is Relevant to High School Calculus This report does not address how calculus is or should be taught in high school, yet it is relevant to high school and middle school teachers and administrators because it opens windows into the situation their STEM-intending students will encounter when they enter college. Mathematics is unique among all disciplines in having created a course, calculus, which is both the lodestar of the K-12 curriculum and the bedrock of post-secondary preparation for science and engineering. These distinct perspectives on this course create much of the discontinuity that students experience as they transition from high school to college. For many high school students, mathematical success is the result of their ability to master a collection of problem types. Learning calculus as a mastery of problem types is a daunting task. The procedures and problems of calculus are sufficiently numerous and complex that acquiring this ability signals a remarkable mental achievement. It may be remarkable, but, by itself, it is not sufficient for success in university-level science or engineering. Students are better prepared for post-secondary mathematics when they have developed an understanding of the undergirding principles which, when accompanied by fluent and flexible application of the concepts and procedures of precalculus mathematics, enable them to understand calculus as a coherent and broadly applicable body of knowledge. In US universities, successful navigation of Calculus I is merely the start. Subsequent courses build on the knowledge base of calculus and require fluency in the language in which calculus is written, the language of functions, limits, and series. Post-secondary faculty want their students to be able to move easily between the right triangle and the circle understandings of trigonometric functions, instinctively knowing which is the most productive approach in a given context. They expect that students are at ease with exponentials and logarithms and can recognize when the binomial theorem or a finite geometric series is in play. The problem is that success in AP Calculus does not require this level of proficiency. It is and only claims to be evidence that one can solve the standard problems of