## The Changing Face of Calculus:
First- and Second-Semester Calculus as College Courses

*By David M. Bressoud*
*This is the second half of a two-part article on the impact of calculus
enrollment trends on how we teach calculus, both in high school and in
college. The first article is
available online and also
appeared in the August/September issue of
FOCUS. This article will appear in the November issue.*

There was a time when it made sense to teach single variable calculus as a year-long course. Most of the students who enrolled for the first semester intended to complete a full year. Nearly all the students who began the second semester came from the first-semester course at that institution. No more. The number of students who take their first calculus course in high school is much higher than the number who take calculus for the first time in college^{a}. At many colleges and universities, the majority of students in Calculus II passed the first-semester course somewhere else, a trend that is accelerating. Large and increasing numbers of the students who take Calculus I do so with no intention of continuing on to the second semester of the course.

In the CUPM Curriculum Guide 2004 [6], the first and foundational recommendation for all departments is to ?understand the student population and evaluate courses and programs.? The CUPM Guide encourages departments to ?determine the extent to which goals of courses and programs are aligned with the needs of students? and ?continually strengthen courses and programs to better align with student needs.? I will argue that the traditional year-long single variable calculus course is badly out of alignment with student needs at most institutions.

**The Need for a New Calculus II**

At colleges and universities on semester systems, the traditional Calculus I is differential calculus with the barest introduction to integration in the last few weeks. Calculus II picks up where Calculus I left off, reintroducing and then developing integral calculus for roughly half a semester. It then moves on to sequences and series and may touch on some topics of several variable calculus such as partial derivatives. Aside from the fact that the material on sequences and series never seems to hang together with the earlier work on differential and integral calculus, it is not a bad course. The problem is that few students take it as intended.

In the fall of 2000, 107,000 students took mainstream Calculus II in a 2- or 4-year college or university in the United States [4]. Spring enrollments were certainly higher. Enrollments in Calculus II have been flat^{b}, so we can estimate that roughly a quarter million students take mainstream Calculus II in college each year. In the spring of 2004, over 150,000 high school students earned a score of 3 or better on one of the Advanced Placement Calculus Exams^{c}. Many more high school students obtained International Baccalaureate or community college credit for calculus.

In many colleges and universities, it is already the case that most of the students who take Calculus II come with credit for Calculus I taken in high school. These students are not well served by the traditional approach to calculus in college. They have seen an entire course of differential and integral calculus and have already studied many of the topics that will be covered in the first half of Calculus II. It makes no sense to pretend that they are barely familiar with integration.

On the other hand, those who have taken the AB syllabus have had the equivalent of a one-semester course, not a full college year. There are significant gaps in their knowledge. Most of these students will not have studied integration by parts, L'Hospital?s Rule, or parametrized motion. Many of them would benefit from more and deeper work on limits, implicit differentiation, differential equations, and the theorems of calculus. They need a Calculus II course, but not the one we traditionally offer.

The need for revising Calculus II becomes even more imperative when we remember that the students who come to this course directly from a high school AP course are among our most mathematically promising. The last thing we should do is to put them into a course that does not build directly on their training and lead them into further mathematics. Calculus II should be a course that they find challenging and exciting, one that helps them see the potential of modern mathematics and makes them want to pursue it. Those colleges and universities that find that many or most of their Calculus II students come with credit for calculus taken in high school should assess how well their curriculum meets the needs of these students and entices them into continuing their study of mathematics.

**The Need for a New Calculus I**

The movement of first-semester calculus into the high school curriculum also has profound implications for how we teach Calculus I in college. Students who begin calculus in college are very different from those of a generation ago. They are far more likely to have weakness in their mathematical preparation. They are also far less likely to aspire to taking a second semester of calculus. The traditional Calculus I course was never designed to stand alone. Now it must.

There are many initiatives under way to supplement Calculus I instruction with ?just in time? refreshment or teaching of pre-calculus topics. These are important, but I want to focus on a different side of the problem with Calculus I that is relevant to the needs of the students at my institution, Macalester College. The largest single group of students taking our Calculus I consists of prospective majors in the biological sciences. They are particularly ill-served by the traditional course.

As part of the preparation for the CUPM Curriculum Guide 2004, the subcommittee on Curriculum Renewal Across the First Two Years (CRAFTY) sponsored the Curriculum Foundations Project, a series of eleven workshops for faculty from partner disciplines such as economics, engineering, and physics. These faculty came together to explain and elaborate upon what they want their students to learn from their mathematics courses^{d}. The workshop for biologists, held at Macalester College in November, 2000, was personally enlightening.

Biology programs face constraints. They usually require two semesters of mathematics. Some institutions require more or less, but because there is so much technical information that has to be taught within four years, few feel they can afford to require more than two semesters of mathematics, and most feel that one semester of mathematics is not sufficient. Many medical schools, a driving force in the biology curriculum, require one semester of calculus. It is not uncommon for biology departments to have a mathematics requirement of Calculus I & II. When the biologists gathered at Macalester, they were unanimous and strong in the opinion that Calculus I & II made no sense as the two mathematics courses their majors should take. In their own words: ?The current mathematics curriculum for biology majors does not provide biology students with appropriate quantitative skills. The field of biology is becoming much more quantitative which will necessitate a change in the mathematics curriculum for biology majors? (p. 16). Biologists need mathematics, but not what we have been teaching.

There are aspects of calculus that biologists do need, though they need far less than we teach. The topics identified at the workshop were ?integration for the purpose of calculating areas and average value, rates of change, optimization, and gradients for the purpose of understanding contour maps.? The most important mathematical tools for biologists lie in data analysis and statistics. Here the list of needed topics is far richer, ?descriptive statistics, conditional probability, regression analysis, multivariate statistics, probability distributions, simulations, significance and error analysis? (p. 16).

The kind of statistical analysis that these students need to be able to do is deep. If they are to understand the statistical tools that they will be using, then they need some familiarity with the concepts of gradients, linear transformations, and eigenvectors. Can they learn all this within the context of Calculus I?

Macalester is among a handful of colleges that believes the answer is ?yes.? We have developed a substitute for Calculus I that does not attempt to duplicate the coverage of the traditional course. Its emphasis is on geometric and conceptual understanding, and its syllabus includes:

**Functions and units: **linear functions, power relations, polynomials, trigonometric functions, exponential functions, logarithms.

**Dynamical systems:** simple systems of discrete difference equations and differential equations.

**The derivative: **graphically, numerically, algebraically; Taylor polynomials to second degree.

**The integral **as anti-derivative and as area.

**Functions of two variables: **contours, gradients, optimization via hill climbing.

**The linear algebra of Ax = b: **vectors, projection, least-squares, subspace, span. Emphasis is on 3-dimensional space.

This is a course that stands on its own merits. Not only our biologists, but also our chemists and economists have praised it for meeting the needs of their students. We have been running it experimentally, a program made possible by funding from the Hughes Foundation. That money is ending. We are too small an institution to be able to afford to run two distinct Calculus I courses. In view of the fact that Calculus II now draws so few students from Calculus I, we have decided that it is the traditional Calculus I that will disappear. For us, Calculus I and Calculus II will become very distinct courses.

These courses will still be connected. Part of the intention of the revised syllabus is to show students the importance and usefulness of mathematics and so encourage them to continue their pursuit of it. But our two courses in single variable calculus will no longer be two halves of a single course. Each is being redesigned around the actual needs of the students who take it, moving them forward in their understanding of and ability to use mathematics, preparing them for and enticing them into the further study of mathematics. This should the goal of every mathematics course, whether taught in high school or college.

There are fundamental shifts now occurring in the needs of the students who take our mathematics courses. Macalester?s solution is built around the needs of this college and its students. What is needed at your college or university will depend on your situation and your students. What you cannot expect is that the curricular solutions that worked a generation ago are still best for today.

**Bibliography**

[1] APCentral, AP Research and Data, http://apcentral.collegeboard.com/program/research/

[2] Bressoud, David M., ?The Changing Face of Calculus: First-Semester Calculus as a High School Course,? FOCUS, September 2004.

[3] Ganter, Susan and William Barker,*Curriculum Foundations Project: Voices of the Partner Disciplines*, Mathematical Association of America, 2004.

[4] Lutzer, David J., James W. Maxwell, and Stephen B. Rodi, *Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States: Fall 2000 CBMS Survey*, American Mathematical Society, Providence, RI, 2002.

[5] Morgan, Rick and Len Ramist,*Advanced Placement Students in College: An Investigation of Course Grades at 21 Colleges*, Educational Testing Service Report No. SR-98-13, 1998.

[5] Pollatsek, Harriet et al,* Undergraduate Programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide 2004*, Mathematical Association of America, 2004.

**Notes**

a) As estimated in the first article of this series [2], there are probably between 500,000 and 600,000 high school students taking calculus each year. Also as shown in that article, the numbers taking first-semester calculus in colleges have been flat or slightly declining for the past twenty years. Total mainstream and non-mainstream Calculus I enrollments in all 2- and 4-year institutions in the fall of 2000 was 384,000 [4]. If we estimate that spring enrollments in Calculus I are two-thirds of fall enrollments, then between 600,000 and 700,000 students take some form of Calculus I in college each year. Roughly half of these, perhaps more, will have taken calculus in high school.

b) 111,000 in fall, 1990, 106,000 in fall, 1995 [4].

c) Not all colleges or universities give credit for a 3 on the AP Calculus exam, but a study [5] of students at 21 universities including Boston College, Carnegie Mellon, William and Mary, Michigan State University, Pennsylvania State University, Stanford, University of Virginia, and Yale showed that those who scored a 3 on the AP Calculus AB exam and started at the university with Calculus II did better in that course than did students who had completed Calculus I at that university.

d) The reports of these workshops, written by faculty in the partner disciplines, are available from the MAA as Curriculum Foundations [3]. Page references that follow refer to this edition.

*David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota. He serves both as Chair of the MAA?s Committee on the Undergraduate Program in Mathematics (CUPM) and as Chair of The College Board?s AP Calculus Development Committee. He has been involved with AP Calculus since 1990?91 when he had the privilege of teaching an AB course at the State College Area High School and of learning how to teach calculus from some great teachers, especially Annalee Henderson. *