Rethinking the First Year

*David M.
Bressoud November, 2007*

In my *Launchings* column from this past May, *Holding
on to the Best and Brightest*, I wrote about what I see as one of the
great challenges facing our profession: the stagnation over the past quarter
century in the number of students majoring in the mathematical sciences. This
stagnation in absolute numbers has resulted in a decrease in the percentage
of bachelor degrees that are in mathematics, reaching an historical low in
2005 of 0.86% [1]. My May column focused on the problem
of holding on to the best of the best, those students who complete the two-semester
sequence of single variable calculus not just while in high school, but even
before they are seniors. In this column, I want to begin addressing the much
larger problem of attracting majors from the vast pool of students who enter
our colleges and universities with an interest in majoring in one of the disciplines
in science, technology, engineering, or the mathematical science (STEM).

Mathematics is a field full of exciting ideas and problems, a field with direct and influential ties to some of the most important current work in biology, economics, and information science. Why doesn’t this message of get through to our students? I contend that they don’t get this message because we don’t make much a point of communicating it.

You don’t have to look very hard at the first-year experience of the students who are most inclined to pursue a STEM major to see what’s wrong. The number of students who graduate from high school and then enter college within a year is about 1.85 million [2]. Of these, about half a million, over a quarter of these entering 17 to 18-year olds, have studied calculus in high school [3]. Considering that there are only about 230,000 bachelor degrees awarded in the STEM majors each year [4], roughly 16% of all bachelor degrees, it is safe to conclude that the vast majority of the traditional students who enter college with the intention of pursing a STEM major are students who studied calculus in high school. This is the pool from which we are most likely to attract math majors.

But what do they see in their first year? Most of them, though they studied calculus in high school, did not do well enough to earn college credit for what they studied [5]. Their first experience of calculus is to retread familiar concepts, but at a faster pace with more intense expectations. Those who do arrive with credit for some calculus are not much better off. Whether they begin with Calculus I, II or III, calculus is almost always taught as a process of mastering techniques for solving certain types of problems, problems whose importance and relevance is often lost to the dominating need to develop facility in the technical aspects of the subject.

There are good books that are designed for entering students
and that do an honest attempt at communicating the excitement of working on
mathematical problems. COMAP’s *For all Practical Purposes* [6],
and Burger and Starbird’s *The Heart of Mathematics* [7]
are two outstanding examples that come immediately to mind. But for most students
who enter college with a serious interest in pursuing a STEM major, such courses
are luxuries that do not fit easily into their curriculum. The pressure is
on to get through calculus as quickly as possible.

The combined result is that many students who might have been enticed to pursue mathematics find themselves attracted elsewhere during this critical first year. We need to rethink how we present our discipline to these students.

There are two basic approaches to rectifying this problem, and we employ both of them at Macalester. One is to make calculus more interesting. The other is to guide prospective math majors into a non-calculus course that is required for and that directly supports the major.

Several of the “reform calculus” texts of the late 1980s and early 1990s accomplished this, beginning first-semester calculus with difference equations and quickly leading into differential equations. This accomplishes many things: It gives students an immediate appreciation for the power and usefulness of calculus. It provides motivation for the definition of the derivative and many of the technical properties of differentiation. It frontloads the one thing we really want students to remember about their calculus class if this is where their mathematics education ends. And in most cases it gives them something that is fresh and different from what they studied in high school.

There are other ways of enlivening calculus. I particularly
like Pomona’s calculus course for those students who arrive with credit
for AP Calculus. The book *Approximately Calculus* [8]
was developed from this course. It includes both non-calculus topics such
as RSA public-key encryption and a fresh approach to calculus, emphasizing
Taylor polynomials and power series as tools of approximations.

What all of these approaches to calculus share is an emphasis on problem-solving around interesting questions. This both engages and holds student interest, and I’ve found it to be a very effective enticement into the study of further mathematics. I have also found that students who have been required to solve non-standard problems that draw on and challenge their conceptual understanding of the ideas of calculus are well equipped to pick up technical skills as they need them.

Can we afford to change our calculus classes in this way? After
all, calculus is a sequence of service courses. Our colleagues in engineering
and the physical and biological sciences have expectations for what their
students will learn from us. Surely, we must devote our efforts to hammering
home those skills that will be needed later. Actually, as revealed in the
*Curriculum Foundations Workshops* [9], when leading
educators in the STEM disciplines were brought together to clarify what their
majors need from their mathematics courses, the ability to “think mathematically”
dominated all else. In calculus, it is the ability to read and write differential
equations that is most valued, not the ability to integrate by parts (useful
though it is). Addressing the issue of integration by parts, facility with
the technique is not what these students most need. Our colleagues want their
students to understand how integration by parts is connected to the chain
rule and the rule for the derivative of a product. It is an awareness of these
connections, of how and why integration by parts works, that our colleagues
value. Across the board, the consistent message from all disciplines was to
teach fewer topics, but that what we teach needs to be taught for understanding.

The other approach to engaging the interest of prospective STEM majors is to offer a challenging non-calculus course that counts toward the mathematics major and that students are encouraged to take in their first year. This often is a course that serves as a broad introduction to discrete mathematics, but it could equally well be a course in number theory, linear algebra, graph theory, geometry, or statistics.

Macalester offers a course in discrete mathematics that also serves to begin the bridge to the proof-intensive courses of the junior and senior mathematics curriculum. With an emphasis on combinatorics and number theory, our course engages students in analyzing problems and synthesizing solutions for which they need to explore unfamiliar topics in mathematics. By requiring them to write up solutions and by critiquing first drafts, requiring rewrites, and holding to high standards, it effectively begins the needed transition to proofs while these students are still in their first year.

This is a class that highlights interesting mathematics. It attracts students who are tired of calculus but still interested in mathematics. Of the 500 students who enter Macalester each year, about 80 of them take this class. We draw not just prospective math and computer science majors but also students interested in physics, chemistry, or economics who have enjoyed mathematics, consider themselves to be good at it, and are intrigued by the eclectic mix of topics in this course. Many of our majors tell us that this was the class that enticed them to pursue mathematics.

Over the next two months, I want to provide more specific illustrations of what we do at Macalester. Next month I will describe one of the core projects from our discrete math class that I have found to be particularly effective at both engaging student interest and getting them to begin thinking and writing as mathematicians. The following month I will provide some details of how we have recast our first class in calculus so that it better meets the needs of the majority of students who take it, most of who have seen some calculus before and almost all of whom, as biology or economics majors, have no intention of continuing with calculus beyond this first class.

[1] David J. Lutzer *et al*, *CBMS
Survey Fall 2005: Statistical Abstract of Undergraduate Programs in the Mathematical
Sciences in the United States*. 2007. Providence, RI: American Mathematical
Society.

NCES Digest of Education Statistics. nces.ed.gov/programs/digest/d06/tables/dt06_251.asp?referrer=report

[2] NCES Digest of Education Statistics. nces.ed.gov/programs/digest/d06/tables/dt06_007.asp?referrer=report

[3] The basis for the half million figure
is explained in the April 2007 *Launchings* column, *The Crisis
of Calculus*. www.maa.org/columns/launchings/launchings_04_07.html

[4] NCES Digest of Education Statistics.

nces.ed.gov/programs/digest/d05/tables/dt05_249.asp

[5] About a third of those who study calculus in high school earn college credit for this course.

[6] COMAP, *For All Practical Purposes*.
7th edition. 2006. W. H. Freeman.

[7] Edward Burger and Michael Starbird, *The
Heart of Mathematics*. 2nd edition. 2004. Key College.

[8] Shahriar Shahriari, *Approximately
Calculus*. 2006. American Mathematical Society.

[9] William Barker and Susan Ganter, *The
Curriculum Foundations Project: Voices of the Partner Disciplines*. 2004.
Washington, DC: Mathematical Association of America. www.maa.org/cupm/crafty/cf_project.html

Access pdf files of the

Purchase a hard copy of the *CUPM
Curriculum Guide 2004* or the *Curriculum
Foundations Project: Voices of the Partner Disciplines*.

Find links to course-specific software resources in
the *CUPM
Illustrative Resources*.

Find other *Launchings*
columns.

David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, and president-elect of the MAA. You can reach him at bressoud@macalester.edu. This column does not reflect an official position of the MAA.