David M. Bressoud January 2010
The highly linear structure of the traditional mathematics curriculum is one of its strengths: One knows what a student is supposed to learn when and what common mastery can be expected of students in later courses. Moreover, the nature of the subject lends itself to such linearity. No one would question that a student needs to be fluent in arithmetic before tackling algebra. Algebra, in turn, is a necessary pre-requisite to the study of calculus that must precede differential equations or analysis.
Nevertheless, we mathematicians do carry this linear progression to unnecessary extremes. There is no reason why calculus must precede linear algebra or introductory number theory. A full course of differential equations requires fluency with the rules of differentiation and integration, and if partial differential equations are to be included, there are concepts from several variable calculus that are pre-requisite, but what about the biology student who would benefit from familiarity with the language and qualitative analysis of systems of differential equations? Does such a student require three semesters of calculus in its full glory before being admitted to the mysteries of differential equations? And for the student for whom statistics is the appropriate college mathematics course, does this student need the same knowledge of algebra as the student preparing to tackle calculus?
These are not academic questions. By building long chains of courses that must be traversed before getting to what is most interesting and pertinent, we often cut students off from their intended careers. Many students successfully complete one or more courses along their college progression and then fail to enroll in the next course of the sequence. They have been lost not because they are incapable of handling the mathematics they will need, but because they tire of courses whose relevance to their goals is unclear, a perception that is compounded by the fact that many of these pre-requisite courses are taught without serious thought about how students learn and what needs to be done to facilitate this. This is the problem of persistence.
I have encountered two separate threads within the past year that bring this issue to the forefront. The first was the release of a study at Arizona State University  that followed five years worth of students who had enrolled in precalculus and had declared majors in engineering, mathematics, or science, majors that required at least one semester of calculus. Students who did well in this course and then took calculus were generally well-prepared: 82% of those who earned a C or higher in precalculus and then took calculus earned a C or higher in their calculus course. The problem was that almost half of these students who earned a C or higher never took the required calculus course, choosing to switch to a major that did not require calculus instead. Even among those who earned an A in precalculus, 43% switched majors rather than taking calculus. Similar if less dramatic patterns held for continuation from first to second semester calculus and on to several variable calculus.
It is important to note that the reasons for leaving were not necessarily attached to the mathematics class, but many of them were directed at mathematics. The dominant problems revolved around the student’s sense of lack of support: issues of advising and career counseling, large classes and inadequate recitation instructors, and faculty who were less approachable than those in other disciplines. It is significant that students did not find that switching majors decreased the amount of work they needed to do. These finding are entirely consistent with the analysis of why students leave science majors in Seymour and Hewitt’s Talking about leaving: Why undergraduates leave the sciences .
This was a wake-up call at ASU and has resulted in greater support for and attention to the teaching of precalculus. I strongly suspect that ASU is not unique.
The second thread involves an issue that is now being tackled by AMATYC (American Mathematical Association of Two-Year Colleges) and by the Carnegie Foundation for the Advancement of Teaching through its Statway program. The problem they are addressing is that students who enter with mathematical skills below the level needed to start with a college-level mathematics course have a low rate of getting to and successfully completing such a college-level mathematics course.
This problem is most pronounced in our two-year colleges. Almost 60% of the students enrolled in mathematics in the fall term in two-year colleges are enrolled in courses variously labeled as pre-transfer level, pre-college level, or developmental. Many of these students require a sequence of two or more courses at this level before they are ready for a college-credit-bearing mathematics course. For these students, often returning adults, life intervenes. It is not unusual to find that among those who successfully complete one course in their sequence, 30% may fail to enroll in the subsequent course. The problem is that this compounds. For students facing a three-course sequence, two pre-college level courses and one college level course, if 70% complete each course with a C or higher and 70% of these enroll in the next class, only 0.7^5, about one in six, of those who start will complete the sequence, a fairly typical success rate.
The sequence that almost all of these students enter is one designed to prepare them for precalculus, even though many of them neither need nor want to study calculus. The college-credit-bearing course that is most appropriate for many of these students is statistics. The task that both AMATYC and Carnegie are now wrestling with is to guide and support the creation of sequences that prepare students for college statistics, decreasing the number of courses they must traverse by identifying and focusing on those mathematical skills most relevant to the study of statistics. At the same time, AMATYC and Carnegie are very cognizant that such a sequence should not preclude the possibility of building on what has been accomplished to complete the preparation for the study of calculus. Shortening the time to completion while not closing off options is a true challenge, but an important one for our community to tackle.
All of this is part of a recurrent theme in my columns: More high schools are teaching college-level mathematics to ever more students, while more colleges are teaching high school-level mathematics to ever more students. These parallel tracks serve few students well, especially when the transition from high school to college involves backing up. One of the great tasks before us to re-engineer both sides of this transition so that most students are prepared for college mathematics when they graduate from high school, and all students can find college mathematics courses that are engaging, relevant, and interesting and, at the same time, open up rather than closing off options.
 Thompson, Patrick W. et al. (2007). Failing the Future: Problems of persistence and retention in science, technology, engineering, and mathematics (STEM) majors at Arizona State University. Tempe, AZ. Office of the Provost.
 Seymour, E. & Hewitt, N.M. (2000). Talking about leaving: Why undergraduates leave the sciences. Westview Press. Boulder, CO.
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