Increasing Access to Advanced Mathematics

Purpose

Mathematics faculty at Mount Holyoke College believed that students not majoring in mathematics would find interesting and applicable to their majors certain topics ordinarily taught in advanced mathematics courses. However, because these courses usual ly had four prerequisites, they were not accessible to non-math majors. This project aimed to add courses to the mathematics curriculum that would allow students majoring in other disciplines to study these advanced mathematical concepts.

Innovative Features

Seven courses containing topics of potential interest to non-mathematics majors were revised so that they required, with one exception, only one or two semesters of prior mathematics courses at the Calculus I level or higher. Courses thus revised inclu ded: Analytic Number Theory: Lie Groups; Mathematical Statistics; Differential Equations with modelling; Polyhedral Differential Geometry; Symmetry, Groups and Geometry; and Theory of Equations.

A prerequisite for several of theses courses is a Laboratory in Mathematical Experimentation ("the Lab") where students work in small groups, with the aid of a computer, on six or seven research topics ranging widely across mathematics. They construct examples, look for patterns, formulate conjectures, and try to support them with proof. The faculty's experience in the Lab with the power of collaborative learning and the exploration of rich examples shaped the seven revised courses, to the benefit of math majors and non-majors alike.

Evaluation

To be sure that the revised courses did not damage the quality of the mathematics major by changing its sequencing or making undue concessions to non-majors, the department invited mathematics faculty from other institutions to sit in on the courses. Though they were originally intended to provide quality control, they also helped in resolving pedagogical and mathematical difficulties. Their reports were taken into account in revising the courses for future offerings.

In addition, students were surveyed by an outside evaluator in each of the three years of the project to determine their attitudes toward mathematics both before and after the courses, their impressions with regard to the adequacy of computer training, the usefulness of the small groups, and the value of other instructional components.

Project Impact

Faculty quickly discovered that given limited resources, if they wanted to institutionalize the new courses, they would have to substitute them for courses already being taught. Thus the department ended up reducing the number of prerequisites that mat hematics majors would need for these advanced courses, in the process making many more courses available to students earlier in the major. This redesign of the major became the major impact of the project, since non- mathematics majors who were supposed to be the principal beneficiaries of this work constituted only five percent of the actual enrollments.

On the other hand, mathematics majors who took these courses discovered that they could, with comparatively little mathematical experience, use computers to take on big, messy problems. This discovery led some to participate in summer research or intern ships, which further built their confidence. The fact that the number of mathematics majors has increased substantially over the time of this project suggests that the new instructional strategies intended for non majors may be of substantial benefit in gaining and retaining majors.

Survey results showed no overall effect on the anxiety level of students, which is not surprising given that the courses were taken primarily by students already committed to a mathematics major. However, half of the students reported a more positive a ttitude toward mathematics, another outcome that correlates with the increase in the number of majors.

Two other surveys, one by a researcher at the University of Connecticut, and the other by the Mathematical Association of America, specifically identified the laboratory course as a primary factor in encouraging women to major in math and the early access to a wide variety of courses as a secondary one.

Lessons Learned

With judicious use of computers, relatively inexperienced students can grapple with complicated and advanced topics in a way that builds their understanding of important mathematical concepts. Early in the major, students appreciate the breadth and div ersity of the discipline and the fundamental and complementary roles of investigation and proof.

The experience of this project led the Mathematics Department at Mount Holyoke to question the highly sequential structure of the mathematics major as it exists at most institutions. They find that students who take courses similar to the ones resultin g from the project before the standard junior-level theoretical courses in algebra and analysis get much more out of these courses and are better able to see connections among them.

The project directors are now convinced that it is possible to build a mathematics major around junior-senior level courses that are accessible to students in other disciplines. They have successfully put into place an array of attractive courses that reduce prerequisites but do not undermine the intellectual rigor of the major. They suspect that the situation in most science majors is similar and that the highly hierarchical nature of traditional curricular models is due more to historical convention than to necessity.

Project Continuation

All but one of the courses, in their revised forms, continue to be offered on a regular basis and are accessible to all students at some time after the freshman year. The syllabi, assignments, teaching notes, etc. have been fully documented, so that al l instructors can teach the courses.

Project Recognition

The Mathematics Department at Mount Holyoke was cited as one of the ten exemplary mathematics departments in the country by the Mathematics Association of America. The visiting committee especially noted as unusual and admirable the very flexible prere quisite structure, which enables non-majors to penetrate deeply into the discipline.

Available Information

Copies of an article about the thinking that underlies the project and the ideas about reducing prerequisites, along with information about the specific courses, are available from:

Donal O'Shea
Harriet Pollatsek
Department of Mathematics, Statistics and
Computer Science
Mount Holyoke College
South Hadley, MA 01075
413-538-2162
E-Mail: doshea@mhc.mtholyoke.edu
E-Mail: hpollats@mhc.mtholyoke.edu

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