The principal conclusion was that the most important task of the first two years is to move students from a procedural/computational understanding of mathematics to a broad understanding encompassing logical reasoning, generalization, abstraction, and formal proof. The emphasis was on the belief that this transition should begin as early as possible in the student’s undergraduate career.
Students in the first two years should have the opportunity to sample a wide variety of mathematics. In the past, calculus and linear algebra formed the mathematical foundation during these first two years. Nobody was suggesting the elimination of these “bricks” in the foundation. Rather, alternatives should be available as well. Suggested possibilities include discrete mathematics, number theory, geometry and knot theory. With many students entering our colleges and universities with Advanced Placement credit in calculus offering these alternatives seems even more appropriate.
Workshop participants recognized that mathematics students are headed in many directions. Some (a relatively small number) are headed to graduate school in mathematics, others are planning a career in the secondary school classroom, while others are headed to jobs in industry. Academic advising takes on a very important role and must be approached thoughtfully. A one-size curriculum does not fit all!
Essential themes should be threaded through different courses. One such theme is the nature of mathematical language and knowledge, which includes basic logic, the role of definitions, statement versus converse, and the nature of proof. Other themes are more specific — the concept of function, approximation, algorithm, linearity, and dimension.
The group also compiled a list of skills with which students should gain proficiency during these first two years. These skills include (but are not restricted to) standard computations, visualization, geometric skills, translating mathematics into words (and vice-versa), recognizing incorrect statements or answers, and communicating mathematics both orally and in writing.
It is important that there be a balance between computational skills, conceptual understanding, theoretical reasoning, and applications. This balance can help ease the transition to subsequent courses.
The role of technology was discussed and it was noted that students should have experience with “appropriate” use of technology. This may involve the use of graphing calculators, but is not restricted to such usage. In particular, students should have some experience with computer algebra systems during the first two years.
Many mathematics departments across the country have recently noticed a decline in the number of mathematics majors. Where do these majors start? They start in the courses during the first two years. Workshop participants agreed that recruitment and retention of mathematics majors is an important concern. A well-designed mathematics curriculum in the first two years can draw in and keep prospective mathematics majors. Suggestions included actively pursuing students who perform well in these early mathematics courses, encouraging them to consider a major in mathematics. Specific recruitment suggestions included presenting topics and activities in these courses that will attract students to mathematics, enriching these courses with a wide variety of applications, and including students in the “life” of the mathematics department. This last suggestion could involve formation of an active math club, possibly a student chapter of MAA or a Pi Mu Epsilon chapter. Anything that would help a student “identify” with the mathematics department will help with retention. This must be done seriously and early in the student’s college experience.
This workshop was just a part of a part of a massive effort by MAA’s Committee on Curriculum Renewal Across the First Two Years (CRAFTY) to determine the interests of partner disciplines. To see all of the reports generated by the Curriculum Foundations Workshops watch for the upcoming MAA Report entitled A Collective Vision: Voices of the Partner Disciplines, edited by William Barker and Susan Ganter.
Herbert Kasube was a participant in the CF Workshop on Mathematics Preparation for the Major at MSRI and is a member of both CRAFTY and CUPM. He teaches at Bradley University.