To understand the relations between undergraduate and graduate education in mathematics, one must understand the niche that mathematics departments occupy in the ecology oi their institutions. It is a question of properly reconciling the two great academic missions: teaching and research. While acknowledging the importance oi research, undergraduate students, parents, and legislators quite rightly place a high premium on the quality of undergraduate mathematics teaching. And while recognizing the fundamental importance of teaching, faculty, postdoctoral fellows, and deans equally rightly place a high value on the conduct of top-notch research in mathematics. Fortunately, excellent programs of teaching and research need not be mutually exclusive. Indeed, in the best of circumstances, they are mutually supportive and they intersect, roughly speaking, in the program of graduate education.
Graduate education is the critical link in the social and intellectual fabric of the mathematics discipline, and it is funded to an overwhelming extent by undergraduate instruction. On one hand, federal funding for graduate education in mathematics has increased dramatically in the last five years. Still, total federal support is proportionally less than that for graduate education in the biological or physical sciences and it remains inadequate to the task. On the other hand, demand for mathematics instruction at the university is so great that it cannot be met by the regular faculty alone; teaching assistants (and postdocs and part-time lecturers) are needed to supplement the instructional capacity of most math departments.
The income derived from meeting the huge demand for mathematics instruction at all levels finances the mathematical enterprise in a public university. Indeed, the income from instruction typically exceeds the cost of maintaining the department; mathematics is an institutional "cash cow." Be that as it may, instruction is the fiscal mechanism that provides far graduate education, postdoctoral training, and a rich intellectual environment for research in mathematics. Exploiting this niche correctly by cultivating the relations among research and graduate and undergraduate education is the key to the health and prosperity of the department and the discipline.
Mathematics at The Ohio State University
The Ohio State University is a large, public land-grant institution committed to excellence in research and teaching. Currently, it is an ambitious and rapidly changing institution that aspires to become a truly great university. In a departure from its history as an open admissions school, Ohio State recently adopted a more selective admissions policy that has dramatically improved the quality and graduation rates oi the undergraduate student body. Still the university quite properly admits students from an enormous variety of backgrounds, which results in a heterogeneous student body, from the point of view of mathematical preparation. But regardless of students' preparation or intended major, successful completion of required mathematics courses means that the department must provide a variety of course options for its many students.
Last year, the department taught approximately 30,000 students at all levels (some students more than once, of course). Of these students, slightly more than 300 were declared mathematics majors, and another 120 were graduate students in the PhD program. Thus, the vastly greater part of the instructional mission is service teaching for other departments. Instruction on this scale is an enormous responsibility that is taken very seriously by the department. The curriculum ranges from remedial mathematics, through the standard undergraduate courses for science and engineering students, to upper division courses for math majors, basic graduate courses, and topics courses that explore the frontiers of current research. In addition to the mathematical preparation of science and engineering majors, undergraduate-level instruction includes the critical tasks of cultivating mathematically gifted high school students, preparing future grade school teachers, educating future professionals in actuarial science, and preparing high performing honors students for graduate study in mathematics. …