Courses

MATH 498:  TOPICS IN MODERN MATHEMATICS:
SCHOOL MATHEMATICS FROM A
COHERENT MATHEMATICAL PERSPECTIVE

Winter, 2010                     Tues & Thurs, 2:30 – 4:00 pm       
Hyman Bass, instructor       hybass@umich.edu

Content:  This is a mathematics course, emphasizing fundamental mathematical structures, developed through reasoning and proof, and with careful attention to mathematical language, notation, and communication.  But the mathematical themes are chosen to afford abundant and illuminating contact with typical topics in the school curriculum, topics that are often presented in ad hoc and disconnected fashion.  These include the structure (arithmetic and geometric) of the number line, the geometric interpretation of the operations of arithmetic, the structure of place value and of place value based arithmetic algorithms, divisibility and divisibility tests, prime factorization, the nature of decimal representations of rational numbers, polynomial and exponential functions, complex numbers, mathematical symmetry, etc.

For the mathematical development, the emphasis will be on a coherent story line of investigation of the mathematical structure of addition and multiplication in the basic number systems: integers, rational numbers, real numbers, complex numbers, and modular arithmetic.  The organizing mathematical structures used will those of a group, and of a ring.  However, these notions will be dealt with mainly concretely, in the familiar number systems, rather than abstractly.

The course has little by way technical prerequisites.  Students should be comfortable with the (continuous) real number line.  They should know elementary properties of polynomial (linear and quadratic) polynomials, and exponential functions.  More importantly, students should be able to flexibly and comfortably think about familiar mathematical objects in novel and general, but simple ways, and to reason fluently across numerical and geometric representations of mathematical ideas.

Course structure:  Each class will involve a mixture of formats – lecture, whole group discussion, student presentation, and small group work.  Students work will be both individual and in small teams (of two or three).  The mathematical tasks will be of diverse kinds: to develop the general theory; to achieve applications of the theory; and to gain practice and skill with the new ideas and techniques being developed.  Student work, both written and oral, will be expected to meet high mathematical standards of clarity, precision, and rigor.

Materials and references:  There is no text for the course, though the following will be a useful reference, in particular for exercises.

“Mathematics for High School Teachers- An Advanced Perspective,” Zalman Usiskin, Anthony L. Peressini, Elena Marchisotto, & Dick Stanley.  Prentice Hall (2003)

Basic materials for the course will be prepared and distributed electronically.