# Colloquium

Colloquium on Education in Mathematics

Speaker: Hyman Bass (University of Michigan)

Title: A vignette of mathematical practices in action

Abstract: Mathematical Practices are, essentially, the things you do when you do mathematics. While they are emphasized in the Common Core, many teachers find it difficult to fully understand what they mean, or what they “look like.” I will a present small piece of mathematics that arose from a question about fractions in third grade. I will use this as a context to illustrate mathematical practices in action. Here is the problem: Suppose that s students want to equally share c cakes.

What is the smallest number, p(c, s), of cake pieces, needed to achieve this fair distribution? We will derive a formula for p(c, s) and describe two different distribution schemes that achieve this, the “linear” and the “Euclidean” distributions. The Euclidean distribution corresponds to the “Euclidean square tiling” of a c x s rectangle R, and we shall see that this square tiling is “isoperimetric” in the sense that it has smallest “perimeter” among all square tilings of R. I will describe a generalized version of this problem that is still open.

Mathematical Practices are, essentially, the things you do when you do mathematics. While they are emphasized in the Common Core, many teachers find it difficult to fully understand what they mean, or what they “look like.” I will a present small piece of mathematics that arose from a question about fractions in third grade. I will use this as a context to illustrate mathematical practices in action. Here is the problem: Suppose that s students want to equally share c cakes.

What is the smallest number, p(c, s), of cake pieces, needed to achieve this fair distribution? We will derive a formula for p(c, s) and describe two different distribution schemes that achieve this, the “linear” and the “Euclidean” distributions. The Euclidean distribution corresponds to the “Euclidean square tiling” of a c x s rectangle R, and we shall see that this square tiling is “isoperimetric” in the sense that it has smallest “perimeter” among all square tilings of R. I will describe a generalized version of this problem that is still open.

Bio: Hyman Bass is the Samuel Eilenberg Distinguished University Professor of Mathematics and Mathematics Education at the University of Michigan. He has served as President of the American Mathematical Society and the International Commission on Mathematical Instruction and as Chair of the National Academy of Sciences’ Mathematical Sciences Education Board, and of the AMS Committee on Education. He is a member of the U.S. National Academy of Sciences, the American Academy of Arts and Sciences, the Third World Academy of Sciences, and the National Academy of Education. In 2006 he received the U. S. National Medal of Science. His mathematical research spans various domains of algebra, notably algebraic K-theory and geometric group theory. His work in education (largely with Deborah Ball) focuses mainly on mathematical knowledge for teaching, and on the teaching and learning of mathematical practices, such as reasoning and proving, and discerning and developing mathematical structure, in K-16 classrooms.