3: A Rich Problem and Its Potential for Developing Mathematical Knowledge for Teaching¹

What makes a rich mathematical task for teachers? For students, rich tasks ask solvers to use non-algorithmic thinking, to find relationships among mathematical ideas, to identify and use relevant knowledge, to apply considerable cognitive effort, and to monitor their own progress (Smith & Stein, 1998). Teacher educators have wondered whether these characteristics are the same or if something different is needed for a mathematical task to be valuable for preservice teachers. In this paper, we share some of our thinking and struggles related to this issue.

Our work is part of a project at the University of Michigan-Dearborn where we are revising the Mathematics for Elementary Teachers curriculum to be problem and reasoning based. (See also Flowers, Kline, & Rubenstein, 2003 and Flowers, Krebs, & Rubenstein, 2006). We build on a modest number of problems that have the potential for developing mathematical proficiency (National Research Council, 2001) and mathematical knowledge for teaching (Ball, 2003; Ball & Bass, 2000; Ball, Bass, Sleep, & Thames, 2005). Although prospective teachers need to know the mathematics they teach as well as the foundations and extensions of that mathematics, research indicates that their knowledge of mathematics is limited (Ball, 1990; National Research Council, 2001). In particular, U.S. elementary teachers, in contrast to their counterparts in other countries, lack a profound understanding of fundamental mathematics (Ma, 1999) and struggle with processes of abstraction, generalization, and justification. Moreover, the recommendations of the National Council of Teachers of Mathematics (2000) require more of teachers than before. When children are asked, as is recommended, to reason, represent, communicate, and build connections among mathematical ideas, their teachers need a strong mathematical foundation to analyze student responses (Conference Board of the Mathematical Sciences, 2001; Mathematical Sciences Education Board, 2001; National Research Council, 2001).