Algebra for Teaching—Some Recommendations for Teacher Preparation



May 14, 2007


Critical Issues in Education Workshop: Teaching and Learning Algebra, MSRI Workshop


Berkeley, California

The development of CME Project began in 1992 with a grant from NSF to develop a geometry course, but the basic principles upon which the program rests evolved over nearly four decades. The basic position of the development team is that the real utility of mathematics—both for students who specialize in mathematically related fields and for those who take other directions—lies in a style of work, a collection of mathematical habits of mind, that mathematicians use to make sense of the world, such as visualization, performing thought experiments, reasoning by continuity or linearity, and mixing deduction with experiment. The basic results and methods of high school mathematics—the Pythagorean theorem, the techniques for solving equations or graphing lines or analyzing data—are the products of mathematics. The actual mathematics lies in the thinking that is used to create and develop these results. It is essential to develop both the results and the thinking. The entire program, from the uses of technology to the design of the problem sets in each lesson, is devoted to helping students become mathematical thinkers as they develop the content knowledge to apply that mathematical thinking competently.

This presentation gives examples of how this philosophy plays out in the context of the program’s development of algebra in high school. There are several mathematical habits that are dominant in algebra: reasoning about and picturing calculations and operations, seeking regularity in repeated calculations, purposeful transformation and interpretation of algebraic expressions, “chunking” (changing variables in order to hide complexity), and seeking and modeling structural similarities in algebraic systems that include the number systems of arithmetic but also include other algebraic systems. The presentation shows how these habits are introduced and strengthened across algebra strand of the program, providing students with general-purpose tools that allow them to solve problems, make and use abstractions, and develop mathematical theories.