A Chinese teacher on how a profound understanding of fundamental mathematics is attained

One thing is to study whom you are teaching, the other thing is to study the knowledge you are teaching. If you can interweave the two things together nicely, you will succeed…. Believe me, it seems to be simple when I talk about it, but when you really do it, it is very complicated, subtle, and takes a lot of time. It is easy to be an elementary school teacher, but it is difficult to be a good elementary school teacher.

SOURCE: Ma, 1999, p. 136. Used by permission from Lawrence Erlbaum Associates.

appropriate ways to respond. A teacher must interpret students’ written work, analyze their reasoning, and respond to the different methods they might use in solving a problem. Teaching requires the ability to see the mathematical possibilities in a task, sizing it up and adapting it for a specific group of students. Familiarity with the trajectories along which fundamental mathematical ideas develop is crucial if a teacher is to promote students’ movement along those trajectories. In short, teachers need to muster and deploy a wide range of resources to support the acquisition of mathematical proficiency.

In the next two sections, we first discuss the knowledge base needed for teaching mathematics and then offer a framework for looking at proficient teaching of mathematics. In the last two sections, we discuss four programs for developing proficient teaching and then consider how teachers might develop communities of practice.

The Knowledge Base for Teaching Mathematics

Three kinds of knowledge are crucial for teaching school mathematics: knowledge of mathematics, knowledge of students, and knowledge of instructional practices.1 These can be seen in the instructional triangle (Box 9–1 in chapter 9 and below).2 Mathematics and students are two of the triangle’s vertices, and instructional practices are the interactions portrayed by the arrows.

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