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Teaching Mathematics for Understanding: Discussing Case Studies of Four Fifth-Grade Teachers

In this article, we discuss teachers' knowledge and beliefs about mathematics and how it is best taught and learned. We consider the complex interrelationships between these beliefs and teachers' classroom practice in the context of four case studies of fifth-grade teachers in California, a state that is trying to effect major changes in mathematics instruction. We argue that instruction is heavily shaped by a teacher's knowledge and beliefs and that meaningful change in instruction will entail fundamental changes in what teachers know and believe. We organize our discussion around three categories of knowledge and beliefs: (a) teachers' knowledge and beliefs about learners, learning, and teaching-what they know and believe about how mathematics and other knowledge is learned and taught and what characteristics of students influence the learning process; (b) teachers' knowledge of mathematics-their knowledge of the mathematics content they teach; and (c) teachers' knowledge and beliefs about mathematics-how they view the nature of mathematics, where it comes from, and how it is useful.