studied how teachers’ knowledge of students’ mathematical thinking is related to how they teach and to how well their students achieve.
From the many examples of misconceptions to which teachers need to be sensitive, we have chosen one: An important mathematical notion that poses a major stumbling block when students are moving from arithmetic to algebra is the role played by “=,” the sign for equality.25 As we discussed in chapter 8, many if not most elementary school children have the misconception that the equality sign is a signal to do something, to carry out the calculation that precedes it.26 The number immediately after the equal sign is seen as the answer to the calculation. For example, in the number sentence 8+4=□+5, many students would put 12 in the box. Children can develop this impression because that is how the notation is often described in the elementary school curriculum and most of their practice exercises fit that pattern. Few teachers realize the degree of their students’ misunderstanding of such sentences.27 Moreover, although most teachers have some idea that equality is a relation between two numbers, few realize how important it is that students understand equality as a relation, and few consider this need for understanding when they use the equals sign.
Knowing classroom practice means knowing what is to be taught and how to plan, conduct, and assess effective lessons on that mathematical content. It includes a knowledge of learning goals as expressed in the curriculum and a knowledge of the resources at one’s disposal for helping students reach those goals. It also includes skill in organizing one’s class to create a community of learners and in managing classroom discourse and learning activities so that everyone is engaged in substantive mathematical work. We have discussed these matters in chapter 9. This type of knowledge is gained through experience in classrooms and through analyzing and reflecting on one’s own practice and that of others.
In the sections that follow, we consider how to develop an integrated corpus of knowledge of the types discussed in this section. First, however, we need to clarify our stance on the relation between knowledge and practice. We have discussed the kinds of knowledge teachers need if they are to teach for mathematical proficiency. Although we have used the term knowledge throughout, we do not mean it exclusively in the sense of knowing about. Teachers must also know how to use their knowledge in practice. Teachers’ knowledge is of value only if they can apply it to their teaching; it cannot be