dialogue that occurs in pair or class situations can help generate self-regulating speech that a student can produce while problem solving. Such helping can also increase the metacognitive awareness of the helper as he or she takes into consideration the thinking of the student being helped.
The framework of How People Learn suggests that classroom environments should at the same time be learner-centered, knowledge-centered, assessment-centered, and community-centered (see Chapter 1). These features map easily to the preceding discussion of the three principles, as well as to the chapters that follow. The instruction described is learner-centered in that it draws out and builds on student thinking. It is also knowledge-centered in that it focuses simultaneously on the conceptual understanding and the procedural knowledge of a topic, which students must master to be proficient, and the learning paths that can lead from existing to more advanced understanding. It is assessment-centered in that there are frequent opportunities for students to reveal their thinking on a topic so the teacher can shape instruction in response to their learning, and students can be made aware of their own progress. And it is community-centered in that the norms of the classroom community value student ideas, encourage productive interchange, and promote collaborative thinking.
Effective teaching and learning depend, however, on balance among these features of the classroom environment. There must be continual connections between the learner-centered focus on student knowledge and the more formal knowledge networks that are the goals of teaching in a domain. Traditional teaching has tended to emphasize the knowledge networks and pay insufficient attention to conceptual supports and the need to build on learner knowledge. Many students learn rote knowledge that cannot be used adequately in solving problems. On the other hand, an overemphasis on learner-centered teaching results in insufficient attention to connections with valued knowledge networks, the crucially important guiding roles of teachers and of learning accessible student methods, and the need to consolidate knowledge. Four such excesses are briefly discussed here.
First, some suggest that students must invent all their mathematical ideas and that we should wait until they do so rather than teach ideas. This view, of course, ignores the fact that all inventions are made within a supportive culture and that providing appropriate supports can speed such inventions. Too much focus on student-invented methods per se can hold students back; those who use time-consuming methods that are not easily generalized need to be helped to move on to more rapid and generalizable “good-enough” methods. A focus on sense making and understanding of the meth-