Even with high standards, exemplary textbooks, and powerful assessments, what really matters for mathematics learning are the interactions that take place in classrooms. The literature on mathematics education, perhaps surprisingly, contains little reliable data about those interactions. Most of the available research evidence consists of reports by teachers of their practice, but an increasing amount comes from systematic observations of lessons. The discussion in this section addresses both types of evidence.

The emphasis in U.S. elementary and middle school mathematics teaching seems to be predominantly on number and operations. Teachers of 93% of the fourth graders and 88% of the eighth graders in the 1996 NAEP mathematics assessment reported that they gave the topic “a lot” of instructional emphasis.^{59} At grade 8, algebra also received a lot of emphasis (for 57% of the students), but that was the only other curriculum strand to receive much attention. Fourth-grade teachers reported giving considerable emphasis to facts, concepts, skills, and procedures (over 90% of the students got “a lot”), with less emphasis on reasoning processes (52%) and even less attention to communication (38%). Eighth-grade teachers’ responses followed a similar pattern, with somewhat less attention to facts, concepts, skills, and procedures (79%). In a recent study comparing schools participating in state initiatives in mathematics and science with schools not involved in such initiatives, elementary school teachers in the initiatives schools spent significantly more time than their counterparts on reasoning and problem-solving activities.^{60}

For decades, mathematics educators have been exhorting teachers to allow children to use *manipulatives*—counting blocks, geometric shapes, and other objects—to support their thinking. The use of manipulatives, however, is not a common classroom practice. In 1996, teachers of 27% of the fourth graders in NAEP reported that their students used counting blocks and geometric shapes at least once a week; 74% used them at least once a month, leaving 26% who seldom if ever used them. Teachers of 8% of the eighth graders said that their students used such manipulatives at least once a week, and teachers of more than half the students reported essentially no use. Data were not available on how this use was connected to mathematical ideas, words, and notations.

Materials such as rulers and calculators are apparently used much more frequently than manipulatives in mathematics teaching. Teachers of almost