The Context: Typical Mathematics Instruction

Research studies provide a clear, consistent picture of typical school mathematics instruction in the United States. What we know is largely derived from two kinds of data and associated research analyses. One type of study that has been carried out over several decades has involved direct observation of classroom teaching (e.g., Stake and Easley, 1978; Stodolsky, 1988; Stigler et al., 1999; Hiebert et al., 2005), and another has used teacher self-report data from surveys (e.g., Weiss et al., 2001; Grouws, Smith, and Sztajn, 2004).

These studies present a remarkably consistent characterization of mathematics teaching in upper elementary school and middle-grade classrooms in the United States: Students generally work alone and in silence, with little opportunity for discussion and collaboration and little or no access to suitable computational or visualization tools. They focus on low-level tasks that require memorizing and recalling facts and procedures rather than tasks requiring high-level cognitive processes, such as reasoning about and connecting ideas or solving complex problems. The curriculum includes a narrow band of mathematics content (e.g., arithmetic in the elementary and middle grades) that is disconnected from real-world situations, and a primary goal for students is to produce answers quickly and efficiently without much attention to explanation, justification, or the development of meaning (e.g., Stodolsky, 1988; Stigler and Hiebert, 1999). As earlier chapters in this volume have indicated, reflecting research evidence regarding how people learn best when the goal is developing understanding (National Research Council, 1999), such pedagogy is at odds with goals aimed at deeper learning and transfer.

Although this pervasive approach to mathematics teaching has not been directly established as the cause of the generally low levels of student achievement, it is difficult to deny the plausibility of such a connection. In response, an array of reform initiatives has been aimed at changing what and how mathematics is taught and learned in American schools. Although reformers have disagreed on some issues, they share the goal of enhancing students’ opportunities to learn mathematics with understanding and hence the attendant goal of promoting teaching mathematics for understanding. These goals reflect a focus on deeper learning in school mathematics.

Evolution of National Standards in Mathematics

School mathematics reform has a long history that cannot be adequately described in the limited space here, so we focus on the most recent

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