student achievement in California demonstrated that when teachers had experienced extended inservice opportunities to learn about mathematics curriculum and instruction, their students’ achievement increased (Wiley and Yoon, 1995). Also, a study of mathematics reform, Quantitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR), program found higher achievement among students whose teachers were involved in a sustained program of curriculum development, in this case, a program that emphasized enhancing teachers’ understanding of strategies, having teachers implement new strategies, and encouraging teachers to reflect on instructional outcomes (Brown et al., 1995).14
In addition, Grouws and Schultz (1996) summarized a series of studies designed to gauge the impact of the University of Wisconsin’s Cognitively Guided Instruction (CGI) research program for mathematics teacher effectiveness. According to Grouws and Schultz, the studies found that providing teachers with knowledge of how students think and opportunities to develop strategies in specific content domains changed teaching behaviors and improved student learning. In one study of first- and second-grade teachers, the CGI group provided teachers with knowledge of young children’s thinking and with strategies for teaching addition and subtraction. Subsequently, these teachers spent more time in their mathematics instruction on problem solving and assessing student thinking than a control group of teachers who received equivalent hours of inservice training. The students of CGI teachers also performed better in some math assessments—higher in problem solving, comparably on computational tasks.
In their review, Grouws and Schultz specifically note a certain type of teacher knowledge, called pedagogical content knowledge (Shulman, 1986). They state, “In mathematics, pedagogical content knowledge includes, but is not limited to, useful representations, unifying ideas, clarifying examples and counterexamples, helpful analogies, important relationships, and connections among ideas. Thus, pedagogical content knowledge is a subset of content knowledge that has particular utility for planning and conducting lessons that facilitate student learning.”
All of the studies cited in this chapter, as well as those cited earlier (e.g., Fetler, 1999), lend strong support to the idea that when teachers receive high-
Information about the QUASAR program is available at <http://www.ed.gov/pubs/math/part6.html>.