consensus, based on research evidence, that the development of mathematical proficiency requires continual instructional opportunities for students to build their understanding of core mathematical concepts, fluency with mathematics procedures, and metacognitive strategies to guide their own mathematical learning. However, there is little empirical evidence to support detailed conclusions about precisely how this is best accomplished.
WHAT DO SUCCESSFUL TEACHERS KNOWABOUT MATHEMATICS AND HOW TO TEACH IT?
There is strong reason to believe that teachers’ knowledge and skills make a difference in their practice (Wenglinsky, 2002; Rockoff, 2004; Rivkin, Hanushek, and Kain, 2005; Clotfelter, Ladd, and Vigdor, 2007). Researchers have tried to disentangle the different kinds of knowledge about mathematics, about students, and about the learning process that teachers use (Ball, Lubienski, and Mewborn, 2001). Research that has searched for connections between straightforward—yet crude—measures of teacher knowledge (such as number of courses taken or degrees earned) and student learning has provided relatively little insight into questions about what skills and knowledge are most valuable for teachers (see Chapter 3). For example, research shows that high school students taught by mathematics majors outperform students taught by teachers who majored in some other field, but that research does not illuminate what it is that the teachers who majored in mathematics do in the classroom (Monk and King, 1994; Goldhaber and Brewer, 1997; Rowan, Chiang, and Miller, 1997; Wilson, Floden, and Ferrini-Mundy, 2001; Floden and Maniketti, 2005).
Beginning in the 1980s, a growing number of scholars looked to more qualitative research—largely based on interviews and classroom observations—to provide richer pictures of the mathematical thinking teachers do when teaching. These studies supported the development of more nuanced descriptions of teachers’ knowledge and skills by illuminating the ways that the mathematical knowledge needed for teaching differs from the mathematical knowledge needed to succeed in advanced courses. The concept of pedagogical content knowledge gave a name to the knowledge of content as it applies to and can be used in teaching (see, e.g., Shulman, 1986; Ball, Lubienski, and Mewborn, 2001).
Mathematicians have always played a critical role in defining the kinds of knowledge and skills that are most useful to mathematics teachers. Several recent publications, including textbooks for aspiring elementary mathematics teachers, studies, and analytic essays, have laid out current thinking (e.g., Parker and Baldridge, 2003; Beckmann, 2004; Milgram, 2005; Wu, 2007). For example, Wu (2007) argued that, at a minimum, teachers of grades 5-12 must be knowledgeable about the importance of definitions, the
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