Using the Performance Assessment for California Teachers to Examine Pre-Service Teachers' Conceptions of Teaching Mathematics for Understanding

Article excerpt

Motivated by a concern about the low performance of American students in mathematics, the National Research Council published the report Adding It Up: Helping Children Learn Mathematics (2001). This report summarizes a core body of research revealing that U.S. students continue to perform poorly in mathematics. While they can carry out straightforward procedures, American students demonstrate limited understanding of mathematical concepts and are unable to apply their knowledge to solve novel problems. This report also explains that the preparation of elementary and middle school teachers falls short of equipping future teachers with the knowledge they need to help students develop mathematical proficiency. Thus, a key challenge for mathematics teacher education is to prepare prospective teachers to teach mathematics for understanding. As Philipp (2008) describes, meeting this challenge is difficult because "teachers lack the depth and flexibility of mathematical understanding and the corresponding beliefs they need to teach for proficiency (NRC, 2001)" (p. 3). Moreover, given the persistence of traditional teaching practices in American teaching, prospective teachers have had few, if any, opportunities to participate in classrooms that promote learning mathematics for understanding (Lortie, 1975; Zeichner & Liston, 1987).

Teacher education programs have adopted several approaches to help future teachers develop the knowledge, skills, and practices for teaching mathematics for understanding. Some examples include using narrative and video cases that illustrate this model of mathematics teaching and engaging future teachers in the analysis of these cases (Hatfield & Bitter, 1995; Lampert & Ball, 1998; Santagata, Zannoni, & Stigler, 2007) and designing courses that integrate the development of pedagogical content knowledge through the examination of mathematics and mathematics pedagogy (Philipp, 2008). Still, we know very little about the particular ways that pre-service teachers have come to understand teaching mathematics for understanding, as well as what practices they perceive will help them accomplish this goal. As Boaler and Humphreys (2005) describe, an important direction of teacher research involves understanding teachers' decision making in the moment of teaching. They write:

Each [pedagogical] move is important and demonstrates the complexity of teachers' work. Such moves also demonstrate the level, or "grain size," at which teaching decisions are made. Teachers are often offered advice that is at a much bigger grain size, such as whether to use group work to have discussions or lecture. We see [...] that teachers need to make decisions that are at a smaller grain size, such as when and how to curtail a discussion, which examples of representation to use, or which students to call upon. The field of educational research has not developed extensive knowledge of the detailed pedagogical practices that are helpful for teachers to learn, yet the difference between effective and ineffective teaching probably rests in the details of moment-to-moment decision making. (p. 53)

The goal of this study is to begin to uncover the detailed ways that pre-service elementary teachers examine and understand mathematics teaching, the grain size at which they do so, and what constitutes evidence of teaching mathematics for understanding for pre-service mathematics teachers. Thus, the central research questions for this study are: (a) What are pre-service teachers' conceptions of teaching mathematics for understanding? and (b) What counts as evidence for pre-service teachers of their practices in the planning, enactment, and reflection on teaching?

This study takes place in the context of the Performance Assessment for California Teachers (PACT), also known as the Teaching Event. The PACT is a standards-based performance assessment designed to measure pre-service teacher learning (Pecheone, Pigg, Chung, & Souviney, 2005). …