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Abstract
This article describes seven teacher knowledge frameworks and relates these frameworks to the teaching and assessment of elementary teachers' mathematics knowledge. The frameworks classify teachers' knowledge and provide a vocabulary and common language through which knowledge can be discussed and assessed. These frameworks are categorized into two classes: content knowledge and content knowledge for teaching. Content knowledge frameworks include Bloom's Taxonomy (1956); Skemp's (1976) Instrumental and Relational Understandings; Hiebert and Carpenter's (1992) Procedural and Conceptual Understandings; Webb's (2002) Depth of Knowledge; and Porter's (2002) Cognitive Complexities. Content knowledge for teaching frameworks includes Schulman's (1986) Type of Teachers Knowledge and Ball's (2000) Mathematical Knowledge for Teaching. The Diagnostic Teacher Assessment of Mathematics and Science (DTAMS), a tool that assesses both mathematics teachers' depth of conceptual knowledge and pedagogical content knowledge, is used to concretely connect the frameworks. The article concludes with examples of quantitative assessments of teachers' mathematics knowledge based on these teachers' knowledge frameworks.
Keywords: mathematics, teacher, DOK, frameworks, knowledge
Overview
The need to both strengthen and increase teachers' mathematics knowledge has led many researchers to design mathematics knowledge frameworks as ways to understand what teachers' mathematics knowledge looks like (Ball, 2002, 2003; Ball & Bass, 2000; Bloom, Englehart, Furst, Hill & Krathwohl, 1956; Hiebert & Carpenter, 1992; Hill & Ball, 2006; NCTM, 2004; Porter, 2002; Shulman, 1986; Petit & Hess, 2005, 2007; Skemp, 1976, 1993; Webb, 1997, 2002). Some frameworks describe teacher knowledge in terms of degree of cognitive knowledge exhibited (Bloom, 1956; Skemp, 1976, 1993; Hiebert & Carpenter, 1992; Webb, 1997, 2002; Porter, 2002) while others describe and classify teachers' pedagogical or specialized content knowledge (Ball, 2003; Shulman, 1986). Such frameworks provide the means to plan for and promote higher order thinking behaviors- first in the teacher, then in the student.
This article describes seven teacher knowledge frameworks and relates these frameworks to (a) the teaching of elementary mathematics and (b) assessment of that teaching through the Diagnostic Teacher Assessment of Mathematics and Science (DTAMS), a tool that assesses both mathematics teachers' depth of conceptual knowledge and pedagogical content knowledge and is used to concretely connect the frameworks. DTAMS items reflect different types of teacher content and pedagogical knowledge, including depths of content knowledge. Depth of knowledge measures three levels...