Mathematical Knowledge for Teaching
Ball and her colleagues (Ball, 1993, 2007; Ball, Hill, & Bass, 2005; Ball & McDiarmid, 1990) have focused on understanding the special ways one must know mathematical procedures and representations to interact productively with students in the context of teaching. Their pioneering work has succeeded in identifying a statistical relationship between this mathematical knowledge for teaching (MKT) and student achievement (Ball, et al., 2005; Hill, Rowan, & Ball, 2005). We extend this work by focusing not only on particular mathematical understandings but also the conceptual structures within which those particular understandings lie. Our reason for this focus is pragmatic:
If a teacher’s conceptual structures comprise disconnected facts and procedures, their instruction is likely to focus on disconnected facts and procedures. In contrast, if a teacher’s conceptual structures comprise a web of mathematical ideas and compatible ways of thinking, it will at least be possible that she attempts to develop these same conceptual structures in her students (Thompson, Carlson, & Silverman, 2007)
Rather than focusing on identifying the mathematical reasoning, insight, understanding and skill needed in teaching mathematics, we focus on the mathematical understandings “that carry through an instructional sequence, that are foundational for learning other ideas, and that play into a network of ideas that does significant work in students’ reasoning” (Thompson, 2008). We refer to these understandings as coherent understandings: powerful, generative “big ideas” from which an understanding of a body of mathematical ideas and its relation to other bodies can emerge.
It is important to note that coherence is not a characteristic of one’s understanding of a particular mathematical idea, for coherence in curricula or students’ understandings depends on the way in which they fit together (Thompson, 2008). This notion of coherence is a challenge to traditional mathematics teacher education efforts that seek to support teachers in “gain[ing] the ability to do the mathematics … and understand[ing] the underlying concepts so they will be able to assist their students, in turn, to gain a deep understanding of mathematics” (Musser, Burger, & Peterson, 2008). When a focus is on coherence, the emphasis is not just on doing and learning “the mathematics,” but rather on developing a scheme of understanding within which a variety of mathematical ideas are connected and that can serve as a conceptual anchor for mathematics curricula and instruction. The research described in this chapter is grounded in the development of coherent mathematical knowledge for teaching.