Mathematical Habits of Mind for Teaching: Using Language in Algebra Classrooms

Abstract

ABSTRACT:The notion of mathematical knowledge for teaching has been studied by many researchers, especially at the elementary grades. Our understandings of this notion parallel much of what we have read in the literature, but are based on our particular experiences over the past 20 years, as mathematicians engaged in doing mathematics with secondary teachers. As part of the work of Focus on Mathematics, Phase II MSP, we are developing, in collaboration with others in the field, a research program with the ultimate goal of understanding the connections between secondary teachers' mathematical knowledge for teaching and secondary students' mathematical understanding and achievement. We are in the early stages of a focused research study investigating the research question: What are the mathematical habits of mind that high school teachers use in their professional lives and how can we measure them? The main focus of this paper is the discussion of the habit of using mathematical language, and particularly how this habit plays out in a classroom setting.Keywords: Mathematical habits of mind, mathematical language, algebra(ProQuest: ... denotes formulae omitted.)Our Philosophy and ApproachBuilding on two decades of prior work, the Focus on Mathematics (FoM] Math and Science Partnership program (MSP) has, over the last decade, developed and refined a distinctive framework for a mathematics-centered approach to developing teacher leaders, and it has built a mathematical community based on that framework. The FoM approach involves teachers, mathematicians, and educators working together in professional development activities. The common thread running through this tightly connected set of activities is an explicit focus on mathematical habits of mind.We define mathematical habits of mind (MHoM] to be the web of specialized ways of approaching mathematical problems and thinking about mathematical concepts that resemble the ways employed by mathematicians (Cuoco, Goldenberg, & Mark, 1997, 2010; Goldenberg, Mark, & Cuoco, 2010; Mark, Cuoco, Goldenberg, & Sword, 2010}. These habits are not about particular definitions, theorems, or algorithms that one might find in a textbook; instead, they are about the thinking, mental habits, and research techniques that mathematicians employ to develop such definitions, theorems, or algorithms. Some examples of MHoM follow:* Discovering the structure that is not apparent at first by experimenting and seeking regularity and/or coherence.* Choosing a useful representation-or purposefully toggling among various representations-of a mathematical concept or object.* Purposefully transforming and/or interpreting algebraic expressions (e.g., rewriting x^sup 2^ -6.x+ 10 as (;t-3)^sup 2^ + l to reveal its minimum value].* Using mathematical language to express ideas, assumptions, observations, definitions, or conjectures.Our work over the past decade has convinced us of the importance of MHoM for students and for teachers of mathematics, particularly at the secondary level. These habits foster the development and use of general purpose tools that make connections among various topics and techniques of secondary school mathematics content; they can bring parsimony, focus, and coherence to teachers' mathematical thinking and, in turn, to their work with students. In this sense, we envision MHoM as a critical component of mathematical knowledge for teaching (Hill, Rowan & Ball, 2005) at the secondary level (i.e., the knowledge necessary to carry out the work of teaching mathematics).We begin this paper by describing the mathematical community that we have built and the impact that it has had on our teachers, in particular, the impact on teachers' mathematical understanding and instructional practices. Then we discuss the research that grew out of our desire to study scientifically how MHoM might be an indicator of teacher effectiveness. …