When the world is not the problem: Real-world contexts in analogies

This paper considers real-world contexts in the teaching of mathematics. An in-depth qualitative analysis of mathematics lessons in two urban high schools explores how real-world contexts serve as sources of analogies for introducing mathematical concepts and procedures. This study identifies and describes the effective discursive practice of elaboration which supports and develops analogies. Elaboration and analogies offer additional means for teachers to "center" (Tate, 2005) instruction on urban students' lived experiences. INTRODUCTION Our focus on teachers' use of real-world contexts emerges from a research and professional development project on culturally relevant mathematics pedagogy (CureMap). CureMap consists of three dimensions: teaching mathematics for understanding; centering instruction on students, their experiences, and their communities; and creating opportunities for students to think critically about and with mathematics (Rubel & Chu, 2012). Real-world contexts (RWCs), defined as objects, situations, or practices that exist independent of the mathematics lesson, span the CureMap framework in multiple ways. For instance, teaching mathematics for understanding includes making explicit connections between school mathematics and out of school phenomena. Similarly, centering instruction on students and their experiences invites the inclusion of RWCs that are local and/or relevant to students. Developing students' ability to be critical with mathematics suggests using mathematics as a lens through which to view and analyze RWCs. The correspondence between problem solving in school and mathematical thinking about a RWC is not always direct. For instance, individuals can be successful in solving mathematical problems outside of school, in well-defined, situated contexts, but then have difficulty with similar, in-school tasks (e.g., Noss, Hoyles, & Pozzi, 2002; Saxe, 1988). Further, when solving problems based on RWCs in school, students frequently do not take realistic considerations into account (e.g., Carpenter, Lindquist, Matthews, & Silver, 1983). For instance, when asked how many two-foot boards can be cut from two five-foot boards, a common response is five, with the reasoning that ten divided by two is five (Verschaffel, De Corte, & Lasure, 1994).