The NCTM's Standards (2000) suggest that a representation is not only a product (a picture, a graph, a number, or a symbolic expression) but also a process, a vehicle for developing an understanding of a mathematical concept and communicating about mathematics. To serve as a vehicle in learning and communication, however, a representation must be personally relevant and meaningful to a student. Therefore, when choosing a representation to explore with a group of students or when reviewing student work, we ought to consider everything that students bring to the classroom. Even at a young age, students come to school with their own, often culturally influenced, valid representations (Lave 1998). Because those representations have been crafted, interpreted, and modified by the students themselves, they become vital to classroom instruction. To dismiss what students bring naturally to the classroom reduces mathematics to a one-way transaction between teacher as expert and student as novice, confirming the notion that a student's own thinking and all that he or she brings to mathematics is marginal at best. By relocating student-generated representations to the center of the instruction, the nature of how students experience mathematics changes dramatically. It reconsiders mathematics as a vibrant dialogue among different but equally valued thinkers. This deliberate approach to the teaching of mathematics, we believe, becomes vital if we are serious about creating greater equity for our students.
Kara Imm is also a doctoral student in Urban Education at the Graduate Center, City University of New York.
Stylianou advises graduate-student researchers Imm and Chae.
The three authors are colleagues at the Urban Math Project and are grateful to the children of Greenwich Village Middle School, a public middle school in lower Manhattan, New York, for allowing them to share their work.