At different points in our careers and in differ- ent ways, we realized that there must be a balance between personal representations and con- ventional representations. This stance gradu- ally became a part of our philosophies and pedagogies. We since have found that focus- ing on the meaning that students make of the mathematics they do and the representations they use is more revealing of their understanding than is looking for our representations in their work. A Notion of Representation Many of us assume that children’s representations are little more than their haphazard recording of mathe- matical work. However, in Principles and Standards for School Mathematics, the National Council of Teachers of Mathematics (2000) suggests that “teach- ers can gain valuable insights into students’ ways of interpreting and thinking about mathematics by look- ing at their representations” (p. 68). This view implies that children’s representations provide clues to the ways that children make sense of the mathematics that they are learning. Holding this view of represen- tations also helps us see representations as paths into dialogues with children about their mathematical thinking; it also helps us view these representations as “bridges . . . to more conventional ones, when appro- priate” (NCTM 2000, p. 68). 196 TEACHING CHILDREN MATHEMATICS It’s Not Just Notation: Jill A. Perry and Sandra L. Atkins Jill Perry, perry@rowan.edu, teaches general and mathematics pedagogy courses to preservice teachers at Rowan University in Glassboro, New Jersey. She is interested in the connections among mathematics, language, literacy, representation, and communication. Sandra Atkins, sandra_atkins@mcgraw-hill.com, is a senior mathematics consultant for Wright Group/ McGraw-Hill in Bothell, Washington. Her interests are in mathematical communication and the relationship of mathematical reasoning to representations. Valuing Children’s Representations M athematics teachers often say to their students, “Show your work,” when what they really mean is, “Show this to me as I showed it to you” or “Show my work.” During our first few years as mathematics teachers, we spent much of our time trying to make our students think as we thought—to use the symbols that we were using in the ways in which we were using them. We find it uncomfortable to admit that during those years we spent little time trying to find out what our students were thinking. We valued the process and product looking like ours more than the process and product being theirs. We expected our students to use conventional representations before we had given them sufficient time to develop mathematical concepts. Copyright © 2002 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.