The focus of NCTM's 2001 Yearbook is one aspect of how students learn mathematics–in particular, how they learn to form abstractions and build mathematical representations of phenomena. It explains how students come to develop mathematical habits of mind, and even how they develop misunderstandings about mathematics. Discussions on the nature and roles of representation, the use of representation tools for gaining insights, various symbol systems used in mathematics, and the role of context in the interplay between modeling and representation make this an excellent resource for teachers who want to make smart decisions about content and pedagogy.
pt. 1. Roles for representations. Systems of representations and the development of mathematical concepts / Gerald Goldin, Nina Shteingold Representation and abstraction in young children's numerical reasoning / Constance Kamii, Lynn Kirkland, Barbara A. Lewis Algebra : what are we teaching / Mark Saul Attending to transparent features of opaque representations of natural numbers / Rina Zazkis, Karen Gadowsky pt. 2. Tools for thinking. The influence of language on mathematical representations / Irene T. Miura The role of tools in supporting students' development measuring conceptions / Michelle Stephan ... [et al.] Promoting the use of diagrams as tools for thinking / Carmel Diezmann, Lyn English Constructing a foundation for the fundamental theorem of calculus / Mary J. Schnepp, Ricardo Nemirovsky Mapping diagrams : another view of functions / Mark Bridger, Maxine Bridger A multirepresentational journey through the law of cosines / Daniel Scher Representations of reversal : an exploration of Simpson's paradox / Lawrence Mark Lesser pt. 3. Symbols and symbol systems. Presenting and representing : from fractions to rational numbers / Susan J. Lamon Representations of patterns and functions : tools for learning / Wendy N. Coulombe, Sarah B. Berenson Promoting multiple representations in algebra / Alex Friedlander, Michal Tabach From Fibonacci numbers to fractals : recursive patterns and subscript notation / Deborah S. Franzblau, Lisa B. Warner Tracing the origins and extensions of Michael's representation / Regina D. Kiczek, Carolyn A. Maher, Robert Speiser pt. 4. The role of context. Listen to their pictures : an investigation of children's mathematical drawings / Kristine Reed Woleck Using literature to invite mathematical representations / Phyllis Whitin, David Whitin Representation in realistic mathematics education / Margaret R. Meyer Charting a visual course to the concept of function / Michal Yerushalmy, Beba Shternberg Teaching mathematical modeling and the skills of representation / Joshua Paul Abrams.