The Roles of Representation in School Mathematics: 2001 Yearbook (Yearbook (National Council of Teachers of Mathematics))

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. Educators And Scholars Discuss The Roles Of Invented And Presented Representations, The Kind The Child Creates Versus The Kind The Teacher Gives To Students, As An Aspect Of The Internalization Of Mathematical Reasoning. Representations Are The Devices Which People Use To Help Them Gain Insights, And By Examining The Interplay Between Modeling And Representation, The Roles For Representation Can Be Defined. 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 [and Others] --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. Albert A. Cuoco, Frances R. Curcio. Includes Bibliographical References. Contents......Page 2 Preface......Page 6 1. Systems of Representations and the Development of Mathematical Concepts......Page 11 2. Representation and Abstraction in Young Children’s Numerical Reasoning......Page 34 3. Algebra: What Are We Teaching?.......Page 45 4. Attending to Transparent Features of Opaque Representations of Natural Numbers......Page 54 5. The Influence of Language on Mathematical Representations......Page 63 6. The Role of Tools in Supporting Students’ Development of Measuring Conceptions......Page 73 7. Promoting the Use of Diagrams as Tools for Thinking......Page 87 8. Constructing a Foundation for the Fundamental Theorem of Calculus......Page 100 9. Mapping Diagrams: Another View of Functions......Page 113 10. A Multirepresentational Journey through the Law of Cosines......Page 127 11. Representations of Reversal: An Exploration of Simpson’s Paradox......Page 139 12. Presenting and Representing: From Fractions to Rational Numbers......Page 156 13. Representations of Patterns and Functions: Tools for Learning......Page 176 14. Promoting Multiple Representations in Algebra......Page 183 15. From Fibonacci Numbers to Fractals: Recursive Patterns and Subscript Notation......Page 196 16. Tracing the Origins and Extensions of Michael’s Representation......Page 211 17. Listen to Their Pictures: An Investigation of Children’s Mathematical Drawings......Page 225 18. Using Literature to Invite Mathematical Representations......Page 238 19. Representation in Realistic Mathematics Education......Page 248 20. Charting a Visual Course to the Concept of Function......Page 261 21. Teaching Mathematical Modeling and the Skills of Representation......Page 279