Carraher & Schliemann (2007)

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Early Algebra and Algebraic Reasoning

Outline of Headings

  • Why Algebraic Reasoning?
  • Rationale and Structure of Chapter
    • The Recent Focus on Algebraic Reasoning in the Early Grades
    • A Decisive Moment
      • Event 1: NCTM's Endorsement
      • Event 2: The RAND Mathematics Study Panel Report
    • Five Key Issues
      • Issue 1: The Relations Between Arithmetic and Algebra
      • Issue 2: Process versus Object
      • Issue 3: The Referential Role of Algebra
      • Issue 4: Symbolic Representation (narrowly defined)
      • Issue 5: Symbolic Representation (broadly defined)
  • A Traveler's Guide to Early Algebra
    • School Algebra and EA
    • EA Versus Pre-Algebra
      • Pre-Algebra Approaches
      • EA Approaches
      • On the Possibilities of EA
      • Parsing (Early) Algebra
    • Algebra Is Latent in the Existing Early Mathematics Curriculum
  • Arithmetic and Numerical Reasoning as an Entry Point Into EA
    • The Field Axioms and Other Properties of Numbers
    • Studies That Introduce Algebra Through Generalizations About Numbers
    • Quasi-Variables
    • Summary: Numerical Reasoning and EA
  • Arithmetic and Quantitative Reasoning as an Entry Point Into EA
    • Quantities, Measures, and Magnitudes
    • Quantitative Thinking and Number Lines
    • Can Students Apply Other Properties of Arithmetic to Magnitudes?
    • Referent-Transforming Properties
    • EA Studies that Focus on Magnitudes and Measures
    • Why Quantitative Thinking Is Unavoidable in EA
    • The Davydov Approach to EA
    • The Measure Up Project
    • Summary: Quantitative Reasoning and EA
  • Arithmetic and Functions as an Entry Point Into EA
    • Can Young Students Reason with Functions?
    • Functions As Rules for Generating Collections of Figures
    • Functions Expressed Through Multiple Representations: The Early Algebra, Early Arithmetic Project
    • Summary: What Can Young Student Learn About Functions?
  • Concluding Thoughts
    • What Kinds of Representations Express Algebraic Ideas?
    • Patterns and Functions
      • Issues Common to Patterns and Tables
    • Is a Scalar Approach Valid?
    • What Goals Are Achievable in the Short Term, Mid Term, and Long Term (for Students and Teachers)?


Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 669–705). Charlotte, NC: Information Age.
address = {Charlotte, NC},
author = {Carraher, David W. and Schliemann, Anal\'{u}cia D.},
booktitle = {Second handbook of research on mathematics teaching and learning},
chapter = {15},
editor = {Lester, Frank K.},
pages = {669--705},
publisher = {Information Age},
title = {{Early algebra and algebraic reasoning}},
year = {2007}