Carraher & Schliemann (2007)

Early Algebra and Algebraic Reasoning

• Authors: David W. Carraher and Analúcia D. Schliemann
• Book: Second Handbook of Research on Mathematics Teaching and Learning
• Year: 2007
• Source: http://www.infoagepub.com/products/Second-Handbook-Research-Mathematics-Teaching-Learning

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)?

Corrolary

APA

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.

BibTeX

@incollection{Carraher2007,
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}
}