# Carraher & Schliemann (2007)

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*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} }