Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers, 4th Edition (Paperback) book cover

Teaching Fractions and Ratios for Understanding

Essential Content Knowledge and Instructional Strategies for Teachers, 4th Edition

By Susan J. Lamon

Routledge

296 pages

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Paperback: 9780367441678
pub: 2020-03-31
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Description

Written in a user-friendly, conversational style, the fourth edition of this groundbreaking text helps pre-service and in-service mathematics teachers build the comfort and confidence they need to begin talking to children about fractions and ratios, distilling complex ideas and translating research into usable ideas for the classroom.

For two decades, Teaching Fractions and Ratios for Understanding has pushed readers beyond the limits of their current understanding of fractions and rational numbers, challenging them to refine and explain their thinking without falling back on rules and procedures they have relied on throughout their lives. All of the material offered in the book has been used with students, and is presented so that readers can see the brilliance of their insights as well as the issues that challenge their understanding. Each chapter includes children’s strategies and samples of student work for teacher analysis, as well as activities for practicing each thinking strategy, designed to be solved without rules or algorithms, using reasoning alone.

The fourth edition of this popular text has been updated throughout and includes new examples of student work, updated artwork, and more.

As with previous editions, an equally valuable component of this text is the companion book MORE! Teaching Fractions and Ratios for Understanding (2012), a supplement that is not merely an answer key but a resource that provides the scaffolding for the groundbreaking approach to fraction and ratio instruction explored here. MORE! includes in-depth discussions of selected problems in the main text, supplementary activities, Praxis preparation questions, more student work, and templates for key manipulatives.

Table of Contents

Preface

  1. Proportional Reasoning: An Overview
  2. Student Strategies

    Introduction

    The Constant of Proportionality

    Reasoning: Beyond Mechanization

    Invariance and Covariance

    Solving Proportions Using K

    Multiplicative Thinking

    Critical Components of Powerful Reasoning

    Getting Started

    Analyzing Children’s Thinking

    Activities

     

  3. Fractions and Rational Numbers
  4. Student Strategies

    New Units and a New Notational System

    The Psychology of Units

    New Operations and Quantities

    Interference of Whole Number Ideas

    Problems with Terminology

    Development of Sets of Numbers

    Kinds of Fractions

    What are Fractions?

    Rational Numbers

    Fractions as Numbers

    Fractions, Ratios and Rates

    Many Sources of Meaning

    Multiple Interpretations of the Fraction

    Activities

  5. Relative Thinking and Measurement
  6. Student Strategies

    Two Perspectives on Change

    Relative Thinking and Understanding Fractions

    Encouraging Multiplicative Thinking

    Two Meanings for "More"

    The Importance of Measurement

    The Compensatory Principle

    The Approximation Principle

    Recursive Partitioning Principle

    Measuring More Abstract Qualities

    Other Strategies

    Activities

  7. Quantities and Covariation
  8. Student Strategies

    Building on Children’s Informal Knowledge

    Quantities Unquantified

    Quantifiable Characteristics

    Discussing Proportional Relationships in Pictures

    Visualizing, Verbalizing, and Symbolizing Changing Relationships

    Covariation and Invariance

    Cuisenaire Strips

    Scale Factors

    Areas and Volumes of Scaled Figures

    Similarity

    Indirect Measurement

    Testing for Similarity

    Mockups and Pudgy People

    Activities

  9. Proportional Reasoning
  10. Student Strategies

    The Unit

    Units Defined Implicitly

    Using Units of Various Types

    Reasoning Up and Down

    Units and Unitizing

    Unitizing Notation

    Flexibility in Unitizing

    Children’s Thinking

    Classroom Activities to Encourage Unitizing

    Visual Activities

    Reasoning with Ratio Tables

    Problem Types

    Ratio Tables

    Increasing the Difficulty

    Analyzing Relationships

    Percent

    Percents as an Instructional Task

    Reasoning with Percents

    Activities

  11. Reasoning with Fractions
  12. Student Strategies

    Visualizing Operations

    Equivalent Fractions and Unitizing

    Comparing Fractions

    Fractions in Between

    Activities

  13. Fractions as Part-Whole Comparisons
  14. Student Strategies

    Part-Whole Fractions: The Big Ideas

    Unitizing and Equivalence

    Problems in Current Instruction

    Fraction Models

    Fraction Strips

    Comparing Part-Whole Fractions

    Discrete Units

    Multiplication

    Partitive and Quotative Division

    Division

    Other Rational Number Interpretations

    Activities

  15. Fractions as Quotients
  16. Student Strategies

    Quotients

    Partitioning as Fair Sharing

    Partitioning Activities

    Children’s Partitioning

    Equivalence

    Simplifying Fractions

    Understanding Fractions as Quotients

    More Advanced Reasoning

    Sharing Different Pizzas

    Activities

  17. Fractions as Operators
  18. Student Strategies

    Operators

    Exchange Models

    Composition

    Area Model for Multiplication

    Area Model for Division

    Compositions and Paper Folding

    Understanding Operators

    Activities

  19. Fractions as Measures
  20. Student Strategies

    Measures of Distance

    Static and Dynamic Measurement

    The Goals of Successive Partitioning

    Understanding Fractions as Measures

    Units, Equivalent Fractions, and Comparisons

    Fraction Operations

    Activities

  21. Ratios and Rates
  22. Student Strategies

    What is a Ratio?

    Notation and Terminology

    Equivalence and Comparison of Ratios

    Ratios as an Instructional Task

    What is a Rate?

    Operations with Rates and Ratios

    Linear Graphs

    Comparing Ratios and Rates Graphically

    Speed: The Most Important Rate

    Characteristics of Speed

    Students’ Misconceptions About Speed

    Average Speed

    Distance – Speed – Time and Graphs

    Activities

  23. Changing Instruction

Student Comments

Why Change?

A Summary of Fraction Interpretations

Central Structures

Characteristics of Proportional Thinkers

Obstacles to Change

Sequencing Topics

Directions for Change

Challenging Problems

 

 

Index

About the Author

Susan J. Lamon is Professor Emerita of Mathematics, Statistics, and Computer Science at Marquette University.

Subject Categories

BISAC Subject Codes/Headings:
EDU000000
EDUCATION / General
EDU010000
EDUCATION / Elementary
EDU029010
EDUCATION / Teaching Methods & Materials / Mathematics