4th Edition

Teaching Fractions and Ratios for Understanding Essential Content Knowledge and Instructional Strategies for Teachers

By Susan J. Lamon Copyright 2020
    294 Pages
    by Routledge

    294 Pages
    by Routledge

    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.


    1. Proportional Reasoning: An Overview

    Student Strategies


    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


    2. Fractions and Rational Numbers

    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 3/4


    3. Relative Thinking and Measurement

    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


    4. Quantities and Covariation

    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


    Indirect Measurement

    Testing for Similarity

    Mockups and Pudgy People


    5. Proportional Reasoning

    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


    Percentages as an Instructional Task

    Reasoning with Percentages


    6. Reasoning with Fractions

    Student Strategies

    Visualizing Operations

    Equivalent Fractions and Unitizing

    Comparing Fractions

    Fractions in Between


    7. Fractions as Part–Whole Comparisons

    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


    Partitive and Quotative Division


    Other Rational Number Interpretations


    8. Fractions as Quotients

    Student Strategies


    Partitioning as Fair Sharing

    Partitioning Activities

    Children’s Partitioning


    Simplifying Fractions

    Understanding Fractions as Quotients

    More Advanced Reasoning

    Sharing Different Pizzas


    9. Fractions as Operators

    Student Strategies


    Exchange Models


    Area Model for Multiplication

    Area Model for Division

    Compositions and Paper Folding

    Understanding Operators


    10. Fractions as Measures

    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


    11. Ratios and Rates

    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


    12. 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


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