Teaching Fractions and Ratios for Understanding

Essential Content Knowledge and Instructional Strategies for Teachers

Susan J. Lamon

• Publisher: Routledge
• Year: 2020
• Print ISBN: 9780367441647
• eBook ISBN: 9781000037333
• Edition: 4
• Topic: Education
• Subtopic: Education General

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.

Chapter 1

Proportional Reasoning: An Overview

Student Strategies: Grade 6

Some middle school children discussed the tree house problem and the responses from students A, B, C, D, and E are given here.

First, solve the problem yourself and explain your reasoning to someone else, then rank the student responses according to the sophistication of their mathematical reasoning.

These people disagree on the height of the tree house. How high do you think it is? Explain your reasoning.

Introduction

For too long, proportional reasoning has been an umbrella term, a catch-all phrase that refers to a certain facility with rational number concepts and contexts. The term is ill-defined and researchers have been better at determining when a student or an adult does not reason proportionally rather than defining the characteristics of one who does. Without appropriate instructional goals, purposeful teaching of the topic was impossible, and proportional reasoning remained an elusive by-product of instruction in fractions. Because elementary and middle school mathematics curricula provide no more than a cursory treatment of rational number ideas, the emergence of proportional reasoning is left to chance. Yet, the fact that most adults do not reason proportionally—my estimate well exceeds 90%—presents compelling evidence that this reasoning process entails more than developmental processes and that instruction must play an active role in its emergence.

Proportional reasoning is one of the best indicators that a student has attained understanding of rational numbers and related multiplicative concepts. While, on one hand, it is a measure of one’s understanding of elementary mathematical ideas, it is, on the other, part of the foundation for more complex concepts. For this reason, I find it useful to distinguish proportional reasoning from the larger, more encompassing, concept of proportionality. Proportionality plays a role in applications dominated by physical principles—topics such as mechanical advantage, force, the physics of lenses, the physics of sound, just to name a few. Proportional reasoning, as this book uses the term, is a prerequisite for understanding contexts and applications based on proportionality.

Clearly, many people who have not developed their proportional reasoning ability have been able to compensate by using rules in algebra, geometry, and trigonometry courses, but, in the end, the rules are a poor substitute for sense-making. They are unprepared for real applications in statistics, biology, geography, or physics, where important, foundational principles rely on proportionality. This is unfortunate at a time when an ever-increasing number of professions rely on mathematics directly or use mathematical modeling to increase efficiency, to save lives, to save money, or to make important decisions.

For the purposes of this book, proportional reasoning will refer to the ability to scale up and down in appropriate situations and to supply justifications for assertions made about relationships in situations involving simple direct proportions and inverse proportions. In colloquial terms, proportional reasoning is reasoning up and down in situations in which there exists an invariant (constant or unchanging) relationship between two quantities that are linked and varying together. As the word reasoning implies, it requires argumentation and explanation beyond the use of symbols ab=cd⋅

In this chapter, we will examine some problems to get a sense of what it means to reason proportionally. We will also look at a framework that was used to facilitate proportional reasoning in four-year longitudinal studies with children from the time they began fraction instruction in grade 3 until they finished grade 6.

Try not to be put off by mathematical language that you will meet in this chapter. In order to be able to talk about proportional reasoning and its associated topics, we need words and symbols, and we are simply not at liberty to change standard mathematical terminology. Don’t expect total understanding right away. This chapter is intended as an overview of the many essential learning experiences and contexts that contribute to proportional reasoning. Everything will make more sense to you as you work through upcoming chapters.

The Constant of Proportionality

The mathematical model for directly proportional relationships is a linear function of the form y = kx, where k is called the constant of proportionality. Thus, y is a constant multiple of x. Equivalently, two quantities are proportional when they vary in such a way that they maintain a constant ratio: yx=k. The constant k plays an essential role in understanding inversely proportional relationships as well. In the mathematical model for inverse proportions, k = xy. In spite of its importance, k is sorely neglected in instruction.

Pedagogically speaking, k is a slippery character, because it changes its guise in each part...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• Chapter 1 Proportional Reasoning: An Overview
• Chapter 2 Fractions and Rational Numbers
• Chapter 3 Relative Thinking and Measurement
• Chapter 4 Quantities and Covariation
• Chapter 5 Proportional Reasoning
• Chapter 6 Reasoning with Fractions
• Chapter 7 Fractions as Part–Whole Comparisons
• Chapter 8 Fractions as Quotients
• Chapter 9 Fractions as Operators
• Chapter 10 Fractions as Measures
• Chapter 11 Ratios and Rates
• Chapter 12 Changing Instruction
• Index