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

ISBN-10: 0415886120

ISBN-13: 9780415886123

Edition: 3rd 2012 (Revised)

Authors: Susan J. Lamon

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Description:

This popular text addresses the urgent need for curriculum materials that cross traditional boundaries to include many of the elements that are integrated in the teaching/learning enterprise: mathematics content, teacher understanding, student thinking, teaching methods, instructional activities, and assessment. The book pushes readers beyond the limits of their current understanding of rational numbers, challenging them to refine and explain their thinking--without falling back on rules and procedures they have relied on throughout their lives. Written in a conversational and easy to understand style, this is not a textbook as much as it is a resource book. An underlying assumption is that facilitating teacher understanding using the same questions and activities that may be used with children is one way to help teachers build the comfort and confidence they need to begin talking to children about complex ideas. Unlike a textbook that is used to study formal theory and then discarded when it comes to putting ideas into practice, the many problems and activities included to facilitate teacher learning are valuable resources for use in elementary and middle school classrooms. Changes in the second editioninclude: *even more student work incorporated in every chapter; *discussion of the connectivity between the topics addressed in the book and the elementary and middle school mathematics curricula; *an increased emphasis on measurement; *expansion of some topics, including number sense, percent, scale factors, similarity, and linear graphs; *clarification of the characteristics of ratio and proportions and how to use these to generate discussion with children; and *content-related interview questions for exploring children's thinking. This book is a valuable resource for researchers and curriculum developers in mathematics education, pre-service and in-service teachers of mathematics, those involved in the mathematical and pedagogical preparation of mathematics teachers, and graduate students in mathematics education. The methods and activities it includes have been tested with students in grades 3-8 and with pre-service and in-service teachers and other adults. This text is accompanied by MORE--a supplement that is not merely an answer key but a resource that includes in-depth discussions of all the problems in the text; develops and extends discussion of the issues, teaching problems, and other considerations raised in the chapters; and contains additional problems--with and without solutions--that instructors may find helpful for assessment purposes.
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Book details

List price: $55.95
Edition: 3rd
Copyright year: 2012
Publisher: Routledge
Publication date: 10/19/2011
Binding: Paperback
Pages: 160
Size: 7.25" wide x 10.25" long x 0.50" tall
Weight: 1.364
Language: English

Preface
Proportional Reasoning: An Overview
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
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 �
Activities
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
Activities
Quantities and Covariation
Student Strategies
Building on Children's Informal Knowledge
Quantities Unqualified
Quantifiable Characteristics
Discussing Proportional Relationships in Pictures
Visualizing, Verbalizing, and Symbolizing Changing Relationships
Covariation and Invariance
Cuisenaire Strips
Scale Factors
Similarity
Indirect Measurement
Testing for Similarity
Mockups and Pudgy People
Activities
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
Percent
Percents as an Instructional Task
Reasoning with Percents
Activities
Reasoning with Fractions
Student Strategies
Visualizing Operations
Equivalent Fractions and Unitizing
Comparing Fractions
Fractions in Between
Activities
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
Multiplication
Partitive and Quotative Division
Division
Other Rational Number Interpretations
Activities
Fractions as Quotients
Student Strategies
Quotients
Partitioning as Fair Sharing
Partitioning Activities
Children's Partitioning
Equivalence
Should We Reduce?
Understanding Fractions as Quotients
More Advanced Reasoning
Sharing Different Pizzas
Activities
Fractions as Operators
Student Strategies
Operators
Exchange Models
Composition
Area Model for Multiplication
Area Model for Division
Compositions and Paper Folding
Understanding Operators
Activities
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
Activities
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
Activities
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
Index
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