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

by Lamon, Susan J.

• ISBN13: 9780805852103
• ISBN10: 0805852107
• Edition: 2nd
• Format: Paperback
• Copyright: 2005-10-26
• Publisher: Lawrence Erlbau

Summary

This popular text, Teaching Fractions and Ratios for Understanding, 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 edition include: *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 emphasize 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.

Table of Contents

Preface	xiii
Fractions and Proportional Reasoning: An Overview	1	(14)
Introduction	3	(1)
The Constant of Proportionality	4	(1)
Reasoning: Beyond Mechanization	4	(1)
Invariance and Covariance	5	(2)
Multiplicative Thinking and Preserving Ratios	7	(1)
Critical Components of Powerful Reasoning	8	(1)
Changing Instruction	9	(1)
Getting Started	10	(3)
Analyzing Children's Thinking	13	(1)
Activities	14	(1)
Reflection	14	(1)
In the Classroom	14	(1)
Fractions and Rational Numbers	15	(14)
New Units and a New Notational System	16	(1)
The Psychology of Units	17	(1)
New Operations and Quantities	18	(1)
Interference of Whole Number Ideas	19	(1)
Problems With Terminology	20	(1)
What Are Fractions?	21	(1)
Rational Numbers	22	(1)
Fractions as Numbers	22	(1)
Many Sources of Meaning	23	(1)
Some Interpretations of the Fraction 3/4	24	(3)
Activities	27	(1)
Reflection	27	(1)
In the Classroom	28	(1)
Relative and Absolute Thinking	29	(10)
Two Perspectives on Change	30	(2)
Relative Thinking and Understanding Fractions	32	(1)
Encouraging Multiplicative Thinking	32	(3)
Activities	35	(3)
Reflection	38	(1)
In the Classroom	38	(1)
Measurement	39	(10)
The Importance of Measurement	40	(1)
Some Principles of Measurement	41	(1)
Measuring More Abstract Characteristics	42	(2)
Children's Thinking	44	(1)
Activities	45	(3)
Reflection	48	(1)
In the Classroom	48	(1)
Quantities and Covariation	49	(15)
Building on Children's Informal Knowledge	50	(1)
Quantities Not Quantified	50	(1)
Quantifiable Characteristics	51	(2)
Discussing Proportional Relationships in Pictures	53	(2)
Visualizing, Verbalizing, and Symbolizing Changing Relationships	55	(1)
Covariation and Invariance	56	(3)
Activities	59	(4)
Reflection	63	(1)
In the Classroom	63	(1)
Reasoning Up and Down	64	(13)
The Unit	65	(2)
Units Defined Implicitly	67	(1)
Using Units of Various Types	68	(1)
Reasoning Up and Down	69	(2)
Activities	71	(4)
Reflection	75	(1)
In the Classroom	76	(1)
Unitizing	77	(11)
Units and Unitizing	78	(1)
Notation	79	(1)
Flexibility in Unitizing	80	(2)
Classroom Activities to Encourage Unitizing	82	(1)
Visual Activities	82	(2)
Activities	84	(3)
Reflection	87	(1)
In the Classroom	87	(1)
Sharing and Comparing	88	(11)
Rational Numbers as Quotients	89	(1)
Partitioning as Fair Sharing	90	(1)
Early Partitioning Activities	91	(1)
More and How Much More	92	(1)
The Richness of Partitioning Activities	93	(1)
Activities	94	(4)
Reflection	98	(1)
In the Classroom	98	(1)
Proportional Reasoning	99	(13)
More Reasoning Up and Down	100	(1)
Problem Types	101	(2)
Ratio Tables	103	(2)
Increasing the Difficulty	105	(1)
Analyzing Relationships	106	(2)
Characteristics of Proportional Thinkers	108	(1)
Activities	109	(2)
Reflection	111	(1)
In the Classroom	111	(1)
Reasoning with Fractions	112	(12)
Comparing Fractions	113	(4)
Fractions in Between	117	(2)
Other Useful Ways of Thinking	119	(2)
Activities	121	(2)
Reflection	123	(1)
In the Classroom	123	(1)
Part--Whole Comparisons with Unitizing	124	(14)
Part--Whole Fractions	125	(1)
Equivalence	125	(2)
The Big Ideas	127	(3)
Comparing Part--Whole Fractions	130	(2)
Multiplication and Division	132	(2)
Activities	134	(2)
Reflection	136	(1)
In the Classroom	137	(1)
Partitioning and Quotients	138	(12)
Rational Numbers as Quotients	139	(1)
Children's Partitioning	140	(3)
Equivalence	143	(1)
Ratios and Rates	144	(2)
Addition and Subtraction	146	(1)
Activities	147	(2)
Reflection	149	(1)
In the Classroom	149	(1)
Rational Numbers as Operators	150	(19)
Operators	151	(2)
Exchange Models	153	(3)
Composition	156	(2)
Area Model for Multiplication	158	(2)
Area Model for Division	160	(2)
Compositions and Paper Folding	162	(1)
Understanding Operators	163	(1)
Activities	164	(4)
Reflection	168	(1)
In the Classroom	168	(1)
Rational Numbers as Measures	169	(12)
Measures of Distance	170	(1)
Static and Dynamic Measurement	170	(2)
The Goals of Successive Partitioning	172	(1)
Understanding Rationals As Measures	173	(1)
Units, Equivalent Fractions, and Comparisons	174	(2)
Addition and Subtraction	176	(1)
Activities	177	(3)
Reflection	180	(1)
In the Classroom	180	(1)
Ratios and Rates	181	(21)
What Is a Ratio?	182	(1)
Notation and Terminology	183	(2)
Equivalence and Comparison of Ratios	185	(2)
Duals and Ratio Arithmetic	187	(2)
Ratios as an Instructional Task	189	(3)
What Is a Rate?	192	(2)
Equivalence Classes	194	(2)
Comparing Ratios and Rates	196	(2)
Activities	198	(3)
Reflection	201	(1)
In the Classroom	201	(1)
Distance-Rate-Time Relationships	202	(10)
Speed: The Most Important Rate	203	(1)
Characteristics of Speed	204	(1)
Students' Biggest Problems with Speed	205	(1)
Average Speed	206	(2)
Distance-Speed-Time and Graphs	208	(1)
Activities	209	(2)
Reflection	211	(1)
In the Classroom	211	(1)
Similarity and Percents	212	(13)
Similarity	213	(1)
Scale Factors	214	(1)
Similarity and Equivalence Classes	215	(2)
Percent	217	(1)
Percents as an Instructional Task	217	(1)
Operating With Percents	218	(1)
Activities	219	(5)
Reflection	224	(1)
In the Classroom	224	(1)
Changing Fraction Instruction	225	(8)
A Summary of Rational Number Interpretations	225	(5)
Central Structures	230	(1)
Conclusions	231	(1)
Ideas for Changing Fraction Instruction	231	(2)
Index	233