Teaching Fractions And Ratios For Understanding: Essential Content Knowledge And Instructional Strategies for Teachers - 2nd edition

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 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 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. 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 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 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. Book Cover......Page 1 Title......Page 4 Copyright......Page 5 Contents......Page 8 Preface......Page 14 1 Fractions and Proportional Reasoning: An Overview......Page 18 2 Fractions and Rational Numbers......Page 32 3 Relative and Absolute Thinking......Page 46 4 Measurement......Page 56 5 Quantities and Covariation......Page 66 6 Reasoning Up and Down......Page 81 7 Unitizing......Page 94 8 Sharing and Comparing......Page 105 9 Proportional Reasoning......Page 116 10 Reasoning With Fractions......Page 129 11 Part–Whole Comparisons With Unitizing......Page 141 12 Partitioning and Quotients......Page 155 13 Rational Numbers as Operators......Page 167 14 Rational Numbers as Measures......Page 186 15 Ratios and Rates......Page 198 16 Distance-Rate-Time Relationships......Page 219 17 Similarity and Percents......Page 229 18 Changing Fraction Instruction......Page 242 Index......Page 250 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 underst Presents a resource that integrates mathematics content, student thinking, pedagogy, activities, and assessment in the realm of rational numbers, including number sense, percentages, fractions, scale factors, similiarity, line graphs, ratio and proportion, and other related topics Like the first edition, this book provides curriculum materials that cross traditional boundaries to include many elements that are integrated in the teaching/learning enterprise: mathematics content, teacher understanding, student thinking, etc.