Intro -- Foreword -- Introduction—The Challenge of Fractions -- Appreciate the Fraction Challenge -- From Natural Numbers to Real Numbers -- A Word on the Word Fraction -- Cognitive Shifts to Consider -- The Rush to Algorithms -- What Can You Expect from This Book? -- CHAPTER 1—Convey the Many Meanings of a/b -- Roberto’s Story -- Recognizing Misconceptions -- Limited Ideas About the Meaning of a Fraction -- Difficulty Conceptualizing a Fraction as a Single Number -- Unpacking the Mathematical Thinking -- The Part-Whole Meaning of a/b -- The Measure Meaning of a/b -- The Quotient Meaning of a/b -- The Ratio Meaning of a/b -- The Multiplicative Operator Meaning of a/b -- The Rational Number Meaning of a/b Embodied by the Number Line -- Targeting Misconceptions with Challenging Problems -- CHAPTER 2—Use Visual and Tactile Models -- Maya’s Story -- Beyond Misconceptions of Fractions -- Limited Repertoire of Fraction Models -- Lack of Connectedness Among Models -- Unpacking the Mathematical Thinking -- Continuous Models -- Discrete Models -- Discussing and Connecting Models -- Targeting Misconceptions with Challenging Problems -- Maya’s Story, Part 2 -- Recognizing Misconceptions -- The Parts Need Not Be Equal -- The Parts Must Be Clearly Delineated -- The Parts Must Have the Same Shape -- The Shaded Regions Must Be Grouped into One Part -- Unpacking the Mathematical Thinking -- The Importance of Equal Parts -- Area, Not Shape, Is the Focus -- Targeting Misconceptions with Challenging Problems -- Maya’s Story: Epilogue -- CHAPTER 3—Focus on the Unit -- Ed’s Story -- Recognizing Misconceptions -- The Whole Is Made of One Piece -- A Fraction Is Smaller Than the Whole, the Unit, or the “1” -- Difficulty Conceiving of or Writing Fractions Greater Than 1 -- Limited Experience with Non-continuous Units -- Unpacking the Mathematical Thinking -- The Unit Is Defining -- Working with a Variety of Units -- Revisiting the Partition and Iteration Process -- Targeting Misconceptions with Challenging Problems -- Two Vignettes -- Linda’s Story -- Jason’s Story -- Recognizing Misconceptions -- Difficulty Going from Part to Whole -- Difficulty Discriminating Between What Is Relevant and What Is Not -- Unpacking the Mathematical Thinking -- A Fraction Is a Relation Between Two Quantities -- Proceeding from Part to Whole -- Infusing Problems with Distractors: Trapping or Stimulating Students? -- Targeting Misconceptions with Challenging Problems -- A Final Note -- CHAPTER 4—Teach the Concept of Equivalence (Not Just the Rule) -- Lisa’s Story -- Recognizing Misconceptions -- Different Fraction Names for the Same Quantity or Number -- Overreliance on Physical Models (3rd Grade and Up) -- Difficulty with Discrete Quantities (3rd Grade and Up) -- Limited Concept of the Equals Sign (4th Grade and Up) -- Rote Application of a/b = (n × a)/(n × b) (4th Grade and Up) -- The Misuse of Language (All Grades) -- A Partial View of the EFA (5th Grade and Up) -- Additive Thinking -- Un.
packing the Mathematical Thinking -- Build on Students’ Informal Experiences with Equivalence -- Cultivate the Equivalence Meaning of Equality -- Explain Equivalence by Connecting Fractions to Multiplication and Division -- Begin with Equal-Sharing Problem Situations -- Model Equivalence Using Different Interpretations of Fractions -- Be Mindful That Models Lead to Concept Building -- Targeting Misconceptions with Challenging Problems -- CHAPTER 5—Compare and Order Fractions Meaningfully -- Nicole’s Story -- Recognizing Misconceptions -- Overreliance on Ready-Made Models -- Difficulty Comparing Fractions Without the Common Algorithm -- Lack of Attention to the Unit -- Inappropriate Whole-Number Reasoning -- Predominance of Additive Thinking -- Unpacking the Mathematical Thinking -- Using Models -- Reasoning with Unit Fractions -- Using the Concept of Equivalence (Common Denominators or Numerators) -- Comparing to Benchmarks -- Using Multiplicative Thinking -- Noticing Patterns -- Looking Ahead: Visualizing the “Cross-Product” Method -- Targeting Misconceptions with Challenging Problems -- CHAPTER 6—Let Algorithms Emerge Naturally -- Vignette 1: Division of a Whole Number by a Fraction -- Vignette 2: Multiplication of a Whole Number by a Fraction -- Recognizing Misconceptions -- Difficulty Seeing Fractions as Numbers -- Rote or Incorrect Application of Algorithms -- Knowing Fractions Means Knowing the Algorithms -- Lack of Fraction Operation Sense -- False Beliefs About the Effects of Operations on Numbers or Quantities -- Lack of Attention to the Unit -- Unpacking the Mathematical Thinking -- Begin with Problem Situations That Students Can Tackle -- Allow Students to Devise Their Own Algorithms -- Revisit Meanings of Addition and Subtraction -- Revisit Meanings of Multiplication and Division -- Emphasize That Relationships and Properties Still Hold -- Highlight Important Changes in Ways of Thinking -- Targeting Misconceptions with Challenging Problems -- CHAPTER 7—Connect Fractions and Decimals -- Denis’s Story -- Recognizing Misconceptions -- Scarce Contact with Decimals in Daily Life -- Lack of Connectedness Between Fractions and Decimals -- Difficulty with Symbol Meaning -- Overreliance on the Money Model -- Poor Understanding of Decimal Magnitude -- Rote or Incorrect Application of Decimal Algorithms -- Unpacking the Mathematical Thinking -- Extending Place Value to Tenths and Hundredths -- The Models We Use Are Important -- Comparing Decimals Meaningfully -- Importance of the Unit -- Sensing Approximate Values -- Making Sense of Operations -- Targeting Misconceptions with Challenging Problems -- CONCLUSION—Moving from Rote to Reason -- Foster These Seven Habits of Mind -- Teach Meanings First, Algorithms Last -- Look Ahead to Ratios, Proportions, Proportional Relations, and Linear Functions -- From Fractions to Ratios -- From Ratios to Proportions -- From Proportions to Proportional Relationships -- From Proportional Relationships to Linear .
Functions -- Concluding Thoughts -- References -- Index -- About the Author -- .
