Оглавление
Kimberly Morrow-Leong. Mathematize It! [Grades 6-8]
What Your Colleagues Are Saying …
Mathematize It! The Book at a Glance
Mathematize It!
Mathematize It!
Contents
List of Videos
Preface
How This Book Can Help You
How to Use This Book
Acknowledgments
Publisher’s Acknowledgments
About the Authors
CHAPTER ONE Introduction Why You Need to Teach Students to Mathematize
Problem-Solving Strategies Gone Wrong
What Is Mathematizing? Why Is It Important?
Focusing on Operation Sense
Using Mathematical Representations
Verbal
Contextual
Concrete
Pictorial
Symbolic
Teaching Students to Mathematize
Building Your Understanding of the Operations and Related Problem Situations
NOTES
Exploring in the Mathematizing Sandbox: A Problem-Solving Model
A Note About Negative Values
Final Words Before You Dive In
Descriptions of Images and Figures
CHAPTER TWO Add-To and Take-From. Thinking About Active Addition and Subtraction
Addition and Subtraction Problem Situations
Sandbox Notes
Students and Teachers Think About the Problems
Finding the Unknown, Three Story Structures
Result Unknown
Change Unknown
Start Unknown
Story Structures: Implications for Teaching
Focus on Take-From Situations
Complicating Things: The Start Unknown Variation
Modeling
Trying a Simpler Case
Distinguishing Computation, Operation, and Operation Sense
Teaching Students to Use Concrete and Pictorial Models
Extending to Negative Values
Students and Teachers Think About the Problems
Reasoning and Notating With Negative Values
Conceptual Models of Integer Contexts
Translation
Relativity
Bookkeeping
Counterbalance
The Importance of Clear Notation
Adjectives and Verbs Are Not the Same
Thinking Ahead to Algebraic Representations
Key Ideas
Try It Out! Identify the Problem Situation
Write the Problem
Branch Out
Reflect
Descriptions of Images and Figures
CHAPTER THREE Part-Part-Whole. Thinking About Part-Part-Whole Situations
Addition and Subtraction Problems Situations
Sandbox Notes
Students and Teachers Think About the Problems
Take-From Action
Part-Part-Whole Relationships
Defining the Part-Part-Whole Situation
Modeling Relationships Versus Action
Writing Equations: Addition or Subtraction
Extending to Negative Values
Students and Teachers Think About the Problems
Representing Part-Part-Whole With Two-Color Counters
Translating Models Into Equations
Improving the Accuracy of Student Models
Blurred Lines Between Part-Part-Whole and Take-From
The Special Case of Both Parts Unknown
Key Ideas
Try It Out! Identify the Problem Situation
Write the Problem
Branch Out
Reflect
NOTES
Descriptions of Images and Figures
CHAPTER FOUR Additive Comparison Another Kind of Relationship. Thinking About Additive Comparison Situations
Addition and Subtraction Problem Situations
Sandbox Notes
Students and Teachers Think About the Problems
Language Can Get Tricky
Extending to Negative Values
Students and Teachers Think About the Problems
A Note About Equations
Knowing When to Add and When to Subtract
Difference, Direction, and Absolute Value
Modeling the Difference When Working With Negative Values
Key Ideas
Try It Out! Identify the Problem Situation
Write the Problem
Branch Out
Reflect
Descriptions of Images and Figures
CHAPTER FIVE Equal Groups Multiplication and Division Two Factors, Different Jobs. Thinking About Equal Groups Situations
Sandbox Notes
Students and Teachers Think About the Problems
Multiplier and Measure Factors
Making Sense of Factors Less than One
When the Measure Factor Is Less Than One
When the Multiplier Factor Is Less Than One
When the Measure Factor Is Less Than Zero
When the Multiplier Factor Is Less Than Zero
A Progression of Equal Groups Understanding
Equal Groups as an Early Understanding
Equal Groups as Repeated Addition
Equal Groups as Ratios or Rates
Thinking Ahead to Exponential Reasoning
Thinking About Equal Groups Division
Students and Teachers Think About the Problems
Partitive Division
Measurement (Quotitive) Division
The Unknowns in Partitive and Measurement Division
Matching Models to Contexts
Tracking the Unit Whole and the Referent Whole
Tracking Wholes in Partitive Division
Tracking Wholes in Measurement Division
Key Ideas
Try It Out! Identify the Factor
Write the Problem
Branch Out
Reflect
Descriptions of Images and Figures
CHAPTER SIX Ratios and Rates Describing Relationships. Thinking About Problem Situations With Ratios and Rates
Multiplication and Division Problem Situations
Sandbox Notes
Students and Teachers Think About the Problems
Defining Terms
Fraction Versus Ratio
Ratio Versus Rate
Proportion or Proportional Reasoning?
Four Contexts of Ratios and Proportional Reasoning Problems
Associated Sets
Well-Chunked Measures
Extensive Versus Intensive Measures
Stretchers and Shrinkers
Part-to-Whole
Percentage: A Special Ratio
Associated Rates
Translating Five Representations
Translating the Five Representations: Try It Out
Key Ideas
Try It Out! Identify the Ratio
Write the Problem
Branch Out
Reflect
Descriptions of Images and Figures
CHAPTER SEVEN Multiplicative Comparison More Than Scale Factors. Thinking About Multiplicative Comparison Situations
Multiplication and Division Problem Situations
Sandbox Notes
Students and Teachers Think About the Problems
Additive Versus Multiplicative Comparison
Constant Ratio
Absolute and Relative Change
When Multiplication Makes a Quantity Smaller
The Dilation Ratio
Extending to Negative Values
Students and Teachers Think About the Problems
Negative Measure Factors and Vectors
Negative Scale Factors and Reflections
Key Ideas
Try It Out! Identify the Problem Situation
Write the Problem
Branch Out
Reflect
Descriptions of Images and Figures
CHAPTER EIGHT Area/Array Two Factors, Same Job. Thinking About Area/Array Situations
Multiplication and Division Problem Situations
Sandbox Notes
Students and Teachers Think About the Problems
Symmetric Versus Asymmetric Multiplication
Representing Fraction and Decimal Division
Extending to Negative Values
Models of Thinking and Models for Thinking
Array as a Mathematical Structure
Benefits of Using an Array Structure as a Tool for Modeling a Problem Situation
Drawbacks of Using an Array Structure as a Tool for Modeling a Problem Situation
Key Ideas
Try It Out! Identify the Problem Situation
Write the Problem
Branch Out
Reflect
NOTES
Descriptions of Images and Figures
CHAPTER NINE Probability and Cartesian Products: Combinatorics. Thinking About Combinatorics Situations
Multiplication and Division Problem Situations
Sandbox Notes
Students and Teachers Think About the Problems
Understanding of Probability Is Based in Intuition
Experiencing Experimental Outcomes in Game Format
Analyzing and Making Sense of Experimental Outcomes
Building Intuition With Games
Using Models to Predict Number of Outcomes
The Path Toward Systematic Thinking
Patterns of Thinking in Making an Organized List
Making Sense of Repeated Multiplication
Key Ideas
Try It Out! Identify the Problem Situation
Write the Problem
Branch Out
Reflect
Descriptions of Images and Figures
CHAPTER TEN Changing How You Teach Word Problems
Getting Into the Mathematizing Sandbox
Eight Shifts in Instruction for Building Students’ Problem-Solving Skills
Do Word Problems for Sense-Making
Treat Context and Computation Separately
Create More and Varied Representations
Explore All the Work Operations Can Do
Add Operation Sense Routines to the School Day
Offer Students Experiences With a Variety of Problem Situations
Listen to Students and Be Curious
Make Time for Mathematizing in the Sandbox
Guidance for Moving Forward: FAQs
Descriptions of Images and Figures
APPENDIX: Situation Tables
Table References
Addition and Subtraction Problem Situations
Multiplication and Division Problem Situations
References
Index
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