Intro -- Acknowledgments -- Contents -- Part I Introduction -- 1 "Mathematics Matters in Education" to Roger E. Howe and to All: An Introduction -- 1.1 Introduction -- 1.2 Structure of the Book -- References -- 2 About Roger E. Howe and His Contributions to Mathematics Education -- 2.1 Introduction -- 2.2 Accomplishments as an Educator -- 2.3 Contributions and Achievements as a Leader -- 2.4 Contributions and Achievements as a Scholar in Mathematics Education -- 2.4.1 Identifying and Articulating Core Ideas and Practices in K-12 Mathematics Curriculum, Especially in Elementary School Mathematics -- 2.4.2 Developing and Improving Mathematics Training for Preservice Elementary Teachers and In-Service Teachers -- 2.5 Summary -- References -- 3 Cultural Knowledge for Teaching Mathematics -- 3.1 Place Value -- 3.2 Units and Numbers -- 3.3 Length and the Number Line -- 3.4 Symmetry -- References -- Part II Knowing and Connecting Mathematics in Teaching and Teacher Education -- 4 The Content Knowledge Mathematics Teachers Need -- 4.1 Introduction -- 4.2 The Two Basic Requirements -- 4.2.1 Five Fundamental Principles -- 4.2.2 Two Caveats -- 4.2.3 Textbook School Mathematics (TSM) -- 4.2.4 The Data -- 4.3 TSM Confronts Mathematical Integrity -- 4.3.1 The Importance of Definitions: The Case of Fractions -- 4.3.2 Other Garbled Definitions in TSM -- 4.3.3 Geometry in Middle School and High School -- 4.3.4 How Coherence and Purposefulness Impact Learning -- 4.4 What Does It Mean to Know a Fact in Mathematics -- 4.5 Professional Development -- 4.6 Pedagogical Content Knowledge (PCK) -- Appendix 1: Applied Mathematics -- Appendix 2: The Existence of TSM -- References -- 5 Knowing Ratio and Proportion for Teaching -- 5.1 Introduction -- 5.2 The Rule of Three -- 5.3 Euclidean Ratio and Proportion -- 5.4 Mathematicians on School Math.

5.5 Euclidean Magnitudes and Measurement -- 5.6 Conclusions -- References -- 6 How Future Teachers Reasoned with Variable Parts and Strip Diagrams to Develop Equations for Proportional Relationships and Lines -- 6.1 Background -- 6.1.1 Equations for Proportional Relationships as Part of a Multiplicative Conceptual Field -- 6.1.2 Reasoning with Quantities -- 6.1.3 Reasoning with Variables -- 6.1.4 Interpreting Fractions -- 6.1.5 Interpreting Multiplication -- 6.1.6 The Variable Parts Perspective for Reasoning About Proportional Relationships -- 6.1.7 The Variable Parts Perspective for Reasoning About Lines in a Coordinate Plane -- 6.1.8 Research Questions -- 6.2 Methods -- 6.3 Results and Discusson -- 6.3.1 How Future Teachers Developed and Explained an Equation for a Proportional Relationship Using the Variable Parts Perspective -- 6.3.1.1 Diana, Kelly, and Jeff's Reasoning -- 6.3.1.2 Ideas, Concepts, and Ways of Reasoning the Future Teachers Used as they Developed an Equation for the Fertilizer Task -- 6.3.1.3 Points of Tension and their Resolution: Referent Unit and Equality -- 6.3.2 How Future Teachers Developed and Explained an Equation for a Line in a Plane Using the Variable Parts Perspective -- 6.3.2.1 Alice, Kelly, Claire, and Diana's Reasoning -- 6.3.2.2 Ideas, Concepts, and Ways of Reasoning the Future Teachers Used as they Developed an Equation for the Line Task -- 6.3.3 Further Discussion of the Future Teachers' Reasoning on the Two Tasks -- 6.4 Conclusion -- References -- 7 Giving Reason and Giving Purpose -- 7.1 Giving Reason and Giving Purpose -- 7.2 Math Studio -- 7.3 Lesson -- 7.3.1 Using a Number Talk to Orient and Foreshadow -- 7.3.2 Using Story Problems to Represent Strategies on a Number Line -- 7.3.3 Observations About the Lesson -- 7.4 Giving Reason and Giving Purpose -- 7.5 Data and Analysis -- 7.6 Planning.

7.6.1 Appraising Students' Past and CurrentMathematical Work -- 7.6.2 Planning the Sum Featured in the Story Problem -- 7.6.3 Responding to Anticipated Student Work -- 7.6.4 Observations About Planning and Its Impact on Instruction -- 7.7 Decomposition of Planning for Connectionsthat Give Reason and Give Purpose -- 7.7.1 Components -- 7.7.2 Identifying Current and New Mathematical Ideas and Evaluating Them -- 7.7.3 Choosing a New Mathematical Idea to Focus on -- 7.7.4 Designing Work that Elicits Current Ideas, New Ideas, and the Connections Between Them -- 7.7.5 Interaction Among Components and Knowledge Used -- 7.8 Cultivating Teaching that Gives Reason and Gives Purpose -- References -- 8 Who Are the Experts? -- References -- Part III Identifying and Structuring Core Ideas and Practices in K-12 Mathematics Curriculum -- 9 Building on Howe's Three Pillars in Kindergarten to Grade 6 Classrooms -- 9.1 Pillar I: A Robust Understanding of the Operations of Addition and Subtraction -- 9.1.1 Situations that Give Meaning to the Operations -- 9.1.2 Levels in Adding and Subtracting Single-Digit Numbers -- 9.2 Pillar II. An Approach to Arithmetic Computation that Intertwines Place Value with the Addition/Subtraction Facts -- 9.3 Pillar III. Making Connections Between Counting Number and Measurement Number -- 9.3.1 Limitations of Length for Showing Place Value and Addition and Subtraction -- 9.3.2 Counting Number and Measurement Number Do Relate Well to Show Multi-digit Multiplication and Division -- 9.3.3 Numbers on the Number Ray Tell Distances from the Endpoint/Origin -- 9.4 Visual Models Are Central Core Ideas and Practices in the CCSS-M and Deserve Attention and Discussion -- References -- 10 Is the Real Number Line Something to Be Built or Occupied? -- 10.1 Two Story Lines of the Number Line -- 10.2 The Construction Narrative.

10.2.1 Affordances of the Construction Narrative -- 10.2.2 Difficulties with the Construction Narrative -- 10.2.2.1 The Whole Number/Fraction Divide -- 10.2.2.2 The Continuum Gap -- 10.3 The Occupation Narrative: Cognitive Premise -- 10.4 Occupation Narrative: Quantity, Unit, Measure, and Number -- 10.5 Some History -- 10.6 Coordinatizing the Geometric Line ("Descartes in Dimension One") -- 10.7 Conclusion: What Is Achieved by This Occupation Narrative of the Number Line? -- References -- 11 What Content Knowledge Should We Expect in Mathematics Education? -- 11.1 Introduction -- 11.2 Concepts and Skills -- 11.3 Geometric Measurements -- 11.4 Fractions -- 11.5 Mathematical Understanding for Secondary Teaching -- 11.6 Conclusions -- References -- 12 Approaching Euclidean Geometry Through Transformations -- References -- 13 Curricular Coherence in Mathematics -- 13.1 Introduction -- 13.2 Coherence of Content -- 13.2.1 Logical Sequencing -- 13.2.2 Evolution from Particulars to Deep Structures -- 13.2.3 Using Deep Structures to Make Connections -- 13.3 Coherence of Practice -- 13.3.1 Using Structure -- 13.3.2 Abstraction -- 13.4 How Do We Achieve Curricular Coherence? -- References -- Part IV Supporting and Engaging Mathematicians in K-12 Education -- 14 Attracting and Supporting Mathematicians for the Mathematical Education of Teachers -- 14.1 Formative Experiences of the Author -- 14.2 Major Lessons Learned in Math Science Partnerships -- 14.3 Attracting Mathematicians to the Mathematical Education of Teachers -- 14.4 Supporting Mathematicians in Work with Teachers -- 14.4.1 General Remarks -- 14.4.2 The Common Core State Standards in Mathematics (CCSS-M) (National Governors Association Center for Best Practices &amp -- Council of Chief State School Officers, 2010) and Related Documents.

14.4.3 Benefits of Broadly Based and Supportive Teams Especially for Summer Institutes -- 14.4.4 Cognitive and Linguistic Issues in Communicating Math -- 14.4.4.1 Quantifier Errors and Omissions -- 14.4.4.2 Tendencies to Overgeneralize or Oversimplify -- 14.4.4.3 Misleading Diagrams: One Example Only -- 14.4.4.4 Technical Terms that Are Suggested by, but Not Coextensive with, Ordinary Usage -- 14.4.5 The Variety of National Systems of Education -- 14.4.6 Connecting Instructors of "Math for Teachers" with the Rest of a Math Department -- 14.4.7 "Elegant Exposition" Versus "Effective Exposition" for Teaching Mathematics -- 14.5 Rewarding Mathematicians for Work with Teachers -- 14.6 Conclusions -- References -- 15 The Contributions of Mathematics Faculty to K-12 Education: A Department Chair's Perspective -- 15.1 Introduction -- 15.2 The Work of Mathematics Departments: An Internal Perspective -- 15.3 The Work of Mathematics Departments: An External Perspective -- 15.4 Evaluating Mathematicians' Work in Math Education -- 15.5 Conclusions -- References -- 16 Supporting Education and Outreach in a Research Mathematics Department -- 16.1 Introduction -- 16.2 Background -- 16.3 Education and Outreach Roles and Recognition -- 16.4 Alternative Scholarship -- 16.5 Training in Educational Research -- 16.6 Mathematics Teacher Education -- 16.7 Teacher Professional Development -- 16.8 Case Study of Successes and Challenges -- 16.9 Developing Support Structures -- 16.10 Conclusions -- References -- Erratum -- Index.

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