Teaching Multiplication with Lesson Study -- Foreword -- Preface -- References -- Acknowledgements -- Contents -- Contributors -- About the Editors -- Chapter 1: Introduction: Japanese Theories and Overview of the Chapters in This Book -- 1.1 Origin of This Book -- 1.2 Overview of Japanese Theories for Designing Lessons -- 1.2.1 Mathematical Thinking and Activity: Aims and Objectives -- 1.2.2 Terminology and Sequences: Extension and Integration -- 1.2.3 Problem-Solving Approach: Not Only a Teaching Method -- 1.2.4 Change Approaches for Developing Students and Teachers -- 1.3 Overview of Chapters in Part I: The Japanese Approach -- 1.4 Overview of Chapters in Part II, Focusing on Ibero-American Countries -- References -- Part I: Japanese Approach for Multiplication: Comparison with other Countries, and Theoretical, Historical, and Empirical Analysis for Lesson Study -- Chapter 2: Multiplication of Whole Numbers in the Curriculum: Singapore, Japan, Portugal, the USA, Mexico, Brazil, and Chile -- 2.1 Comparison of Curricular Standards' Descriptions for Introducing Multiplication in Different Countries -- 2.2 Comparison of the Assigned Grade Levels for Multiplication -- 2.2.1 Range of Digits -- 2.2.2 The Meaning of Multiplication -- 2.2.3 The Definition of Multiplication -- 2.2.4 Multiplication Tables -- 2.2.5 Use of Algorithm or Column Method for Multiplication -- 2.2.6 Comparing the Results with Previous Research -- 2.3 Questions for Later Chapters -- References -- Chapter 3: Problematics for Conceptualization of Multiplication -- 3.1 Definitions of Multiplication and Their Meanings in Situations in School Mathematics -- 3.1.1 The Concept of Multiplication in Pure Mathematics in Relation to School Mathematics -- 3.1.2 Multiplicative Situations, Expression, and Translations -- 3.1.2.1 Origin of Written Situations.

3.1.2.2 In Situations of Geometry with Proportionality -- 3.1.2.3 In Situations with Quantities and Definition by Measurement -- 3.1.2.4 Contradictions between Repeated Addition and Situations with Quantities -- 3.1.2.5 Using the Situation of Multiplication Only for the Attribute of the Object -- 3.1.2.6 In the Situation of Area, As for Extension to Decimals and Fractions -- 3.1.2.7 In the Situation of Tree Diagrams -- 3.1.2.8 Seeing the Tree Diagram as an Operator -- 3.1.2.9 Activity of Elementary School and Cartesian Product -- 3.1.2.10 In Situations of Splitting as for Partitive Division -- 3.1.2.11 Another Usage: Splitting in Relation to the Distributive Law -- 3.1.2.12 Limitations of Every Model for Multiplication -- 3.1.2.13 Conceptual Fields for Multiplication -- 3.2 Problems with Multiplication that Originate from Languages -- 3.3 European Languages and Their Historical Usages -- 3.3.1 The Transition in Chile -- 3.4 Final Remarks -- References -- Chapter 4: Introduction of Multiplication and Its Extension: How Does Japanese Introduce and Extend? -- 4.1 The Introduction of Multiplication Using the Japanese Approach -- 4.1.1 The Way to Initiate the Situation for Multiplication Before Repeated Addition in the Japanese Approach -- 4.1.1.1 Repeated Addition and Challenges to Difficulty -- 4.1.1.2 Use of the Multiplicand and Multiplier for Students to Think of Division Situations by and for Themselves -- 4.1.1.3 Commutativity and Order in Expression -- 4.1.1.4 Differences in the Multiplier and Multiplicand in an Array and a Block Diagram -- 4.1.1.5 Revisiting Which Notation Is Better and Why -- 4.2 Preparation for Multiplication in the Japanese Curriculum and Textbooks -- 4.2.1 Preparation for Introduction of Multiplication in the First Grade -- 4.2.1.1 Composition and Decomposition of Cardinal Numbers for Binary Operations.

4.2.1.2 Counting by Twos or by Fives as the Base for the New Unit to Count -- 4.2.1.3 Polynomial Notation -- 4.2.1.4 Production of Tentative/Arbitrary Units -- 4.3 Proportionality for Extension of Multiplication -- 4.3.1 Introduction of Proportional Number Lines and Their Adaptation for Extension -- 4.3.2 Extension of Multiplication by Using Proportional Number Lines -- 4.3.3 Partitive and Quotative Divisions Using Multiplication -- 4.3.4 Relationships Among the Rule of Three, Multiplication, and Division -- 4.3.5 From Division to Ratios and Rates Using the Multiplicative Format -- 4.4 Various Meanings of Fractions Embedding the Meanings of Division Situations -- 4.5 Further Challenges to Distinguish Additive and Multiplicative Structures -- 4.5.1 Redefinition of Proportionality at Junior High School -- 4.6 Final Remarks -- References -- Chapter 5: Japanese Lesson Study for Introduction of Multiplication -- 5.1 Lesson Study for the Introduction of Multiplication -- 5.1.1 Lesson Study on the Meaning of Multiplication, by Mr. Natsusaka -- 5.1.1.1 Description and Plan of the Lesson Being Investigated -- 5.1.1.2 A Public Lesson (Open Class) by Mr. Natsusaka -- 5.1.1.3 Post-Open Class Discussion -- 5.1.2 Lesson Plan on Applying the Meaning of Multiplication After Learning the Multiplication Table, by Mr. Tsubota -- 5.2 Evidence to See Any Number as a Counting Unit -- 5.3 Comparison of the Japanese and Chilean Approaches -- 5.4 Final Remarks -- References -- Chapter 6: Teaching the Multiplication Table and Its Properties for Learning How to Learn -- 6.1 Revisiting the Japanese Educational Principle -- 6.2 A Survey of Appropriate Grades to Introduce the Multiplication Table -- 6.3 The Multiplication Table in Japanese Textbooks for Learning How to Learn -- 6.3.1 Developing Multiplication Tables for the Rows of 2, 5, 3, and 4.

6.3.2 Transferring the Responsibility for Construction and Memorization of the Multiplication Table -- 6.3.3 Extension of the Multiplication Tables of 6-9 and 1 -- 6.3.4 Properties of the Multiplication Table for Discovering the World of Multiplication with a Sense of Wonder -- 6.4 Memorizing the Multiplication Table as a Cultural Practice -- 6.4.1 Using the Cards -- 6.4.2 Using Area-Array Cards -- 6.4.3 Using a Notebook and Journal Writing at Home -- 6.5 The Sense of Wonder in the Multiplication Table -- 6.5.1 Focusing on Beautiful Patterns with a Sense of Wonder and Appreciation -- 6.5.2 Preparing a Problematic: "Why" -- 6.5.3 How to Begin the Class? -- 6.6 Final Remarks -- References -- Chapter 7: The Teaching of Multidigit Multiplication in the Japanese Approach -- 7.1 Diversity of Column, Algorithm, and Vertical Form Methods for Multiplication -- 7.1.1 Historical Illustration of Diversity -- 7.1.2 Revisiting the Confusion Between the Multiplier and Multiplicand, and the Need to Differentiate Them -- 7.1.3 Terminology for Teaching Column Multiplication -- 7.2 Lesson Study for Introducing Multiplication in Vertical Form -- 7.2.1 Lesson Study Video Introducing Vertical Form -- 7.2.2 Mr. Muramoto's Objectives for This Class -- 7.2.3 Description of Actual Lesson Episodes -- 7.2.4 Criteria for Formative Assessment in the Lesson Plan -- 7.3 Annex for Sect. 7.2: Excerpts of the Lesson Plan by Mr. Muramoto, Illustrating Why and How a Japanese Teacher Prepares School-Based Lesson Study -- 7.3.1 Maruyama Elementary School Mathematics Group Vision and Mathematics Lesson Study Group's Goals -- 7.3.1.1 Actual Setting of the Students in Maruyama -- 7.3.1.2 Research Theme for Lesson Study -- 7.3.1.3 Focal Points for Kyozaikenkyu (Preparation of Teaching Materials According to the Objective/Research on the Subject Matter) for Implementation of the Research Theme.

7.3.1.4 Thinking About Assessments That Help Students to Be More Precise in Their Problem-Solving Processes -- 7.3.2 Support for Other Teachers in School to Improve Students' Learning -- 7.3.2.1 Necessary Communication with Other Teachers -- 7.3.3 To Promote Human Character Formation with Strong Hearts and Minds, Students Who Acquire This Kind of Competency Can Participate in the Classroom in the Following Ways -- 7.3.3.1 Planning Consistent Development of Proficiency in Logical Thinking -- 7.3.4 Survey of Students for Preparation and Challenges -- 7.3.5 Exploring Topics That Students Learn in the Third Grade -- 7.3.6 Challenging Issues for the Lesson Study Group with Viewpoints -- 7.3.6.1 Viewpoint 1: Teaching Material to Connect Unknown Content with Learned content -- 7.3.6.2 Viewpoint 2: Knowing the Significance of Own Ideas Through Comparison with Others' Understanding -- 7.3.6.3 Viewpoint 3: Prepare the Task Sequence with Formative Assessments -- 7.3.7 Unit and Lesson Plans -- 7.4 Multidigit Multiplication in Vertical Form: Task Sequence for Extension and Integration in the Case of Gakko Tosho -- 7.4.1 Task Sequence for Extension -- 7.4.1.1 Task 1: Extension by Students -- 7.4.1.2 Task 2: 4 × 30 -- 7.4.1.3 Task 3: 21 × 13 -- 7.4.1.4 Tasks 4 and 5: With Carrying and with 0 -- 7.5 Final Remarks -- References -- Part II: Ibero and Ibero-American Contributions for the Teaching of Multiplication -- Chapter 8: An Ethnomathematical Perspective on the Question of the Idea of Multiplication and Learning to Multiply: The Languages and Looks Involved -- 8.1 Introduction -- 8.2 Alternative Modes -- 8.2.1 Project Learning -- 8.2.2 Thinking of Multiplication Through Research Scripts -- 8.3 Multiplication: Tables with Polygons -- 8.4 Multiplication using Art and Technology -- 8.4.1 Multiplication Using the Calculator.

8.5 Some More Ideas About Learning and Teaching of Mathematical Knowledge.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2023. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.