Knowing and Teaching Elementary Mathematics by Ma, Liping -- Read -- Imperial Library of Trantor

Index

STUDIES IN MATHEMATICAL THINKING AND LEARNING Contents Author’s Preface to the Anniversary Edition Series Editor’s Introduction to the Anniversary Edition A Note about the Anniversary Edition

PERMISSIONS ACKNOWLEDGMENTS

Foreword Acknowledgments Introduction Chapter 1 Subtraction With Regrouping: Approaches To Teaching A Topic

THE U.S. TEACHERS’ APPROACH: BORROWING VERSUS REGROUPING

Construing the Topic Instructional Techniques: Manipulatives

THE CHINESE TEACHERS’ APPROACH: “DECOMPOSING A HIGHER VALUE UNIT”

Knowledge Package and Its Key Pieces Manipulatives and Other Teaching Approaches

DISCUSSION

Making Connections: Consciously Versus Unconsciously Models of Teachers’ Knowledge of Subtraction: Procedural Understanding Versus Conceptual Understanding Relationship Between Subject Matter Knowledge and Teaching Method: Can the Use of Manipulatives Compensate for Subject Matter Knowledge Deficiency?

SUMMARY

Chapter 2 Multidigit Number Multiplication: Dealing With Students’ Mistakes

THE U.S. TEACHERS’ APPROACH: LINING UP VERSUS SEPARATING INTO THREE PROBLEMS

Reasons for the Mistake Teaching Strategies

Procedural Conceptual

Relationship Between Subject Matter Knowledge and Teaching Strategy

THE CHINESE TEACHERS’ APPROACH: ELABORATING THE CONCEPT OF PLACE VALUE

Interpreting the Mistake

Distributive Law The Place Value System Place Value and the Distributive Law

Knowledge Package Teaching Strategies

Explanation and Demonstration Students Find the Problem Tr. Chen’s Approach

DISCUSSION

“Conceptual Understanding”: Not a Simple Story Knowledge Package and Its Key Pieces Relationship Between Subject Matter Knowledge and Beliefs: Is the Intent of Teaching for Understanding Enough?

SUMMARY

Chapter 3 Generating Representations: Division By Fractions

THE U.S. TEACHERS’ PERFORMANCE ON CALCULATION THE CHINESE TEACHERS’ PERFORMANCE ON CALCULATION

Making Sense of the Algorithm Alternative Computational Approaches

Alternative I: Dividing by Fractions Using Decimals4 Alternative II: Applying the Distributive Law Alternative III: “You Don’t Have to Multiply”

THE U.S. TEACHERS’ REPRESENTATIONS OF DIVISION BY FRACTIONS

The Mathematical Concepts that the Teachers Represented

Confounding Division by with Division by 2 Confounding Division by with Multiplication by Confusing the Three Concepts No Confusion, But No Story Either Correct Conception and Pedagogically Problematic Representation

Dealing with the Discrepancy: Correct Computation Versus Incorrect Representation An Inadequate Understanding of Procedure Impedes Creating a Representation Can Pedagogical Knowledge Make Up for Ignorance of the Concept?

THE CHINESE TEACHERS’ APPROACH TO THE MEANING OF DIVISION BY FRACTIONS

The Models of Division by Fractions

The Measurement Model of Division: “Finding How Many There Are in ” or “Finding How Many Times is of ” The Partitive Model of Division: Finding a Number Such That of It is Factors and Product: Finding a Factor That Multiplied by Will Make

Meaning of Multiplication by a Fraction: The Important Piece in the Knowledge Package The Representations of the Models of Division by Fractions

Rich Topics Representing the Partitive Model Several Stories With a Single Subject

DISCUSSION

Calculation: How Did It Reveal Teachers’ Understanding of Mathematics? “A Concept Knot”: Why It is Important Relationship Between Teachers’ Subject Matter Knowledge and Their Representations

SUMMARY

Chapter 4 Exploring New Knowledge: The Relationship Between Perimeter And Area

HOW THE U.S. TEACHERS EXPLORED THE NEW IDEA

Teachers’ Reactions to the Claim Teachers’ Responses to the Student

HOW THE CHINESE TEACHERS EXPLORED THE NEW IDEA

Teachers’ Approaches to the Problem A Map of How Teachers’ Exploration Was Supported Teachers’ Responses to the Student

DISCUSSION

Attitude Toward the Discipline: Promoter of Teachers’ Mathematical Inquiry Being Acculturated to Mathematics: Should It be a Feature of Mathematics Teachers? Relationship Between Teachers’ Subject Matter Knowledge and Positive Responses to Students’ Proposals: How Can a Mathematical Inquiry be Promoted and Supported?

SUMMARY

Chapter 5 Teacher’ Subject Matter Knowledge: Profound Understanding Of Fundamental Mathematics

A CROSS-TOPIC PICTURE OF THE CHINESE TEACHERS’ KNOWLEDGE: WHAT IS ITS MATHEMATICAL SUBSTANCE?

To Find the Mathematical Rationale of an Algorithm To Justify an Explanation with a Symbolic Derivation Multiple Approaches to a Computational Procedure: Flexibility Rooted in Conceptual Understanding Relationships Among the Four Basic Operations: The “Road System” Connecting the Field of Elementary Mathematics

KNOWLEDGE PACKAGES AND THEIR KEY PIECES: UNDERSTANDING LONGITUDINAL COHERENCE IN LEARNING ELEMENTARY MATHEMATICS AS FUNDAMENTAL MATHEMATICS PROFOUND UNDERSTANDING OF FUNDAMENTAL MATHEMATICS SUMMARY

Chapter 6 Profound Understanding Of Fundamental Mathematics: When And How Is It Attained?

WHEN IS PROFOUND UNDERSTANDING OF FUNDAMENTAL MATHEMATICS ATTAINED?: WHAT THE PRETEACHING GROUPS KNEW ABOUT THE FOUR TOPICS

Differences Between the two Chinese Preteaching Groups Differences Between the U.S. Teachers and the two Chinese Preteaching Groups Differences Between the Chinese Teachers and the two Preteaching Groups

PROFOUND UNDERSTANDING OF FUNDAMENTAL MATHEMATICS: HOW IT IS ATTAINED

Studying Teaching Materials Intensively Learning Mathematics From Colleagues Learning Mathematics from Students Learning Mathematics by Doing It

SUMMARY

Chapter 7 Conclusion

ADDRESS TEACHER KNOWLEDGE AND STUDENT LEARNING AT THE SAME TIME ENHANCE THE INTERACTION BETWEEN TEACHERS’ STUDY OF SCHOOL MATHEMATICS AND HOW TO TEACHIT REFOCUS TEACHER PREPARATION UNDERSTAND THE ROLE THAT CURRICULAR MATERIALS, INCLUDING TEXTBOOKS, MIGHT PLAY IN REFORM UNDERSTAND THE KEY TO REFORM: WHATEVER THE FORM OF CLASSROOM INTERACTIONS MIGHT BE, THEY MUST FOCUS ON SUBSTANTIVE MATHEMATICS

Appendix References Bridging Polarities: How Liping Ma’s Knowing and Teaching Elementary Mathematics Entered the U.S. Mathematics and Mathematics Education Discourses

INTELLECTUAL ROOTS, PROFESSIONAL PREPARATION, COLLEGIAL SUPPORT FOR THE BOOK

Academic and Professional History Bearing Upon the Book Financial and Collegial Supports

THE RANGE OF IMPACTS

Impact of Book and Math Wars The Book’s Impact or Lack of It on Preparation of Mathematics Teachers Impact of the Book on Teachers’ Professional Development Impact on Policy Impact on the Public International Impact

THE NATURE OF AND REASON FOR THE STUDY’S IMPACT A SMALL NARRATIVE ABOUT A LARGER DISCOURSE CHANGE REFERENCES APPENDIX

Response to “Bridging Polarities: How Liping Ma’s Knowing and Teaching Elementary Mathematics Entered the U.S. Mathematics and Mathematics Education Discourses” Author Index Subject Index

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