Knowing_and_Teaching_Elementary_Mathemat (1)

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viii Contents5. Teachers’ Subject Matter Knowledge: Profound Understanding Of 92F undamental MathematicsA Cross-Topic Picture of the Chinese Teachers’ Knowledge: What Is 92Its Mathematical Substance?Knowledge Packages and Their Key Pieces: Understanding97L ongitudinal Coherence in LearningElementary Mathematics as Fundamental Mathematics 99Profound Understanding of Fundamental Mathematics 101Summary 1056. Profound Understanding Of Fundamental Mathematics: When And 107How Is It Attained?When Is Profound Understanding of Fundamental Mathematics 107Attained?: What the Preteaching Groups Knew About the Four TopicsProfound Understanding of Fundamental Mathematics: How It 110Is AttainedSummary 1217. Conclusion 123Address Teacher Knowledge and Student Learning at the Same Time 124Enhance the Interaction Between Teachers’ Study of School125M athematics and How to Teach ItRefocus Teacher Preparation 127Understand the Role That Curricular Materials, Including Textbooks, 127Might Play in ReformUnderstand the Key to Reform: Whatever the Form of ClassroomInteractions Might Be, They Must Focus on Substantive Mathematics128Appendix 131References 133New to the Anniversary Edition: Fang and Paine’s “Bridging P olarities:How Liping Ma’s Knowing and Teaching Mathematics Entered the U.S.Mathematics and Mathematics Education Discourses”New to the Anniversary Edition: Ma’s Response to “Bridging P olarities” 158Author Index 160Subject Index 164138
Author’s Preface to the Anniversary EditionI must admit that I did not expect this book to be a success. About twelve years ago, when Idecided to devote my effort to turning my doctoral dissertation into a book, my motivationwas very simple. I did not feel good thinking that my dissertation, which had taken so muchof my passion and work, would sleep forever on the shelf in the Stanford School of EducationLibrary. My dissertation should not merely be a “test” for me in earning a degree; itshould serve others, in particular, classroom teachers. Indeed, this simple feeling pushedme to undertake the creation of this book. Now, ten years after its publication in 1999, over70,000 copies have been sold.A few years ago, I paid a visit to a beautiful east coast city. On Saturday, in the middleof my morning walk, I was stopped by a blocked street. The excited people standing on thesidewalk told me that they were waiting for the annual cross-city footrace to pass by. A fewminutes later, the procession began. Runners—male, female, young, old, smiling, serious,alone, accompanied by friends and family—covered the wide street in front of me. Theykept on coming. It took about twenty minutes for all the runners to pass the place whereI stood. “There are so many people!,” I exclaimed. “This year we have about 40,000 runners,”someone replied. The number struck me—it was the number of copies of my bookthat had been sold. Only when I physically saw 40,000 people, was I aware of its significance.I was moved—if the book benefited that many people, it was worthwhile work.In the fall of 2003, I received a package mailed from Korea—two copies of my book inKorean were included. Last year, mathematics education scholars in China and Chile beganto translate the book into Chinese and Spanish. I am told that before the end of 2009, theChinese translation of the book will have been published.It is not a surprise that the book has drawn attention from different countries—after all,it is based on a cross-national comparative study. Moreover, as my doctoral adviser LeeShulman pointed out, I myself, as a scholar in education, have been nurtured by Chineseand American learning environments. Indeed, a cross-national perspective may help us tosee what might be otherwise unnoticed. Now, I invite you to look at two pictures of mathematicseducation in the U.S. and China. Both illustrate aspects of education in the twocountries, but the scenes they illustrate are one hundred years apart.The first picture is sketched in this book, which was published in 1999. A group ofAmerican teachers were interviewed and asked questions about four topics of elementaryschool mathematics. The same questions were asked of a group of Chinese teachers. Thecomparison of their responses constitutes the main content of this book.Compared with their U.S. counterparts, the Chinese interviewees were significantly moreproficient in the procedural aspect of all four topics, particularly in the more advanced topicsof division by fractions and area and perimeter. All the Chinese teachers attained thecorrect answer when computing , while only about half of the U.S. teachers did. Nota single Chinese teacher showed any hesitation in using formulas to find the perimeter and
Page 2 and 3: Knowing and TeachingElementary Math
Page 4 and 5: Knowing and TeachingElementary Math
Page 6 and 7: To Jianfeng, Sushu, and John,and to
Page 10 and 11: xAuthor’s Preface to the Annivers
Page 12 and 13: xiiAuthor’s Preface to the Annive
Page 14 and 15: xivSeries Editor’s Introduction t
Page 16 and 17: ForewordLee S.ShulmanThe Carnegie F
Page 18 and 19: xviii Forewordbe tests that assess
Page 20 and 21: AcknowledgmentsAbout 30 years ago,
Page 22 and 23: xxii AcknowledgmentsI would like to
Page 24 and 25: xxiv Introductionthe measurement mo
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Page 28 and 29: xxviii IntroductionChapters 1 throu
Page 30 and 31: Chapter 1Subtraction With Regroupin
Page 32 and 33: Subtraction with Regrouping: Approa
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Multidigit Number Multiplication: D
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Chapter 3Generating Representations
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For some reason it is in the back o
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Generating Representations: Divisio
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We also encourage students to solve
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of math.” She said she used to ha
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been changed. Now what is known is
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While the U.S. teachers tended to u
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Exploring New Knowledge: The Relati
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Teachers’ Subject Matter Knowledg
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105These four properties are interr
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Chapter 6Profound Understanding Of
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Profound Understanding of Fundament
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Chapter 7ConclusionAs I said at the
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Conclusion 125as necessarily preced
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Conclusion 127Time is an issue here
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Conclusion 129evidence, students wo
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AppendixFIG. A.1. The U.S. teachers
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ReferencesArmstrong, B., & Bezuk, N
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References 135Kaput, J., & Nemirovs
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References 137Stein, M., Baxter, J.
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Fang and Paine’s “Bridging pola
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Liping Ma’s “Knowing and Teachi
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Author Index161Ferrucci, B., 150, 1
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Author Index163Simon, M., xxviii, x
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Subject Index165commutative law, 39
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Subject Index167knowledge,see also
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Subject Index169teacher preparation
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