Knowing_and_Teaching_Elementary_Mathemat (1)

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ForewordLee S.ShulmanThe Carnegie Foundation for the Advancement of TeachingThis is a remarkable book. It is also remarkably easy to misunderstand its most importantlessons. Liping Ma has conducted a study that compares mathematical understanding amongU.S. and Chinese elementary school teachers as it relates to classroom teaching practices.What could be simpler? What could one possibly misconstrue? Let me count the ways.• ThisbookappearstobeacomparativestudyofAmericanandChineseteachersofmathematics,but its most important contributions are not comparative, but theoretical.• Thisbookappearstobeaboutunderstandingthecontent of mathematics, rather than itspedagogy, but its conception of content is profoundly pedagogical.• Thisbookappearstobeaboutthepracticeofmathematicsteaching,butitdemandsahearing among those who set policy for teaching and teacher education.• Thisbookappearstobemostrelevanttothepreservicepreparationofteachers,butitsmost powerful findings may well relate to our understanding of teachers’ work and theircareer-long professional development.• Thisbookfocusesontheworkofelementaryschoolteachers,butitsmostimportantaudience may well be college and university faculty members who teach mathematics tofuture teachers as well as future parents.I shall try to clarify these somewhat cryptic observations in this foreword, but first a briefbiographical note about Liping Ma.Liping became an elementary school teacher courtesy of China’s Cultural Revolution.An eighth-grade middle-school student in Shanghai, she was sent to “the countryside”—inher case a poor rural village in the mountainous area of South China—to be re-educatedby the peasants working in the fields. After a few months, the village head asked Liping tobecome a teacher at the village school. As she has described it to me, she was a Shanghaiteenager with but eight years of formal education struggling to teach all the subjects to twoclasses of kids in one classroom. Over the next seven years, she taught all five grades andbecame principal of the school. A few years later, she would be hired as the ElementarySchool Superintendent for the entire county.When she returned to Shanghai filled with curiosity about her new calling, she found amentor in Professor Liu, who directed her reading of many of the classics of education—among them Confucius and Plato, Locke and Rousseau, Piaget, Vygotsky, and Bruner.Professor Liu eventually became president of East China Normal University where Lipingearned a master’s degree. She longed to study even more, and to pursue her further educationin the United States. On the last day of 1988, she arrived in the United States to studyat Michigan State University.
ForewordAt Michigan State University, she worked with, among others, Sharon Feiman-Nemserand Suzanne Wilson in teacher education, with Deborah Ball and Magdalene Lampert inmathematics education, and with Lynn Paine in comparative education. She participatedin the development and analysis of a national survey of elementary teachers’ mathematicalunderstandings, and marveled at the general misunderstandings that persisted amongthe U.S. teachers. They struck her as quite unlike the teachers she had come to know inChina.After a few years, her family chose to live in California, and Liping was admitted tothe doctoral program at Stanford University to complete her coursework and dissertation.I served as her advisor and the Spencer Foundation awarded her a dissertation-year fellowshipto complete the study that forms the basis for this book. This support, along with continuedhelp from Michigan State, made possible her travel to China to collect the data fromChinese teachers. After completing her Ph.D., Liping was awarded a two-year postdoctoralfellowship to work with Alan Schoenfeld at Berkeley, where she continued her research andwhere her dissertation was transformed into this superb book.What are the most important lessons to be learned from this book? Let us return to thelist of misconceptions I presented earlier and discuss them more elaborately.This book appears to be a comparative study of American and Chinese teachers ofmathematics, but its most important contributions are not comparative, but theoretical Theinvestigation compares Chinese and American teachers and the Chinese, once again, knowmore. What could be simpler? But the key ideas of this book are not comparisons betweenAmerican teachers and their Chinese counterparts. The heart of the book is Dr. Ma’s analysisof the kind of understanding that distinguishes the two groups. Chinese teachers arefar more likely to have developed “profound understanding of fundamental mathematics.”To say that they “know more” or “understand more” is to make a deeply theoretical claim.They actually may have studied far less mathematics, but what they know they know moreprofoundly, more flexibly, more adaptively.This book appears to be about understanding the content of mathematics, rather thanits pedagogy, but its conception of content is profoundly pedagogical. Liping Ma set outto account for the differences in content knowledge and understanding between U.S. andChinese elementary teachers, but her conception of understanding is critical. She hasdeveloped a conception of mathematical understanding that emphasizes those aspects ofknowledge most likely to contribute to a teacher’s ability to explain important mathematicalideas to students. Thus, her stipulation of four properties of understanding—basic ideas,connectedness, multiple representations, and longitudinal coherence—offers a powerfulframework for grasping the mathematical content necessary to understand and instruct thethinking of schoolchildren.This book appears to be about the practice of mathematics teaching, but it demands ahearing among those who set policy for teaching and teacher education. Policy-makershave become frantic in their insistence that future teachers demonstrate that they possessthe knowledge of subject matter necessary to teach children. We are about to see tests ofcontent knowledge for teachers proliferate among state licensing authorities. These cannotxvii
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 8 and 9: viii Contents5. Teachers’ Subject
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 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
Page 26 and 27: xxvi Introductionschool or college
Page 28 and 29: xxviii IntroductionChapters 1 throu
Page 30 and 31: Chapter 1Subtraction With Regroupin
Page 32 and 33: Subtraction with Regrouping: Approa
Page 54 and 55: Multidigit Number Multiplication: D
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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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Knowing_and_Teaching_Elementary_Mathemat (1)

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