“Both sides in the math wars claim Dr. Ma as their own. Districts have distributed her book to teachers. Its broad appeal offers some hope for common ground in math education…. We will continue fights over whether children should be taught arithmetic rules or theory. What Dr. Ma shows is that we need both.” —TheNewYorkTimes “Ma’s book is a significant contribution to mathematics education because it begins to tackle the important and complex question of ‘what is mathema- tical knowledge for elementary teaching.’ In doing so, she helps us to understand elementary mathematics as a complex and demanding subject that is to be taken seriously.” —ContemporaryPsychology “A stealth hit for math junkies on both sides of the ‘math wars,’ and a must read for anyone interested in solving the problems of public schools.” —ChristianScienceMonitor “This book supports the need for, and indeed the educational benefits of, changing professional teaching conditions for U.S. teachers…. It provides some food for thought for everyone involved in improving mathematics education. And it supports the necessity that even at the elementary school level, students can, and should, learn challenging mathematics.” —NationalCouncilofTeachersofMathematicsNewsBulletin “Must reading for those who call for more mathematics and those who champion reform pedagogy in teacher education.” —CHOICE “Thecontributionsofthisbookaremultifaceted….Thisbookisanexcellent resource and will interest anyone involved in teaching preservice teachers, as well as researchers concerned with teachers’ knowledge of content and methods.” —TeachingChildrenMathematics Knowing and Teaching Elementary Mathematics The 20th anniversary edition of this groundbreaking and bestselling volume offerspowerfulexamplesofthemathematicsthatcandevelopthethinkingof elementary school children. Studies of teachers in the U.S. often document insufficient subject matter knowledge in mathematics. Yet, these studies give few examples of the knowl- edge teachers need to support teaching, particularly the kind of teaching demanded by reforms in mathematics education. Knowing and Teaching Elemen- tary Mathematics describes the nature and development of the knowledge that elementary teachers need to become accomplished mathematics teachers, and suggests why such knowledge seems more common in China than in the United States, despite the fact that Chinese teachers have less formal educa- tion than their U.S. counterparts. Along with the original studies of U.S. and Chinese teachers’ mathematical understanding, this 20th anniversary edition includes a new preface and a 2013 journal article by Ma, “A Critique of the Structure of U.S. Elementary School Mathematics” that describe differences in U.S. and Chinese elemen- tary mathematics. These are augmented by a new series editor’s introduction and two key journal articles that frame and contextualize this seminal work. Liping Ma earned a Ph.D. from Stanford University, following a masters degree in education from East China Normal University. After a term as a senior scholar at the Carnegie Foundation for the Advancement of Teaching, she is now an independent scholar. She served as a member of the National Mathematics Advisory Panel from 2006 to 2008. Studies in Mathematical Thinking and Learning Alan H. Schoenfeld, Series Editor Artzt/Armour-Thomas/Curcio Becoming a Reflective Mathematics Teacher: A Guide for Observation and Self-Assessment, Second Edition Artzt/Armour-Thomas/Curcio/Gurl Becoming a Reflective Mathematics Teacher: A Guide for Observation and Self-Assessment, Third Edition Baroody/Dowker (Eds.) The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise Boaler Experiencing School Mathematics: Traditional and Reform Approaches to Teaching and Their Impact on Student Learning Carpenter/Fennema/Romberg (Eds.) Rational Numbers: An Integration of Research Chazan/Callis/Lehman (Eds.) Embracing Reason: Egalitarian Ideals and the Teaching of High School Mathematics Cobb/Bauersfeld (Eds.) The Emergence of Mathematical Meaning: Interaction in Classroom Cultures Cohen Teachers’ Professional Development and the Elementary Mathematics Classroom: Bringing Understandings to Light Clements/Sarama Learning and Teaching Early Math: The Learning Trajectories Approach Clements/Sarama/DiBiase (Eds.) Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education English (Ed.) Mathematical and Analogical Reasoning of Young Learners English (Ed.) Mathematical Reasoning: Analogies, Metaphors, and Images Fennema/Nelson (Eds.) Mathematics Teachers in Transition Fennema/Romberg (Eds.) Mathematics Classrooms That Promote Understanding Fernandez/Yoshida Lesson Study: A Japanese Approach to Improving Mathematics Teaching and Learning Greer/Mukhopadhyay/Powell/Nelson-Barber (Eds.) Culturally Responsive Mathematics Education Hulbert/Petit/Ebby/Cunningham/Laird A Focus on Multiplication and Division: Bringing Research to the Classroom Kaput/Carraher/Blanton (Eds.) Algebra in the Early Grades Kitchen/Civil (Eds.) Transnational and Borderland Studies in Mathematics Education Lajoie Reflections on Statistics: Learning, Teaching, and Assessment in Grades K-12 Lehrer/Chazan (Eds.) Designing Learning Environments for Developing Understanding of Geometry and Space Li/Huang (Eds.) How Chinese Teach Mathematics and Improve Teaching Ma Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States, Anniversary Edition Martin Mathematics Success and Failure Among African-American Youth: The Roles of Sociohistorical Context, Community Forces, School Influence, and Individual Agency Martin (Ed.) Mathematics Teaching, Learning, and Liberation in the Lives of Black Children Petit/Laird/Marsden A Focus on Fractions: Bringing Research to the Classroom Petit/Laird/Marsden/Ebby A Focus on Fractions: Bringing Research to the Classroom, Second Edition Reed Word Problems: Research and Curriculum Reform Remillard/Herbel-Eisenmann/Lloyd (Eds.) Mathematics Teachers at Work: Connecting Curriculum Materials and Classroom Instruction Romberg/Fennema/Carpenter (Eds.) Integrating Research on the Graphical Representation of Functions Romberg/Carpenter/Dremock (Eds.) Understanding Mathematics and Science Matters Romberg/Shafer The Impact of Reform Instruction on Mathematics Achievement: An Example of a Summative Evaluation of a Standards-Based Curriculum Sarama/Clements Early Childhood Mathematics Education Research: Learning Trajectories for Young Children Schliemann/Carraher/Brizuela (Eds.) Bringing Out the Algebraic Character of Arithmetic: From Children’s Ideas to Classroom Practice Schoenfeld How We Think: A Theory of Goal-Oriented Decision Making and its Educational Applications Schoenfeld (Ed.) Mathematical Thinking and Problem Solving Senk/Thompson (Eds.) Standards-Based School Mathematics Curricula: What Are They? What Do Students Learn? Sherin/Jacobs/Philipp (Eds.) Mathematics Teacher Noticing: Seeing Through Teachers’ Eyes Solomon Mathematical Literacy: Developing Identities of Inclusion Sophian The Origins of Mathematical Knowledge in Childhood Sternberg/Ben-Zeev (Eds.) The Nature of Mathematical Thinking Stylianou/Blanton/Knuth (Eds.) Teaching and Learning Proof Across the Grades: A K-16 Perspective Sultan & Artzt The Mathematics That Every Secondary Mathematics Teacher Needs to Know Sultan & Artzt The Mathematics That Every Secondary School Math Teacher Needs to Know, Second Edition Watson Statistical Literacy at School: Growth and Goals Watson/Mason Mathematics as a Constructive Activity: Learners Generating Examples Wilcox/Lanier (Eds.) Using Assessment to Reshape Mathematics Teaching: A Casebook for Teachers and Teacher Educators, Curriculum and Staff Development Specialists Wood/Nelson/Warfield (Eds.) Beyond Classical Pedagogy: Teaching Elementary School Mathematics Zaskis/Campbell (Eds.) Number Theory in Mathematics Education: Perspectives and Prospects Knowing and Teaching Elementary Mathematics ’ Teachers Understanding of Fundamental Mathematics in China and the United States th 20 Anniversary Edition Liping Ma Thirdeditionpublished2020 byRoutledge 52VanderbiltAvenue,NewYork,NY10017 andbyRoutledge 2ParkSquare,MiltonPark,Abingdon,Oxon,OX144RN RoutledgeisanimprintoftheTaylor&FrancisGroup,aninformabusiness ©2020Taylor&Francis TherightofLipingMatobeidentifiedasauthorofthisworkhas beenassertedbytheminaccordancewithsections77and78ofthe Copyright,DesignsandPatentsAct1988. Allrightsreserved.Nopartofthisbookmaybereprintedor reproducedorutilisedinanyformorbyanyelectronic,mechanical, orothermeans,nowknownorhereafterinvented,including photocopyingandrecording,orinanyinformationstorageor retrievalsystem,withoutpermissioninwritingfromthepublishers. Trademarknotice:Productorcorporatenamesmaybetrademarksor registeredtrademarks,andareusedonlyforidentificationand explanationwithoutintenttoinfringe. FirsteditionpublishedbyLawrenceErlbaumAssociates,Inc.1999 SecondeditionpublishedbyRoutledge2010 LibraryofCongressCataloging-in-PublicationData Acatalogrecordhasbeenrequestedforthisbook ISBN:978-0-367-44394-8(hbk) ISBN:978-0-367-44395-5(pbk) ISBN:978-1-003-00944-3(ebk) TypesetinBaskerville byIntegraSoftwareServicesPvt.Ltd. To Jianfeng, Sushu, and John, and to Cathy Kessel, with deepest love and appreciation