Knowing and Teaching Elementary Mathematics · Knowing and Teaching Elementary Mathematics ... The reviewer will refer to the book ... was persons, which children

time to consider pre-college educationform an intuition thatit would help the sit-uation if teachersknew more mathe-matics. If these math-ematicians get moreinvolved in mathe-matics education,they are likely to besurprised by how lit-tle this intuitionseems to affect theagenda in mathemat-ics education reform.

Partly this nonin-terest in mathematical expertise reflects an attitudewidespread among educators [Hi] that “facts”, andindeed all subject matter, are secondary in im-portance to a generalized, subject-independentteaching skill and the development of “higher-order thinking”. Concerning mathematics in par-ticular, the study [Be] is often cited as evidence forthe irrelevance of subject matter knowledge. Forthis study, college mathematics training, as mea-sured by courses taken, was used as a proxy for ateacher’s mathematical knowledge. The correla-tion of this with student achievement was foundto be slightly negative. A similar but less specificmethod was used in the recent huge Third Inter-national Mathematics and Science Study (TIMSS) ofcomparative mathematics achievement in forty-odd countries. For TIMSS, U.S. students demon-strated adequate (in fourth grade) to poor (in

SEPTEMBER 1999 NOTICES OF THE AMS 881

Book Review

Knowing and Teaching Elementary Mathematics

Reviewed by Roger Howe

Knowing and Teaching Elementary Mathematics:Teachers’ Understanding of FundamentalMathematics in China and the United StatesLiping MaLawrence Erlbaum Associates, Inc., 1999Cloth, $45.00, ISBN 0-8058-2908-3Softcover, $19.95, ISBN 0-8058-2909-1

Notation: The reviewer will refer to the bookunder review as KTEM.

For all who are concerned with mathematicseducation (a set which should include nearly every-one receiving the Notices), KTEM is an importantbook. For those who are skeptical that mathemat-ics education research can say much of value, it canserve as a counterexample. For those interested inimproving precollege mathematics education inthe U.S., it provides important clues to the natureof the problem. An added bonus is that, despitethe somewhat forbidding educationese of its title,the book is quite readable. (You should be gettingthe idea that I recommend this book!)

Since the publication in 1989 of the Curriculumand Evaluation Standards by the National Council ofTeachers of Mathematics [NCTM], there has been asteady increase in discussion and debate about re-forming mathematics education in the U.S., includ-ing increased attention from university mathe-maticians (cf. [Ho]). Many mathematicians who take

Roger Howe is professor of mathematics at Yale Univer-sity. His e-mail address is [email protected].

Acknowledgments: I am grateful to A. Jackson, J. Lewis,R. Raimi, K. Ross, J. Swafford, and H.-H. Wu for useful com-ments, and to R. Askey for bibliographic help.

rev-howe.qxp 7/12/99 12:38 PM Page 881

882 NOTICES OF THE AMS VOLUME 46, NUMBER 8

twelfth grade) mathematics achievement[DoEd1–3]. To analyze whether teacher knowledgemight help explain TIMSS outcomes, data onteacher training was gathered. In terms of collegestudy, U.S. teachers appear to be comparable totheir counterparts in other countries [DoEd1–3].

How can this intuition—that better grasp ofmathematics would produce better teaching—ap-pear to be so wrong? KTEM suggests an answer. Itseems that successful completion of college coursework is not evidence of thorough understandingof elementary mathematics. Most university math-ematicians see much of advanced mathematics asa deepening and broadening, a refinement andclarification, an extension and fulfillment of ele-mentary mathematics. However, it seems that it ispossible to take and pass advanced courses with-out understanding how they illuminate more ele-mentary material, particularly if one’s under-standing of that material is superficial. Over thepast ten years or so, Deborah Ball and others [B1–3] have interviewed many teachers andprospective teachers, probing their grasp of theprinciples behind school mathematics. KTEM ex-tends this work to a transnational context. The pic-ture that emerges is highly instructive—and sober-ing. Mathematicians can be pleased to have at lastpowerful evidence that mathematical knowledgeof teachers does play a vital role in mathematicslearning. However, it seems also that the kind ofknowledge that is needed is different from whatmost U.S. teacher preparation schemes provide, andwe have currently hardly any institutional struc-tures for fostering the appropriate kind of under-standing.

The main body of KTEM (Chapters 1–4) pre-sents the results of interviews with elementaryschool teachers from the U.S. (23 in all) and China(72 in all). The U.S. teachers were roughly evenlysplit between experienced teachers and beginners.Ma judged the group as a whole to be “above av-erage”. In particular, although “math anxiety” isrampant among elementary school teachers, thisgroup had positive attitudes about mathematics:they overwhelmingly felt that they could handlebasic mathematics and that they could learn ad-vanced mathematics. The Chinese teachers werefrom schools chosen to represent the range ofChinese teaching experience and expertise: urbanschools and rural, stronger schools and weaker.

The teachers’ grasp of mathematics was probedin interviews organized around four questions. Insummary form, the questions were as follows:

1) How would you teach subtraction oftwo-digit numbers when “borrowing” or“regrouping” is needed?

2) In a multiplication problem such as123× 645, how would you explain whatis wrong to a student who performsthe calculation as follows?

123

×645

615

492

738

1845

(The student has correctly formed thepartial products of 123 with the digitsof 645, but has not “shifted them to theleft”, as required to get a correct an-swer.)

3) Compute 1 3

412

. Then make up a story

problem which models this computa-tion, that is, for which this computationprovides the answer.

4) Suppose you have been studyingperimeter and area and a student comesto you excited by a new “theory”: areaincreases with perimeter. As justifica-tion the student provides the exampleof a 4× 4 square changing to a 4× 8rectangle: perimeter increases from 16to 24, while area increases from 16 to32. How would you respond to this stu-dent?

These questions are in order of increasing depth.The first two involve basic issues of place-value dec-imal notation. The third involves rational num-bers and also involves division, the most difficultof the arithmetic operations. It further requires“modeling” or “representation”—connecting a cal-culation with a “real-world” situation. The lastproblem, which was originally stated in terms ofperimeter and area of a “closed figure”, poten-tially involves very deep issues. Even if one re-places “closed figure” with “rectangle”, as all theteachers did, one must still compare the behaviorof two functions of two real variables.

On sheepskin the American teachers seemed de-cidedly superior to the Chinese: they all were col-lege graduates, and several had MAs. The Chineseteachers had nine years of regular schooling, andthen three years of normal school for teachers—in terms of study time, a high school degree. How-ever, measured in terms of mastery of elementaryschool mathematics, the Chinese teachers came outbetter.

The rough summary of the results of the inter-views is: the Chinese teachers responded more or



less as one would hope that a mathematics teacherwould, while the American teachers revealed dis-turbing deficiencies. In more detail, on the first twoproblems, all teachers could perform the calcula-tions correctly and could explain how to do them,that is, describe the correct procedure. However,even on the first problem, fewer than 20% of theU.S. teachers had a conceptual grasp of the re-grouping process—decomposing one 10 into 10ones. By contrast, the Chinese teachers over-whelmingly (86%) understood and could explainthis decomposition procedure. On the second prob-lem, about 40% of the U.S. teachers could explainthe reason for the correct method of aligning thepartial products, while over 90% of the Chineseteachers showed a firm grasp of the place valueconsiderations that prescribe the alignment pro-cedure.

On the third problem, a gap appeared even atthe computational level: well under half of theAmerican teachers performed the indicated cal-culation correctly. Only one came up with a tech-nically acceptable story problem. Even this onewas pedagogically questionable, since the unitsfor the answer (31

2) was persons, which childrenmight expect to come in whole numbers. The Chi-nese teachers again all did the calculation cor-rectly, and 90% of them could make up a correctstory problem. Some suggested multiple prob-lems, illustrating different interpretations of divi-sion.

On the fourth problem, the U.S. teachers did ex-hibit some good teaching instincts, and most,though not all, could state the formulas for areaand perimeter of rectangles. However, when itcame to analyzing the mathematics, they were lostat sea. Although most wanted to see more exam-ples, over 90% were inclined to believe that the stu-dent’s claim was valid. Some proposed to looksomething up in a book. Only three attempted amathematical investigation of the claim, and againa lone one found a counterexample. The Chineseteachers also found this problem challenging, andmost had to think about it for some time. After con-sideration, 70% of them arrived at a correct un-derstanding, with valid counterexamples. Of the30% who did not find the answer, most did thinkmathematically about the problem, though notsufficiently rigorously to find the defect in thestudent’s proposal.

The contrast between the performances of thetwo groups of teachers was even more dramaticthan this summary reveals. Some Chinese teach-ers gave responses that more than answered thequestion. They sometimes offered multiple solu-tion methods. In the integer arithmetic problems,some indicated that, if the student was havingtrouble here, it meant that something more fun-damental had not been learned properly. Thesecomments point to a deeper layer of teaching cul-

ture that simply does not exist in the U.S. For ex-ample, American teaching of two-digit subtractionis usually based on “subtraction facts”, the resultsof subtracting a one-digit number from a one- ortwo-digit number to get a one-digit number. Theseare simply to be learned by rote. The Chinese basesubtraction on these same facts, but they refer tothis topic as “subtraction within 20” and treat itas one to be understood thoroughly, since they re-gard it as the link between the computational andthe conceptual basis for multidigit subtraction. Inanswering question 3, some Chinese teachers sug-gested that the given problem was too easy and of-fered harder ones. Also, the Chinese teachers werecomfortable with the algebra that is implicitly in-volved in performing arithmetic with our stan-dard decimal notation—for example, many ex-plicitly invoked the distributive law whendiscussing multidigit multiplication. No suchawareness of the algebraic backbone of arithmeticwas shown by the American teachers.

In these first four chapters, KTEM also discussesissues of teaching methods. Without going into de-tail about this, I will report that the same limita-tions that teachers showed in giving a conven-tional explanation of a topic also prevented themfrom getting to the conceptual heart of the issuewhen using teaching aids such as manipulatives.

Thus, KTEM suggests that Chinese teachers havea much better grasp of the mathematics they teachthan do American teachers. The hard-nosed mightask for evidence that this extra expertise actuallyproduces better learning. Since Ma’s work did notextend to a simultaneous study of the students ofthe teachers, KTEM cannot address this question.However, the substantial studies of Stevenson andStigler [SS] do document superior mathematicsachievement in China. (The Stevenson-Stigler pro-ject provided part of the motivation for Ma’s work.)KTEM itself also provides some evidence of supe-rior learning in China and of a sort directly relatedto the knowledge of teachers, as indicated in theinterviews. The four interview questions were pre-sented to a group of Chinese ninth-grade studentsfrom an unremarkable school in Shanghai. They all(with one quite minor lapse) could do all the cal-culations correctly and knew the perimeter andarea formulas for rectangles. Over 60% found acounterexample to the student’s claim about areaand perimeter, and over 40% could make up astory problem for the division of fractions in ques-tion 3. These Chinese ninth-grade students demon-strated better understanding of the interview prob-lems than did the American teachers.

One should also entertain the possibility that Mawas overly optimistic in judging her group of Amer-ican teachers to be “above average”. However, thisrating is broadly consistent with evidence from amuch larger set of interviews conducted by Deb-orah Ball [B1–3] and also with the study [PHBL] of



over two hundred teachers in the Midwest. In thatstudy, for example, only slightly over half the sub-jects could provide an example of a number be-tween 3.1 and 3.11. The portion of satisfactory re-sponses to questions testing pedagogicalcompetence was considerably smaller. The resultsof KTEM are also consistent with massive informaltestimony from serious workers in professional de-velopment for teachers. The remarkable thing isthat this problem—the failure of our system to pro-duce teachers with strong subject matter knowl-edge and the negative impact of this failure—is notmore explicitly recognized. Furthermore, solvingthis problem is not a major focus of mathemati-cal education research and of education policy. Ihope that KTEM will provide impetus for makingit so.

KTEM gives us new perspectives on the problemsinvolved in improving mathematics education inthe U.S. For example, it strongly suggests thatwithout a radical change in the state of mathe-matical preparedness of the American teachingcorps, calls for teaching with or for “understand-ing”, such as those contained in the NCTM Stan-dards, are simply doomed. To the extent that theydivert attention from the crucial factor of teacherpreparedness, they may well be counterproductive.KTEM also indicates that claims that the tradi-tional curriculum failed are misdirected. The tra-ditional curriculum allowed millions of people tobe taught reliable procedures for finding correctanswers to important problems, without eitherthe teachers or the students having to understandwhy the procedures worked. At the same time,students with high mathematical aptitude couldlearn substantially more mathematics, enough tosupport various technical or academic careers.This has to be counted a major success.

However, times have changed. The success ofthe traditional curriculum has fostered a mathe-matically based technology, which in turn has cre-ated conditions in which that curriculum is nolonger appropriate. There are at least two reasonsfor this. First, we have cheap calculators that willdo (at least approximately) any calculation of theelementary curriculum (and much more) with thepush of a couple of buttons. These machines aretypically much faster and more reliable than we arein doing these calculations. We also have “computeralgebra” systems that will do more kinds of cal-culations than any single human knows how to do.It has always been one of the strengths of mathe-matics to seek reliable and systematic methods ofcomputation, which has often meant creating al-gorithms. Anything that has been algorithmizedcan be done by a computer. Automation of calcu-lation means that actually performing a calculationis no longer a problem working people usuallyhave to worry about.

At the same time, it means that calculation ismuch more prevalent than before. Hence, peoplehave to spend more time determining what calcu-lation to do. That is the second reason that math-ematics education needs to change. My daughterwas a solid mathematics student but had no en-thusiasm for the subject and did not expect touse it in whatever career she might choose. Nowshe works in management consulting, and shefinds that her high school algebra comes in handyin creating spreadsheets. Simply learning compu-tational procedures without understanding themwill not develop the ability to reason about whatsort of calculations are needed. In short, to func-tion at work, people now need more understand-ing and less procedural virtuosity than they did ageneration ago. (Who knows what they will needin another generation!)

The good news from KTEM is that there is noserious conflict between procedural knowledgeand conceptual knowledge: Chinese teachers seemto be able to develop both in their students. (Thisis another intuition of most mathematicians I knowwho have been studying educational issues: itshould be the case that procedural ability and con-ceptual understanding support each other. TheChinese teachers had a traditional saying to de-scribe this learning goal: “Know how, and alsoknow why.”) The bad news is that our currentteaching corps is not capable of delivering thiskind of double understanding: we can only rea-sonably ask them for procedural facility. Let us beclear that this is not a matter of teachers lackingcertification or teaching outside their specialty,which are both frequent problems that aggravatethe situation. The certification procedures, theteaching methods courses, most college mathe-matics courses, the recruitment processes, theconditions of employment, most current teacherdevelopment—none of these is geared to ensuringthat U.S. mathematics teachers have themselves theunderstanding needed to teach for understanding.In short, virtually the whole American K–12 math-ematics education enterprise is out of date.

How might the U.S. create a teaching corps withcapabilities more like those of the Chinese teach-ers? To begin to answer, we should try to be pre-cise as to what the differences are between the twogroups. From the evidence of KTEM, I would listthree salient differences:

1. Chinese teachers receive better early training—good training produces good trainers, in a virtu-ous cycle.

2. Chinese mathematics teachers are specialists.Making mathematics teaching a specialty can be ex-pected to increase the mathematical aptitude of theteaching corps in two ways: it reduces the man-power requirements for mathematics educationby concentrating it in the hands of the mathe-



matically most qualified teachers, and it raises theincentives for mathematically inclined people to be-come teachers. Beyond its recruitment implica-tions, it means that Chinese teachers have moretime and motivation for developing their under-standing of mathematics. This self-improvementis amplified by a social effect: specialization cre-ates a corps of colleagues who can work togetherto deepen the common teaching culture in math-ematics. Thus, making mathematics teaching aspecialty works in multiple ways to increase thequality of mathematics education.

3. Chinese teachers have working conditions whichfavor maturation of understanding. U.S. teachersspend virtually their whole day in front of a class,while the Chinese teachers have time during theschool day to study their teaching materials, towork with students who need or merit special at-tention, and to interact with colleagues. New teach-ers can learn from more experienced ones. All canstudy together the key aspects of individuallessons, an activity they engage in systematically.They can also sharpen their skills by discussingmathematical problems. Stevenson and Stigler [SS]have observed that time for self-development is ageneral feature of mathematics education in EastAsia, which, to go by TIMSS [DoEd1–3] as well as[SS], has the most successful systems of mathe-matics education in the world today.

The combination of training, recruitment, andjob conditions that prevails in China helps producea level of teaching excellence that Ma calls PUFM,“profound understanding of fundamental mathe-matics”. PUFM and how it is attained is the concernof Chapters 5 and 6. It is important to understandthat PUFM involves more than subject matter ex-pertise, vital as that is; it also involves how to com-municate that subject matter to students. Educa-tion involves two fundamental ingredients: subjectmatter and students. Teaching is the art of gettingthe students to learn the subject matter. Doing thissuccessfully requires excellent understanding ofboth. As simple and obvious as this propositionmay seem, it is often forgotten in discussions ofmathematics education in the U.S., and one of thetwo core ingredients is emphasized over the other.In K–12 education the tendency is to emphasizeknowing students over knowing subject matter,while at the university level the emphasis is fre-quently the opposite. (This cultural difference maywell be part of the reason some university math-ematicians have reacted negatively to the NCTMStandards. The emphasis on teaching methodsover subject matter is prominent in the recom-mendations and “vignettes” of this document.)Both these views of teaching are incomplete.

What educational policies in the U.S might pro-mote the development of a teaching corps in whichPUFM were, if not commonplace, at least not ex-

Getting the Mathematics to the StudentsMa’s notion of “profound understanding of fundamental

mathematics (PUFM)”, involves both expertise in mathemat-ics and an understanding of how to communicate with stu-dents. Teacher Mao, one of the teachers Ma identified as pos-sessing PUFM, eloquently expressed the need for both typesof understanding:

I always spend more time on preparing a class than onteaching, sometimes three, even four times the latter. I spendthe time in studying the teaching materials; what is it that Iam going to teach in this lesson? How should I introduce thetopic? What concepts or skills have the students learned thatI should draw on? Is it a key piece on which other pieces ofknowledge will build, or is it built on other knowledge? If it isa key piece of knowledge, how can I teach it so studentsgrasp it solidly enough to support their later learning? If itis not a key piece, what is the concept or the procedure it isbuilt on? How am I going to pull out that knowledge and makesure my students are aware of it and the relation betweenthe old knowledge and the new topic? What kind of review willmy students need? How should I present the topic step-by-step?How will students respond after I raise a certain question?Where should I explain it at length, and where should I leaveit to students to learn it by themselves? What are the topicsthat the students will learn which are built directly or indi-rectly on this topic? How can my lesson set a basis for theirlearning of the next topic, and for related topics that they willlearn in their future? What do I expect the advanced studentsto learn from this lesson? What do I expect the slow studentsto learn? How can I reach these goals? etc. In a word, one thingis to study whom you are teaching, the other thing is to studythe knowledge you are teaching. If you can interweave thetwo things together nicely, you will succeed. We think aboutthese two things over and over in studying teaching materi-als. Believe me, it seems to be simple when I talk about it, butwhen you really do it, it is very complicated, subtle, andtakes a lot of time. It is easy to be an elementary schoolteacher, but it is difficult to be a good elementary schoolteacher.

I would like to highlight the concern in Teacher Wang’s state-ment for the connectedness of mathematics, the desire tomake sure that students see mathematics as a coherentwhole. This is certainly how mathematicians see it, and to usit is one of the major attractions of the field: mathematicsmakes sense and helps us make sense of the world. For me,perhaps the most discouraging aspect of working on K–12educational issues has been confronting the fact that mostAmericans see mathematics as an arbitrary set of rules withno relation to one another or to other parts of life. Many teach-ers share this view. A teacher who is blind to the coherenceof mathematics cannot help students see it.

—R. H.



tremely rare? This question is discussed in Chap-ter 7, the final chapter of KTEM. I would like to addmy own perspective on the issue. The differences(1), (2), and (3) listed above suggest part of the an-swer.

Differences (2) and (3) are primarily matters ofeducational policy. No revolution in Americanhabits is required to create mathematics special-ists or to give them opportunity for study and col-legial interaction. What is mainly required is po-litical will.

Regarding difference (2), the manpower con-siderations which favor mathematics specialists be-ginning in the early grades are much stronger inthe U.S. than in China. The U.S information soci-ety has much higher demand for mathematicallyable people than does the predominantly ruraleconomy of China. Hence, schools face much heav-ier competition for mathematically competent per-sonnel, and every policy that could lower theirmanpower requirements or improve their com-petitive position would benefit mathematics edu-cation. The difference in technological level alsomakes the need for coherent mathematics educa-tion greater in the U.S. than in China. Simply par-titioning the present cadre of elementary teachersinto math specialists and nonmath would alreadyoffer the average child a better-qualified (elemen-tary) math teacher while relieving many others ofwhat is now an onerous duty, all without raisingoverall personnel requirements. Some educatorshave for some time been calling for mathematicsspecialists even in the elementary grades [Us]. Per-haps the evidence from KTEM that having teach-ers who understand mathematics can make a dif-ference already in the second grade (the usualtime for two-digit subtraction) can convince edu-cation policymakers to heed this call.

Regarding difference (3), testimony from inter-views of teachers with PUFM indicates that havingtime for study and collegial interaction is an im-portant factor in developing PUFM. Such timewould be most productive in the context of math-ematics specialists—both study and discussionwould be more focused on mathematics. Sched-uling this time might be more controversial thancreating specialists because it requires resources.In fact, in East Asia classes are larger than here,so a given teacher there handles about the samenumber of students as does a teacher in the U.S.[SS].The improvement in lessons promoted by studyand interaction with colleagues seems to morethan make up for larger class size. There is cur-rently in the U.S. a call to reduce class size. On theevidence of KTEM and [SS], I believe that the re-sources required for such a change would be bet-ter spent in eliminating difference (3).

What will be hardest is eliminating difference(1), that is, establishing in the U.S. the virtuous

cycle, in which students would already graduatefrom ninth grade or from high school with a solidconceptual understanding of mathematics, a strongbase on which to build teaching excellence. I ex-pect that movement in that direction will, at leastat the start, require massive intervention fromhigher education. New professional developmentprograms, both preservice and in-service, thatfocus sharply on fostering deep understanding ofelementary mathematics in a teaching context willneed to be created on a large scale. Current uni-versity mathematics courses will not serve; asKTEM makes clear, the needs of teachers at pre-sent are of a completely different nature from theneeds of professional mathematicians or techni-cal users of mathematics, for whom almost allcurrent offerings were designed.

I would recommend that these programs bejoint efforts of education departments and math-ematics departments to guarantee that the twopoles of teaching, the subject matter and the ped-agogy, both get emphasized. These departmentshave rather different cultures, and developing pro-ductive working relationships will not be a simpletask; but with sufficient backing from policymak-ers who understand the current purposes andneeds of mathematics education and the shortfallbetween current capabilities and these needs, somebeneficial programs should emerge.

While the greatest need for improvement isprobably at the elementary level, middle school andsecondary teachers should not be neglected in thenew professional development programs. Un-doubtedly they know more mathematics than thetypical elementary school teacher, but they toomust have suffered from the lack of attention tounderstanding during their early education. More-over, they need to deal with a larger body of ma-terial than do elementary teachers.

There is also the issue of texts. The Chineseteachers have materials, texts, and teaching guidesthat support their self-study. American texts tendto be lavishly produced but disjointed in presen-tation [Sc, DoEd1-3], and the teacher’s guides donot help much either. Thus, the intervention pro-grams should also work to create materials whichwill help teachers both learn and transmit a co-herent view of mathematics. Eventually, thesemight be the basis for new texts.

At least at the start, these programs should bemultiyear in scope, both so that teachers who donot have the favorable working conditions of Chi-nese teachers can nevertheless refresh and pro-gressively improve their understanding of math-ematics and so that those teachers who do obtainsuch working conditions can get to the level whereself-directed study can be a reliable mode of im-provement. One of the most outmoded ideas in ed-ucation is that a teacher can reasonably be ex-pected to know all that he or she needs to know,



of subject matter or teaching, at the start of work.Continued study, especially of subject matter,since teaching itself will provide plenty of oppor-tunities for learning about children, should be-come the norm. If a program of this sort is im-plemented successfully, it should gradually becomeless necessary. The step-by-step improvement ineducation provided by teachers with better un-derstanding and the gradual deepening of teach-ing culture by teachers interacting collegiallyamong themselves should allow elaborate devel-opment programs to shrink and eventually disap-pear or to shift to study of more sophisticatedtopics, becoming, in subject matter at least, morelike standard college mathematics courses. Thiswould constitute truly satisfying progress in oursystem of mathematics education. However, it willrequire great effort and resolve to achieve.

In summary, KTEM has lessons for all educa-tional policymakers. Legislators, departments of ed-ucation, and school boards need to understand thepotential value in creating a corps of elementary-grade mathematics specialists who have sched-uled time for study and collegial interaction. Uni-versity educators need to understand teacher train-ing in mathematics as a distinct activity, differentfrom but of comparable value to training scientists,engineers, or generalist teachers. I believe thatthese mutually supportive changes would give usa fighting chance for successful mathematics ed-ucation reform.

References[B1] D. BALL, The Subject Matter Preparation of Prospec-

tive Teachers: Challenging the Myths, National Cen-ter for Research in Teacher Education, East Lansing,MI, 1988.

[B2] ———, Teaching Mathematics for Understanding:What Do Teachers Need to Know about the SubjectMatter?, National Center for Research in TeacherEducation, East Lansing, MI, 1989.

[B3] ———, Prospective elementary and secondary teach-ers’ understanding of division, J. Res. Math. Ed., 21(1990), 132–144.

[Be] E. BEGLE, Critical variables in mathematics education:Findings from a survey of empirical literature, MAAand NCTM, Washington, DC, 1979.

[DoEd1] U.S. Department of Education, Pursuing excel-lence: A study of U.S. fourth-grade mathematics andscience achievement in an international context, U.S.Government Printing Office, Washington, D.C., 1997.

[DoEd2] U.S. Department of Education. Pursuing excel-lence: A study of U.S. eighth grade mathematics andscience teaching, learning, curriculum, and achieve-ment in an international context, U.S. GovernmentPrinting Office, Washington, D.C., 1996.

[DoEd3] U.S. Department of Education, Pursuing excel-lence: A study of U.S. twelfth-grade mathematics andscience achievement in an international context, U.S.Government Printing Office, Washington, D.C., 1998.

[Hi] E. D. HIRSCH, The Schools We Need and Why We Don’tHave Them, Doubleday, New York, 1996.

[Ho] R. HOWE, The AMS and mathematics education: Therevision of the “NCTM Standards”, Notices of theAMS 45 (1998), 243–247.

[NCTM] Curriculum and Evaluation Standards for SchoolMathematics, National Council of Teachers of Math-ematics, Reston, VA, 1989.

[PHBL] T. POST, G. HAREL, M. BEHR, and R. LESH, Intermedi-ate teachers’ knowledge of rational number con-cepts, E. Fennema, T. Carpenter, and S. Lamon (eds.),Integrating Research on Teaching and LearningMathematics, SUNY, Albany, NY, 1991, pp. 177–198.

[Sc] W. SCHMIDT et al., A Summary of Facing the Conse-quences: Using TIMSS for a Closer Look at UnitedStates Mathematics and Science Education, KluwerAcademic Publishers, Dordrecht, 1998.

[SS] H. STEVENSON and J. STIGLER, The Learning Gap, WhyOur Schools Are Failing and What We Can Learnfrom Japanese and Chinese Education, PaperbackReprint edition, Touchstone Books, January 1994.

[Us] Z. USISKIN, The beliefs underlying UCSMP, UCSMPNewsletter, no. 2 (Winter 1988).


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Knowing and Teaching Elementary Mathematics · Knowing and Teaching Elementary Mathematics ... The reviewer will refer to the book ... was persons, which children