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AUTHOR Addington, Susan; Clemens, Herbert; Howe, Roger; Saul, Mark TITLE Four Reactions to "Principles and Standards for School Mathematics." PUB DATE 2000-10-00 NOTE 10p. PUB TYPE Journal Articles (080) JOURNAL CIT Notices of the AMS; v47 n9 pp1072-1079 Oct 2000 EDRS PRICE MF01/PC01 Plus Postage. DESCRIPTORS Elementary Secondary Education; *Mathematics Curriculum; *Mathematics Instruction; *National Standards IDENTIFIERS *National Council of Teachers of Mathematics

ABSTRACT This essay presents four reactions to the National Council of Teachers of Mathematics (NCTM) "Principles and Standards for School Mathematics" (PSSM) .The four respondents include three university mathematics professors and one teacher. The first author describes the publication as refreshingly free of mathematics education jargon, carefully organized, and written with significant input from mathematicians. The second author states that PSSM is a responsible, insightful, balanced, and inspiring document. The third author acknowledges the strengths of the document but argues that there is too much emphasis on curriculum and insufficient emphasis on teacher training. The final author imagines the document being read 100 years after its publication and concludes that it is an important, if flawed, document, particularly in the area of geometry.(MM)

Reproductions supplied by EDRS are the best that can be made from the original document. Four Reactions to "Principals and Standards for School Mathematics."

Susan Addington Herbert Clemens Roger Howe Mark Saul

U.S. DEPARTMENT OF EDUCATION PERMISSION TO REPRODUCE AND Office of Educational Research and Improvement DISSEMINATE THIS MATERIAL HAS BEEN EDUCATIONAL RESOURCES GRANTED BY INFORMATION CENTER (ERIC) LI This document has been reproduced as received from the person or organization S. Addington originating it. 1:1 Minor changes have been made to improve reproduction quality. TO THE EDUCATIONAL RESOURCES Points of view or opinions stated in this INFORMATION CENTER (ERIC) document do not necessarily represent official OERI position or policy. 1

2 BEST COPY AVAILABLE Four Reactions to Principles and Standards for School Mathematics Susan Addington, Herbert Clemens, Roger Howe, and Mark Saul

In April 2000 the National Council of Teachers of Mathematics (NCTM).,published Principles and Standards for School Mathematics (PSSM). This document is an updated and unified version of the school mathematics standards issued by the NCTM over several years starting in 1989. The Notices invited four individuals to write short pieces describing their reactions to Principles and Standards, these pieces appear below. The Notices has published two other pieces about PSSM: a news item, "Updated NCTM Standards Released", June/July 2000, pages 683-684; and an article, "Principles and Standards for School Mathematics: A Guide for Math- ematicians", by Joan Ferrini-Mundy, September 2000, pages 868-876. Allyn Jackson

Susan Addington standards and every school district chooses its own curricula (sometimes several). Do the updated NCTM standards provide a good The U.S. has some current urgent problems. blueprint for reforming K-12 mathematics edu- Politicians, noticing that citizens now care about cation in the United States? education, have instituted quick fixes such as high- Yes. But reforming K-12 mathematics education stakes standardized tests, with budgets cut, prin- in the U.S. is like turning the Titanic around with cipals fired, and students failed if they do not a canoe paddle or perhaps like herding a million measure up. Most large states are suffering dire cats. teacher shortages, especially in urban schools; Despite the lack of national directives, the U.S. mathematics is one of the subjects suffering the K-12 educational system is amazingly uniform most. Most of the population thinks that learning and deeply set in its ways. The Third International mathematics is learning terms and practicing pro- Mathematics and Science Study documented how cedures, as do most elementary school teachers mathematics teaching in the U.S. is overwhelmingly and some middle and high school teachers. Even "a mile wide and an inch deep," covering dozens if a consensus were reached about what to teach of topics superficially. In a 1999 book, Stigler and and how, it takes time for teachers to learn new Hiebertl characterized U.S. mathematics teaching material and adjust to new teaching methods. as "learning terms and practicing procedures." I Three times through a lesson is a good rule of have found this to be accurate in most of the class- thumb for mastering a new teaching technique; for rooms I have visited, despite some notable excep- most K-12 teachers this means three years. And tions. districts, for the most part, have not accepted the fact that teachers need time every day to discuss On the other hand, the long history in the United and reflect on teaching and mathematics with their States of local control of schools and Americans' colleagues. general distrust of directives from the federal gov- So how could a bunch of mathematics teachers ernment mean that every state has its own get all these cats on the same track and steer the Susan Addington is associate professor of mathematics at oceanliner of U.S. education away from the ice- California State University, San Bernadino. Her e-mail bergs? address is [email protected]. In fact, the 1989 NCTM Standards were the first 1James W Stigler and James Hiebert, The Teaching Gap: such set of subject-matter standards developed by Best Ideas from the World's Teachers for Improving Ed- a professional organization; standards in other ucation in the Classroom, Free Press, 1999. subjects soon followed. The NCTM standards were

1072 NOTICES OF THE AMS VOLUME 47, NUMBER 9 very influential on state departments of educa- tion, school districts, grant agencies, and textbook How To Obtain Principles and Standards publishers. These organizations appreciated the Principles and Standards for School Mathematics is avail- fact that a knowledgeable group had organized the able on the World Wide Web, on paper, and on CD-ROM. subject and laid out general recommendations. In The Web versionisavailableathttp :// the absence of any federal standards, it became vir- standards.nctm.org/.A PDF version may be tually mandatory for textbook publishers to claim purchased onlineathttp://www.nctm.org/ that their books adhered to the NCTM standards, standards/.Paper and CD-ROM versions may be despite the fact that some of the books were the ordered on the Web athttp://www.nctm.org/ same old thing with cosmetic changes. Many (prob- standards/buyonl i ne/or by telephone at 800- ably not most) teachers began to rethink their 253-7566 (from overseas, 1-703-620-9840, extension 2601). teaching methods, and many districts tried new The postal address for orders is: curricula. National Council of Teachers of Mathematics "Math wars" and backlashes notwithstanding, I 1906 Association Drive expect the NCTM standards will continue to serve Reston, VA 20191-9988 as national guidelines. Since we may never have a national curriculum, this is the closest we will get The book (#719) is $45 (NCTM members $36), the CD- to national standards. The NCTM is a respected pro- ROM (#736) is $30 (members $24), and the PDF fessional organization, and it has done a careful version is $30 (members $24). Those purchasing job on the Standards. paper copies of the document can also request a free CD- The updated version of the Standards, released ROM. For orders under $50 there is a shipping and han- this year under the title Principles and Standards dling charge of $7. Discounts are available on larger or- for School Mathematics, was written with significant ders; consult the NCTM for further details. input from mathematicians; this was not the case A. J. with the 1989 Standards. Stung by misunder- standings of recommendations such as "decreased same material and more is on the Web; eventually attention to problems by type" and accusations of the Web version will include a search engine, more removing calculation and proof from the curricu- E-examples, video clips, and other supporting ma- lum, the NCTM has bent over backwards to be clear terial. The Web version is especially convenient for about its recommendations, supporting them with novices in mathematics education: one does not explicit research citations, sample problems, have to invest the time or money to order a paper vignettes of lessons, and samples of student work. copy. Principles and Standards is intended to be read Chapter 8, "Working Together to Achieve the Vi- by many different groups: mathematics teachers sion", outlines what must be done and by whom and supervisors, local and state educational ad- to successfully reform mathematics education. ministrators, mathematicians, politicians, parents, Much of the vitriol in the "math wars" that broke and business and community leaders. out in the 1990s comes from deeply held beliefs The document is refreshingly free of mathe- about mathematics education: that traditional Eu- matics education jargon. The writing is nonpedan- clidean geometry is the best, or the worst, way to tic and readable by any interested, educated per- teach mathematical reasoning; that paper-and- son. Readers will find little gems of mathematics pencil calculation is the key to, or a major imped- lessons, such as the young child who discovers, iment to, understanding numbers; that calcula- using a calculator, that one cannot ever get to 100 tors should not be used until paper-and-pencil by counting by 3s, then uses other mathematical arithmetic has been mastered or that they should props to explain why. replace paper-and-pencil arithmetic. Mathematics The graphic design is clean and restful to the education research does not give definitive an- eye. Unlike in many K-12 mathematics textbooks, swers to these questions, and many thoughtful there are no extraneous elements. In addition to teachers would agree with parts of each opinion. text there are mathematical illustrations, diagrams We need to set aside our differences and get down and graphs, reproductions of student work, ref- to the urgent business of convincing the American erences to "E-examples" (interactive activities or citizenry that teachers and schools need respect video clips on the Standards Web site), and pho- and financial support, convincing administrators tographs of classrooms showing ways of organiz- that teachers need continuing professional devel- ing a mathematics class other than by having stu- opment, convincing current and future teachers dents sit in rows of desks, listening or writing. that knowing more mathematics is a vital profes- Principles and Standards is carefully organized: sional requirement, and convincing students that the edges of pages are color-coded by grade band, it is possible and important to understand math- with the ten standards listed down the side as ematics. tabs. So it is easy to find, say, "Geometry" in grades Reading this careful document from the NCTM 9-12, or "Reasoning and Proof" in pre-K-2. The is a good start.

OCTOBER 2000 NOTICES OF THE AMS 1073

4 Herbert Clemens concepts and the modeling of proce- dures used in solving problems. (PSSM, Principles and Standards for School Mathemat- page 20) ics (PSSM) seems pretty good to me! Who can argue with a document that sets forth, in a very clear and The "math wars" that have flared up yet again in well-organized manner, the consensus agenda for and outside our profession are encapsulated in the the subject matter to be learned in the K-12 cycle? word "thus" in the above quotation from PSSM. Who can argue with the carefully prepared and doc- Traditionalists: The above implication is a com- umented classroom vignettes to illustrate student plete non sequitur, dangerous in the hands of thinking and activity? Certainly no one with much all but the most mathematically sophisticated classroom experience. of school teachers. On the other hand, who can not argue with the Constructivists: Learning procedures without suggestions for the teaching of the subject matter conceptual understanding is pointless and ul- and the tentative indications of mathematics ed- timately useless. Anyway, machines can do ucation research included in PSSM to bolster those procedures far better than humans. suggestions? And that argument is precisely the What is perhaps underappreciated in such de- point. There is no consensus on teaching mathe- bates is the fact that rote learning is not the enemy matics, and perhaps in the best of all possible of mathematical understanding, but an essential worlds, there need not be one. In any case, each vehicle to it. For example, many if not most stu- individual teacher and user of PSSM will have to dents cannot develop an intuition for abstraction critique, accept, reject, incorporate, and adapt and symbolic manipulation without first correctly what is put forth in the document as it pertains to manipulating a lot of symbols. Perhaps one of the how teachers do their jobs. So if there is a danger principal disappointments in the "math wars" has related to PSSM, it does not seem to me to be a dan- been the failure of the participants to acknowledge ger in the document itself. The danger is that the that appreciation of this fact is shared by both individual user, for whatever reason, cannot or sides. will not read the document critically; the danger PSSM begins to recognize the synergy between is greatest with regard to teaching methods. A rote training and conceptual learning. But, as is nat- teacher can read it and "check off" mathematical ural for a document prepared and vetted by a large topics, but cannot read it and "check off" teach- cross-section of the professional community, it ing methods, even though I firmly believe that reflects the fact that our community has a long way there are many teachers and mathematics super- to go to get the balance right. The evolution of the visors around the country who will do just that. reading wars finally led to the recognition of the In fact, PSSM sets a very high standard for the necessary balance and constructive tension be- individualization of teaching and learning, per- tween "phonics" and "whole language". It is the haps a higher standard than the U.S. market will same issue in mathematics. bear at this point in history. The last time the bar Of course we all believe that teachers should was raised too high, the result was the so-called teach with conceptual insight and clarity and "new math", which cannot be judged a success by should use all the tools at their disposal to con- any measure. PSSM reflects much that is good in vey the mathematical essence clearly and persua- our attempts to rethink and to yet again reform sively to as many students as possible. But let us mathematics education. But in spite of all that has not forget that training "by rote" is a less danger- been learned and in spite of the protestations of ous weapon in the hands of a teacher of limited the finest researchers and teachers, it remains a mathematical preparation and understanding than fair question to ask whether PSSM falls prey to the are attempts to foster understanding in others same fundamental mistake of raising the bar too that one does not have oneself. high. It is not a question of what PSSM says, but Proof is a very difficult area for under- of how it will be read by teachers and administra- graduate mathematics students. Per- tors. haps students at the secondary level Which brings me to PSSM and the "math wars": find proof so difficult because their ...most of the arithmetic and algebraic only experience with writing proofs has procedures long viewed as the heart of been in a high school geometry course, the school mathematics curriculum can so they have a limited perspective.... now be performed with hand-held cal- Reasoning and proof should be a con- culators. Thus, more attention can be sistent part of students' mathematical given to understanding the number experience in prekindergarten through grade 12. Reasoning mathematically is Herbert Clemens is professor of mathematics at the Uni- a habit of mind, and like all habits, it versity of Utah. His e-mail address is clemens@ must be developed through consistent math . utah .edu. use in many contexts. (PSSM, page 56)

1074 NOTICES OF THE AMS VoLume 47, NumBER 9

5 If mathematical reasoning and proof are so dif- fields of competing ideologies and interests, yet the ficult for undergraduate mathematics majors, is the framework it lays out is mature, insightful, de- same not true one hundred times over for most el- tailed, and even-handed. I hope only that those who ementary and middle school teachers? And if that read and use PSSM do so with as much thorough- is the case, how on earth can we ask teachers to ness, responsibility, thoughtfulness, and critical foster these skills in their students, except perhaps spirit as the writers brought to their task. with vehicles (not yet invented) that work in a "teacher-proof" context? Teachers by and large are "kid people", not "math people". The "math peo- ple" are busy at an easier job, earning four times Roger Howe as much at some dot-com enterprise down the I would encourage AMS members who read the street. Let us be very careful about asking teach- PSSM to keep in mind the larger picture. Elements ers for more than we have any right to expect. We of the larger picture include: run the risk of disrespecting the profession and a) Here is a document that says that it is the its high level of skill and dedication and of foist- ideas in mathematics that count. Four of the ing upon it the responsibility for remedying a de- ten standards are about ideasproblem solv- ficiency that we as a society cause by the reward ing, reasoning and proof, communication, structure we have designed and perpetrated. It is and connections. of course essential that a document like PSSM lay I would be happy for the U.S. to commit to math- out a vision and goals. But it is also important to ematics education with a heavy emphasis on ideas, guide each teacher of mathematics to find where especially since he or she as an individual fits into the enterprise b) PSSM explicitly corrects the lesson taken by and give contexts for relatively successful teach- many from the 1989 Standards that ability ing that fit an individual teacher's own mathe- actually to do arithmetic does not matter. matical knowledge and ability. On this point, per- PSSM states clearly that skill at computation haps PSSM still reflects a bit of a "one-size-fits-all" is a goal. In pre-K-2, students should learn mentality. In fairness, however, there are indica- "addition facts"the sums of all pairs of tions throughout PSSM that show that we are be- single-digit numbers. In 3-5 they should also ginning to understand and grapple with this prob- learn the multiplication tables and "develop lem. fluency" with whole-number arithmeticall Of course, when it comes to my own area, geom- four operations. In 6-8 they should "develop etry, I must admit that I cringed when I read: and analyze algorithms for computing with Geometric ideas are useful in repre- fractions, decimals and integers, and de- senting and solving problems in other velop fluency in their use." There is even a areas of mathematics and in real-world statement on page 153 that students should situations, so geometry should be in- know the multiplication tables by the end of tegrated when possible with other fourth grade. areas. (PSSM, page 41) This does not mean that PSSM is a perfect doc- ument. A mathematically attentive reader will find It seems to me that the integration of geometry with items that jar at all levels and in most parts of the other areas (indeed, the disappearance of it as a document, at least the ones that I looked at. Here separate course in some schools) is quite naturally are a few samples. On page 144, lines 4-7, the in- the enemy of logical development. It is really hard dicated calculation appears to produce a fifth- to track a concept and its logical development grader 2 meters tall and an adult with a height of when it pops up periodically only in service to 2.5 meters. On a higher level, on page 314, it is ex- some other part of mathematics or application. To plained how to set up a coordinate system to pro- me it would be far better for all concerned if high duce a manageable proof that the medians of a tri- schools did a lot less in the way of advanced place- angle are concurrent. However, the "best", or ment calculus courses and spent more time on mathematically most natural, proof of this is co- deepening understanding of "elementary" mathe- ordinate-free, with vectors, which are one of the matics like geometry and algebra/numerics. This recommended techniques, along with coordinates, change would be especially beneficial for "mathe- in the first bullet of the third geometry expecta- matically gifted" students, another group whose tion (page 308). Or, just across on page 315, why needs PSSM begins to recognize. is so much effort expended with no attempt to for- All in all, I would like to take my hat off to the mulate the precise relation between a transfor- writers, reviewers, and others involved in the prepa- mation and a matrix: that the k-th column of the ration of PSSM. It seems to me to be a responsible, matrix is the image of the k -th standard basis insightful, balanced, and inspiring document. The problems of teaching and learning mathematics it Roger Howe is professor of mathematics at Yale Univer- addresses are fundamental ones fraught with mine- sity. His e-mail address is howe@math .yal e . edu.

Ocroest 2000 NOTICES OF THEAMS 1075

6 vector? On a still higher level, a main expectation b) Separate certification and substantially of the "Geometry" standard for 9-12 is the use of more training for the intermediate level (4-8). ti.ansformations and symmetry to analyze and ex- c) Specially designed courses built to connect plore mathematical situations. However, on page important mathematical ideas to classroom 300 in the "Algebra" standard, there is a discus- experience. sion of "exponential functions" that is summa- d) Particularly for elementary teacher can- rized by this sentence: didates, instruction adapted to their state of This type of exploration should help preparedness. students see that all functions of the 2) Sustained in-service training focused on mathematics, including night and summer courses formf(x) = abx + cshare certain prop- and teacher institutes. erties. 3) Continued learning on the job, by self-study Although changing the sign of a and changingb and collegial interaction. to 1 Ibare mentioned in the discussion, there is no Point (3) will probably require considerable sup- hint that any function in this family can be trans- port, at least at first. The in-service training of formed to any other by simple transformations of point (2) could provide some of this support. Sup- the coordinates (e.g., of the type x °ix + fi ,and port could also come from mathematics resource similarly for y). This would seem to violate the teachers or mathematics specialists in elementary "Connections" standard. schools. I believe that it would be a mistake to conclude Developing a system with these features re- from items like these that PSSM is a document quires mutually reinforcing action by educational that should be discounted. Such items are proba- policymakers, schools of education, and mathe- bly nearly unavoidable given the way it was created: matics departinents. The essential enabling role will by a committee of twenty-seven, chosen to repre- fall to educational policymakers. To make the pro- sent a diversity of points of view on mathematics grams of study in points (la) and (lb) viable, cer- education. Most of its readers will not notice such tification requirements will have to be beefed up, things. especially at the middle school level. Probably also Another aspect of the larger picture is that PSSM there would need to be some fairly probing ex- is primarily concerned with curriculum. Curricu- amination for certification. Time will have to be pro- lum is the aspect of mathematics education on vided in the teacher's work week for the activities which mathematicians tend to focus. However, I be- of point (3), and incentives to ensure participation lieve that curriculum is a secondary issue in U.S. in activities of point (2) will likely also be needed. mathematics education today. The primary issue Given the appropriate certification policies and is the capacity of the teaching corps. A superb working conditions, education schools and math- curriculum is beside the point if it makes demands ematics departments would then have to cooper- that a teacher cannot meet. A potentially superior ate to create the courses of point (1) and the pro- curriculum may in practice be inferior if imple- grams of point (2), and these efforts would also mented incorrectly. Conversely, an excellent require appropriate support. Steps in this direction, teacher can make up for a lot of deficiencies in a in Connecticut since the late 1980s and recently textbook or program. I feel that the main question in Puerto Rico, have had encouraging results. facing U.S. mathematics education is how to cre- Rather than debate at great length about curricu- ate a teaching corps that can deliver instruction that lum, let us get serious about enabling teachers to will promote understanding, reasoning, connec- develop the mathematical skills they need to teach tions, and communication in mathematics, as well well. as technical skill and broad knowledge. A main finding of mathematics education research during the past decade is that doing even elementary mathematics right involves substantial mathe- Mark Saul matical judgment and understanding, as well as an Normally, one must not pass on information of- ability to listen to students and adapt instruction fered anonymously, but the present situation is to their needs. We do not currently know how we something of a special case. can reliably produce teachers with this combina- A colleague at a large state university discovered tion of skills. It seems clear that it will take sus- the manuscript transcribed below in a container tained and purposeful effort. I would guess that that might be described as a metal shoe box. It it would need at least the following combination seems to be a message from the future. Is it this, of ingredients. or a fraternity hoax? I offer this decision to the 1)More and better mathematical training of prospective teachers. This has several aspects: Mark Saul is an associate editor of the Notices and teaches a) More mathematics courses for primary in the Bronxville School District. His e-mail address is (K-3) candidates. [email protected].

1076 NOTICES OF ME AMS VOLUME 47, NumBER9 reader, together with the very real issues the text accommodations be made as needed raises. to promote access and attainment for all students.... Achieving equity re- May 15, 2098 quires a significant allocation of human and material resources...(PSSM, pages A Brief Communication to the History 12-14). Division of the American Mathematical Society Why did the authors feel impelled to include such obvious and commonplace assertions? An This communication describes a manuscript re- historical reading will give us the answer. At the cently uncovered in the archives of the Lappan turn of the millennium, the United States found it- Memorial Library. As you know, the Great Comet self the strongest and wealthiest country in the of 2065 wiped out all electronic records, ushering world and in the history of the world. Yet its wealth in a Little Dark Age, from which we are just now was distributed with significant inequalities. No bal- recovering. The present manuscript represents a ance had yet been struck between the economic in- rare glimpse into a world over which history has centive necessary for productivity and a broad in- drawn a curtain. clusion of the entire population in that productivity The manuscript is entitled Principles and Stan- to protect it from the ravages of social divisions. dards for School Mathematics (PSSM) and is a prod- This was the great historical task of the twenty-first uct of the National Council of Teachers of Mathe- century. matics (NCTM), an organization that of course still Other "principles" are likewise dated. The odd- exists. The document begins with a "Statement of est thing, however, about this section of the doc- Intention": ument is that it treats almost nothing specific to This document is intended to mathematics. The principles stated mostly apply, with only slight adjustments, to the learning of any set forth a comprehensive and co- other subject. herent set of goals for mathematics for Let us turn to the role of the mathematician in all students... ; education at the time, a role that was not yet well defined. Some mathematicians involved them- serve as a resource for teachers, ed- selves directly in teaching precollege students and ucation leaders, and policy-makers... ; even contributed towards an understanding of teaching and learning by doing educational re- search. Many more mathematicians were involved guide the development of curriculum in preparing "preservice" teachers, a process that frameworks, assessments, and in- was then rather haphazard (the current document structional materials; acknowledges but does not detail the problems in this area). Still others assumed the role of critic, stimulate ideas and ongoing conver- commenting on the work of others without per- sations.... (PSSM, page 6) forming any themselves. No one of these roles is This primary source offers a wealth of information today thought inappropriate, but no one of them about the era that produced it, too much to treat dominates any individual mathematician's activi- thoroughly in the present brief communication. We ties as it did in the past. will therefore focus on one aspect: the role played This lack of balance between research and out- by mathematics education in American society of reach was a grave fault in the structure of the the time and particularly the role played by math- American mathematical community. Even at that ematicians in education. time there were models, such as those in Eastern The document first lays out several basic tenets Europe, where mathematicians were much more that we now take for granted. These are contained closely and successfully integrated into the edu- in a set of six overarching "principles" about the cational efforts of the nation. learning and teaching of mathematics, many of The main part of the document, the ten "Stan- which sound dated. For example, the "Equity Prin- dards", tells us more about this problem. Some of ciple": the labels on these standards are indeed strange. Today, we would think of algebra as essentially the The vision of equity in mathematics study of binary operations, so that a "Number and education challenges a pervasive soci- Operation" standard would grow, as the grade lev- etal belief in North America that only els get higher, into a standard for algebra. But it some students are capable of learning turns out that "Numbers and Operations" dis- mathematics.... Equity does not mean cusses mostly the relationship between operations that every student should receive iden- and algorithms, then a sore point in education. At tical instruction; instead it demands the time, many people erroneously assumed that that reasonable and appropriate a mere mastery of algorithms would somehow

Ocrosur 2000 NOTICES OF THE AMS 1077 induce an understanding of arithmetic. It seems and Euclid's set of axioms the only way to de- that this special standard was needed to address scribe geometry. Euclid's was a towering accom- this error. plishment which has already lasted more than two Even with the burden of this historical baggage, millennia. We can continue to honor this accom- the wording of this standard is strikingly modern. plishment while extending it and finding alterna- There are important attempts to use findings from tives. the nascent field of cognitive psychology, especially During the twentieth century the wisdom of for the early years. There are concrete checkpoints separating the studies of geometry and axiomat- and benchmarks offered, both in understanding ics became clear. For example, until about the era and in computational ability. There are fascinat- of the PSSM, geometry in elementary school con- ing ideas offered about the use of technology. It sisted largely of the learning of nomenclature. In is interesting that even with the few resources de- middle school, geometry was used as a platform veloped at the time and the small bits of knowl- to do algebra, if it was used at all. Geometry was edge accumulated, the authors got right many started in earnest only in high school, because it things that have been confirmed over time. It seems was felt necessary to go about that study ax- that a collective intuition about education some- iomatically. The "Geometry" standard in PSSM re- times proves accurate, a lesson we might take in flects a sea change in this view of the subject, de- our own time. scribing a unified approach to the subject from The "Number and Operations" standard for early childhood through high school. grades 9-12 shows most clearly the need for the This standard, too, betrays the flaws of the cul- input of the mathematician. Much is made of the ture that produced it. For example, mention is study of vectors and matrices and the exploitation made (pages 235, 313) of how the introduction of of parallels in structure between these objects and coordinates links algebra with geometry, but other various number systems. But the discussion con- useful links between these two branches of math- tains significant gaps. The next generation of math- ematics are not mentioned. Another connection ematicians was able to work more closely with ed- with algebra is missed when the composition of uca tors and examined more closely the geometric transformations is brought up (page mathematical and cognitive structures involved 314). Oddly, no mention is made of the noncom- here to find a pedagogy that reflects these struc- mutative nature of this operation or its relation- tures. ship to matrix multiplication. The "Algebra" standard is not that at all, but a It seems important to note that the lack of such hodgepodge of mathematical ideas packaged and mathematical connections cannot be attributed to labeled for convenience. Throughout the grades the organizational problems. The document describes study of algebra is confounded with the study of an elaborate system of "Association Review functions, of models, of pattern, and of several Groups"people in mathematics, statistics, and other important mathematical concepts related to other fields who offered input. There are also but not part of algebra. One might well imagine the mathematicians listed among the authors of the lumping and splitting that resulted in the some- document. times strange taxonomy of the PSSM. The authors' Thus it is not an organization at fault, but a cul- intuitions are mostly correct but have not been fo- ture. However, the culture seems to have been cused through the lens of the mathematician. more resilient than this one document shows. The There remains in this document a trace of a ten- process of setting down a single set of goals sion that permeated the decade that preceded it. brought to the attention of the mathematical com- Many educators had decided, largely without evi- munity exactly the gaps that seem so obvious to dence, that mathematics could be made more "in- us now. Thus the document contributed as much teresting" or "exciting" through the use of what was in its flaws as in its successes. termed "real-world applications". However, this The historical record includes a number of ar- document is fairly balanced and includes a good tifacts of the controversy that surrounded math- mix of elegant "pure" mathematics and more ematics education at the time. In some cases both rough-and-tumble applications. educators and mathematicians tried to think of The standard called "Geometry" is in much bet- PSSM as a curriculum in itself. We now know that ter shape, perhaps because this branch of mathe- there are good reasons why curriculum should matics has the longest and most successful history not be set on this high level, but on a lower level. of any as a classroom subject. Even today we can A paper curriculum changes when it is imple- think of ourselves as just coming out from under mented in ways that are dependent on such fac- the shadow of Euclid. Geometry in the schools was tors as the teachers implementing the curriculum, for centuries synonymous with axiomatics, and the students' social environment, or the form of there remain today mathematicians "married" to assessment of the curriculum. Especially in a set- Euclid. For example, some think that straightedge ting as diverse as American society, we have found and compass are the only viable construction tools it more effective to implement refined versions of

1078 NOTICES OF m AMS VOLUME 47, NUMBER 9 documents such as PSSM rather than impose cur- riculum from a level of management that cannot react to a close observation of the intended audi- ence. Another bone of contention seems to have been the mathematical accuracy of the document. Even today there are mathematicians who will inspect documents such as this one for errors, like some- one attending a concert only to hunt for wrong notes. Such flaws do not significantly affect the strength of the document. A curriculum is not a mathematical argument. The latter has the struc- ture of a balloon: a single hole renders it useless. But a document guiding educational practice can hold up despite a few weak points or even down- right errors. There are, however, deeper flaws, where im- portant mathematical questions go begging. For ex- ample, on page 292 we have: "...students can learn to appreciate vectors as a means of simultane- ously representing magnitude and direction." Does this really lie at the heart of the vector concept? Or on page 297: "Students should...study the be- havior of polynomial, exponential, rational, and pe- riodic functions." These categories are not paral- lel: perhaps students should also acquire a taste for Greek, Italian, Chinese, and orange food. These flaws are more important, and more worthy of the attention of the mathematician, than small slips in logic or exposition. To conclude this communication I would like to offer the suggestion that this artifact from the past can teach us something about our own en- deavors. As we see how our forebears struggled and erred, we must wonder what will be thought of our own efforts one hundred years hence. For we are not ready for The Definitive Document in educa- tion, and it is not clear that such a document is even possible. A historical perspective on this PSSM, as well as on our own work, will allow us to recog- nize our own errors and use them, as well as our triumphs, to guide our next steps.

OcroBER 2000 NOTICES OF THE AMS

1 0 1/ 04/15/2002 13:16 19153519823 ADDINGTON OR DENNIS PAGE 01

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