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INSTITUTIONPUB DATENOTE

AVAILABLE FROM

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JOURNAL CITEDRS PRICEDESCRIPTORS

ABSTRACT

DOCUMENT RESUME

TM 028 838

Raimi, Ralph A.; Braden, Lawrence S.State Mathematics Standards: An Appraisal of Math Standardsin 46 States, the District of Columbia, and Japan.Thomas B. Fordham Foundation, Washington, DC.1998-03-0071p.; For other state standards reports, see TM 028 835-37and CS 216 412.Thomas B. Fordham Foundation, 1015 18th St., N.W., Suite300, Washington, DC 20036; phone: 202-223-5452; fax:202-223-9226; toll-free phone: 888-TBF-7474; World Wide Web:http://www.edexcellence.netCollected Works Serials (022) Reports Evaluative

(142)

Fordham Report; v2 n4 1998MF01/PC03 Plus Postage.*Academic Achievement; Core Curriculum; Elementary SecondaryEducation; Knowledge Level; Mathematics; *MathematicsInstruction; State Programs; *State Standards

The Thomas B. Fordham Foundation has commissioned studies ofstate academic standards in five core subjects. This is the fourth of thesestudies, focusing on state standards for mathematics. For this evaluation ofmathematics standards, researchers developed nine criteria under the fourareas of: clarity, content, reason, and negative qualities. These criteriawere applied to the standards documents of 46 states and the District ofColumbia, and standards for Japan were reviewed for comparison purposes. Theremaining four states either had no standards or did not make current draftsavailable. Only three states received a grade of "A," and only nine receiveda grade of "B." More than half received either a "D" or an "F." The principalfailures of these documents stem from the mathematical ignorance of thewriters of the standards, sometimes compounded by carelessness and sometimesby a faulty educational ideology. The average mathematics teacher can be ledto a better grasp of the material that should be taught and the way to teachit than the writers of the standards seem to believe. The Japanese standardsdocument, exemplary in most respects, falls short in the category of reason.Notes are presented for each state and Japan. An appendix lists the documentsreviewed. (Contains one table.) (SLD)

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An Entry from the Diaries of John Adams (1735-1826)

June 1, 1756.

Drank Tea at the Majors. The Reasoning of Mathematics is founded on certain and infallible Principles. Every Word theyUse, conveys a determinate Idea, and by accurate Definitions they excite the same Ideas in the mind of the Reader that werein the mind of the Writer. When they have defined the Terms they intend to make use of, they premise a few Axioms, orSelf evident Principles, that every man must assent to as soon as proposed. They then take for granted certain Postulates,that no one can deny them, such as, that a right Line may be drawn from one given Point to another, and from these plainsimple Principles, they have raised most astonishing Speculations, and proved the Extent of the human mind to be morespacious and capable than any other Science.

3

tV'ordham MeportVol. 2, No. 3 March 1998

State Mathematics StandardsAn Appraisal of Math Standards

in 46 States, the District of Columbia,and Japan

byRalph A. Raimi andLawrence S. Braden

4

4OORDHAM

THOMAS B.

UNDATIONOUTSIDE THE BOX

EXPERT ADVISORY COMMITTEE

Henry Alder, Ph.D., Professor Emeritus of Mathematics, University of California, Davis, California

Harold Stevenson, Ph.D., Professor of Psychology, University of Michigan, Ann Arbor, Michigan

TABLE OF CONTENTS

List of Tables

Foreword by Chester E. Finn, Jr.

Executive Summary

iii

vii

I. The Scope of this Study- 1

The Documents Themselves

II. What Standards Are 3

III. The Standards Movementin Recent Years

4

IV. Judgment, Criteria and the 6Ratings: An Overview

V. Definitions of the Criteria 8

VI. Examples for the Criteria 12

VII. Rating the States: A Tableof Results

18

VIII. Notes, State-by-StateAlabama 23

Alaska 23

Arizona 24Arkansas 24California 25

Colorado 27Connecticut 27Delaware 28District of Columbia 28Florida 29Georgia 29Hawaii 30Idaho 30

Illinois 31

Indiana 31

Kansas 32Kentucky 32Louisiana 33Maine 33

Maryland 34Massachusetts 35Michigan 36Mississippi 36Missouri 37Montana 37Nebraska 38New Hampshire 38New Jersey 39New Mexico 40New York 40North Carolina 41North Dakota 42Ohio 42Oklahoma 43Oregon 43Pennsylvania 44Rhode Is Ian, 45South Carolina 46South Dakota 47Tennessee 48Texas 50Utah 51Vermont 51Virginia 52Washington 53West Virginia 53Wisconsin 54Japan 55

Appendix: List of DocumentsReviewed 5 7

6

TABLES

National Report Card ix

Table 1 20Numerical Ratings for the States

FOREWORD

The Thomas B. Fordham Foundation is pleased to pre-sent this appraisal of state mathematics standards by RalphA. Raimi of the University of Rochester and Lawrence S.Braden of St. Paul's School.

This is the fourth such publication by the Foundation.In July 1997, we issued Sandra Stotsky's evaluation of stateEnglish standards. In February 1998, we published exami-nations of state standards in history and geography. Sciencefollows.

Thus, we will have gauged the states' success in settingstandards for the five core subjects designated by the gover-nors and President Bush at their 1989 education "summit"in Charlottesville. The national education goals adoptedthere included the statement that, "By the year 2000,American students will leave grades four, eight, and twelvehaving demonstrated competency in challenging subjectmatter including English, mathematics, science, historyand geography." Although other subjects have value, too,these five remain at the heart of the academic curriculumof U.S. schools.

All are critically important, to be sure, but mathematicshas special significance in today's debates about boostingthe performance of U.S. students by setting ambitious stan-dards for their academic achievement.

Mathematics is, of course, the third of the "three R's."Practically nobody doubts its central place in any seriouseducation, its intellectual significance, or its practicalvalue. Math is ordinarily the second subject (after reading)that young children encounter in primary school. "Mathaptitude" constitutes half of one's S.A.T. score. And it wasin no small part the weak math performance of Americanyoungsters on domestic and international assessments thatled us to understand that the nation was at risk. (Because itis universal, because it is sequential and cumulative, andbecause its test questions are easy to translate, mathematicshas long been the subject most amenable to illuminatingcross-national comparisons of student performance.)

Math also blazed a trail into the maze of national stan-dards. Even as the Charlottesville summit was convening,the National Council of Teachers of Mathematics(NCTM) was putting the finishing touches on its reportentitled Curriculum and Evaluation Standards for SchoolMathematics. In the ensuing decade, that publication andits progeny have had considerable impact on U.S. educa-tion, not least on the state math standards reviewed in thefollowing pages. I have no doubt that, of all the "nationalstandards" set in the various academic subjects, these havebeen the most influential. Indeed, I have heard policy mak-ers declare that what America needs in other academicsubjects are counterparts to the "NCTM math standards."

It is vital to understand, however, that the NCTM'smission was notand today is notthe codification of tra-ditional school mathematics into clear content andperformance standards. Rather, NCTM's main project wasto transform the teaching and learning of mathematics inU.S. schools.

The effects of that hoped-for transformation on statemath standards are abundantly clear in this appraisal.Some readers may judge that the states should go furtherstill to transform their expectations for students and teach-ers in the direction set forth by NCTM. Others will judgethat they have gone much too far already. In any case, it'snoteworthy that today, nine years after it was unveiled,"NCTM math" no longer commands the public consensusthat it once appeared to have. California, for example,recently adopted new statewide standards that could fairlybe termed a repudiation of the NCTM approach.

The important thing to know about the present docu-ment is that we did not ask its authorsa distinguisheduniversity mathematician and a deeply experienced schoolmath teacherto grade the states on how faithfully theirstandards incorporate the NCTM's model for math educa-tion. Rather, we asked them to appraise state standards interms of their own criteria for what excellent math stan-dards should contain.

Advised by two other nationally respected scholars, theauthors did precisely that. They developed nine criteria(under four headings) and then applied them with greatcare to the math standards of 46 states and the District ofColumbia. (The remaining four states either do not havepublished standards or would not make their current draftsavailable for review.) For comparison purposes, the authorsalso describe Japan's math standards and apply their criteriato these.

The results are sobering. Only three states (California,North Carolina, and Ohio) earn "A" grades, and just nineget "B's." Those 12 "honor" grades must be set alongside 16failing marks (and seven "C's" and 12 "D's").

The results differ markedly from those of the recentCouncil for Basic Education (CBE) appraisal of the "rigor"of state math standards at grades 8 and 12. The CBE studybegins with a list of performance standards expressed in 51clauses (or "benchmarks") for the 8th grade and 30 for the12th. These clauses are largely drawn from the NCTMstandards of 1989. The state documents under study werethen scanned for those 81 demands, which, when present(and weighted by their closeness to the template clauses),were counted up for a total score.

The present document does not begin with a list of thiskind, and similarity to the NCTM standards was not a

desideratum. The criteria used by Braden and Raimi arewell described within the report itself, and include not onlyanalyses of the "academic content" expressed or implied,but also qualities of exposition and taste affecting the stan-dards' usefulness.

In view of the ferment in American math education andthe continuing lackluster performance of U.S. youngstersin this key discipline, we must take notice of the findingsreported herein. While state math standards are in manycases too new for them fairly to be held responsible forpupil attainment in this discipline, it appears that thesedocuments, which were supposed to improve the situation,in most cases will not help and in many instances appear tobe symptoms of the very failure they were intended torectify.

To be sure, excellent math education continues in someclassrooms and schools. State standards are not supposed toplace a ceiling on how much is taught and learned. Butthey are meant to serve as a floor below which schools andteachers and children may not sink. As we learn fromMessrs. Raimi and Braden, in many states today that floorseems to have been confused with the muddy excavationthat ordinarily precedes construction.

We are grateful indeed to both authors for the rare ener-gy, thoroughness, and mathematical insight that theybrought to this arduous project. Raimi is professor emeritusof mathematics at the University of Rochester and formerchairman of the math department (and graduate dean) atthat institution. His scholarly specialty is functional analy-sis, and he has had a lifelong interest in effectivemathematics teaching. Braden has taught mathematics andscience in elementary, middle, and high schools for manyyears in Hawaii, in Russia, and now in New Hampshire. Heis a recipient of the Presidential Award for Excellence inScience and Mathematics Teaching. He holds a bachelor'sdegree in mathematics from the University of Californiaand an M.A.T. in mathematics from Harvard.

We also thank the two distinguished scholars whoadvised the authors throughout. Henry Alder is professoremeritus of mathematics at the University of Californiaand a former president of the Mathematical Association ofAmerica. He has been a member of the California StateBoard of Education and recently served on the committeeto rewrite that state's mathematics framework. HaroldStevenson is professor of psychology at the University of

Michigan, a 1997 recipient of the American PsychologicalAssociation's Distinguished Scientific Award, and can fair-ly be termed America's foremost authority on Asianprimary/secondary education and its comparison with U.S.schools and students. Among many publications, he co-authored The Learning Gap, a pathbreaking analysis ofelementary education in Asia and the United States. Hehas a particular interest in the standards, curricula, andpedagogy of mathematics, which discipline has been thefocus of many of his comparative studies, and has beendeeply involved with the Third International Mathematicsand Science Study (TIMSS).

In addition to published copies, this report (and itscompanion appraisals of state standards in other subjects)is available in full on the Foundation's web site:http://www.edexcellence.net. Hard copies can be obtainedby calling 1-888-TBF-7474 (single copies are free). Thereport is not copyrighted and readers are welcome to repro-duce it, provided they acknowledge its provenance and donot distort its meaning by selective quotation.

For further information from the authors, readers cancontact Ralph Raimi by writing him at the Department ofMathematics, University of Rochester, Rochester, N.Y.14627, or e-mailing rarm @db2.cc.rochester.edu. LawrenceBraden can be e-mailed at lbraden @sps.edu. The ThomasB. Fordham Foundation is a private foundation that sup-ports research, publications, and action projects inelementary/secondary education reform at the nationallevel and in the vicinity of Dayton, Ohio. Further informa-tion can be obtained from our web site or by writing us at1015 18th Street N.W., Suite 300, Washington, D.C.20036. (We can also be e-mailed through our web site.) Inaddition to Messrs. Raimi and Braden and their advisors, Iwould like to take this opportunity to thank theFoundation's program manager, Gregg Vanourek, as well asstaff members Irmela Vontillius and Michael Petrilli, fortheir many services in the course of this project, andRobert Champ for his editorial assistance.

Chester E. Finn, Jr. PresidentThomas B. Fordham FoundationWashington, D.C.March 1998

9

Almost every one of the 50 States andthe District of Columbia have by nowpublished standards for school mathemat-ics, designed to tell educators and thepublic officials who direct their work whatought to be the goals of mathematics edu-cation from kindergarten through highschool. They are generally given as"benchmarks" of desired achievement asstudents progress through the grade levelsto graduation, though sometimes theyinclude guides to pedagogy as well. Thepresent report represents a detailed analy-sis of all such documents as were available, 47 in all,though it has only space to offer rather abbreviated judg-ment of their value and a rating of their comparativeworth. Grades of A, B, C, D, or F were given to each state,based on an analysis of the contents according to criteriaand grade levels as described early in the report. Somecomments on each State conclude the report.

On the whole, the nation flunks. Only three statesreceived a grade of A, and just nine othersa grade of B. More than half receivegrades of D or F, and must be counted ashaving failed to accomplish their task.

EXECUTIVE SUMMARY

On the whole, the nation

flunks. Only three states

received a grade of A,

and just nine others a

grade of B.

The grading is described below, but itshould be understood that anything lessthan an A should be unacceptable.

A state, after all, is not a child to begraded for promise or for effort; the failureof a state to measure up to the best cannotbe excused for lack of sleep the nightbefore the exam. The failure of almostevery State to delineate even that whichis to be desired in the way of mathematicseducation constitutes a national disaster.

Even if the states' standards documentswere exemplary, there would remain a problem of imple-mentation. The public usually hears of the problems ofschools as questions of funding, of discipline, and evensometimes of teacher preparation or recruitment, but itgenerally imagines that their intellectual goals are clear.For basketball players and musicians the goals are indeedknown. But for elementary and secondary education in theUnited States today, there are no such agreements in placeregarding its essential core: its academic program. This isespecially so in mathematics, as the standards under reviewhere illustrate.

The authors of this report believe it unconscionablethat, in writing these standardsthese documents of pure

intent, whose success depends only on theefforts of experts already in placesomany states are so remiss in their duty.

As we have seen it, the principal fail-ures stem from the mathematicalignorance of the writers of these stan-dards, sometimes compounded bycarelessness and sometimes by a faultyeducational ideology. We are convincedthat the average math teacher can be ledto a better grasp of both the material thatshould be taught at various grade levelsand the manner in which it should be pre-

sented, than the writers and editors of these documentsimagine.

Our criteria for judgment were four: Clarity of thedocument's statements, and sufficient Content in the cur-riculum described or outlined in the text, were our first twodemands. Third, since deductive reasoning is the backboneof mathematics, we looked to see how insistently that qual-ity (denominated Reason) was to be found threaded

through all parts of the curriculum.Finally, we assessed whether the docu-ment avoided the negative qualities thatwe called False Doctrine and Inflation.These four major criteria, some of thembroken down into sub-criteria, were indi-vidually graded and the scores combinedfor a single total.

The most serious failure was found inthe domain of Reason. There is visible inthese documents a currently fashionableideology concerning the nature of mathe-matics that is destructive of its properteaching. That is, mathematics is todaywidely regarded (in the schools) as some-thing that must be presented as usable,

"practical," and applicable to "real-world" problems atevery stage of schooling, rather than as an intellectualadventure.

Mathematics does indeed model reality, and is miracu-lously successful in so doing, but this success has beenaccomplished by the development of mathematics itselfinto a structure that goes far beyond obvious daily applica-tion. Mathematics is a deductive system, or a number ofsuch systems related to one another and to the world, asgeometry and algebra are related to each other as well as tostatistics and physics; to neglect the systematic features ofmathematics is to condemn the student to a futile exercisein unrelated rule memorization. Most of the standards

The failure of almost

every State to delineate

even that which is to be

desired in the way of

mathematics education

constitutes a national

disaster.

10

documents we have read, for all that theyclaim to foster "understanding" above rotelearning, lack the qualities that wouldlead their readers, America's teachers, inthe desired direction.

This lack of logical progression, seenespecially in what passes for geometry andalgebra in the grades from middle schoolupward, is visible in the lack of clarity ofthe documents. It is also visible in theiradvocacy of the use of calculators andcomputers in the early grades, wherearithmetic and measurement as ideasshould rather be made part of the stu-dent's outlook, by his learning throughmuch experience and practice the nature of the numbersystem. Learning to calculate, especially with fractions anddecimals, is more than "getting the answer"; it is an exer-cise in reason and in the nature of our number system, andit underlies much that follows later in life. Only a personignorant of all but the most trivial uses of calculation willbelieve that a calculator replacesduring the years of edu-cationmental and verbal and written calculation. Adultshave need of calculators, and indeed computer programs,for computing their income taxes and doing their jobs. Butthe educational needs of children are quite different.

Content was the most successful part of these docu-ments. This country has a traditional curriculum from thepoint of view of content, and many states at least mentionmost of it, including such recent additions as statistics and

There is visible in these

documents a currently

fashionable ideology

concerning the nature of

mathematics that is

destructive of its proper

teaching.

6

a1

probability. However, much has been lost,especially from the Euclidean geometrythat was so large a part of a high schoolprogram 50 years ago; and the fragmenta-tion of the curriculum into too manydifferent "threads" has also diluted thetraditional curriculum.

The enterprise of writing standardsgoes hand-in-hand with the improvementof classroom practice, and there is nodoubt that teachers of the next few years,seeing the inadequacy of most of what wehave surveyed, will themselves offer sug-gestions for improvement. Members ofthe public, too, are often dissatisfied with

vague education, led by vague standards, and they, too, willbe heard. We believe the exercise of writing these docu-ments is worthwhile, and we wish more states took itseriously enough to put their best talent to work on them.

In particular, the "best talent" must include not onlymembers of the school establishment and state depart-ments of education, but also persons knowledgeable in theuses of mathematics and the creation of new mathematics.That is to say, scientists (including statisticians, engineers,and applied mathematicians) and research mathematiciansfrom the mathematics departments of the universities.These two communities have been most noticeably absentfrom the first rounds of standards construction, and futureimprovement is not possible without them.

Cab )

NATIONAL REPORT CARDState Math Standards

I Alabama( Alaska

IArizonaC Arkansas

I California( Colorado

Connecticut

( Delaware1 District of Columbia

(. Florida

[ Georgia( Hawaii

Idaho

C Illinois

IIndianaC Iowa

IKansas( Kentucky

Louisiana

( MaineMarylandMassachusetts

MichiganMinnesota

Mississippi

( Missouri

1 Montana( Nebraska

INevada( New Hampshire

INew JerseyCNew Mexico

New York

( North CarolinaINorth Dakota

( OhioOklahoma

(1 OregonPennsylvania

( Rhode Island

ISouth Carolina( South Dakota

Tennessee K-8

( Tennessee 9-12

Tennessee average

( TexasUtah

( VermontI Virginia

( WashingtonI West Virginia

" WisconsinI Wyoming

( Japan

11.5

7.512.2

L

2.516.05.44.77.8

4.74.8

11.5

1.3

2.26.88.3

L

4.55.21.3

3.53.01.3

1.3

6.2

9.22.5

11.3

I 14.2

5.613.5

3.95.35.51.0

4.51.5

2.5x_14.0

8.25 1

r 10.6 1

11.7

L 8.3 1

11.8

2.512.5

7.4

15.0

B

AD

D

C

D

D

B

F

F

D

C

ND

D

F

F

F

F

F

NB

F

F

ND

CF

B

AD

AF

D

D

F*

A*CB

B

CB

F

B

C

NA

I) California

CJapanINorth Carolina

( Tennessee 9-12

OhioWest Virginia

Arizona( Mississippi

I Virginia

( UtahIAlabama

( GeorgiaNew York

( TexasNew Jersey

( IndianaVermont

CTennessee averageIDelaware

( AlaskaWisconsin

Illinois

New Hampshire( North Dakota

Pennsylvania

ColoradoOregon

( KentuckyFlorida

( ConnecticutDistrict of Columbia

( KansasISouth Carolina

( OklahomaI Maine

( MarylandNebraska

( Missouri

IArkansas( New Mexico

Tennessee K-8

Washington

Idaho

( MontanaSouth Dakota

HawaiiLouisiana

Massachusetts

Michigan( Rhode IslandI Iowa

( MinnesotaNevada

( Wyoming

16.015.014.2 A14.0 A*13.5 A12.512.211.9 B

11.8 B

11.7 B

B11.5

11.5

11.3 B

10.6 B

9.2 C8.38.3 C

8.25 C

7.8 C

L 7.5 C

7.4 C

6.86.25.6 D

5.5 D

[ 5.45.3

D1-11 5.24.8 D

4.7 D

4.7 D

1

4.5 D

4.5 D

3.9 F

3.5 F

3.0 F

3.0 F

2.7 F

2.5 F

L 2.5 F

2.5 F*

1

2.5 F

2.2 F

1.5 F

1.5 F

1.3 F11-

1.3 F

1.3 1 F

1.3 F

1.0 F

NNNN

g@d1218nen-'000@en-MoDo@-BY-1a,DoCis.6* Partial Grades

12

)

I. THE SCOPE OF THIS STUDY: THE DOCUMENTS THEMSELVES

The authors were commissioned by the Thomas B.Fordham Foundation to study and compare the "standards"or "frameworks" for school mathematics as published bythe 50 States and the District of Columbia, and to gradethem according to a set of uniform criteria representing theseveral purposes or qualities such documents should have.

Ralph A. RaimiProfessor Emeritus

An initial judgment was needed even to solicit the correctpublication, and we began by sending 51 letters, identicalexcept as to the addressee. The text of the letter defineswhat sort of state publication we intended to study. Here isone of them:

Department of MathematicsUNIVERSITY OF ROCHESTER

Rochester, New York 14627Tel (716) 275-4429 or 244-9368 FAX (716) 244-6631 or 442-3339

Email: RARM@db2.cc.rochester.edu

13 July 1997

State of ArkansasDepartment of Education4 State Capitol MallLittle Rock, Arkansas 72201-1071

Office of the Superintendent or Director:

I am conducting for the Fordham Foundation (1015 18th St. NW, Suite 300, Washington, D.C. 20036) astate-by-state analysis of the mathematics curriculum and assessment standards of those States that publish aformal, public Framework or Standards for mathematics at the levels K-12. I write this letter to ask you to sendme what your state publishes of this nature.

Since the 50 States (and D.C.) have different ways of going about these matters it is hard fot me to namethe documents I want. I know that in California, for example, they have a "framework," while New York pub-lishes separate pamphlets with titles such as "Three Year Sequence for High School Mathematics (I, II, andIII)," i.e., a detailed curriculum guide as specified by the Regents. Other states might have a set of sampleexaminations by grade, which define the content by implication.

However you do it, I am interested in the guidelines by which local school districts in your state knowwhat the state education department considers appropriate (or mandatory, if such is the case) in mathematicsinstruction and testing at all levels from kindergarten to high school graduation.

If there is a charge for these documents, please let me know and I will send in payment a check made outaccording to your instructions.

If there is some other office I should address this request to, please either forward my letter there or let meknow the proper office to write to, or to phone or fax.

You can communicate with me by telephone, fax or email; all relevant addresses are found on theletterhead above.

Thank you for your attention.

Sincerely yours,<signed>Ralph A. Raimi

13

About 30 states and the District of Columbia respondedby sending printed or photocopied documents. Three orfour states invited us to download such a document from aWeb page of the state's education department, and a fewreplied explaining that the standards (or frameworks)documents we wanted were in the process of revision andnot available. (The District of Columbia will hereafter beincluded as a "state" when we speak of the states generical-ly.) Alaska wrote saying it printed such a document butdoes not send it out of state, so we secured a copy from afriend who lives there. We later discovered that Iowa,which did not respond at all, does not publish anythinganswering our description. In most other cases of non-response we nonetheless located a Web page containingthe document freely available to the public, and weobtained it. During the writing of this report, other statescompleted and sent us their latest drafts, or published themon their Web pages, but Minnesota, Nevada, andWyoming do not have their current drafts in quotableform. Other states having their standards in draft form areincluded in this study, though we have made clear thatthey are drafts, not yet adopted. In all, we have ended withdocuments from 46 states and the District of Columbia,and for comparative purposes one similar document fromJapan. All states are named in the Appendix to this report,where enough bibliographical information is given to makeclear just what we used for our commentary and where itwas obtained, or by what state office published, and when.We hope we have not included any document that hasbeen superseded by the time of publication of this report.Of this, however, we cannot be certain.

It can be seen from the variety of titles in theAppendix that there is no uniform nomenclature, thewords "Standards" and "Frameworks" often being usedmore or less interchangeably, along with "Curriculum

1

Content Guide," etc. Certain States do define a difference.Alaska, California, Colorado, New Jersey, Pennsylvaniaand South Carolina are among those clear in this regard,publishing one document called Standards and anothercalled Framework. In this usage, a "framework" will usuallyinclude the "Content Standards" or refer to it as a sub-sidiary document, but will also include guidelines forpedagogy, assessment, and other structural features of themathematics program. In cases where a second document(usually a "framework") is in our possession but contributesnothing to what we are judging, we may omit its mentionin the Appendix. Where the distinction or dual publica-tion makes a difference in the interpretation of our results,the second document will be listed. Also, some of thestandards we received are printed in company with stan-dards for science and other subjects; we shall of course bereporting only on the mathematics sections, and suchremarks introductory to the whole document as arerelevant to mathematics.

Except where we are actually naming a document forreference we shall refer to all these publications genericallyas "standards." Whatever they are named, they are eitherthe state's official answer to the description given in ourletter of solicitation reprinted above, or what we construeas such an answer from the state's publications on itsInternet web pages. As will appear in the description of thecriteria we used to judge these documents, the presence ofnon-curricular information in documents labeled "frame-work" will make no difference in our ratings, sincepedagogical and assessment advice and procedures are notpart of what we consider in this study, except as theirpresence might cast additional light on the content (andsometimes "performance") standards that are the realsubject of this report.

4

II. WHAT STANDARDS ARE

Standards are intended as a statement of what studentsshould learn, or what they should have accomplished, atparticular stages of their schooling. They are thus inferen-tially a guide to curricula, instruction and examinations,but are always something less than a curriculum outline bywhich detailed textbooks or examinations could be com-pletely envisioned. They are also less than complete guidesto instruction in another way, in that they do not intend,except incidentally, to prescribe or suggest actual pedagogi-cal procedure. Detailed lesson guides, examinations, andprescriptions as to pedagogy are in most states left to localschool districts, or published separately from the standardsdocument.

A few states do give statewide tests at certain gradelevels, and have designed their standards documentsaccordingly, and it appears that more states are planning todo this in the future; even so, a typical state standards doc-ument is intended to be compatible with any of a numberof different textbook choices and methods of instruction. Itis only intended to assure a uniformity of outcome, asmight be measured by a statewide examination if there wasone. Where such examinations are not mandated, a state'sstandards are still intended as serious advice to the localschool districts.

i;

While the standards documents under review usuallyomit prescriptions concerning pedagogy, but rather attemptto state simply what should be known (however arrived at)by students at each stage of their progress, there are docu-ments that do importantly include pedagogical advice. In afew cases this advice is considered by us despite the abovedisclaimers, but only to the degree that this extra informa-tion casts essential light on the meaning of the contentstandards themselves.

For example, a state might prescribe the use of calcula-tors in the teaching of "long division," and with suchfirmness that we are forced to conclude that the usual "longdivision" algorithm is not, or is hardly, to be taught at thatpoint. In such a case, while the standards might also statethat 6th grade students (say) should "divide and multiplynumbers in decimal notation," we cannot credit the docu-ment with demanding that proficiency with the standardalgorithm is intended unless it is so stated. Similarly withsample examination questions, classroom scenarios, etc.,which are not part of our assessment except as they clarifythe content demands. Our own interest is in the curricu-lum itself, or (more accurately) its intended result, asdictated or implied by the standards as published.

15

III. THE STANDARDS MOVEMENT IN RECENT YEARS

The best known document of the sortunder review is not a publication of astate at all, nor of the U.S. government. Itis the 1989 publication of the NationalCouncil of Teachers of Mathematics(NCTM) called Curriculum andEvaluation Standards for SchoolMathematics, and generally referred to as"The NCTM Standards." Since 1989, thisdocument has been the single most influ-ential guide to changes in the nation'sK-12 mathematics teaching, and in thecontents and attitudes of our best-sellingtextbooks.

The NCTM is a non-governmentalprofessional association, founded in 1920, which hasbecome one of the principal voices of its profession. Othervoices are the two teachers' unions (NEA and AFT), the51 state education departments themselves, all of which allhave specialists in school mathematics, the U.S.Department of Education and, less officially, the facultiesand deans of the major schools of education.

The NCTM publishes a monthly magazine, TheMathematics Teacher, and also other, more specialized jour-nals, including a journal of mathematical educationresearch. Its membership consists almost entirely of schoolteachers, professors in university schools of education, andeducation administrators and officials at all levels, and farexceeds the combined membership of the professionalmathematicians' organizations. NCTM also publishesmany other guides, yearbooks, and research reports, and itconducts national and regional meetings at which profes-sional information is exchanged.

The most obviously missing voice in this listing of thoseinfluential in school mathematics today is that of themathematics profession itself, as it might be represented bythe three major professional organizations: The AmericanMathematical Society (AMS), the MathematicalAssociation of America (MAA), and the Society forIndustrial and Applied Mathematics (SIAM). The MAA isdevoted to the advancement of the teaching of mathematicsat the college level, whereas the other two are principallyinterested in scientific research and publication. All threetake an interest in the health of the profession itself; butnone of them has traditionally considered school educationat the K-12 level an interest of more than marginal priori-ty. While a few members of NCTM are professors ofmathematics in university mathematics departments, it isdoubtful that one in a hundred of those persons who callthemselves "mathematician" or belong to the American

The most obviously

missing voice in this

listing of those influential

in school mathematics

today is that of the

mathematics profession

itself.

1

Mathematical Society, also belongs toNCTM or reads The Mathematics Teacher.

Yet a serious attempt to bridge the gulfof interests and influence between theworld of K-12 mathematics education andthe world of professional mathematics wasmade during the era of "The New Math,"roughly the period 1955-1970, the effec-tive bridge then having been thecurriculum-writing projects and "teachers'institutes" financed (mainly) by theNational Science Foundation at the time.This passed without lasting success, and itappears that, while a few mathematicianshave always been present at councils con-

cerned with school mathematics, or in grassroots projectsof educational experimentation, or in textbook-writingteams, and while as professors they have been among theteachers of those who later become school teachers andprofessors of education, their influence on curricula andclassroom practice has been negligible over most of thepresent century, including today. Professors of education,even mathematics education, are members of a "secondculture" almost as distinct from the world of mathematicsas C.P. Snow's literary culture was from that of his friendsthe physicists.

This separation is a cultural phenomenon more pro-nounced in the United States than in most Europeancountries and Japan, and it is not really a necessary con-comitant of the different professional responsibilities of thetwo groups. Indeed, another "bridging" effort on the part ofthe mathematical societies and NCTM is being madetoday, especially in connection with the movementtowards standards for school mathematics, but it is onlybeginning. The results will be impossible to assess for sometime to come.

The 1989 NCTM Standards, which was the work ofNCTM and not the mathematics profession, is comparableto almost all of the state "standards" volumes under studyin this report, and in many cases is their acknowledgedancestor; but it is not completely so. It is much longer thanany of them (except for the New Jersey "framework"), andcontains much advice on pedagogy, including "vignettes,"i.e., imagined dialogues or classroom conversations, illus-trating the way NCTM expects its recommendations mightplay out in practice. Its influence is manifest in most statestandards, even where they confine themselves to curricu-lar content. Its educational philosophy, usually briefly andinadequately referred to as "constructivism," and its cate-gories of mathematics curriculum: e.g., "mathematics as

problem-solving," "mathematics as communication,""mathematics as reasoning," "measurement," "numbersense," etc., are all echoed, often strongly and by explicitcross-reference, in the state documents.

The NCTM Standards offers 13 such categories (eachcalled a "standard") for Grades K-8, and 14 for Grades 9-12. Mostly the list is the same for the two levels, but thereis some variation: Some titles which are applicable to K-8(e.g., "whole number computation") may become obsoleteat the 9-12 level, while a few appear at 9-12 ("trigonome-try") that could not reasonably have been part of theearlier work. The educational philosophy of NCTM fol-lows the maxim of Jerome Bruner, an influentialpsychologist of education, who said that there is no subjectthat cannot be presented to a child of any age whatsoeverin some intellectually respectable form. From Bruner's psy-chological theories arises the doctrine of the "spiralcurriculum," according to which learning is best construct-ed by the individual in stages, each subject returned to

again and again, but each time at a higher "cognitivelevel."

Thus, "Problem-solving," "Geometry," and "Reasoning"are titles applicable to lessons at all levels. "Statistics," foranother example, is now a staple of kindergarten mathe-matics, though exemplified mainly by exercises of adata-gathering sort. ("Find out the favorite ice-cream fla-vors of your classmates.") And the rubric "Algebra" canalso find its way into the earliest grades under the alternatetitle of "patterns" (often construed quite literally).

A second list of rubrics for mathematical accomplish-ment in the schools is provided by the periodic NationalAssessment of Educational Progress (NAEP), in which K-12 mathematical progress is measured by a Federal agencyunder the headings: 1. Number Sense, Properties andOperations; 2. Measurement; 3. Geometry; 4. DataAnalysis, Statistics and Probability; and 5. Algebra andFunctions. Some states have organized their standards withattention to these categories.

17

IV. JUDGMENT, CRITERIA, AND THE RATINGS: AN OVERVIEW

Whether this or that rubric is represented at each gradelevel will be of little interest to the present report, inwhich we shall rather pay our attention to the topics thatactually appear, and to what depth of knowledge or under-standing is demanded in each appearance. Whether thosetopics we count appropriate appear under one or anotherpsychological (or philosophical) rubric will not matter.This is fortunate, as the rubrics are not always strictly com-parable among states. The age of the child is.

Teachers and school districts looking for practical guid-ance tend to ask what to teach in the 7th grade, forexample, and what book to choose for that grade level,rather than to ask what sequence of topics in probability,say, should obtain in the K-12 progression as a whole, orwhether their state succeeds in classifying what is accom-plished in each grade into a list of independent rubricscovering mathematics comprehensively. It is, in fact, curi-ous that the NCTM, and hence the states, have spent somuch effort in this direction, whose principal consequenceappears to have been the fragmentation of the mathemat-ics curriculum during each year of schooling into amultitude of themes, or "threads," with consequent loss ofdepth in each, not many of Bruner's "higher cognitive lev-els" having been in fact reached by the spiraling. Thisphenomenon has been characterized by one knowledgeablecommentator, concerning the United States' poor perfor-mance on the 1996 Third International Mathematics andScience Survey (TIMSS), who remarked that the typicalAmerican curriculum is "a mile wide and an inch deep."The same can be observed from our children's performanceon the NAEP mathematics tests.

In contemplating each state's standards, we shall ignoremost theoretical debates and simply ask the text to havethis function: We imagine a newly arrived teacher in aremote district, of average knowledge and experience inteaching the mathematics traditionally taught at the gradelevel he works in. He has access to textbooks and othermaterials, but does not have previous experience in theschools of that particular state. He wishes to construct asyllabus and choose appropriate books (and perhapstechnical tools, such as graph paper, compasses, computers,and calculators) in order to set up lesson plans andexaminations.

This (imagined) teacher is not for this purpose askingthe state for instruction in pedagogy, or even in mathemat-ics; he only wants to know what his state demands thestudent should know of mathematics by grades 4, 8, and12; or perhaps by end of kindergarten, grade 3, grade 6,grade 9 and grade 12; or even grade-by-grade.

The standards published by the state should answer thisquestion.

Choosing textbooks and deciding the mixture of lec-ture, homework, group work, and classroom discussion arenot usually, under the American tradition of local control,the decision of the state at all; but it has increasinglybecome the custom that the framework for results and evena list of the desired results themselves should be ofstatewide uniformity, at least approximately.

Our judgments of how the state has complied with thisrequest, our ratings of the documents we call "standards,"will be based on four major investigations:

1. Our estimate of the success the printed document willhave in plainly telling this imagined teacher what hewill need to know to satisfy the requirements of thestate's citizens as to content;

2. Our estimate of how well the state's criteria, as we havebeen able to understand them, match our own standardsof what schoolchildren should know, and our own expe-rience of what they can be expected to knowanddemonstrate knowledge ofby the grade levels named;

3. Our estimate of how well the document as a wholedemands that the students' command of the "topics"taught and learned include the understanding of themathematical reasoning and unifying structures that distin-guish such mastery from the simple gathering ofinformation; and

4. Our estimate of whether the document, whatever suc-cess it may otherwise enjoy in clarity and sufficiency ofcontent, at the same time injures its own purpose by ask-ing things it should not, or giving advice it should not,or by the exhibition of ignorance, carelessness, or pre-tension.

These four descriptions are summaries of our categoriesof judgment, but still need detailing: Just what propertiesshould worthwhile standards display? We have isolatednine criteria for judgment, but, following the four pointsenumerated in the preceding section, we group them underthe following four short titles:

(I) Clarity(II) Content(III) Reason(IV) Negative Qualities

We shall explain these criteria in two stages: first, bygiving them definitions (Section V); and secondly, byillustrative examples (Section VI). Following this exposi-tion we shall present the ratings of the states in the form ofa matrix of numerical grades (Section VII).

The general explanation of why a state receives a gradeit does, under any of the criteria, will be implicit in our

descriptions of the criteria. Section VII, the numericalscores, will nonetheless be introduced by a further explana-tion of our ratings. The table of the grades themselves (i.e.,the Ratings), brief as it is, represents the goal, and is thecondensation of the judgment of the authors after study ofthe documents in light of the criteria.

19

DEFINOTO

I. Clarity refers to the success the doc-ument has in achieving its own purpose,i.e., making clear to the (imagined)provincial teacher what the state desires.Clarity refers to more than the prose.(Furthermore, some of the other evils ofbad prose will have a negative category oftheir own in the fourth group, NegativeQualities.)

(A) Firstly, of course, the words and sen-tences themselves must beunderstandable, syntactically unam-biguous, and without needless jargon.

1s eel F THE XIIIIITER

Ds the State asking K-12

instruction in mathematics

to contain the right

things, and in the right

amount and pacing?

(B) Secondly, what the language says should be mathemat-ically and pedagogically definite, leaving no doubt ofwhat the inner and outer boundaries are, of what isbeing asked of the student or teacher.

(C) And thirdly, the statement or demand, even if under-standable and completely defined, might yet ask forresults impossible to test in the school environment,whether by a teacher's personal assessment or astatewide formal paper-and-pencil examination. Weassign a positive value to testability.

Thus, the first group, Clarity, gives rise to three criteria:

I-A Clarity of the languageI-B Definiteness of the prescriptions givenI-C Testability of the lessons as described

II. Content, the second group, is plain enough inintent. Mainly, it is a matter of what might be called "cov-erage," i.e., whether the topics offered and the performancedemanded at each level are sufficient and suitable. To thedegree we can determine it from the standards documents,we ask, is the State asking K-12 instruction in mathematicsto contain the right things, and in the right amount andpacing?

Here we shall separate the curriculum into three parts(albeit with fuzzy edges): Primary, Middle, and Secondary.It is common for states to offer more than one 9-12 cur-riculum, but also to print standards describing only the"common" curriculum, or one intended for a "school-leav-ing" examination in grade 11 or so. In such cases we shallnot fault the document for failing to describe what it hasno intention of describing, though we may enter a notementioning the omission. Other anomalies will give rise to

notes in Section VIII, which follows theactual ratings.

We cannot judge the division of con-tent with year-by-year precision becausevery few states do so, and we wish ourscores to be comparable across states. Asfor the fuzziness of the edges of the threedivisions we do use, not even all thosestates with "elementary," "intermediate,"and "high school" categories divide them

I by grades in the same way. One popularscheme is K-6, 7-9, and 10-12 for "ele-

mentary," "intermediate" and "high"schools, while another divides it K-5, 6-8, and 9-12. Incases where states divide their standards into sufficientlymany levels (sometimes year-by-year), we shall use the firstof these schemes. In other cases we will merely accept thestate's divisions and grade accordingly. Therefore, Primary,Middle, and Secondary will not necessarily mean the samething from one state to another. There is really no need forsuch precision in our grading, though of course in anygiven curriculum it does make a difference where topics areplaced.

Thus, the second group, Content, gives rise to threecriteria:

II-A Adequacy of Primary school content (K-6,approximately)

II-B Adequacy of Middle school content (or 7-9,approximately)

II-C Adequacy of Secondary school content (or 10-12,approximately)

In many states, mathematics is mandatory through the10th grade, while others might vary this by a year or so.Our judgment of the published standards will not takeaccount of what is or is not mandatory in each state; thus,a rating will be given for II-C whether or not all studentsin fact are exposed to part or all of it. (Some standards doc-uments, as in the case of the District of Columbia, onlydescribe the curriculum through grade 11, and we adjustour expectations of content accordingly.)

The difficult question here is to define "adequacy" ofcontent. This can only be done by a listing of some sort.The authors of this report have in their minds a standard ofwhat is possible and desirable to be taught at each level ofschool mathematics, and it is this standard of ours againstwhich the adequacy of the states' standards is measured. Toset this "standard" forth in abstract terms, by definitionrather than enumeration, is, we believe, impossible. We

ro20

shall therefore attempt to do it by tworather indirect means, as will be explainedin Section VI, Examples for the Criteria.

III. Reason, the third "group," willhave only the one entry: Reason.

Civilized people have always recog-nized mathematics as an integral part oftheir cultural heritage. Mathematics is theoldest and most universal part of our cul-ture, in fact, for we share it with all theworld, and it has its roots in the mostancient of times and the most distant oflands.

The beauty and efficacy of mathemat-ics both derive from a common factor that distinguishesmathematics from the mere accretion of information, orapplication of practical skills and feats of memory. This distinguishing feature of mathematics might be calledmathematical reasoning, reasoning that makes use of thestructural organization by which the parts of mathematicsare connected to each other, and not just to the real worldobjects of our experience, as when we employ mathematicto calculate some practical result.

The essence of mathematics is its coherent quality, aquality found elsewhere, to be sure, but preeminently here.Knowledge of one part of a logical structure entails conse-quences which are inescapable, and can be found out byreason alone. It is the ability to deduce consequences whichotherwise would require tedious observation anddisconnected experiences to discover, that makes mathe-matics so valuable in practice; only a confident commandof the method by which such deductions are made canbring one the benefit of more than its most trivial results.

Should this coherence of mathematics be inculcated inthe schools, at the level K-12, or should it be confined toprofessional study in the universities? A recent report (17June 1997) of a task force formed by the MathematicalAssociation of America to advise the NCTM in its currentrevisions of the 1989 Standards argues for its earlyteaching:

People untaught in

mathematical reasoning

are not being saved from

something difficult; they

are, rather, being

deprived of something

easy.

One of the most important goals of mathematicscourses is to teach students logical reasoning. This is afundamental skill, not just a mathematical one .

[Teachers] should recognize its theoretical nature,which idealizes every situation, as well as the utilitarianinterpretations of the abstract concepts . . .

It should be recognized that the foundation of mathe-matics is reasoning. While science verifies through

co)

observation, mathematics verifies throughlogical reasoning. Thus the essence ofmathematics lies in proofs, and the dis-tinction among illustrations, conjecturesand proofs should be emphasized . . .

If reasoning ability is not developed in thestudents, then mathematics simplybecomes a matter of following a set ofprocedures and mimicking examples with-out thought as to why they make sense.

(This task force, which continues its workin 1998, is headed by Kenneth Ross,former President of the MathematicalAssociation of America.)

Even a small child should understand how the memo-rization of tables of addition and multiplication for thesmall numbers (one through 10) necessarily produces allother information on sums and products of numbers of anysize whatever, once the structural features of the decimalsystem of notation are fathomed and applied. At a moreadvanced level, the knowledge of a handful of facts ofEuclidean geometrythe famous Axioms and Postulates ofEuclid, or an equivalent systemnecessarily imply (forexample) the useful Pythagorean Theorem, the trigono-metric Law of Cosines, and a veritable tower of truthsbeyond.

Any program of mathematics teaching that slights theseinterconnections doesn't just deprive the student of thebeauty of the subject, or his appreciation of its philosophicimport in the universal culture of humanity, but even atthe practical level it burdens that child with the apparentneed for memorizing large numbers of disconnected facts,where reason would have smoothed his path and lightenedhis burden.

People untaught in mathematical reasoning are notbeing saved from something difficult; they are, rather, beingdeprived of something easy.

Therefore, in judging standards documents for schoolmathematics, we look to the "topics" as listed in the "con-tent" criteria not only for their sufficiency at this and thatgrade level, and not only for the clarity and relevance oftheir presentation, but also for whether their statementincludes or implies that they are to be taught with theexplicit inclusion of information on their standing withinthe overall structures of mathematical reason.

A state's standards will not rank higher by the Reasoncriterion just by containing a thread named "reasoning,""interconnections," or the like, though what we seek mightpossibly be found there. It is, in fact, unfortunate that somany of the standards documents we examined contain a

21

thread called "Problem-solving and MathematicalReasoning," since that category often slights the reasoningin favor of the "problem-solving," or implies that they areessentially the same thing. Mathematical reasoning is notfound in the connection between mathematics and the"real world," but in the logical interconnections withinmathematics itself.

Since children cannot be taught from the beginning"how to prove things" in general, they must begin withexperience and facts until, with time, the interconnectionsof facts manifest themselves and become a subject of dis-cussion, with a vocabulary appropriate to the level.Children then must learn how to prove certain particularthings, memorable things, both as examples for reasoningand for the results obtained. The quadratic formula, thevolume of a prism, and why the angles of a triangle add to astraight angle, for example. What does the distributive lawhave to do with "long multiplication";why do independent events have proba-bilities that combine multiplicatively?Why is the product of two numbers equalto the product of their negatives?

(At a more advanced level, the reason-ing process they have become familiarwith can itself become an object of con-templation; but except for the vocabularyand ideas suitable and necessary for dailymathematical use, the study of formallogic and set theory are not for K-12classrooms.)

We therefore shall be looking at thestandards documents as a whole to deter-mine how well the outlined subjectmatter is presented in an order, or a wording, or a context,that can only be satisfied by including due attention to thismost essential feature of all mathematics.

disagreements among schools of experts on some of thesematters. Indeed, our choice of the name "false doctrine" forthis category of our study is a half-humorous reference toits theological origins, where it is a synonym for heresy.Mathematics education has no official heresies, of course;yet if one must make a judgment about whether a teaching("doctrine") is to be honored or graded zero, as we arerequired to do in the present study, deciding whether anexpressed doctrine is true or false is a mere necessity.

The NCTM, for example, officially prescribes the earlyuse of calculators with an enthusiasm the authors of thisreport deplore, and the NCTM discourages the memoriza-tion of certain elementary processes, such as "longdivision" of decimally expressed real numbers, and thepaper-and-pencil arithmetic of all fractions, that we thinkessential, that should be second-nature before the calcula-tor is invoked for practical uses. We must assure the reader

that, while we differ with the NCTM onthese and other matters, our own view isnot merely idiosyncratic, but also hasstanding in the world of mathematicseducation, as can even be seen in some ofthe documents under review. And even

Mathematical reasoning

is not found in the

connection between

mathematics and the

"real world," but in the

logical interconnections

within mathematics itself.

C.

IV. Negative Qualities, the fourth group, looks for thepresence of unfortunate features of the document thatinjure its intent or alienate the reader to no good purpose.Or, if taken seriously, will tend to cause that reader to devi-ate from what otherwise good, clear advice the documentcontains. We shall call one form of it False Doctrine, aphrase that almost explains itself, but will need someexamples. The second form will be called Inflation becauseit offends the reader with fruitless verbiage, conveying nouseful information.

Under False Doctrine, which can be either curricular orpedagogical, is whatever text contained in the standardswe judge to be injurious to the correct transmission ofmathematical information. As with our criteria concerningContent, our judgments can only be our own, as there are

were that not true, it is still our duty tomake our own stand and to make it clear.We cannot simultaneously credit twoopposing positions on what should belearned.

While in general we expect standardsto.leave pedagogical decisions to theteachers (most standards documents doso, in fact), so that pedagogy is not ordi-narily something we are rating in the

present study, there are still in some cases standards con-taining pedagogical advice that we believe undermineswhat the document otherwise recommends. Advice againstmemorization of certain algorithms, or a pedagogical stan-dard mandating the use of calculators to a degree weconsider mistaken, might appear under a pedagogical rubricin a standards document under consideration. Here is oneof the places where our general rule not to judge pedagogi-cal advice fails, for if the pedagogical part of the documentgives advice making it impossible for the curricular part, asexpressed there, to be accomplished properly, we must takenote of the contradiction under this rubric of FalseDoctrine.

Two other false doctrines are excessive emphases on"real-world problems" as the main legitimating motive ofmathematics instruction, and the equally fashionablenotion that a mathematical question may have a multitudeof different valid answers. Excessive emphasis on the "real-world" leads to tedious exercises in measuring playgrounds

22

and taking census data, under headingslike "Geometry" and "Statistics," in placeof teaching mathematics. The idea that amathematical question may have variousanswers derives from confusing a practicalproblem (whether to spend tax dollars ona recycling plant instead of a highway)with a mathematical question whose solu-tion might form part of such aninvestigation. As the first (January 21,1997) Report of the MAA Task Force on the

NCTM Standards has noted,

Students are sometimes

urged to discover truths

that took humanity many

centuries to elucidate.

It should also be recognized that results in mathematicsfollow from hypotheses, which may be implicit orexplicit. Although there may be many routes to a solu-tion, based on the hypotheses, there is but one correctanswer in mathematics. It may have many components,or it may be nonexistent if the assumptions are inconsis-tent, but the answer does not change unless thehypotheses change.

Again, constructivism, a theoretical stance commontoday, has led many states to advise exercises in havingchildren "discover" mathematical facts, or algorithms, or"strategies." Such a mode of teaching has its values, incausing students better to internalize what they have there-by learned; but wholesale application of this point of viewcan lead to such absurdities as classroom exercises in "dis-covering" what are really conventions and definitions,things that cannot be discovered by reason and discussion,but are arbitrary and must merely be learned.

Students are also sometimes urged to discover truthsthat took humanity many centuries to elucidate, thePythagorean theorem, for example. Such "discoveries" areimpossible in school, of course. Teachers so instructed willnecessarily waste time, and end.by conveying a mistakenimpression of the standing of the information they mustsurreptitiously feed their students if the lesson is to come toclosure. And often it all remains open-ended, confusingthe lesson itself. Any doctrine tending to say that tellingthings to students robs them of the delight of discoverymust be carefully hedged about with pedagogical informa-tion if it is not to be false doctrine, and unfortunately such

doctrine is so easily and so often giveninjudiciously and taken injuriously thatwe deplore even its mention.

Finally, under False Doctrine must belisted the occurrence of plain mathemati-cal error. Sad to say, several of thestandards documents contain mathemati-cal misstatements that are not meremisprints or the consequence of momen-tary inattention, but betray genuineignorance.

Under the other negative rubric, Inflation, we speakmore of prose than content. Evidence of mathematicalignorance on the part of the authors is a negative feature,whether or not the document shows the effect of this igno-rance in its actual prescriptions, or contains outrightmathematical error. Repetitiousness, bureaucratic jargon,or other evils of prose style that might cause potential read-ers to stop reading or paying attention, can render thedocument less effective than it should be, even if its clarityis not literally affected. Irrelevancies, such as the smugglingin of political or trendy social doctrines, can injure thevalue of a standards document by distracting the reader,again even if they do not otherwise change what it essen-tially prescribes.

The most common symptom of irrelevancy, or evidenceof ignorance or inattention, is bloated prose, the making ofpretentious though empty pronouncements, conventionalpieties without content. Bad writing in this sense is a verynotable defect, though not the greatest, in the collection ofstandards we have studied. Some examples will appearbelow.

We thus distinguish two essentially different failuressubsumed by this description of pitfalls, two NegativeQualities that might injure a standards document in waysnot classifiable under the headings of Clarity and Content:Inflation (in the writing), which is impossible to make useof; and False Doctrine, which can be used but shouldn't.How numerical scores will be assigned under these head-ings will be explained in Section VII below; here we signalonly their titles, as used in the Table of Ratings there:

IV-A False DoctrineIV-B Inflation

23

VI. EXAMPLES FOR THE CRITERIA

Note: The examples given in this section are taken fromstate standards as named, but it must not be imagined thatthese are the only states, or the only quotations, that couldhave been used to the same purpose. Each example, forgood or bad, can be matched by many others, from manyother states.

Criterion I: Clarity

I-A. A standard should be clear:

Demonstrate understanding of the complex numbersystem (Arkansas, "Number Sense" strand, grades9-12);

rather than unclear:

Connect conceptual and procedural understandingsamong different mathematical content areas(Washington, Standard 5.1, Benchmark 2 - grade 7,p.65).

"Conceptual understandings" and "procedural under-standings" might mean something, likewise "different . . .

content areas," but "connecting" the two former "among"the latter is just too hard to understand, or make use of as aguide to classroom activity.

I-B. A standard should be definite:

Given a pattern of numbers, predict the next two num-bers in the sequence (Alaska, Framework, "Reasoning,"Level 1: ages 8-10);

rather than indefinite:

Apply principles, concepts, and strategies from variousstrands of mathematics to solve problems that originatewithin the discipline of mathematics or in the realworld (Alaska Framework, "Problem-solving," Level 3:ages 16-18).

The first example is extremely definite, though as a les-son it might be mathematically questionable. The second("apply principles . . . to solve problems") is exemplary inits indefiniteness. An indefinite standard describes some-thing a teacher has no way of completing ("bringing toclosure" is a common phrase for the process of completinga lesson), even if it is quite easy to know if that particularlesson is relevant to the standard.

0E3

The wording in this example also illustrates a commonfailing in definiteness shared by many states at all levels:the use of the word "problem" without a hint of the natureof the problem. Mathematics is preeminently the scienceof problem-solving, but the "problem" of adding twelveand seven (grade 1) is of a different order from the "prob-lem" of determining the dimensions of a rectangle of givenperimeter and area. Every 7th grade teacher in a certainstate might understand exactly what is meant by "problem"when the standard for grade 7, under "patterns," uses thatword; but this understanding is parochial, an accident ofthe curriculum and culture traditional in that place andtime. Our hypothetical teacher from out-of-state will notknow this meaning. When the authors of the presentreport also cannot discern what is meant, such a standardmust be counted inadequate.

The quoted second example from Alaska is actuallyeven worse than this, since it occurs under the rubric"Problem-solving," and ought rather to elucidate that termthan assume its meaning is already clear. If the rubric hadbeen "algebra" we would at least know that algebra prob-lems were meant, though this knowledge would still beinsufficient to define a standard of accomplishment.

I-C. A standard should be testable:

Analyze spatial relationships using the Cartesian coordi-nate system in three dimensions (New York, "Four yearsequence, Modeling," p.27);

rather than ineffable or not testable:

Students [will] utilize mathematical reasoning skills inother disciplines and in their lives (New Jersey Standard4.4, by the end of grade 8).

It should be plain that the second standard, while desir-able as a long-range goal for the teaching of mathematics,is not something one can do more than conjecture about; ateacher has little way of finding out whether it has beenaccomplished. On the other hand, there are many tests forwhether the student can "analyze spatial relationships,"whatever interpretation one wishes to put on the phrase.And there are many. The standard concerning "spatialrelationships" is not very definite, or clear, but whatever itis, it is testable.

Even though "clear," "definite," and "testable" are notthe same thing, they are a family of qualities that tend togo together. SAandards that fail any of these three testsannounce thZi4 Ives to the reader as somehow unclear,

24

albeit in different ways. We therefore group them together,and can think of no better overall title for the groupingthan Clarity.

The idea here is that our imagined teacher in a stateoffering standards highly rated by us for each entry underClarity should be in no doubt about what to explore,explain and examine, how to judge whether a book coversthe ground described, or what to test the children upon ateach stage in their progress. A poor score, on the otherhand, signals a standards document from which he canglean little or no help at all on what the state expects ofhim in his daily tasks, and will have to rely on his owninsight and experience to guess at what it might be, eventhough the intended content of the state curriculum mightbe adequate and clear in the mind of the state's educationaladvisors.

Criterion II: Content

Content will be rated separately for the categoriesPrimary, Middle, and Secondary, in most cases meaning K-6, 7-9, and 10-12, with a year's leeway at either end of eachsegment, according to the way each state divides its con-tent standards. Our grades will observe the state's owndivisions, and therefore will be applicable to slightly differ-ent segments of the K-12 program for different states. Todecide whether a state is asking its children to learn theright things at the right times, and enough such things, andnot an unreasonable amount either, the judge must have aset of standards of his own. The authors of this report doindeed have them, though for purposes of judging otherswe must be rather flexible, there being more than one wayto go about securing a given result by the end of theschooling process.

Much as we would like to print here a complete exem-plary curriculum that conforms to our own standards, thetask would be prohibitively long (as every state well appre-ciates that has appointed a committee to do just that) andthe resulting document would overwhelm the presentreport. Nor is it possible for a few sentences from a samplestandards document to illustrate sufficiency, which must bejudged by its entire extent. Hence, we shall not offer anyquotations to serve as good and bad examples, as we do forcriteria groups I, III, and IV. Just the same, since the readerdeserves some notion of where we stand on some of thecrucial topics that must form part of every mathematicaleducation, it is worth stating a few desiderata briefly:

We believe the traditional arithmetic of fractions anddecimals should be complete by grade 6, but with moreunderstanding of logical connections than was customary50 years ago and with many more applications than the

traditional storekeeping and mensuration skills. We wishthe middle grades to introduce geometry and algebra andthe logic of equations and their application, and not be the"review of elementary math, plus ratios" that has beencommon in recent years. We wish the secondary curricu-lum to be mathematics as the mathematics professionunderstands it: not a collection of rules for algebra,trigonometry, graphing and the like, but an organized bodyof knowledge, albeit mainly of algebra, geometry and theelementary functions, with application to human affairsclearly distinguished from the inner logic of the mathemat-ics itselfand both of them fully represented.

To describe a curriculum in so few words is, of course,insufficient; but we might point to several of the state doc-uments as models. None of them is a model in all respects,but the reader may deduce from the scores (in Part VII, orin the Notes on that State in the following section) whichstates describe the content we consider sufficient. They areall publicly available (see Appendix). While not the high-est overall, Alabama, Arizona, and Tennessee (at the 8-12level) are high-ranking for content, and California, NorthCarolina, and Ohio are good models in both content andthe other categories. We have also included high scores forJapan, which presents a document much different from anyof the American states, but which is exemplary in contentwhen rightly read, and interesting in other regards as well.We shall have more to say about the Japanese standards inthe Notes on the States (Part VIII). There is no singlescale for content; what is left out of one listing might sim-ply have been omitted in order that something else mightbe included, so that states with equal scores will not haveidentical intentions concerning content. This, too, is a rea-son for not trying to include a model curriculumdescription here.

Criterion III: Reason

There is no single place, grade level, or strand, whereone can find whether a state standards document exhibitsthe guidance being graded under this criterion. The mereinclusion of a category of instruction labeled"Mathematical Reasoning" or the like is no guarantee thatits contents serve the purpose. To the contrary, evidence ofthe demand for reasoning is more often found elsewhere, ifat all, inextricably bound up with the mathematical con-tent. Quoted examples can only indicate in part the toneof the document as a whole, and it is the whole whichgives rise to our rating. At the risk of being unfair to thetwo documents as a whole we shall nonetheless give a pairof such examples, one exhibiting a poor incorporation ofthe ideal of fostering mathematical reasoning, and the

2 5,

other a good and clear indication of the kind of lesson thatis designed to teach such reasoning and give the student anappreciation of the coherence of mathematics.

Vermont, which has a thread named "MathematicalProblem Solving and Reasoning," lists 17 specifics, someunder the grade 9-12 column, some under the grades 5-8column, and some under the grades K-4 column. Here aresix of them (p. 7.4):

Students [shall](a) Solve problems by reasoning mathematically with

concepts and skills expected in these grades.(b) Create and use a variety of approaches, and under-

stand and evaluate the approaches that others use;determine how to break down a complex probleminto simpler parts; extract pertinent informationfrom situations.

(c) Formulate and solve a variety of meaningful prob-lems.

(d) Formulate and solve meaningful problems in manykinds of situations using grade-related mathematicalconcepts and reasoning strategies.

(e) Extend concepts and generalize results to othersituations.

(f) Work to extend specific results and generalize fromthem.

That these instructions are vague and uninformative isevidenced by the fact that the reader would be hard put todecide which grade level, K-4, 5-8, or 9-12, any of these sixspecifications is intended to apply to. It is therefore clearthat there is no progression of logical skill, or problem-solv-ing skill, or theoretical understanding, to be deduced fromthese six statements. In fact, (a) and (e) are listed under K-4; (b) and (c) under 5-8, and (d) and (f) under 9-12.Notice that (a) and (d) are nearly identical, yet (a) is forK-4 and (d) is for 9-12. The main difference appears to bethat in 9-12 the "problems" are to be meaningful, an ill-defined idea having no relevance to either problem-solvingor reason.

Indeed, any of the six instructions can be construed toapply to any levels whatever, including the editors of amathematical research journal such as the Proceedings of theAmerican Mathematical Society. And the editor of even thatjournal might hesitate before grandly asking an author to"extend concepts and generalize results to other situa-tions," as was here prescribed for children at the level K-4.

Surely the Vermont authors had something more specif-ic in mind, but it does not come across in the frameworkthey actually wrote. Some demands for mathematical rea-soning do occur in other parts of Vermont's Framework, tobe sure, notably in the section headed "Mathematical

Understanding," where certain required skills are men-tioned; and all of this must be taken into account in givingan overall score for Reason. A high score would requirethat the intended lessons in reasoning, or making connec-tions, be part of a sufficient number of the specific contentdemands throughout the document as to make plain whatlessons in reasoning are intended; but Vermont's section,quoted (in part) here, that includes the word "reasoning"fails to do so. It contains exhortations, not standards.

On the other hand, Virginia does show the proper quali-ty clearly enough in some places to make it convenient toquote a selection by way of contrast:

(Grade 2) The student, given a simple addition or sub-traction fact, will recognize and describe the relatedfacts which represent and describe the inverse relation-ship between addition and subtraction (e.g., 3 =

7, .... , 7 - 3 = ).

(Grade 5) Variables, expressions and open sentenceswill be introduced. . . .

(Grade 6) The student will construct the perpendicularbisector of a line segment and an angle bisector, using acompass and straight-edge.

(Geometry) The student will construct and judge thevalidity of a logical argument consisting of a set ofpremises and a conclusion. This will include .. . identi-fying the converse, inverse and contrapositive of aconditional statement. .. .

(Algebra II) The student will investigate and describethe relationships between the solution of an equation,zero of a function, x-intercept of a graph, and factors ofa polynomial expression.. ..

These examples from Virginia's standards are, with oneexception concerning pure logic, simple instructions as tocontent, but presented in a manner that carries the propermessage of the unity of mathematics, and the reasoning bywhich its parts are held together. Five such examples donot, of course, amount to a curriculum in mathematicalreasoning, and in fact Virginia is, like most states, some-what lacking in these qualities overall; but these fiveshould serve as illustration of the standard desired; whereasmere exhortation, to "make connections," "use . .. a vari-ety of reasoning strategies," and the like, are not sufficientguidance, and simply are not usable in the absence of a ref-erent.

An instruction of the sort that should appear more oftenis also found in The District of Columbia's (Mathematics)

26"00

Curriculum Framework, where on page M-5, under "What astudent should know and be able to do by the end of Grade11," appears the following:

5. Explain the logic of algebraic procedures.

This is not completely stated, and so its intent must beinferred; but most of the documents under review don'teven go this far in reminding the reader that the logic of aprocedure is essential to its intelligent use.

A more complete statement of the same desideratummay also be found in Pennsylvania's standards draft for the6th grade level "Algebra and Functions" strand. Studentshere are to:

1. write verbal expressions and sentences as algebraicexpressions and equations, and solve them, graph themand interpret the results in all three representations.

1.1 . . . write and solve one-step linear equations in onevariable.

1.2 . . . write and evaluate an algebraic expression for agiven situation using up to three variables.

1.3 . . . apply algebraic order of operations and the com-mutative, associative and distributive properties toevaluate expressions and justify each step in the process.

Again, Pennsylvania's proposed "Standards" offers, atthe grade 11 level,

Prove two triangles or two polygons are congruent orsimilar using algebraic and coordinate as well as deduc-tive proofs.

The Pennsylvania instruction is a little vague, especiallyfor "polygons," in that it does not specify the sort ofhypotheses that are likely to have been in place when theproof was to be done; but the idea is plain, the lesson valu-able, and the word "prove" a delight to encounter.

Criterion IV: Negative Qualities

IV-A False DoctrineA standard must not offer advice which, if followed,

will subvert instruction in the material otherwisedemanded:

First ExampleFor 7th grade, under Standard: 5070-07 ("The students

will develop number concepts underlying computation and

estimation in various contexts by performing the opera-tions on a pair or set of numbers."), Utah asks as #3 of"Skills and Strategies,"

Develop an algorithm for multiplication of commonfractions and mixed numbers by using models or illustra-tions; explain your reasoning.

This suggestion is misleading in that "develop" impliesthat the conventions concerning the multiplication offractions are somehow to be discovered by the students, andindeed then justified by them. In fact the very definition ofmultiplication for fractions is conventional, not naturaland "discoverable." It is a triumph of the development ofmathematics that the rule for the multiplication of frac-tions is both a consistent extension of the multiplicationand division in the ring of integers, and at the same timeinterpretable as an operation concerning measurements ingeometry and partitioning in finite sets. That the defini-tions and results are independent of the "fraction"representation of the rational numbers involved, i.e., of thedenominator chosen, is another complication. As with anyother convention or fact, this cannot be discovered, thoughits invention and its rationale should be thoroughly taught.

That 7th-grade students should participate in discussionsof these matters is of course to be desired, but a teacherwho feels obliged to set up a classroom environment inwhich students "develop an algorithm . . ." will be wastingtime, and fooling either himself or his students; while pos-sibly neglecting the truly interesting features of the systemof rational numbers and its possible uses.

Second ExampleArizona, in "Patterns, Algebra, and Functions," grades

9-12, asks students to analyze the effects of parameterchanges in formulas defining functions, "using calculators."

This curriculum obviously intends the student to under-stand the effects of parameter changes, but believes the useof (graphing) calculators sufficient for the purpose. It is notsufficient for the purpose. Attention to the symbolicexpressions, individual point calculations and point plot-ting, are a necessity. True, calculators are a valuable aid,even a necessity, in serious use with complicated data inscience and in business; but that is life, not instruction,and the proper use of calculators must follow, not precede,the logical, conceptual understanding of the effects of para-meter changes. A standard that instructs teachers toconfine themselves to calculators in exercises, where theywill not suffice for instructional purposes, is destructive ofthe lesson intended in that part of the curriculum.Omitting the phrase "using calculators" might have madethis standard a good one, for the desideratum (to under-

27

stand the effect of parameter changes) is clear and valu-able; but the method of getting there should rather havebeen left to the teacher. There doubtless is a place for cal-culators somewhere in there, but this Arizona pedagogicalinstruction preempts the choice in an unfavorable direc-tion.

The widespread prescribing of calculators in the earlygrades is also destructive of learning, for the algorithmsconcerning fractions and decimal multiplication and divi-sion are not mere "19th Century skills" now renderedobsolete by technology, but when properly taught inculcate"number sense" and the ability to estimate; and serve asessential preparation for the more sophisticated operationsof later years, such as the algebra of rational functions.

Third ExampleAlabama, in its general introduction to the detailed list

of standards that follows, has a section headed "Use ofManipulatives," which contains this directive:

Throughout their schooling, students should beinvolved in activities in which manipulatives are usedto aid in conceptual and procedural understanding.. ..Use of manipulatives helps . . . clarify algorithms .

Using manipulatives also richly illustrates the connec-tion between concrete experiences and abstractmathematics.

In the early childhood years, the use of manipulatives,such as blocks, Cuisinaire rods, and surely the abacus, cangive a child the basis in experience that must underlie theabstractions of mathematics; and the world has used themsince the dawn of mankind. But "throughout" their school-ing is an error, and has moreover led to an unnecessarilylarge manipulatives industry allied to textbook publishing,whose products decorate the displays at teachers' confer-ences and professional journals.

Mathematics is not raw experience; it is an analogue forexperience. As it becomes more sophisticated, as with alge-braic equations representing physical relationships, it mustnecessarily supplant that which can be manipulated; other-wise, the lesson of algebra is lost. A teacher who takesseriously the Alabama instruction as given will find himselflooking for ways to employ such curiosities as "algebratiles" in teaching the solution of equations, and literal bal-ances for the understanding of just those methods thatwere designed to render literal balances unnecessary.Rather than look for ways to use manipulatives, teachersshould strive for ways to wean students from their use, littleby little, as a potential bicyclist must be weaned from train-ing wheels.

IV-13 inflationA standards document must not employ language

whose purpose can only be to fill paper, nor must it sug-gest a profundity impossible for the level in question,especially if the indications are that the author doesn'tunderstand the words being used. Such language weak-ens the respect the user of the document should have forvalid parts of the document, even where it does notresult in outright error or false doctrine.

Bad writing in this sense differs from vagueness or otherfailures of clarity in that it is not simply mistaken, incom-plete or obscure, but in that it is pretentious and cannot betaken seriously, or is empty of content altogether. If it doesinclude a definite instruction, that instruction is generallyimpossible of execution, at least at the level of instructionin question.

Example 1In Michigan's Model Content Standards for Curriculum

(Mathematics) , Standard 8 states:

Students draw defensible inferences about unknownoutcomes, make predictions, and identify the degree ofconfidence they have in their predictions.

Then the first item under this standard, identicallyworded and repeated at each of the levels primary, middleschool, and high school, is

Make and test hypotheses.

As these four words constitute a short description of theentire science of statistics (and indeed of all science), this"standard" is impossibly broad, hence useless. Later itemsindicate certain particulars, e.g., "Make predictions anddecisions based on data, including interpolations andextrapolations." This is broad enough, but does includesome clues (e.g., "interpolations") limiting the intendedlessons, though even here much the same wording is usedat all three levels. This particular statement can be called,generously, "too vague," but "Make and test hypotheses" isworse, and is no guide to either curriculum or instruction.

Example 2Oklahoma's "Priority Academic Student Skills," at the

grade 2 level, under the rubric "Mathematics asConnections" requires that the student will

A. Develop the link of conceptual ideas to abstractprocedures.

B. Relate various concrete and pictorial models ofconcepts and procedures to one another.

28

C. Recognize relationships among different topics inmathematics.

D. Use mathematics in other curriculum areas.E. Use mathematics in daily life.

Now, B, D, and E are understandable, even though theydo not say anything very specific. Children in grade 2 typi-cally learn the names and notations for integers up to athousand, and how to add and subtract some of them, per-haps making change at a grocery store; they learn somegeometric language and so on. It is hard to see why theircurriculum should be subjected at this level to a rubric suchas "Mathematics as Connections" at all, except that thereare theorists who insist that all categories by which mathe-matics may be classified should obtain at all levels.Condemned to follow out this theory for grade 2, theauthors could hardly avoid having to invent empty itemssuch as A and C to inflate the list.

Example 3.Hawaii's Essential Contents ("Geometry and Spatial

Sense," Grades 9-12) claims to prescribe

Non-Euclidean geometries; andHyperbolic and elliptical geometries.

This shocking entry perhaps springs from its authors'having seen a popularization of these topics somewhereand deciding it to be a pleasant sort of thing to talk about

in school. But, like capsule biographies of famous mathe-maticians and magazine articles about Fermat's LastTheorem, popularizations of arcane mathematical theoriesare not curriculum. A serious lesson in non-Euclideangeometries is impossible at this stage of schooling. EvenEuclidean geometry is slighted in Hawaii's EssentialContents, and non-Euclidean geometry can no more beunderstood in the absence of Euclid's structure than nightcan be understood in the absence of day.

(There is a possibility, evidenced in some advertising wehave seen, that the [Euclidean] properties of the sphere areregarded as "non-Euclidean geometry" by some manufac-turers of manipulatives to be used in the schools. If that isthe origin of the mention of "non-Euclidean geometry" inseveral of the state standards, it is a poor use of the phrase,and as a lesson in geometry is inferior to the deductiveEuclidean geometry it appears to replace.)

Again, under Hawaii's rubric "Patterns andRelationships" we find that

Students develop algebraic thinking through . . . [amongother things] . . . Topological concepts.

In a curriculum guide that nowhere mentions an axiom,theorem, or deductive argument, this is also unrealistic,even apart from the problematic association of "algebraicthinking" at the high school level with such "topologicalconcepts" as might possibly be found there. Are these"Essential Content" items for real? No. They are Inflation.

29

VII RATING THE STATES: A TABLE OF RESULTS

Every state is listed in the followingtable for the sake of completeness, thoughseveral of them are ungraded, marked "n"rather than with one of the numbers 0, 1,2, 3, or 4, because (except for the specialcase of Tennessee) they do not have astandards document at all (Iowa) orbecause they have standards in process ofrevision, with no draft publicly availablefor comment (Minnesota, Nevada, andWyoming). The State-by-State Notes(Section VIII) that follow the table ofgrades will explain the case of Tennessee,which has two sets of scores but with asingle final (average) grade for its twodocuments.

Giving grades to states differs somewhat from givinggrades to students in school or college. In grading students,teachers are dealing with youngsters taking several courses,not all of which interest them greatly. The students arepressed for time on examinations, and during the week aswell, and are sometimes fearful, ill, or troubled by personalproblems. We wish to encourage students to improve, orreward them for performance we know about but which isnot present in the written work they present. In conse-quence, we give many grades of "A" for less thanexemplary performance. The world rather expects that10% of a class (or 5%, or 15%) will receive grades of A(i.e., 4 grade points).

A state publishing standards, however, is not an anxiousyoungster pressed for time. A state has time and moneyand the ability to secure expert advice. It can consultbooks and other states. It should be without psychologicalproblems or learning disabilities. This particular "assign-ment," the writing of standards, is a professionalresponsibility, and any performance short of exemplarymust be judged so, and announced so, even if the top gradewere received by none of them.

Thus, while we have used the scale 4, 3, 2, 1, 0 in moreor less the usual way, "4" representing the best availableand "0" representing the least useful, these figures must beseen simply as a ranking of value, five degrees of excellencebeing about as many as we can distinguish. It was only aftertabulating all the numerical scores that we decided whichtotal scores should accord with the more familiar scheme ofA, B, C, D, Fgrades which are given as the right handcolumn of the chart.

The grading for Negative Qualities might seem a bitcurious, grades of 4 being awarded for the absence of FalseDoctrine, or of Inflation, and 0 for those states having the

The collapse of deductive

reasoning as a

desideratum in American

school mathematics is the

single most discouraging

feature of the study of

these documents.

most; but, since a total score was neededfor computing the final grade of A, B, C,D, or F, it seemed convenient to scalenegations positively, in order to be able touse additions only to arrive at a total.Otherwise some states would have endedwith negative scores. However, thismethod of scoring negative qualitieswould give a state publishing a standardsdocument containing no words at all agrade of 4 points, because of its perfectionin achieving the absence of inflation andfalse doctrine; and as it turned out thiswould have been just sufficient to earn aD. Thus, the 16 States that received a

final evaluation of "F" would have scored better, or as well,in our final tally if they had turned in a blank paper. (Withnegative scores in part of the document, it seems, as inMinkowskian geometry, there is simply no way to avoid atleast some anomaly.)

General ImpressionsThe collapse of deductive reasoning as a desideratum

in American school mathematics is the single most dis-couraging feature of the study of these documents. Thesecond, strikingly evidenced by the paucity of grades of 4 atthe Primary level for Content (Criterion II), and for FalseDoctrine (Criterion IV(B)), was the enthusiasm withwhich many states have embraced the recent doctrine thatthe algorithms for multiplication and division of fractionsand decimals are obsolete and can be replaced by calcula-tors. Indeed they are, for technicians who need thenumerical answers for practical everyday purposes; butinstructional purposes demand otherwise. We did notdemerit the prescribing of technology per se, at the primaryor any other level, but we did downgrade its prescription inplaces where its use obscures a mathematical lesson, orblunts a mathematical perception, that can only be con-veyed in its absence. And there are many such places, fromthe decimal calculations of the 4th grade to the solution oflinear systems in the 12th. Most failures found in thesestandards documents require more detail of presentation,and an effort is made in the Notes (Section VIII) todescribe a representative selection of them.

In addition to all the states we have named below, oneother Standards is graded for comparative purposes: Japan,which appears at the end of the listing. The Japanese stan-dards document is a translation of a publication of theJapanese Ministry of Education, and is listed in theAppendix, with information on its provenance. This

Kom30

document, while brief, is exemplary in most respects. Evenso, it falls short of our maximum in the category of Reason(III), perhaps only due to deficiencies in the translation, orto a cultural difference that makes it seem unnecessary tothe Japanese to mention the matter sufficiently often whendescribing content. We must judge only by what we canread, however. (See also the Note on Japan in Section VIIIbelow.)

There are nine scores (of 0 to 4) for each state, but forevaluation purposes there are but four Categories, I, II, III,and IV (as described above), for each of which an averageis struck before the four averages are added for a total scorefor the state. That is, we are weighting equally each of thecriteriaClarity, Content, Reason, and NegativeQualitieseven though some are split into more subheadsthan others. Thus, 16 is the highest possible total score.

31

TABLE 1. NUMERICAL RATINGS FOR THE STATES

Nig CLARITY GCOTEIU REASON QUALITIES MI GRADELanguage

I (A)Reference

I (B)Testability

I (C)Hem.

II (A)Middle

II (B)Second. Reasoning Abs. of False Doctrine

II (C) III IV (A)Abs. of Inflation

IV (B)

Alabama 3 .

AL (avg) II

-.

3.0

C Alaska 1 0

AK (avg) 11 0.7Arizona 2133 444 2 1

I AZ (an') 2.7 I 4.0 II 2.0 II

C Arkansas 0 0 0 0 0 0 1

1 AR (avg) o.o I o.o 1.0 II

C-California 4 4 4

CA (avg) II 4.0 4.0 II 4.0 II

C Colorado 0 1 II 1 1 2 2 1

CO (avg) II 0.7 1.7 II 1.0 II

( Connecticut 0 f 1 --1-- 1-- ) ---i- 1

1 cr (avg) II 0.7 1,0 II 1.0 II

C Delaware 1 1 I-- 1 412 2

1 DE (avg) II

,1.0 2.3

,.

2.0

( District of Columbia

1 DC (avg) I

Florida 2 1 1, 1 2 2 2 0

1 HI (avq)

1-1 o

4

103.3 0 0 1 0 0

11 0.0 0.3 II 0.0 II

I_ FL [oval 2.0 II 0.0 II

( Georgia 3 3 4 1

GA (avg) II 1.0 II

( Hawaii

( Idaho 1 -,, 1 If 0 1 11- 2 ',I 0 0

I ID (avg) II 0.7 1.0 II 0.0 II 0.5C Illinois

IL (avg) II

2 ,,, 0 II, 2 I 2 II 2 2 1 1

1.3 2.0 11 I1.0

1 2 11

2.5

II 11.5 II B I4.0 II 1.0 II 354 1 2 I 3 2 )

2.3 II 2.0 II 2.5 II 7.5 II C

4.J

3 )3.5 II 12.2 II B

3 0 if1.5 II 2.5 II F

4 40,

4.0 II 16.0 II A2 2 i

2.0 111_ 5.4 11 D

2.0 II 4.7 II Dd

2

3

2.5II

,

2

II 7.8 C r, 0 1 0 0 0 , 1 , , 3 ..

II 4.7 II D0.7 0.0 II 1.0 I 3.0

1 2-1.5 II 4.8 II D

,

.1 2

3.7 3.3

3

2

4

3.5 II 11.5 II B

1.0 II 1.3 11 F

1

3

C Indiana .. 4 3 j 3 3 -II 2 hi-- 1 I 0 I 2 II 4

IN (avg) II 3.3 2.0 I 0.0 II 3.0f f----- 7-- -.1--- -,--( Iowa 1 n n n n n L. n ji n 1 n _11

C Kansas 2 2 1 2 2 , 0 0 0 I

KS cLas) II 1.7 1.3 II 0.0 II 1.5

(Kentucky 1 1., L 2 2 . 0 I 3

KY (avg) II 1.0 1.7 0.0 171 2.5

0 -11 0 11 2C Louisiana 0

II 0.3 I 0.0 11 0.0 II 1.0I LA (avg)

( Maine 0 0 0 2 1 0 r 3

ME av 0.0 1.0 II 0.0 II 2.5

II 2.2

1

11

11-

II 6.8

1

I

2 \,'

II F

Maryland

\I MD (avg) II

i Massachusetts 1

1 MA (04 II

r\- Michigan 0

I MI (avg) II

( Minnesota n:-.Mississippi 4

1 MS (mg) II

0

II

Missouri

I MO (avg)

Montana

I MT(avg)

1 1

0.7 II

1 1 0

1.0 II

0 1 1

0.3 II

n n n

3 3

3.7

1 1 2

0.7 II

0 0 0 1

I I 0.0

0 2

1.3 _11, 0.0 II 1.0

0 0 00.3 II o.o II 0:0

1 1 0 01.0 II 0.0 II don n n n n

4 . 4 2 3 2

3.7 II 2.0 II 2.5

. .

0

2.0

1.0

2

0.0

II 0.0 II

0

0

II 3.0 II F

II 1.3 II

0-

11 1-1111.9 II B

0

0.0 II 2.7 11 F

0

0.5 1.5 II F

32

TABLE 1. NUMERICAL RATINGS FOR THE STATES (continued)

@I= CONTENT REASON QUALITIES GRADE

Language

I (A)Reference

I (B)Testability

I (C)Elem.

II (A)Middle

II (B)Second.

II (C)Reasoning Abs. of False Doctrine

III IV (A)Abs. of Inflation

IV (B)

Nebraska

I NE (mg)

li 0

II 0.0

I

II

0 11 1 i

1.0

1

1.0 1.011

3.0

Nevada

New Hampshire

[ NH (avg)

Tr n n 72 L 2

1.7

IJI

n n il n '1.,-

1 2 A 2 d

II 2.0

n

2

[ r.-. n n-'l

6.2 Il

N

1[ I

I 1.0 I

r2

I.3

1.5 I D

New Jersey

NJ ay.rl 2 1, 1 I

1 2.0

3 , 3 11 4 1,

II 2.7

1 r 2

2.0

2 --IL

3

2.5 I 9.2 I C

New Mexico

[ NM (avg)

L'---0

II 0.0 011

II 1.0

1 0-

0.03

..

1.5 I[ 2.5 II F

B

New York

1 NY (avq)

1-1r- 1 11

)., LI

1.3

2 . 2 3 71

3.0Iii

4 3

3.0

4 1 4

4.0 II 11.3 'il

[North Carolina

I NC (avq)

If 4

4.0L 4 3 1 4 ':i.

II 3.7

4 3

3.0

ii 44i

3.5 II 14.2 I ANorth Dakota r 2 1r

-11 ii 1 3 [ 3, 1 0 2 2

1-

FWD (avg) 1.3 2.3 0.0 [ 2.0 5.6 -I

13.5 IL

D

A _I

Ohio

I OH (avq)

irl- 4 'L. 2

II 3.0

11 4 1 4 I1

II 4.0 .

4 3 1iru

4

3.0 I 3.5 II

2

C

E OklahomaI OK (avg)

C Oregon

I OR (avg)

1-\, Pennsylvania 2

1 PA (avg) I

( Rhode Island 0

I RI (avg) II

C-South Carolina 2

1 00.7

1

2 1 1 ii

1.3 II

A 1 If 2

1.7 II

0 -II- 0il... ..

0.0 II

II

SC (avg) II 1.1

1

3 II

South Dakota 0 0 0 0I SD (avg) II

-0.0

II

{',--Tennessee (K-8) 0 ,, 0 7 0,I TN (K-8) (avg) II 0.0 II

1.0

3

2.7

3

2 0

0.0 I

00.0o

0.0

o0.0

1 n

1.5

3

0

1.5

2.0

2

1.5

0.0

0.5

1.5

0

1.0T ennessee (9 -12) 4 11

TN (9-12) (avg) 4.0*Partial grades; Tennessee Average for both documents

11

n

4.0

3 3

3.0 3.0

/

II .9

II 5.3 II D I

11 5.5 II D

[I 1.0 II F

II 4.5 II D17"

11 1.5 II

II 2.5 II F"

14.0

8.25

A'C

Texas

I TX (avg)

( UtahI UT (avg) II

Vermont

c(,,,g) II

C Virginia1 VA (avg)

Washington

[NA (avg) II

H West VirginiaI WV (avg) II

( Wisconsin

I WI (avq)( W yoming(\' Japan

JN (avg)

II

II

4 3

3.1

3

4 2

2.7

2 1 1

1.3..

4...

4.0

0_ _

0

0.04 3

3.3,

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4.0

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II

II

II

3

3.0

1 33.0

4 4

3.0

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3 3

3.0 II

3

3:5 II 10.6 II B

3

3.0 11 11.7 II B

II 8,3 II C

3.5 11 11.8 B I

0

1.5 II 2.5 II, F I

3.5 II 12.5 II B I

3.0 II 7.4 II C IJ

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11 15.0 II A ri

33

VIII. NOTES, STATE-BY-STATE

The following Notes are not intended as full explanations of the numerical scores tabulated in Part VII above and repeat-ed here state-by-state. Those explanations are implicit in the numbers themselves, coupled with the description of thecriteria given earlier in this report. To review all the details that entered the scores would require more space than is possible.

In particular, the quotations are not necessarily drawn from or based on the most typical or characteristic features of thedocument in question. On the whole, they are selected to provide, in their ensemble, a glimpse of some of the detail, here andthere, that no summary or averaging procedure can provide. Since these notes mainly (though not entirely) exhibit weak-nesses rather than strengths in the documents mentioned, we expect that, taken together, they provide additional insightinto what, by our criteria, is the overall failure of the current efforts at writing useful state standards for school mathematics.That certain particular criticisms appear under the headings of particular states does not mean they are peculiar to the Statein which they appear. Many are applicable to other states, and the objects of these criticisms tend to occur in much the sameformsometimes in exactly the same formin the standards documents of other states.

Alabama

The Content is one of the most com-prehensive, and is mostly, though notalways, direct in language. A typical gooditem: "Describe characteristics of plane andsolid figures using appropriate terms.Examples: round, flat, curved, straight" (1stgrade, p. 18). Too often, on the otherhand, occurs something like this:"Describe, extend, and create a wide vari-ety of numeric and geometric patterns"(5th grade, p. 46). This is not a "learningobjective" at all, and even as a pedagogicaldevice is too broad and open-ended.

There is a lot of content in the coursedescriptions, though including some old-

fashioned items that really should beretired as unilluminating and time-wasting,such as Descartes' Rule of Signs andSynthetic Division. The document as awhole slights deductive reasoning, which isprincipally mentioned in the geometrycourses, but not carefully outlined eventhere, where its use is most traditional. It issurely evidence of this lack of attention tological structures that the authors assignthis impossible task: "Use the FundamentalTheorem of Algebra to solve polynomialequations" (p. 35) which mistakes the logi-cal status of an existence theorem. Yet thedocument as a whole is one of the best.

Alaska

The Framework (1) is extremely vague,offering examples of lack of clarity, andalso of inflation: "By strategically applyingdifferent types of logics, students will learnto recognize which type of logic is beingused in different situations and respondaccordingly." ("Different types of logic" is,we believe, a reference to induction anddeduction, about which much is madethese days, without much effect. And"respond accordingly" couldn't be vaguer.)On page 4-12, under "benchmarks" forMath Content Standard A (Content ofMath), at the 16-18 year-old level, the fol-lowing appears: "A student would be ableto .. . explore linear equations, nonlinearequations, inequalities, absolute values,vectors and matrices." "Exploration" is nota standard, and "absolute values" is curi-

ously misplaced in this list. Clarity has suf-fered here, and more than clarity.

At the grade 3-5 level, under "ProblemSolving," readers are told, "Evaluate therole of various criteria in determining theoptimal solution to a problem." This is notonly unclear, but is Inflation. For highschool: "Recognize how mathematicschanges in response to changing societalneeds." (We believe mathematics is eternaland unchanging. There certainly are manythings that vary in response to social pres-sures, but the document puts it badly.) TheStandards (2) partly makes up for the defi-ciencies in the Framework (1), especially inits avoidance of inflated language, but itoutlines a program lacking in sufficientcontent, especially at the secondary level.

34

arirt(fritirRYSIETOIRITIIIECIAMICA

Alabama

1. CLARITY 3.0

II. CONTENT 4.0

III. REASON 1.0

IV. NEGATIVE QUALITIES 35

TOTAL SCORE (out of 16) 115

GRADE

COPOCIMernalAlaska

I. CLARITY 0.7

II. CONTENT 1.3

III. REASON LO

IV. NEGATIVE QUALITIES 25

TOTAL SCORE (out of 16) 75

GRADE C

Arizona

The "performance objectives" are verybriefly stated, and, while the reader has tomake some difficult inferences, they areapparently demanding and good. Also,Arizona lists extra objectives for honor stu-dents, a useful feature. Sometimes,however, ambition outruns the language.For example, under "Geometry 4M -P4,""PO 3: State valid conclusions using givendefinitions, postulates and theorems" (p.21). This is inflated and indefinite, as noneof the surrounding text asks for knowledgeor skills that would give it substance.Taken seriously, so vast an instructionshould be broken down into numerousgraded demands, describing one or moreyears of algebra or geometry. (Curiously, noparticular subject matter is mentioned

here.) Thus, Reason is badly outlined as athread in an otherwise comprehensive doc-ument; it is put into one corner of thecurriculum, as it were. As for Inflation,here is an example of unreality at the"honors" level in high school:"Demonstrate technical facility with alge-braic transformations, including techniquesbased on the theory of equations." The"theory of equations" is ill-defined today,and as understood 50 years ago concernedthings about real polynomials that did notreally translate into "techniques" concern-ing transformations. Curiously enough, thisstandard is also found verbatim in Idaho's"algebra" strand, at the 9-12 level (seebelow).

Arkansas

The telegraphic style of this Frameworkmay conceal a good program in manyschool districts, but it says so little that itcannot be of muchuse. "Use technolo-gy" is a sentencethat occurs repeated-ly and oftenpointlessly. Standard5.2.9, "Use mathe-matical reasoning tomake conjecturesand to validate andjustify conclusionsand generalizations,"is no more helpfulthan to say, "Usemathematics."Standard 5.2.16,"Apply algebraicprocesses to non-algebraic functions,"is opaque. Standard 2.1.6 (grades 9-12),"Explore non-Euclidean geometries," isunreasonable where Euclidean geometry isalready too little explored.

(In fact, "non-Euclidean geometry" intoday's high schools, where it is mentioned

at all, sometimes designates sphericalgeometry, which is quite Euclidean. If thisis what is meant, it should be stated; if this

is not what is meant, it isinflation, since what thehistory of mathematicscalls "Non-EuclideanGeometry" is any of a"Use mathematical

reasoning to make

conjectures and to validate

and justify conclusions

and generalizations," is

too more helpful than to

say, "Use mathematics."

number of sophisticatedaxiomatic systems thatdiffer from the Euclideansystem only in postulat-ing alternatives to thefamous parallel postu-late.)

The best feature isthe document's relativelack of outright FalseDoctrine, but even in itsbrevity there is a greatdeal of inflation, e.g.,"Visualize algebra as a

bridge between arithmetic and higher levelmathematics" (p. 9). This is neither clearnor definite nor testable, nor yet an item ofcontent nor of reasoning. Yet it is labeled a"learning expectation."

35

fr.SZITWrirE RTEIPIOIRiTIECTATIVOS

Arizona

I. CLARITY 1.7

II. CONTENT 4.0

III. REASON

IV. NEGATIVE QUALITIES 3.5

TOTAL SCORE (out of 16) hji

GRADE

Arkansas

I. CLARITY 0.0

II. CONTENT 0.0

III. REASON 1.0

IV. NEGATIVE QUALITIES 1.5

TOTAL SCORE (out of 16) Z.5

GRADE

California

The Standards is a scant 37 pages inlength, and is classified by grade level fromK-7 and then by subject headings (the sub-jects are strangely called "disciplines"),from Algebra I (in Grade 8) through cours-es which prepare the student for APCalculus and AP Statistics. The writing isalways terse and to the point.

At the start of each grade (K-7), theexpectations for that grade are summarizedin a hundred words or so, permitting arapid and accurate overview of the whole.The details follow by rubric, the same forall grade levels:

i Number Senseii Algebra and Functionsiii Measurement and Geometryiv Statistics, Data Analysis, and

Probabilityv Mathematical Reasoning

Although naming a rubric "Algebra andFunctions" is stretching things at the lowerlevels, there is in general no undue multi-plicity of rubrics, e.g. a strand labeled"Calculus" which at least one other statemysteriously included in its frameworkright down to the first grade."Mathematicalreasoning" is theonly one here whosepresence might bequestioned, for itsdemands are of arather generalnature, and somecould be considered"inflation" by theauthors of this reportif they were notthoroughly exempli-

the per unit cost)."Again, under "Mathematical

Reasoning," grade 3, we find that "Studentsuse strategies, skills and concepts in findingsolutions." So often admonitions of thissort are simply left hanging as emptyexhortations, but here follow six specifica-tions, e.g., "express the solution clearly andlogically using appropriate mathematicalnotation and terms and clear language, andsupport solutions with evidence, in bothverbal and symbolic work." A tall order,perhaps, but conveying (in passing) anoth-er important point: In speaking of "the"solution, the phrasing insists that mathe-matical problems have a single solution.Here the "reform" philosophy of what itsopponents sometimes have called "fuzzymath" is firmly rejected.

In grade 4, instead of reading"Investigate the relation between the areaand the perimeter of a rectangle" (a popu-lar, though confusing, entry in many statestandards, and in any case "investigate" isnot a content standard), we find (1.2) "rec-ognize that the rectangles having the samearea can have different perimeters," and(1.3) "understand that the same numbercan be the perimeter of different rectan-

gles, each having adifferent area." It is alsorefreshing to observe, ingrade 4 "Number Sense"and in the Glossary, thatHere the "reform"

philosophy of what its

opponents sometimes

have called "fuzzy math"

is firmly rejected.

fled in the contentstandards of theother rubrics.

Each rubric is headed by one or moregeneral admonitions which would alsotend to be labeled not "Definite" or"Testable" were it not that their subhead-ings explain exactly what is meant. Under"Algebra and Functions," grade 3, forexample, students are to "...represent sim-ple functional relationships" (which isvague to say the least), but then theprovincial teacher imagined in the Criteriais immediately told what that means interms of content, e.g. "solve simpleproblems involving a functional relation-ship between two quantities (e.g., findthe total cost of multiple items given

mathematical educatorsin California know whatprime numbers are andtell us carefully. (cf.Pennsylvania, below.)

Scientific calculatorsare mandated for the firsttime in grade 6, but onlyfor good reason, and afterthe essential propertiesof the real number sys-

tem have been assimilated through handcalculations. (Japan introduces these elec-tronic aids in grade 5, while mostAmerican states demand their use from thebeginning.) By the end of grade 5,California students are to be able to dolong division with multiple-digit divisorsand to represent negative integers, deci-mals, fractions, and mixed numbers on anumber line. By the end of grade 7 theygraph functions, use the Pythagoreantheorem, evaluate algebraic expressions,organize statistical data, and in general areprepared for high school algebra andgeometry.

36

iSITYAVVIE RiiiPTOIRITNIC/A1RIDi

California

I. CLARITY 4.0

II. CONTENT 4.0

III. REASON 4.0

IV. NEGATIVE QUALITIES 4.0

TOTAL SCORE (out of 16) 16

GRADE A

The years 8-12 are described by subject:Algebra I and II, Geometry, Probabilityand Statistics, Trigonometry, LinearAlgebra, Mathematical Analysis, and asthe "Advanced Placement" subjects ofstatistics and calculus. The Standards is notprescriptive in its pedagogy. How theteacher, or the textbook, goes about thejob is left to the discretion of the teacheror school district. A table is provided sug-gesting placement of this material by year,so that integrated curricula are possible bythe same standards as the subjects woulddemand when taken in the form of courses.It is clear that a course-by-course programwould place Algebra I in the 8th grade,Geometry in the 9th, and Algebra II in the10th, completing the state-mandated cur-riculum for graduation.

Algebra I names 25 items, some neces-sarily mechanical and some with welcomeattention to logical structure, includingknowing the quadratic formula and its proof"Practical" rate problems, work problems,and percent mixture problems are to bestudied as well as the (impractical?)Galilean formulas for the motion of a parti-cle under the force of gravity.

In Geometry, students must be able toprove the Pythagorean theorem and muchelse, including proofs by contradiction, andclassical Euclidean theorems on circles,chords, and inscribed angles. "Geometry"also includes the basic trigonometry ofsolving triangles, and the properties of rigidmotions in the plane and space.

Mathematical Induction is introducedin Algebra II, which material is apparentlyintended for the 10th grade or earlier.Experience will have to show if this andcertain other ambitious demands are reallypossible, or should rather be left for the fol-lowing years and a volunteer audience.

Other topics are either preparation for col-lege work or are (these days) college-levelwork themselves (de Moivre's Theorem,the Binomial Theorem, Conic Sectionswith foci and eccentricities, etc.). We arenot told how much of this program must betaken by all California students, or whetherthe courses named Trigonometry andStatistics are intended as options.

There is no mention of "shop math,""finite math," or "business math" (etc.),directed at the non-college bound student,or remedial courses for older students whohave failed earlier. Perhaps theMathematics Framework (see below),required by California law to be revised for1999, will address these issues. The authorsof this report did not downgrade California(or any other States) for this sort of omis-sion, when the core curriculum is adequateand well stated. We do, however, expectthat California will in due course find itadvisable to add something appropriate toits "elective" curriculum at the high schoollevel, as does, for example, Tennessee,which offers a branching of alternate tracks.

If teachers and textbooks can be foundto carry it through properly, this Standardsoutlines a program that is intellectuallycoherent and as practical for the non-sci-entific citizen as for the future engineer.Whatever of "real-world" applicationschool mathematics can have, is foundhere, set upon a solid basis of necessaryunderstanding and skill. Initial reaction tothe adoption of this document included awidespread apprehension that this "returnto basics" represented an anti-intellectualstance: rote memorization of pointless rou-tines instead of true understanding of theconcepts of mathematics. The opposite istrue. One can no more use mathematical"concepts" without a grounding in fact andexperience, and indeed memorization anddrill, than one can play a Beethoven sonatawithout exercise in scales and arpeggios.

There is always a danger that intellectu-ally challenging material, be it in music,literature, or mathematics, will in thehands of ignorant teachers, or bowdlerizedtextbooks, become reduced to pointlessdrills. The history of American schoolmathematics in the 20th century has large-ly been a chronicle of conceding defeat inadvance, teaching too little on the groundsthat trying for more will fail. California isto be commended for taking up the chal-lenge head-on, and announcing itsintention in the clearest terms in itsContent Standards.

It is the more curious, then, that theadoption of this Standards has been attend-

ed by an extraordinarilybitter public debatecentering on their char-acterization as areactionary documentdiscouraging, ratherthan demanding, the"real understanding" ofmathematics. An earli-er version of thisStandards had beencomposed by a specialCommission onStandards whichworked through most of1997 on standards forEnglish and mathemat-ics, to be approved bythe State Board ofEducation. Theappointment of theCommission was itselfextraordinary, and the consequence of pub-lic dissatisfaction with current teaching ofcore academic subjects.

California by law publishes aFramework for mathematics instructionevery seven years. Past Frameworks includ-ed standards of the sort under review here,along with pedagogical and administrativeinformation, and the most recent was pub-lished in 1992 in the midst of enthusiasm,on the part of the school administrators, forthe point of view represented nationally bythe NCTM Standards of 1989, and oftencalled "reform." (The "reform" trend hadbeen visible in California even earlier.)Two foci of opposition soon appeared:"HOLD" in Palo Alto and"Mathematically Correct" in San Diego,both citizens groups (including mathemati-cians and engineers) publishing web pagesdesigned to persuade readers that the"reform" represented by the 1992Framework and its progeny, should be dis-carded in favor of something usually(though simplistically) called "traditional."In particular, they pointed to what theysaid were deteriorating scores of Californiachildren on national tests.

In a word, the anti-reform campclaimed "Johnny can't add," but insteadspends his school time measuring play-grounds and talking it over with hisclassmates; and he uses a calculator whenasked to multiply 17 by 10. Thus the new,ad hoc Commission on Standards wasappointed, quite apart from the legallymandated Framework committee, toineffectadjudicate this controversy. TheCommission, a citizens' commission notintended to be expert in mathematics (new

One can no more use

mathematical "concepts"

without a grounding in

fact and experience, and

indeed memorization and

drill, than one can play

a Beethoven sonata

without exercise in scales

and arpeggiOs.

standards for othercore subjects werealso part of itscharge), took advicefrom experts of itsown choosing andended sharply divid-ed. It voted by a largemajority in favor of adocument that it sub-mitted to the StateBoard of Educationon October 1.

The Board, whichhas final say, heardmuch public testimo-ny, includingopposition expressedby mathematicians,and rejected the draft.The Board used that

draft, however, as abeginning for the very substantial revisionit ultimately approved in December of1997. That revision, which is the Standardsreviewed here, was mainly prepared by agroup of mathematicians at StanfordUniversity, and its publication has generat-ed more public controversy than anythingseen earlier. Apart from segments of thepublic, two groups of professionals are nowin contention: the mathematics educationcommunity, or a vocal part of it, againstthe mathematicians' community, or a vocalpart of that.

(In the meantime, the Board has thetask of reconciling the mathematics stan-dards implicit in the new Framework,expected to be published in 1998, with therevised Standards under review here.)

Newspaper reports of the controversymake it apparent that those opposed to theBoard's revisions, and who wish the Boardto return to the document submitted tothem by the Commission in October,include the California Superintendent ofPublic Instruction and at least one high-ranking official of the National ScienceFoundation. This party portrays this revisedStandards as a return to the failures of pastyears, a document devoted to the mindless,pointless manipulation of outdated algo-rithms. The authors of this report believesuch a characterization is mistaken, andthat the mathematicians who participatedin the final revision had no such intention,and their product no such resultexcept aspoor teaching might make it so. It is to thebetter mathematical education of teachersthat California (and the rest of us) mustlook for improvement of result.

Colorado

The best feature of the Content stan-dards is the listing of topics additional tothose demanded of all students. UnderFalse Doctrine is the prescription of calcu-lators and computers for whole numberarithmetic at the K-4 level. The entire texttakes up only 16 pages, but then is followedby an Index occupying three pages, surelyinflation of a sort. The text is often vague,indefinite, or carelessthus at page 12 thereader finds this phrase: "solving real-worldproblems with informal use of combina-tions and permutations"; and in theGlossary, "Algebraic Methods" are definedas "the use of symbols to represent numbers

and signs to represent their relationships"(p. 22). As to the former, why "informal"?And by the Glossary definition, algebraicmethods could be exemplified by "2+3=5,"since 2,3, and 5 are symbols and + and =are signs representing their relationships.But "algebra" means something more thansymbolism. On page 14, this carelesslyplaced item for grades 5-8 occurs: "Solveproblems using coordinate geometry."Taken seriously, this is something done in12th grade pre-calculus courses, or in col-lege, unless "problem" means somethingquite trivial. Such carelessness must becounted False Doctrine.

StATE REPORT CARD

Colorado

I. CLARITY 0.7

II. CONTENT 1.7

III. REASON 1.0

IV. NEGATIVE QUALITIES 2.0

TOTAL SCORE (out of 16) 54

GRADE

Connecticut

This August 7, 1997 document ismarked "Second Draft," but will requireexpansion and definiteness of reference tobe of any value. Reason is pushed to onecorner of the "Geometry" pageat the9-12 level, and among five other itemswhere readers are told, "Develop anunderstanding of an axiomatic systemthrough geometric investigations, makingconjectures, formulating arguments andconstructing proofs." This recapitulation ofthe thousand-year long development ofancient Greek mathematics is unlikely in ahigh school, and should be replaced by arealistic set of standards for deductivegeometry, if that is what is meant. As writ-ten it is too ambitious. On the other hand,the following, from level 5-8, is too child-ish: "Use real-life experiences, physicalmaterials and technology to construct

meanings for the whole numbers...."By grade 5 the abstract notion of a wholenumber should long have been in mind,without blocks and calculators.

The Framework is too brief to containmuch False Doctrine or Inflation. Still,standard 4, "ratios, proportions and per-cents," runs through all grades; under it, atgrades 5-8, we find the following: "usedimensional analysis to identify and findequivalent rates"; and this, at grades 9-12:"Use dimensional analysis and equivalentrates to solve problems." Not muchprogress there. The entire page, of whichthe two quoted items form about 20 per-cent, is inflated by the strain of findingenough words to justify the existence ofstandard 4 altogether, for it surely does notdeserve equal billing with (say) standard 9,"Algebra and Functions"and even

38

'Connecticut

I. CLARITY

II. CONTENT

III. REASON

IV. NEGATIVE QUALITIES

GRADE

REPORT 1M1

0.7

1.0

1.0

2.0

TOTAL SCORE (out of 16) 41

Standard 9 omits geometric series, thequadratic formula, and the binomialtheorem.

Delaware

The 1995 text is brief, though augment-ed by pedagogical suggestions, and it coversK-10 only, as a guide to statewide testing atthe grade 10 level. An "Appendix A" hasbeen added to the state's Web site, givingan indication, all too brief, of what will beexpected at grades 11 and 12, and includ-ing a welcome announcement that the"logical framework" of algebra, and deduc-tive proofs in general, will form animportant part of the program. However,this part is not yet as fully elaborated as themain text covering K-10. Here, where thestandards themselves are vague, which isoften, the suggested instructional activitiesgenerally illustrate a very minimal content.Projects, keeping journals, reporting to theclass, and "real-life" applications are toooften emphasized above intellectual con-tent. The constructivist stance leads to

instructions concerning student "explo-ration," sometimes producing mystifyingopen-ended demands, e.g., "Examine therelative effect of operations on rationalnumbers" ("Number Sense," grades 6-8).Relative to what? Which operations? Whyrational numbers?

Where items are less vague, they candescribe a valuable curriculume.g., for"Geometry" at grades 6-8, "Use a compassand straight edge as tools for basic geomet-ric constructions" (p. 51).

If this were accompanied by a lesson inthe reasoning behind the validity of somesuch constructions, it would prepare for thelogical analyses indicated for the grade 11-12 levels. But the very use of the word"tools" for the ruler-and-compass backboneof Euclid's geometry generates a misunder-standing of that ancient branch of

rat hi faittlIrMEIRTONtiTinfAlinDi

Delaware

I. CLARITY

II. CONTENT

III. REASON

IV. NEGATIVE QUALITIES L5

1.0

7-5

LO

TOTAL SCORE (out of 16) 7.8

GRADE C

mathematics. There is very little of Reasonthreaded through the performanceindicators.

District of Columbia

There is here an occasional good idea,but such ideas are few. On page M-13(grade 11 level), the reader discovers this:"Illustrate that a variety of problem situa-tions can be modeled by the same type offunction." As a summary of what can belearned from several years of progress inalgebra, this request is exemplary, but therest of the document simply does not pre-sent or even outline such a program.(Idahosee belowposts the samedemand, verbatim but for the word "recog-nize" in place of "illustrate.") On page M-9occurs this example of Inflation: "Throughthe use of technology, students can experi-

ence a richer set of algebra experiencesthat allow them to investigate algebraicmodels at a conceptual level through repre-sentations in terms of graphs, tables,polynomials and matrices." Richer thanwhat? Algebraic models of what? Withoutdefinition, such sentences do not guideinstruction. And "matrices" at the 11thgrade level are a disproportionate demand,where algebra does not include the qua-dratic equation or binomial theorem.While the D.C. document doesn't containmuch under the heading NegativeQualities, it also contains very little that ispositive.

39

STATE REPORT CAR

District of Columbia

I. CLARITY

II. CONTENT

III. REASON

IV. NEGATIVE QUALITIES

0.7

0.0

1.0

3.0

TOTAL SCORE (out of 16) 4.1

GRADE 9

Florida

We consider the two documents (seeAppendix) together: (1) The Sumhine StateStandards is filled with examples and teach-ing strategies as well as content advice. It iswell printed and easy to use, but thoseparts covering grades 6-12 have beensuperseded by (2), Florida CourseDescriptions, which describes the materialby course titles rather than by "threads."The general tone of the Course Descriptionsis consonant with that of the Standards;however, it features an overemphasis onthe instrumental ("real-life") purposes anduses of mathematics, even in "honors"courses presumably directed at collegepreparatory students.

Thus, on page 3 of "Course 1206320 -Geometry Honors," we see, at MA.B.3.4.1,"Solve real-world and mathematical prob-lems involving estimates of measurements,including length, time, weight/mass, tem-perature, money, perimeter, area andvolume, and estimate the effects of mea-

surement errors on calculations." This istypical, and among other things rathercareless in classifying money problemsunder "geometry." There is indeed very lit-tle geometry in this course, from either theEuclidean or the analytic standpoint.Some other failures: On page 112, we read,"Recognizes, extends, generalizes, and cre-ates a wide variety of patterns andrelationships using symbols and objects."This is not clear or definite, and is inflated.(The associated example is actually simple,and worthy of more explicit description.)On page 50, the Performance DescriptionMA.A.1.4.3a, "determines whether calcu-lated numbers are rational or irrationalnumbers"; it is exemplified by a contrived"real-world" situation concerning when anautomobile's braking distance is given by arational or irrational number. Real-worldphysics has not yet achieved the means ofdetermining whether a measurement is oris not that of an irrational number. Thus,

I. CLARITY 1.3

II. CONTENT L0III. REASON 0.0

IV. NEGATIVE QUALITIES 1.5

TOTAL SCORE (out of 16) 4$

GRADE

here is an example of straining after "real-life" ends by presenting a false scientific, ifnot mathematical, doctrine. Finally, onpage 48 Example M.A.A.1.3.3a is a multi-ple-choice question with all its answerchoices wrong: carelessness again.

Georgia

On page 1 we find that "[c]alculatorsand computers are essential tools for learn-ing and doing mathematics at all gradelevels." This is not so. The Georgia "CoreCurriculum" is labeled "Draft," and is cer-tainly one of the better ones; but itunfortunately avoids mention of thedeductive structures of mathematics evenin places where it would be natural as, forexample, in "Geometry": "States andapplies the triangle sum, exterior anglesand polygon angle sum theorems." Whynot ask for their proof, too? The proofs arenot difficult, and are more enlighteningthan the results themselves. It is an oppor-tunity missed. In another place concernedwith geometry, the student is expected "to[use] tools such as compass and straight-edge, paper folding, tracing paper, mira orcomputer to construct congruent segments,angles. . .." This reduces ruler and compassto tools like the others, unrelated to thedeductive structure of the subject. As tools,they are certainly inferior to computerprintouts, but this is irrelevant to theinstructional value that should be their

purpose. This January, 1997 draft is not yeta completed document, especially indescribing the organization intended for itsadvanced offerings, which are numerousand rich in content, and include AdvancedPlacement courses in statistics as well ascalculus. It is mainly the standard K-12curriculum implied by these standards thatlacks sufficient attention to Reason,overemphasizes the uses of technology, andcarries the message in all courses thatmathematics is preeminently for directpractical use. Almost all the demands areclear and definite, but indefinitenessappears more often than it should. In"Discrete Mathematics," the student"solves problems that relate concepts toother concepts, and to real-world applica-tions, using tools such as calculators andcomputers." Probably the authors hadsomething definite in mind here, but thewords do not convey it. Furthermore, toomuch is expected of technology here if cal-culators are expected to help "relateconcepts to other concepts." This is some-thing that should be done in the mind.

40

STATE R PO/RT CARD

Georgia

I. CLARITY 3.7

II. CONTENT 3.3

III. REASON 1.0

IV. NEGATIVE QUALITIES 3.5

TOTAL SCORE (out of 16) 115

GRADE

Yet much that is written is definite andgood. For example, at the K-5 level,"Determines the missing number or symbolin addition or subtraction number sen-tences" describes a necessary abilityexactly, and "uses skip-counting as readi-ness for multiplication" pays properattention to the pacing of the teaching ofarithmetic.

Hawaii

In the Essential Content document (2),there appears under "Content" in the"Statistics/ Probability" thread, the head-ing "Probability in real-world situations";to its right, under the correspondingPerformance rubric, are three entries, ofwhich the third is "Solve real-life problemsusing statistics and probability" (p. 64).This is not an enlightening amplificationof the heading. Both documents are of thisvague nature. In the Performance Standardsdocument (1), on page S-17, occurs the 9-12 instruction, "Geometric and spatial

explorations include: . . . Non-Euclideangeometries, and hyperbolic and ellipticalgeometries." This is unbelievable wherethe documents do not otherwise mentionaxioms or proofs. And it cannot meanspherical geometry (see Note on Arkansasabove), because, while spherical geometryis related to one form of non-Euclideangeometry, it is not at all hyperbolic. Veryfew statements in either document are defi-nite enough to qualify as False Doctrine, orany doctrine at all.

Idaho

Grades 5-6 were plainly written by adifferent team than the others, and aremuch better than all the rest. We countgrade 6 as part of "middle school" for scor-ing purposes here; hence, this part ofContent receives a better grade than therest, which is inadequate on all counts.The total organization of the Content Guideand Framework generates some careless rep-etitionsfor instance, fractions anddecimal calculations are prescribed in near-ly identical terms at all grades from Kthrough 4. Surely not "decimals" in kinder-garten? And one "performance objective"at 1st grade level is that "[a]lt students willreflect on and clarify their thinking aboutmathematical ideas and situations." This isnot testable.

Of the 9-12 level the reader is told,"Every mathematics course in Grades 9-12will address objectives from each of thefourteen standards included in this frame-work. In order to accomplish this, tediouscomputations and graphical representa-tions by pencil and paper and pencil drill

must be de-emphasized to the point thatthe use of technology (calculators, comput-ers, etc.) will be used to perform these tasksat all levels of mathematics." Yet Idaho isone of the few states to include "geome-tries" [plural] in its high school curriculum,asking all students to "develop an under-standing of an axiomatic system throughinvestigating and comparing variousgeometries." The unreality of this prescrip-tion is discussed above, where Arkansasand Hawaii make a similar demand."Geometries" cannot be understood in theabsence of the understanding of at leastone example, preferably the Euclidean,something Idaho does not sufficiently offer.To speak in such grand terms, of "geome-tries," is Inflation. Under "Algebra" thisstandard appears: "demonstrate technicalfacility with algebraic transformations,including techniques based on the theoryof equations," and under "functions," "rec-ognize that a variety of problem situationscan be modeled by the same type of func-tion." The first of these statements is

41

STATE EPORT CARD

Hawaii

I. CLARITY 0.0

II. CONTENT 0.3

III. REASON 0.0

IV. NEGATIVE QUALITIES 1.0

TOTAL SCORE (out of 16) 11

GRADE F

STATE REPORT CARD

Idaho

I. CLARITY 0.7

II. CONTENT 1.0

III. REASON 0.0

IV. NEGATIVE QUALITIES 0.5

TOTAL SCORE (out of 16) Li

GRADE F

incoherent. (It also occurs verbatim in oneof Arizona's "honors" performance objec-tives.) The second, which catches one'sattention with the awkward and obscurephrase, "the same type of function," alsooccurs verbatim on page M-13 of theDistrict of Columbia's Framework.

IllinoisUnder Goal 7, dealing with measure-

ment, at 7.C.4a, is an example ofsomething clear and good: "Make indirectmeasurements, including heights and dis-tances, using proportions (e.g., finding theheight of a tower by its shadow)" is a plainand direct entry relevant to the Goal.Indefinite or obscure entries are muchmore numerous, however. For example,under "Number Sense," for late highschool, the reader finds, "Determine thelevel of accuracy needed for computationsinvolving measurement and irrationalnumbers" (p. 19). Again, 6.A.4, for earlyhigh school, states, "Identify and apply theassociative, commutative, distributive andidentity properties of real numbers, includ-ing special numbers such as pi and squareroots"; this is quite incoherent, apart fromthe indefiniteness of "apply." "Calculatorsand computers" are apparently mandatedin grades K-4 for teaching "number sense,"among other things (Standard C, see p.

18), though the itemization is obscure onthis point. This is either lack of clarity orFalse Doctrine. Certainly the standardalgorithms for the decimal system are notprescribed. Here is excess verbiage: "Usegeometric methods to analyze, categorizeand draw conclusions about points, lines,planes and space." This can only mean"Learn Euclidean geometry," and indeedthe document does call for the traditionaltwo-column Euclidean proof format at9.4.4c. But 9.4.4c is hard to take seriouslyin that it is confined to a single itemamong 18 such, the rest regarding geome-try as mainly a practical and empiricalstudy; and there is not enough time left todo all that is implied by "two-columnproofs." There is, in general, a paucity ofdemand for deductive reasoning in otherparts of the curriculum, too, though a wiseteacher might read some such demand intothe rather indefinite instructions that aregiven. The Learning Standards are avowedly

tSVT.J7i CARD

Illinois

I. CLARITY 1.3

IL CONTENT L0III. REASON 1.0

IV. NEGATIVE QUALITIES 25

TOTAL SCORE (out of 16) 62

GRADE

intended for all students, but appear to be acompromise: over-ambitious for some andneglectful of others, and vague enough topermit many sorts of teaching and manydepths of curriculum.

Indiana

The writing in this MathematicsProficiency Guide is very good, direct andplain, and only sometimes indefinite, aswhen phrases like "in a variety of ways" areused. The content, however, suffers froman unusually heavy commitment to "real-life" applications as the rationale for alllessons. On page 207, "Constructive learn-ing should be incorporated into almostevery mathematics lesson" apparentlyrequires the state to omit all too manymathematics lessons, as our ratings forContent indicate. The logical structuresof algebra and geometry are among theomissions.

Even the announced commitment to"real-world" examples is sometimes beliedby a predisposition to theory, as when atable of heights attained by an upwardsthrown baseball turns out to be a theoreti-cal set of heights attained by the Galileanformula for falling objects, and not a set of

measurements as advertised.There is also mathematical error, as in

the "topology lesson" on page 154: "Givestudents a piece of paper with a circle thesize of a dime. Cut out the circle. Can aquarter be made to pass through this holewithout tearing the paper?" This is not alesson in topology. Topology is, of course,not a K-12 subject, but if the word is men-tioned it ought to be mentioned correctly.On the same page, "The formation of a seashell depicts many golden ratios which areintertwined" is very badly stated. There isonly one golden ratio, not that this fact isthe only error in the quoted statement.However, the fact that the author herecannot or will not distinguish between amathematical abstraction (the goldenratio) and its applications or exemplifica-tions is characteristic of the view ofmathematics inculcated by the entireProficiency Guide. These are but a few

irATIANTieilittENOTIKT

IndianaCYARD

1. CLARITY

II. CONTENT

III. REASON

IV. NEGATIVE QUALITIES

3.3

L0

0.0

3.0

TOTAL SCORE (out of 16) 8,3

GRADE C

examples of the plethora of "activities" and"real-world" applications that take up mostof the space in this long document, inwhich actual demands for mathematicalcontent take but a small space.

42

Kansas

This Standards is very well organized:the "Outcomes" are listed and classified bysubject thread and grade level early in thedocument, then spread out, one "outcome"to a page, followed on that page by a num-ber of examples or exercises illustratingtheir import, for they are rather general inlanguage and tone. But the illustrations areoften trivial and illustrate very little, orthey are carelessly written, sometimes evensuggesting lessons destructive of theannounced purpose. On page 74 are giventwo matrices named A and B, with instruc-tions to multiply to get AB and then BA,and draw some lesson therefrom. (This is acalculator exercise.). But the desired "les-son," which appears to concern thecommutative law for real numbers, accord-ing to the top of the page, cannot be hadbecause BA does not exist, i.e., the matri-ces cannot be multiplied in that order.People who know matrix multiplicationonly from calculator exercises, even writ-ten correctly, cannot learn the main

lessons to be learned from matrix multipli-cation in school.

A more striking example of the inap-propriate emphasis on technology occurs atthe 8th grade level: "Use a computer pro-gram that generates Pythagorean triples.(Note: Many text-books have BASIC lan-guage computer programs already writtento do this. The student could enter theprogram and run it.) What are the first 10Pythagorean triples? (i.e., the 10Pythagorean triples with the smallestunique whole numbers)"

The reader is immediately impelled towonder whether the triple (20,21,29) "pre-cedes" or "follows" (9,40,41), but this isincidental. The real fault here is that usinga previously written BASIC programteaches a student no more aboutPythagorean triples than would theteacher's writing some of them on theblackboard. A "lesson" of this sort is oneconsequence of a standards document'sinsistence on "appropriate technology" and

eavn ru-IPC;)1707 cirvTlo

Kansas

I. CLARITY 1.7

II. CONTENT 1.3

III. REASON 0.0

IV. NEGATIVE QUALITIES 1.5

TOTAL SCORE (out of 16) 45

GRADE

manipulatives at all levels. Algebra tiles,which Kansas recommends for the 10thgrade, are another. The 10th grade is atime when one hopes such "trainingwheels" have long been left behind. Here,instead, Reason is left behind. False doc-trine.

Kentucky

Kentucky's companion document,Transformations, is not listed in theAppendix because it is a guide to instruc-tion rather than a standards document ofthe sort we find in the Core Content com-mented on here. The latter is organized toshow, for each thread (such as "algebraicideas"), the items labeled "concepts,""skills," and "relationships," at each of thelevelselementary (K-5), middle (6-8),and high school (9-11)or more accurate-ly, what is expected by the time ofassessment, which will be during the 5th,8th, and 11th grades. The list is brief andgenerally inadequate. As a guide to assess-ment it would have to be augmented bymany specifics, and perhaps experiencedKentucky teachers already know them; butthe purpose of a Core Content guide is notsimply to be a general reminder of whatone already is doing. Nor is such a guidevery helpful to new teachers.

A typical "skill" listing for "algebraicideas" at the grade 5 level is, "Studentsshould be able to find rules for patterns,extend patterns, and create patterns"

(p. 6). At grade 11, they should be able to"solve and graph a variety of equations andinequalities." It is not possible to deducefrom statements like this whether the con-tent at each grade level is rich or poor, andwe have rated Content accordingly, i.e.,only by as much as we have been instruct-ed by the text. Some instructions are moredefinite, and this pattern should be fol-lowed elsewheree.g., "Students shouldunderstand arithmetic and geometricmeans." This is good as far as it goes, but atsuch places, where the Core Content is defi-nite, it is meager.

While a vague instruction might some-times imply more content than our scoresgive credit for, one also sometimes findsthat the vagueness is actually impenetra-ble, as when students are to understand"how ratio and proportion can be used toconnect mathematical ideas." Ratio andproportion are relationships that in schoolmathematics connect numbers, or lengths,etc., not mathematical ideas. This must becalled Inflation. Again, 11th grade studentsare to understand "order and equivalence

43

asinta opoorag; ecH2o IKentucky

I. CLARITY 1.0

II. CONTENT 1.7

III. REASON 0.0

IV. NEGATIVE QUALITIES isTOTAL SCORE (out of 16) 51

GRADE 9

relations," and "how numbers in the realnumber system relate to each other." Werethe phrasing more comprehensible and def-inite, these might describe some ratheradvanced college work. The abstract ideaof "equivalence relation," for example, andany application of that idea, are found diffi-cult by many advanced collegeundergraduates.

Louisiana

It is possible to find a meaningful anduseful item here and there. At the 5-8level, "demonstrating a conceptual under-standing and applications [sic] ofproportional reasoning (e.g., determiningequivalent fractions, finding a missing termof a given proportion)" is perfectly plainand definite, and suited to the level. Butconsider Benchmark K-4, "Data Analysis":

"Demonstrating the connection of dataanalysis, probability and discrete mathe-matics to other strands and real-lifesituations." It is not possible to believe thisis asked of the same K-4 students, less than10 years old, who in another part of thesame standards document are "identifyingand drawing lines and angles and describ-ing their relationships to each other and tothe real world," something much more rea-

sonable. At every level of this document,there is such a mixture of the trivial andthe impossibly general or sophisticatedoropaque. Even the Glossary cannot be ofreal use to anybody; it defines "CoordinateGeometry" as "geometry based on the coor-dinate system," and "Magnitude" as "size orlargeness." There is much else that is nothelpful, including this: "Due to the rapidgrowth in technology, the amount of infor-mation available is accelerating so rapidlythat teachers are no longer able to impart acomplete knowledge of a subject area" (p.1-2). Even without technology, the conclu-sion that more is knownabout anythingthan teachers can impart was true in thetime of Plato. Such platitudes do not con-tribute to the purpose of a standardsdocument, and are Inflation.

Maine

Under the Content Standard,"Students understand and demonstratethat ideas are more powerful if they can bejustified," is the following PerformanceIndicator: "[Students should understand]that proving a hypothesis false (i.e., thatjust one exception will do) is much easierthan proving a hypothesis true (i.e., truefor all possible cases)." The lesson that asingle counter-example falsifies a conjec-ture ("conjecture" is better than"hypothesis" here) is an important one,worth teaching. However, to cast proof as ameans of making an idea more "powerful,"rather than merely true, is not to make amathematical statement at all, and perhapsto make a false statement, as the history ofdemagoguery shows. The teaching ofReason is ill served in general in thesestandards, as Maine's rating for Reasonindicates.

This Curriculum Framework is for math-ematics and science both, and gives themathematical reader a chance to observeanother field (science) in passing. On page

4, "Understand that matter can be neithercreated nor destroyed" is a doctrine of great18th century importance, and was learnedby children as late as 1900 in just thoseterms, but it is now known to be mistaken,and its exceptions have had important20th century implications. With mathe-matics, on the other hand, the Mainedocument has it the other way round: the(apparent) exceptions to old doctrine arebeing emphasized beyond their desserts.Deductive thinking, given us by ancientGreece, is here replaced by a "contentstandard" urging that "students usedifferent methods of thought to justifyideas," and one performance indicatorunder this standard advises that students"use intuitive thinking and brainstorming."These are things we do perforce; mathe-matics should teach us to govern theseimpulses, channel them, and recognizetheir pitfalls. Also under the general head-ing of Reason is "make inductive anddeductive arguments to support conjec-tures." This is about as much detail as the

44

STATE REPORT CARD

Louisiana

I. CLARITY

II. CONTENT

III. REASON

IV. NEGATIVE QUALITIES

0.0

0.0

1.0

TOTAL SCORE (out of 16) 15

GRADE F

egitrAvanwEilwartratirosMaine

I. CLARITY

II. CONTENT

III. REASON

IV. NEGATIVE QUALITIES

0.0

1.0

0.0

IS

TOTAL SCORE (out of 16) 15

GRADE F

document offers.The illustrative, mainly "real-world,"

examples throughout the document do notoften have much mathematical content.To advertise reasoning is not the same as toteach it in contexts where its value is visi-ble. There is not much context here tomake this possible.

Maryland

The Framework items of student "expec-tations" occupy only pages 18-23 of this 42page pamphlet, the rest being introduction,educational philosophy, and other com-mentary. There aresix extremely gener-al Goals. Forexample, Goal 2 is"to develop anunderstanding of thestructure of mathe-matical systems:concepts, properties,and processes." Eachgoal has four, five orsix "subgoals." Forexample, Subgoal2.3 is to "[u]nder-stand concepts,properties, andprocesses of geome-try," which is notmuch advancementin specificity. Finally,there are from twoto 13 "expectancies"under each subgoal.These are also gen-eral, and not keyedto particular gradelevels. Expectancy 2.3.7 is to "explore geo-metric ideas such as topology, analyticgeometry, and transformations." This is asspecific as the Framework gets, whichmeans that it is of no practical value. Thisparticular example was chosen to showthat it can also be inflated and unrealistic:topology is no subject for school mathe-matics anyway, and this should be knownto the person who wrote this "expectancy."Most of the Framework is of this level of

generality and sometimes unreality, andwhile it also contains some advice abouthow local schools can use it in construct-ing curricula, it is not really a guide of the

sort envisioned in thecriteria for the ratings asdescribed earlier in thisreport.

What value there isin the package of twodocuments in our posses-sion is mainly in TheHigh School Core LearningGoals (document (2) inthe Appendix), which ismore definite, but quiteundemanding of content.It also has goals andexpectations, e.g., underGoal 2, "Geometry,Measurement andReasoning," Expectation2.1 states that "the stu-dent will represent andanalyze two- and three-dimensional figures usingtools and technologywhen appropriate." Oneof the "indicators" underthis Expectation (2.1.3)

is that "[t]he student will use transforma-tions to move figures, create designs, anddemonstrate geometric properties." Clearlythis indicator is compatible with whatevergeometry course the local district mightwant to create. Despite the vagueness,there is False Doctrine in the overemphasison calculators and computers, and the"real-world" applications, examples ofwhich in the Core Learning Goals over-whelm whatever intellectual content the

The most that can be said

about change is that we

now have some mathe-

matical processes that

were not earlier known;

but the ones taught in

high school have mostly

been around for a long,

long time, and the world

still has need of people

who understand them.

45

STATE RIEPOAT

Marylandc D

I. CLARITY 0.7

II. CONTENT 1.3

III. REASON 0.0

IV. NEGATIVE QUALITIES 1.0

TOTAL SCORE (out of 16) 3.0

GRADE

text might attempt to imply.The two documents have the ambition

to provide a philosophical background forcurriculum and teaching, but do not end bysaying anything that can be used, and byoccasionally saying what is not so, as whenthe author of the Core Learning Goalsmakes this statement: "With the change intechnologies, the mathematical processeschange" (p. 1). How can this be, when onpage 15 is pictured a house with a patio,inviting the student to find some optimaldimensions? This problem and its solutionhave not changed in 5000 years; though wenow have a more convenient notationthan did the ancients, the mathematicalprocesses are the same. The most that canbe said about change is that we now havesome mathematical processes that were notearlier known; but the ones taught in highschool have mostly been around for a long,long time, and the world still has need ofpeople who understand them.

Massachusetts

The Framework is nicely laid out andprinted, with excellent photographs as wellas diagrams, but it is pedagogically andmathematically incoherent, and lacking incontent. It says that manipulatives shouldnot be confined to the early grades, andindeed it illustrates their use at all levels,but manipulatives are poor preparation forproofs "by mathematical induction" in the9th and 10th grades (p. 78). Any collegemathematics teacher will testify to the dif-ficulty of teaching proofs by induction tocollege mathematics students, and the ideathat this should occur in 10th grade isunrealistic, suggesting that "mathematicalinduction" has been confused with therather vague notion of "inductive reason-ing" in the mind of the author.Furthermore, the placing of mathematicalinduction under the heading of "Geometryand Measurement" is also incoherent.

At grade level 11-12, under"Geometry and Spatial Sense," we havethis incoherent Learning Standard: "Usevectors, phase shift, maxima, minima,inflection points, and precise mathematicaldescriptions of symmetries to locate anddescribe objects in their orientation" (p.81). The other standard in this section is"deduce properties of, and relationships

between, figures from given assumptions,"which is coherent but vague. Oppositethese two, under "Example of StudentLearning," is a picture of a parallel pair ofPlexiglas rectangles held in place by somebolts, and the instruction to dip this into asoap solution to illustrate Steiner points fornetworks. This model has nothing to dowith the two standards, and, except for its"popular science" appeal, has no educa-tional value at this level of mathematicalinstruction.

At grades 5-8, students are to "engagein problem-solving, communicating, rea-soning, and connecting to . . . develop andexplain the concept of the PythagoreanTheorem." This kind of verbiage is inflat-ed; the thing is a theorem, not a concept ofa theorem, and "communicating" and"connecting" etc., are indicative of the sortof time-wasting richly repeated in otherparts of the Framework, where "Examples ofStudent Learning" are given. All this in adocument that doesn't ask for the essentialskills of factoring of polynomials, or thequadratic formula, or the binomial theo-rem. Page 28 contains this: "Move awayfrom the notion that basics must be mas-tered before proceeding to higher-levelmathematics." For students of ordinary

46

STATAE REPORT CARD

Massachusetts

I. CLARITY 1.0

II. CONTENT 0.3

III. REASON 0.0

IV. NEGATIVE QUALITIES 0.0

TOTAL SCORE (out of 16)

GRADE F

ability this is the road to "mathematicsappreciation," not mathematics. Indeed,"mathematics appreciation" seems to bethe main theme of the Framework whichtalks of Fibonacci numbers and Nautilusshells, soap films and Steiner points, frac-tals, notions whose mathematical contentis generally beyond the school mathemat-ics level, and which can only serve asamusements. The document as a whole isnot serviceable for the purposes outlined inthe introduction to the criteria, as givenearlier in this report.

Michigan

The educational philosophy given inthe introductory pages of the ModelContent Standards is clear and definite:These standards are to be consistent withthe "constructivist" view of the education-al process, "firmly grounded in the work ofJohn Dewey.... In other words, it is nolonger sufficient to simply know mathe-matical facts; learners must be able tounderstand the concepts behind them andto be able to apply them to problems andsituations in the real world." However, itis not possible to determine from thestandards as written what are the "mathe-matical facts" desired, insufficient thoughthey might be, let alone the way to "under-stand the concepts." The items are vague,and are more exhortations than standards,or are incoherent.

Under Content Standard 9, concerningnumber systems, at the High School level,students are to "develop an understandingof irrational, real, and complex numbers."There is something mathematically offbeatabout the progression, "irrational, real, andcomplex," where the historical construc-tion runs "rational, real, and complex"; and

a mathematician's unease increases twopages later, where students are to "developan understanding of the real and complexnumber systems and of the properties ofspecial numbers including i, e, and conju-gates." Conjugates? Again, this is anincoherent list, collected without thoughtof the pedagogical sequence it is intendedto suggest, and perhaps without under-standing of the non-parallel qualities ofthat particular listing.

A typical example of a standard isContent Standard 3: "Students developspatial sense, identify characteristics anddefine shapes, identify properties anddescribe relationships among shapes."Below the standard are specifications, somefor elementary, some for middle school,and some for high school. Here is one, formiddle school: "Generalize the characteris-tics of shapes and apply theirgeneralizations to classes of shapes." Peoplewho do not understand mathematics mightbe intimidated by the technical words"generalize" and "classes of shapes" intobelieving this sentence was written inEnglish, or (alternatively) into believing

iiSitrAii/EINRIEIP.101RITMICSAIROMI

Michigan

I. CLARITY 0.3

II. CONTENT 1.0

III. REASON 0.0

IV. NEGATIVE QUALITIES 0.0

TOTAL SCORE (out of 16) 1.3

GRADE F

that mathematics is not really supposed tobe written in English, but either deductionwould be incorrect.

The principal faults of the entire docu-ment are its brevity and its vaguenessastandards document cannot afford bothwhich render it an unsuitable guide toinstruction or choice of curriculum.

Mississippi

This Curriculum Structure is well writtenand rich enough in content to serve itspurpose, though it is sprinkled throughoutwith an excessive number of references to"real-world" applications, and is corre-spondingly short on instruction concerningthe structures of mathematics (i.e., relativeto our criterion, Reason). Page 51 containsthis dangerous doctrine, though oneobserved by most other states withoutexplicit mention: "Even though proofremains an important component in thegeometric [sic] course, a shift from the tra-ditional two-column deductive proofremoves proof as the primary focus of thecourse to one in which the student pro-vides informal arguments either orally or inwriting." More happy, however, is the factthat Mississippi even mentions that proof(deductive proof) is important at all.

The exaggerated devotion to "real-life"applications sometimes leads to inflatedclaims, or verbiage, e.g., "The student will

understand the role of application of matri-ces in connection with conic sections indescribing real-life phenomena" (p. 65).Apart from the fact that no high schoolstudent is in a position to relate the mathe-matical properties of the conic sectionswith real-life phenomena, and that the useof matrices in their analysis is quite anadvanced branch of algebra, this particularinstruction is found under the goal con-cerned with statistics and probability, withwhich it has no discernable connection.(The conic sections are indeed the paths ofcomets and planets, but analyses of thisdepth cannot be done without someknowledge of calculus, which is notexpected here, and is not particularly aidedby the use of matrices anyway.)

Another contrived "real-life" connec-tion is the following: "Solve problemsinvolving factors, multiples, prime andcomposite numbers; include concepts ofcommon factors and multiples and prime

4 7

onpooa Otano IMississippi

I. CLARITY 3.7

II. CONTENT 3.7

III. REASON LO

IV. NEGATIVE QUALITIES L5

TOTAL SCORE (out of 16) 11.9

GRADE

(

factorization (expressed using exponents);include real life applications of these con-cepts" (p. 33). The entire instruction isadmirable except for the last clause, whichis impossible.

Missouri

By making some inferences that mightor might not be consistent with the intentof the rather unclear wording of this entiredocument, one can deduce that Missouriintends a fair amount of mathematical con-tent, which gives this Framework a betterscore for Content than under the other cri-teria. The document is well organized, notonly by strand (e.g., "Data analysis,Probability and Statistics") and grade level(K-4, 5-8, and 9-12), but by "What all stu-dents should know," "What all studentsshould be able to do," and "Sample learn-ing activities." But these carefullyconstructed categories are not often filledwith valuable information.

Under "Mathematical Systems andNumber Theory," Grades 9-12, all studentsshould be able to "select and apply appro-priate technology as a problem-solving toolto achieve understanding of the logic ofalgebraic and geometric procedures" (p.64). However, the Framework does notmention axioms and theorems with proofs

as another way (i.e., in addition to "tech-nology") of achieving this understanding.There is at this point a reference to NCTMStandard 14 which, upon examination,shows no mention of technology in thisconnection. The implied technologyapparently includes "algebra tiles," whichare mentioned on page 64, too, thoughthey are better described as manipulatives.This is False Doctrine by our criteria, andan avoidance of the sort of mathematicalinstruction that should be implied by thetitle "Mathematical systems and numbertheory" that heads the page.

Other instructions on the same page areless mischievous because less definite:"Compare and contrast the real numbersystem and its various subsystems," and"extend understanding and application ofnumber theory concepts." Such languagecharacterizes the Framework; here is anoth-er example taken almost at random:"Experiences should be such that studentsuse discovery - oriented, inquiry -based and

ArE it+ EP'/OTIPtirtrAN-C153

Missouri

L CLARITY

II. CONTENT

III. REASON

IV. NEGATIVE QUAUTIES

0.1

LO

0.0

0.0

TOTAL SCORE (out of 16) Li

GRADE F

problem-centered approaches to investi-gate and understand mathematics" (p. 9).One can hardly select textbooks on thebasis of this sort of instruction withoutalready knowing from other sources what isto be taught.

Montana

The major part of the cited documentconcerns institutional matters: accredita-tion, teacher education, and general goals(and visions) for education in science andmathematics. The "MathematicsCurriculum Standards" occupy page 57 andpart of page 58, and are mostly too generalto be of usee.g., "include the study of

trigonometry." Brief and general as theyare, the listed items exhibit a disinclina-tion to ask that anything be overtly taught,suggesting instead "explorations" and"experiences," as in the instruction forgrades 5-8 to "include explorations of alge-braic concepts and processes."

48

tietrAitrinitTifP CWRMontana

I. CLARITY 0.0

II. CONTENT 1.0

III. REASON 0.0

IV. NEGATIVE QUALITIES 05

TOTAL SCORE (out of 16) 15

GRADE

Nebraska

Page viii states this directive: "Themathematics/science learner will demon-strate mathematical and scientific literacyin a global environment." At page D-5(high school geometry) occurs the "mea-surable performance" demand: "Use thedeductive nature of geometry to solveproblems." This demand is, as is the style ofthis Framework, followed by an entry underthe follow-up rubric, "A closer look," urg-ing real-life experiences to "enhance theunderstanding of modeling as a problem-solving tool." Finally, there are the "sampleinvestigations" to close out this instruc-tion: The first is a treasure hunt, and thesecond concerns the decorating and fur-nishing of a room. All told (and "global

environment" or not), none of this helps"demonstrate mathematical and scientificliteracy." The idea that deduction, alleged-ly asked for in this "measurable process," isa mental process is lost in all these "experi-ences." Another example occurs on pageD-11: "Technology . . . must be used tobuild the basic notion of a function." Herethe false doctrine is more urgent.Technology is no longer an aid, but anecessity, or so it is said. But the basicnotion of a function was understood wellbefore the technology required here wasinvented. Indeed, reasoning about func-tions helped develop technology, not theother way round.

New Hampshire

This Framework is easy to handle andread, for it spends a bare four pagesexplaining what it is about, and then goesabout it. The "Societal goals" are excellent,e.g., "All students will develop strongmathematical problem solving and reason-ing abilities" (p. 3). Its page on "Howstudents learn mathematics" quite properlyemphasizes students' engagement with thematerial, and does not consider "real-life"applications the only path to knowledge.However, it repeatedly prescribes manipu-latives for students at levels that should bebeyond that sort of thing (False Doctrine),as at "Spatial Sense," on page 19, wherethe "proficiency standard" for End of Grade10 says, "Use manipulatives, and/or coordi-nate geometry to explain properties oftransformations. . . ." At this level, geome-try should be presented conceptually, andindeed logically, rather than as a continua-tion of the wooden block constructions ofkindergarten.

By the end of grade 6, this 30 page doc-ument uses three lines under "NumberOperations" to ask students to "demon-strate an understanding that when dividingtwo whole numbers that are greater thanone, the quotient will be smaller than the

dividend." Such a detail is jarring in alist which, a few lines earlier, rather moregrandly asks 3rd graders to "explain therelationship among the four basicoperations."

Another curiosity, unrelated to ournumerical scores under the criteria listedabove, occurs at algebra level 7-12, wherestudents are to, "[m]odel and solve prob-lems that involve varying quantities withvariables, expressions, equations, inequali-ties, absolute values, vectors, and matrices"(p. 25). This sentence is a bit awkward,and would attract the attention of anymathematician by its inclusion of the misfitphrase "absolute values," which is simplynot parallel with the other items named ina list already too-long. The word "with" isalso strange here, but apparently means"using," as in New Jersey's Standard 4.13for 12th grade algebra, which reads,"Model and solve problems that involvevarying quantities using variables, expres-sions, equations, inequalities, absolutevalues, vectors and matrices," which (bywhat cannot be a mere coincidence)repeats the same list in the same order.Believing there must be a common originto this rather strange listing, the authors

49

STATE REPORT CARD

lebraska

I. CLARITY

II. CONTENT

III. REASON

IV. NEGATIVE QUALITIES

0.0

1.0

1.0

1.0

TOTAL SCORE (out of 16) 3.0

GRADE F

IESITTAITJESERTEIPIO1RITIICIATIen

New Hampshire

I. CLARITY 1.7

II. CONTENT i.0III. REASON 1.0

IV. NEGATIVE QUALITIES 1.5

TOTAL SCORE (out of 16) 6.2

GRADE

went to the NCTM 1989 Standards, whichis the acknowledged model for so many ofthe states' standards, and found on page150 a similar, if simpler sentiment:"Represent situations that involve varyingquantities with expressions, equations,inequalities, and matrices." The NCTMlist, however, omits the misfit phrase"absolute values" and the word "vectors"that both New Hampshire and New Jerseychose to include at identical points in theiridentical listings.

New Jersey

The Framework (document [2], seeAppendix) is a very ambitious and verylong commentary on the Content Standards(document [1], which appeared a little ear-lier). A teacher who follows the outerimplications of theFramework's manysuggestions could verywell prepare a studentto bypass the first yearat MIT, while others,using other textbooksequally consistentwith these standards,would be able toavoid much thatshould be mandatoryin any school system.More than moststates, New Jerseyseeks to embed math-ematics in a culturalframe of reference,which is a worthyeffort in the hands ofwell-educated teach-ers; but the classicalcontent of mathemat-ics, and its backboneof deductive reason-ing, without whichno amount of culturalframework can reallybe understood, areoften slighted in these standards.

No proof of the Pythagorean theorem isdemanded, yet the student is expected to"explore applications of other geometriesin real-world contexts" (Standard 7, grades9-12). While this is not explained in thestandards, the Framework gives as an exam-

ple of "other geometries" the thoroughlyEuclidean geometry of figures drawn on asphere. (There really are other geometriesthan the Euclidean, such as projectivegeometry; and the discovery of hyperbolic

geometry in the 19thcentury was a milestonein the history of mathe-matics.) Without agood background in the

A teacher who follows the

outer implications of the

Framework's many

suggestions could very

well prepare a student to

bypass the first year at

MIT, while others, using

other textbooks equally

consistent with these

standards, would be able

to avoid much that should

be mandatory in any

school system.

Euclidean deductivesystem, or of more ana-lytic geometry than anyhigh school is likely tooffer, such "non-Euclidean" studies aremore "math apprecia-tion" thanmathematics.

The emphasis on"real-world" applica-tions and activities isextreme, and one mustoften guess at the intel-lectual content implied.In other places, thereare admirable instruc-tionse.g., "Describeand apply proceduresfor finding the sum of afinite arithmetic seriesand for finite and infi-nite geometric series,"which would be even

more admirable if the word "prove" hadappeared somewhere in that sentence.There is some attention to Reason in theNew Jersey standards, but not enough,which is one concomitant of the falling offof Content in the high school years.

50

STATE REPORT CARD

New Jersey

I. CLARITY 2.0

II. CONTENT 1.7

III. REASON 2.0

IV. NEGATIVE QUALITIES 2.5

TOTAL SCORE (out of 16) 91

GRADE C

N ew Mexico

One good thing about these standards isthe forthright statement at the outset that"proficiency in English is of the highestimportance." This is immediately qualified,however, by supporting "the use of the stu-dent's primary, or home language, asappropriate . . . while the student acquiresproficiency in English." There is not muchto object to here, but there is also notmuch guidance, as to when and how muchand how.

Similarly, Standard 1 ("Problem solv-ing") states that "students will usemanipulatives, calculators, computers, andother tools, as appropriate, in order tostrengthen mathematical thinking, under-standing, and power to build onfoundational concepts." This instruction isgiven verbatim for each of the levels K-4, 5-8, and 9-12. But the evasive "asappropriate" renders this recommendationnugatory, just as in the earlier recommen-dation concerning bilingual education.Teachers are looking to the state's stan-dards exactly to find out what isappropriate and when; it is not for the stateto leave the essence of its recommenda-tions undefined. Instructions so heavilyqualified as to be interpretable at will must

count as Inflation. And if the interpreta-tion is the use of manipulatives in highschool, or calculators at K-4 when arith-metic begins, it is False Doctrine.

The language of the Standards is some-times garbled by excessive devotion togeneralities. Standard 2 asks that students"use mathematics in communication"where the most likely interpretation, asevidenced in the associated benchmarks, isthat "use communication in mathematics"is closer to the intention. The NCTM1989 Standards has a thread called"Mathematics as Communication," whichis probably closer yet. A similar inversiontakes place in Standard 3, which states that"students will understand and useMathematics in Reasoning," apparently amisprint for ". . . use reasoning in mathe-matics," since at least two of the associatedbenchmarks concern the use of reason inobtaining mathematical results.

Even so, the specifics are vague, hereand throughout the document. Under"Number systems and number theory," forexample, appears the line, "develop andanalyze algorithms." Specifications thisgeneral cannot serve as a guide to teachingor curriculum. More general yet, indeed

s1TTAIT7E11 iffETPXOTititirt7ATRIN

N ew Mexico

I. CLARITY

IL CONTENT

III. REASON

IV. NEGATIVE QUALITIES

0.0

1.0

0.0

15

TOTAL SCORE (out of 16) 25

GRADE F

breathtaking, is this one, under Standard 4,for grade Level 5-8: "Students will applymathematical thinking and modeling tosolve problems in other curriculum areassuch as employability, health education,social studies, visual and performing arts,physical education, language arts, and sci-ence; and describe the role of mathematicsin our culture and society." Among otherthings, this particular skill is hardlytestable.

N ew York

The best feature of this booklet, inwhich only 16 pages are devoted to mathe-matics, is the absence of inflated verbiageand unrealistic expectations, and theabsence of downright bad advice. However,the brevity of the mathematics section ofthe "mathematics, science, and technolo-gy" standards permits extremely varyinginterpretations as to content. On page 26 itasks for proofs by mathematical induction,for example, a difficult topic at the highschool level, and it asks for the analysis ofinfinite sequences and series. These arequite demanding and specific. Yet at theintermediate level the requirement thatstudents "develop appropriate proficiencywith facts and algorithms" (this comesunder mathematical "operations" as arubric) permits too much leeway.Sometimes only a teacher already experi-enced in what New York means by certainterminology will catch the drift of a too-

condensed demande.g., at high schoollevel, "Students . . . model and solve prob-lems that involve absolute value, vectors,and matrices" (p. 27). Any mathematicianwould be puzzled at this juxtaposition ofvectors and absolute values. And, in thesame list of performance standards, "repre-sent problem situations using discretestructures such as finite graphs, matrices,sequences, and recurrence relations" seemsto imply a lot of content, but it does sosomewhat vaguely, in view of the phrase"represent problem situations," which doesnot quite say what it probably means. Doesit mean "solve problems," and if so, whatsorts of problems? Here is a place wherethere is white space on the page, in whichillustrative exercises would be valuable; butin general, the illustrative examples arefew, and most of them are on the wholemuch less demanding than any reasonableinterpretation of the text.

51

STATiE REPORT CAR

N ew York

I. CLARITY

II. CONTENT

III. REASON

IV. NEGATIVE QUALITIES

1.3

3.0

3.0

4.0

TOTAL SCORE (out of 16)

GRADE

North Carolina

There is very little to complain of inthis generally excellent document, fromwhich other states could learn. It is careful-ly done. It has over 500 pages, but they areoccupied with practical information ofexactly the sort the new teacher in townwould need to know. Only pages 13-61 arerequired to outline the content, but it isdone briskly, by topic and grade level. Mostof the rest of the book reiterates the con-tent standards and couples them with"sample measures,"which are test ques-tions or classroomexercises. These canserve a teacher eitheras class material or,most important, asclarification of theimport of the stan-dards themselves.

Not everythingclear and definite isnecessarily good,however. On page 55,competency goal 5.2,"Use synthetic division to divide a polyno-mial by a linear binomial," betrays a ratherold-fashioned origin for this Algebra II cur-riculum. Synthetic division is a superbexample of an unnecessary algorithm, onewhich, unlike the "long division" algo-rithm for the 5th grade in the hands of agood teacher, does not offer any insightinto the nature of the process. Such time-wasting is unusual in this curriculum,

4W4f,

There is very I

however.There is nothing dated about 7.11:

"Explore complex numbers as solutions toquadratic equations" (p. 55). Filled in withgood teaching, the instruction is excellent(though "explore" ought not to be called a"competency goal"), but will this rathergeneral "exploration" lead to learning? Onpage 472 this goal is repeated, with some"sample measures," among them, "Find aquadratic equation that has these roots:

5+2i and 5-2i"; and,"solve the quadraticequation .. ." [here anexample requiring thequadratic formula isittle to

complain of in this

generally excellent

document, from which

other states could learn.

eta

given]. However, hereas in all too many otherplaces, the inner struc-tures of mathematicsare ignored in favor of atelegraphic crispness. Itwould seem exemplaryto say, "Here is theequation; find theroots," and then, "Here

are the roots; find the equation." But thereis more that can and should be said: Whatif the given roots are not complex conju-gates of each other? Does the sum of theroots have the same connection to thecoefficients in the complex case as the real?What good is a quadratic equation?

It is hard to introduce such considera-tions into a list of contents, and most stateshave gone overboard trying to do so, gener-

C39

)1,

;,TOTAL SCORE (out of 16) 141

A

ally to the disadvantage of the contentsthemselves. Yet it can be done, and theplaces where North Carolina's scores areless than 4 are indications of such minorfailures.

Reason, for example, gets a good work-out in the geometry course, but somewhatless in the algebra and trigonometry sec-tions, where it could have been injectedwith little extra effort. The excessive num-ber of threads in the elementary gradesleads to a diffusion of content. And in the"sample measures" sections that follow thecontent listings, a disproportionate amountof space is devoted to the 9-12 segment ofthe curriculum. Elementary teachers, whoare generally not specialists in mathemat-ics, need more guidance, year-for-year, thanthose in the more advanced years.

North Dakota

The organization is excellent, each pagecontaining one standard (e.g., "NumberTheory" or "Geometric Concepts") andone of the three grade levels K-4, 5-8, or 9-12. Then "Benchmarks/Performancestandards" give some particulars, and arefollowed by examples of "specific" knowl-edge that support them. Finally, there are"performance activities," exercises or class-room projects, to illustrate the knowledgeto be acquired, and how.

It is to be expected that the standards arequite general, but the benchmarks are oftennot much more particular, while the "exam-ples of specific knowledge" are usually listsof vocabulary items with little indication ofhow much is to be known concerning each.For example, under "Data Analysis" at the5-8 level, two of the benchmarks are "evalu-ate arguments that are based on statisticalclaims" and "display and use measures ofcentral tendency and measures of variabili-ty." As written, these could as easily beplaced at the 9-12 level, or in college, andindeed some of them do reappear at 9-12.The reader has little idea what progressionin knowledge is contemplated as the studentgrows older, i.e., what is written is not suffi-ciently definite and is not testable.

Finally, the associated activities often

are trivial, and not up to the intellectuallevel suggested by the benchmarks, bothhere and elsewhere. In the present case,one of them reads as follows: "Studentswork with partners to measure one anoth-er's height in centimeters. These heightsare recorded on the board. Each student isto find the mean, mode, range, and medianof the heights." As this could be done withany set of numbers, nothing was gained bythe time consuming use of teams and yard-sticks. (In the following activity, by theway, where predictions are called for at asomewhat higher intellectual level, thedata set is hypothetical.)

Much of the document is good, but theintellectual level deteriorates in the highergrades. There are brave words e.g., in 9-12 algebra all the elementary functions, allof them, are to be "investigated," but it isnot clear how deeply or to what end. Onecannot determine from these standardswhether students learn the binomial theo-rem or the proof of the quadratic formulaor Pythagorean theorem. The geometrycourse in particular shows little logicalstructure or coherence. Yet "Fermi prob-lems" are mandated at page 35,"discovering fractals" at page 29, and "cod-ing theory" at page 31. "Fractals" and such

North Dakota

I. CLARITY 1.3

II. CONTENT

III. REASON 0.0

IV. NEGATIVE QUALITIES LO

TOTAL SCORE (out of 16) 56

GRADE

things as "conjecture" are defined in theGlossary, but Fermi problems and codingtheory are not. (The Glossary containssome careless errors, too.) At the grades 5-8 level students are to "solve problemsusing coordinate geometry" (p. 21). This isnot believable, unless by "problem" ismeant something like plotting a pointwhose coordinates are given. Some of theperformance activities that illustrate thissection on geometry require the use of"multi-link cubes," marshmallows, tooth-picks and a "geoboard."

Ohio

Although it features extensive (attrib-uted) quotations from the NCTM 1989Standards, this document's prescriptionsare usually more definite and comprehen-sive. From the 10th grade onward, eachmandate is followed by a section headed,"In addition, college-intending studentswill . ..," the section containing extra ormore advanced material. Ohio is thereforeamong that minority of states that recog-nize in their standards that it is not wickedto offer some students opportunities notevery student might desire or be able to use.

The organization is by strand and level,the levels being keyed to what are appar-ently state-wide "proficiency tests" given atthe 9th and 12th grades, as well as someearlier standardized achievement tests atgrades 4, 6, and 8. Most items are intro-duced with "The student will be ableto . . .," and the rest is brief, e.g., "identifycommon shapes in the environment" (atthe K level), or "identify parallel lines, per-pendicular lines, and right angles ingeometric figures and the environment" at

the grade 4 level. The wide margins havespace for clarifying comments here andthere, and suggested exercises or illustra-tions.

The standards take less space thanappears in this over-200 page book, as theyare repeated, first classified by strands andthen by grade level. In three pages, then,one can see what is expected by grade 5 inall subjects. Alas, one of the "subjects" inthe early grades is Strand 2: "Problem-solv-ing strategies," which contains largelyvacuous demands, such as "validate andgeneralize solutions to problems" (p. 133).Still, even in Strand 2 there may occur avaluable reminder of something not everystate demands, e.g., (still at 5th gradelevel), "Read a problem carefully andrestate it without reference to the originalproblem" (p. 33).

In Negative Qualities, Ohio is slightlydowngraded in the category False Doctrinefor its too frequent use of "explore" whennaming something a student should be ableto do. Anyone can explore, after all, and a

53

Ohio

I. CLARITY 3.0

II. CONTENT 4.0

III. REASON 3.0

IV. NEGATIVE QUALITIES 3.5

TOTAL SCORE (out of 16) 13.5

GRADE A

teacher using these standards might wellslight the direct transmission of knowledge,and demand for its exhibition, in favor ofopen-ended discussion, some of which isvaluable in its place, of course, but theplace of which should be better definedthan it is in this document.

Oklahoma

These Priority Academic Student SkillsMathematics employ the categories (orthreads) of the NCTM 1989 Standards,except at the kindergarten level, where itwisely recognizes their inapplicability, andsimply lists quite reasonably what in math-ematics should be done there. It wouldhave done well to continue in this vein, forthe NCTM threads, "Problem Solving,""Number Sense," "Patterns .," etc., oftenencourage the invention of content itemsthat make little sense in many grade levels.In the early grades, however, the items areoften clearly and definitely stated, as at thegrade 1 level, "Use models to constructaddition facts to 10." By the 9-12 levels,however, such a statement as "Recognizethe connections between trigonometry,geometry and algebra" is surely inflated,and surely untestable, as is "Understandthe connections between trigonometricfunctions and polar coordinates, complexnumbers and series" (p. 49), somethingthat (though also vaguely stated anduntestable) could only be appropriate in acalculus course, so far as "series" is con-cerned. (This last list of things to be

"connected" also appears verbatim in theSouth Carolina Framework, page 144.) Vastand inappropriate generalities of this sortmust leave a teacher wondering about hisown competence, when it is really theauthor of the lines who is prescribingimpossibilities.

One curiosity: The resemblancebetween these core skills and the NCTMStandards is reinforced by the unattributedquotation, on page 48, of a line thatappears on page 150 of the NCTMStandards: "Represent situations thatinvolve variable quantities with expres-sions, equations, inequalities andmatrices," a strikingly awkward formula-tion that appears in a slightly varied andeven more awkwardly augmented versionin the New Jersey and New Hampshirestandards.

As to Content and Reason, there is inthe present document no mention of thePythagorean theorem, or of any particularaxiom or theorem connected with geome-try, which is still said to be studied "fromseveral perspectives" and whose "founda-tions (e.g., postulates, theorems)" are to be

Oklahoma

I. CLARITY 0.7

II. CONTENT 1.7

III. REASON 0.0

IV. NEGATIVE QUALITIES 15

TOTAL SCORE (out of 16) as

GRADE F

understood "through investigation andcomparison of various geometries" (p. 48).The surrounding text does not supportsuch ambition, even if it were possible atthe high school level. Nor is the logic ofalgebraic procedures delineated either byprescription or by implication. The generalvision of mathematics offered is lacking inboth substance and coherence, and is poor-ly expressed as well.

Oregon

The "Oregon Standards" named in theAppendix is a very brief portion of a verybrief document of that name, and by itselfdoes not serve the present purpose.However, the state sent, in answer to ourletter of inquiry, the Test Specifications forStatewide Examinations at the 5th, 8th and10th grade levels, along with sample testsat those levels and a Teachers' SupportPackage. Taken together, our hypotheticalnewcomer to Oregon has here the equiva-lent of the more usual form of curriculumstandards for guidance; and we felt entitledto rate the results along with those of otherstates which provided the more usualstandards or framework documents, recog-nizing that the materials at hand areintended for all students, and only up tothe 10th grade level.

The best quality to be observed (espe-cially in the Teacher's Support Package) isthe encouragement given students to write

their answers to questions in good Englishsentences. However, the content demand-ed is disappointing, and almost nothing isasked for that answers our criterion con-cerning Reason. The grade 10 examinationbooklet states (under "Measurement,")that "students will be given necessary for-mulas to complete problems" in theexamination room (p. 6). The eligible con-tent includes all the common figures,including circles. In other words, the basiccompetence demanded of Oregon highschool graduates includes none of the com-mon mensuration formulas, let aloneanything of theoretical import. At thesame level, "Geometry" contains an exer-cise in recognizing which of four drawingsdepicts a pair of perpendicular lines andwhich does not, surely an exercise in merevocabulary, and that of a 5th grade level.The Pythagorean theorem is mentioned,but not in any significant context.

54cin

triarRiElPIOIRMICTAIRIDi

Oregon

I. CLARITY 1.3

II. CONTENT 1.0

III. REASON 1.0

IV. NEGATIVE QUALITIES 2.0

TOTAL SCORE (out of 16) 5.3

GRADE

C-7Altogether, an undemanding curriculum isimplied by both the very brief "standards"and the more extensive but excessivelysimple Test Specifications.

Pennsylvania

While the "Proposed AcademicStandards" examined is a provisional docu-ment ("proposed" by the Department ofEducation for approval by the Governor'sAdvisory Commission on AcademicStandards, that is, with further approvalsbefore becoming official), the introductorycommentary says that "the suggested stan-dards contained in this report are theproduct . . . of more than a year of work byeducators and others from all acrossPennsylvania, the comments and sugges-tions of literally hundreds of citizens fromall walks of life. . . ." The Commission didnot write the document, but "the membersof the Commission feel the standards beingrecommended here are rigorous, measur-able, applicable, and understandableexpectations of what students should knowand be able to do." Also, "TheCommission was diligent in its efforts toremove from the standards educational andprofessional jargon."

The result, as evidenced by a compari-son between some items in this documentand their correspondents in an earlier andcontroversial standards document of 1993,is indeed less jargon, but not, we believe,what the state hoped for. The Glossarydefines "prime" so as to include "one" as aprime, and states that "1, 2, 3, 5, 7, 11, 13,17, and 19 are prime numbers" (p. 27)."Gradient" is also incorrectly defined, andthe definition of "limit" exhibits a familiarobscurity. The deficiencies in the standardsthemselves are congruent with these obser-vations. The introduction states: "Whilemathematics is a very interesting andenjoyable field to study for its own sake, itis more appropriately used as a tool to helporganize and understand information fromother disciplines." We submit that this is afalse dichotomy, and that the use of mathe-matics is appropriate everywhere.

The standards themselves are divided

by grade levels: 3,5,8, and 11 marking thelevels of what seem to be proposed state-wide assessments. They are also divided bythreads, e.g., "Number Systems ...,""Problem-solving ...," "Probability ...," etc.

The attempt to avoid jargon is palpa-ble and often successful, so that the itemsare brief. Thus, the reader finds, under"Number Systems" at grade 3, "Use draw-ings, diagrams, or models to show theconcept of fraction as part of a whole"; atgrade 5, "Explain the concepts of primeand composite numbers"; at grade 8, "Usemodels to represent operations on positiveand negative numbers"; and at grade 11,"Convert between exponential and loga-rithmic forms." Important vocabulary itemsare italicized where they appear, which isitself a guide to the intention of the phraseor sentence.

Yet jargon of a sort remains, inherentin the way the ideas are expressed and notonly in the words. "Convert between expo-nential and logarithmic forms," once itsmeaning is penetrated, says no more thanthat the definitions of "exponent" and"logarithm" should be understood. Theawkwardness of the "convert between"phrasing here derives from a long traditionof confused jargon whose usefulness wasfinally obliterated by the modern concep-tion of the exponential functions and theirinverses. In 1940 (already anachronistical-ly) the schools taught that "a logarithm isan exponent.. . ." This is not actually mis-taken, but it is, or was, the beginning of atangled sentence that most high schoolstudents of that era failed to understand.The current attempt, quoted above, may beshorn of most of that tangled wording, butit still betrays the lack of change of con-cept. By grade 11 a student should eitherlearn nothing at all about logarithms, orshould learn about inverse functions at thesame time, to be able to place the subject

55

frkrilitiftelVETP-107.11fiTirtrAllitla

Pennsylvania

L. CLARITY

II. CONTENT

III. REASON.

IV. NEGATIVE QUALITIES

1.7

0.0

15

TOTAL SCORE (out of 16) 55

GRADE

,

in its reasonable setting, a setting not rec-ognized by the quoted item.

In short, the content implied by theentire document is less than what studentscan and should learn at each grade level;even if the jargon used earlier has beenimproved, the deficiencies once made easi-ly visible by such language too oftenremain. On the other hand, the text some-times announces a startlingly ambitiouscurriculum expectation, as where it asksgrade 11 students for proofs by "mathemat-ical induction." The average grade 11student will not learn this unless thorough-ly educated in other forms of mathematicalreasoning and syntax at earlier stages of thecurriculum, and in the context of definitesubject matter, such as algebra and geome-try. But the entire thread containingmathematical reasoning (p. 7) exists in iso-lation from the others, which show littleevidence of demand for reasoning withinsubstantive contexts. Here thePennsylvania document exhibits what isprobably the most common failing of allthe standards documents this report hasstudied.

Rhode Island

This attractively packaged Frameworkhas a small poster-sized full text of thestandards folded into its inside front cover,all of which is repeated in the Frameworkproper, along with commentary and peda-gogical notes. The notes contain aphilosophical view that partially explainsthe vagueness of the standards themselves,but do not explainwhat in the way ofmathematicaldemands upon thestudents they are sup-posed to represent.

Sample entries:Under "Algebra"(grades 5-8), "Identifyand justify an appro-priate representationfor a given situation."Under "Reasoning"(grades K-4), "Makelogical conclusionsabout mathematics."Under "Problem -Solving" (grades 11-12 ) , "Usesophisticated as well as basic problem-solv-ing approaches to investigate, understand,and develop conjectures about mathemati-cal concepts." None of these is amplifiedanywhere in the document.

As philosophical background, which wecount False Doctrine: On page 15 are listedfeatures of "the traditional classroom"opposite a corresponding list for "the learn-ing community," which Rhode Islandintends shall replace it. Opposite "Teacherknows the answer" is "More than one solu-tion may be viable and teacher may not

p.

have it in advance." Opposite "Thinking isusually theoretical and 'academic"' is"Thinking involves problem solving, rea-soning, and decision-making." We believethinking involves all that is named onboth sides of this apparent dichotomy andmore, and deplore this denigration of theo-ry and the academy. And while real-world

problems often have nosingle answer, mathe-matical problems do.

As to the relativeabsence of Content in

,ZY:7P

a betterThere must be

compromise between the

dry lecture and student-

directed brainstorming

than what is implied by

this Framework.

this Framework, theauthors of this reportare not saying that nomathematics is beingtaught in the RhodeIsland schools, or thatthis Framework intendssuch a thing. But whatis written in the"Process and ContentStandards" is so vaguethat it is impossible to

tell. In K-4, it is literally not clear at all ifthe student really knows how to add, sub-tract, multiply and divide, or just has an"appreciation" for those things. What partof these four operations shall childrenmemorize? What algorithms are taught,and what schemes are left to their ownconstruction? The Framework offers noanswer, but the teachers across the stateneed one.

The Constructivist stance elaboratedthroughout the Framework would, if dili-gently followed, prevent a full curriculumfrom taking hold, since there is not time

56Pig

411ritirbittalt g lu

Rhode Island

I. CLARITY 0.0

II. CONTENT 1.0

III. REASON 0.0

IV. NEGATIVE QUALITIES 0.0

TOTAL SCORE (out of 16) 1.0

GRADE F

(

for the students to construct all therequired knowledge for themselves. Itmight be that Rhode Island does notintend it so, but the vignettes contributedby teachers to illustrate ideal lessons arguethat they, at least, are taking the idea seri-ously, so much so that one of them (at ajunior high school) announced that,among other things, "Students also discov-ered Euler's formula, F+V-2=E." In truth,the formula is simple enough for childrento understand, even if its proof is haz-ardous; but to say the children "discovered"it, or to have made them believe so, has itsown dangers. True discovery is not taughtby surreptitiously feeding answers. Theremust be a better compromise between thedry lecture and student-directed brain-storming than what is implied by thisFramework.

South Carolina

The Standards is a revised version of therelevant portions of a Framework, pub-lished earlier and containing much besidescontent standards, some of it illuminatingthe intention behind the Standards, whichis the principal subject of this evaluation.In the Standards are found fairly demand-ing curriculum indications, especiallyconcerning numerical and algebraic con-cepts for the early grades (p. 22), andconcerning numerical systems at the mid-dle school level (p. 38). On page 66("Geometry," high school) occurs a moretypicali.e., ambiguousdemand:"Deduce properties of and relationshipsbetween figures from given assumptions by:using logical reasoning to draw conclusionsabout geometric figures from given assump-tions." The suggested method (the onlyone mentioned there) uses the same wordsas the thing it is supposed to be a methodfor, and so does not add anything. But thereal ambiguity is the question of the depthof investigation asked for. It could be any-thing from "math appreciation" to afull-scale course in Euclid. Items of thiskind, which are numerous, render unclearwhat students are asked to accomplish, andmight even render inaccurate our appraisalin this report of what "content" is beingrated by our numerical scheme.

On the other hand, the instructions (p.60) concerning the desired facility (grade

12 level) in manipulating algebraic expres-sions, and solving equations andinequalities, are well-stated and illustratedby examples, though the content appears abit thin. Binomial theorem? Quadratic for-mula? Proof of either? We don't know. Toadd to the uncertainty, the phrase "usingappropriate technology" appears repeatedlythroughout the entire document, in everycontext. The teacher is thus deprived ofnecessary guidance as to the balancebetween what is to be internalized by theintellect and what is to be consigned tomachinery.

The educational philosophy expressedin the Framework, though not a subject ofnumerical evaluation in this report,includes this curious observation, whichmight illuminate the vision of past practicein mathematics education that underliesthe recommendations in the presentStandards: "Mathematics should be a disci-pline that helps [students] to make sense ofthings, not a discipline that is arbitrary anddevoid of meaning" (p. 11). We believethat the presumption that people, and inparticular teachers, have in the pastthought the latter is unwarranted.

Under False Doctrine of another sort isthe instruction on page 10 of the Standardsvolume, concerning geometry and spatialsense for primary grades students:"Students at the primary level are not

57

WAiltirairRTETPWRISouth Carolina

I. CLARITY 1.3

IL CONTENT LiIII. REASON 0.0

IV. NEGATIVE QUALITIES 05

TOTAL SCORE (out of 16) 45

GRADE

expected to know these words," It is notclear which words are in question, but thechildren are, in the sentence following,expected to do things with tessellations,symmetry, congruence, similarity, scale,perspective, angles, and networks; whyshould they be deprived of an adequatevocabulary? Grades 7 and 8, on the otherhand, sometimes apparently call for toomuch, or too advanced, material: fractals,Fibonacci, the Golden Mean, and permu-tations (e.g., to calculate nPr, albeit withtechnology). At that level, fractals canonly be "math appreciation," and nPr willnever be assimilated.

South Dakota

The introduction tells of the purposesof standards, in particular, that they are notto be construed as a curriculum, but thatthey "make an effort to describe the 'bigideas' or core content which should belearned by all students... Content standardsprovide guidelines for the development ofcurriculum by listing important areas ofcontent which must be addressed."

Further, "These standards compelteachers not only to look at the mathemat-ics taught but at what is learned, and atwhy it is learned." And, "Checklists ofconcepts and skills are no longer the pri-mary component of the mathematicscurriculum." The examples of standardsand the more particular "benchmarks" list-ing what a student should know or do atparticular grade levels, to be quoted below,will indicate that these purposes are notmet by the present document.

There are six standards, (here) brieflylabeled algebra, geometry, measurement,numbers, functions, and statistics. Eachstandard is printed at the top of a new pageand followed by a "Rationale," a paragraphtelling of its importance and place in thecurriculum, and mentioning such things associetal needs, real-life experiences, con-nections across the curriculum, our dailylives, etc. Finally, the "benchmarks" offerwhat specifics the document contains; theyare all prefaced by "Students will...." Thequotations given below will mention thegrade level intended for each. We hope thevacuity of each of them as a guide for

teachers or for curriculum developmentwill speak for itself.

Model number relationships using avariety of strategies (3.4);Analyze and describe situations thatinvolve one or more variables (9-12);Compare and contrast spatial relation-ships using geometric figures (3-4);Use models, manipulatives, and comput-er graphics software to build a strongconceptual understanding of geometryand its connection to other mathemat-ics strands, science and art (5-8);Use models, manipulatives, and comput-er graphics software to build a strongconceptual understanding of trigonome-try and its connection to othermathematics strands, science and art(9-12);Measure quantities indirectly usingtechniques of algebra, geometry, ortrigonometry (9-12);Utilize the real number system todemonstrate problem-solving strategies(5-8);Determine and describe the effects ofnumber operations on real numbers withvarying magnitudes (9-12).

Many of the words used in these state-ments, and all the others, are marked asbeing defined in the Glossary (page M 14).Some of the definitions are misleading, atthe least. "Number Systems" is defined as"systems of numbers including, but not

58

ATI R E R CAR

South Dakota

I. CLARITY

II. CONTENT

III. REASON

IV. NEGATIVE QUALITIES

0.0

0.0

0.0

1.5

TOTAL SCORE (out of 16) 15

GRADE F

limited to, complex, discrete, and binarynumber systems." But "binary" is a way ofwriting real numbers, not a system of num-bers; the given definition is analogous tosaying that whole numbers, fractions, andRoman numerals are three kinds of numbersystems. It is downright destructive ofmathematical understanding to teach chil-dren that a typographical artifact, such asbinary notation, is a fundamental mathe-matical system, like the complex numbers.

One more quotation before we leave:A "benchmark" on page M 11, grade level9-12, states that "students will demonstratethe concept of limit using mathematicalmodels." We have not been able to deter-mine the meaning of this instruction.

Tennessee

The Framework for grades K-8 is a 1996document while the Curriculum Frameworkfor grades 9-12 is dated 1991. Tennesseestates that the latter is now under revision.Since they are so different in every way,obviously written by people with vastly dif-ferent purposes, we are giving Tennesseetwo separate sets of ratings. Averagingthem does not honestly reflect our estimateof either.

The K-8 Framework begins with four"Process Standards": Problem Solving,Communication, Reasoning, andConnections, each incorporated into eachof seven "Content Standards": "NumberSense and Number Theory"; "Estimation";"Measurement and Computation";"Patterns"; "Functions and AlgebraicThinking"; "Statistics and Probability";and "Spacial Sense and GeometricConcepts." Then there is a set of fivelearning "expectations," and a Glossary forterms that appear in the document.

Some idea of the educational philoso-phy underlying the entire Framework maybe had from the Glossary entries. Theauthors warn that these definitions are notcomplete, or designed for student use, butthat "the definitions given are restricted toonly the common mathematical meaning."It is further explained that "the terms anddefinitions included in this Glossary wereproduced by the Mathematics CurriculumFramework Committee for Grades K-8." Hereare a few of these definitions:

Algebraic thinkingthinking skills whichare developed by working with problemswhich require students to describe, extend,analyze, and create a variety of oral, visual,and physical patterns (such as ones basedon color, shape, number, sounds) from reallife and other subjects such as literatureand music.

Equationtwo mathematical expressionsjoined by an equals sign.

Model(verb) to show or illustrate a con-cept or problem by using physical objectswith manipulations of these objects; to usesimpler or more familiar objects and situa-tions to explain a new concept, to solve agiven problem, or to demonstrate under-standing of a concept.

The first of these definitions("Algebraic thinking") shows the marks ofthe deplorable current fashion emphasizingthe primacy of "real-life" applications of

mathematics to such a degree that mathe-matics is deliberately confused with theworld it seeks to model. Except for "think-ing skills," a nebulous phrase in itself, thereis absolutely no reference here to anythingremotely mathematical, symbolic, ordeductive, let alone algebraic.

The definition of "equation" is amusing;it is a definition of what an equation lookslike on the printed page, perhaps, butseems not to understand that an equationis a certain sort of statement, a sentencewritten in English. The sad state of publicunderstanding of mathematics can betraced to the early acquisition of such"definitions," which can only lead tomeaningless manipulation of symbols,if that.

The definition of "model" shows a mis-understanding of the role of mathematicsin modeling phenomena; the given defini-tion totally reverses the process, betrayingthe notion in the mind of the writer thatthe mathematics is the difficult thing, andthe real world the simpler, by which themathematics is to be elucidated. This maybe true for very young children, whoseunderstanding of arithmetic must of coursebe rooted in experience, but suggests thatthe authors of these Frameworks wish toconvey an understanding of mathematicsand its relation to the real world that stopsat that point. Certainly, they are ignoringthe current use of "model" in mathematics:as a verb it refers to the use of mathematicsto picture real phenomena, while as a nounit refers to the mathematical structure thatserves in place of the reality. This defini-tion has it backwards.

A study of this Glossary does not causeone to anticipate much mathematical con-tent in the Framework itself, for all that itlists an enormous number of rubrics bywhich to classify what happens in gradesK-8. There are also some goals: "Learningto value mathematics"; "Becoming confi-dent in one's own ability"; "Becoming amathematical problem solver"; "Learningto communicate mathematically"; and"Learning to reason mathematically." Yetwith all these rubrics, there turn out to beonly 12 pages of non-repeated text materialin this 46 page Framework. These conve-nient pages order the items by grade level,with the mathematical categories listedwithin each level. Here is the place, if any,to give substance to the philosophy andcategories of the surrounding texts. We arewilling to tolerate vagueness in a title orcategory description, provided the text

Fc)59rr fl

irSiTrAittilfitTETICOWrirlik Rita

Tennessee (K-8 only)

I. CLARITY

II. CONTENT

III. REASON

IV. NEGATIVE QUALITIES

0.0

15

0.0

1.0

TOTAL SCORE (out of 16) j5

GRADE F

tratrAwiEsiguipiormaregrai

Tennessee (9.12 only)

I. CLARITY

II. CONTENT

III. REASON

IV. NEGATIVE QUALITIES

4.0

4.0

10

3.0

TOTAL SCORE (out of 16) 14.0

GRADE A

ultimately makes plain what the general-ization describes. In the K-8 Framework,however, most generalizations are not latermade more specific.

A typical "learning expectation" fromthe K-8 document concerns "Patterns,Functions and Algebraic Thinking" for theGrade 6-8 level: "Describe, extend, ana-lyze, and create a wide variety of patternsin numbers, shape, and data using a varietyof appropriate materials, including manipu-latives and technology."

The phrases, "a variety of ..." and "awide variety of . . .," as emphatic modifiersof perfectly understandable plurals, are fre-quent enough in many standardsdocuments, but more frequent in this K-8Framework than usual. They add nothingto the statement but words; they are infla-tion.

Coming to the level Grades 3-5, now,under "Number Sense and NumberTheory," we find that students should

Articulate and model the relationshipbetween fractions and decimal notationfor a rational number;Select a notation and justify its appro-priateness for a particular situation;Demonstrate an understanding ofdecimals by extending whole numberplace value concepts;Use geometric models to illustrateproperties of whole numbers andfractions and their operations.

Here the second expectation is opaqueto the outsider, including the professionalmathematician. The fourth expectation isindefinite enough to be the basis for eithera good lesson or a poor one. The first andthird expectations are appropriate but rep-etitious; one cannot do the first withoutthe understanding demanded in the third.

The 9-12 Framework is a document ofcomparable length, though without all therubrics and repetitions of the other; henceit contains more. It outlines one sequence,"Arithmetic 9" (remedial), "AppliedMathematics I" (also remedial), and"Applied Mathematics II" (partly remedi-al) as evidently designed for thenon-college bound students; another, con-sisting of "Algebra I," "Algebra II," and"Unified Geometry," is the standard pre-college curriculum that has traditionallyoccupied grades 9, 10, and 11 in manyparts of the country.

There are descriptions of yet othercourses ("Pre-algebra," for example) whichfit into alternate tracks, all of which areoutlined in such a way that students show-

ing unusual proficiency, or difficulty, have akey to an appropriate next step in a differ-ent track, much as one would find in acollege catalogue. As a whole, the 9-12document spends very little space on"overarching" category listings. Algebra isalgebra, geometry geometry and trigonom-etry trigonometry. There is no talk of"patterns," no "spatial sense and geometricconcepts," and no "mathematics as com-munication."

For example, under Content for"Algebra II" we have a section on the com-plex numbers, with the direction, "Simplifycomplex fractions involving complex num-bers." This is something that can hardly bedone by people who have scanted compu-tation with ordinary real fractions in favorof decimal representation and machines.The instruction is clearly stated and sub-stantial in content, unlike so much in theK-8 listings. Other typical 9-12 demands,from various courses and levels are

Construct a line parallel to a given line;Transform expressions with rationalexponents into simplest radical formand conversely;Solve simple exponential and logarith-mic equations;Classify a system of equations in threevariables as consistent, inconsistent,dependent, [or] independent;Use the Gaussian elimination method.

This last skill can be made to seemunnecessary and old-fashioned to a studentwith a suitable calculator and the mandate

to use it on all occasions. But it is only as"foolish" as learning multiplication tables.The Gaussian elimination is like walking,where a calculator is more like an airplane.But one cannot fly an airplane next door.

Not everything old-fashioned is good,though, and certain truly pointless old top-ics appear here and there in Tennessee's9-12 Framework. "Synthetic division" forexample, is a lavender-scented relic of for-gotten ballrooms and will surely notsurvive the revision now under way.Descartes' Rule of Signs is another fossilwhich is seldom accompanied by an under-standable rationale in school instruction,and at best gives insights of little value.(And despite all the changes of the past 50years, Tennessee is not the only state toretain these two topics.) Also, there aresome important things missing in the pre-sent 9-12 curriculum because they havealways been missing; precedent is notalways a good guide.

Thus, in our Table of Ratings in SectionVII we omit grading the K-8 Framework for"Content" at the "Secondary" level, andwe omit grading the 9-12 Framework for"Content" at the two earlier levels, assign-ing "n" for "no grade" in those places.Then we average what is left, giving fourtotals for each document as if it were astate unto itself. At the end there is thegrade of C, corresponding to an overall,though misleading, "average" for all ofTennessee.

Texas

This document is particularly lucid andeasy to read, though our copy lacks pagenumbers. It takes the curriculum grade-by-grade for K-8, and by course titlesthereafter, a logical choice, and one whichavoids the fragmentation of the later cur-riculum so common in "integrated" highschool mathematics courses. The specifica-tions of what is demanded at each leveland each subject heading are introducedwith useful summaries telling the readerwhere the document is and where it isgoing; and then the specific items areindeed specific, as they should be. Thecontent is better than average, with theimportant omission of almost everythinghaving to do with mathematical reasoning.There are other omissions concerning con-tent, sometimes requiring the reader toinfer itfor example, that the quadraticformula is to be proved and not just usedfrom memory is inferred from the mentionof "completing the square" as a method ofsolving a quadratic equation. In pre-calcu-lus there appear to be no trigonometricidentities, and the binomial theorem getsonly brief mention; in these cases we inferless rather than more.

While the word "reasoning" appearsfrequently, it most often refers to makingconnections between the real world and itsmathematical models, rather than to thelogical connection between mathematicalstatements. Thus, "uses pictures or modelsto demonstrate the Pythagorean theorem,"and "use the Pythagorean Theorem tosolve real-life problems," ("Geometry," 8thgrade) are not followed up by a demand forits proof, or even its placement in an orga-nized mathematical systemnot evenwhere axiomatic systems come under dis-cussion in grades 9-12. There the coursedescription leaves unclear what sorts ofproofs, if any, will be produced or learned.The "multiplicity of approaches" to geome-try here does not outline a coherent courseof study. For example, "In a variety of ways,the student develops, applies and justifiestriangle similarity relationships, such asright triangle ratios, trigonometric ratios,and Pythagorean triples" gives a confusedmessage. Pythagorean triples are ofalgebraic interest and teach little aboutgeometry, and "... develops, applies, andjustifies triangle similarity relationships,such as ... Pythagorean triples" doesn't

61

es,Airekturep:Oitalatts'isT01

Texas

I. CLARITY 3.1

II. CONTENT 3.0

III. REASON 1.0

IV. NEGATIVE QUALITIES 3.5

TOTAL SCORE (out of 16) 10.6

GRADE

quite make sense.The geometry course, little as its

description implies as to content, goes sofar as to suggest that non-Euclidean geome-tries are to be studied; this won't do, wheneven Euclidean geometry gets such shortshrift. Texas is not the only state whosestandards make such a recommendation(see Notes for Arkansas and Hawaiiabove).

Utah

The organization of this documentemphasizes the continuation of strandsthrough several grade levels to such adegree that particular instructions aresometimes mistakenly placed. ThusStandard 5000, is intended for kinder-garten, 5010 for grade 1, 5020 for grade 2,etc. For each grade level there are strands,e.g. Strand 12 having to do with fractionsand decimals. Thus 5030-12 names aninstruction for the 3rd grade level concern-ing fractions and decimals, and 5000-12 thesame but for kindergarten. The followinginstruction might be appropriate for 5040-12, and is given there: "Develop conceptsof fractions, mixed numbers, and decimals."But the same sentence appears also at5000-12, 5010-12, 5020-12, and 5030-12,making it appear that decimal fractions area feature of kindergarten instruction andthen repeated each year for the next fouryears in identical terms. It is careless of thewriters to appear to suggest this.

Each strand, at each grade level, beginswith very general "objectives," which can-not be of much valuee.g., 5040-07begins with "Develop meaning for theoperations by modeling and discussing arich variety of problem situations." (This is4th grade level, for the strand concernedwith what ordinary people call the arith-metic of whole numbers.) However, afterthe objectives come the "Skills and

Strategies" section, offering particulars. Inthe case of 5040-07, one of them is"Demonstrate through the use of manipu-latives that multiplication and division areinverse operations . . . (3X4=12; 12/4=3),"which could not be clearer or more defi-nite. This is general throughout thedocument, and a good program is defined.Even Reason is well treated from time totime, as in 5350-14, "Recognize the appli-cations of field properties in solvingequations and inequalities," a recognitionnot often enough referred to in most states'standards.

On the negative side, the document isentirely too long and repetitious, the sameinstruction occurring in the same wordsagain and again, so that the reader will beunable to decide the proper placement ofskills, or unable to see the desired progres-sion of skill as students progress throughthe grades. This is a species of Inflation,and another is the curious jargon thatappears repeatedly, as if the processes ofarithmetic were more mysterious than theyare. In 5350-14 it is asked that students"solve problems with real numbers usingVenn diagrams" when all that is meant (webelieve) is that students should recognizethe inclusion relations between, say, the setof rationals and the set of reals. The word"strategy" turns into a quite unacceptabletechnical term in 5010-07: "Recognize and

TS"..TWirEeRtEIPIOTRITIMMR D

Utah

L CLARITY 2.7

II. CONTENT 3.0

III. REASON 3.0

IV. NEGATIVE QUALITIES 3.0

TOTAL SCORE (out of 16) 11,7

GRADE

employ the strategy that division by zero isundefined. (You do not divide by zero.)"The parenthetical command carries themessage, after all, without any strategy.The following item in the same "skills andstrategies" section says, "Recognize andemploy the strategy that when zero is a fac-tor, the product is zero." This is no more orless a fundamental fact of multiplicationthan that 3X4=12, and it certainly is not a"strategy." Inflation again. A careful prun-ing of about half the words, or maybetwo-thirds of them, in these core standardswould make a much better programdescription, for the scatter-shot listings doharbor very good curriculum choices.

Vermont

Only three pages are given to mathe-matics, yet the generalities, while oftenindefinite, portray an adequate curriculumup to the high school level. The highschool descriptions, however, permit toomany interpretations. The high school stu-dent is to "understand the basic structuresof number systems," for examplepossiblya large order and possibly not, dependingon what is meant. On the other hand,arithmetic at K-4 and at 5-8 are reasonablyoutlined, and include negative numbers, allrational numbers, and the mention of non-repeating decimals. The entire document istoo brief to give much detail, yet some par-ticular indicators outline a strong primary

school program. Negative numbers, forexample, are introduced in level K-4. Theweakest thread is "Problem-Solving andReasoning," especially at the high schoollevel, where the text is impossibly general,and indeed inflated, e.g., "Work to extendspecific results and generalize from them;and gather evidence for conjectures andformulate proofs for them; understand thedifference between supporting examplesand proof." That last clause is excellent; itis a pity that its points of reference are nothinted at. One does not make proofs; onemakes proofs about things, and those thingsshould be named in a sufficient standardsdocument.

®2

MSTITAITIEWE ITEM itratrallifig

Vermont

I. CLARITY 1.3

II. CONTENT 3.0

III. REASON 1.0

IV. NEGATIVE QUALITIES 3.0

TOTAL SCORE (out of 16) 83

GRADE

VirginiaThis standards document is so well

written, definite, and free of jargon andinflation, that one is disappointed at themodesty of its expectations. Despiterepeated disclaimers that "technologyshall not be regardedas a substitute for astudent's . . . profi-ciency in basiccomputations," thereis excessive emphasison technologythroughout, e.g.,under Trigonometry":"Graphing utilities . .

. provide a powerfultool for solving/verify-ing trigonometricequations andinequalities" (p. 22).If "solving/verifying"is related to "prov-ing," this is not good advice. If onlyverification is wanted, it is probably easierto ask the teacher. On page 10, at grade 4,standard 4.8 prescribes calculators for prod-ucts of two three-digit numbers, at grade 5any divisor of more than two digits will callfor calculator assistance (standard 5.5, p.11), and at grade 6 and presumably there-after, adult status having been reached, anydivisor of more than one significant figurewill demand calculators (standard 6.6, p.13).

At the high school level, in "AlgebraII," linear systems are only to be solved bymatrix inversion using calculators, and theentire mathematical meaning of this partof algebra is thereby bypassed, including

such practical applications as the recogni-tion of linear dependency and the analysisof systems whose matrices are not square(p. 21). These are not particularly sophisti-cated matters, but they are fundamental. If

linear algebra is to betaught at all, in even sosimple a matter as dis-covering theintersection of three

This standards document

is so well written, definite,

and free of jargon and

inflation, that one is disap-

pointed at the modesty of

its expectations.

C

planes in space, thelogic of the eliminationof variables is of greaterimportance than theproduction of a list ofnumbers, which can befound by looking at theanswer book, after all.Thus, Virginia is ofdivided mind in itsadvocacy of Reason, foron page 20 it also says,

"The student will construct and judge thevalidity of a logical argument consisting ofa set of premises and a conclusion . . ." andgoes on in some detail. When, as in thecase of the set of linear equations just men-tioned, a golden opportunity arises forusing a deductive argument, the state miss-es the opportunity.

An "Advanced Placement" calculuscourse is described, but the rest of the highschool curriculum as outlined is not a suffi-cient preparation for this level of work. Inparticular, the lack of needed exercise inmathematical reasoning in most of the cur-riculum, as the example of linear algebraindicates, is also visible in the treatment ofgeometry.

33

Virginia

I. CLARITY 4.0

II. CONTENT 2.3

III. REASON 2.0

IV. NEGATIVE QUALITIES 3.5

TOTAL SCORE (out of 16) 11.8

GRADE

One unusually good feature of thesestandards is the repeated instruction con-cerning speech and vocabulary, that thestudent should use technical words fluentlyand correctly, and explain his work. Agood exercise, and a model for other goodexercises, is found on page 13, item 6.3,"The student will explain orally and inwriting the concepts of prime and compos-ite numbers." Again, under "Patterns,Functions, and Algebra," 7th grade level,"The student will use the following alge-braic terms appropriately in written and/ororal expression: equation, inequality, vari-able, expression, term, coefficient, domain,and range" (p. 16). However, in speakingmathematically here the authors shouldhave followed their own advice, and usedthe mathematical "or," which means thesame thing, we believe, as their "and/or."

Washington

Standard 1 ("The student understandsand applies the concepts and procedures ofmathematics") has five components, ofwhich 1.2 ("Understand and apply con-cepts and procedures from measurement")occupies page 57. Now component 1.2 hasthree sub-components, one of which,named "Approximation and Precision" hastwo entries at each benchmark (i.e., gradelevel). The second entry in each bench-mark is

At grade 4: estimate to predict and deter-mine when measurements are reasonable,for example, estimating the length of theplayground by pacing it off.

At grade 7: use estimation to obtain rea-sonable approximations, for example,estimating the length and width of theplayground to estimate its area.

At grade 10: use estimation to obtain rea-sonable approximations, for example,estimating how much paint is needed topaint the walls of a classroom.

This sort of repetition is commonthroughout this nearly vacuous document,and is probably intended, by its slight vari-ations, to suggest a progression of skills asthe student grows older, but in few casesdoes it say anything instructive. In the casejust quoted the three exercises are at leastwell-defined; more often they are so gener-al as to defy interpretation. An example, atStandard 3 ("The student uses mathemati-cal reasoning"), where component 3.3,headed "Draw Conclusions and VerifyResults," prescribes at the 4th grade level,"reflect on and evaluate procedures andresults in familiar situations"; at the 7thgrade level, "reflect and evaluate on [sic]procedures and results in new problemsituations"; and at the 10th grade level,"reflect on and evaluate procedures andresults and make necessary revisions"(p. 63).

This document contains standards forother subjects besides mathematics, whichtakes up only 12 pages. It is marked"Washington State Commission onStudent Learning, APPROVEDFebruary

40SiTiAiTiiirRkE10,101RiTinfidiRTEa

West Virginia

I. CLARITY IS

II. CONTENT 3.7

III. REASON 20

IV. NEGATIVE QUALITIES 15

TOTAL SCORE (out of 16) 125

GRADE

26, 1997," but the interior pages aremarked "Work in progress." Since the pre-sent edition does contain some content, asfor example in "Geometry" (p. 59), it ispossible that the impulse behind the pre-sent adherence to vague generalities is notfinal state policy, and that future editionswill contain even more that a teacher ortextbook committee can actually use.

West Virginia

This document is intended, amongother things, as a guide to statewide assess-ment, and places in boldface, for thatpurpose, certain crucial items. The lan-guage is everywhere direct and usuallyquite definite, e.g., for 1st graders, "Giventwo whole numbers whose sum is 99 orless, estimate the difference, and find thedifference using various methods of calcu-lation (mental computation, concretematerials, and paper/pencil)" (p. 46). Suchclarity is maintained throughout, though itsometimes makes the document sound likethe table of contents, or sometimes exercis-es, of a series of textbooks. At the highschool level, in "Algebra II," standard A2.9asks the student to "perform basic matrixoperations and solve a system of linearequations using the inverse matrix method.Graphing calculators will be used to per-form the calculations." This is definite andclear, but not advisable, except as part of amore comprehensive treatment of systemsof linear equations, as has been comment-ed above (see the note for Virginia).

Furthermore, clarity and definiteness arenot always present; one of the boldfaceinstructions at the high school level is"solve problems using non-routine strate-gies" (p. 179). Even though this instructionoccurs within the course description"Algebra II," it is not possible to tell whatis meant; yet the state intends to examinestudents on this matter.

There is a thread called "Computer/Technology" that appears at every gradelevel, K-8, and is apparently designed tomake sure children learn to use calculatorsor computers. It is unfortunate that thesetopics appear within the "mathematics"portion of the Instructional Goals andObjectives, where they might give theunwary teacher the impression, sometimesmistaken, that these sometimes sociologi-cal exercises are designed to augmentmathematics lessons. "Identify work pro-duced by using technology as intellectualproperty and thus protected [by] copyrightlaws" (grade 7, p. 121) is plainly not math-ematics. This entire thread should be

6 4

s rem., E R #E1Is OrlitardAlitiiiii

Washington

I. CLARITY ao

II. CONTENT 1.0

III. REASON 0.0

IV. NEGATIVE QUALITIES 1.5

TOTAL SCORE (out of 16) 2.5

GRADE

thought through again as it is related to themathematics curriculum. A distinct course,perhaps called "Computer Science," teach-ing children how to use computers andhow to behave in their presence, would beanother matter, of course.

Wisconsin

This 1997 "Standards" is a draft; thefinal version will serve as a syllabus forstatewide tests; page xi states, "If contentdoes not appear in the academic standards,it will not be part of a WSAS test." Somethings that do not appear are the quadraticformula, the binomial coefficients, a geo-metric series, a Euclidean proof, a conicsection, an asymptote. The curriculum, inother words, is weak; even the Pythagoreantheorem comes in for only casual men-tions, and the trigonometric functionsappear only in the mention of "trigonomet-ric ratios" in right triangles, as an incidentin geometry to be mastered by the 12thgrade level.

On page 27, "Geometry and its study ofshapes and relationships is an effort tounderstand the nature and beauty of theworld. While the need to understand ourenvironment is still with us, the rapidadvance of technology has created anotherneed: to understand ideas communicatedvisually through electronic media. Forthese reasons, educated people in the 21stCentury need a well-developed spatialorder to visualize and model real-world

problem situations."Such a definition reduces geometry to a

sort of empirical science in the service ofgraphic art, and shows nothing of thenature of the deductive structure whichlaunched modem science. On page 29, theideas of geometry are confused with theartifacts of "analytic geometry," as if slopeand intercepts were geometric objects.

While the definition (called"Rationale") for geometry misses the pointof mathematics as a model for structuralrelations, there is not in fact muchInflation or False Doctrine in this docu-ment. It is praiseworthy that, on page 22, itis warned that "the tools of technology arenot a substitute for proficiency in basiccomputational skills." Yet the document isevasive on this point, as where it is advisedthat "selecting and applying algorithms foraddition, subtraction, multiplication, anddivision" is given equal billing with "usinga calculator" (p. 25). If one wants ananswer to the question of whether childrenare taught "long multiplication," theanswer will not be found here.

The document is similarly unclear on

65

ncnin nnrx-InsT et:\ op 1Wisconsin

I. CLARITY

II. CONTENT

III. REASON

IV. NEGATIVE QUALITIES

0.0

3.0

TOTAL SCORE (out of 16) 74

GRADE C

many other points, to the degree that onecannot really determine the suggested con-tent at most levels. In particular, whilelogic is mentioned, there is no evidencethat logical argument is ever associatedwith substantive mathematical informa-tion; all subject matter mentioned ispresented without regard to the connec-tions of one mathematical idea withanother.

Japan

Japan has a delightful consistencythroughout its document. In the primarygrades, each grade starts with a list ofobjectives for that grade, followed by"Content" ("Numerical Calculations,""Quantities and Measurements," and"Geometrical Figures"), followed by a fewparagraphs entitled "Remarks concerningContent." In grade 3, the students "learnhow numbers are set on the abacus("soroban"), and to use it in simple addi-tion and subtraction"(p. 9). In grades 5 and6, "Quantitative Relations" is added to thelistings under "Content," followed by theusual "Remarks concerning Content."Here there is no misguided notion of hav-ing 9 or 10 "strands" running through all ofthe grades (some American standards doc-uments consider that computing areas inthe 1st grade should form part of a strandcalled "calculus.") Inthe middle school(grades 7.9) eachgrade has exactly thesame format:"Objectives,""Content,""Remarks." All gradelevels are refreshingto read; they are mod-els of "clear, definite,testable," and nograde runs more thana few pages. In grades10-12, the format isrepeated, but by subject matter rather thanby grade level. There is also an "honors"track for those who are academicallyinclined, as well as a third track, very rigor-ous, for those whose interests lie in scienceand engineering. At the end, there are verydefinite guides about the sequence of allthe courses, taken separately or in parallel.

All this is achieved in a mere 47 pages,less than a tenth of the longest Americanframework, though "framework" is proba-bly a good description of the Japanesedocument as well, for it does contain peda-gogical hints of importance. In particular,consider this quotation from the "SecondGrade Content":

To enable children to develop theirabilities to use addition and subtractionthrough getting deeper understandingof them.

which is followed by, inter alia

To understand that addition and sub-traction of 2- and 3- digits numbers areaccomplished by using the basic facts ofthese operations for 1-digit numbersand to know and use them in columnform.

It is typical of the Japanese standards toincorporate, as this example does, theessence of the reasoning into the impliedlesson plan associated with the content tobe conveyed. That is, "adding in columns"is not merely a practical necessity made

obsolete by the elec-tronic calculator, asmany educators nowbelieve, but is (whenproperly taught, of

Calculators are not used

before the 5th grade, to

give children the chance

to understand the decimal

system first.

course) an elucidationof the nature of thedecimal system, and anillustration of howmathematical reasoningproceeds from minimalinformation to whatappears to be enor-mously more. In thiscase, the 9X9 addition

table memorized by the end of the 2ndgrade enables a child, provided with a bitof understanding of the nature of the sys-tem, to construct a table potentiallyinfinite in scope. The writer of this itemshows this understanding clearly, and manyother examples can be quoted to show thesame quality.

Items of content are clearly paced fromgrade to grade, with lists of vocabulary tobe acquired by the end of the year given asfurther indication of what the year is sup-posed to accomplish. By the 3rd gradechildren are to understand decimal frac-tions, graphs, angles, "radius," the sphere.By the end of grade 4, they are to know

6 6

*SilerAtiTlEiRieftirOTRffarrtR

Japan

I. CLARITY

IL CONTENT

III. REASON

IV. NEGATIVE QUALITIES

4.0

4.0

3.0

TOTAL SCORE (out of 16) 15.0

GRADE A

Hiles for rounding off approximations,adding fractions with common denomina-tor, multiplying and dividing two-digitnumbers, and by the end of the 5th, con-gruence, the making of a statistical graph,least common divisors and multiples, and"to know that the result of division ofwhole numbers can always be representedas a single number using fractions." Thislast quotation also carries a lesson in con-cept of "number" which most Americancurricula either avoid or take for granted.

Calculators are not used before the 5thgrade, to give children the chance tounderstand the decimal system first. One ofthe stated "objectives" at the grade 9 levelincludes, "[to] acquire the way of mathe-matically representing and coping with,and to enhance their abilities of mathe-matically considering things, as well as tohelp them appreciate the mathematicalway of viewing and thinking." This much,taken alone, is vague (though poetic), butis followed by specifics enough: factoring,the quadratic equation, functional rela-tions, the Pythagorean theorem, and so on,all presented in such a way as to follow"the mathematical way of viewing andthinking," much more than in such a wayas to convince students of the "real-life"applications. Real life comes in its ownplace, a few years later.

APPENDIX: LIST OF DOCUMENTS REVIEWED

AlabamaAlabama Course of Study, MATHEMATICS, MathematicalPower K- 12

(Alabama State Department of Education Bulletin 1997, No.4)50 North Ripley St., P.O. Box 302102Montgomery, AL 36130-2101

Alaska(1) Math/Science Framework (1996)(2) Mathematics Performance Standards (1997)Department of Education801 W. Tenth StreetJuneau, AK 99801-1894

Arizona.Mathematics Performance Objectives (March 11, 1997; adoptedAugust 26, 1996)Arizona Department of Education1535 West JeffersonPhoenix, AZ 85007

ArkansasMathematics FrameworkGeneral Education DivisionArkansas Department of EducationFour State Capitol Mall, Room 304ALittle Rock, AR 77201-1071(Fromhttp://arkedu.k12.ar.us/wwwade/sections/curframe/frame.htm;July 22, 1997)

CaliforniaThe California MATHEMATICS Academic Content Standards(Prepublication Edition, February 2, 1998) for Grades K-12California State Board of Educationhttp://www.cde.ca.gov/board/board.html

ColoradoModel Content Standards (June 8, 1995)Colorado Department of EducationDenver, CO, 80203

Connecticut"Mathematics Curriculum Framework," Second Draft,August 7, 1997Connecticut Department of Education165 Capitol AvenueRoom 305, State Office BuildingHartford, CT 06106-1630

DelawareMathematics Curriculum FrameworkVol. 1 Content Standards (1995, revised March 1, 1996)Delaware Dept. of EducationDover, DE 19903-1402(also http://www.dpi.state.de.us/dpi/standards/math)

District of ColumbiaMathematics-Science-Technology Curriculum Framework, Grades K-12, Revised Edition (undated, but post-1989)District of Columbia Public Schools415 12th Street, N.W., Suite 1209Washington, D.C. 20004-1994

Florida(1) Sunshine State Standards and Instructional Practices, Mathematics

(May 29, 1996)(2) Florida Course Descriptions, Grades 6-12, pp151-263 (1997)Florida Department of EducationCapitol Building, Room PL 08Tallahassee, FL 32301

Georgia-"Mathematics Quality Core Curriculum," Draft Revision, Edition2 ("provided by the Georgia School Improvement Panel,February, 1997")CD ROM available from the Georgia Department of EducationTelephone (404) 657-7411

Hawaii(1) State Commission on Performance Standards (Final Report,

June 1994)(2) Essential Content (December, 1992)Department of EducationOASIS/Systems Group641 18th Avenue, Room V201Honolulu, HI 96816-4444orHawaii Department of Education1390 Miller Street, #307Honolulu, HI 96813

IdahoK-12 Mathematics Content Guide and Framework (1994)(in three parts: K-4, 5-6, and 7-12)http://www.state.id.us

IllinoisIllinois Learning Standards (July 25, 1997)Illinois State Board of Education100 North First StreetSpringfield, IL 62777

IndianaMathematics Proficiency Guide (Spring 1997)Indiana Department of EducationState House, Room 229Indianapolis, IN 46204-2798(also http://doe.state.in.us)

Iowa(Iowa apparently does not intend to publish a Standards orFramework of the sort that is under review in this report.)

JapanMathematics Program in Japan (Kindergarten to Upper SecondarySchool)Japan Society of Mathematical Education (JSME), January, 1990(Excerpt from the National Courses of Study, Revised by theMinistry of Education)Published by: Japan Society of Mathematical EducationPrivate Postbox No.18, Koishikawa Post OfficeTokyo, Japan

KansasMathematics Curriculum Standards (Revised July 1993, ReprintedOctober 1996)Kansas State Department of Education120 SE 10th AvenueTopeka, KS 66612(also http://www.ksbe.state.ks.us)

KentuckyCore Content for Mathematics Assessmenthttp://www.kde.state.ky.us/caa/CCA/CCM1.htmlKentucky Department of Education500 Mero StreetFrankfort, KY 40601

LouisianaContent Standards Foundation Skills (May 22, 1997)http://www.doe.state.la.us/os2httpd/public/contents.mframe.htm

MaineCurriculum Framework for Mathematics and Science(undated, but post-1995)Maine Department of Education23 State House StationAugusta, ME 04333-0023

Maryland(1) MathematicsA Maryland Curriculum Framework (1985)(2) High School Core Learning Goals, Mathematics (September

1996)Maryland State Department of Education200 West Baltimore StreetBaltimore, MD 21701-7595

MassachusettsMathematics Curriculum Framework (December 1995)Commonwealth of MassachusettsDepartment of Education350 Main StreetMalden, MA 02148-5023

MichiganModel Content Standards for Curriculum, including Academic CoreCurriculum Content Standards (July 25, 1996)http://cdp.mde.state.mi.us/ContentStandards/Mathematics/

Minnesota"K-12 Mathematics Framework" (Draft chapters, 1997)"Please do not quote, copy, or cite." Our copy was mailed to usfrom:

"Sci-Math, MN"638 Capitol Square550 Cedar StreetSt. Paul, MN 55101

MississippiMathematics Curriculum Structure (1995)Mississippi Department of Education550 High Street, Room 501Jackson, MS 39201

MissouriMissouri's Framework for Curriculum Development in Mathematics,K-12 (1996)Missouri Department of Elementary and Secondary Education205 Jefferson StreetJefferson City, MO 65102

MontanaFramework for Improving Mathematics and Science Education (1996)Montana Office of Public InstructionPO Box 202501Helena, MT 59620-2501

NebraskaMathematics and Science Frameworks for Nebraska Schools(March 6, 1994)Nebraska Department of Education301 Centennial Mall SouthP.O. Box 94987Lincoln, NE 68509

Nevada"We are in the process of developing new math standards duringthe 1997-1998 school year."

New HampshireK-12 Mathematics Curriculum Framework (February 1995)New Hampshire Department of Education101 Pleasant StreetConcord, NH 03301

New Jersey(1) Core Curriculum Content Standards for Mathematics

(1995, revised 1996)http://www.state.nj.us/njded/education.htm

(2) New Jersey Mathematics Curriculum Framework (1996)New Jersey Mathematics Coalitionhttp://dimacs.rutgers.edu/nj_math_coalition/framework

New MexicoContent Standards with Benchmarks for Kindergarten Through 12thGrade (Fall 1996)New Mexico Department of Education300 Don GasparSanta Fe, NM 87501-2786

68

New YorkLearning Standards for Mathematics, Science and Technology(March 1996)New York Education DepartmentEducation Building111 Washington AvenueAlbany NY 12234

North CarolinaStandard Course of Study and Grade Level Competencies,Mathematics K-12 (1992, 1993)North Carolina Department of Public Instruction301 N. Wilmington St.Raleigh, NC 27601-2825

North DakotaMathematics Curriculum Framework Standards and Benchmarks(Revised 1996-1996; Draft in progress March 24, 1997)North Dakota Department of Public InstructionState Capitol Building, 11th FloorBismarck, ND 58505-0440

OhioModel Competency-Based Mathematics Program (November 1990)Ohio Department of Education65 South Front Street, Room 810Columbus, OH 43215-4183

OklahomaPriority Academic Student SkillsMathematics (March 1997)Oklahoma State Department of Education2500 N. Lincoln BoulevardOklahoma City, OK 73105-4599

Oregon(1) Standards (January 1997)

(Mathematics pages are 9-12 and 29,30)(2) Oregon Statewide Mathematics Assessment, Test Specifications

Grade 3, Grade 5, Grade 8, Grade 10 (1997)(3) Sample Tests for (2)(4) Mathematics Teacher Support Package (October, 1996)Oregon Department of Education255 Capitol St. NESalem, OR 97310-0203

Pennsylvania"Proposed Academic Standards for Mathematics 'for theGovernor's Advisory Commission on Academic Standards"'Undated but clearly 1997, taken fromhttp://www.cas.psu.EDU/PDE.HTML on 8/14/97Pennsylvania Department of Education333 Market StreetHarrisburg, PA 17126-0333

Rhode IslandMathematics Framework K-12 (October 1995)Department of Education225 Westminster StreetProvidence, RI 02903

South Carolina(1) Mathematics Framework (November 1993)(2) Mathematics and Academic Achievement Standards

(November 1995)Curriculum Framework Office1429 Senate StreetColumbia, SC 29201

South DakotaMathematics Content Standards (Approved June 17, 1996)Division of Education Services and Resources700 Governors DrivePierre, SD 57501-2291(Note: A letter of August 18, 1997 states that "South Dakota is inthe process of rewriting the Content Standards.")

Tennessee(1) Mathematics Framework/ Grades Kindergarten Through Grade

Eight (October 11, 1996)(2) Mathematics Curriculum Framework, Grades 9-12

(November 15, 1991)Tennessee Department of EducationAndrew Johnson TowerNashville, TN 37243-0375

TexasTexas Essential Knowledge and Skills for Mathematics ("Chapter111," to be implemented by September 1, 1998, with Chapter C,9-12, "effective September 1, 1996")Texas Education Agency1701 North Congress AvenueAustin, TX 78701-1494http://www.tea.state.tx.us/telcs

UtahCore Curriculum/ Mathematics Units (September 19, 1996)http://www.uen.org/cgi-bin/websql/lessons/query_lp.hts?corearea=2&area=1

VermontVermont's Framework of Standards and Learning OpportunitiesScience, Mathematics and Technology Standards (1996)Vermont State Board of Education120 State StreetMontpelier, VT 05620-2501http://www.state.vt.us/educ/stand/smtstand.htm

VirginiaStandards of Learning for Virginia Public Schools (June 1995)Commonwealth of Virginia Department of Education101 N. 14th StreetRichmond, VA 23219

WashingtonEssential Academic Learning Requirements/Mathematics (February26, 1997)Commission on Student LearningRoom 222, Old Capitol BuildingOlympia, WA 98504-7220

69

West VirginiaInstructional Goals and Objectives for West Virginia Schools(September, 1996)West Virginia Board of Education1900 Kanawaha Blvd. E.Charleston, WV 25305-0330http://access.k12.wv.us/dshafer/pmat9-12.htm

Wisconsin"Model Academic Standards for Mathematics" (Draft, 1997)Wisconsin Department of Public InstructionP.O. Box 7841Madison, WI 53707

WyomingStandards are in progress and were not available for review atpublication time.

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