munity itself laughing at the follies committedrnin the name of promoting a betterrnunderstanding of mathematics.rnTo take an example, the language ofrnthe “theory of sets” had been basicrnamong mathematicians for 100 years,rnand can ease enormously the path tornmuch that people find perplexing inrnschool. Anyone should be able to learnrnenough about sets and this vocabulary inrna verv few hours to permit him to understandrnan honestly presented course ofrnhigh school mathematics, including allrnthe traditional material and more; hisrnsavings in time will have exceeded thosernfew hours a hundredfold, and in understandingrnimmeasurably. SMSG introducedrnset theory into its first books,rnwhich as it happened were for the highrnschool level. Later books, written forrngrade school years, also introduced thernsubject of sets, hoping later to make usernof it when revised high school booksrnwere written. During SMSG’s short life,rntherefore, a chapter on sets appeared atrnthe opening of every vear’s textbook, unfortunatelyrnmaking it appear as if setsrnwere the be-all and end-all of New Math.rnThis redundancy was copied into therncommercial texts of the time as well, andrnteachers leaped on it to the neglect ofrnmore prosaic matters, like getting a correctrnanswer in arithmetic.rnEasy as it looked, teachers didn’trnalways get the notion of “set” straightrnthcmseUes, and could teach the mostrnegregious confusions as truth. One textbookrnlesson plan suggested that thernteacher, as an example, distinguish thernsubset “boys” from the subset “girls” (inrnthe set “this class”) by asking the boys tornstand, and then the giris to stand, and sornon; one teacher I heard about then askedrn”the set of boys” to stand up. But whilernboys, being human, can stand, sets cannot.rnSo fine a distinction may be meaninglessrnto a third-grade teacher, or to anyonernwho has never made real use of it;rnbut if exactly that distinction is notrnmade plain, and into a habit of mind andrnspeech, the notion of set is valueless inrnlater mathematical reasoning.rnOn the other hand, SMSG and its imitatorsrnwere also guilty of some pointlessrnpedantrv, ridiculous even if logically correct:rn”Write the numeral that names thernnumber solving 3x – 7 = 8,” for example.rnThat’s not even English. If you actuallyrnask a mathematician to write down hisrnphone number, he will cheerfully handrnyou a numeral without a moment’s hesitationrnor apology. He can make the distinction,rnsure, but he only does it when itrncounts.rnJust the other day I heard an agingrnacademic say that Marxism hasn’t failed,rnbecause it hasn’t been tried—not anrnoriginal trope, for we have heard thernsame of Christianity for ages. HadrnSMSG really been tried? The mass ofrnAmerican teachers—and children—rnwere not, in the end, exposed to, letrnalone taught, what the SMSG mathematiciansrnprescribed. But to plead thusrnis only to evade responsibility.rnOliver Wendell Holmes once wroternthat the American Gonstitution is an experiment,rn”as all life is an experiment.”rnExperimental scientists like Holmes understandrnthat reality is not to be pushedrnaround, neither by nine old men nor bv arnprestigious bunch of mathematical geniusesrnwith a pipeline to the U.S. Treasury.rnTheir prestige was unchallenged,rntheir genius without peer, and theirrnpipeline of pure gold; but the realitiesrnoverwhelmed them. The cadre of teachersrnalready out there had preexistingrninterests and capabilities, the publicrnpatience was shorter than experimentsrnthat could lose a generation of children,rnand the educational experts, the FEB,rnwas gathering its strength for the politicalrnbattle that finally turned the pipelinernback in their direction.rnToward the end, Begle wrote, “I see littlernhope for any further substantial improvementsrnin mathematics educationrnuntil we turn mathematics education intornan experimental science, until wernabandon our reliance on philosophicalrndiscussion based on dubious assumptions,rnand instead follow a carefully constructedrnpattern of observation and speculation,rnthe pattern so successfullyrnemployed by the physical and natural scientists.”rnBegle himself died a disappointedrnman six years later, though hernhad continued after SMSG to work brilliantlyrntoward a proper study of mathrnematics education. His disappointmentrnwas for the future more than for SMSG,rnbecause he foresaw correctly that PEBsponsoredrnresearch in education wouldrnnot follow his sensible, if unexciting, prescription.rnMeanwhile, the PEB, having takenrnback the schools, resumed educating itsrnfuture leaders with exactly the “philosophicrndiscussion based on dubious assumptions”rnBegle had warned of. It wasrnthe education of teachers that Begle hadrncome to see as the truly intractable problem.rnSMSG, for all its faults, could solvernthe problem of choosing, pacing, andrnstating an excellent curriculum; anotherrnten years’ experimentation would surelyrnhave removed what Duren called its “excessivernenthusiasm for logical language,”rnfor example. But the SMSG institutesrnhad been hopelessly inadequate to therntraining of teachers, and the PEB is perforcernin charge of the next generations.rnThere is no market in sight for even arnperfect SMSG curriculum.rnThe textbooks today are again notrnwritten by mathematicians, and indeedrnshow no sign of SMSG influence. Theyrnhave eliminated the “set theory” thatrnthey had hailed in 1975, and they arernquite silent about numerals; so muchrnis to the good. On the other hand, theyrncontain even less mathematics than theyrndid in 1955, except that at the collegepreparatoryrnupper levels some of them,rnintended for superior students andrnteachers, are a great deal better. Thernbooks for grades one to eight come packagedrnfor teachers with mountainousrn”Teachers’ Guides,” in which the mathematicsrnis swamped into insignificancernby the instructions on engaging thernattention and improving the self-esteemrnof the students.rnThe general mathematical literacy,rnnot notably improved by SVISG, hasrncontinued its decline under PEB managementrnas well. Developmental psychology,rnnot mathematics, informs thernseminar rooms of the schools and thernteachers colleges, while at the higher levelsrnthe research journals of the PEB arernfilled with what almost every mathematicianrntoday would condemn as beingrnat least a waste of time.rnPerhaps an example is in order. Thernanthology Perspectives on Research in EffectivernMathematics Teaching, publishedrnby the National Gouncil of Teachers ofrnMathematics, contains these insights:rnAs opposed to the contextboundrnascription of meaning in e’erydayrnlanguage use, scientific theories arernpresumed to rest upon the strictrnuse of their technical terms. Researchersrnoften pick the labels (thernwords, the ‘signifiers’) for their keyrncategories following a contiguityrnrelation between the concept (thern’signified’) they have in mind andrnone specific of the many facets ofrnmeaning ascribed to the word inrneveryday u s e . . . . The constructionrnof a metatheory capable of executingrnthe critical comparison of com-rnJANUARY 1996/41rnrnrn