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Algebra in S Hool Algebra d Igebraic Thinkin in S hool Math m ti s Seventi h Yearbook Carole E. Greelles Seventieth Yearbook Editor Arizona State University Mesa, Arizona Rheta Rubenstein General Yearbook E'ditor University o.f ]I/fichigan-Dearborn Dearborn, Michigan NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS 1 istory of Algebra in th Scho I urriculum Jeremy Kilpatrick Andrew Izsak flthere is a heaven/or school subjects, algebra H)ill never go there. It is the one subject in the curricuhul1 that has kept children .Ironz finishing high school, /j~onl developing their special interests' and /;~07n enjoying rnuch of their h0711e study work. It has caused 1110re jCllnily ro"Vvs, ,nore tears, 7110re heartaches, and nzore sleeples5' nights than any other school sul~ject. -Anonynlous editorial writer [ca. 1936J N THE United States and Canada before 1700, algebra was absent not only fronl I the school curriculu1l1 but also fro111 the CUITicululll of the early colleges and sCIllinaries. That situation changed during the eighteenth and nineteenth centuries as colleges and universities across North Anlerica began to offer courses in alge­ bra. In January 1751, when Benjanlin Franklin's acade111Y was established, the new master Theophilus Grew offered "Writing, Arithnletic, Merchants Accounts, Alge­ bra, AstTon0111Y, Navigation, and all other branches of Mathenlatics" (Overn 193 p. 373), and "algebra to quadratics" continued to be pali of the freshnlan curricululll after the acaden1Y becanle the University of Pennsylvania in 1779. Algebrcl is first nlentioned as being in the Harvard curriculunl in 1786 but was probably taught 111LlCh earlier, perhaps as early as 1 (Cajori 1890, p. By 1742, Yale freshnlen were studyin_g algebra along with arithnletic. As of 1814, in what appears to have been the earliest collegiate 111athen1atics course in Canada, students at King's College in Windsor, Nova Scotia, were being taught "Euclid and Wood's algebra" (Archibald and Charbonneau 1995, p. 16). College algebra vIas in those days what had been called since the tinle of Isaac Newton ,specious arithmetic or 1.f71h)prr;:ol rJ) A it11111ofir' llIP':1n;na th~t it pynrpc:c:::prl C:::\Jrnhnlicilllv r1l1pc; fnr nner . 0- ALGEBRA AND ALGEBRAIC THINKING IN SCHOOL MATHEMATICS with any species of quantity. Students learned to manipulate expressions and solve sin1ple equations with numerical coefficients. Most rules were given 'vvithout proof, factoring was 01111tted, and negative quantities 'vvere avoided as far as possible, being of son1ewhat questionable status. By 1820, Harvard had decided to require algebra for adn1ission, and COIU111- bia, Yale, and Princeton followed suit in 1 1, 1846, and 1 respectively (Jones 1967, p. 50; Overn 1937, p. 374). Candidates for the 1846 freshn1an class at Yale, for example, were told that they would be examined jn ele1l1entary algebra "pre­ ceding quadratic equations" (Diane E. Kaplan, personal cOn1n1LIl1ication, July 21, 2006). In Canada, McGill University had opened its doors officially in 1821, and as of 1857 and probably before, the n1atriculation examination included "Arithrlletic; Algebra, to Quadratic Equations; Euclid's Elen1ents, 3 books" (Carolyn Kieran, per­ sonal con1rnunication, June 26, 2(06). When the Poly technique Montreal opened in 1873 (as the Ecole Poly technique), some algebra was most certainly offered, but it is not clear what, if any, was required for entrance. The catalog for 1878---1879 says explicitly that algebra \vas taught, together with functions ane! an introduction to differential calculus, in the first two years of the three-year curriculu111 (Louis Charbonneau, personal C0111111Un icatioll, July 12, 2006). Algebra Enters the School Curriculum In 1827, by passing An1erica's first high school law, Massachusetts 111ade the teaching of algebra, geometry, and surveying n1andatory in the high school of every town with 500 f3n1ilies or 1110re. As that hst of subjects suggests, algebra, as well as other branches of nlathenlatics, "was originally introduced into secondary edu­ cation in An1crica for practical rather than disciplinary reaS011S and because of its appJ ications to surveying and navigation rather than for the purpose of 111ecting a college entrance requiren1cnt" (Overn 1937, p. 374). As the nineteenth century progressed, however, a course in algebra Vias in­ creasingly required for college entrance, and algebra "was n10vcd fr011'1 the colI to the secondary schools with little or no n10dification" (Osborne and Cross\vhite 1970, p. 158). Thus algebra was introduced into the school curriculun1 for disparate purposes: as vocational preparation or as academic preparation. In subsequent de­ cades, debate and even discord \vould arise over the appropriate en1phasis of school algebra and to whon1 the subject should be CCHl1peting Conceptions of hoal A1!Sebra Generalized Arithmetic For n10st of the nineteenth century, while mathen1aticians such as Vv'illian1 Rowan Han1iIton, George Boo Ie, Arthur Cayley, and James Joseph Sylvester were =S A HISTORY OF ALGEBRA IN THE SCHOOL CURRICULUM 5 developing the foundations of n10dern algebra, school algebra ren1ained an exten­ f, sion and generalization of school arithn1etic built largely by induction on a base of nU111erical quantities (lnd operations on then1. A survey of U.S. algebra textbooks publishedfronl 1 0 to 1928 revealed that throughout that period, n101'e than hal f 1- the exercises \vere given over to algebraic techniques (factoring, roots, powers, and funda111ental operations), with the next an1ou11t of attention given to equa­ e, tions and fornlulas (Chateauneuf 1929~ sec also Osborne and Crosswhite 1970, p. e- 159). Influenced by nineteenth-century faculty psychol (the n1ind is conlposed 1, of separate faculties or powers) and the correspondi ng educational l1lode I of nlent:d as discipline (drill and repetition are the best \vays to strengthen young minds and cul­ c; tivate nlen10ry), textbook authors and teachers stepped up the con1plexity and dif­ ?r­ ficulty of algebra exercises, particularly during the years fj-onl 1880 to 1910 (Overn cd 1937, p. 376). According to David Eugene S1111th (1926), factoring in particular lut "began to occupy an undue an10unt of space in the closing quarter of the nine­ 79 teenth century~' (p. 10). An1Y Olive Chatcauneufnoted that in the 1890s attention to on techniques for nlanipulating algebraic expressions reached a crescendo, occupying J 1S son1e 64 percent of all textbook exercises (1929, p. 151). Writing in the First Yearbook of the National Council of Teachers of Math­ ernatics (NCTM), Sn1ith (1926, p. 3) described the elen1entary algebra of 1900 as consisting of a large amount of abstract J11z111ipulation ofpolynoIllials, including long problems he in the III ult i plication and clivisi on 0 f integral and fractional expressions, \;vi th ex- ~ry tended work in the ilnding roots, in factoring, in lowest cOllllllonmultiple, and ell in highest common factor, ane! with equ~11ly manipulations of cOlllplex lu­ fr~lctions and radicals. Simultaneous linell- equations extended to fOLlr ;llld more its UIlkIlowns, and si m ultaneous quadratics of the trick variety were in evidence. lIe went on to characterize the teaching of algebrZl in 1900: The subject \\/as usually taught as ifit wereZl purely mathematicll discipline, UI1- 111- related to life except as life might enjoy the mean' Valuable ZlS the teacher might feel it to be, the majority of pupils looked UPOIl it as a fairly inter- way 0 f nowhere. (p. 20) ate By the end of the nineteenth century, the practical value of studying if it had ever been pron1inent, appears to have faded. n(tional Thinking Meanwhile, decade from 1880 to 1890 had seen ll1Llny schools ill Europe begin to lllake the function concept the core of secondary school n1athen1atics (Nordgaard 1928, p. 70), using it to streamline the curriculum, unify the branches ofn1athenlatics, correlate n1athematics "with science, introduce students to 111ath enlatical theory, and provide 11lore applications." As a result, calculus becan1e the an1 obvious extension and cullllination of tIle study of functions, their graphs, and the ALGEBRA AND THINKING IN IViATHEMAflCS propl,.:rtics of CLlnl'S. ll1 J l)():2, 10 IKe bCC~lrlll;? "the hl'sf cnulltry III wurld include \vork ill the cileLllus ~lS (1 rcgll!~lr ~llld required pdrt of tile curriculum ill h scculld~l ~·;ch()()l (p. 77). III 11)0"+. li.\ Klein proposed (h~!t "the fUllction i gr~lphictlly relJ1Tselltcd hOLlle! J'llrn1 the cClltr~!lll()til)J1 oflll:l1h 111(ltil>~1l k(\l'hi ~llllL ~lS d flatur:!l cUllSequl'llce, the clements orihe calculus should be included ill the curriculum uf oIl IdSS [hi -' school {quULed by l'Jordg~l~lrd, p. ~\ I J. The fulluwing ye~lL ~lt d cOllfcrellcc ill fv1c r(ll 10, Itdy the rcfurms proposed by [( icin wcre ~ldupted by the (jcrm(ll1 iety of N:ltllr(tl ieniists. It \VLlS (it th:H lllecti thdt thc c\,pressiolljifllkli(}flolcs nl'llk('1l (fUIll'tiollLll thillking) W~l>; cnill,..'d. K Ic i 11 \ c II d 0 r S C III e 11 t u f t 11 e fu II c t i U 11 COil Ce p tin !1 LI c II Ccds e COil dar y s c 11 no I 11l(1theill~lfics droLllld thc world. but ill the LJllilcd St1tCS dlld CLlrJ:HLl it \V~lS ;1p­ j1clrclltly very much u minor illllucllce. Ch;1tcLll!IlCLlf's (1 --)()) ~;u ur U.S. 1- ('men!;] ~1I 1';1 texthooks showed th:!! exerc'iscs 011 phs he lll~lill pl~lCC (h;l~ i'lillCtil-)rlS dppe:lrcd ~l\CLlged kss th;111 :1 tenth of I perccllt duri the lli)ll'iL'clllh centLiry ~il1d h;ld !lot rC;lchec! .5 pcrcenl by I ~. ReIther th;m ((lki rUlll'lion as ;[ l! Jl i fy i COIl C cpt, the III C m b c r s () r the C () 11 fe 1'1..' Ill' C 0 11 f'v1 u t h l' 11 W 1i C sur t h ceo m - mittec or Tell (j\,I;ltioll;ll Educ1tinn:t1 !\ssoci(ltioll IS9--:t, pp.
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