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. lOS ) held proposed tll;1t a br;l be trc;ltcd ~1S llcrajizcd :lrithmctic ill thc middle des tu provi inductive prepdratiull for its formal introdLictiul1 in ninth grude. The cOlllmittce
III e III b c r.s e 11 d e)1' sed t 11 c c q II d t i 0 ll, all d 11 U t (h C fLl Jl C t ion, ;1 S des e r v i 11 g " C S pc C i :II emphasi (p. 111) -<1 vicw that prcv~ljlcd throughout the flrst deC~l of (he twenticth century. \Villi;1ll1 Betl (1 ()l()) Jloted thelt "drter llluch COllt a new :Illel gre(ltly reducecl ;llgehrL1]l ram is l:lkillg form" (p. 163) in the junior hi school. He CJ(lilllCd thu( there W()S cOllsideLlhlc LlI1CerLlinty ;1S to hnw lllLlch (11- icntioll shuulel lic giVCll tn iT~lclitioJl~t1 word prohlems, ~llld therc \\,:lS ~I cOlltillui dcb(ttc over thc i'ullctioll concept. III his view, illstCdc! of reo lli7i1lg (II 1':1 :lrOUlld thc i'ullctiull concept. (cxtbook :lLltllors h~ld rctaillcd :tll thc (lId lll;ltcri;r1 :l1lci simply tried "'to giue ()Il :1 fl'\V liutches of fUllctioll:ll \'v()rk. it is this dOl/hie bur ck':Il .. ~lllc! /l()! thl~ fUllction icie() (lS ~~llch. \\Ilich is c(lLl~~illg lllelllY .SCC()lllLlry tC;ll'll(~'rS tll vicw the ilCW I'rogrLlm us :1 sort oj' IlO) (p. I ()]).
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d c l' : I d c' 11' () III I ~~ () () t () I C)"+ () s :I 'IV ~ III l' 11 U rIll U II S g r ()\\ t II i 11 U. S. h i h s c h() () I clll'ulllllellh. li"Ulll (lun percent ur tile rourtcell- tu se\cntc,... 'll-yearol ;11- tl...'Jldill sch()ul ill I :~l)() tu illIJl'C th~lll 7] perccllt by ! i)-W (.f~1111C~~ ;Inc! TY:ll'k Il)X]).
Lilrullllll'llt gn)\\th ill Call:ldLl durill thc samc pcri()d \\L1.S ~lpp:lrClltly >;cc()llcl ullly to ri1dt of the United St(ltcs ( ldill 200 I: L~I7CTCI()pl!di[( ()j CUIlUc/U I ). DLirill th:lt period. enrollmcnts ill ~l Ilrst grew but then to shrink, dt least ill till...:
United S(dtL'S (scc JOIlCS ~llld Co\)()rcl I c) p. . (ur t'llru!llllcllt d~tt~l jJ"Olll 1i)()U 1\) 1(55). III li~(){). mure th~ll1 percellt of S. h' school students \\crc uki dl . (lild th~!t IHllllhl.:'r illLrc~1 to ~tllll()st 57 ent by ]l) In. /\J \\~lS CS A HISTORY OF ALGEBRA IN THE SCHOOL CURRICULUM 7
becon1ing a nlajor source of failurc in school, however, and enrollments began to er decline as it Vv'3S increasingly made an elective ratherth3n a required subject. Enroll nlents in algebra decreased steadily after ]910, lling to around 30 percent by 1940 and below percent by the 1950s. In contrast, more recent enrollment llunlbers III are 11ll1ch higher. The National Assessment of Educational Progress, for CX~lIl1ple, Ile revealed that in 1999 1110re than 90 percent of sevel1teen-ye~1r-olds had taken at least III one course in algebra (Can1pbell, IIo111bo, and f\1 azzeo 2000, p. 63). In 2004, 29 percent ofthirteen-year-olds were taking algebra, prin1arily in eighth grade, and the percentage of seventeen-year-olds \vhosc hi n1athen18tics course was second year algebra had increased frOJl1 37 percent in 1978 to 53 percent in 2004 (rcrit, Moran, and Lutkus 2005, pp. 56,58). Herbert Kliebard and Barry Franklin (2003) noted that the declines in enroll 111cnt in high school l11athematics in the first h~llf of the twentieth century vvere part of a general n10ven1ent in which the U.S. school curriculum was revised to the ideal of social efflciency: the doctrine that "the job the schools first ~lnd fore 11- most was to train children and youth fc)r their predicted adult rol (p. 405; for a fuller treatn1ent of the effects of social efficiency argun1ents on school nlatheIllat d ics, see Stanic 1986). Edward Lee Thorndike (1923) and his students exanlined how algebra tasks were rated in in1portance by college teachers of science and the ul way in which topics in algebra were used in high school textbooks and Encyclope he dio Britannico articles. They concluded, "Algebra is a useful subject, but its utility varies enorn10Llsly" (p. 89). In particular, teachers should pay n10re attention to reading and using algebraic symbolisn1, forn1ulas, ~1l1d graphs and give less atten It- tion to teaching students such skills as solving quadratic equations or worki with or the binomial theorem. Arguments that algebra was of little v~lIue
to the average student. Critics like David Sneddon claimed that "(J I ta ught in ial Anlerican high schools is a nonfunctional and there1~)re nearly valueless subject for lr- 90 per cent of all boys and 99 per cent of all . rl (quoted by Reeve] 936, p. I). TS Describing the "progress in algebra" in thenrst quarter of the twentieth cen- tury, David Sn1ith (1926) acknowledged that the purpose orteaching a bra "a quarter of a century seems to have been to make mathematicians; the purpose today is to 111ake well-informed American c (p. 20). That purpose, he "consists in givi to everyone a idea oCthe meani ofal lJra. to- wit h Cl l! S t' I lie a t ion s \v hie h eve r yon cis Iike 1y t C) III e c f ' (p. 21).
Scho I Algebra in the NeVI tvlath Era The new n1ath era, which In n1id 1950s and ended in the lllicl 19705. jO witnessed a sea ill schoo I al general i zeel ari th lllet i c to system i c structure and proof. A 11lain impetus j~)r this change was to prepare students enled
1 1 "l"\ r ~ 1 ticipated by such mathen1aticians as Eric Ten1ple Bell (1936), who argued that alge bra provided a less complicated structure than geoll1etry and ought to be presented to students in its abstraction, a bypothetico-decluctive systenl open to multiple in terpretations. He demonstrated how the field postulates could be L1sed to convey algebraic ideas, an approach that was pursued in textbooks published during the new math era. For example, the authors of the School Mathematics Study Group's Firs! COllr,)'e in A Igcbra chose to introduce the properties of arithnletic operations and then to state and prove theorems based on those operations, beginning with a theorem on the uniqueness of the additive inverse (Pollak 1965). Another exanlple is presented by the Algebra I book f1'on1 the so-called Ball State project (Bnlll1fle1, Eicholz, and Shanks 1961), which after introducing four postulates, proves that if x (J ({ h, then x h. This axiomatic c1pproach had been encouraged by Saunders Mac Lane (1957), who wrote enthusiastically, "The proofs of algebraic theorems are neater and easier than those of geonletry! T'raditionally, high school geometry is said to be the subject where logic can best be learned. Algebra would be a better place!" (p. 100). As it turned out, students had trouble seeing the I . behind the algebraic theorems, and teachers struggled to convince then1 that the proofs were necessary and important. Rene Thol11 (1973) sun1nlarized one view of the problenl when he observed that meaning is 1110re important than rigor in nlathematics and that al though algebra has an extren1ely rich syntax, "'the 'n1eaning' of an algebraic sY1Tlbol is established with difficulty or is non-existent" (p. 207). He notee! "the heuristic poverty of algebra, where each new difilculty presents itselflike a wall which neces sitates entirely new nlethods ifit is to be surmounted" (p. 208). ForThonl, axiolllat ics should appear at the end of algebra instruction, not the beginning. Four points (2 through 5) of the ColI Entrance EXLlmination Board's (1959) nine-point program callle to characterize the approach to elementary al uncler- taken duri the new math era (p. 33):
2. Understandi of the nature and role of deductive reasoni . n algebra, as well as in 3. Appreci ati on of nlathel~na tica 1 structures ("patte r115" )~~for eXcllll p] e, properties of natural, rational, real, and conlplcx numbers 4. Judicious use ofuni . varialJles, functions, and relations 5. Treatment of inequalities al wit 11 u~l t ion s
A s part 0 f thi s a pproac 11, various terms \vere laced or redefined. [vI L1C h 0 f tbe new helped unravel some of the nlysteries of elenlentary al which bad never been clear about distinctions arnong such ternlS as literol numbers, Ifllknoll'71s, )'{{r/obles, and COllstOlltS. Tbe ternlinology of litend nllmber and lirend equation was dropped; equations and inequalities became open . a van longer defll1ed as "a quantity that varies"-was said to a symbol to be replaced cs A HISTORY OF ALGEBRA II~ THE SCHOOL CURRICULUM 9
by nallles of set elenlents, usually I1u1l1bers; and a function 'vvas defined as a set of ordered nU111ber pairs having certain properties. n- The effort to nl0dify the algebra curriculunl during the new n1~1th en.1 had S0111C :y lasting effects. For exanlp]e, inequalities are still included with the study of equa 1C tions, Joci are c0111nlonly defined as sets of points rather than as 1110ving points, and trigono111etry c011tinues to be introduced by nleans of real-valued periodic func
tlS tions. Because the n1athenlaticians directing the refornl beli that what was a good n1athematics for "coll capable" students was also good for students who are less able, two-year courses were constructed that would treat the same topics as the one-year course (Beglc 1969). Unfortunately, however, l11any two-year courses in introductory algebra tended to treat the content in a rcductionistic Llshion, e111- ploying nUll1erOLlS repetitious exercises and giving little attention to reasoning or complex problell1 solving. Nonetheless, as the new math era faded into the back-to basics lTIOVement of the 1970s, with algebra instruction returning to an emphasi on 11lc.mipulating SY111bols and solving sin1ple equations, the idea ren1aincd alive that a larger percentage of high school students 111ight profit frol11 the study of introduc
Lie tory algebra if the course were extended over n10re than a year.
School AI~ebra in the Era of Standards
01 Algebra l-en1ained a prominent source of failure in high school during the 1970s and 1980s (Moses and Cobb 2001), and so when NCTM ( 1989, 2000) began :Ie its Standards-based refonll efforts, school algebra canle under heavy scrutiny once ~s .- 1t- again. fn recent years n1athen1atics educators have pursued a set.of interconnected chan with respect to algebra that are at least as complex and that have as 9) reaching inlpl ications as those the reform has pursued v/ith respect to any other branch of school n1athen1atics. We consider next these changes \;\Iith respect to sev ~r- eral of the Standards that NeTlv1 has put forth.
Approaches to AI~ebra In the late 1 970s and 1980s, research on students' understanding of algebraic 111an ipu lations and of the function concept (e.g" Kieran 1992; Leinhardt, Zasl and Stein 1990) helped 11lathematics educators clarify why so few students \vel-e learning algebra well. Some difficul' such as students' ten to interpret the equal Sl as a cOn1n1Cl1ld to compute an answer, that arith- 11lCtic instruction were contributing to 'r difficulties with aJ Others, such as students' difflculties in correctly identifyi particular examples of ations as functions, that a revision of the algebra currlculUlll \vould be required if
7S, shldents were to understand ide3s of function. The late and 3brupt introduction
~) 1l algebra in isolation fron1 other branches of nlathematics and fron1 applications to no other disciplines apparently contributed to students' difficulties. In NCTM 098 9) cn~;:]tpd ~ P~ttprn~ ilnn R p Llti nn~ II i ne; Standard for ki ndenlarten throu2h fourth 10 ALGEBRA AND ALGEBRAIC THINJ(li~G IN SCHOOL MATHEMATICS
grade, a Patterns and FUllctions Standard (lnd (In Algebra Standard for hfth through eighth gracie, and Algebra and Functions Standards for ninth through twelfth grade. NCTM's (2000) subseqLlent decision to establish a Content Standard for algebra fron1 prekindergarten through twelfth grade, though it presented some problenls of interpretation at the earlier grades, was generally vvelcomed by n1athen1atics educa tors. !Vleanwhile, researchers and curriculun1 developers eXaml1l1 ap- proaches that would ~lIlow students to connect algebra more clearly to their previolls study of mathematics and to other domains. These approaches, S0111e of which hae! been considered in earlier eras, included algebra as (a) generalized arithn1ctic, (h) a problen1-solving tool, (c) the study of functions, ane! (d) n1odel' (e., Bednarz, Kieran, and Lee 1996). For several reasons, a nunlber of curriculum 10pmel1t efforts focused on algebra as it might arise from attention to functions of quanti ties embedded in problcn1 situations. One reason was a desirc to move ~l\vay fi'OJll the 111tlllipubtion of sYIl1bols ill expressions and equations and toward an approach that students would fj nd more 1l10tivating. A second reason was the perception that a focus on functions of quantities would allow learners to use their understanding of problenl situations to develop conceptual understanding. Mathematics educators saw a shin in emphasis frOll1 equation solving to the study of functions as a shift CrC)ll11l1erllOrizing procedural rules toward n1aking sense ofproblcrll situations. Us ing problems about varying quantities as an entry point to the study of fUllctions, rather than the set-theoretic definition, allowed recent refC)rn1-orientecl curricula , Star, Herbel-Eiscllll1~1I11l, and Smith 2000; van Reeuwijk and Wijers 1997) to introduce the stuciy of functions in the Illiddle grades. (See Daniel Ch~1zan, this vol ume, for further discussion of equation- and functions-based approaches ill recent U.S. curricula.) A third stimulating a functions-based approach to al was the avail- (1 b iii ty 0 f incrcasi ngly powerful and a ff(Jrdab Ie desktop and hanclh e IcI C01l1P u ters that made possible new activities, particularly ones that lllade use ofnlultiplc connected representations, inclucli (see Romberg, Fellncma, and Carpenter 1993, es pecially chzlpters . Before computers were widely available, graphs \vere tin1e consLlming to create and awkward 10 manipulate. A typi task 'vvas to IJroclucc a by substituti v~dues into an al ic expression and plotti resulting ordered pairs. New soft\varc linked graphs to other representations offullctions and to problem situations in a vari of For example, students could manipulate anyone of three tations (all algebraic tabular, or IC on), :Jnd the computer woul cl upda te the other tV/O (e ,IVI oschkovi c 11, Sc hoe nfe ld, and Arcavi 1993). Or students could examine connections graphs and problem s itua ti ons. instance, COll1 p uters attached to sen so rs co ul d graph d ist:ln c e, speed, temperature, or other physical quantities \vith respect to tin1e (Mokros and Tinker 1987). Such activities allo\ved students to 11lanipulate the problem situation and see :5 A HISTORY OF ALGEBRA IN THE SCHOOL CURRICULUM 11
the resulting changes in graphs and to exanline functions that could not be re:ld ily expressed in closed algebraic for111. New softvv'are also nlade possible ~lctivities in which conlputers executed syn1bolic manipulations, allowing students to elevote n10re attention to problen1-s01ving strategies and investi conjectures (St~lr,
]- Herbel-Eisenn1ann, and Snlith 2000).
J Attention to Students' Thinking Efforts to pron10te standards in school rnathenlatics are causing teachers and .IS ld researchers to look beyond how students perforn1 symbol manipulations and h) errors they 111ake to how they think about algebraic concepts. Over the last half century, n1athenlatics educators have nlade not only in conceiving new 11t instructional approaches that make usc of increasingly available technology but also ti - in understanding how students think about algebra. In 111uch of the research in the 1970s and 1980s, researchers examined students' construction, n1anipulatio11, and interpretation of algebraic and graphic representations and catalogued their errors, which researchers often characterized as bugs or 111isconceptions (e , Kieran 1992; Leinhardt, Zaslazsky, and Stein 1990). A subsequent generation of research cxan1- ined 1110re closely students' reasoning with representations~ and several general fmdings enlerged. One was that students often attend to representational features that teachers and researchers nlight not notice. For instance, although anyone expe rienced with linear functions focuses on the slope and .v-intercept when connecting lS, y l71X -+- b to its graph, learners do not necessari1y appreciate the asymmetric role Jl ~l to that the intercepts play. They nlay see the x-intercept as sal ient and try to inc lude it J1- in the algebraic representation along \vith the y-intercept (e.g., Moschkovich 1998; Schoenfeld~ Sn1ith, and Arcavi 1993). A second finding concerned the process by ~n t which students 11lake connections between representltions and problem situ::ltions. Simply telling students about correspondences between features of a representa l i 1- lat tion----such as l7l and b in y 177 X a problem situation is 'cally insuf ficient. Novices coordinate representations and problem situations by refIning their understan' of both ( ,fZS{lk 2004; Meira 1995). Such results reveal n10re subtlety and nuance in students' with ons than previoLls re search on functions has found. The research has nlade clear that teachi and learn ing \vith ll1ultiple representations linked to one another and to problen1 situations is mel necessarily very conlplex. ate ~ebra r Frolll the outset, the NCTM (1989, 2(00) S/ondords efforts the to 111ake schoo 1 n1athen1atics--anci al j 11 particular-avai lable to all students. eIll T\vo argU111ents have led 111athelll3tics educators to consi how to teach ideas re d, lated to algebra to a 111uch broader ranQe of students. The first arQUlllcnt has to do ~ ~ - ~ with econonlic opportunity and equal citizenship. Algebra is often referred to as a see rrrltpl(ppnpr tn '1 ('nllpCT(> ('(111('~ii()n !lnrl thp C;-HC'fTc) C)uch education alTords. NCTI\1 ALGEBRA {\ND THII'-lI (2()O()) drticuLitcd dll uily Principle the il1lpOrl~lllC[_\ uf high cx tions ~llld stro sLllJport fur all students ~llld cxplicitly stated th~ll (lll studCllLs should le~lrn ~l t)f~l.':\ i tlwt ma(hem~\tic~d litcr~lcy is civil righh i>;slI:;.' u!'uur tillll:., P"ohert f\!1c)se.'-) ~llld Ch~irlcs Cnbb, h .. (2()Ul) likcn intruductioll uf cumputcrs controlled by bolic nt~ttions tu the introduction ufthe mcch~lllic~lJ cottUll pickl'r: Just ~IS (he I~ltlcr c Llhor (km~lJlds ill [-jcLI!fure duri tIll.:' ]l)-l-Os ~Illd ll)5()s, the cUll1pLllCr i~; currcntly ch~lIlgi Llbol- deIlldnds in indust [)~!l1icl Ch:l- 1~111 (20()()) and Robert l\/1oses (lI/Juscs atld Cobb 20n I) dcsi appn)~lclles to ~dgel)r~l intcllc1c:d to Cll llliddle ~1l1d hi school students' intcrl's1 and tu llldkc the content more accessible. Both <-lpproaches :Ut' h~lscd un problems dr~l\\'11 from studcnts'lives. CThe seccH1l1 drgument Iws to do with rccollccptLl~l1i/i clcrnCl1tlrv \~ch()ul llldthcm:llics in \v~lyS tlwt hetter prcp~lrc students for the form:l] study of' ~lIgehr:!. A I'll ndclllll'Il ta I q LIe sti em is til e e\ tcn t to \vh ic [1 the stud y 0 f ~\ri t h rnet i c Gin pru\' idc :lll ~tdCqU:ltc round~ltioll fur the suhsequellt study or~tlgebr~l. /\ h~lS oftcn bcen il1- tcrpretcd ~\S gCllc/'u/i:'L'd uri/hille/ie', but th~\t term has tlkel1 on ill lc~lSt two lnl~::lJlill (Jill' llle:llli ~lrises from the perspective tll:lt br~l involves problems in \vhich SOllle numbers in arithmetic cumpLlLltiollS (Ire repl:lced by sylllb()I~). 'fllat perspec tive can :llllplify discontinuities between students' experiences in solving prublems with arithmetic and with br:l. Llini such discuntinuities, Carolyn Kier:ll1 (I ed th~lt (tlgcbr~\ is difllcult 1'01' students because the l-cprcscnt~lti()IlS ~lre abstract ~llld lJCCllUSC the required opefcltions. i~tljy those relati qUdntitic's in \vord-problcll1 situ;1tions, conflict with operations :~tudc\lts have k~lrl1ccl tu usc through years of mUdejing \vith llrithmetic: Students need (0 .sLlhtrctct \vhcrc uncc they cldded, ~lnd to dividc \vhere once they multiplied. ;\ contrasli llle:llli I'nr j' :lrithmetic CJllph:1Si/cs oppurtunities J() r s t LI d c II t s t () e In cr~lli/illg i\ctivities \vhilc they drc sludyi :!rithlllctic. ROlllUlo Lins (tile! .I~lmcs !<:lput (2(JU~) offered cl dcilniliul1 th~d Clh:UlllPds~;e:-; sllch dctivi ;lS well ciS lllore COllVClltioll:t1 cqu:ltion soh'ill : .lIgl'hruic Ihinkill,r.:, inv()lves ({/) c\cliberclte rcllizcltiun ~lnd c Ills inL'lude j \ i il (\ 11 c! I~ \ - deduci I ~;()I1lC ilcr~ll Il'S Ufl1Ulll () f~)r ~111 "hule numbcrs. The 'eli rcnt Illcani fur lized ~trithnlL'liL' impl rather dil'fcrcllt approdche~i to drilhmctic irl.~1ructiotl ~!nd further Ii tinn and in\csti tion. CJ r a I appnXlc e)ll Ilcrali;:i e\jlericl1ccs with \\huic numbers ~llld counli rj~lrb~!r~l DelL! (200-+) rc:purb 011 the jVk~lSllrc Up projcct. which is e\~llllilli ICS A HISTORY OF ALGEBRA IN THE SCHOOL CURRICULUM 13 early elen1e11tary school students' capacities to con1pare lengths, areas, and volumes j 1(1 qualitatively. First graders introduce and use letters to express such con1parisons. Ill.?, For instance, if A B, then B A, ""-1 + C B + C, and A -- C B C. A central crs question is hovi well such experiences prepare students for subsequent equation :on solving llsing for111al algebra. ,nd 11.1- ~ssues in School A~ge ra ;to lice We close \1v'ith three issues that are underrepresented in current research on ~)n1 the teaching and learning of algebra and that to be addressed by teachers and research ers. 001 Tensions for Teachers .A As noted by Kieran (1992) and tlelen Doerr (2004), researchers have pai d little an attention to teachers' knowledge and practices with respect to the teaching of alge 111- bra. The few studies that have exan1ined prospective and practicing teachers' subject 111atter and pedagogical content knowledge have e111phasized their conceptions and jeh rnisconceptions about the function concept. Doerr concl uded that ex isting research ce- suggests that teachers' understanding of functions is nlO1'e procedural than con ceptual and is not well connected. ()ther research has focused on teachers' use of ran arithn1etic and algebraic strategies when solving equations and on their predictions are about students' perforn1ance on equation-solving tasks presented in symbolic anci word problen1 forn1ats (Nathan and Koedinger 2000a, 2000b). Results indicate that use teachers think of algebra prin1arily as a set of procedures for solving equations and nee are not well prepared to take other approaches to algebra, including those that n1akc extensive use of technology. This finding in1plics, in turn, that teacher education at tics all grade levels is critical if approaches intended to n1ake the subject accessible to a ti c. brooder fonge students are to 1110ve fron1 into classroon1s across the United and Canada. yes sed Fractions as Foundations uc- Research that exan1ines arithn1etic foundations for still focuses to a extent on interpretations of the equal' and whole-number arithn1etic. Little :hat attention has been given to the role th:lt reasoning about fractional qUcllltitics can ex- play in learning to reason 'vvith OilS. ons playa central (( role in al creason HI-lsi vVu (2001) that the most useful fcnn1 ply of prealgebra is the deve]opn1ent of con1putational fluency and the thorough study of fractions and fraction arithn1etic. He contends that students who are not cOll1fort able con1puting v.,;ith nun1bers will be disposed to n1anipulate symbols, that ; to the explanation of fraction con1putation procedures provides a natural entree into syn1bol use, that solving equations requires working with fractions, and that un derstanding fractions is nrercCluisite to understanding slones of linear functions. ALGEBRA AND ALGEBRAIC THINKING IN SCHOOL MATHEMATICS To Wu's argull1ents, we add three more: (1) Understanding the representation of f}"actions on the nunlber line is necessary if learners are to understand graphs in the Cartesian pbne, including interpolation between given points; (2) the study offrac tions and the study of algebra address S0111e of the san1e big ideas, as, for eXcullple, to add fractions or to add olgebraic ternlS one hos to ensure that all addends are expressed in the san1e units; and (3) fraction onci whole-number arithnletic provide opportunities for learners to develop multip] icative structures and an understanding of the cbstributive property, both of which are central to \vorking \vith algebraic expressions ond equations. Symbol Manipulation A111id all the attention . recently to functions-based clJlproaches to algebra, the role that algebraic lllanipulation call play in the learning of algebra has been downplayed, ot least in the United States. The introduction of handheld calculators capable of symbolic manipulation has led to debates about which, if any, syn1bolic nlanipulation capacities are still important f~)r students to develop. In other coun tries, there has been a growing recognition that through synlbolic manipulation stu dents can develop a deeper understanding of the nlathematical objects with which they work (e.g., Artigue 2002; Kieran 2004). Concluding Observation To the casual observer, algebra might appear to be a stalwart of the school cur riculum, and in 111any respects, it is. But as this brief history nlakes clear, school algebra has varied in the content taught, the purposes for which it has been included, and the students to whon1 it has been offered. If anything has been constant, school algebra has too often been seen as a source of diffIculty and failure--a gauntlet to be run rather than territory to be claimed. At one tinle in the United algebra becanle no more than an elective, but that seems unlikely to happen ·n. Whether algebra \vi II remain a separate course or courses, however, is unclear. III Europe and Asia, olgebra is often integrated into a con1prehensive curricuh!l11 that includes ge01T1etry and other branches of 1110then1atics. Recent trends in North that algebra \vill increasingly be i into learners' experiences earlier and nlore fully. ]fthe ofnlaking comprehensible, usefuL and a pleasure to learn is to be will continue worki to develop to relatcclto tions and functions, approaches that nlay eli nlarkeclly from their own experi- enees when they learned the subject. ICS A HISTORY OF ALGEBRA IN THE SCHOOL CURRICULUM 15 REFERENCES he Archibald, Thomas, and Louis Charbonneau. "Mathematics in Canada bej~)re 1945: A lC Preliminary Survey." 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