8 Proportional Reasoning

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Lesh, R., Post, T., & Behr, M. (1988). Proportional Reasoning. In J. Hiebert & M. Behr(!s.) Number Concepts and Operations in the Middle Grades(pp. 9"#118). Reston,$% La'rene rlba* & +ational onil o- Teahers o- Mathe*atis.

Proportional Reasoning

Rihar! LeshI%T /0ste*s

Tho*as Postni2ersit0 o- Minnesota

Merl0n Behr +orthernIllinois ni2ersit0

Proportional reasoning is a -or* o- *athe*atial reasoning that in2ol2es a sense o-o#2ariation an! o- *ltiple o*parisons, an! the abilit0 to *entall0 store an!proess se2eral piees o- in-or*ation. Proportional reasoning is 2er0 *honerne! 'ith in-erene an! pre!ition an! in2ol2es both 3alitati2e an!3antitati2e *etho!s o- thoght.

In or o'n researh 'e ha2e onsi!ere! the essential harateristis o- proportionalreasoning to in2ol2e reasoning abot the holisti relationship bet'een t'o rationale4pressions sh as rates, ratios, 3otients, an! -rations. This in2ariabl0 in2ol2esthe *ental assi*ilation an! s0nthesis o- the 2arios o*ple*ents o- these

e4pressions an! an abilit0 to in-er the e3alit0 or ine3alit0 o- pairs or series o-sh e4pressions base! on this anal0sis an! s0nthesis. It also in2ol2es the abilit0 togenerate sess-ll0 *issing o*ponents regar!less o- the n*erial aspets o-the proble* sitation. This perspeti2e has not been ni2ersall0 e*plo0e! b0 theresearh o**nit0.

%ll persons 'ho sol2e a proble* in2ol2ing proportions !o not neessaril0 seproportional reasoning. In -at, one ol! notie si*ple n*ber relationships (sine% is three ti*es B, 5 *st be three ti*es 6) or se a rote algorith* sh as ross*ltipliation. To sol2e proportions o- the t0pe %7B 476, st!ents are o-ten taght

the ross *ltipliation *etho! %6 4B 'here 4 %67B: 0et researh an!e4periene ha2e onsistentl0 sho'n that this *etho! is (1) poorl0 n!erstoo! b0st!ents (Post, Behr, & Lesh, 1988), (;) sel!o* a


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ornerstone o- all that is to -ollo'. This hapter !issses this onstrt -ro* bothperspeti2es in!iating 'hat 'e belie2e to be transition *ehanis*s an! st!entbeha2iors. e then e4plore re3ire*ents -or a o*pter base! *o!el -or thesoltion o- proportion relate! proble*s an! !isss the on!itions n!er 'hih this*o!el is able to pro!e reasonable soltions. Ainall0, 'e raise 3estions as to thei*pliations 'hih this *o!el *ight ha2e -or -tre researh 'ith hil!ren.

Pre2ios atte*pts to assess proportional reasoning abilit0 (arpls, Plos, & /tage,

198"a, 198"b: +oelting, 198Ca, 198Cb) ha2e -ose! largel0 on in!i2i!alresponses to *issing 2ale proble*s. Those st!ents 'ho 'ere able to ans'ersess-ll0 the n*eriall0 to be a protot0pe -orproportional reasoning tas>s, e2en thogh the reasoning in2ol2e! !oes not -it thee3ation %7B 76 bt instea! -its the e3ation %4B 46. In -at, Piagetiansha2e arge! that an earl0 phase in hil!ren@s proportional reasoning apabilitieso-ten in2ol2es s haraterie! b0 the s0ste* o-relationships *5 are onsi!ere! to be proportion relate! tas>s ('hih o- orsethe0 are), e2en thogh the t'o si!es o- the e3ation are not s0**etri. Tas>sharaterie! b0 the s0ste* (a7b)5 n *st be !i--erentiate! -ro* theproportional sitations.

% *aGor goal o- this hapter 'ill be to sort ot an! lari-0 aspets o- proportionalreasoning 'hih preser2e the inten!e! *athe*atial *eanings o- this ter* an!'hih researh has sho'n to be e!ationall0 or ps0hologiall0 signi-iant. %nothergoal 'ill be to i!enti-0 neglete! areas o- researh on proportional reasoning.

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SOME IMPORTANT TYPES OF PROPORTIONAL REASONING TASKS

ritial insights n!erl0ing *an0 o- the *ost basi onepts in siene,*athe*atis, an! e2er0!a0 proble* sol2ing o-ten onsist o- reogniing si*ilarpatterns or strtral si*ilarit0 in t'o !i--erent sitations. Bease proportionalreasoning !eals 'ith one o- the *ost o**on -or*s o- strtral si*ilarit0, it iso-ten lin>e! to so*e o- the *ost i*portant ele*entar0 bt !eep onepts at the

-on!ation le2el o- *an0 areas o- siene or *athe*atis. %s state! earlier, 'ebelie2e proportional reasoning is both the apstone o- ele*entar0 arith*eti an!the ornerstone o- all that is to -ollo'. It there-ore opies a pi2otal position inshool *athe*atis (an! siene) progra*s.

%s eah !o*ain o- >no'le!ge ses this basi reasoning para!ig*, it ten!s to be*o!i-ie! in sbtle 'a0s to -it the peliar nee!s o- the !isipline. Aor e4a*ple, in*i!!le shool arith*eti, slightl0 !i--erent -or*s o- proportional reasoning arerelate! to so*e o- the -ore*ost oneptal #entere! instrtion an! researh.

1. Missing 2ale proble*s %7B 76 'here three 2ales (inl!ing oneo*plete rate pair) are gi2en, an! the goal is to -in! the *issing part o- theseon! (an! e3i2alent) rate pair.

2. o*parisons proble*s %7B N O 76 'here all -or 2ales are gi2en, an!the goal is to G!ge 'hih is tre%7B 76 or %7B 76 or %7B Q 76

3. Trans-or*ation proble*s

(a) !iretion o- hange G!g*ents%n e3i2alene is gi2en o- the -or* %7B 76.Then, one or t'o o- the -or 2ales %, B, , or 6 is inrease! or !erease! b0a ertain a*ont, an! the goal is to G!ge 'hih relation ( , Q, or ) is tre-or the trans-or*e! 2ales.(b) trans-or*ations to pro!e e3alit0%n ine3alit0 is gi2en o- the -or* %7B 76.Then, -or one o- the -or 2ales %, B, , or 6, a 2ale -or 4 *st be -on! sothat, -or e4a*ple (%4)7B 76.

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(a) e3i2alent n*bers or ratios o- n*bers, -or e4a*ple, 17" ;7 "79

(b) e3i2alent e4pressions ha2ing both *easre*ent nits an! n*bers,-or e4a*ple, >ilo*eters 7 *eters 1CCC 7 1 ##Q 1 >ilo*eter 1CCC*eters

() e3i2alent e4pressions in2ol2ing relations an!7or operations as 'ell asn*bers an! nits o- *easre-or e4a*ple, -eet7; seon!s " -eet per seon! ;.C=FF *iles per

hor(!) e3i2alent e3ations, -or e4a*ple, ;75 5718 ##Q 5; ;418 ##Q

5 'here the trans-or*ations preser2e so*e i*portant properties'hile hanging others.

In arith*eti, e3als signs () an sall0 be interprete! as *eaning es so*ething a2ariable. Aor e4a*ple

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(a) In *an0 si*ple proportions, sh as "7447;E, the 2ale o- 4 annot2ar0: 0et this !oes not *ean that 4 is not a 2ariable. Here, thei*portant thing to >no' abot 4 is that it is an n>no'n 'hih an be*aniplate! sing rles si*ilar to those that appl0 to >no'nn*bers.

(b) /o*eti*es e2en -i4e! onstants, sh as S representing pi, an beassigne! a range o- 2ales (e.g., ;;7E or ".1= or ".1=1F9;F"F)

!epen!ing on the le2el o- preision that is hosen as appropriate in agi2en sitation. et, this !oes not *ean that S is a 2ariable.

/o the 2ale o- a s0*bol *a0 be -i4e!, 0et the s0*bol is a 2ariable: or the 2ale o-a s0*bol *a0 be able to 2ar0, 0et the s0*bol is a onstant. ?2erl0 si*plistie4planations, sh as treating 2ariables as thogh the0 'ere si*pl0 things that an2ar0, are bon! to reslt in on-sion -or hil!ren.

Transformations and invariance.To generate a set o- ratios e3i2alent to "79, aseries o- trans-or*ations an be se! to no'n 2ale -or 4 in a proportion li>e "7447;E, a series o- trans-or*ations an be se! to *ap one entire e3ation toanother"74 47;E ##Q 4 ;E"74 ##Q 5; 81 ##Q 4 9. %n!, -or an0 o- the pree!ing>in!s o- trans-or*ations, isses relate! to le2els o- e3i2alene arise. %n0 ti*e anobGet is trans-or*e!, so*e in-or*ation is lost or gaine!: the isse is 'hether thealtere! in-or*ation is o- interest. hih properties re*ain in2ariant an! 'hih !onotN Aor e4a*ple, t'o e3ations *a0 be onsi!ere! e3i2alent -or an0 o- the-ollo'ing reasons. (Reall that in the pre2ios setion a si*ilar list 'as generate!-or e4pressions.)

(a) The0 ha2e the sa*e trth 2ale (e.g., both tre or both -alse).(b) The0 ha2e the sa*e soltion set (e.g., sa*e 2ales satis-0 both).() ?ne is a si*pli-ie! -or* o- the other.(!) The0 an be trans-or*e! into one another sing spei-ie! operations.

hen e!ators treat trans-or*ations as thogh


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The n!erstan!ing that an e3ation (as a 'hole) represents an algebrai s !ealing 'ith trans-or*ations an! in2arianeha2e been neglete! in the researh literatre on proportional reasoning, e2enthogh Piaget (Piaget & Inhel!er, 19F) an! a -e' others ha2e e*phasie! itsi*portane in ertain t0pes o- onser2ation tas>s.

?ne reason that trans-or*ation tas>s ha2e been neglete! is their ten!en0 toin2ol2e !0na*i ations 'hih are !i--ilt to portra0 in te4tboo> or paper#an!#penil testing -or*ats. % seon! -ator is the arti-iial o2ere*phasis on


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Pre"proportional reasoning.I- the *ost ritial harateristi o- proportionalreasoning is reogniing s haraterie! b0 the e3ations% B 4 or % # B 4 # 6. That is, the tas>s that these e3ations !esribeten! to be natrall0 re!ible to non#proportions o- the -or*s P 4 or P 4 #6, 'here P is a !iretl0 perei2able 3antit0.

I- hil!renDs inlinations to se a!!ition to sol2e proble*s haraterie! b0 %7B476are not 2er0 reliable in!iators o- proportional reasoning, then 'h0 ha2eresearhers (largel0 !e2elop*ental ps0hologists) re-erre! to


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% se2enth gra!er 'as gi2en a ;4" retangle (see Aigre ;) an! as>e! to s are relegate! to an algorith*isoltion. /o, in general, s haraterie! b0 a relationship bet'een t'o rational e4pressions that is,%7B76. e belie2e, ho'e2er, that this is a neessar0 bt not a s--iienton!ition. ?ther t0pes o- sitations also re-let tre proportional reasoning abilities.

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Aigre ;. nlarging a retangle

Transitions from pre"proportional reasoning to proportional reasoning.2en thoghPiaget an! other !e2elop*ental ps0hologists o-ten spea> o- proportional reasoningas thogh it 'ere a global abilit0, or a *ani-estation o- a general ogniti2estrtre, it appears that the e2oltion o- proportional reasoning is haraterie! b0a gra!al inrease in loal o*petene (e.g., Lesh, Post, an! Behr, 198E: Torniaire& Plos, 198F: arpls et al.. 198"a 198"b). Proportionalit0 is initiall0 *astere! ins*all an! restrite! lasses o- proble* settings. o*petene is then gra!all0e4ten!e! to larger lasses o- proble*s.

In this setion 'e sggest that the pree!ing


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(a) /oltions to it 2ali!ate! !e2elop*ental se3enes in proportionalreasoning, 'hih ha2e been h0pothesie! b0 Piaget (Inhel!er &Piaget, 19F8: Piaget, Urie, /e*ins>a, & Bang, 198), +oelting(198Ca, 198Cb), arpls et al. (198"a, 198"b), arpls an! Peterson(19EC), Hart ( 1981), an! *an0 others.

(b) /oltions sho'e! ho' *ehanis*s 'hih ha2e been sho'n to-ailitate general oneptal e2oltion *a0 also pla0 i*portant roles

in proble* sol2ing.

Be-ore 'e onsi!er a no'n 2ale. Ho'e2er, as 'e pointe! otearlier in this hapter, sess in the se o- this strateg0 is a relati2el0'ea> in!iator o- proportional reasoning.

Piaget et al. (198) re-erre! to this stage as


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natre, bt the0 !o not neessaril0 realie that the0 nee! to onsi!erthe onstantl0 inreasing !i--erene bet'een the relate! ter*s o-eah rate pair, that is, o- eah ratio. %or!ing to Piaget, pre#proportionalit0 is broght abot b0 oor!inating -ntions, 'hereasproportionalit0 is base! on re2ersible operations. The *ain !i--erenebet'een -ntion relate! thin>ing an! operation#relate! thin>ing isthat the -irst t0pe is essentiall0 non#re2ersible: that is, gi2en a hange

in one o- the -or 2ariables in a proportion, the hil! annoto*pensate b0 hanging one o- the re*aining 2ariables.

(e) PiagetDs en b0st!ents 'or>ing either in!i2i!all0 or in grops o-three, raging -ro* *i!!le shool throgh a!lts. Thegrop !esribe! in this setion 'ere a2erage abilit0se2enth gra!ers.

Problem#Are! Ainle0 began teahing 1C 0ears ago in

enter2ille. He an! his ne' bri!e rente! an! apart*entat "188 Main /treet -or V;FC per *onth. He alsoboght a ne' $ rabbit -or V=FCC. His starting salar0'as V1=,CCC. This 0ear, Are!@s brother, To*, alsobegan teahing in enter2ille. He an! his ne' bri!erente! the 2er0 sa*e apart*ent that Are! ha! rente!1C 0ears earlier, onl0 no' the rent 'as V="C per*onth. He too boght a ne' $ rabbit: the prie 'asno' V89CC. ?ther prie !i--erenes an be -on! in theatalogs an! ne'spapers that 0o ha2e been gi2en.hat shol! To*@s rrent starting salar0 be, so that it'ill be e3i2alent to Are!@s salar0 -ro* 1C 0ears agoN

Aigre ". The in-lation proble*.

(retrn to te4t re-erene)

In general, aor!ing to Piaget, a!olesentsD proportional reasoning !e2elops -ro*(1) a global o*pensator0 strateg0 (o-ten a!!iti2e in natre) to (;) a *ltipliati2estrateg0 'ithot generaliation to all ases to (") a -inal -or*lation o- a la' o-proportions. Ho'e2er, in atte*pts to 2eri-0 PiagetDs theor0, it has been note! that

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the le2el o- reasoning that a hil! ses is o-ten not onsistent aross tas>s or e2en'ithin a gi2en tas> (e.g., 'hen the n*ber relationships or pereptal !istratorsor onte4tal 2ariables are hange! slightl0). 2en thogh the stages that Piaget!esribes ha2e pro2en to be 3ite robst -or !esribing hil!renDs beha2iors on agi2en tas>, 2ariabilit0 aross tas>s is o-ten 3ite large. This has been re-erre! to ase the one !esribe! belo' sall0 too> or st!ents bet'een ;C an! =C*intes to pro!e. The soltion !esribe! here is t0pial o- those pro!e! b0 orst!ents, 'ho 'ere a2erage abilit0 *i!!le shoolers throgh a!lts. /t!entsgenerall0 'ent throgh -i2e !istint reoneptaliation 0les !ring their =C*inte session. +otie the si*ilarit0 bet'een the stages in their soltion an! thestages note! b0 Piaget an! others.

Conceptuali$ation %#The st!entsD -irst oneptaliation o- the proble* se!a!!iti2e reasoning base! on onl0 a biase! sbset o- the gi2en in-or*ation. Thegrop sbtrate! to -in! prie !i--erenes -or pairs o- pres*abl0 o*parable ol!an! ne' ite*s. Bt onl0 a -e' ite*s 'ere onsi!ere!: these 'ere si*pl0 the onesthat 'ere notie! -irst, sh as the ar an! one or t'o ite*s in the atalog.Arther*ore, nothing 'as sai! e4pliitl0 to in!iate ho' these sbtrate!!i--erenes 'ol! allo' a ne' salar0 to be !eter*ine!.

Conceptuali$ation h0 !i! the st!ents shi-t to a seon! oneptaliation o- theproble*N T'o possible e4planations are sggeste!. (1) %s the0 'or>e! ot !etailsrelate! to their -irst oneptaliation, the0 began to reognie !i--ilties that ha!at -irst been ignore! (e.g., l0 reognie! as -oolish, it ser2e! a 2er0positi2e -ntion: it intro!e! a *ltipliati2e 'a0 to thin> abot relationshipsbet'een ol! pries an! salaries an! rrent pries an! salaries.

Conceptuali$ation '#The st!entsD thir! oneptaliation o- the proble* se! apattern reognition an! repliation t0pe o- pre#proportional reasoning. The st!entsnotie! a pattern o- ite*s 'hose prie inreases 'ere (appro4i*atel0) a si*pleinteger ratio (i.e., appro4i*atel0 a -ator o- ;). /o the0 gesse! that To*Ds salar0shol! be appro4i*atel0 t'o ti*es Are!Ds ol! salar0. This insight sho'e! real

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pro*ise: ho'e2er, the ne' larit0 o- thoght that it intro!e! enable! thest!ents to notie that, -or e4a*ple, so*e ite*s ha! atall0 !erease! in priee2en thogh *ost o- the* ha! inrease!.

Conceptuali$ation (#The st!entsD -orth oneptaliation se! a tre*ltipliati2e proportion, bt it 'as still base! on onl0 a biase! sbset o-in-or*ation: that is, the st!ents began spea>ing abot e proble* sol2ing a entral part o- the shool*athe*atis rril* is the ontribtions that sh e4perienes *a>e tohil!renDs n!erstan!ings o- other entral onepts, *an0 o- 'hih in trn arerelate! to an! in2ol2e proportional reasoning.

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PROPORTIONAL REASONING AND RATIONAL N!M"ER CONCEPTS

In earlier setions, 'e sggeste! that researh on proportional reasoning shol!pa0 greater attention to tas>s in2ol2ing (1) !0na*i trans-or*ations, (;) *ore thana single representation s0ste*, (") *easre*ent nits as 'ell as n*bers, an! (=)*ore than a single t0pe o- rational e4pression (rate, ratio, 3otient, -ration). ?nthe other han!, 'e belie2e that the ter* en onsi!erabl0 !i--erentpoints o- 2ie' abot the *ost essential harateristis that !istingish rates an!ratios.

$ergna! (198", this 2ol*e) -ollo's a tra!ition establishe! b0 the earl0 Uree>srates are !e-ine! as ha2ing to !o 'ith 3antities in t'o !i--erent *easre spaes(e.g., "C *iles7F hors), 'hereas ratios are !e-ine! as ha2ing to !o 'ith 3antities'ithin a single *easre spae (e.g., 1F oo>ies71C oo>ies). The Uree>s -a2ore!this perspeti2e bease the0 'ere partilarl0 intereste! in 'a0s that earlier'hole#n*ber#base! *easre*ent onepts -lo'e! into the !o*ain o- rationaln*bers an! rational e4pressions. $ergna!, being a !e2elop*ental ps0hologist,no !obt -a2ors this perspeti2e -or si*ilar reasons.

apt, L>e, Pohols>0, an! /a0er (198) an! /h'art (198", this 2ol*e) -ollo' atra!ition si*ilar to Uass. This position is re-lete! to!a0 b0 Are!enthal (19E"),Lebes3e (19), hitne0 (198a: 198b) an! other *athe*atiians 'ho ha2ebeen less onerne! abot reoniling rational n*ber onepts 'ith lo'er or!er'hole#n*ber#base! onepts an! *ore onerne! abot -in!ing lin>s to higheror!er topis relate! to 2arios t0pes o- -ntions an! *ore o*ple4 *easrespaes. Theirs is base! on a *athe*atis o- 3antit0 as ontraste! 'ith the *ore

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o**onl0 se! *athe*atis o- n*ber. apt an! /h'art begin 'ith a!istintion bet'een t'o basi t0pes o- 3antities

(a) 4tensi2e 3antities inl!e e4a*ples sh as F *iles, F !egrees(te*peratre), F !egrees (angle), or F ser2ings (-oo!). 4tensi2e3antities tell en the troble to pro2i!e rigoros !e-initions 'hih highlight *an0tas> harateristis that are e!ationall0 or ps0hologiall0 signi-iant: this isbease a *athe*atiianDs obGeti2e is sall0 to -os on strtral si*ilaritiesbet'een tas>s rather than on the ps0hologial !istintions bet'een the*. /o -or*an0 tas> harateristis 'hih are ps0hologiall0 signi-iant, there !o not e4istorrespon!ing


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are also apples. %ll apples an! oranges are X-ill in the blan>Y. P%TDs librar0 is then*o!i-ie! to inl!e the ne' in-or*ation gaine! -ro* the st!entDs response.

hen P%T is in


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ho'e2er, is *ore o*-ortable 'ith the interpretation o- rates being restrite! tonit rates. (;) Mathe*atiall0, a 3antit0 sh as


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Aigre F.Integer n*ber line labele! sing nits o-

*iles#per#hor.

(retrn)

Aigre .Measre spae -or*e! b0 the ross#pro!t

P4B.

(retrn)

Aigre E.Mapping -ro* t'o *easre spaes P an! B in to a thir! *easre

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spae /.

(retrn)

%t an e2en higher le2el o- abstration, all -or basi t0pes o- rational e4pressions

(-rations, rates, ratios, an! 3otients) an be represente! sing a single*athe*atial *o!el, a ho*ogeneos spae onsisting o- "4" *atries. ?bGets,relations, operations, an! trans-or*ations an all be represente! as *atries 'ithinthis single 2etor spae. Aro* an intiti2e point o- 2ie', this is not srprisingbease, -or e4a*ple, e2en thogh -rations an! rates both re-er to single3antities, both also i*pliitl0 in2ol2e o*parisons bet'een t'o 3antities. Aor-rations (e.g., "7= pia), the o*parison is oneale! 'ithin the


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(")The stan!ar! rles -or a!!ition sall0 !o not *a>e sense -or ratios(i.e., or!ere! pairs o- 3antities). Aor e4a*ple, 'hat 'ol! it *ean to


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'ith translations 'ithin an! a*ong 2arios *easre*ent s0ste*s an!representation s0ste*s.

%lthogh !e2elop*ental ps0hologists ha2e ten!e! to spea> o- proportionalreasoning as thogh it 'ere a global abilit0 relate! to so*e general ogniti2estrtre, *athe*atis e!ation researh learl0 sho's that the e2oltion o-proportional reasoning is haraterie! b0 a gra!al inrease in loal o*petene.It is or perspeti2e that researh haraterie! b0 loal oneptal !e2elop*ent

'ill pro2i!e ritial insights abot the *ehanis*s b0 'hih hil!ren e2ol2e tohigher or!er -or*s o- reasoning. Proportional reasoning see*s to be an espeiall0pro!ti2e researh area -or st!0ing this pheno*enon.

REFERENCES

Behr. M., Lesh, R., Post, T, & /il2er. . (198"). Rational n*ber onepts. In R. Lesh& M. Lan!a (!s.),*c0uisition of mathematics concepts and processes (pp. 91#1;). +e' or> %a!e*i Press.

Behr, M., ahs*th. 1., Post, T, & Lesh, R. (198=). ?r!er an! e3i2alene o-rational n*bers % linial teahing e4peri*ent.1ournal for Research inMathematics ducation, 1F, ";"#"=1.

Be>, +. ( 198). 2ariables affecting seventh grade student3s performance andsolution strategies on proportional reasoning ,ord problems.npblishe! !otoral!issertation, ni2ersit0 o- Minnesota.

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%+?L6UM+T/

This researh 'as spporte! in part b0 the +ational /iene Aon!ation n!er grant+o. 6P#8=Et51EE. %n0 opinions, -in!ings an! onlsions e4presse! are those o-

the athors an! !o not neessaril0 re-let the 2ie's o- the +ational /ieneAon!ation.(top)
http://www.cehd.umn.edu/ci/rationalnumberproject/88_8.html#tophttp://www.cehd.umn.edu/ci/rationalnumberproject/88_8.html#top

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8 Proportional Reasoning

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