Conviction and explanation in the context of geometry ,,r

s.At.TydskrOpvo€dk., 1992,12(4)

ftivarc Sccto Educario! Council ("RISEC).1992. post sch@t

&d No I of

Education Paper

lne Fiucadonal Renewal SraLc$/. Posjiion 1992. Joh&ndsburg: PRISEC.

Scoq L.B. & B@kn&n, J.L.lg9a. Medaveranteaa .likheidvn oa.lavJsbesluiheninE . Ialeiding tot aspeLte die ,adeertrtehel. Joh.nncsbug: lix PaFia. '@ Suid Afrila (Republlek) (RS A). ]9AI. Wer op tegni*e katte res , ,r. 10.1vdz /98/. PrcLoria: Siaalsdrukker. Vdenigins van Tegniese Kolleges (YTK). I990a. Die dadslelling van 'n eie onddwysfokus vi reglicse kollega

wat onds Wet 104 vm 1981 Eso.red, mct die gevolglike eic poslalaerhoudilgsrorm. Ongepubliscqde menorddnn. Prcroria: Komit@ vm Te8nicse KolleSchoofde Vere. nding van TeBniac KolleS* (VTK,. l9q0b. /.8ri',.

ko eges ti/ bercepson tetuts die rcgte k6e. Inli8r '8.ba.rurc Prclorr: Kom,Le vo tegnic*

vd N. 1984. OFningsiedc oD die e6le jaNergaderins van die Vq.niginS va Tegniqe Kolleges. 27

Viljoen, G.

Julie 1984.

Conviction and explanation in the context of geometry M.D. de Villiers F@ulty ol Educadon, Univcsity of Durbd-Wesrville, privdb Bag X540Ot, Durbd, 4ilOO Rcpubtic of South AAi@ Accepted

Jul,

1992

Quaharivc cmpni.al rcscarch i.djcared lha! atdougb rhc najdiry of pupih' needs tor pqsonal onviclion werc satisfied b], qu$i emlirical means, lhey nevcnhcl.$ exhibncd an tndependenr nced for explanation which semed b b. only sadslicd by some son ot logico deducrive esunenr. putits'n@ds in rhis resp@r d.l fteir fulfitnent de ampdcd with lhose ol nalhem.ricians. A cae is made rhar rhe erptdaLry funclion of p.oof is in many cascs nor only lorentially ndc medlngtul ro pupits, thar fie leriticatlon tuncrion, but atso inbll@lualy nore honesr Kwaharicwe empnic.c lavo6ing hcL ddo! gedui dar hoewel dic neeste leertinse se behocftes an lqsoolike sekerheid op 'n kw6i cnpiriese wysc bcvredig is, hulte nogtans ,n onairanHike b€hocfte ae vciklding vcrooD wa!

slynbad slcss deur die ccn of dder losicsneduktiewe rsunenr bcvrcdig kd word. Lerlinge se behoefbs in die vcrbdd en hullc bevredigins word vdgelyk ner di€ van wkkundigcs. ,n Saak word uirgemaak ilar die vqHdings_ furksie vo bewvsvosing ln baic scvalte nie ner porcnsieel ncd betekerisvol s die verifiiiringsfuntsie vir leerlingc is nie, mad ook intellekru@l ccniker.

lntroduction The problcm rhar pupils have wilh perceiving a need for prmf is well'lmow. ro al1 hlgh-sch@l te&hers and is idenrificd q.fiu-t ct.eoriun in all a-.arondt r6edch 6 a n.-oFoblcm in the lcaching ol prool Who has nor ye! expcricnced frus'Iation when confronred by pupils askins

,,r)

R%ddin8 ftc fiBt quesdon, Bell (19?6:24) dsi8ned rhe rollowinS

rhre

funcrions to a

prool

L

Verification (usrificarion or convicrion); conccm€d wjrh lhe trurh of a proposition. 2. Illuminarion (explanadoD)i conveying isisht inlo why a proposition is true.

do we have

3. Systematjzatio{ lhe dganizarion of rcsuhs inro a d€ducrivc A@ording ro Afanasjcwa in Frcudenrhat (1958:29) pupits' problems wnh proof should nor sinply be arFjbuted to sLo* cognitivc developmcnt (e.9. aD inabtliry ro rson logicatty), bur al.o 1,1.r rhey r.! ro' .cr'tfe /r/r;,r,neding. p-?oce md uselulncss) of p.oot In facr, several srudies bavc shoqr that very youn8 childrcn de quilc capabte of toSical reasoning

also fDlfils a fuction of discovery dd,/or corrnunicalion (De Villien, 1990a&b). Li lhis .nicte, howevq, an jnvesligatio ot pupils' cognilivc needs with resp@r ro conviction ard cxplana_ tion wiftin fie @ntexr of geomelry de reporred.

anempts by resedchers ro reach lo8ic ro pupits have frequcmly

In

jn

si ations which arc real and rudnir&tut ro thm (Wa ine ron, 1974; Hewson. 1977i Donatdson, 1979). Furrhemore,

trovidcd no sralisrically siSnificanr differerces in pupits' pcrtom.\e and rtFe.iar:or ^t pruuf (Deer. tabqi Vueter, 1975). Morc rhan ariylhing else,

jl

seens rhar the fu.damenlal

issue at hand is he aplropriale /lcarridrion of fic various fmcdons of proofm pupils. This raiscs lwo qucsrions which eere invesrigalcd (De Vit-

rysrcm ol drioms. majo' con.eprs dd Lheorem. Besides the above funcdo6. proof for a rnaftemadcian ofren

Pupils' cognitive need tor convtc on s dies by Sni$ (198?) and De Vi ie$ & Njismc (1987) ir

was found that only for 1qo of rhe 2 000 pupils involved, proof

fte functioo of convictiory'justificano.To sain qualitarive infomatlon aboul pupils'n@ds for @ftai y dd how lhcy hem.eivc\ trolld (hoJsc ro obkin persunat conviclion, a teaching expdimdr was c@ducted dd a sdies of inrwiews had

held wilh Srd

6 ro

10 pupils durinS 1987

liers,1990a): 1. Whar /ua.lirrr does proof have wjrhin marhemarics irself wtuch cd potendally be ulilircd in the marhcmatics ctassr@m to make pr@f a morc meaningfut acriviry?

purtosc pupils were placed

2. Whal coSnidvc needr do pupils sponraneously exhibir, or cd be crealcd, so ihar lhc fucrionalily of pr@f can be il

reJriag. This finding was

luslrated in the fulfilmcnr of lhosc needs?

dd

1988. For this

in new or

relatively unkrcM problem situatioN. Tbe following findinS was madq rtu mJbtitt of pApils spaitaEo6l! satisfied theb need Jor penonal

co^victiot

it suh situtio6 bt reMt

1. Thjrry our

of

oJ qu6i enpiical

b6ed on fic fotlowing resulbj 32 Std 7 pupils in the reaching expeim

spontaneously indicared

fiat firnher

r

quasi empiricat testing

465

s.Afi.J.Educ., 1992,144)

(i-e cmsruclion

and measuremenr) would satisfy

fteir n@d

lor "earrnry wirh rc'Dql lo lhe lollowrnS 8@mcrric (onJq rure: 'If lhe midpoints of lhe adjacent sides of a quadnlate ral ABCD de conscculively connected, llen a parallele

srm EFGH is fomcd.' TFical pupil rcsponses were (all respomcs are frely raNlated from Afdkaalls): 'Draw a million md tesr' 'Issuc a law that the whole populalion should draw least two quadrilaterals to lst the rcsll!' 'l€l a class of schml children each &aw a vdiery

quadiilatcrals and

Ie$ lhe result.

If fiqe

a1

of

are NO

excepdons. lt wiil always bc possible.'

of l1 Std 6 to Sld 10 pupils intewiewed, also sponhneously oblained cerlajnty wirh resp@t Io the above

2. Eigh! ou!

cojecure by means of costrrclion ard metusuemen! of a nubcr of differdt quadrilarerals (five of $ese pupils 'vcre from SId 8 ro 1O)- Ody lhrcc pupils $erefore cxplicirly chose deductlon as a melhod

of verification.

3. In another inteNic*, thr€ ou ol seven Std I and 10 pupils obtaincd conainty exclusively by means of consrtucdon ud mcasurement wilh resper ro lhcir own (visuauy fomulatd) conjccNres $aI lhe adjacent usles and diagonals of a sivcn iss@les trapczim were equal. For exmple, cosidcr lbe followiDg exrac! lrom fte irlewjew with Lam (Sld 10): l: 'Are you lm% ccrtain ftat the angles and diaSonals are equal?'

Li 'l

abou! lhc angles. but I'm not so sure abou! lhe diagmals ... pdhals reasoably ccnain

I:

rhinh

l'm cefiain

'How will you make dead sue whefter thcy

arc

rcally equal?'

L: 'By drawing$em and ftedsurins.' These lhre pupils also did nor display m addiLiorol need for thc deduclivo vcrificarion of these resuhs, as lllustrated by thc following cxlract from lhe in!fliew wilh Llnette (Srd 10) who had convinccd hcrscll by consmcdon od

I: 'lf I would

now Bive you a pr@f ... would that make you more cerlain that ir is always Irue ... or de you ar the moment sufficie.lly satisfied hat n is always

L: (with

I:

emphasis as indlcated)

'l

am presenlly satisficd,

since I obwcd ir uyre$ md ncasucd i! n)i"f. I am feeling salisficd bc.ausc I did it uretl. 'You hd!e no nccn rhar I srve )o- a poof ro cunvrn..

4.

Thc other student was also of thc opinion lhat a mathemali ciar would not, like hinself, nake a coNtruclion ro test a definjtior (or conjecture), brt ody.use lo8ical deducdm ('he would sit here wnb m 4 a.d a d piwe it'). In moLhcr inle.view. all five the Sld ? lo Std 9 pupils chos coNlructior dd measurem4t to evaluate md oblain cer lainty abour fte validity of the following rwo €onjectDies: 'iscribcd ansles in a emi circlc dc 9f'; and 'ansles isqibed on the smc chord dc cgul.' In the above cxullcs, pupils did nor always use accuale constredon ud mcasuomenl bul somedmes orny rough drawings which lncy thcn simply evaluated visually (c.9. the first cxample). The above resulls woc also nol isolated

obseralios, bur has regularly been obsewed in vario$ orhd sltualions wilh diflerenl g@netric conjectues. Of course. rhe finding based on lhee results is not ncw and is confimed by cnpincal research on tE Vm Hiel€ 'hcory (Usiskin,1982: Burgd & Shaughnssy. 1986). Pupils' cognltlve need lor €xplanation and its fullilmenl In rhis regdd fte followinS rsults were oblained: 1. Despite thcir cefiainry wjth respect 1o the 6nj@lure lhal a paralleloSrm will always be fonned by conleclin8 the mldpolnts of fie sides of any quadnl eral, all the Sld 7 pupils in the aforcmcntioncd lcaching expcimenr exhibiled an independent need for dpldndrion, which was illu.slrated by rcsponding lositiveiy to the question: 'You have now all convinccd yourself that this conj@tute is irue, buL would you likc to know why iI is true?' The cl6s then foud fte given deductive expldalionquite salisfaclory.

2. All eigh,Srd 6 lo I0 p.prl' who l-ad cho$n Io bc quasi dpirical tcsling !o obtain cmvicli@ wilh rcspect lo the previous conjecture showed an additional ned for explam ti@. For exmple, consider the following represenlalivc cxrract from the intewiew wilh VicB (SId8): I:'Arcyou now dead sure fiar ir will alwaysbe Eue?' V: Yes. now l'm ce ain.'

l'Willa

loglcal proolnake you more cenain?' V: 'Ycs, but... norreally.I'm quite ccnain.' I: 'Doyou rafier havc a n@d to hrow wrr it is lrle?'

I:

'Doyou really wmt to krow wry?'

V: 'Ycs. quile... l'm inquisiriveabou!

il.'

Afrcr providins a proof, fie pupils wcrc lhcn whcrhcr ir sairfied fteir nee.d for cxplanalio. In al

asked cascs.

thc rcaction was posilive, allhough some lupils reqlcstcd

L: 'No, I convinccd nyscll' (soudins vcry fim). Two of the other four pupils qere also asked Io evaluate cenain ahemadve definidons for lhc isos.eles rapezia. In both cascs rhcy chosc quasi{npirical lcsling rarher thd dednctive pi@l For imtancc, Manin (Sld 9) rcaoted as follows when asked how he wonld make sue whclhcr any cyclic quadritateral with equal diagonals was an isoscclcs trapezium oi not: M:'One could draw a circle with lwo chords of cqual ld8rh and inGrsecring, and rher conn*tin8 the cnd points and measuinS re base m8les. md il the base

dglcs dc not cqrlal, lhcn il is nor an iso$eles

trape

!ha! the

lrmf

be repeated again before they mswcred in

fte

armadve. no fdher need lor deductive verinca don. $e thr@ pupils who had used cotrtruction md mca lurendl wilh iespecl to the Sivd isoseles trapezium slill cxhibilcd a need ftr expldalion which had nol b@n

:1. Despite displayinS

satisfied by lheir quasi-empirjcal approach- As before. &en needc lo- e\plddr'on werc $en sari.fre.d by lhc Plc*nrrrior of proofs by the interviewer.

4. Simildly, fie five lupils in ihe inteniew about thc afore' mentioned circlc lheorcms sdll exhibircd a nccd for expla' nation *hich had not be€n sadsfied by their quasi€mpirical

amroachi a n@d which was thcn only satisficd by the

\

!.oducdon

considd

of a

inlerriew wilh

l:

logico-deductive argumenr. For cxample, reprcsentarive exrac! from lhe

ftc followi.! lmarr

(Srd

9)l

Huq would ynD mdkc enai4 ... Or dc you dtready dead sure IhaI ftosc uSles

de

also cquat?'

L: 'No'

l:

'Now how would you

]our"fnake

swe

il you wcnt

d

urgenr need

home this aftcmoon? Suppose you had

to make surc: how would lo! make sure? Whar would yol do? ... Now I wmr your honesr opinion. I do not want whar you nay lhirLk I m cxpetinS of personal opinion.' - I need L: 'I would pcrhaps maLe a largcr drawinS md

you

I:

your

work

more accurably! 'You would not pdhaps rry !o produce a proof for ir

L:'Yes, if I mate largcr drawi.Es, and fic dslcs coDrinue to come oul cqual, rben you could prove rhal

if

you have a chord comected ro rwo poinrs on

rhe

circle. lhcn thoe argles wjll bc e{ual'

L 'Do you leel

tha!

if you havc

accuraleiy and mcasured rhcm,

drawn

enouSh

md sullose

you

would s@ lhat they werc always equal, woukL lhat be sufficientproof ihar ir is always true? Rencmbq I am

you opnrion.' 'Ycs, if I have d@c it quire a number of rimet Do you havc a need ro s@ a lroof for ir?'

asking you

L:

I:

L:'No'

I:

I:

tullilmenl hobably mosr reachers of mathemalics betide !ha! a prcof for thc mathematician provides absolnre cenainry and rhat ir is therefme rhe absolure aurhority in lhe etablishmenr of rhc \dlorry of a conjerru'c. The) qm Lo hold nc rur,c vredescribed by Davis md Hcrsh (1986:65) rhar behi.d each rh@ren in the mathcmati.al heraruc there srands a scqucrce of lo8ical treslomatioN noving from hlpolhesis lo conclu .on. absol-rcly conprencnsole ard incturabty gud.ireeing !ru!h. Howcvc!, this view is false. Pr@f is .ot necessarily a prqcqnisite for conviclion nor does il gudanree L-urh: @ rh. conlrary, covictlon is probably far norc frequendy a prerequi sire lor thc finding of a proof. As polnted our by Beil (1976:24) lhe view rhaL Lhc main funcrim of proof 6 thar of vcrificariorvconvicrion .avoids ronciJ,ldrion ot lhe rel n"rJrc ol prml sin,e Lovicrion in maihemlli.s is oftcn obtained 'by qlire olhcr meDs than rhat of following a losicaL ploof'. Rescarch marhemariciaff for iNtmce seldon scrutinjze rhe published proofs of resulrs in derail, bul arc rarhd led by the ertablished aurhoriry of lhc au$or, thc !cs!in8 of sl@ial cases and u i.format evalualion wherher 'the nerhods md resul! fir in. seen rcasonable' (Davis

& Hersh,

1986:67).

Polya (1954:8.{) also sfesscs thar convictioD ofren lr€edes a pr6l 'Whcn you have sarisfied yourscli that rhe ireorcm is lrxe, you stan provin8 it.'

'BuI do you pdhaps wonder lr,t i! is rruc? Do you perhaps have a ne4d ro k.ow rr) those lwo angles de equal, rather fian just knowing that rhey are

L: 'Ycs,I

Mathemallclans' cog

tyJ(' qro.dt . tooz.;] c, ^[. nltlve need fot convicflon and lls

would vcry much likc to knoe

wr' it is rrue'

'Really?'

ln snuadons like lhe above wherc convjction aclLrally providcs lhc notivarion for a lroof, tbe funcdoD of such a pr@f lor the mathmaricim clearly camor be Lhar of lcrificatiodcovicrion, bur bas b be lookcd for in rernrs of orher functiois ol proof, lot exataplc, dptanatian.\n ta.t. a very high level ol conviction may somerimcs be reached cvcn in lhc

The interviewer lhcn Save a lo8icat ]]rmf ot rhc resrh in letm ofthe lexLcritr mSteol a trianSle.lh@rcm. I: 'Now docs thls argument explain b you why ir is tue? Does it sarisfy your need lor cxplanarion? ... Docs lt saristy you comllcrcly?' L: 'Yes. il satisfies mc ... I fiint ir is sufficienr,fsound ing quirc sarisfied).

Althou8h

Lmarr (S!d 9)

clearly had no ned for a losical p.oof within the conrcxr of verificalion. he ncvcnheless displaycd u indepcnde.t nccd for extlanarion, which was ften satisfied by a loSical proof. On fie basis of rhe pr@ed ing rcsulLs fte followlng gcncral finding was rhereforc made. oll thz pupiLs \9h. ha.C ca eirced hetutete.t bJ qu6i enpiiaL testihg stiLl ethibited a need Jor erptcna tion, which seened rb be satisJie.t b sane soft .[ k)sica ded ctite ugunvnL h shonld, however, also bc point.d our fta! somc of these lupils did indicale Lhar rhe givo loSico deduclive dSumcnt had also funher nrcreascd lhcir coni; dence/certainry, ahhoDgh it had aiready b@n very high duc to their quasi empirical approrch. Furhcr research is also

n€essary on differenr r'?es of argumenLs, lor example, in fomal/fonnal, analytical/g@mctricat^ransformatioral. and how rhcy satislypupil's ne€.ds tor explanadoD.

abscncc of a l)roof. For inslance, in fici! discussion ot the 'heuristic cvidence' in suppmr of the srili uproved rwin

prime pair th@rcm md rhc lamous Ricmmn h)?orhcsis, Davis ud Hersh (1983:369) conclLdc i\ar lhis evtdence is ,so stronA lhal it cadcs convictlon cvcn wihout rigorous proot,

Mathematicians' cognit've need

lor explanation and

its fulfllment Ahhough ir is

possiblc to achjeve quilc a hjgh lcvcl ot in rhe validiry of a conj€iure by mens of quasi empiical verificarion (e.g. accurare conslrucrions and mcduernerr; umoical substitution),rhis genera|y provtdes no satislaclory explmation wr) ir may be rue. I! mqely confirm Lhat ;r is rruc and, evcn rhough the considcnrion oi morc and nore exmples may i.creaso one's confidoce even mo.c, lI gives no psycholoSical sarislaoruy sense of ilumina tion. lhar is. m iLsighr or undersrandinS imo how it is rhe conscquence of orher fdrlllar rcsults. For iLsrance, dcspjte the conv cine hcuistic evidcDce in suplofl of lhc eslier rnenti@cd Ricmann hypothesis. mlhemadci s srjl have a buming nc€d for explmation as sLared by Davis dd Hersh confidenco

(1983:368): A proof \rould bc a way of understdding pr) rhc Rieriam conjccrure is lruc, which is sonethins nore than jusr knowing from convnrcing hcurisric rcasoning rhat ir F true.

S At.J.Dduc.. 1992.12(4)

Similarly Gale (1990:4) also reently emphasized wirh rcfe' r@ce to Fci8cnbaM's cxpcrimental discoveris jn fraclal geometrf ftat the function of lheir eventual proofs was ftat of e\plrnaLiondd nor Lhar o'vs,ficat;onar alr. Thcrcforc. in mosl cascs when lhc resulls concmcd are mruirivcly 5clt cvidcfl dJo! s-ppodcd by ,ovincins qua\i-

empirical or heujsdc evidence, fte function of piof for mathematicim is certainly noL that of vcrification, bul rather $.r of expldduon. l F not a que.rion ol rd-krn8 surc. bur raihd a question ol 'explainlng why Furthcnnoie, f