bab 6 - structure and insight problem solving

8/6/2019 Bab 6 - Structure and Insight Problem Solving

1/26

Structure and InsightProblem S lv ng

.n

L ng e r e C do s Hol Con nc n i u t d t t hin m: e a l t c rein , fulw y u ; Y

ho i E wa eve op ng a t e r h al p in edt h punders d ng s r f or b l s lv a hi king nl

the !R sy l s s. Al gd ng N lf f n ul im u h s B w llere on e ed th an ful e

n ; o Am ri nsy lo sl h ere t ll e1y n"1f f f mp ss ons. Th d ke kno le n ! t co p ex an nb nd f bll and nd and ngbut h ments re ained f ed u nebnne s f w .h tou ht, ll now ed e on t d l o: f

E ro n he920s e al ych lo y se f h f nd t ydff eap gci ing ex imn l ath he l ni

ld not ly x l s l theor nOuen ed f A nl o year f erwa nd y Is re iving en d ntn M y

o bl l n

h te e nt i !nt odue n


2/26

G alP i pl sa a129. )fa gcstn up m h thetea i r b-/ \ v:1

GtST L PRINCIPLES AND SOM M HEM IEXA PLES' 1i t ltth .y ou of E ro d ion f psyc ologyat ccehc om n g !1r o ( v u ' f m xpu k b sourc of y G

y - b y KO , K fa d 'Nhe er- wr d dy the hum i d (a l i co ing se sat o a

r rd n! j r 7ln princ . h t, r th r thn cy kin n for t on of un r t ndin s T y

t s i tcn v ry y fe hu u d u r h xi t n fty t m n z g pr nc e C nt al to t eiry wthe a g oJ w s r I wh c y b o r t oof yd : n u b yev a ed

r t1 c t o k d w z o p u pr .c5SC 1 :t . hu t \ ul b

: tl'f m r ly Um at n f l i u n h.T pr nt d ;, l r t r du ct f A

p y ' t i m c el d th t the er e v r rou ht so unitU t x ri of prciving, thin dIh

m th n ju t th u o its const tu nt stimul re wasnatural ten ency fort r i to's e stru ture n h s orh r er e tions There a pr uretowh f rm or ge t lts" in the environ e an this was affectedby

cifi ru f c i - )So e of th g t lt notions re exe i e\n Fig 6 Notice the co f gur t o of ots (a) see Lu hins&Luch ns19 pic lreactio to thp tte )for s a ia on ha . Bu if o e adds t o ore dots aweh v on in (b , th patt f r a tr ng e w th the topmost dot functio ing at o\ if on ra t t ree b t o an s two a vein c

re ult i r c an e in whi h ou pin o nte ott e nterI(d)i r iv : erely e her l to not er ual patte the hexa on Tpl x d r how th id ntity an f n tion of a prceptu. i55 t, ch ithIt 5u oundin t tO yh d i r yd di g on i r la it o her dotsrou i Tnt t c : inw yh ng r

ur h r r: do ot think of wh t e see n ig6 as er ya cof do. A g t lt th r . Urp r pJiof the dotsi i ebyur kI th m M nh; w.r g izM i


3/26

130 6. STRUCTUREAND INSIGHT IN PROBLEM SOLVING

/ 1 - ) 1 v ~J( 1t/ J .(vJ, 1. ,, _/ . ~ ~it ~ , / . j , f l 'J!lt,,,.

, \ I

0 0 0 0

Ia) (b) (c) (d)

FIG.. 6.1 Dominance or perer::--tual or,anlt.alion. A dot Uhc: empty circle:) 15ui.dcnctd di""ertntly ,ecordin' In chc soundin1 eoniUL

#

. ; ~ / . }

triangle, squnrc. and hexngon. Agnin, the perceiver d o c ~not merely r c g i ~ t e rindividual d o t ~ :he or she brings to the experience an orgnni1.ing princirle thatmake!l the whole perception add up to more th:1n the !lum or i t ~ p n r t ~ .

Ge:o;talt p!lychologist!lcited numerous demou!ltratlon!l or t h i ~phenomenon ineveryday experience. Mu!ilc wa!l a good exnmplc, b r : : n u ~ cmclnc.fy w n ~cnmposed of many Individual notes, but when those Individual n o t e ~were t r n n l ' p o ~ e dton new key, lhcy rctnincclthclr chnrncter n ~ n mclncly. It w n ~their i n t c r r r . l a t i o n ~ ,or the pattern or the note!l, that the li!ltencrperceived and nnt the i n t l i v i t l u n l t o n c ~ .The !ltrwturc contributed by the !1UhjccllveellJ'Ierlcnccnr I ~ C t i . ' C p t l o nw n ~furlhlfdemon:o;trated inthe phenomenon of applHtnt motion (Wertheimer, 1923, t r n n ~ lated and reprinted in 1938). A movie film w u mAde Uf" nf thou!lnnd!l or !'itillframes, but when these were pre!lented In r a . ~ tsuccession. one perceived the !lumor t h ~ Individual frames as a moving picture. As a f i n : ~ lexample, notice theshapes in Fig. 6.2. These are broken or Incomplete s h a p e ~ .but, rnthcr thnnseeing them as a collection of connected lines and curves, we perceive them aswhole forms with ga.ps and additional piece!. Perception tends to :o;c'ck " c l o ~ u r c "in such figures. The tens1on created by the v l ~ u a lincongruity is re!lolved into theperception of a unified whole.

Although concerned Initially with perceptual phenomena ~ u c hn ~ we hnvedemonstrated, the attention of gestult psychC'Ioglstseventually f O < : u ~ e don a more

V general problem, the nature of thinking and problem solving. The gcstnlti!il$came to believe that thinking and perceptual proceues were governed by thesame ba!lic principles. They sugge!lted that the way paltcm!l in vilmnl nnd nuditory arrays were registered by the perceiver might be very much like the waythoughts were organized by the thinker. In other words, the ps)chological field(the "inner !ipace in which cognition o ~ e u r r c d )might be !iubjcct tn the !'nmc


4/26

Gutalt Principles and Some Mathematical Examples 131

tendency In~ c c

tstrucl\trc. lf so. the,., thinking too would be affcctctl hy context.1111d illc1mgruitic11 u m u n ~ldc.:n!l would liC.:ck "cquilibriu111" in pun.: stntctmulfnrll\!1 (llll 11111\)llJ! nr c l n ! ~ U r cin perception).

lll.l'iNirl 111111 ',,,.,/tll'ttl St/'111'/111'1', One f111.'\lll 11f ) l C ~ I n l l~ 1 1 1 d i ~ "111 fll'\lhklllsolving wns the :phenomenon of insight. The role of i n s i ~ h tin problem solvingViliS lttttlldllccd .to tlu.: g c . ~ l l l l l b t l \lurgcly thr11111::h the: e f f m l ~11f \ V I l l l ' ~ t l l \ ~l(llhkr(I IJ25). who had worked clo!icly with Wcnhcimer on the cnrly e x p e r i m e n t ~inperception. Ki\hlcr had o c c : ~ s i o nto observe closely the behavior of n captivecolony of chinip.anz.ccs over a number of years. He especially noted their cffomtu. ~ o l v c

cvcryd;ly prohlcms, suchall

trying '


5/26

1 32 6. STRUCTURE A"'O INSIGHT IN PROBLEM SOLVING

seemingly random activity. To cite nnother example, an o ~ wns given tw o sti


6/26

Gestalt Principles and Some Mathematical Examples 133

shown them h11w 111 l'Onl>truct u line (drop n pcrpcndlculnr) from the upperlclt-hund ~ u i i u ! rllliCh 111111It funned u 'JO-degree ungle with the buscL l'hcy werethen to mcnl>urc the new line and multiply it by the length of the bnse to find theunswcr (sec Pig. 6.3. top). Using this standard algorithm, the children wereSIICl'CMllrully l'lltiiJUitlng the ltrCIIll u r IIUII\Y f'rnctlcc rlgurc:l' when Wenhclmcr~ ; t ~ p p c dtu thl' front nf the room and posed 11disturbing problem. He sh\>wed themthe 1 1 u r u l l c l o ~ r u mtlcplc:tellnl lhc buthllll ur Fig. 6.3 (un "upcndl!d" vcrslon orthe nne nt l.hl 'top of the fi[lurc), and 1 1 . ~ k c c ithem In find illi nrcn.

The childrcti.'s rcnclions were mixed. Some declared "no fair," saying theyhud not bcdn t'aught how to do that kind of problem; others said their altitude! imcs-busc ntgi,rithm liimply would nol W N k on that kind of figure; still othersliimply gave up and refused to consider the problem at all. The difliculty was thatwhen lhc diildrcn dropped a perpendicular from the top left-hand comer, as they

, I

S t ~ t u l ~ r . d,Jigurithm

: t . 7~ . L J . . l T__..... I

b"e

r ' , .,

Wtrthtimttf'.l,. ~ J ~ r l l t l o g f l ~probl1m

FIG; 6.3 Findin& the area or a rarattcloaram usin& the standard alcorithm.Children were c o n r u ~ e dwhen rrlyinJ il to' Wcnhcimcr's rroblcm n,urt.(Adartcd rrom Wenhcimcr. 1 9 4 ~ / 1 9 5 9 . 1


7/26

1 34 6. STRUCTURE AND INSIGHT IN J'ROBLEM SOLVING

hnd been taught, the line ended up somewhere to the left or the bn!le, ! i ( ) thnl thestandard fommln didnot seem to apply (!tee Fig. 6.;\., bottom).

Let us analyze the parallelogram problem (rom the point of view nf mnthemati-cal structure. What do the children actually do when they drop n perpendicular.m ~ a s u r c . :it, 1111tlrnultlrly by t h ~h:tsc'! n , ~ ynrply n gcncrnlulgorilhlll for culc.:ululiul!the nren nf n foursidcd f i ~ t u r e ;thnt is, m u l t l r l y l n ~ tbnsc times nhitudc i ~ n shm1n11method for dividing the figure inton number of uniform ~ q u u r c snnc.J ~ o u n t i n gthemto determine the number or "square" incheli. Understanding the nature of t h i ~algorithm. one should have no trouble figuring out how to apply it to squnrcs orother rectangles. But the algorithm's r c l a t i o n ~ h i pto the parallelogram i$ not ~ oobvious, at least not from the standpoint of Its perceptunl Impact. The humantendency hi perceive thing!! 11!1 nrgnnit.c:dWh


8/26

.,

: .... ---Gestalt Principles and. SomeMathematlcal Examples 135

children been given correct rules tv follow, their strict adherence to an algorithm~ l i g h thave yicided performances indistir.guishable from those of children who

. I

4 n ~ e r s t n o d t h ( u n d e r l y i n gmathematic:ll structure.Further. in the foregoing example it Is not clear that the gestalt tendency to

perceive things as mgnnited wholes necessarily facilitated problem solving. Infuct, we saw that the perceptual wholeness of the parallelogram might interfere\Vith n o t i d n ~i ~ egap-extra relationship that transformed the figure into a rec-tnngle io whic" the standard fonnula for area applied. In this and other gestaltd e m o n ~ t r a t i o n ~ ~the perceptual structures and the true structures of the problemsC!ften seem to ~ i efor the mind's attention. ':'his presents a persistent difficulty ininterpreting g c ~ t a l ttheory with respect to problem solving. Nevertheless, Wertheimer finds. 'i\inplc c v i ~ c n c ef


9/26

13 6 6. STRUCTURE AND INSIGHT IN PROBLEM SOLVING

The carpenter l a u r . n ~ . \Vhy don't you think? Must you count them out. one hynne?''

Oenr reader. if Yt'U were the apprentice. wh:ll would ynu tlu'!If Y


10/26

Gestalt Principles and Scme Mathematical Examples 137

lctlious cumputation of successive ndtlilior.s, the problem now appears to yield toa simple t w o - s t ~ pmultiplication and dlvision prc.cess-Cinding the number of thesquares in the ;whoh: rc.:tangle and then dividing by 2 to find the number ofo u t s i d l ~pnnl!ls .in the staircase.

How due !I this intuition relate to the problem of summing the numbers from I to100'! lnsteuu or computing I + 2 + 3. etc., as the carpenter's apprentice mighthnve done, Wertheimer suggests we bring the. problem into the visual domain totry to understa.nd its structural properties.

I 2 ) 4

L ~91 98 99 100

J :LJ JI

Note thut t h ~numhcr!l cnn he puircd iu!luch 11 \I!UY thut e111:h pnir !lUIIIllttt the llnmc

numhcr. Nnl4 .nlsu the number of pnirs th;s procedure wuuld yield nnd therelation lll' the.numbcr of pairs to the total number of elements In the series. Fiftypairs, c11ch summing to 101. gives us the answer, SOSO.

Another diagrammatic approach \1. cnhcimer points out also invol vcs pairinghut is strulturnlly different from the approach above. It is directly analogous tothe s t r n t q ~ ys\gltcstcd for snlvinlt the !ltnircnsc prnhlcm. We i m n ~ i n cn complcmcntnry series uf idcnticaf numhcrs in l'rder to ~ i m p l i f ythe computation.

I f 3 4 97 98 99 100100 99: 98 97

. j .4 3 2 I

Now we have , ~ v ocomplete series of I to 100. They are arranged in pairs, each of~ h i c hs u m ~lt.l 101. Ortehundred pairs or 101 equal 10,100: then since we beganby doubling t h ~ !series, we divide the product in half to find the desired sum,5050. . . . .

Of course ~ ; cneed not write out all the pairs in the manner shown above. Thepoint of these' demonstrations is simply to capture the problem structure intuitively. Once ~ ' { c\lrlderstand the principle or pairing. and the invariability of thepair sums, we:c:an even invent shortcut formulas to describe the operations wehave perfor,mcd on the series. The results arc two versions or Gauss's formula forthe ! 'Urn or :u\' even-numbered series, either 1


11/26

138 6. STRUCTURE AND INSIGHT IN PROBLEM SOLVING

saw the series problem. only that the lntuitle: behind h:s f c ~ n u l awas nwn:e u ~ i l y g r u ~ r c t lwhen the prohlcm w n ~rccu!Ct in v l ~ t w lnr dinJ!rnmnwtk tc1111s.W e r t h e i m e r ~d e m n n ~ t r n t i o n ~hnve ~ r e n tintuitive " ~ " ~ " c n l .Vel nne q u c ~ t i o n ~whether all or even most mathcmatknl problems cnn he set up ln visuall1.1ihkfu r r ta . Mnthemntlcnl l l t t l l l ' l l l r t ' l11 nu t nlwn)'!l, r':trnllt'lrtl hy !IJl:tlhti ! l t tmtmc. Itmny be no accident, thererore, that mnny t'f Wertheimer's pcd:tr.nsknl exnmpksarc drawn from geometry. whkh deuls with !


12/26

The Process of Productive Thinking 139

prbductivc thinki.ng in Wertheimer's demonstrations In connection with the pnrnllclogrum und ~ o ; a r p c n h : r ' l lupprenticc problems. But we arc left wondering how1 1 u ~ ht h l n k i n ~ tpri,ceell5, b there n 11tep-hy11tcp procc1111we t'Rn follow In the mind

ol' thep r o h l ' - ' 1 1 1 ~

solver'/ If problem structure .is as Important as Wertheimer!eliVoH


13/26

140 6. STRUCTURE AND INSIGHT IN PROBLI:M ~ O L V I N G

of the problem through ~ u c h~ t r a t e g i c ~as lWnl u n a l y ~ i sund problem rcformulation.

In current cognitive r ~ y c h n l o g y ,nr. irnportnnt dllltlnltinn ill drnwn hctwccnprocessing thnt i!l ''bottom-up" and "top-down." ClnriOcntlon nf thlll d i ~ t i n ~ t i o nIICCfl1,. In urtll'f hrc:niiiiC the twn I Y f ' < ' ~ur r r m . t ' ~ ~ i n al l l l i U ~ I " ~ I c l l r r r t ~ n l' ' l ' f l t l l l h ; h r ~to instruction. If processing proceed5 from the bottom up, thut i!i, from !ipc:clficchnracteristlc!i ur the mutcrlub. then cmphn!lls.in tc:tchlng !ihould he placed nnbasic number facts and simple calculntiotis before moving " u p ~ to nbstrnctmathematical concepts. tn contrast, top-down r r o c e s s i n ~ tsuggcm an appronch tomathematics instruction thnt initially emphasizes the logic nnd !itructure of mnthematics and only later insists on the details of computational algorithms nnd othersuch components of specific problems. tn practice, it is often difficult to sortoutbottom-up from top-down processes; in matllematlcal performnh.ccs the processes appear to interact c:onstantly. Nevertheless. the di!itinction is interestingpedagogically because it s u g g e . ~ t salternative instructionnl foci.

Duncker on Problem Solving

Among gestalt psychologists, Karl D11ncker ( 1945) mo!it explicitly pur!;ucd thedi!ilinctions between bnttom-up nnd tnp-dnwn p r n c c : s s i n ~nn


14/26

T h ~Proc:c:;a of Produc.lve 7hlnklng 141

He noted that a problem could be solved "from above" by refonnulating theproblem 11u thut u particular .clan of solutions would be sought. This type of~ o l u l i u ndt'l"l\'r\tll'dun whnt Ounckcr cnlled an "nnnlyld5of the CMnicl, .. lhnt i ~ .ligu;in!! uut.whnt wu!l wrong, what needed to be changed. "Analvsls of f"nls,"~ , ! , I l lhe r"'"i ,ur 1 1 u h ~ l l u urrum u b u v ~ .l n v u l v t ~rocudull un whllt the pt'Oblcmrcully dcmondi:d. so as to overcome the nonnal tendency to become fixed onpunkulur 1 1 1 1 ~ : ~uf solution u u e m p t : ~ .Solutions could ulso come "from below"through notking features of the task and allowing those features to suggestpossible sulutiuns. This type of solution depended upon an "analysis of materials," that is. noting what was present and what could be used.

Dunckcr's .hypotheses arc llliwratcd in "talking aloud1 ' protocols of adultstrying In find ~ nanswer to the question, "Why are all six-place numbers of theform 267,267.:591.591; 1 1 2 ~ 1 1 2divisible by 137" The protocol of one indi-viduaf was as follows:

'! .

I. Arc t h ~triplet$ t ~ ~ e m s e l v e s~ r h u p sdivisible by 13?2. b _ t h c r ~p c r h : ~ p ssome soti of rule here about the sumo( the digits, as there is

with djvisibility by 97J. The t ~ i n gmust roll'ow rrom a hidden common principle:of structure-the

fir:r;i lrtplet is 10 times the 5econd, 591,59lls 591 multiplied by 11, no: byli>l. (c: So7) No: by 1001. Is 1001 divisible by 137 (Total duration 14m i n u t e ~ )[Duncker, 1945, p. 31J.. .

.,

According tu.Dunckcr the third solution strategy, looking for a hidden commonprinciple, ~ r u ~ 1 1nut of nn unuly11111orthe ttoal. Whnt the subject has to discover isthe {tl'ncrn_l.pri.hciplethat allnumbers of the rorin.abcabc have the number 1001os n f111.:tur .ltKH bcl.ngdivisible by 13. The rcfonnulation of the problem by thisAuhjcc:t t ' r t ' i l t < - ~n Jtnnl uf findlna n hidden cnmmon principle, 11cttlng up "connlct" that s c r ~ e l llo direct ind motivate activity to find that common principle.Thllt CUll he cliuructcriY.t:tl llll an lllh!IUJll At Aolullun rrom ubovc. Dut we see lhlllthe clue thot leads to solution is actually a suggestion from below: The factorI00 I is d i s c o ~ ~ r e dby noticing something about the nature of the six-digit numbers, namely, 'the consistent relationship of the first triplet to the second. The

divisibility of .iOOI by 13 is the only remaining discovery needed for problemsolution, and this is quickly verifJed.Jn an experiment devised to test theefficacy of suggestions from "above I ' and

"below,". Duncker gave different groups of subjects different hints while theyworked on t h ~"13" problem. The h i n t ~that markedly improved the likelihoodorsolution (producing n SO% sollllion r:1te) were: "The numbers nre divisible by1001" and "1001 is divisible by 13." Both arc suggestions from below in thesense that they set the subject on the track of the number 100 I

1what it is divisible

by, and what is divisible by it. A suggestion to analyze the goal ("Look for amore fundamentalcharacter from which the divisibility by 13 becomesevident'')did not help and neither did general statements about ~ h eproperties of division


15/26

142 6. STRUCTURE AND INSIGHT IN PROBLEM SOLVING

(c .g., "If a common divisor of numbers is divisible by 13. then they arc nildivisible by 13"). Although bringing attention to the number 1001 was whatworked In t h i ~problem, Dunlkcr found thnt the number need nnt hnvc hccnmentioned explicitly. hnd the experi111en1 hcen "el up In such o wny as to fndlitalci t \ tlilll.'ovcry. When the prnhlc111 w a ~p r c ~ c r t l c l ltilling M l l t ' t c s ~ i v c n t l l l l h c r . ~nscxnmplcs-"\Vhy 11re s i x - d i ~ d tnumbers of the form 276.276; 277 .277; 27R.27Rulways divisible! by 1 3 ' ! " - r n o ~ ts u b j e c t ~suhtrnctcll s u c ~ . c s s l v cnumbers fromeach other (11gain, an attempt to find the hidden underlyin(!. principle). t h u ~arriving at the number 1001.

Although Duncker expressed a preference for solutions from above bnscd onanalysis of goals ar:d analysis of connlct-hc called such solutionr. ' ' o r b . ~ . - : : n ~< ~ p p o s e dto "mcchnnlc:ul"-thc del nih uf hb ~ t u d i c sfall to Clltuhlish the ~ : u p c r i o r ity of one strategy over the other. Jt seems more tueful to acknowledge theeffects of both goal analysis and analysis of materials and to appreciute theircomplex interactions. lnJJ.apter 8, we examine these same processes from theperspective of modem cognitive psychology. We d e ~ c r i b eexperiments spcc:ifically designed to tease out the differential effects or task materials and generalsolution strategies. and we relate these effects to modch or human informationprocessing.

A Gestalt View of Learning

The principles set forth by Wertheimer and Duncker. among other g e ~ t n l t i ~ t ~ .~ h c dlight on some complex tt!lpects uf human t h i n k i n J t ~ l h corgnnii".ntion ofperception and problem solving. But although both were i n t e r c . ~ t e din fmteringelegant thinking. and ulthuugh their demom;trntlun!C nppcored rdevant to ttitching. their work did not extend very far into the renlm nf instruction. Kntnnn( 1940/1967) c n ~ lthe gcstah concern w i t ~structure nnd menning Into n moretraditional experimental fonn by S)'lltemnticafly comparing gestult prindplc!l nndtraditional learning theory a ~ they explain how people learn to r ~ r f ' o r mt n . < k . ~ .Like Wertheimer, Kntona l : ~ b e l e dv : ~ r i o u stypes oflenrning either "senseless" or"meaningful." By 'cnllelcss lean-.ing, he mennt role memori1.ntion. Mcantngfullrarnirje, N leamir g by Jnderst:!nding, was b : ~ s e dupon organi1.ing a set ofs t r u c t u r : ~ l l yrelated I d e a ~or components.

To test the differential effects o( t h e ~ etwo types of lcnmlng, Kntona dc!iigncda series of studies to be conducted in several st3$C!i, The {!.enerul idea w n ~ to lookfortaskll that could be taught by two or more methods b n ~ c don different thcorctien! urientntion!;. Ench method brought about lcnmlng In the short run us mensured by pretests and p o s t t ~ s t s .First, experimental suhjccts were p r e t e ~ t ~ < . lto besure they were not already competent at the task 10 be t ~ t u g h LDifferent groups ofexperimental subjects were then given ehher instnlction based on rotc mcmorl7.ntion or instruction that stressed the prin:iples underlying the task!'. In the instructionnl conditions thnt s t r ~ s s e dmeaning. some g r o u p ~hnd tu detect the p r i n c i p l e ~t h c m ~ e l v c snnd some groups were given explicit d c ~ c r i p t i o n sof the prinl'iplcs.


16/26

' '

Tho Procou of Productlvo Thinking 143

At the end of lhe period of instruction, all subjects could usually do the task,because the various teaching procedures were carefully chosen and applied toensure learning, The crucial comparison of teaming types came at a subsequent

stage. At this comparison stage, Katona tested sutjects to see how well theycould to the task a month Inter (retention), He also examined how learning onetMk contrihuted'to executing other t a ~ k ~that shared some characteristics with theone taught (transfer).

In one experiment, for example, Katona (1940/1967) asked people to learn alengthy scric:; of numbers, such as I 4 9 1 6 2 5 3 6 4 9 6 4. Three differentgroups were g i ~ e nthree different sets of instructions a.s f o l l o ~ s :

Oruup I, 'Recite these numbers 5lowly three times, for example, "one hundred'furtynine or une hundred sixty-two. , , . "

< rnup ~ . l h : t ~ l l t h l 1 1llluwly l l" yuu may lr.nuw II cumrlctcly and rrc:clacly: "The; ~ m ~ ~national p r o d u ~ tof the \Jnitc:d States last year w a ~

.\ 14,1J 16,l.Sl,MIJ.64."Omup :\. Try to team the following series (i.e., no special instructions were

given ihis group, just tht. printed list or numbcn).

Group 3 subjects pondered the series briefly and then appeared to notice, or" d i ~ c o v ~ . : r . "a ~ e r t n i npattern in the numners: I 4? 16 25 36 49 64. The other twogroups followed the specific instructions biven.

Directly following the c ~ m p l e t i o nof this given task, all groups could recitethe list of digits virtually without err\lr. But during the comparison stage, dif

ferences emerged among the groups. Asked a week later whether they stillremembered the series, Group I suid the question was " u n f a i r " ~Group 2 remembered n partial answer, such as "the GNP was about Sl4 billion"; butGroup 3 was _qble to r e m ~ m b e rthe list perfectly and could in fact extend theseries cven'further (e.g., 81 100 121. ), Thus. Katona reasoned, there was aqualitative ~ l i.well as :tuantltative difference between the kinds of learning engaged in by the three groups.

Notice t.hat ~ o r n etype of grouping was induced by each of the three forms ofinstruction. Tht kind used by Oroup I was clearly irrelevant and even interferedwith noticing t ~ epattern or the series. Oroup l 's partial answer wu a sensible

response to the: memory question, given the context established by the originalinstructlun11. tn:ract, thl:; may be int.:rprctcd as a structure-based response; olbeita response .to structure of a different kind than that or a number series. Oroup 3,the " m c n n i n g f ~ l "lenming group, WM the mo:>t ~ u c c e s s f u lin the memory tt!.llk.Having d i s . G o . y ~ r e dthe principle underlying the series, or In some cases havingbeen tuld thc principle, they grouped the numbers along structural lines, consistent with the u ~ d e r l y l n gmathematical organization of the series. This group alsodem,mstratcd t'tansfer in their ability to extend the series.

Katona's experiments were an attempt to -prove that learning did not conststmerely of memorizing a set or associations or a procedure. Learning could also

mcnn r < ; o r g n n i ~ : i n ginfonnotion so nli to fonn a structure that had the power to


17/26

144 6. STRUCTURE AN D INSIGHT IN PROBLEM SOLVING

explain other similarly structured problems. In Katona's view. this accounted forthe trnnl'fcr of knowleege to new siruations. Finding the "problem l'trueturc - theprinciple underlying the problem-not only made It easier to do 5imilar p r o b l e m ~but al!io e n a b l e ~one to rrcmt.rtrttct the 5olution long nfler the initinl expmure tnthe problem tusk. T h i ~w n ~b e c n u ~ ethe rcorgnni1.ution thut lll.'l'lllllpanlctl meaningful learning rrovidcd la5ting p r i n c i p l e ~to guide rceon5tructinn. In mcmori 7-ing lists of digits, for exomplc, J ) C r ~ o n swho le:mctlthe princ.:lplcs for gcncr:1tin!!the sedes were able to reconstruct and extend them indennitcly. The innuence ofgestalt thinking is clear in Katona's cxpl3nation!i for such_phenomcna: Fncts thatnrc organized into a structured whole are retained as part of thnt whole. enchbeing remembered because or its plnce within thnt structure.

Unllcd Cln n ~ e r i c ~ur lllmllor e x p e r i m e n t ~using c:nrd tricks lind mah:hst i dpwhlcms,.KIItonn (1940/1967) rcoc:hcd scvcrnl c n n d u ~ l n n ~rel!nrclinJt the natureof meaningful learning:

h ) Lcarninr by memorizing is different l"roccss from learning hy u n t l c r ~ t n n t J i n g :(b ) learning by understanding involves substantially the ~ : ~ m ep w c e s ~: 1 ~ d u e ~prohlcm solving-the discovery o ( 1 principle: (c) both prol>lcm stllving anti mcnnin3fullcarning consist l"rimarily in chang'nt:. or or!!anizing. the matcrinl. Thl roleof organization is to establish. discover. or understand an Intrinsic r c l a t i n n ~ h i pIpp.5J.S4J,

Note that Katona. in his time. w:ts reacting a g a i n ~ !n very doctr'nnirc S- :-'-theory .. wcr Wcr1heirTcr nnd the oth:-r g ~ r . t a l tp ~ y c h o l o g i ~ t ~ .h c n c ~ :hi!i condcmnatiiH1

of lenrning by memorizing, which he apparently equnted with r(ltc teaming. T h i ~argument has softened with time. and most psychologists today recognize that Hsharp dichotomy between rote and meaningful teaming is not warranted, thntmemorl;dng I ~ not necessarily an altrrnativr to undcrlitnnding. I ndecd. rurrc ntwork on memory m a k e ~it clear th'nt mcmori1.ing i ~ an active p n K c ~ ~ .dependenton organizing principles very much like those Katonr p r o p o l i c ~ .Research showsthat a tendency to organize information as a means to more efficient rncrnoryincreases with age, and this Increase C'.>rrelates with better performnnce on memory tasks (Kreutzer, Leonard, & Flavell, 1975). In t!tls connection, it is interesting to note that people In Oroup 3 or Katona 11 number ~ e r i c ~tnlik nctunlly hnd

less to remember than t h ~other groups, b e c a u ~ cthey on Jy had t


18/26

The Procllss of Productive Thinking 145

interest to note the parallels between Katona's conclusions and those arising fromthe extensive rc!lcarch on "discovery learning." Discovery has often been proposed as the b e ~ tway to teach new concepts in mathematics and other subject: ~ r c a s .

The strategy is to make available to children all the relevant mat


19/26

146 6. STnUCTURE AND INSIGHT IN PROBLEM SOLVING

being encouraged to notice the underlying mathcmntlcnl muctures thnt mndeprobler.u meaningful. In 5ome nf K a t o n a ~c x ~ r i m e n t sthe mcnningfullcnmingRriHIJl.dld not hnvc to (li\Covcr lhc \ l n d c : r l y i n ~ trrinclrlc: nr ~ t n r < . t u r c :fur tht"lllselves. l n ~ t e a dthey were shown the principle di:-ectly nnu then given a charH.'l' to~ t u t l yIt urtd upply it 111 new prohlcrll'l, Thcllc: lltutlcnlll !lid not pcrforru rttlll'ltdifferently from ~ t u d c n t swho discovered the principle for t h e r n s e l v c ~ .

A ~ t u d yby G a g n ~and Brown' 19fll 1 nlso sugg.ests thnt the content rather thnnthe meth::1d of instruction m:\y account for the apparent superiority of discoveryteaching. In this study teeMge boys were taught to obtain formulns for the ~ U i l 1 ! 'of specific number series. A "rule-and-example" group was ~ h o w nthe formulasand .then led through a number of examples In which they identified terms aridpracticed finding the numerical values of the series using the formuln. "Discovery" and "guided-discovery" groups, on the other hand, were nsked to ckrivcthe formula for the sum of each series and were given increasingly. explicit hintsto help them achieve th.: solution. Both the rule-and-example nnd the guideddiscovery programs proceeded in small. carefully liequenced steps bn5ed ongeneral principles of programmed learning. In the test 5ituations. the g r o u p : ~werepresented with n number serie5 that they had not seen before nnd were r c q u i r ~ dtoderive the formula for its sum. The guided-discovery group wali most s u c c c ~ l i f u lin the novel problem-solving tasi


20/26

Implications of Gestalt Thinking for Instruction 147

that highlight tlu!ir vnrious interrelntcd components and promote insight into theirunderlying ~ i r u ~ t u r c l l .Although Wertheimer c.Jid not SI'\CII lc out In so manywnttl!l, w(' tn) !lnr


21/26

1118 6. STRUCTURE AND INSIGHT IN PIWBLEM SOLVING

find


22/26

Implications of Gestalt Thinking for Instruction 14 9

d i ~ ~ c o v e r yof the: underlying structures, in the gestalt sense, of the problems to beMi:vcd. Jtoly:e pruvldcll a ~ c tof llpcdfic qucstlunll ur steps to follow in working onn :rrnhlem. c n l ~of which could be charn.:tcrlzed all lruristic, a word of Greeko ~ l g i nm c u n i n ~ . u ; , . ; , , ~to d / ~ < ' O \ ' c r .The value of heuristics Is tunt they aliow nwuulcl-hr p n h l ~ h~ o l v c rIn rtunt"d ~ y ~ t c m n t l c n l l ytnwnrcf imijlht, inMcnll 11flc'aving prudu"tive thinking" to chance or to the gifted few who nre quick to secpwhlc111 ~ t r u , : t u r con their ,,wn .

. Accmding l\l Polya (1945/1957). problem solving may be divided into fourstages. nnmcly, understanding the problem, devising a plnn for finding the solutiun, earryintt tlUt the plan, and looking back tJ verify the procedure and checkthe rcllult. These four lltngc-s, with detailed questions t.nd hints relevant to each.nrc listed in' fig. 6.5. Let us examine them in the context of a specific problem:

A stunt nwtorcyclist plans to r i d ~his tike the length of a tightrope stretched from. the up;>er' ldt-hand ctrner or the back or an auditorium to the lower right-hand

c.'umer ur thc'front or the auditorium. The d i m e n s i o n ~or the audilorium arc 100 by60 hy 30 fc:cL He nccd5 to know how long a t i ~ h t r o p ehe should bring to span thedistance he plans to ride. How would you help him find out7

Now, inwttinc 11 lib.th grader. Drian. trying to follow Polya 's steps. ll'har i.r tire1111J.:nm"? ':'The l e n ~ t ho ( the tightrope." IVhat arr th data? "A room I00 by60 feet nml .10 ,rcct h i g ~ . "IVIrat is tlw emu/don.' Here Brian would probablywant In dr:ew n: figure and label the knowns and unknowns nppropritucly, ns inl ~ i ~ .h A ' : ' l ' h l ~c:onditl


23/26

Flrt.Yu u fte,.. tu ' ' " t l , t t ~ t r l

' " ' l 'oblrm.

S..cond.Find tnt connection bet wren

lh t !1111 an d lh t lll'lknown.Vo v may be ob 1iged

to con,lc1tr aailiary problemsif tn lmmel'i tt e connection

tannot be lovnd,You should obuin eventually

1 p/ 1 , of the solution.

Thld ,C ~ r r you t your plan.

Fourth.

E x 1 m i n ~tht , , lutlon obtained.

li N ) f. nS T 1\NOING THE PROn l.EM~ I t ir 111e unnowrt) ~ I t " ' the d1r1.> H-111 /r t i t ~conrlitlonlh II rnHihl" In urhly th r.nntlltlnn1 11 th r.nntlilitln ' " " " ' ' " " 'to dflttmine tne unkl'lown? Or h II l n i U I I i c i ~ n t ?Or r ~ d u n d ~ n t '

Or contradictoty1OttW figure. Introduce rultthlft rHHtlinn. s ~ , , , , , ~ Y ~ t l n v l" ' ' " of the condition, CM you write them down I

DEVISING A PLAN

H 1 v ~vn u l ~ ' e nit ho!lott7 Ot l u v ~vo u , , . ~ n'" " ' ~ ' " "ntnlol""' " 'a stiqt\tly different lorm1

Do yo v now a " ' ' t td prohf11n) Do vn u ktlow a tk-.nrem thAtcould b ~ uwluP

L )I)#; at rh t unknown/ And try to think of a llmiliar prohlemhaving the um e or a slmil11 unknown.Htrt Is 1 problem r t l l t td to youn and solved b ~ l o r t .C o : ~ l dvouuse It'! Could yo u utt Its result1 Could vo u ute Its method'Should yo u Introduce some a u ~ e i l l a r yelemrnt in order 10 1\'ukeiu utt pouibl t 1

Could yo u restate the problerr.? ..:ould you restate it stilld i f f e r e n t l y ~Go back to definitions.

If yo u cannot solve the proposed problem try to solve lir11 somerelated problem, Could vo u Imagine 1 more c e t ~ ' i h l erelatedproblem? A more oeneral problem? A more specltl problem1 Antntl('?ovs prcblem? Could \'l'\1 solve 1 part of the problfm? Keeponly 1 p u t ol t h ~condition, drop the othftr pArt: ho w fat Is th,.unknown then determined, ho w can It vary? Coul:f yo u derivetome thing uttlul frt>m tl., data? C o u l ~yo u think of othtr, dltftapproprlart to determine th t vnl:nooA:\? Could 'you change th eunknown or th t da1a, or both If nec:P.sury, se that tke new mknown and tilt new dt l l ar t nearer to tach other? Old yo u u \ ~all tilt data? Did yo u utt th whole condition? Htve yo u tAkenInto Iecount all eutntlal n o t l ~ n tln\'ulvfltl In thll prcltlf!m1

CAFIAVING OUT THE PLM'f ~ r r y l n S '..,,,, voJr plan of the solution, chtck tlch l t l !p. Cn vo utee clearly tkat ihe ttep it c o t r e e t ~Can vo u prove thai itis correct?

LOOKING BACK

Can yo u chk th l rtlult? Can yo u check th e argvment1Can you derive the rttuft differently? Can yo u tee It af a glance?Can you use the result, or the method, for some other problem?

FIG. 6.5 Polya's s t a ~ e sof rrohlern !tnlvln,. (From Polya, 1 9 ~ 7 .Cnryri,ht 1 9 ~ 7by rnnceton Unlveuity Pn:H. Rerrlnte\1 by (lem11sslon.)

15,)


24/26

Implications of Gestalt Thinking for Instruction 1 1

., ,.

I

Tightrope tLength of auditorium IWidth of auditorium wHeight ol auditorium h

cI 100w GOh 30t 7

FIG. 6.6. 'Labeling unknowns and drawing a figure facilitates visualizing a solu-

tion procel.!urc ror the tigl.t:opc problem.

so!v.e the p r ~ b l e m .Then, provided he chooses appropriate algorithms or procedures, all that temains is for him to compute and check the resu Its.

Notice h


25/26

152 6. STRUCTURE AND INSIGHT INPROBLEM SOLVING

i n t u i t i o n ~tell u ~ the v i ~ u a land lipntinl n ~ p e c t l iof l'imple m n t h e m n t i c ~: ~ r eimportant.

With relipcctto the p r o c e ~ ~of human problem ~ ; o l v i n g ,Polya ( 1962) hns s:dd:

Solving prohlrnH ,, lhr ~ ! ' < ' : I r kn\hlrvrmrnl ur intrllip,rrwr, nncllrllrlllr,rtH'f' h lhr5flecific gift of man. The ability to gn round nn o b ~ t : u : l c ,to undcrt:tke nn indifl'l'tl ' o u r ~ ewhtrc 1111 direct l'nurse prr.,cnt5 l t ~ c l f .r n l s c ~the clever nnlmnlnhovc the dullone. raises mnn far above the most clever a n i m a l ~ .and men of talent above theirfellow men fVol. I, p. IIRI.

'The anility tel go round nn o b ~ t n c l e ,to undertnke nn indirc


26/26

References 153

The procrss of achieving insight, although largely unexplored by gestaltsyr.hologists. was analyzed in part by Duncker. Duncker drew a distinctionctwccn processing from above, Starting with anaJy1iiS of goals and problemcformulation, and processing from below, starting with the analysis of problemnaterials. The ~ n r , n edistinction is made in current anr.lyses of cognitive perfornance, although in practice the interaction between ,the two types of processes,ather than their differences, may be more amenable tQ :;tudy.

Attempting to;develop a gestalt theory of lea:niog, Katona suggested thatneaningfullcarning. like problemsolving, depended on a presentation that eithernadc explicit or. allowed the Ieamer to discover the underlying mathematical;tructure. If the principles underlying the content of learning and problem solving.vcre u n d c r ~ t u u d ,' ~ o l u t i o n scould be r t ~ c o n m u c t e d ,extended, nnd remembered.:n contrast, rote learning appeared to limit the Ieamer's ability to remember and~ c n c r u l l z eocw lcarnins. These contentions were similar to those found in the

:nore recent litcra'rureon discovery learning. A review of that literature suggested:hat discovery is i1ot a unitary concept and is hard to cxpHcate experimentally.Studies purportini to lest the efficacy of the method confounded discovery withthe opportunity fu npprehend underlying $lructures of material to be learned.

The g e : ; t ~ ! tview of problemsolving is thilt insight grows outof an understanding of t h ~ p r o h l ~ ' i nus u whok nnu pf the rclntion of th\! p a r t ~lo the whole.lnnuencr.d by. gestalt theory, Polya has developed hints that encourage the problem solver h>: rcct>nsider the g o a l ~ ;of the problem, search memory (or similarproblems solved, before. and analyze the materials or givens of the p r o b l e ~ .These hints may, be helpful in promoting the problem reformulation and goal

analysis that appear to facilitate the emergence of insight.Gestalt psychqlogists have givenus intuitively appealing demonstrations ofthe organization of thinking and perception, and they foreshadowed many of theconcerns of t ~ d a y ' scognitive psychologists. Of themselves, however, the demonstrations arc not clear with respect to many of the processes we hope toinfluence t h r Q u g l \ ~ i n s t r u c t i o n .The nature of mental representations and the processes by whkh1problems wre formulated and solution strategies chosen areissues that rci.1uh'(; clarification. The:;e topics arc receiving further treatment incurrent a n n l y ~ c s~ t 'problem-solving behavior, and we return to them in Chap-ter 8. ;

.~ .

/ ~

I ' '! REFERENCES:

1 1 ~ 1 1 .1\. T. Mr11 11/mrrtlirmnlitj, Ntw Yurk: ~ i n 1 1 1 nnnd ~ k h u ~ l t r .tY17.Cmnhach, L. J. i'hc ioglc or cJ.pcrimcnuon di5covcry. In L. S. Shulman & E. R. Keislar (Ells.).

/.rllnt/11/l, /1.\' '"",;trr,l': A l'fitlntlllf'l'"'''"' C : h k n ~ u :Rnnll M ~ N a l l y ,IYMi.Duncker. K. On rrohlem-5olvlng.Psydwloxiral MIIIIIIRTfl{'hJ, 194S. JR(270). 1-tl2.( i l ! J ( h ~I R. M ~ llrliwn. L. T. Sumc r n ~ : t n r 5in the flrtlj!rnmmlnllor Cflnt'CfiiUnllcllminj:. l f l l t r l l l l / "/

l : ' . r p t r i ~ r , t a lf's.\;JtoloN)', 1961 ; 62 (4). 3 D 321.

bab 6 - structure and insight problem solving

Documents