1987 Metcalfe & Wiebe 1987 Intuition in Insight and Noninsight Problem Solving

7/30/2019 1987 Metcalfe & Wiebe 1987 Intuition in Insight and Noninsight Problem Solving

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Memory & Cogni tion1987, 15(3), 238-246Intuition in insight and noninsightproblem solving

JANET METCALFEIndiana University, Bloomington, Indianaan d

DAVID WIEBEUniversity of British C olumbia, Vancouver, British C olumbia, CanadaPeoples metacognitions, both before and during problem solving, may be of importance inmotivating and guiding problem-solving behavior. These metacognitions could also be diagnos-tic for distinguishing among different classes of problems, each perhaps controlled by different

cognitive processes. In the present experiments, intuitions on classic insight problems were com-pared with those on noninsight and algebra problems. The findings were as follows: (1) subjectivefeeling of knowing predicted performance on algebra problems but not on insight problems;(2) subjects expectations of performance greatly exceeded their actual performance, especiallyon insight problems; (3) normative predictions provided a better estimate of individual perfor-mance than did subjects own predictions, especially on the insight problems; and, most impor-tantly, (4) the patterns-of-warmth ratings, which reflect subjects feelings of approaching solu-tion, differed for insight and noninsight problems. Algebra problems and noninsight problemsshowed a more incremental pattern over the course of solving than did insight problems. In general,then, the data indicated that noninsight problems were open to accurate predictions of perfor-mance, whereas insight problems were opaque to such predictions. Also, the phenomenology ofinsight-problem solution was characterized by a sudden, unforeseen flash of illumination. Wepropose that the difference in phenomenology accompanying insight and noninsight problem solv-ing, as empirically demonstrated here, be used to define insight.

The rew arding quality of the ex perience of insight maybe one reason why scientists and artists alike are willingto spend long periods of t ime thinking about unsolvedproblems. Indeed, creative individuals often actively seekout w eaknesses in theoretical structures, areas of un-resolved conflict, and flaws in conceptual system s. Thistolerance and even questing for problem s carries the riskthat a particular problem m ay have no solution or that theinvestigator may be u nable to uncover it . For instance,i t was thought for many y ears that Eucl ids fi fth postu-late m ight be derivable even though no one w as able toderive it (see H ofstadter, 1980). The p ayoff for success,how ever, is the often noted "discovery" expe rience forthe individual and (perhaps) new know ledge structures forthe culture. This special mode of discovery may bequalitatively different from more routine analyticalthinking.Bergson (1902) differentiated between an intuitive modeof inquiry and an analytical mode . Many other theo ristshave similarly emph asized the importance of a me thodof direct appe rception, variously called restructuring, in-

This research was supported by Natural Sciences and EngineeringResearch Council of Canada Grant A-0505 to the first author. Thanksgo to Judith Goldberg and Leslie K iss for experime ntal assistance. Thanksalso go to W . J. Jacobs. Please send correspondence about this art icleto Janet Metcalfe, Department of Psychology, Indiana Umversgy,Bloomington, IN 47405.

tuition, illumination, or insight (Adams, 1979; Bruner,1966; Davidson & Sternberg, 1984; Dominowski, 1981;Duncker, 1945; Ellen, 1982; Gardner, 1978; Koestler,1977; Levine, 1986; Maier, 1931; Mayer, 1983; Polya,1957; Sternberg, 1986; Sternberg & Davidson, 1982;Wallas, 1926). Polanyi 0958) noted:

We m ay describe the obs tac le to be overcome in solv-ing a problem as a "logical gap," and speak of thewidth of the logical gap as the measure of the in-genuity required for solving the problem. "Illumi-nation" is then the leap by which the logical gap iscrossed. I t is the plunge by w hich w e gain a footholdat another shore of reality. (p. 123)

Steruberg (1985) said that "significant and exceptionalintellectual accomplishment--for example, major scien-tific discoveries, new and important inventions, and newand significant understandings of major literary, philo-sophical, and similar work--almost always involve [sic]major intellectual insights" (p. 282). Arieti (1976) statedthat "the experience of aesthetic insight--that is, of creat-ing an aesthetic unity--is a strong emotional ex-perience .... The artist feels almost as if he had touchedthe universal" (p. 186). Although these major insightsare of crucial importance both to the person and to theculture, their unpredictable and subjective nature presentsdifficulties for rigorous investigation, Sternberg and

Copyright 1987 Psychonom ic Society, Inc. 238


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INT UIT IO N IN PRO BL EM SO LV ING 239Davidson (1982) sugg ested that solving small insight puz-zles may serve as a model for scientific insight. We shalladopt this approach in the present paper.Despite the impo rtance attributed to the process of in-sight, there is little em pirical evidence for it. In fact, Weis-berg and Alba (1981a) claimed correctly that there w asno evidence whatsoever (see also We isberg & Alba,1981b, 1982 ). Since that time, tw o studies investigatingthe m etacognitions that precede and accompany insightproblem solving have provided some data favoring theconstruct. In the first study (M etcalfe, 1986a), feeling-of-knowing pertbrmance was com pared on classical in-sight problems and on general information memory ques-tions (Nelson & Narens, 1980). In the problem -solvingphase of the study, subjects were given insight problem sto rank order in terms of the likelihoo d of solution. Onthe m em ory half of the study, trivia questions that sub-jects could not answer immediately (e.g., "What is thename of the villainous people w ho lived underground inH. G. Wellss book The Time Machine?") were orderedin terms of the likelihood of remembering the answerson the second test. The m emo ry part of the study wasmuch like previous feeling-of-knowing experiments onmem ory (e.g . , Gruneberg & Monks, 1974; Hart , 1967:Lovelace, 1984: Nelson, Leonesio, Landwehr, & Narens,1986; Nelson, Leonesio, Shimamura, Landwehr, &Narens, 1982; Schacter, 1983). Metcalfe found that thecorrelation betw een predicted solution and actual solutionwas not different from zero for the insight problem s,although this correlation, as in other research, was sub -stantial for the mem ory qu estions. Metcalfe interpretedthese data as indicating that insightful solutions could notbe predicted in advance, which would be expected if in-sight problems were solved by a sudden "flash of illumi-nation." However, the data m ay have resul ted from adifference between problem solving in general andm em ory retrieval, rather than a difference betwee n in-sight and noninsight problem-solving processes.In a second study (Metcalfe, 1986b), subjects were in-structed to provide estimates of how close they w ere tothe solutions to problems every 10 sec during the problem -solving interval. These estimates are c alled feeling-of-warm th (Simon, New ell, & Shaw, 1979) ratings. If theproblems were solved by w hat subjectively is a sudden flashof insight, one would expect that the warmth ratings wouldbe fairly low and constant until solution, at which pointthey wo uld jump to a high value. This is what was foundin the experiment. On 78% of the problem s and anagramsfor wh ich subjects provided the correct solut ion, theprogress estimates increased by no more than 1 point, ona 10-point scale, over the entire solution interval. On thoseproblems for which the w rong answer was given, however,the warmth protocols showed a more incremental pattern.Thus, it did not appear to be the case that there w ere nocircumstances at all under which an incremental patternwould appear. It appeared with incorrect solutions.However, in that study, the incremental pattern may havebeen attributable to a special decision-making strategy,

rather than to an increm ental problem-solving proce ss.Thus, wh ether noninsight problems show a warmth pat-tern different from that of insight problem s is still unclear.A straightforward comparison of the warmth ratingsproduced during solution of insight and noninsightproblems is, therefore, important and has not been at-tempted previously. Simon (1977, 1979) provided severalmo dels that apply to incremental problems such as al-gebra, chess, and logic problems. Basically, Simon et al.(1979) propo sed that people are able to use a directed-search strategy in problem solving (as opposed to an ex-haustive search through all possibili t ies, which in m anycases would be impossible) because they are able to com-pare their present state with the goal state. If a move makesthe present state m ore like a goal state (i.e., if the persongets "warm er"), that move is taken. Simon et al. (1979)provided several think aloud proto cols that sugg est thatthis "funct ional" or "me ans-end" analysis of reducingdifferences can be applied to a wide range of analyticalproblems. They noted that the Logic Theorist (a computerprogram that uses this heuristic) "can almost certainlytransfer without modification to problem solving intrigonometry, algebra, and probably other subjects suchas geom etry and chess" (p. 157). If this mo nitoring pro-cess is guiding human problem solving, then the warmthratings sh ould increase to reflect subjects increasing near-ness to the solutions. Of cou rse, if insight problem s aresolved by some nonanalytical, sudden process, as previ-ous research suggests, we w ould expect to find a differ-ence, depending on problem type, in the warm th pro-tocols.The experim ents described below explored the metacog-nitions exhibited by subjects on insight and on noninsightproblems. Experiment 1 co mp ared warmth ratings dur-ing the solution of insight problems w ith those producedduring the solution of noninsight problems. The nonin-sight problem s were the type that have been analyzed andmodeled by programs that use a functional analysis. Thus,we expected to find that subjects warmth ratings wo uldincreme nt gradually over the course of the prob lem -solving interval. We expected that warmth ratings on theinsight problems would, in contrast, increase rapidly onlywhen the solution was given. Experiment 2 u sed algebraproblems rather than the m ul t is tep problems that havebeen m odeled with search-style programs. Algebra prob-lems m ay be m ore characteristic of the sorts of problemspeop le solve daily than are (at least some of) the mu ltistepproblems used in Experiment 1. As noted above, how-ever, because the m eans-end search strategy shou ld beapplicable to algebra problems, increm ental warmth pro-tocols were expected. For these reasons, as well as theiravailability, algebra problem s we re wo rth investigating.In addition to examining warmth ratings during the courseof problem solving, Experiment 2 also investigated otherpredictors of perform ance: subjects feeling-of-knowingrankings, normative predictors of performance, and sub-jective estim ations of the likelihood of success. We ex -pected that noninsight problems w ould show m ore incre-


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240 METCALFE AND WIEBEmental warmth protoco ls than wo uld insight problems.We also expected that people w ould have more accurateme tacognitions (about how we ll they w ould be able tosolve problems and which problems they would be ableto solve) for the noninsight than for the insight problem s.

E X P E R IM E N T 1MethodSubjects. Tw enty-six volunteers were paid $4 for a 1-h sessionof problem solving. Seven of these subjects either produced no cor-rect answers on one of the insight or the noninsight problems orproduced correct answers immediately, so that no warmth protocolscould be ob tained for the solution interval. Thus, 19 subjectsproduced usable data.M a t e ri a ls . T e n problems, p r o v i d e d o n 3 5 i n . cards, were givenin random order to the subjects for solution one at a time. Half oft h e s e prob lems w ere nonins igh t p rob lems, and half were insightproblems. The noninsight problems were designated as such be-cause past literature had labeled them as multistep problems or be-c a u s e t h e y h a d b e e n a n a l y z e d b y i n c re m e n t a l o r s e a rc h m o d e l s s u c has those of K arat (1982) or Simon (1977, 1979 ). The noninsightproblems are reproduced in Appendix A . T h e insight problems werec h o s e n b e c a u s e they had been considered to be insight problemsby other authors or by the sources from which they were taken.However, we felt free to eliminate problems that in our previousexperiments (Metcalfe, 1986a, 1986b) h a d b e e n designated by s u b -jects as "grind-out-the-solution" problems rather than ins ightproblems. Our criterion for calling a problem an insight problemwas not well defined. This lack of definition may well be one rea-s o n t h a t r e s e a r c h on insight has p r o g r e s s e d so s lowly. W e s h a l l re t u r nto this point m our conclusion. The insight problems we u sed arereproduced in Appendix B .P r o c e d u r e . T h e subjects were told that they would be asked tosolve a number of problems, one at a time. Once they h a d t h e a n -swer, they were to write it down so that the experimenter couldascertain whether it wa s right or wrong. If the experimenter hadany doubt about the correctness of the answer, shc asked the sub-ject for c larification before proceeding to the next problem. Dur-ing the course of solving, the subjects were asked to provide warmthratings to indicate their perceived nearness to the solution. Theseratings were m arked by the subject with a slash on a 3-cm visualanalogue scale on which the far left end was "cold," t h e f a r rightend was "hot," and intermediate degrees of warmth were to beindicated by slashes in the middle range. Altogether, there were40 lines that could be slashed for each problem (to allow for them a x im u m a m o u n t o f ti m e t h at a s u b j e c t w a s p e r m i t te d t o w o r k o na given problem); these lines were arranged vertically on an an-s w e r s h e e t . T h e subjects were t o l d t o put their first rating at thefar left end of the visual analogue scale. They then w orked theirw a y d o w n t h e s h e e t marking warmth r a t i n g s a t 1 5 - s e c intervals,which were indicated by a click given by the experimenter. Be-cause it requires less attention on the part of the subject, is non-symbolic, and apparently is less distracting and intrusive, this visual-analogue-scale technique for assessing w a r m t h is superior to theMetcalfe (1986b) technique of writing down numerals.ResultsThe probabili ty level of p _< .05 was chosen for s ig-nificance. The increments in the w armth ratings were as-sessed in two w ays. First, the angle subtended from thefirst rating to the last rating before the rating given w ith

the answer in a particular protocol for a particular problemwas m easured. (We refer to this henceforth as the "an-gular warm th"). Second, the difference between the firstslash or rating and the last rating before the rating givenwith the answer w as measured. (We refer to this as the"differential warm th"). Th ese two m ethods can yielddifferent results because angular warmth , unlike differen-tial warmth, varies according to the total time spent solv-ing the problem. For exam ple, consider two p rotocols ,both of w hich start at the far left end of the scale and bothof which have the warmth imm ediately before the answergiven as a slash in the exact center of the scale. Let thefirst protocol have a total time o f 1 m in and the seco nda total time o f 2 min. Whe n these two are ranked accord-ing to differential warm th, they will be tied, or be c onsi-dered to be equ ally increm ental. Wh en they are rankedaccording to angular warm th, however, the first protocolwill be said to be more incremental than the second. Thus,the differential warmth measure considers the total solvingtime (w hatever it is) to be the unit of analysis, whereas theangular warmth measure gives an indication of the incre-ment in warmth per unit of real time. We could not decidewhich m ethod was more appropriate , so we used both.The correctly solved problems were separately rank or-dered from greatest to least on each of the angular warmthand the differential warmth m easures, for each subject.Then a G oodman and Kruskal gamm a correlat ion (seeNelson, 1984, 1986), com paring the rank orderings ofthe increment in warmth (going from m ost incrementalto least) and problem type was com puted. These gamm aswere treated as summary data scores for each subject. Apositive correlation (which is what was expected) indi-cates that the noninsight problem s tended to have moreincreme ntal warm th protocols than did the insight prob-lems . The overall correlation on the angular warrnth mea-sure was .26, which is significantly different from zero[t(18) = 2 .02, M S e = .32]. The o verall correlation on thedifferential warmth measure was .23, which was also sig-nif icantly different f rom zero [t(18) = 1.63, M S e = .37,by a one-tailed test].Thus, the warmth protocols of the insight problems inExperiment 1 show ed a more sudden achievement of so-lution than did those of the no ninsight problem s. This isprecisely wh at was expected given that the insight prob-lems involved sudden illumination and the noninsightproblem s did not.

E X P E R I M E N T 2MethodSubjects. Seventy-three University of British Co lumbia studentsin introductory psy chology participated in exchange for a smallbonus course credi t . To al low assessment of performance on thefeeling-of-knowing tasks detailed below, it w as necessary that thesubjects correctly solve at least one insight and one algebra problem,and that they miss at least one ~nsight and one algebra problem .Twenty-one subjects failed to get at least one algebra or one in-


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INTUITION IN PROBL EM SOLVING 241sight problem correct, and so were dropped from the analyses. Foursubjects got a l l the algebra problems correct and were dropped.This left 48 sub jects who provided usable feeling-of-knowing data.For warm th-rating data to be usable, i t was necessary that thesubjects get at least one ~nslght and one algebra problem co rrectwith at least three warmth ratings. Thirty-nine subjects providedusable data for this analys~s.

Materials. The materials were classical insight problems (repro-duced in Appendix B) and algebra problems selected from a highschool algebra textbook ( reproduced in Appendix C) . The insightproblems were selected, insofar as possible, to require little cogni-tive work other than the critical insight. Weisberg and Alba (1981 b)argued against the idea of insight because they found that provid-ing the clue or "insight" considered necessary to solve a problemdid not ensure problem so lution. Sternberg and Davidson (1982 )p o i n t e d o u t that this failure of the clue to result ~n immediate problemsolution may have occurred not because there was no process ofinsight, but rather because there were a number of additionalprocesses involved in solving the problem as well as insight. Inan attempt to circumvent such additional processes, we tried to useprob lems tha t were m inimal .Procedure and Design. The subjects were show n a series of in-sight or algebra problems, one at a t ime, randomly ordered w ithininsight or algebra problem-set b lock. I f they knew the answer tothe problem, either from previous experience or by figuring it outimmediately, the problem w as eliminated from the test set. Oncefive unsolved problems (either insight or algebra, depending on orderof presentation condition) had been accum ulated, the experimenterarranged in a circle the 3 5 in. index cards on which the p roblem swere type d and asked the subjects to rearrange them into a line goingf rom the p rob lem they thought they were m ost l ike ly to be ab leto solve in a 4-min interval to that which they were least likely tobe able to solve. This ranking represents the subjects feelings thatthey w il l know (or feel ing-of-knowing) order ing. The f ive cardswere reshuffled, and the subjects were asked to assess the proba-bility that they could solve each problem. The cards were shuffledagain and then presented one at a t ime for solution. Every 15 secduring the course of solving, the subjects w ere told to indicate theirfeeling of warm th ( i .e. , their perceived closeness to solution) byputting a slash through a line that was 3 cm long, as in Experi-ment 1. T he subjects we re not told explicitly to anchor the first slashat the far left of the scale, but they tended to do so. Altogethe r,there were 17 l ines that could be s lashed for each problem . Thesubjects continued through the set of five test problem s until theyhad either written a solution or exhausted the time on each. Thenthe procedure was repeated with the o ther set of problems (ei therins ight or a lgebra). The order of problem set ( insight or a lgebra)was counterbalanced across subjects. The subjects were tested in-dividually in 1 -h sessions.

ResultsWarmth ratings. The gam mas com puted on the angu-lar warmth m easures indicated that the insight problemsshow ed a less incremental s lope than did the algebraproblem s: Mean G = .35, which is significantly greaterthan zero [t(37) = 3.10, M S e = .49] . Gamm as computedon the differential warmth measure showed the same pat-tern: Mean G = .32, which is also significantly greaterthan zero [t(37) = 2.56, M S e = .581.Figure 1 p rovides a graphical representation of the su b-jects warmth values for insight and algebra problems dur-ing the minute before the correct solution was given. Thehistograms in Figure 1 contain data from all subjects whohad ratings in the spec ified intervals. To c onvert the visual

analogue scale to a numerical scale, the 3-cm rating lineswere divided into seven equal regions, and a slash occur-ring anywhere w ithin one of these regions was given theappropriate numerical warm th value. Thus, ratings of 7could occur before a solution was given because the sub-jects could, and did, provide ratings that were almost, butnot quite, at the far right end of th e scale. The trends ofthe distributions, over the last minute o f solving time, go-ing from the bottom to the top panel in Figure 1, tell thesame story as the angular and differential warmth m ea-sures: There w as a gradual increment in warmth with al-gebra problem s but little increasing warmth w ith the in-sight problems.Feeling of knowing on ranks. A G oodman and Kruskalgamm a correlation was comp uted between the rank or-dering given by the subject and the response (correct orincorrect) on each problem, for each of the two sets ofproblem s. Then an analysis of variance was perform edon these scores; the factors w ere order of presentationof prob lem block (ei ther algebra fi rs t or insight fi rs t--between subjects) and problem type (algebra or insight--within subjects). There was a significant difference ingamm a between the algebra problems (mean G = .40)and the insight problems (mean G = .08) [F(1,46) =6.46, MS e = .77]. The correlation on the algebraproblems was greater than zero [t(46) = 4.6, M S e =.36], whereas the correlation for the insight problem s wasnot [t(46) = . 8, M S e = .47]. This latter result replicatesMetcalfe (1986a). Thus, it appe ars that the subjects fairlyaccurately predicted which algebra problems they w ouldbe able to solve later, but were unable to predict whichinsight problem s they w ould solve.Feeling of knowing on probabilities. The problemsin each set were ranked according to the stated probabil-i ty that they w ould be solved, and another gam m a wascomp uted on the data so arranged. Because the co rrela-tion cannot be computed if the identical probabilities aregiven for all problem s (in either set), 4 subjects had tobe elim inated from this analysis, leaving 44 sub jects. Be-cause there cou ld be ties in the prob ability estimates, andbecause there could be some inconsistency between therankings and the estimates, the results are not identicalto those p resented above. As before, the difference be-tween the algebra problem s (mean G = .40) and the in-sight problem s (mean G = . 15) was significant [F(1,42)= 2.8, M S e = .99 one-tailed]. The correlation for thealgebra problem s was significantly greater than zero[t(42) = 3.82, M S e = .48], whereas the co rrelation forthe insight problem s was not [ t (42) = 1.2, M S e = .65].This analysis is consistent with the analysis conducted onthe ranks.Calibration. To co mp are the subjects overall abili ty(Lichtenstein, Fischoff, & Phillips, 1982) to p redict howwell they w ould perform on the insight versus the algebraproblems, the mean value was computed for the five prob-ability estimates (one for each problem ). This mean w ascompared with the actual proportion of problems that eachsubject solved correctly. Both the predicted performance


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242 METCALFE AND WIEBE

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INSIGHT ALGEBRA

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L = 6-~ Ec~i~i~ ~_ _i,_~ ....... lo~T ~ 10 ................. o1 2 3 4 5 6 7 1 2 :3 4 5 6 7WARMTH RATING WARMTH RATING

Figure 1. Frequency histograms of warmth ratings for correctly solved insight and algebraproblems in Experiment 2. The panels, from bottom to top, give the ratings 60, 45, 30, an d15 sec before solution. As sho wn in the top p anel, a 7 rating was always given at the time ofsolution.

and the actual perform ance were be tter on the algebraproblem s than on the insight problem s [F(1,46) = 54.67,M S e = . 11]. Previous research had show n that subjectsoverestimated their ability m ore on insight problem s thanon m emory questions (Metcalfe, 1986a), and we though tthey they w ould perhaps overestimate m ore on the insightproblems than on the algebra problems. Predicted per-

formance on the insight problems w as .59, whereas ac-tual performance w as only .34. Predicted performancewas .73 on the algebra problems, whe reas actual perfor-mance w as .55. The interaction showing that there wasgreater overestimation on the insight than on the algebraproblems was significant [F(1,46) = 3.18, M S e = .08,one-tailed]. This result is fairly w eak. Not only is the in-


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INTUITION IN PROB LEM SOLV ING 243teraction significant only by a one-tailed test, but also,insofar as the actual performance differed between the in-sight and algebra prob lem s, the interaction involvingpredicted performance could be eliminated by changingthe scale. Despite these hedges, the result suggests thatpeople may overestimate their ability more on insight thanon algebra problem s. None of the interactions with orderof set was significant.Pe rsonal versus normative pred ictions. We lookedat normative predictions because it was po ssible that therewas no, or a very diffuse, underlying difficulty structure(or a restricted range in the probability correct) with theinsight problem s, and hence the zero feeling-of-know ingcorrelations could simply reflect that lack of structure,or range. In addition, there is the interesting possibili tythat the normative predictions of problem difficulty arem ore accurate at predicting individual behavior in par-ticular situations than are sub jects self-evaluations. Nel-son et al. (1986) found such an effect with m emo ry re-trieval. If this we re the case in problem solving as w ell,then the experimenter w ould in theory be able to predictbetter than a person him -- or herself whether that personwould solve a p articular problem.The problem s were ranked ordered in terms of thei rdifficulty by com puting across subjects the probability ofsolution for each. Although ideally difficulty sho uld havebeen computed from an independent pool of subjects, thiswas no t cost effective. Thu s, there is a sm all artificialcorrelation induced in this ranking because a subjects ownresults made a 2.1%, rather than a 0% contribution tothe difficulty ranking. To see whether normative rankingwas better than subjective judgment as a predictor of in-dividual problem-solving performance, two gammas werecom pared. The first was based on the normative rankingagainst the individuals performance, and the seco nd wasbased on the subjects own feeling-of-knowing rank or-dering against his or her performance. T he norm ativeprobabilities were a m uch better predictor of subjects in-dividual perform ance than their own feelings of know -ing. The norm ative correlation for the insight problem swas .77; for the algebra problems, it was .60. T hese corre-lations indicate that there was sufficient range in thedifficulty of the problem s (both insight and algebra) thatoverall frequency co rrect was a go od predictor of in-dividual performance. The zero feeling-of-knowing corre-lation, discussed earlier, is therefore probably not attrib-utable to a restricted range of insight-problem difficulty.The interaction between ow n versus normative gamm asas a function of problem type w as significant IF(1,46) =10.13, M S e = 1.13]. Table 1 gives the me ans. The ideathat subjects may have privileged access to idiosyncraticinform at ion that m akes them especially able to predicttheir own performance was overwhelmingly wrong in thisexperiment.

DISCUSSIONThis study shows that there is an empirically demon-strable distinction between problems that people have

Table 1Mean Gamma Correlations Between Personal and NormativePredictions and Actual Performance for Insight andAlgebra Problems in Experiment 2Type of Prediction

Personal Norm ativeInsight .08 .77Algebra .40 .60

though t were insight problem s and those that are gener-ally considered not to require insight, such as algebra ormultistep problems. The above experiments showed thatpeoples subjective metacognitions were predictive of per-formance on the noninsight problem s, but not on the in-sight problem s. In addition, the warmth ratings that peo -ple produced during noninsight problem solving show eda more incremental pattern, in both exp eriments, than didthose problems that were preexperim entally designatedas involving insight. Th ese findings indicate in a straight-forward m anner that insight problems are, at least sub-jectively, solved by a sudden flash of illumination; nonin-sight problems are solved more incrementally.A p ers is tent problem has b locked the s tudy of theprocess of insight: How can we ascertain when w e aredealing with an insight problem? Le t us now propo se asolution. Given that the warmth protocols differentiate be-tween problem s that seem to be insight problems and thosethat do not, we m ay use the warmth protocols themselvesin a diagnostic manner. If we find problem s (or indeedproblem s for particular individuals) that are accom paniedby step-function warmth protocols during the so lution in-terval , we m ay define those problems as being insightproblems for those peop le. Thus, we propose that insightbe defined in terms of the antecedent phenom enology thatm ay be monitored by metacognitive assessm ents by thesubject. Adopting this solution may have interesting(although as yet unexplored) consequences. Perhaps theunderlying processes involved in solving an insightproblem are qualitatively different from those involvedin solving a noninsight problem. It m ay (or m ay not) bethat contextual or structural novelty is essential for in-sight. Perhaps there is a class of problems that provokeinsights for all people. But pe rhaps insight varies with thelevel of skill within a particular problem-solving domain.If so, we m ight be able to use the class of problems thatprovoke insight for an individual to denote the individualsconceptlaal developm ent in the domain in question. Per-haps this person-problem interaction will provide someoptim al difficulty level for m otivating a person and there-fore have pedagogical consequences. Insight problemsm ay be espec ially challenging to people, and their solu-tion distinctly pleasurable. Of course, m any other possi-bilities present them selves for future consideration. Theprocess of insight has heretofore be en virtually opaqueto scient ific scrut iny. Differentiating insight p roblemsfrom other problems by the phenomenology that precedessolution may facilitate illumination of the process ofinsight.


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24 4 M E T C A L F E A N D W l E B ER E F E R E N C E S

ADAMS, J. L. (1979). Conceptual blockbust ing. New York : Nor ton.AmE~, S. (1976). Creativi ty: The mag ic synthesis. New York: BasicBooks.BERGSON, H. (1902). An introduction to metaphysics. New York :BRUNER, J. (1966). On knowing: Essays for the lef t hand. Cambridge:Harvard University Press.DAVIDSON, J. E., & STERNBERG, R. J. (1984). The role of insight inintellectual giftedness. Gifted Child Qua rterly, 28, 58-64.DEBoNo, E. (1967). The use of lateral thinking. New York: PenguinDEBoNo , E. (1969). The mechanism of mind. New York: Penguin.DOMINO WSra, R. L. ( 1981). Comm ent on an examination of the allegedrole of fixation in the solution of insight problem s. Journal of Ex-perimental Psychology: General, 110, 199-203.DUNCrd~R, K. (1945). On problem solving. Psychological Monographs,58(5, Whole No. 270).ELL Er~, P. (1982). Direction, past exp erience, and hints ~n creativeproblem solving: Reply to We~sberg and Alba. Journal of ExperimentalPsychology: General, 111, 316-325.Flxx, J. F. (1972). Mo re games for the superintell igent . New York"Popular Library.GARDSER, M. (1978). Aha.t Insight. New York : F reeman.GRUNEBERG, M. M., & MOr~KS, J. (1974). Feeling of knowing In cuedrecall. Acta Psychologica, 38, 257-265.HART, J. T. (1967). Memory and the memory-monitoring process. Jour-nal of Verbal Learning & Verbal Behavior, 6, 685-691.HOFSTADTER, D. R. (1980). GOdel, Escher, Bach: An eternal golden

braid. New York: Vintage Books.KARAT , J. (1982). A model of problem solving with incomplete con-straint knowledge. Cognitive Psychology, 14, 538-559.KOESTLER, A. (1977). The act ofcreaUon. London: P~cadoo.LEWN~, M. (1986). Principles of ef fect ive problem solving Unpubhshedmanuscript, State University of New York at Stonybrook.LICHT ~NST~IN, S., FISCHOFF, B., & PraILL~S, L. D. (1982). Cahbra-t~on of probabilities: The state to the art to 1980. In D. Kahneman,P. Slovic, & A. Tversky (Eds.), Judgment under uncertainty. Heuris-tits and biases. Cambridge: Cambridge University Press.LOV ELACE, E. A. (1984). Metamem ory: Monitoring future recallabil-ity during study. Journal of Experimental Psychology: Learning,Memory, & Cognit ion, 10, 756-766.LUCHINS, A. S. (1942). Mechanizataon in problem solving. PsychologtcalMonographs, 54(6, Whole No. 248) .MAIER, N. R. F. (1931). Reasomng in hu mans. II. The solution of aproblem and ~ts appearance ~n consciousness. Journal of Campara-t ive Psychology, 12, 181-194.MACER, R. E (1983). Thinking, problem solving, cognttion. New York:FreemanMETCAL FE, J. (1986a). Feeling of knowing in mem ory and problemsolving. Journal of Experimental Psychology: Learning, Memory. &Cogni t ion, 12, 288-294.METCA LF~, J. (1986b). Premo nitions of insight predict impending er-ror. Journal of Experimental Psychology: Learntng, Memory, & Cog-nition, 12, 623-634.NELSON, T. O. (1984). A com parison of current measures of the ac-curacy of feeling of knowing prediction. Psychological Bulletin, 9 5 ,109-133.N[LSON, T. O. (1986). RO C curves and m easures of d~scriminat~onaccuracy: A reply to Swets. Psychologtcal Bullenn, 100, 128-132.NELSON, T. O., LEONESlO, R. J., LANDWEH R, R. S., & N AR EN S, L.0986). A com parison of three predictors of an individuals memo ryperformance: Th e individuals feehng of know ing vs. the normativefeeling of know ing vs. base-rate item difficulty. Journal of Expenmen-tal Psychology: Learning, Memory, & Cognition, 12, 279-287.NELSON, T. O., LEONESIO, R. J., SHIMAMURA, A P . , LANDWEHR,R. F., & NAR~NS, L. (1982). Overlearning and the feeling of know-ing. Journal of Experimental Psychology: Learning, Memory, & Cog-nition, 8, 279-288.

NELSON, T. O., & NARENS, L. (1980). Norms of 300 general-lntormataonquestions: Accuracy of recall, latency of recall, and feeling-ot-knowingratings. Journal of Verbal Learning & Verbal Behavior, 19, 338-368POLANYI, M. (1958). Personal knowledge: Towards a post-crtticalphilosophy. Chicago: University of Chicago Press.POLYA, G. (1957), How to solve it. Princeton, NJ: Princeton Univer-sity Press.RESTLE, F., & DAvis, J. H. (1962). Success and speed of problem solwngby ~ndividuals and groups. Psychological Review, 69, 520-536.SCnACXEa, D. L. (1983). Feeling of knowing in episodic memory. Jour-nal of Experimental Psychology: Learning, Memory, & Cognition,9, 39~54.SIuoN, H. (1977). Models of discovery. Dordrecht, The Netherlands.Reidel.SIUON, H. (1979). Models of thought . New H aven" Yale UniversityPress.SIt, ON, H., NEWELL, A., &SnAW, J. C. (1979). The process of crea-tive thinking. In H. Simon (Ed), Models of thought (pp. 144-174).New H aven: Yale University PressSTEaNaER6, R. J . (1985) . Beyond IQ. Cambridge, MA. CambridgeUniversity Press.SXERr~aER~, R. J. (1986). Intelligence applied. San Diego: Harcourt,Brace. Javonovlch.STERNBERG, R. J., & DAVIDSON, J. E. (1982, June). The nund of the

puzzler. Psychology Today, 16, 37~4.TRA V E RS, K . J., DALTON, L. C., BRUNER, V. F., & TAYLOR, A. R.(1976) . Using advanced algebra. Toronto: Doubleday.WALLAS, G. (1926) . The art of thought. New York. Harcourt.WEISBERG, R. W., & ALl, A, J. W. (1981a). An exam ination of the al-leged role of "fixation" in the solution of several "insight" problems.Journal of Experimental Psychology: General, 110, 169-192.WEISBERG, R. W., & ALBA, J. W. (1981b). Gestalt theory, insight andpast expertence: Reply to Dominowski. Journal of Experimental Psy-chology: General, 110, 193-198.W~ISaEa~, R. W., & AL aA, J. W. (1982). Problem solwng is not likeperception: More on gestalt theory. Journal of Experimental Psychol-ogy: General, 111, 326-330.

A P P E N D I X AIncremental Problems1. If the puzzle you solved before yo u solved this one washarder than the puzzle yo u solved after you solved the puz-zle you solved before you solved this one, was the puzzleyou solved before y ou solved this one harder than this one?(Restle & Davis, !962)2. Given containers of 163, 14 , 25, and 11 ounces, and a sourceof unlimited water, obtain exactly 77 ounces of w ater. (Lu-chins, 1942)3. Given state:

Goal state:


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A P P EN DIX A (C on t in u e d )Allowable mo ves: Move only one disc at a t ime; take onlythe top disc on a peg; never place a larger disc on top of asmaller one. (e.g., Karat, 1982; Levine, 1986)4. Three people play a game in which one person loses and twopeople win each round. The one who loses must double theamount of m oney that each of the other two players has atthat t ime. T he three players agree to play three games. A tthe end of the three games, each player has lost one gameand each person has $8. What was the original stake of eachplayer? (R. Thaler , personal comm unicat ion , September1986)5. Next week I am going to have lunch with m y friend, visitthe new art gallery, go to the Social Security office, and havemy teeth checked at the dentist. My friend cannot meet meon Wednesday; the Social Security office is closed weekends;the dentist has office hours only on Tuesday, Friday, andSaturday; the art gallery is closed Tu esday, Thursday, andweekends. What day can I do everything I have planned?(Sternberg & Davidson, 1982)

INT UIT IO N IN PR OB LEM SO LV ING 2 45link costs 3 cents. She only has 15 cents. How does she doi t? (Experiment 2 ) (deBono, 1967)8. Without l i f ting your pencil from the paper show how youcould join all 4 dots with 2 straight l ines. (Experim ent 2)(M. Levine, personal communication, October 1985)

9. Show how you can divide this figure into 4 equal parts thatare the same size and shape. (Experim ent 2) (Fixx, 1972)

A P P E N D I X BInsight Problem s1. A prisoner was at tempting to escape from a tow er. Hefound in his cell a rope w hich was half long enough to per-mit him to reach the gro und safely. He divided the rop e inhalf and tied the two p arts together and escaped. How couldhe have done this? (Exp eriments 1 and 2) (Restle & Davis,1962)2. Water l i l ies double in area every 24 hours. At the b egin-ning of summ er there is one water lily on the lake. It takes60 days for the lake to become c ompletely covered w ithwater l i l ies. On which day is the lake half covered? (Ex-perime nts 1 and 2) (Sternberg & Davidson, 1982)3. If you h ave black socks and brown socks in your drawer,

mixed in a ratio of 4 to 5, how m any socks will you haveto take out to make sure that you have a pair the same color?(Experim ents 1 and 2) (Sternberg & Davidson, 1982)4. The triangle shown below points to the top of the page. Showhow you can move 3 circles to get the triangle to point tothe bottom of the page. (Experiments 1 and 2) (deBono,1969) oO O

0 0 00 0 0 0

5. A landscape gardener is given instructions to p lant 4 spe-cial trees so that each one is exactly the same distance fromeach of the others. How is he able to do i t? (Experiments1 and 2) (deBono, 1967)6. A m an bought a horse for $60 and sold it for $70. Thenhe bought i t back for $80 and sold i t for $90. How muchdid he make o r lose in the ho rse trading business? (Exp eri-ment 2) (deBono, 1967)7. A w oman has 4 pieces of chain. Each piece is m ade up of3 l inks. She w ants to join the pieces into a single closedloop of ch ain. To open a l ink costs 2 cents and to close a

10 . Describe how to cut a hole in a 3 5 in. card that is bigenough for you to put you r head through. (Experiment 2)(deBono, 1969)11 . Show h ow yo u can arrange 10 pennies so that you have 5rows (lines) of 4 pennies in each row. (Experiment 2) (Fixx,1972)12. Describe h ow to p ut 27 animals in 4 pens in such a w aythat there is an odd number of anim als in each pen. (Ex-periment 2) (L. Ross, personal comm unication, December1985)

A P P E N D I X CM ath Problem s (Taken from T ravers, D alton,Bruner, & Taylor, 1976)1. (3x2+2x+ 10)(3x)2. (2x+y)(3x-y) =3. Factor:

1 6 y 2 - 40yz + 25 z24. Solve for x:

X


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24 6 M E T C A L F E A N D W l E B EAppendix C (Continued)

1. (3x2+2x+10)(3x) =2. (2x+y)(3x-y)=3. Factor:

16y2- 40yz + 25z24. Solve for x:24x5. 18x2 + -3x

6. Factor:x~+6x+9

7. Solve for x:1/sx+10 = 25

-6x2y43xSy3

lo. ~/~3 =11. Solve for x, y, and z:x+2y-z = 132x+y+z = 8

3x-y = 2z = 1

13. Solve for m:m-3 m-2

2m 2m + 114. ~ =15. Solve for a and b:

3a+6b = 52a-b= 1

17. (~~-) (~~) =18. (a2)(a) =

(a~ )19. -( a )20. Find cos0

-0

(Manuscript received May 19, 1986;revision accepted for publication September 2 5, 1986.)

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1987 Metcalfe & Wiebe 1987 Intuition in Insight and Noninsight Problem Solving