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The role of analogy in transfer between similar problem states
COGNITIVE
6, 436-450
PSYCHOLOGY
The
STEPHEN
Role
K.
( 1974)
of Analogy in Transfer Similar Problem States’
REED, Case
GEORGE Western
W.
ERNST,
Reserve
AND
Between
RANAN
BANERJI
University
The study investigated the effect of transfer between two problems having similar (homomorphic) problem states. The results of three experiments revealed that although transfer occurred between repetition of the same problems, transfer occurred between the Jealous Husbands problem and the Missionary-Cannibal problem only when (a) Ss were told the relationship between the two problems and (b ) the Jealous Husbands problem was given first. The results are related to the formal structure of the problem space and to alternative explanations of the use of analogy in problem solving. These include memory for individual moves, memory for general strategies, and practice in applying operators.
Most people would agree that our attempts to solve problems are greatly influenced by our previous attempts to solve problems. After emphasizing that plans are important for guiding human behavior, Miller, Galanter, and Pribram (1960, pp. 177-178) argue that the major source of new plans is old plans. According to their argument, plans are usually remembered and not created. What is remembered, however, is more likely to be an abstract metaplan capable of generating a large number of different plans than the detailed operations of a single plan. More recently, Simon and Newell ( 1971) have identified sources of information which can influence how a person attempts to solve a problem. These include (1) the task instructions, (2) previous experience with the same task, (3) previous experience with analogous tasks, (4) general programs stored in LTM which can be applied to a wide range of tasks, (5) programs stored in LTM for combining information obtained from the external environment with information already stored in memory, and (6) the course of problem solving itself. Our particular interest focuses on analogous tasks as a source of information for solving similar problems. ‘This research was supported by NIMH Grant MH-23297 to the first author. We thank James Edwards and Richard Smook for testing Ss and preparing the data for analysis. Requests for reprints should be sent to Stephen K. Reed, Department of Psychology, Case Western Reserve University, Cleveiand, Ohio 44106. Copyright All rights
436 @ 1974 by Academic Press, Inc. of reproduction in any form reserved.
ANALOGY
IN
PROBLEM
II
437
SOLVING
02
M
FIG. 1. Problem Husbands problems.
space
of legal
moves
for
the
Missionary-Cannibal
and
Jealous
Our specific objective is to study the role of analogy in transfer between problems with similar problem states. A prerequisite for showing that two problems have similar problem states is to define the problem states for each problem. Figure 1 shows the legal problem states for the Missionary-Cannibal problem which can be stated as follows: “Three missionaries and three cannibals having to cross a river at a ferry, find a boat but the boat is so small that it can contain no more than two persons. If the missionaries on either bank of the river, or in the boat, are outnumbered at any time by cannibals, the cannibals will eat the missionaries. Find the simplest schedule of crossings that will permit all the missionaries and cannibals to cross the river safely. It is assumed that all passengers on the boat unboard before the next trip and at least one person has to be in the boat for each crossing.”
The first sionaries on on the left right bank;
number in each cell of Fig. 1 specifies the number of misthe left bank; the second number, the number of cannibals bank; the third number, the number of missionaries on the and the fourth number, the number of cannibals on the
438
REED,
ERNST
AND
BANERJI
right bank. The numbers connecting the problem states show how many missionaries (first number) and how many cannibals (second number) are transferred in the boat. The search proceeds from the initial state (33/O) to the goal state (00/33). Figure 1 illustrates that the shortest path requires 11 moves and that with the exception of states B and L, there is no choice of moves if one is going to continue to progress toward the goal. In order to compare the structure of two problems, one would like to have a measure of similarity such that given two problems of the same type, one could specify how they are similar. However, this may be a difficult objective unless the problem states of the two problems can be formally related. Such a relationship exists between the MissionaryCannibal problem and the Jealous Husbands problem. The latter problem can be stated as follows: “Three jealous husbands and their wives having to cross a river at a ferry, find a boat but the boat is so small that it can contain no more than two persons. Find the simplest schedule of crossings that will permit all six people to cross the river so that none of the women shall be left in company with any of the men, unless her husband is present. It is assumed that all passengers on the boat unboard before the next trip, and at least one person has to be in the boat for each crossing.”
The two problems are related by the fact that Fig. 1 also shows the legal states of the Jealous Husbands problem if husbands are substituted for missionaries, wives are substituted for cannibals, and husbands and wives are paired as couples. The latter constraint is not necessary in the Missionary-Cannibal problem since neither the missionaries nor the cannibals need be individually distinguished. The relationship between the two problems can be formalized as a one-to-many (homomorphic) mapping from the Missionary-Cannibal problem to the Jealous Husbands problem. For example, moving two missionaries corresponds to moving any of three possible pairs of husbands since all husbands are not equivalent. But only one of the three possible moves may be legal, so there is a greater constraint on moves in the Jealous Husbands problem. The purpose of the following experiments was to investigate whether there would be significant transfer between the two problems. Would performance improve significantly on one problem if Ss had first solved the other problem? We would not expect that Ss would recognize the exact relationship between the two problems (unless instructed) but the close correspondence between problem states should assure some commonality in the operations used to solve each problem. For example, in both problems Ss are required to move people and evaluate the
ANALOGY
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PROBLEM
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legality of their moves. A secondary issue concerns whether the greater constraint on moves in the Jealous Husbands problem is sufficient to make it more difficult in spite of the close similarity of the two problems. EXPERIMENTS
Three experiments were conducted in order to explore the role of analogy in problem solving. All three experiments tested whether the prior solution of a problem would lead to a significant reduction in the total time, the total number of moves, or the total number of illegal moves in solving a second problem. All experiments required that Ss solve both problems within a 30-min time limit. The S’s were undergraduates enrolled in psychology courses at Case Western Reserve University. They received additional grade points for their service. None of the Ss knew how many problems they would have to solve. Both the Jealous Husbands (JH) and the Missionary-Cannibal (MC) problems were used in each of the experiments. The Jealous Husbands problem was carried out through the utilization of six wooden figures of males and females which were separated into couples by colors (blue, green, and red). Six coins were used for the Missionary-Cannibal problem. The cannibals were pennies and the missionaries were dimes. The Ss in Expt I were required to solve both the Missionary-Cannibal and Jealous Husbands problems. The description of each problem was identical to the description given in the previous section and no attempt was made to instruct the Ss on the relationship between the two problems. Of the 97 Ss tested, 68 solved both problems within the time limit, 33 Ss starting with problem JH and 35 Ss starting with problem MC. Since the results of Expt I revealed that there was no significant transfer between the two problems, Expts II and III were designed to test conditions under which transfer might more likely occur. Experiment II tested whether transfer would occur between repetitions of the same problem by asking each S to solve the same problem twice, instead of two different problems. Twenty-five Ss solved problem JH twice, 25 Ss solved problem MC twice, and 4 Ss were unable to complete the experiment within a 30-min time limit. The instructions were the same as in Expt I and SS had no prior knowledge that they would have to solve the same problem twice. Experiment III was designed to test whether transfer would occur between the Jealous Husbands and Missionary-Cannibal problems if SS were told the relationship between the two problems. The procedure was identical to the procedure used in Expt I, with the exception that the instructions for the second problem included an additional paragraph that described how the second problem was related to the first
440
REED,
problem. Ss who solved the following additional
ERNST
AND
problem MC information:
BANERJI
as the second
problem
received
The easiest way to solve the problem is to take advantage of your correct solution to the Jealous Husbands problem. The solution of the Missionary-Cannibal problem is the same as the solution of the Jealous Husbands problem if one substitutes a missionary for a husband and a cannibal for a wife. Whenever you moved a husband previously, you should now move a missionary, and whenever you move a wife previously, you should now move a cannibal. It does not matter which missionary or which cannibal you move since missionaries and cannibals are not paired as husbands and wives were in the previous problem.
Ss who solved problem ing information:
JH as the second problem
received
the follow-
The easiest way to solve the problem is to take advantage of your correct solution to the Missionary-Cannibal problem. The solution of the Jealous Husbands problem is the same as the solution of the MissionaryCannibal problem if one substitutes a husband for a missionary and a wife for a cannibal. Whenever you moved a missionary previously, you should now move a husband and whenever you moved a cannibal previously, you should now move a wife. It may make a difference, however, which husband or which wife you move since a husband and wife are paired as couples in this problem.
Of the 75 Ss tested, 56 solved both problems within the time limit, 28 Ss starting with problem JH and 28 Ss starting with problem MC. After the experiment, Ss answered the following question: “There are several possible strategies one could use to solve the second problem. The strategies vary from strategy 1 which is totally dependent on the solution of the first problem to strategy 4 which is totally independent of the solution to the first problem. Please indicate to the experimenter which strategy best describes your own approach to solving the second problem.” The four alternatives are listed in Table 2. A tape recorder operated continuously throughout the problem-solving sessions and the E verbally described each move as the S made it. The E corrected “illegal” moves which would result in a missionary being eaten or a wife being left in the presence of another man without her husband. When illegal moves occurred, the E explained why the move was illegal and asked the S to retract the move. The Ss were not asked to give a verbal protocol of their thought processes since the only required data was the sequence of moves made by each S and the time between moves. This information was obtained after the problem-solving session by replaying the tape and using a stop watch to record the time between moves. The recorded time did not include the time required by E to point out and explain an illegal move when an illegal move occurred.
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RESULTS
Transfer The data for each S was summarized by recording the number of illegal moves, the number of total moves, and the time required to solve each problem. The results from Expt I were used in a two-factor analysis of variance where one factor was the type of problem (MC or JH) and the second factor was the problem order (JH-MC or MC-JH). As expected, the problem order did not have a significant effect for any of the three measures. The total number of moves, the total number of illegal moves, and the total time required by Ss in solving both problems was uninfluenced by whether problem JH or problem MC was presented first. The question of primary interest is whether there was transfer between the two problems which would be suggested by a significant Order X Problem interaction. This interaction was significant only when the performance measure is the number of illegal moves (F( 166) = 6.72, p < .05). For problem JH, Ss averaged 4.4 illegal moves when it occurred first (JH,) and 3.3 illegal moves when it occurred second ( JH, ). For problem MC, Ss averaged 3.1 illegal moves when it occurred first (MC,) and 2.0 illegal moves when it occurred second (MC,). The significant interaction reflects the combined effect of the reduction in illegal moves on problem JH and the increase in illegal moves on problem MC when the order is changed from JH-MC to MC-JH. In order to determine whether there was a significant change for each individual problem, a t-test was used to compare JH, with JH, and MC, with MC,. The results were insignificant for both the Jealous Husbands problem and the Missionary-Cannibal problem. The Order x Problem interaction was not significant when either the total number of moves (F( 166) = 0.14, p > 0.25) or the total time (F( 1,666)= 1.04, p > 0.25) was used as a performance measure. Table I shows the means for each condition and the values of t-tests and confidence intervals for those conditions which resulted in significant F-ratios. The failure to find significant transfer between the two problems raises the question of why such transfer failed to occur when the two problems were both transportation problems with homomorphic problem spaces. One possible answer is that Ss did not have a very good memory for how they solved the first problem. In this case, Ss &ould show little transfer even if they knew the exact correspondence between two problems. A minimal condition for expecting transfer would be to demonstrate that transfer occurs between two identical problems. If Ss were asked to solve the same problem twice, would they remember enough
252 355
246 403
283 304
MC JH
MC JH
MC JH
162 301
184 222
230 317
Trial 2
* p 5 .05 for a one-tailed ** p 5 .Ol for a one-tailed *** .95 confidence interval F-ratios.
Trial 1
Problem
Total
or JHi-JH2.
50-292 -
- 33-157 105-267
-
Confidence interval***
test. test. for MC-MC,
3.49** n.s.
1.34 4.76**
-
t
time
-
Experiment 16.5 15.8
18.0 19.1
II
I
Confidence
intervals
Experiment III 15.5 1.55 16.5 .58
-
19.8
19.6
17.8 17.3
Experiment 17.3 -
18.3
t
moves
Trial 2
Total
TABLE 1 of the Transfer
Trial 1
Summarv
and t-tests
-0.72-5.32 -2.06-3.70
-
-
Confidence interval
Results
are shown
3.5 4.1
2.2 4.3
2.0 3.3
Trial 2
for conditions
3.1 4.4
Trial 1
1.34 1.08
t
moves
resulting
2.71** ns.
Illegal
in significant
.31-2.19
-0.54-2.74 -1.06-3.14
Confidence interval E
E 3
“E
ANALOGY
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443
about their first solution to improve their performance? This question was investigated in Expt II. The results from Expt II were used in a two-factor analysis of variance where one factor was the type of problem and the second factor was the effect of practice (initial test or retest). The effect of practice was significant when time (F( 1,48) = 16.67, p < .Ol) and illegal moves (F( 1,48) = 9.98, p < .Ol) are dependent variables but not when the total number of moves (F( 1,48) = 3.57, p > .05) is the dependent variable. Subsequent tests showed that the reduction in solution time was highly significant for problem JH but was significant at only the .lO level for problem MC. The reduction in illegal moves was significant for both problems. Table 1 shows these results in more detail. The results suggest that Ss do remember enough about how they solve a problem to reduce their solution time when asked to solve the problem again. The failure to find significant transfer in Expt I may therefore be due more to a failure to recognize or make use of the analogy than to a failure of memory. The purpose of Expt III was to investigate whether describing the exact relationship between the Missionary-Cannibal and Jealous Husbands problems would lead to significant transfer. The data analysis for Expt III was identical to the data analysis for Expt I. Differences in the order of presentation caused a significant main effect when solution time was used as a performance measure (F( 1,54) = 4.11, p < .05) suggesting that the amount of transfer was greatly dependent on which problem occurred first. There was no transfer from the Missionary-Cannibal problem to the Jealous Husbands problem since solution time for JH, (304 set) and JH, (301 set) were almost identical. In contrast, there was substantial transfer from the Jealous Husbands problem to the Missionary-Cannibal problem. The average solution time was 283 set for MC, and 162 set for MC,. The differential effects of transfer were also shown for illegal moves. The number of illegal moves increased slightly from JH, (4.1) to JH, (4.3) but declined significantly from MC, (3.5) to MC, (2.2). The total number of moves declined slightly for both problems, but both changes were nonsignificant. Table 1 shows these results. In an attempt to obtain additional information, we asked Ss in Expt III to pick one of four different strategies which best described their degree of reliance on their solution of the first problem. The results indicate that most Ss felt their solution of the second problem was only minimally influenced by their solution of the first problem. The majority of Ss chose strategy 3 which specified only occasional use of memory for their first solution (Table 2). The choice of problem strategy was
444
REED,
Number
of Subjects
ERNST
AND
TABLE 2 Who Reported
BANERJI
Using
Each
Strategy
Strategy 1. I remembered the correct sequence of moves for solving the first problem. I then repeated the “same” sequence of moves in the second problem by using the correspondence between (a) husbands and missionaries and (b) wives and cannibals. 2. I remembered most of the moves for solving the first problem and used this as a basis for solving the second problem. I occasionally made moves, however, without any reference to the first problem. 3. I occasionally used my memory for the first problem as a basis for making moves. I usually attempted to solve the second problem as an independent problem without reference to the first problem. 4. I did not use my memory for t,he first problem at all. I solved the second problem as an entirely separate problem.
Subjects 0
7
32
17
not influenced by problem order. Four of the Ss who solved problem MC as the second problem picked strategy 2, 16 picked strategy 3, and 8 picked strategy 4. Three of the Ss who solved problem JH as the second problem picked strategy 2, 16 picked strategy 3, and 9 picked strategy 4. A one-factor analysis of variance was used to test whether there were performance differences on the second problem among the three groups of Ss who reported different strategies. The analysis was done separately for each of the two problems (MC and JH) and for each of the three performance measures (total moves, illegal moves, and total time). None of the six tests revealed a significant performance difference as a function of reported strategy. Individual
Moves
In addition to this more general analysis, each S’s performance in Expt I was analyzed for the number of moves he made between adjacent legal states and for the average time per move. Table 3 shows the results for both problems. The moves are labeled to correspond to the search space in Fig. 1. “Forward moves” are moves that are made toward the goal. For example, Ss solving problem JH made an average of 1.3 moves from state E to state F and took an average of 36 set to make that move. “Backward moves” are moves that are made away from the goal. Ss solving problem JH made an average of 0.3 moves from state F to state E and took an average of 7 set to make that move. The time taken to move from one legal state to another legal state includes the time taken to make an illegal move but does not include the time taken by E to correct the illegal move. If a S took IO set before
ANALOGY
Moves
AB BC BIj CE LIE: EF FG GH HI IJ JK KL LM LN MO NO
PROBLEM
445
SOLVING
TABLE 3 Between Legal States for the Missionary-Cannibal and Jealous Husbands Problems
Forward
moves
Backward
Number
Move
IN
Time
Move
JH
MC
JH
MC
0 0.9 0.6 0.8 0.6 1.3 1.3 1.4 1.3 1.0 1.0 1.0 0.4 0.6 0.4 0.6
0 0.8 0.5 1.0 0.6 1.1 1.3 1.4 1.4 1.1 1.0 1.0 0.3 0.7 0 s 0.7
0 11 14 13 14 36 11 32 41 35 22 11 3 3 2 2
0 10 11 11 11 25 8 21 24 23 10 4 3 2 2 2
BA CB r)B EC El> FE GF HG IH JI KJ LK ML NL OM ON
moves
Sumber
Time
JH
MC
JH
MC
0 0 3 0.2 0.3 0.2 0.3 0 3 0.4 0.3 0 0 0 0 0 0 0
0 0.2 0.2 0.3 0.2 0.1 0.3 0 4 0.4 0.1 0 0 0 0 0 0
0 8 7 36 33 7 51 50 38 0 0 0 0 0 0 0
0 17 17 28 59 8 31 33 35 18 0 '0 0 0 0 0
making an illegal move and then an additional 5 set before making a legal move, his total time would be 15 sec. It should be noted that the average number of forward moves between two legal states is exactly 1.0 more than the number of backward moves between the same two states for that part of the problem space in which there is only one permissible backward move (states F-L). Whenever a S makes a backward move between two states, he will have to make an extra forward move between the same two states in order to solve the problem. It should also be noted that when only a single backward move exists, a S can only make a backward move by canceling his last forward move. If more than one backward move exists (such as at state E), a S may make a backward move which does not cancel his last forward move (for example, by making the forward move CE and the backward move ED). The results revealed that the average number of moves between each problem state is nearly identical for the two problems, The main difference between problems is the amount of time required to move between states. The most difficult state transitions occur in the middle of the problem. As expected, return moves in which the boat is brought back
446
REED,
Effects Expt
of Transfer
MC
JH
BC BD CE DE EF FG GH HI IJ JK KL LM LN MO NO
1 5 3 5 4 5 9 1 4 2 0 0 1 0 0
+3
Expt MC
BANERJI
4 Forward
0
time
1 12 2 1 22 4 12 12 15 8 4
8 10 25 7 20 6 21 4 8
+2
more
MC
+9
2 0 16 10 +1 6 9 0 0 0 0 0
Timea Expt
JH
+3
Move
II
3
8 4 12 6 5 9 12 $3 16 8 0 2 0 1 Ss took
AND
TABLE on Reducing
I
Move
a A + indicates
ERNST
III JH
Average
+4 +4
0 2 4 5 12 3 9 3 3 8 4 +1 2 0 0
3 3 +3 +1 +4
+12 +26 6 2 0
+2
fl 4 0 0
on the second
6 1 0
$1 0 0
problem.
to the original bank are fairly rapid except when it is necessary to bring two people back (HI). Backward moves also take a considerable amount of time suggesting that a backward move is usually made only after a failure to find a legal forward move. In order to test whether differential amounts of transfer occurred at different problem states, we calculated the reduction in time to make each forward move when the problem was presented as the second problem. Table 4 shows the results for the three experiments. The numbers under column “MC” indicate the difference in move time between MC, and MC, and the numbers under column “JH” indicate the difference between JH, and JH,. The numbers under the column labeled “Average” indicate the mean reduction in move time, averaged across the two problems and three experiments. Problem
Diferences
A secondary objective of the study was to examine whether there would be performance differences on the two problems. The two problems have corresponding problem states, except for the added constraint of pairing husbands and wives in the Jealous Husbands problem. The results from the three experiments revealed that this additional constraint was sufficient to increase significantly solution time and the num-
ANALOGY
IN
PROBLEM
SOLVING
441
ber of illegal moves on the Jealous Husbands problem, but had no effect on the total number of moves. In comparing the MissionaryCannibal problem with the Jealous Husbands problem, solution time increased from 242 to 336 set (F ( 1,66) = 10.37, p < .Ol) in Expt I, from 216 to 313 set (F( 1,48) = 6.11, p < .05) in Expt II, and from 223 to 302 set (F( I,54) = 7.57, p < -01) in Expt III. The number of illegal moves increased from 2.6 to 3.9 (F( 166) = 9.34, p < .Ol) in Expt I, from 2.5 to 3.6 (F( 1,47) = 5.77, p < .05) in Expt II, and from 2.9 to 4.2 (F(1,54) = 9.86, p < .Ol) in Expt III. The total number of moves increased from 18.0 to 19.7 (F( 1,66) = 1.46, p > .lO) in Expt I, from 17.3 to 17.5 (F < 1) in Expt II, and from 16.5 to 16.9 (F < 1) in Expt III. All three findings are consistent with the formal relationship between the two problems. The greater restriction on the number of permissible moves caused by the nonequivalence of all husbands and all wives made it more difficult to evaluate the Iegality of a move. Ss therefore required more time to find a legal move and were more likely to make an illegal move. However, the basic structure of the task environment was the same for both problems since the space of legal moves and problem states is identical for both problems except for the number of combinations which can represent each move and problem state ( Fig. 1) . The total number of moves required to solve each problem therefore did not differ. It should also be noted that the total number of moves did not differ significantly between JH, and JH, or between MC, and MC, in any of the three transfer experiments. This finding may also depend upon the specific structure of the problem space. First, both problems can be solved in only 11 moves so there would be a greater chance for reduction in problems which require a greater number of moves. Second, there are no “blind alleys” in either problem which could result in a succession of legal moves leading away from the correct solution path. Hayes (1965) f ound in his study of communication networks that the number of moves did increase as the length of blind alleys increased. Analogy may be useful in reducing the number of moves if it could be used to avoid entering blind alleys. DISCUSSION
The main aspects of the data are the failure to obtain transfer when Ss were not told the relationship between the two problems (Expt I); the significant transfer which occurred when Ss re-solved the same problem ( Expt II); and the asymmetrical effects of transfer when the Ss were told the relationship between the two problems (Expt III). Since
448
REED,
ERNST
AND
BANERJI
the primary effect of transfer was to reduce solution time, it may be useful to consider a possible set of prerequisites for reducing solution time. First, Ss must recognize that the current problem is analogous to a previous problem. Second, Ss must be able to retrieve some information from memory regarding the solution of the analogous problem. Third, they must be able to translate these operations into the operations of the current problem. Fourth, the translation must either define a unique operation or reduce the number of operations that would otherwise have to be considered. And finally, the total time to retrieve, translate, and use analogous information to find an operator should be less than the total time to find the same operator without using information from the previous problem. The asymmetrical effect of transfer found in Expt III is most likely due either to the second or fourth prerequisites. The instructions indicated the exact relation between the two problems (first prerequisite) and no transfer effects were found in Expt I when this information was not given. The instructions specified the translation rules and the substitution of a wife for a cannibal and a husband for a missionary in solving JH, would seem to be as easy as the substitution of a cannibal for a wife and a missionary for a husband in solving MC, (third prerequisite). However, as pointed out previously, only the translation from JH, to MC, defines a unique move because of the many-to-one mapping prerequisite therefore predicts between problem states. The fourth greater transfer from problem JH to problem MC, as found in Expt III. The second prerequisite could also predict asymmetrical transfer if Ss remembered the solution for problem JH better than the solution for problem MC. There is some evidence for this in Expt II. Although Ss required less time in resolving both problems, the reduction was much greater for problem JH, perhaps indicating better memory for the solution of that problem. At first glance, this may seem unreasonable since the longer solution time usually found for problem JH suggests that it is a more difficult problem. However, problem JH did not require more moves than problem MC and perhaps the greater amount of time spent at each problem state resulted in a better memory trace. This explanation of transfer proposes that Ss remember at least some of the moves for the first problem and translate the moves to apply to the second problem. The explanation can account for (1) the failure to obtain transfer in Expt I because Ss did not know the translation and (2) the transfer found in Expt II because no translation was necessary. It could also account for the fact that transfer occurred for the JH-MC order but not for the MC-JH order in Expt III because the translation
ANALOGY
IN
PROBLEM
SOLVING
449
defines a unique move only for the first order or because of differential memory for the two solutions. The major weaknesses of this explanation are that (1) Ss report that they rely only minimally on their memory for the solution of the first problem, (2) it cannot account for transfer from one part of the problem space to another, and (3) Ss showed little improvement on certain very difficult moves (at state H). Transfer from one part of the problem space to another was previously found for the Missionary-Cannibal task (Thomas, 1974). Preliminary practice on the second half of the problem did not lead to better performance on the second half of the complete problem, but did lead to better performance on the first half of the complete problem. These results are consistent with the view that transfer occurred at a more global level than remembering individual moves. Thomas argued that there are four significant “cognitive changes” that occur in solving the Missionary-Cannibal problem and that a theory of problem solving should be formulated at the level of cognitive changes rather than external moves. Examples of strategies which reflect cognitive changes include (1) move the missionaries over first, (2) move the cannibals over first, (3) isolate the missionaries and cannibals, and (4) balance the missionaries and cannibals. The use of general strategies is consistent with the results of Expts I and II since Ss would more likely use the same strategies if they knew the relationship between the two problems. It is also consistent with Ss’ reports that they rely only minimally on their memory since memory for a few strategies (balance missionaries and cannibals, move all missionaries first, etc. ) is simpler than memory for a sequence of moves. It could predict transfer from one part of a problem to another but only if the same strategy was relevant to both parts. Its main weakness would seem to be to account for the asymmetrical effects of transfer found in Expt III since the translation of a strategy should not depend on which problem came first. It is easy to translate a strategy such as “isolate missionaries and cannibals” into “isolate husbands and wives” although the constraints of legality would make it more difficult to reach the subgoal in the second case. Another more global explanation is that the S becomes more efficient at applying certain operators that are used throughout the problem such as testing the legality of moves. This explanation could account for SS reporting a minimal reliance on memory and for transfer from one part of the problem space to another part. It is less clear how it would account for the failure to achieve transfer in Expt I (since essentially the same operators are involved) and for the asymmetrical effects of transfer in Expt III.
450
REED,
ERNST
AND
BANERJI
In conclusion, the role of analogy in problem solving is a complex issue and we are unable to propose a detailed theory. We have been able to show a formal relationship between two problems, present some experimental data, postulate some possible relationships between the data and the formal structure of the problem space, and suggest some processes which can determine whether transfer occurs between problems or within a problem. Further research on analogy should be revealing, particularly research using protocol analysis or research testing specific predictions. Further investigation of related issues, such as the role of memory in problem solving (cf. Greeno, 1974) or the effectiveness of subgoals (cf. Hayes, 1966)) should also further our understanding of analogy. REFERENCES GREENO, J. G. Hobbits and arcs: Acquisition of a sequential concept. Cognitive Psychology, 1974, 6, 270-292. HAYES, J. R. Problem topology and the solution process. Journal of Verbal Learning and Verbal Behavior, 1965, 4, 371-379. HAYES, J. R. Memory, goals, and problem solving. In B. Kleinmuntz (Ed.), Probkm solving: Research, method, and theory. New York: Wiley, 1966. MILLER, G. A., GALANTER, E., & PRIBRAM, K. H. Plans and the structure of behavior. New York: Holt, Rinehart & Winston, 1960. SIMON, H. A., & NEWELL, A. Human problem solving: The state of the theory in 1970. American Psychologist, 1971, 26, 145-159. THOMAS, J. C., JR. An analysis of behavior in the hobbits-arcs problem. Cognitive Psychology, 1974, 6, 257-269. (Accepted
January
15, 1974)
6, 436-450
PSYCHOLOGY
The
STEPHEN
Role
K.
( 1974)
of Analogy in Transfer Similar Problem States’
REED, Case
GEORGE Western
W.
ERNST,
Reserve
AND
Between
RANAN
BANERJI
University
The study investigated the effect of transfer between two problems having similar (homomorphic) problem states. The results of three experiments revealed that although transfer occurred between repetition of the same problems, transfer occurred between the Jealous Husbands problem and the Missionary-Cannibal problem only when (a) Ss were told the relationship between the two problems and (b ) the Jealous Husbands problem was given first. The results are related to the formal structure of the problem space and to alternative explanations of the use of analogy in problem solving. These include memory for individual moves, memory for general strategies, and practice in applying operators.
Most people would agree that our attempts to solve problems are greatly influenced by our previous attempts to solve problems. After emphasizing that plans are important for guiding human behavior, Miller, Galanter, and Pribram (1960, pp. 177-178) argue that the major source of new plans is old plans. According to their argument, plans are usually remembered and not created. What is remembered, however, is more likely to be an abstract metaplan capable of generating a large number of different plans than the detailed operations of a single plan. More recently, Simon and Newell ( 1971) have identified sources of information which can influence how a person attempts to solve a problem. These include (1) the task instructions, (2) previous experience with the same task, (3) previous experience with analogous tasks, (4) general programs stored in LTM which can be applied to a wide range of tasks, (5) programs stored in LTM for combining information obtained from the external environment with information already stored in memory, and (6) the course of problem solving itself. Our particular interest focuses on analogous tasks as a source of information for solving similar problems. ‘This research was supported by NIMH Grant MH-23297 to the first author. We thank James Edwards and Richard Smook for testing Ss and preparing the data for analysis. Requests for reprints should be sent to Stephen K. Reed, Department of Psychology, Case Western Reserve University, Cleveiand, Ohio 44106. Copyright All rights
436 @ 1974 by Academic Press, Inc. of reproduction in any form reserved.
ANALOGY
IN
PROBLEM
II
437
SOLVING
02
M
FIG. 1. Problem Husbands problems.
space
of legal
moves
for
the
Missionary-Cannibal
and
Jealous
Our specific objective is to study the role of analogy in transfer between problems with similar problem states. A prerequisite for showing that two problems have similar problem states is to define the problem states for each problem. Figure 1 shows the legal problem states for the Missionary-Cannibal problem which can be stated as follows: “Three missionaries and three cannibals having to cross a river at a ferry, find a boat but the boat is so small that it can contain no more than two persons. If the missionaries on either bank of the river, or in the boat, are outnumbered at any time by cannibals, the cannibals will eat the missionaries. Find the simplest schedule of crossings that will permit all the missionaries and cannibals to cross the river safely. It is assumed that all passengers on the boat unboard before the next trip and at least one person has to be in the boat for each crossing.”
The first sionaries on on the left right bank;
number in each cell of Fig. 1 specifies the number of misthe left bank; the second number, the number of cannibals bank; the third number, the number of missionaries on the and the fourth number, the number of cannibals on the
438
REED,
ERNST
AND
BANERJI
right bank. The numbers connecting the problem states show how many missionaries (first number) and how many cannibals (second number) are transferred in the boat. The search proceeds from the initial state (33/O) to the goal state (00/33). Figure 1 illustrates that the shortest path requires 11 moves and that with the exception of states B and L, there is no choice of moves if one is going to continue to progress toward the goal. In order to compare the structure of two problems, one would like to have a measure of similarity such that given two problems of the same type, one could specify how they are similar. However, this may be a difficult objective unless the problem states of the two problems can be formally related. Such a relationship exists between the MissionaryCannibal problem and the Jealous Husbands problem. The latter problem can be stated as follows: “Three jealous husbands and their wives having to cross a river at a ferry, find a boat but the boat is so small that it can contain no more than two persons. Find the simplest schedule of crossings that will permit all six people to cross the river so that none of the women shall be left in company with any of the men, unless her husband is present. It is assumed that all passengers on the boat unboard before the next trip, and at least one person has to be in the boat for each crossing.”
The two problems are related by the fact that Fig. 1 also shows the legal states of the Jealous Husbands problem if husbands are substituted for missionaries, wives are substituted for cannibals, and husbands and wives are paired as couples. The latter constraint is not necessary in the Missionary-Cannibal problem since neither the missionaries nor the cannibals need be individually distinguished. The relationship between the two problems can be formalized as a one-to-many (homomorphic) mapping from the Missionary-Cannibal problem to the Jealous Husbands problem. For example, moving two missionaries corresponds to moving any of three possible pairs of husbands since all husbands are not equivalent. But only one of the three possible moves may be legal, so there is a greater constraint on moves in the Jealous Husbands problem. The purpose of the following experiments was to investigate whether there would be significant transfer between the two problems. Would performance improve significantly on one problem if Ss had first solved the other problem? We would not expect that Ss would recognize the exact relationship between the two problems (unless instructed) but the close correspondence between problem states should assure some commonality in the operations used to solve each problem. For example, in both problems Ss are required to move people and evaluate the
ANALOGY
IN
PROBLEM
SOLVING
439
legality of their moves. A secondary issue concerns whether the greater constraint on moves in the Jealous Husbands problem is sufficient to make it more difficult in spite of the close similarity of the two problems. EXPERIMENTS
Three experiments were conducted in order to explore the role of analogy in problem solving. All three experiments tested whether the prior solution of a problem would lead to a significant reduction in the total time, the total number of moves, or the total number of illegal moves in solving a second problem. All experiments required that Ss solve both problems within a 30-min time limit. The S’s were undergraduates enrolled in psychology courses at Case Western Reserve University. They received additional grade points for their service. None of the Ss knew how many problems they would have to solve. Both the Jealous Husbands (JH) and the Missionary-Cannibal (MC) problems were used in each of the experiments. The Jealous Husbands problem was carried out through the utilization of six wooden figures of males and females which were separated into couples by colors (blue, green, and red). Six coins were used for the Missionary-Cannibal problem. The cannibals were pennies and the missionaries were dimes. The Ss in Expt I were required to solve both the Missionary-Cannibal and Jealous Husbands problems. The description of each problem was identical to the description given in the previous section and no attempt was made to instruct the Ss on the relationship between the two problems. Of the 97 Ss tested, 68 solved both problems within the time limit, 33 Ss starting with problem JH and 35 Ss starting with problem MC. Since the results of Expt I revealed that there was no significant transfer between the two problems, Expts II and III were designed to test conditions under which transfer might more likely occur. Experiment II tested whether transfer would occur between repetitions of the same problem by asking each S to solve the same problem twice, instead of two different problems. Twenty-five Ss solved problem JH twice, 25 Ss solved problem MC twice, and 4 Ss were unable to complete the experiment within a 30-min time limit. The instructions were the same as in Expt I and SS had no prior knowledge that they would have to solve the same problem twice. Experiment III was designed to test whether transfer would occur between the Jealous Husbands and Missionary-Cannibal problems if SS were told the relationship between the two problems. The procedure was identical to the procedure used in Expt I, with the exception that the instructions for the second problem included an additional paragraph that described how the second problem was related to the first
440
REED,
problem. Ss who solved the following additional
ERNST
AND
problem MC information:
BANERJI
as the second
problem
received
The easiest way to solve the problem is to take advantage of your correct solution to the Jealous Husbands problem. The solution of the Missionary-Cannibal problem is the same as the solution of the Jealous Husbands problem if one substitutes a missionary for a husband and a cannibal for a wife. Whenever you moved a husband previously, you should now move a missionary, and whenever you move a wife previously, you should now move a cannibal. It does not matter which missionary or which cannibal you move since missionaries and cannibals are not paired as husbands and wives were in the previous problem.
Ss who solved problem ing information:
JH as the second problem
received
the follow-
The easiest way to solve the problem is to take advantage of your correct solution to the Missionary-Cannibal problem. The solution of the Jealous Husbands problem is the same as the solution of the MissionaryCannibal problem if one substitutes a husband for a missionary and a wife for a cannibal. Whenever you moved a missionary previously, you should now move a husband and whenever you moved a cannibal previously, you should now move a wife. It may make a difference, however, which husband or which wife you move since a husband and wife are paired as couples in this problem.
Of the 75 Ss tested, 56 solved both problems within the time limit, 28 Ss starting with problem JH and 28 Ss starting with problem MC. After the experiment, Ss answered the following question: “There are several possible strategies one could use to solve the second problem. The strategies vary from strategy 1 which is totally dependent on the solution of the first problem to strategy 4 which is totally independent of the solution to the first problem. Please indicate to the experimenter which strategy best describes your own approach to solving the second problem.” The four alternatives are listed in Table 2. A tape recorder operated continuously throughout the problem-solving sessions and the E verbally described each move as the S made it. The E corrected “illegal” moves which would result in a missionary being eaten or a wife being left in the presence of another man without her husband. When illegal moves occurred, the E explained why the move was illegal and asked the S to retract the move. The Ss were not asked to give a verbal protocol of their thought processes since the only required data was the sequence of moves made by each S and the time between moves. This information was obtained after the problem-solving session by replaying the tape and using a stop watch to record the time between moves. The recorded time did not include the time required by E to point out and explain an illegal move when an illegal move occurred.
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RESULTS
Transfer The data for each S was summarized by recording the number of illegal moves, the number of total moves, and the time required to solve each problem. The results from Expt I were used in a two-factor analysis of variance where one factor was the type of problem (MC or JH) and the second factor was the problem order (JH-MC or MC-JH). As expected, the problem order did not have a significant effect for any of the three measures. The total number of moves, the total number of illegal moves, and the total time required by Ss in solving both problems was uninfluenced by whether problem JH or problem MC was presented first. The question of primary interest is whether there was transfer between the two problems which would be suggested by a significant Order X Problem interaction. This interaction was significant only when the performance measure is the number of illegal moves (F( 166) = 6.72, p < .05). For problem JH, Ss averaged 4.4 illegal moves when it occurred first (JH,) and 3.3 illegal moves when it occurred second ( JH, ). For problem MC, Ss averaged 3.1 illegal moves when it occurred first (MC,) and 2.0 illegal moves when it occurred second (MC,). The significant interaction reflects the combined effect of the reduction in illegal moves on problem JH and the increase in illegal moves on problem MC when the order is changed from JH-MC to MC-JH. In order to determine whether there was a significant change for each individual problem, a t-test was used to compare JH, with JH, and MC, with MC,. The results were insignificant for both the Jealous Husbands problem and the Missionary-Cannibal problem. The Order x Problem interaction was not significant when either the total number of moves (F( 166) = 0.14, p > 0.25) or the total time (F( 1,666)= 1.04, p > 0.25) was used as a performance measure. Table I shows the means for each condition and the values of t-tests and confidence intervals for those conditions which resulted in significant F-ratios. The failure to find significant transfer between the two problems raises the question of why such transfer failed to occur when the two problems were both transportation problems with homomorphic problem spaces. One possible answer is that Ss did not have a very good memory for how they solved the first problem. In this case, Ss &ould show little transfer even if they knew the exact correspondence between two problems. A minimal condition for expecting transfer would be to demonstrate that transfer occurs between two identical problems. If Ss were asked to solve the same problem twice, would they remember enough
252 355
246 403
283 304
MC JH
MC JH
MC JH
162 301
184 222
230 317
Trial 2
* p 5 .05 for a one-tailed ** p 5 .Ol for a one-tailed *** .95 confidence interval F-ratios.
Trial 1
Problem
Total
or JHi-JH2.
50-292 -
- 33-157 105-267
-
Confidence interval***
test. test. for MC-MC,
3.49** n.s.
1.34 4.76**
-
t
time
-
Experiment 16.5 15.8
18.0 19.1
II
I
Confidence
intervals
Experiment III 15.5 1.55 16.5 .58
-
19.8
19.6
17.8 17.3
Experiment 17.3 -
18.3
t
moves
Trial 2
Total
TABLE 1 of the Transfer
Trial 1
Summarv
and t-tests
-0.72-5.32 -2.06-3.70
-
-
Confidence interval
Results
are shown
3.5 4.1
2.2 4.3
2.0 3.3
Trial 2
for conditions
3.1 4.4
Trial 1
1.34 1.08
t
moves
resulting
2.71** ns.
Illegal
in significant
.31-2.19
-0.54-2.74 -1.06-3.14
Confidence interval E
E 3
“E
ANALOGY
IN
PROBLEM
SOLVING
443
about their first solution to improve their performance? This question was investigated in Expt II. The results from Expt II were used in a two-factor analysis of variance where one factor was the type of problem and the second factor was the effect of practice (initial test or retest). The effect of practice was significant when time (F( 1,48) = 16.67, p < .Ol) and illegal moves (F( 1,48) = 9.98, p < .Ol) are dependent variables but not when the total number of moves (F( 1,48) = 3.57, p > .05) is the dependent variable. Subsequent tests showed that the reduction in solution time was highly significant for problem JH but was significant at only the .lO level for problem MC. The reduction in illegal moves was significant for both problems. Table 1 shows these results in more detail. The results suggest that Ss do remember enough about how they solve a problem to reduce their solution time when asked to solve the problem again. The failure to find significant transfer in Expt I may therefore be due more to a failure to recognize or make use of the analogy than to a failure of memory. The purpose of Expt III was to investigate whether describing the exact relationship between the Missionary-Cannibal and Jealous Husbands problems would lead to significant transfer. The data analysis for Expt III was identical to the data analysis for Expt I. Differences in the order of presentation caused a significant main effect when solution time was used as a performance measure (F( 1,54) = 4.11, p < .05) suggesting that the amount of transfer was greatly dependent on which problem occurred first. There was no transfer from the Missionary-Cannibal problem to the Jealous Husbands problem since solution time for JH, (304 set) and JH, (301 set) were almost identical. In contrast, there was substantial transfer from the Jealous Husbands problem to the Missionary-Cannibal problem. The average solution time was 283 set for MC, and 162 set for MC,. The differential effects of transfer were also shown for illegal moves. The number of illegal moves increased slightly from JH, (4.1) to JH, (4.3) but declined significantly from MC, (3.5) to MC, (2.2). The total number of moves declined slightly for both problems, but both changes were nonsignificant. Table 1 shows these results. In an attempt to obtain additional information, we asked Ss in Expt III to pick one of four different strategies which best described their degree of reliance on their solution of the first problem. The results indicate that most Ss felt their solution of the second problem was only minimally influenced by their solution of the first problem. The majority of Ss chose strategy 3 which specified only occasional use of memory for their first solution (Table 2). The choice of problem strategy was
444
REED,
Number
of Subjects
ERNST
AND
TABLE 2 Who Reported
BANERJI
Using
Each
Strategy
Strategy 1. I remembered the correct sequence of moves for solving the first problem. I then repeated the “same” sequence of moves in the second problem by using the correspondence between (a) husbands and missionaries and (b) wives and cannibals. 2. I remembered most of the moves for solving the first problem and used this as a basis for solving the second problem. I occasionally made moves, however, without any reference to the first problem. 3. I occasionally used my memory for the first problem as a basis for making moves. I usually attempted to solve the second problem as an independent problem without reference to the first problem. 4. I did not use my memory for t,he first problem at all. I solved the second problem as an entirely separate problem.
Subjects 0
7
32
17
not influenced by problem order. Four of the Ss who solved problem MC as the second problem picked strategy 2, 16 picked strategy 3, and 8 picked strategy 4. Three of the Ss who solved problem JH as the second problem picked strategy 2, 16 picked strategy 3, and 9 picked strategy 4. A one-factor analysis of variance was used to test whether there were performance differences on the second problem among the three groups of Ss who reported different strategies. The analysis was done separately for each of the two problems (MC and JH) and for each of the three performance measures (total moves, illegal moves, and total time). None of the six tests revealed a significant performance difference as a function of reported strategy. Individual
Moves
In addition to this more general analysis, each S’s performance in Expt I was analyzed for the number of moves he made between adjacent legal states and for the average time per move. Table 3 shows the results for both problems. The moves are labeled to correspond to the search space in Fig. 1. “Forward moves” are moves that are made toward the goal. For example, Ss solving problem JH made an average of 1.3 moves from state E to state F and took an average of 36 set to make that move. “Backward moves” are moves that are made away from the goal. Ss solving problem JH made an average of 0.3 moves from state F to state E and took an average of 7 set to make that move. The time taken to move from one legal state to another legal state includes the time taken to make an illegal move but does not include the time taken by E to correct the illegal move. If a S took IO set before
ANALOGY
Moves
AB BC BIj CE LIE: EF FG GH HI IJ JK KL LM LN MO NO
PROBLEM
445
SOLVING
TABLE 3 Between Legal States for the Missionary-Cannibal and Jealous Husbands Problems
Forward
moves
Backward
Number
Move
IN
Time
Move
JH
MC
JH
MC
0 0.9 0.6 0.8 0.6 1.3 1.3 1.4 1.3 1.0 1.0 1.0 0.4 0.6 0.4 0.6
0 0.8 0.5 1.0 0.6 1.1 1.3 1.4 1.4 1.1 1.0 1.0 0.3 0.7 0 s 0.7
0 11 14 13 14 36 11 32 41 35 22 11 3 3 2 2
0 10 11 11 11 25 8 21 24 23 10 4 3 2 2 2
BA CB r)B EC El> FE GF HG IH JI KJ LK ML NL OM ON
moves
Sumber
Time
JH
MC
JH
MC
0 0 3 0.2 0.3 0.2 0.3 0 3 0.4 0.3 0 0 0 0 0 0 0
0 0.2 0.2 0.3 0.2 0.1 0.3 0 4 0.4 0.1 0 0 0 0 0 0
0 8 7 36 33 7 51 50 38 0 0 0 0 0 0 0
0 17 17 28 59 8 31 33 35 18 0 '0 0 0 0 0
making an illegal move and then an additional 5 set before making a legal move, his total time would be 15 sec. It should be noted that the average number of forward moves between two legal states is exactly 1.0 more than the number of backward moves between the same two states for that part of the problem space in which there is only one permissible backward move (states F-L). Whenever a S makes a backward move between two states, he will have to make an extra forward move between the same two states in order to solve the problem. It should also be noted that when only a single backward move exists, a S can only make a backward move by canceling his last forward move. If more than one backward move exists (such as at state E), a S may make a backward move which does not cancel his last forward move (for example, by making the forward move CE and the backward move ED). The results revealed that the average number of moves between each problem state is nearly identical for the two problems, The main difference between problems is the amount of time required to move between states. The most difficult state transitions occur in the middle of the problem. As expected, return moves in which the boat is brought back
446
REED,
Effects Expt
of Transfer
MC
JH
BC BD CE DE EF FG GH HI IJ JK KL LM LN MO NO
1 5 3 5 4 5 9 1 4 2 0 0 1 0 0
+3
Expt MC
BANERJI
4 Forward
0
time
1 12 2 1 22 4 12 12 15 8 4
8 10 25 7 20 6 21 4 8
+2
more
MC
+9
2 0 16 10 +1 6 9 0 0 0 0 0
Timea Expt
JH
+3
Move
II
3
8 4 12 6 5 9 12 $3 16 8 0 2 0 1 Ss took
AND
TABLE on Reducing
I
Move
a A + indicates
ERNST
III JH
Average
+4 +4
0 2 4 5 12 3 9 3 3 8 4 +1 2 0 0
3 3 +3 +1 +4
+12 +26 6 2 0
+2
fl 4 0 0
on the second
6 1 0
$1 0 0
problem.
to the original bank are fairly rapid except when it is necessary to bring two people back (HI). Backward moves also take a considerable amount of time suggesting that a backward move is usually made only after a failure to find a legal forward move. In order to test whether differential amounts of transfer occurred at different problem states, we calculated the reduction in time to make each forward move when the problem was presented as the second problem. Table 4 shows the results for the three experiments. The numbers under column “MC” indicate the difference in move time between MC, and MC, and the numbers under column “JH” indicate the difference between JH, and JH,. The numbers under the column labeled “Average” indicate the mean reduction in move time, averaged across the two problems and three experiments. Problem
Diferences
A secondary objective of the study was to examine whether there would be performance differences on the two problems. The two problems have corresponding problem states, except for the added constraint of pairing husbands and wives in the Jealous Husbands problem. The results from the three experiments revealed that this additional constraint was sufficient to increase significantly solution time and the num-
ANALOGY
IN
PROBLEM
SOLVING
441
ber of illegal moves on the Jealous Husbands problem, but had no effect on the total number of moves. In comparing the MissionaryCannibal problem with the Jealous Husbands problem, solution time increased from 242 to 336 set (F ( 1,66) = 10.37, p < .Ol) in Expt I, from 216 to 313 set (F( 1,48) = 6.11, p < .05) in Expt II, and from 223 to 302 set (F( I,54) = 7.57, p < -01) in Expt III. The number of illegal moves increased from 2.6 to 3.9 (F( 166) = 9.34, p < .Ol) in Expt I, from 2.5 to 3.6 (F( 1,47) = 5.77, p < .05) in Expt II, and from 2.9 to 4.2 (F(1,54) = 9.86, p < .Ol) in Expt III. The total number of moves increased from 18.0 to 19.7 (F( 1,66) = 1.46, p > .lO) in Expt I, from 17.3 to 17.5 (F < 1) in Expt II, and from 16.5 to 16.9 (F < 1) in Expt III. All three findings are consistent with the formal relationship between the two problems. The greater restriction on the number of permissible moves caused by the nonequivalence of all husbands and all wives made it more difficult to evaluate the Iegality of a move. Ss therefore required more time to find a legal move and were more likely to make an illegal move. However, the basic structure of the task environment was the same for both problems since the space of legal moves and problem states is identical for both problems except for the number of combinations which can represent each move and problem state ( Fig. 1) . The total number of moves required to solve each problem therefore did not differ. It should also be noted that the total number of moves did not differ significantly between JH, and JH, or between MC, and MC, in any of the three transfer experiments. This finding may also depend upon the specific structure of the problem space. First, both problems can be solved in only 11 moves so there would be a greater chance for reduction in problems which require a greater number of moves. Second, there are no “blind alleys” in either problem which could result in a succession of legal moves leading away from the correct solution path. Hayes (1965) f ound in his study of communication networks that the number of moves did increase as the length of blind alleys increased. Analogy may be useful in reducing the number of moves if it could be used to avoid entering blind alleys. DISCUSSION
The main aspects of the data are the failure to obtain transfer when Ss were not told the relationship between the two problems (Expt I); the significant transfer which occurred when Ss re-solved the same problem ( Expt II); and the asymmetrical effects of transfer when the Ss were told the relationship between the two problems (Expt III). Since
448
REED,
ERNST
AND
BANERJI
the primary effect of transfer was to reduce solution time, it may be useful to consider a possible set of prerequisites for reducing solution time. First, Ss must recognize that the current problem is analogous to a previous problem. Second, Ss must be able to retrieve some information from memory regarding the solution of the analogous problem. Third, they must be able to translate these operations into the operations of the current problem. Fourth, the translation must either define a unique operation or reduce the number of operations that would otherwise have to be considered. And finally, the total time to retrieve, translate, and use analogous information to find an operator should be less than the total time to find the same operator without using information from the previous problem. The asymmetrical effect of transfer found in Expt III is most likely due either to the second or fourth prerequisites. The instructions indicated the exact relation between the two problems (first prerequisite) and no transfer effects were found in Expt I when this information was not given. The instructions specified the translation rules and the substitution of a wife for a cannibal and a husband for a missionary in solving JH, would seem to be as easy as the substitution of a cannibal for a wife and a missionary for a husband in solving MC, (third prerequisite). However, as pointed out previously, only the translation from JH, to MC, defines a unique move because of the many-to-one mapping prerequisite therefore predicts between problem states. The fourth greater transfer from problem JH to problem MC, as found in Expt III. The second prerequisite could also predict asymmetrical transfer if Ss remembered the solution for problem JH better than the solution for problem MC. There is some evidence for this in Expt II. Although Ss required less time in resolving both problems, the reduction was much greater for problem JH, perhaps indicating better memory for the solution of that problem. At first glance, this may seem unreasonable since the longer solution time usually found for problem JH suggests that it is a more difficult problem. However, problem JH did not require more moves than problem MC and perhaps the greater amount of time spent at each problem state resulted in a better memory trace. This explanation of transfer proposes that Ss remember at least some of the moves for the first problem and translate the moves to apply to the second problem. The explanation can account for (1) the failure to obtain transfer in Expt I because Ss did not know the translation and (2) the transfer found in Expt II because no translation was necessary. It could also account for the fact that transfer occurred for the JH-MC order but not for the MC-JH order in Expt III because the translation
ANALOGY
IN
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SOLVING
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defines a unique move only for the first order or because of differential memory for the two solutions. The major weaknesses of this explanation are that (1) Ss report that they rely only minimally on their memory for the solution of the first problem, (2) it cannot account for transfer from one part of the problem space to another, and (3) Ss showed little improvement on certain very difficult moves (at state H). Transfer from one part of the problem space to another was previously found for the Missionary-Cannibal task (Thomas, 1974). Preliminary practice on the second half of the problem did not lead to better performance on the second half of the complete problem, but did lead to better performance on the first half of the complete problem. These results are consistent with the view that transfer occurred at a more global level than remembering individual moves. Thomas argued that there are four significant “cognitive changes” that occur in solving the Missionary-Cannibal problem and that a theory of problem solving should be formulated at the level of cognitive changes rather than external moves. Examples of strategies which reflect cognitive changes include (1) move the missionaries over first, (2) move the cannibals over first, (3) isolate the missionaries and cannibals, and (4) balance the missionaries and cannibals. The use of general strategies is consistent with the results of Expts I and II since Ss would more likely use the same strategies if they knew the relationship between the two problems. It is also consistent with Ss’ reports that they rely only minimally on their memory since memory for a few strategies (balance missionaries and cannibals, move all missionaries first, etc. ) is simpler than memory for a sequence of moves. It could predict transfer from one part of a problem to another but only if the same strategy was relevant to both parts. Its main weakness would seem to be to account for the asymmetrical effects of transfer found in Expt III since the translation of a strategy should not depend on which problem came first. It is easy to translate a strategy such as “isolate missionaries and cannibals” into “isolate husbands and wives” although the constraints of legality would make it more difficult to reach the subgoal in the second case. Another more global explanation is that the S becomes more efficient at applying certain operators that are used throughout the problem such as testing the legality of moves. This explanation could account for SS reporting a minimal reliance on memory and for transfer from one part of the problem space to another part. It is less clear how it would account for the failure to achieve transfer in Expt I (since essentially the same operators are involved) and for the asymmetrical effects of transfer in Expt III.
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REED,
ERNST
AND
BANERJI
In conclusion, the role of analogy in problem solving is a complex issue and we are unable to propose a detailed theory. We have been able to show a formal relationship between two problems, present some experimental data, postulate some possible relationships between the data and the formal structure of the problem space, and suggest some processes which can determine whether transfer occurs between problems or within a problem. Further research on analogy should be revealing, particularly research using protocol analysis or research testing specific predictions. Further investigation of related issues, such as the role of memory in problem solving (cf. Greeno, 1974) or the effectiveness of subgoals (cf. Hayes, 1966)) should also further our understanding of analogy. REFERENCES GREENO, J. G. Hobbits and arcs: Acquisition of a sequential concept. Cognitive Psychology, 1974, 6, 270-292. HAYES, J. R. Problem topology and the solution process. Journal of Verbal Learning and Verbal Behavior, 1965, 4, 371-379. HAYES, J. R. Memory, goals, and problem solving. In B. Kleinmuntz (Ed.), Probkm solving: Research, method, and theory. New York: Wiley, 1966. MILLER, G. A., GALANTER, E., & PRIBRAM, K. H. Plans and the structure of behavior. New York: Holt, Rinehart & Winston, 1960. SIMON, H. A., & NEWELL, A. Human problem solving: The state of the theory in 1970. American Psychologist, 1971, 26, 145-159. THOMAS, J. C., JR. An analysis of behavior in the hobbits-arcs problem. Cognitive Psychology, 1974, 6, 257-269. (Accepted
January
15, 1974)