Children's Analogical Reasoning about Natural Phenomena

JOURNAL OF EXPERIMENTAL CHILD PSYCHOLOGY ARTICLE NO.

67, 90–113 (1997)

CH972394

Children’s Analogical Reasoning about Natural Phenomena Sabina Pauen and Friedrich Wilkening University of Tu¨bingen, Germany This report investigates children’s analogical reasoning in a physics task, using an analogy generated by the children rather than by the experimenter. A total of 127 elementary school children took part in three related studies. Children learned to predict the behavior of a balance scale. Later, they were asked to solve a force interaction problem. Two versions of the balance scale training were devised: version A suggested an incorrect solution to the target problem (negative analogy), and version B suggested a correct solution to the target problem (positive analogy). In Study 1, 9- to 10-year-olds showed spontaneous transfer in both training conditions. In Study 2, 7-year-olds did not show any transfer in the positive analogy condition. Study 3 revealed that the lack of transfer in younger children was not due to a failure either to notice the analogy or to perform the mapping. Instead, 7-year-olds transferred only selected aspects of the correct solution. q 1997 Academic Press

Analogical reasoning has been of special interest to philosophers and scientists ever since Aristotle. To build an analogy means to compare two given incidences on the basis of their relational structure (Gentner, 1983; Reed, 1987; Ross, 1989; Vosniadou, 1989). For example, the relation between a chocolate and a melted chocolate is equivalent to the relation between a snowman and a melted snowman (Goswami, 1991), just as some relations between elements of our solar system are similar to relations between parts of a single atom (Gentner, 1989). Both comparisons—though differing largely in complexity—are analogical. Analogies can help people to understand a new problem by allowing them to compare the unknown problem’s relational structure to that of a familiar one (Oppenheimer, 1956) and to form general problem schemata by extracting the common causal relations of two given incidences while disregarding other details (Gick & Holyoak, 1983; Holyoak,

This research was fully supported by a doctoral fellowship of the Volkswagen foundation (Germany). We thank Elisabeth Spelke and Gretchen Van de Walle for helpful comments. Correspondence concerning this article and reprint requests should be addressed to Sabina Pauen or Friedrich Wilkening, Psychologisches Institut, Universita¨t Tu¨bingen, Friedrichstr. 21, 72072 Tu¨bingen, Germany. Electronic mail may be sent via Internet to sabina.pauen@ uni-tuebingen.de, or to [email protected]. 90 0022-0965/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.

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1984). Empirical evidence shows that analogies play a key role in adults’ learning about natural phenomena (Gentner & Gentner, 1983; Novick, 1988; Spiro, Feltovich, Coulson, & Anderson, 1989). Based on such findings, one might speculate that this may be the case for children as well. Developmental studies on this issue are rare, however. One reason seems to be that children younger than 10 years old have long been thought of as lacking the basic cognitive capacities to reason by analogy (Gallagher & Wright, 1979; Levinson & Carpenter, 1974; Lunzer, 1965; Piaget, Montangero, & Billeter, 1977). Only recently have developmental psychologists suggested that ‘‘analogy as a learning mechanism is a crucial factor in knowledge acquisition at all ages’’ (Brown, 1989, p. 370). New evidence comes from three different lines of research: (a) studies that use variations of the classical transfer paradigm (e.g., Goswami, 1991; Goswami & Brown, 1989), (b) studies that use metaphor interpretation tasks (e.g., Dent & Rosenberg, 1990; Gentner, 1977, 1988, 1989; Johnson & PascualLeone, 1989; Vosniadou, 1987), and (c) studies on complex analogical problem solving. This third line of research is of particular interest in the present context. Studies on analogical problem solving typically use the following general method: One experimental group is familiarized with a problem and its solution (base analog) and is later asked to solve a second problem that shares causal relations with the first, but involves different agents and objects. The critical test is whether the trained group uses the solution provided by the analogy more often than a control group that lacks such training. In most studies using this method, spontaneous transfer rates (i.e., transfer rates without an explicit hint to use the analogy) are surprisingly low (Brown, 1989; Gick & Holyoak, 1980, 1983; Holyoak, Junn, & Billman,1984). Only with help from the experimenter do many children notice the analogy and transfer the relevant solution. In addition to giving explicit hints, such help can include increasing the surface similarity between elements of the base and the target problem (Gentner & Toupin, 1986; Holyoak, Junn & Billman, 1984), highlighting the goal structure of both problems (Brown, Kane, & Echols, 1986), setting up a ‘‘learning set’’ to look for analogies (Brown & Kane, 1988; Brown, Kane, & Long, 1989; Crisafi & Brown, 1986), or presenting multiple analogies (Chen & Deahler,1989; Gholson, Eymard, Morgan & Kamhi, 1987; Gick and Holyoak, 1980). Even though children perform better when given such aids, spontaneous transfer still appears to be the exception rather than the rule (Brown, 1989). Why may this be the case? Laboratory studies on the development of children’s analogical reasoning skills typically use base analogs chosen by the experimenter rather than by the child. This procedure has one main disadvantage: If a child’s representation of the base analog and/or the target problem differs from that of adults, she may not consider the proposed analogy a useful tool for solving the task at hand. This could lead to low spontaneous transfer rates despite the presence of

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well-developed abilities to reason by analogy. Studies on conceptual change (e.g., Carey, 1991) indicate that children’s naive theories about the world may indeed differ substantially from those of adults. Several authors have thus pointed out that transfer abilities should only be tested by using analogies that refer to causal structures familiar to children of a given age range (Brown & Kane, 1988; Goswami & Brown, 1989). This may imply intensive pretesting in order to make sure that all causal relations relevant to the critical analogy are indeed understood and represented in the same way by children and adults. Such methodological problems can be avoided by exploring analogies that are spontaneously generated in natural settings. One knowledge domain in which observations of this kind have been made is biology. Carey (1985), for instance, told four-year-olds that people have ‘‘spleens’’ and asked whether various other animals and objects have spleens as well. From her findings, she concluded that preschool children use a ‘‘person analogy’’ to guide their inductions (see also Inagaki, 1990; Inagaki & Hatano, 1987, 1991). Since Carey did not intend to study analogical thinking in the first place, her conclusions regarding children’s analogical thinking are based on indirect evidence, only. Studies testing transfer abilities in laboratory settings have the advantage of allowing the experimenter to vary systematically critical aspects of the task. Studies exploring analogical transfer in natural settings have the advantage of referring to examples chosen by children. A promising way to investigate children’s analogical reasoning about natural phenomena may thus be to combine both approaches and to investigate transfer abilities in a laboratory task by using an analogy that children had previously generated on their own. A first attempt to follow this rationale is provided by the studies reported below. In a recent study on children’s knowledge about vector addition, Pauen (1996) observed that a substantial proportion of elementary school children felt reminded of a balance scale when trying to solve a force table problem. Pauen presented children with combinations of two unequal forces pulling at one target object. The target object was fixed to the center of a circular platform, and the platform was surrounded by a barrier with an opening. The forces were weights positioned on small plates. These plates hung from two cords, thereby pulling the target toward the edge of the table (see Fig. 1). In order to make the task more interesting for young children, they were being told a story in which a King (target object) got tired of skating on a frozen lake (platform) and wanted to be pulled into his royal bed, which was standing at the shore (box behind opening in the barrier). Children were asked to rotate the barrier so that the King would move into his royal bed. Underneath the barrier, there was a hidden angle scale, allowing for quantitative measurements of children’s responses. In physical terms, two forces were acting on the object simultaneously and

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FIG. 1. Force table task: Two forces represented by different numbers of weights positioned on plates pulled at a target. The target was fixed to the center of a platform surrounded by a barrier with an opening. Participants were asked to rotate the opening to a position that would allow the target to slide into a little box behind.

continuously. The number of weights on both plates provided proportional information about the amount of both vectors, and their direction was determined by the angle between the cords. Nine different vector addition problems of this kind were presented, and no feedback was given during the entire experimental session (i.e., the target was never released). Solving this kind of problem, the large majority of first- to third-graders (80–85%) and 45% of the fourth graders erroneously predicted that the object would always be pulled straight in the direction of the stronger component, a response pattern referred to as the one-force-only rule. The correct integration rule (target moves somewhere between the bisector of the angle and the stronger force) requires children to consider information about the number of weights on both plates instead of just looking at which plate carries more weights. This rule was used by 5% of the second-graders, and 10% of the third- and fourth-graders, respectively. Most children using the one-force-only rule argued that the plate loaded with more weights would go down and pull the target object straight behind. Such a pattern of motion characterizes the functioning of an ordinary balance scale. Many participants actually called the force table a balance scale, thus suggesting that the frequent use of the one-force-only rule in the WeightPulling study can be attributed to the use of a false analogy generated by the children themselves.

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The three studies to be reported next tested whether predicting the behavior of a balance scale before solving the force table problem would lead to spontaneous analogical transfer. In Study 1, one group of fourth-graders took part in an analogical training with an ordinary balance scale, thus suggesting transfer of the wrong solution (one-force-only rule; negative analogy); another group of fourth graders took part in an analogical training with a non-ordinary balance scale, thus suggesting transfer of the correct solution (integration rule; positive analogy). In Study 2, second-graders were presented with the positive analogy, only. Since fourth-graders but not second-graders showed spontaneous transfer of the correct solution, Study 3 investigated potential reasons for the lack of transfer in younger children. STUDY 1

Following the design of traditional transfer studies in the analogical problem solving context, children first received training with a balance scale, during which they learned to apply a specific rule to predict the scale’s behavior for different weight constellations. Two independent groups of children were familiarized with two differently constructed balance scales: One group was taught to use the one-force-only rule, and one was taught to use the integration rule. The experimenter then introduced the force table task. Teaching two different rules allows us to test whether the one-force-only rule observed in children solving the force table problem was really triggered by a false analogy. If the balance scale actually serves as an analogical model, then training with both kinds of balance scales should have an effect. Alternatively, children may have alluded to the ordinary balance scale as another situation to which the one-force-only rule applies post hoc. In this case the group that was taught to use the integration rule should not show any transfer. We expected children of both experimental groups to apply the corresponding rule (either the inappropriate one-force-only rule or the appropriate integration rule) to the force table task more often than children without any prior analogical training. A third group of untrained children served as a control group. Method Participants. A total of 51 fourth-graders took part in the study. Fourthgraders were chosen because they had exhibited the greatest variability in their rule choice for the force table task. Group A (one-force-only rule training) consisted of 15 children (mean age: 10;4; range: 9;11 to 10;7), and Group B (integration rule training) consisted of 16 children (mean age: 10;2; range: 9;10 to 10;5). Group C, with 20 fourth graders (mean age: 9;11; range: 9;10 to 10;3), served as a control group. All participants were recruited from an elementary school in Marburg, a small university town in Germany. They came from a middle class economic background. Males and females were equally represented in each group.

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Apparatus. The force table was identical to that used by Pauen (1996; see Fig. 1). Because developmental studies have shown that surface similarities may enhance the likelihood of spontaneous analogical transfer (e.g., Gentner & Landers, 1985; Gentner & Toupin, 1986; Holyoak & Koh, 1987), the balance scale resembled the force table apparatus in its physical appearance. Figure 2 shows the two balance scales used for the experimental training of Group A and Group B, respectively. In each case, a beam was suspended between two upright standing metal bars fixed on a heavy wooden platform. Two plates loaded with weights hung down from the beam at equal distances from the middle, where a pointer was fixed. The pointer pointed straight up whenever the beam was in balance, and it followed the beam’s rotation when different numbers of weights were pulling at both sides. The maximum pointer deflection (about 507 to each side) was determined by the maximum turn of the scales. The pointer rotated in front of a stationary disk with a dark rim that was attached to one of the two upright standing metal bars. To accentuate the pointer’s position of maximum deflection, two cords framed the 1007 angle on the disk. A clamp could be attached in any position at the rim. It was used to predict the movement of the pointer. The two balance scales used in each training condition appeared identical. The only difference concerned the suspension of the beam which was hidden behind the disk and invisible to participants. The balance scale used for Group A was constructed so that the center of gravity and the axis of rotation of the beam fell together in one place (as for an ordinary balance scale). Under these conditions, the beam and the pointer rotate toward the stronger force and do not stop until the heavier plate reaches the bottom (see Fig. 2a). Children seeing this balance scale learned that the stronger force alone—regardless of the proportion of weights and the degree of deflection—determined the movement of the target object, a response consistent with the one-force-only rule. In the balance scale used for Group B, the center of gravity was located below the axis of rotation (swing boat suspension). As a result, the beam again rotates toward the stronger force, but the degree of deflection depends on the size of both forces (see Fig. 2b). The larger the difference between the forces, the more the beam turns toward the heavy side. Under these conditions, correct predictions of the pointer’s movement are consistent with the integration rule, holding for the force table situation.1 Procedure. In order to familiarize children with the basic mechanism of 1

This does not mean, however, that the balance scale situation provides a perfect analogy to the force table problem in all respects. In the case of the balance scale, a physics expert would predict the pointer’s movement based on knowledge about the location of the center of gravity. In the case of the force table, the movement direction of the target object would be predicted based on knowledge about the principles of vector addition. Different physical laws thus seem crucial for an expert interpretation of the balance scale and the force table problem.

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FIG. 2. Two different versions of a balance scale were used for an analogical training of two different rules. (a) One-force-only rule: The center of gravity and the axis of rotation fall together in one place. The beam and the pointer rotate towards the stronger force until the heavier plate reaches the bottom. (b) Integration rule: The center of gravity was located below the axis of rotation (swing boat suspension). The degree of the pointer’s deflection depends on the size of both forces. The heavier plate never reaches the bottom.

the balance scale, the experimenter pressed down one plate and showed the child that the pointer rotated until it covered the framing cord and the plate reached the bottom. During the training session, children were told to put a certain number of weights on both plates while the experimenter kept the

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beam from turning. In the next step, they were asked to fix the clamp in the position on the rim where they expected the pointer to stop its rotation. The child then saw whether the clamp was placed correctly. Once the pointer had reached its final position, the child was asked to explain the event, and the experimenter provided feedback. The analogical training consisted of two parts. During the first part, four different weight proportions were presented in a fixed order (1:1, 1:2, 1:3, 1:6). After that, the experimenter verbally summarized the pointer’s behavior. Group A was told: ‘‘If one plate holds more weights than the other, the pointer moves all the way down to the heavier side and stops right above the cord.’’ Group B was told: ‘‘The more weights we put on the heavier plate, the further down to that side the pointer moves. The pointer will only reach the cord if one plate carries no weight at all.’’ During the second part of the analogical training children received the same four weight proportions. This time, the heavy side changed (1:1, 2:1, 3:1, 6:1) and the experimenter made no comments but reported children’s predictions on a protocol sheet. This phase of the training served to put children in a situation as similar to the experimental setting of the force table problem as possible. Following the training, the experimenter introduced the force table task (for details, see Pauen, 1996). Children were asked to predict the movement of the target object for nine different combinations of weight proportions (1:2, 1:3, 1:6) and angles between both cords (457, 757, 1057) presented in random order. No hints were given to compare the balance scale situation to the force table problem. The balance scale remained visible (on a side table) during the entire experimental session. The instructions and the procedure in the force table task were identical for both experimental groups (A, B) and the control group (C), who had received no balance scale training. Results Analogical training. Both experimental groups learned the appropriate rule for predicting the movement of either balance scale (ordinary or swing-boat suspension). All children who had been taught the one-force-only rule during the first part of the analogical training consistently predicted that the pointer would stop right above the cord at the heavy side, regardless of the given weight proportion of both forces during the second part of the training. All children who had learned the integration rule consistently predicted that the pointer would stop somewhere between its starting position and the cord on the heavy side, depending on the given weight proportion. Rule choice at the force table task. Based on their responses to the force table problem and the criteria defined by Pauen (1996), children were assigned to one of four distinct rule categories. Participants using two rules in alternation needed to be differentiated from those applying only one rule. Probability analysis suggest that the same alternative should be chosen on more than seven of nine trials in order to identify a one-rule response pattern (p Å .039,

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PAUEN AND WILKENING TABLE 1 Percentage of Fourth-Graders in Different Training Conditions (A, B, C) Using Different Rules at the Force Table Task Training condition

Rule

A One-force-only rule training (N Å 15)

B Integration rule training (N Å 16)

C No analogical training (N Å 20 a)

One-force-only Integration Two-forces considered Nonidentifiable

86.7 0.0 0.0 13.2

31.3 50.0 6.3 12.5

45.0 10.0 35.0 10.0

a

Data taken from Pauen (1996).

õ a Å 5%). Accordingly, an individual was assigned to a single rule category only if she had applied the same rule at least eight times. Those who consistently predicted that the King would be pulled straight in the direction of the stronger force were classified as following the one-force-only rule. Those who consistently predicted that the object would move in a direction somewhere in between the bisector of the angle and the stronger force were classified as following the integration rule. Children who alternated between both solutions were classified as following a third rule, the two-forces-considered rule. Other response patterns were classified as nonidentifiable. Table 1 shows the percentages of children using each of the defined rules in all three conditions. Children who had seen the one-force-only rule operating in the balance scale task (Group A) used the same rule in the force table task significantly more often than the control group, x2(1, N Å 35) Å 4.71, p £ .05, continuity corrected. Similarly, children who had learned the integration rule in the swing-boat suspension version of the base analog (Group B) used this rule to solve the target problem significantly more often than the control group, x2(1, N Å 36) Å 5.24, p £ .05, continuity corrected. In both experimental groups the percentage of children answering in accordance with the two-forces-considered rule, thus vacillating between the one-force-only and the integration rules, was significantly lower than in the control group, Fisher’s Exact Test (Group A, N Å 35) p £ .05 and (Group B, N Å 36) p £ .05.2 Nonidentifiable response patterns were found about equally often in all three groups.

Discussion The results clearly show that children’s responses to the target problem varied systematically with the solution provided by the analogical model: The 2

Fisher’s Exact Test was used when more then 25% of the cells contained expected frequencies lower than 5.

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specific rule that predicted the pointer’s behavior during the balance scale training was most frequently applied to the force table task in both experimental groups. This strongly suggests that children were actually engaged in analogical reasoning when trying to solve the target problem. Although our results show that the balance scale provides a good base analog for the force table problem, why should this be the case? On the level of perceptual similarity, the sight of scales loaded with weights and the color identity of central elements to be mapped might have triggered the memory of an ordinary balance scale. However, the analogy cannot be a purely perceptual one. On the level of structural similarity, both problems share the fact that two forces are pulling at one target object, hence influencing its direction of movement. In order to map the pointer’s movement onto that of the King and to transfer different kinds of rules, children had to consider this structural similarity. At the same time, they needed to ignore several structure preserving differences (Holyoak, 1984): (1) The pointer and the King differ substantially in their global shape. (2) They perform different kinds of movement (pointer rotates, King moves forward). (3) Fixing a clamp at the rim of an upright standing disk and turning the opening in a barrier around a large platform are rather distinct activities required by the children. (4) The instruction for the balance scale task is neutral whereas the cover story for the force table task provided a fairy-tale context. The finding of high spontaneous transfer rates despite many surface differences between both tasks thus seems surprising. It should be noted, however, that the children who took part in Study 1 were already 9 to 10 years old. This leaves open the possibility that spontaneous analogical transfer requires the attainment of a certain stage in children’s development of analogical reasoning abilities (e.g., Piaget et al., 1977). Alternatively, one could argue that spontaneous analogies can be observed in children of any given age (e.g., Brown, 1989). This controversy raises the question whether younger children would show similar transfer performance in the present task context. Study 2 tested this hypothesis. STUDY 2

In the study described by Pauen (1996), first- to third-graders adhered to the inappropriate one-force-only rule in the force table situation to a much greater extent than the fourth-graders; about 80% of the second-graders applied this rule. Hence, there seems to be too little room for a significant increase in the rate with which the one-force-only rule could be applied to the force table problem as an effect of training with an ordinary balance scale. However, training with the second version of the balance scale (swing-boat suspension) could increase the use of the integration rule on the target problem in second graders. Accordingly, Study 2 only included a balance scale training for the integration rule. We hypothesized that second-graders transfer the integration rule to the force table problem.

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Method Participants. A sample of 20 second-graders (9 boys, 11 girls; mean age: 7;5; range 7;1 to 7;9) participated in the analogical training (Group B). A second sample of 20 second-graders (10 boys, 10 girls; mean age: 7;8; range 7;6 to 7;11) served as a control group (Group C). All children came from a white middle class economic background. They were recruited from an elementary school in Marburg, Germany. Material and procedure. The balance scale with the swing boat suspension (Fig. 2b) and the force table were identical to those of Study 1. The procedure was the same as that used in Study 1. Results The balance scale training proved to be effective; during the second half of the familiarization trials, children consistently applied the integration rule to the balance scale task. However, when trying to solve the force table problem, 70% of the children still answered according to the one-force-only rule. A similar percentage of the control group (80%) followed the same rule. Only four children (20%) of the experimental group integrated both forces correctly, as did one child (5%) of the control group. The remaining 10% (15% under control conditions) showed nonidentifiable response patterns. In contrast to our hypothesis, analogical training did not increase the number of children using the integration rule significantly, Fisher’s Exact Test (N Å 40): p Å .17, one-tailed. Discussion In contrast to the fourth-graders, the majority of the second-graders failed to transfer the integration rule to the force table problem, despite efficient learning of this rule. Several reasons for this lack of transfer in younger children seem possible. Perhaps children just memorized the correct solution for specific weight proportions in the balance scale problem, but did not understand the general solution principle as summarized by the experimenter. Alternatively, the analogical training might have been too short to weaken children’s idea of how balance scales usually work (that is, in accordance with the one-force-only rule). Another possibility is that children needed a more explicit hint to compare both problems in order to transfer the integration rule. Or, they might have had problems mapping all relevant elements of both situations. Finally, they may have needed more time to think about both problems in order to understand their analogical structure. Instead of giving more explicit hints to use the given analogy, as most studies on children’s analogical reasoning do upon obtaining negative results, Study 3 explores possible explanations for the lack of transfer in Study 2. STUDY 3 The lack of transfer observed among second-graders in Study 2 may be attributed to difficulties in learning or memorizing the integration rule, to

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general problems concerning analogical reasoning abilities, or to a combination of both. Study 3 employs a revised version of the task to address these issues. This revised task allowed us to compare the effects of an elaborated analogical training on different transfer measures for younger and for older children. Children’s spontaneous comments and their answers to explorative interview questions were included in the data analysis. Using this additional information, we were able to determine (a) how well second- and fourthgraders memorized various parts of the integration rule across different occasions, (b) to what extent children of both age groups spontaneously noticed the analogy between the balance scale and the force table task, (c) which elements of both problems they mapped onto each other, and (d) what kind of relational information was transferred by younger and by older children, respectively. In order to check for the possibility of delayed transfer, the force table task was presented twice with an interval of one day. This also enabled us to test the stability of potential transfer effects. Method Participants. A total of 36 children, 18 second graders (mean age of 7;9; range 7;0 to 7;11) and 18 fourth-graders (mean age of 9;7, range: 9;1 to 9;11) from an elementary school in Marburg (Germany) took part in the study. Males and females were equally represented in both samples. Material. The same balance scale with the swing boat suspension (see Fig. 2b) and the same force table apparatus (see Fig. 1) were used as in Study 2. Procedure. The set of training trials was increased by adding the weight proportion of 0:6. This case was introduced to demonstrate that the pointer could only reach the maximum deflection mark when there was no weight pulling at the opposite side. Children had to make predictions for the following sequence of trials: 1:1, 1:2, 1:3, 1:6, 0:6, 1:6. Visual and verbal feedback was provided after each trial. After the presentation of the first series of trials, the experimenter summarized the pointer’s behavior by stating a rule that consisted of three different aspects: ‘‘When one plate holds more weights than the other, the pointer will always move to the heavier side.’’ (Part 1) ‘‘The more weights we put on the heavier plate, the further the pointer rotates towards the heavier side.’’ (Part 2) ‘‘If there are any weights on the lighter side, the pointer will never rotate to the maximum position on the heavy side. The pointer will stop right at the cord (maximum deflection) only when one plate holds no weights at all.’’ (Part 3). Following these explanations children were again shown the same set of trials as before, but this time the side with more weights was changed. Unlike in Study 2, children were given visual and verbal feedback during this part of the training as well. From the second trial on, the experimenter asked the child why the pointer did not stay in the middle, why it had rotated further down than before, and why it had (not) reached the cord. These questions encouraged the child to verbalize all three parts of the general rule, thereby

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actively recalling them. If the child could not answer one of these questions, the experimenter repeated that part of the solution. In order to test the success of the balance scale training, a third series of 10 trials was presented. All unequal weight proportions appeared twice in a random order with each side of the scale carrying more weights in one of the two trials. The children could see the actual movement of the pointer but did not get any verbal feedback from the experimenter. The position of the clamp was noted for each test trial. At the end of the training session, the child was asked: ‘‘Can you explain the rules of the balance scale game? How did you know where the pointer would stop? What did you have to consider in order to put the clamp into the right position?’’ Later, the experimenter specifically asked for different parts of the integration rule: ‘‘What happens if both plates carry the same number of weights? What happens if one plate carries more weight than the other? Does it matter how many weights we put on the heavy plate? What happens to the pointer when there are six weights on this plate and only one on the other? When does the pointer stop right at the cord?’’ These questions not only served to help children recall different parts of the integration rule; they also served to highlight the goal structure of the balance scale task in order to increase the likelihood of analogical transfer (e.g., Brown, Kane, & Echols, 1986). Following the analogical training, the force table task was presented, as in the previous studies. Again, no explicit hints were given to compare both situations. The session ended with an explorative interview in which the experimenter asked the following questions: (1) ‘‘What are the rules of the King game?’’ (2) ‘‘What are the rules of the balance scale game?’’ (3) ‘‘Do you think that both games were similar to each other, or do you think they were different?’’ (4) ‘‘What was similar?’’ (5) ‘‘What was different?’’ If the child did not mention all parts of the integration rule while answering Question 2, the experimenter repeated the questions that had been asked immediately after the balance scale training. This allowed us to determine which aspects of the rule were recalled spontaneously, which parts were only mentioned in reaction to explicit questions, and which parts were not recalled at all. Before the child left the room, she was informed that the King game would be repeated the next day. The next day the child was again asked to solve the force table task and to answer questions (1) and (2) about the rules of the King game as well the balance scale game. The balance scale was also visible on a table nearby, but was not mentioned by the experimenter. Results Effectiveness of the balance scale training. Children’s predictions for the 10 test trials presented at the end of the balance scale training enabled us to judge whether they had learned to use the integration rule. If the clamp was always positioned on the heavier side, we concluded that the child had learned

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Part 1 of the integration rule. If the position of the clamp systematically varied with the number of weights on the heavy plate, we concluded that Part 2 of the integration rule had been learned as well. If the child attached the clamp right above the cord on the heavy side only for weight proportions of 0:6 or 6:0, we concluded that Part 3 of the integration rule had been learned. All 36 children in both age groups consistently responded according to all three parts of the integration rule when predicting the movement of the pointer. Additional data concerning the effectiveness of the balance scale training were provided by children’s answers to Question 2 (‘‘What are the rules of the balance scale game?’’) immediately after the training (1), after the first presentation of the force table task (2), and after the second presentation (3). Two independent judges coded whether a specific part was mentioned spontaneously (2 points), mentioned only after explicitly being asked by the experimenter (1 point), or was not mentioned at all (0 point) by a given individual. The mean intercorrelation of all codings between both judges was r Å .87 (lowest coefficient: r Å .70). Cases of different scorings were discussed and resolved in agreement. A detailed description of the coding criteria for various parts of the integration rule can be found in Appendix 1. In each age group, two complete verbal transcripts are missing due to problems with the original tape recording in each age group. Therefore, the remaining analyses refer to data of 16 second-graders and 16 fourth-graders. Table 2 shows the results. A comparison of the second graders’ performance on the first (1) and last (3) occasion showed that they remembered Part 1 of the integration rule only slightly less on occasion 3 than on occasion 1, McNemar Test (N Å 16), p § .05. However, Part 2 and Part 3 of the integration rule showed a significant decrease across time for this age group, McNemar Test (N Å 16) for Part 2: p £ .05, for Part 3: p £ .05. Fourth graders mentioned all three parts about equally often across occasions, McNemar Test (N Å 16), p § .05. These data only concern children’s spontaneous performance, however. When explicitly asked about different aspects of the integration rule, every child in both age groups was able to repeat all three parts on all three occasions, thus indicating that no part of the integration rule had actually been forgotten. Taken together with the data on the balance scale learning test, these results demonstrate that the analogical training was effective. After the training, second- and fourthgraders not only applied the integration rule consistently, but they also were able to remember and to recall verbally all parts of this rule on the next day. However, second-graders showed a considerable decline in their spontaneous mentioning of Part 2 and Part 3, whereas fourth-graders did not. Spontaneous comparison of the base analog and the target situation. In order to determine the number of children in both age groups who spontaneously noticed parallels between the balance scale and the force table task, two independent judges decided whether each child had made such a remark before the experimenter had explicitly asked about the similarities of the

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TABLE 2 Percentage of Second- and Fourth-Graders Who Spontaneously Recalled Different Parts of the Integration Rule on Three Successive Occasions Occasion

Rule part

1 After analogical training (integration rule)

2 After 1st presentation of force table task

3 After 2nd presentation of force table task

Second-graders (N Å 16) a

1 2b 3c

81.3 75.0 75.0

68.8 56.3 37.5

68.8 31.3 12.5

Fourth-graders (N Å 16) a

1 2b 3c

50.0 75.0 43.8

50.0 62.5 50.0

50.0 81.3 62.5

a

Target moves more to the side of stronger force. Movement of target is determined by exact weight proportion. c Weaker force has some influence on movement. b

two tasks (Question 3 of the explorative interview). An example of such a spontaneous remark is given by Tovel (9;8), who explained how she knew where to put the King’s bed in the force table task: ‘‘it’s the same as with the balance scale.’’ Both judges identified the same comments in 85% of all cases. Unclear cases were discussed and solved in agreement. The majority of children in both age groups spontaneously mentioned parallels between the balance scale and the force table task and noticed the analogy (56% of second-graders and 75% of fourth-graders). This age difference was not significant, x2 (1, N Å 32) Å 0.6, p § .05. Mapping performance. Potential differences between the mapping performed by second- and fourth-graders were analyzed by counting the number of children in each age group who mapped elements crucial for noticing the analogy. These crucial element pairs were: (1) weights–weights, (2) pointer– King, (3) clamp–box. A pair of elements was considered to be mapped by a given child if she explicitly related them to each other within one sentence either during her attempts to solve the force table task or in response to one of the explorative questions ‘‘What was similar? What was different?’’ For example, one child responded: ‘‘there was a King standing in the middle and here was a pointer’’ (Miriam, 7;8), and another said ‘‘the bed of the King moves and the clamp does the same’’ (Johannes, 7;6). Two independent judges identified the same pairs in 88% of all cases. Unclear cases were

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ANALOGICAL REASONING ABOUT NATURAL PHENOMENA TABLE 3 Number of Second- and Fourth-Graders Who Mapped Specific Pairs of Objects in an Improved Analogical Training of the Integration Rule

Pairs of mapped objects

Second-graders (N Å 16)

Fourth-graders (N Å 16)

Weights–weights Pointer–king Clamp–box

15 13 12

16 16 16

discussed and resolved in agreement. Table 3 shows the number of secondand fourth-graders who made valid mappings of different element pairs. Most second-graders and all fourth-graders mapped the central elements of both tasks. None of the differences between the age groups was significant, not even the largest one for clamp–box, x2 Å 2.39 (1, N Å 32), p § .05, continuity corrected. Rule choice at the force table task. In addition to verbal protocols, children’s predictions at the force table provided a crucial measure of transfer performance. The response pattern of each child was analyzed in terms of rules (see Study 1). Figure 3 shows how many children used the one-force-only, two-forces-considered, or integration rule after the improved analogical training of the integration rule during their first session with the force table (Group B/). To allow for a better comparison of the results obtained under different conditions, data from the control groups and the analogical training groups participating in Study 1 (fourth graders) and Study 2 (second graders) are also presented. Figure 3a demonstrates that second-graders responded consistently across conditions. Most of these younger children followed the one-force-only rule. Few other response patterns were observed. The number of second-graders using the integration rule did not differ significantly between the improved analogical training group and the control group, Fisher’s Exact Test (N Å 36), p § .05, one-tailed. For fourth graders (see Fig. 3b), the same comparison yielded a significant result, x2 (1, N Å 36) Å 15.35, p £ .05. More than 70% of the fourth graders used the integration rule after the improved analogical training. This represents an increase of 20% over the short balance scale training condition (Group B) and an increase of 60% over the control group. These findings suggest that only the fourth-graders profited from the more extensive analogical training. When the children were given the force table task a second time, 16 of 18 second-graders and 16 of 18 fourth-graders applied the same rule that they had used on the previous day, thus indicating high stability in performance. Rule transfer from the balance scale to the force table task. Children’s spontaneous comments in the force table task and their responses to the first

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FIG. 3. Percentage of children (second- and fourth-graders) using different rules under control conditions (C), after a short analogical training of the integration rule (B), and after an improved analogical training (B/) of the integration rule: One-force-only rule (target moves in the direction of the bisector of the angle between both forces), bisector rule (target moves in the direction of the bisector of the angle between both forces), integration rule (target moves to a position between the bisector and the stronger force), two-forces-considered rule (children varied between the oneforce-only and the integration rule), nonidentifiable (all other response patterns).

explorative question (‘‘What are the rules of the King game?’’) gave important additional information concerning their transfer performance. Two independent judges coded which children transferred different parts of the integration rule to the force table task during the first session (see Appendix 1 for a description of the coding criteria). The mean intercorrelation between all

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TABLE 4 Percentage of Children Who Transferred Different Parts of the Integration Rule after an Improved Analogical Training of the Integration Rule

Rule part

Second-graders (N Å 16)

Fourth-graders (N Å 16)

1a 2b 3c

93.8 0.0 18.8

87.5 75.0 81.3

a

Target moves more to the side of stronger force. Movement of target is determined by exact weight proportion. c Weaker force has some influence on movement. b

codings of both judges was r Å .89 (lowest score: r Å .85). Table 4 shows the results. Almost all second- and fourth-graders transferred Part 1, stating that the King would move in the direction of the stronger force. In contrast, Part 2, indicating that the number of weights on the heavier plate is crucial for predicting the movement direction of the target object, was transferred by no single second grader but by 12 of 16 fourth-graders. A similar age difference was obtained for Part 3. Whereas 13 of 16 fourth-graders remarked that the target object could only move exactly in the direction of the stronger force if there was no force pulling in a different direction, only 3 of 16 secondgraders argued as such. Both age differences were significant; Part 2: x2 (1, N Å 32) Å 19.2, p £ .05; Part 3: x2 (1, N Å 32) Å 11.6, p £ .05. Discussion Study 3 tested various possible explanations for the lack of analogical transfer observed in younger children. Despite the effectiveness of the balance scale training, the large majority of second-graders did not transfer the integration rule but used the one-force-only rule in the force table task. In contrast, most fourth-graders transferred the solution provided by the analogical model. These findings replicate the results of Study 1 and Study 2. In addition, we found that analogical transfer to the force table task remained remarkably stable until the next day for both age groups. Given that second-graders as well as fourth-graders applied the integration rule successfully in the balance scale task, it seems unlikely that limited information processing abilities account for the lack of transfer in second-graders. Children of this age had no problems considering the proportion of weights on both plates in order to predict the pointer’s movement correctly. Furthermore, it seems unlikely that general problems concerning the ability of the children to reason by analogy account for the observed age differences. Most second-graders noticed the analogy between the balance scale and the force table task without any help

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from the experimenter. They also performed the mapping between all major elements of both tasks. The only striking age difference between both age groups concerns the organization of knowledge about the balance scale in relation to the target problem. The fact that the target object moves in the direction of the stronger force (Part 1 of the integration rule) appeared to be most important for the large majority of second-graders. They spontaneously mentioned this when asked about the rule of the balance scale game. Part 2 and Part 3, on the other hand, were not forgotten but were spontaneously recalled with decreasing frequency on the second and third testing occasions. For the large majority of second-graders, information neither about exact weight proportions (Part 2) nor about the influence of the weaker force on the target object (Part 3) was considered to be very important for describing the pointer’s behavior. In contrast to younger subjects, most fourth-graders spontaneously mentioned Part 2 and Part 3 of the integration rule on all three testing occasions. The same age differences were present in verbal transfer rates: Most older children explained their predictions at the force table task with reference to Part 2 and Part 3 of the integration rule, whereas most younger children focused mainly on Part 1. These age differences could be crucial for explaining the lack of transfer observed among younger children. Given the ability to recall all parts of the integration rule when asked by the experimenter, why did most second-graders (and some fourth-graders) focus on different aspects of the analogical solution than did the majority of fourth-graders (and some second-graders)? One possible explanation is that those who showed no positive transfer did not consider all parts of the integration rule to be equally relevant to the force table problem. Instead, they might have actively recalled and transferred only knowledge about the balance scale which seemed consistent with their naive interpretation of the force table problem. In line with this interpretation, we observed that children whose performance at the force table was atypical for their age group also showed atypical performance in spontaneous recall and verbal transfer of various parts of the integration rule: Second-graders who already did find the correct solution to the force table problem (n Å 2) spontaneously mentioned Part 2 as well as Part 3 of the integration rule even on the last testing occasion, and they verbally transferred Part 3 of the integration rule. Fourth-graders who still applied the one-force-only rule in the force table situation (n Å 3) mentioned neither Part 2 nor Part 3 of the integration rule until the third testing occasion, and they only transferred Part 1 to the force table problem. Yet another factor that might have contributed to the lack of transfer mostly in younger children concerns participants’ representation of the balance scale situation. Verbal protocol data suggest that during the analogical training many younger children paid special attention to the upward and downward motion of the plates and tried to understand how a given weight proportion caused the movement. On the target problem, these children might have

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transferred what they knew about the movement of the plates rather than their knowledge of the pointer’s movement. Accordingly, they expected the King to be pulled straight behind the heavier plate that moves down—a prediction that is perfectly consistent with the one-force-only rule. Both explanations suggest that most second-graders and some fourth graders may have represented the balance scale and the force table task differently. They did not lack the ability to encode the proper solution, and they were clearly engaged in analogical reasoning. An interesting goal for a future study could be to design a specific training for younger children that would change their representation of the base analog. Assuming the effectiveness of such a training, second- as well as fourth-graders should be able to transfer the integration rule from the balance scale situation to the force table problem. GENERAL DISCUSSION

Three studies tested children’s analogical transfer in a physics problem solving task. Second- and fourth-graders first took part in an analogical training with a balance scale and later worked on a force table problem. Independent samples of fourth-graders were provided with different solutions to the balance scale problem. In contrast to the majority of findings reported in the literature (e.g., Brown, 1989; Brown, Kane & Echols, 1986; Goswami, 1992), children of this age spontaneously transferred both kinds of solutions in all relevant aspects. Second-graders, on the other hand, only transferred aspects matching both their prototypical concept of a balance scale and their naive assumptions about the functioning of the force table. Consistent with earlier findings on the development of reasoning about the balance scale situation (Siegler, 1976; Wilkening & Anderson, 1982), secondgraders determined which plate carried more weights in order to predict the movement of the target object. Even though children of this age group have learned to consider information about the exact proportion of weights pulling on both sides of the beam during the analogical training, this experience did not induce a general change in their representation of the balance scale situation. This was indicated by the sharp drop in the number of second-graders who spontaneously recalled this aspect of the solution on the day following the training session. This suggests that younger children may need to be mentally prepared in order to recall and transfer all aspects of the more advanced integration rule. Further support for this hypothesis is provided by the results of the fourthgraders. Whereas a substantial proportion of the children in the control group used the one-force-only and the integration rules in alternation, the percentage of those using the two-forces-considered rule dropped significantly in both analogical training conditions. Based on these findings one might speculate that the base analog helped them to make up their minds which solution might be appropriate for the force table problem. It should be noted, however, that the between-subject design does not allow any final conclusion with

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respect to the question whether children using the two-forces-considered rule were indeed more likely to show analogical transfer than others. Taken together, the presented findings seem to suggest that transfer is only likely to occur if children are prepared to accept the analogical solution (i.e., the integration rule) as a plausible solution to the target problem. Studies testing children’s general ability to reason by analogy therefore need to provide a base and a target problem for which the causal structures necessary to notice the interesting analogy are salient in the children’s representation of both problems. An anecdote from our own study illustrates how children’s ways of representing natural phenomena can differ from those of adults, and how these differences can lead to the application of rather unconventional analogies: While working at the force table task, one second-grader explained why the plate with only one weight on it would move up. He said: ‘‘It’s actually very simple: One weight is quite small. Small things are not as heavy as large things. That’s why they go up in the air. I know it from birds— they are small, they don’t weigh much, and they can fly.’’ This example demonstrates that children’s causal understanding and similarity judgments can differ from those of adults, despite the fact that they are generally able to reason by analogy. This has important methodological implications for empirical studies of children’s developing analogical reasoning abilities. If the base analog has been chosen by the experimenter and if younger children show poor transfer performance, at least two alternative interpretations seem possible: Either children have general difficulties when reasoning by analogy, or the proposed analogy is meaningless to them. In the latter case, a lack of transfer does not imply a lack of analogical reasoning ability. (Imagine a child experimenter who first explains to you that birds can fly because they are small and don’t weigh much and then confronts you with the force table situation, expecting you to use the one-force-only rule as a proof of your ability to reason by analogy!) Although it may seem impossible to separate clearly the ability to reason by analogy from the ability to understand specific causal relations, this is not necessarily the case. More valid conclusions concerning the development of analogical reasoning abilities can be provided by studies that use analogical examples based on causal relations which are part of children’s representation of the base as well as of the target problem. In sum, the presented evidence supports the general hypothesis that young children generate analogies to explain natural phenomena. Developmental psychologists who want to explore children’s understanding of biology, physics, or other domains should thus be aware of the possibility that spontaneous analogical thinking might be involved. At the same time, they should be prepared to find children making rather unconventional analogical comparisons. These unconventional comparisons may well allow new insights into the child’s understanding of causal relations within the given domain.

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APPENDIX 1

Criteria for Coding Verbal Responses to Question 2 of the Explorative Interview Part 1 of the integration rule was considered to be used when a child said that he had to find out where more weights were pulling, or which plate would move down/up. Part 2 of the integration rule was considered to be used (a) when the child claimed that the movement of the target object varied with the number of weights on the heavier plate, (b) when he compared two specific weight proportions and pointed out that the target object would move further to the heavy side in one of both cases, or (c) when the child associated gradual judgments (a little bit, a lot) concerning the pointer’s movement behavior in an appropriate way with specific weight proportions. Part 3 of the integration rule was considered to be used when a child mentioned that the pointer could not reach the cord on the side where the heavy plate was pulling because the other plate still had a weight on it, or when he stated that the pointer would only reach the cord if no weight pulled on the other side. REFERENCES Brown, A. L. (1989). Analogical learning and transfer: What develops? In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 367–412). Cambridge: Cambridge Univ. Press. Brown, A. L., & Kane, M. J. (1988). Preschool children can learn to transfer: Learning to learn and learning by examples. Cognitive Psychology, 20, 493–523. Brown, A. L., Kane, M. J., & Echols, C. H. (1986). Young children’s mental models determine analogical transfer across problems with a common goal structure. Cognitive Development, 1, 103–121. Brown, A. L., Kane, M. J., & Long, C. (1989). Analogical transfer in young children: analogies as tools for communication and exposition. Applied Cognitive Psychology, 3, 275–293. Carey, S. (1985). Conceptual change in childhood. Cambridge, MA: MIT Press. Carey, S. (1991). Knowledge acquisition: Enrichment or conceptual change? In S. Carey & R. Gelman, (Eds.), The epigenesis of mind: Essays on biology and cognition. Hillsdale, NJ: Erlbaum. Chen, Z., & Daehler, M. W. (1989). Positive and negative transfer in analogical problem solving. Cognitive Development, 4, 327–344. Crisafi, M. A., & Brown, A. L. (1986). Analogical transfer in very young children: Combining two separately learned solutions to reach a goal. Child Development, 57, 953–968. Dent, C., & Rosenberg, L. (1990). Visual and verbal metaphors: Developmental interactions. Child Development, 61, 983–994. Gallagher, J. M., & Wright (1979). Piaget and the study of analogy: Structural analysis of items. In J. Magary (Ed.), Piaget and the helping professions (Vol. 8, pp. 114–119). Los Angeles: University of Southern California. Gentner, D. (1977). Children’s performance on a spatial analogical task. Child Development, 48, 1034–1039. Gentner, D. (1983). Structure-mapping: A theoretical framework for analogy. Cognitive Science, 7, 155–170. Gentner, D. (1988). Metaphor as structure-mapping: the relational shift. Child Development, 59, 47–59.

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Gentner, D. (1989). The mechanism of analogical learning. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 199–241). Cambridge: Cambridge Univ. Press. Gentner, D., & Gentner, D. R. (1983). Flowing waters or teeming crowds: mental models of electricity. In D. Gentner & A. L. Stevens (Eds.), Mental models (pp. 99–129). Hillsdale, NJ: Erlbaum. Gentner, D., & Landers, R. (1985). Analogical reminding: A good match is hard to find. Proceedings of the International Conference on Systems, Man, and Cybernetics. Tucson, AZ. Gentner, D., & Toupin, C. (1986). Systematicy and surface similarity in the development of analogy. Cognitive Science, 10, 277–300. Gholson, B., Eymard, L. A., Morgan, D., & Kamhi, A. G. (1987). Problem solving, recall and isomorphic transfer among third-grade and sixth-grade children. Journal of Experimental Child Psychology, 43, 227–243. Gick, M. L., & Holyoak, K. J. (1980). Analogical problem solving. Cognitive Psychology, 12, 306–355. Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15, 1–38. Goswami, U. (1991). Analogical reasoning: What develops? A review of research and theory. Child Development, 62, 1–22. Goswami, U. (1992). Analogical reasoning in children. Hillsdale, NJ: Erlbaum. Goswami, U., & Brown, A. (1989). Melting chocolate and melting snowmen: Analogical reasoning and causal relations. Cognition, 35, 69–95. Holyoak, K. J. (1984). Analogical thinking and human intelligence. In R.J. Sternberg (Ed.), Advances in the psychology of human intelligence (Vol. 2). Hillsdale, NJ: Erlbaum. Holyoak, K. J., Junn, E. N., & Billman, D. O. (1984). Development of analogical problem-solving skills. Child Development, 55, 2042–2055. Holyoak, K. J., & Koh, K. (1987). Surface and structural similarity in analogical transfer. Memory and Cognition, 15, 332–340. Inagaki, K. (1990). Young children’s use of knowledge in everyday biology. British Journal of Developmental Psychology, 8, 281–288. Inagaki, K., & Hatano, G. (1987). Young children’s spontaneous personification as analogy. Child Development, 58, 1013–1020. Inagaki, K., & Hatano, G. (1991). Constrained person analogy in young children’s biological inference. Cognitive Development, 6, 219–231. Johnson, J., & Pascual-Leone, J. (1989). Developmental levels of processing in metaphor interpretation. Journal of Experimental Child Psychology, 48, 1–31. Levinson, P. J., & Carpenter, R. L. (1974). An analysis of analogical reasoning in children. Child Development, 45, 857–861. Lunzer, E. A. (1965). Problems of formal reasoning in test situations. In P. H. Mussen (Ed.), European research in child development. Monographs of the Society for Research in Child Development, 30 (Serial No. 100). Novick L. R. (1988). Analogical transfer, problem similarity, and expertise. Journal of Experimental Psychology: Learning, Memory, and Cognition, 14, 510–520. Oppenheimer, R. (1956). Analogy in science. American Psychologist, 11, 127–135. Pauen, S. (1996). Children’s reasoning on the interaction of forces. Child Development, 67, 2728–2742. Piaget, J., Montangero, J., & Billeter, J. (1977). Les correlats. In J. Piaget (Ed.), L’abstraction re´flechissante (pp. 115–129). Paris: Presses Universitaires de France. Reed, S. K. (1987). A structure-mapping model for word problems. Journal of Experimental Psychology: Learning, Memory, and Cognition, 13, 124–139. Ross, B. H. (1989). Remindings in learning and instruction. In S. Vosniadou, & A. Ortony (Eds.), Similarity and analogical reasoning (pp-438–469). Cambridge: Cambridge Univ. Press.

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Siegler, R. S. (1976). Three aspects of cognitive development. Cognitive Psychology, 8, 481– 520. Spiro, R. J., Feltovich, P. J., Coulson, R. L., & Anderson, D. K. (1989). Multiple analogies for complex concepts: Antidotes for analogy-induced misconceptions in advanced knowledge acquisition. In S. Vosniadou, & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 498–527). Cambridge: Cambridge Univ. Press. Vosniadou, S. (1987). Children and metaphors. Child Development, 58, 870–885. Vosniadou, S. (1989). Similarity and analogical reasoning: a synthesis. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 1–18). Cambridge: Cambridge Univ. Press. Wilkening, F., & Anderson, N. H. (1982). Comparison of two rule-assessment methodologies for studying cognitive development and knowledge structure. Psychological Bulletin, 92, 215–237. RECEIVED: January 27, 1997;

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