7.2.1. Analogy: encoding, retrieval and mapping

The most striking manifestation of human analogical abilities appears through the capacity to detect common abstract relations across situations showing distinct appearances. In the words of Gick and Holyoak (1983):

A “deep” analogy, of the kind that arouses our admiration, is an analogy between remote situations in which essential causal relationships are maintained (p. 8).

Some researchers have even proposed that a comparison can only be considered an analogy when the situations being compared are different in terms of their surface and only similar at the level of their structure (Gentner and Smith 2013). These analogs are said to be “superficially dissimilar”, or from different semantic domains (e.g. physics, economics, cuisine, and so on). The surface of a situation refers to the objects (e.g. the sun) and the objects’ attributes (e.g. the sun is yellow). As for the structure, it qualifies a system of abstract relations that is in play between these objects (e.g. the planets revolve around the sun because the sun is heavier than the planets) (Gentner 1983). An analogy ideally makes it possible to emphasize the structure of a target situation through its similarities with a source situation. For example, Rutherford’s analogy between the solar system and the atom has suggested that, just like the planets and the sun, electrons revolve around the nucleus because it is heavier than they are (Gentner 1983, 1989).

In order to better understand the mechanism of analogy, the latter has been broken down into different sub-processes, operating in succession. A systematic distinction is made between the analogical retrieval stage, which consists of recalling a source situation from memory and the mapping stage, during which the two representations are aligned to establish their correspondences (Holyoak 1985; Gentner 1989). The mapping stage, which is generally considered as the corner stone of analogical reasoning (Gentner 1983; Holyoak and Thagard 1995; Gentner and Smith 2013), follows either from the retrieval of a source, or from the simultaneous presentation of the two analogs, which can typically be the case in educational contexts. It consists of establishing a structural alignment, that is, in determining the structural correspondences between the two analogs.

Experimental work has shown that analogies are not always successfully implemented. Indeed, the structural similarities between two situations are not always detected by the participants in the absence of surface similarity. These difficulties have been documented in adults when the source representation must be spontaneously retrieved from memory (Gick and Holyoak 1980; Gentner et al. 1993; Trench and Minervino 2015). They have also been identified when the two representations to be matched are presented jointly, primarily in children (Gentner and Toupin 1986; Gentner 1988), but also in adults to a lesser extent (Ross 1987, 1989).

The origins of the failure to process analogies have generally been located in a process that precedes that of retrieval and mapping, which is the construction of the representations, also called situation encoding (Gentner 1988; Hammond et al. 1991; Dunbar 2001; Gentner et al. 2003). Encoding is a crucial stage since it influences the nature of the similarities that can or cannot be detected between two representations (Chalmers et al. 1992).

More specifically, drawing analogies between superficially dissimilar analogs requires the encoding to be set at a certain level of abstraction. The abstraction must be neither too low – it would prevent from going beyond the surface similarities – nor too high – which would lead to relevant structural similarities being missed (Gick and Holyoak 1983). Research shows that prior knowledge is a main factor setting the level of abstraction of the encoding. Hence, studies documenting the influence of familiar knowledge on encoding, and as a consequence, on analogical retrieval and mapping, will be reviewed in the remainder of this section.

7.2.2. Prior knowledge and encoding

Research from various domains suggests that prior knowledge has a determinant role for the properties that are encoded (Sander 2000). Indeed, it has been proposed that familiar concepts influence encoding through a structural alignment process similar to that involved in more classical cases of analogy between two specific situations (Bassok 2001; Hofstadter 2001). In this case, it is the structure of a concept in memory, rather than that of a specific source situation, which is mapped to the target situation and guides its interpretation (Hofstadter and Sander 2013). A similar viewpoint has been advanced in the problem-solving literature, putting forward the idea that abstract concepts, such as schemas, can be mapped to a given situation in the same way as a specific analog is (Chi et al. 1981; Gick and Holyoak 1983). These concepts refer to abstract structures that can be instantiated with the specific elements of the situation at hand (Holyoak 2012). In line, Gentner and Kurtz (2005) proposed that the membership of a certain type of categories – called relational categories – is mainly defined by a common structure. The use of such concepts would promote the encoding of relations shared by different situations (Forbus et al. 1995; Holyoak 2012). For example, using the concept of a barrier to describe a wall, a fence, or even poverty or a lack of education, emphasizes that an abstract structure x is an obstacle to y. Thus, processing an abstract encoding critically depends on the possibility to align a target situation with relevant prior knowledge.

Empirical support for this idea comes from studies documenting content effects in problem solving. Notably, Bassok has tested the possibility that the mathematical structure of a problem is interpreted through a process of semantic alignment with familiar concepts evoked by the situation (Bassok 2001). This process has been highlighted in the interpretation of addition and division problems. For addition problems, the hypothesis was that the symmetric roles of addends with respect to their sum (due to commutativity) are aligned with the symmetric roles of two subsets, with respect to their common taxonomic category (e.g. tulips and daffodils). Regarding division problems, the asymmetric roles of the dividend and divisor would be aligned with objects sharing an asymmetric semantic relation (e.g. tulips and vases share a contain asymmetric relation). Bassok et al. (1998) tested this hypothesis by asking participants to construct addition or division problems involving two objects that were taxonomically related and shared a contain relation. The results supported the semantic alignment hypothesis since participants tended to construct problems maintaining a correspondence between the semantic relations of the objects and the mathematical relation involved in the arithmetic operation. The influence of semantic alignment on the interpretation of a problem statement was more directly tested by Martin and Bassok (2005). Participants had to solve division word problems that involved objects related either by a symmetric (e.g. cupcakes and cookies) or an asymmetric (e.g. travelers and cars) semantic relation. The results showed that participants had better performance when the objects described preserved an asymmetric semantic relation, suggesting that they do rely on a semantic alignment to represent the structure of arithmetic word problems.

In the same spirit, studies documenting differences in difficulty between isomorph versions of the Tower of Hanoi problem also illustrate that the problem’s structure is interpreted through familiar knowledge. In the classical version of the Tower of Hanoi problem, three vertical poles are arranged from left to right. In the initial state, the first pole presents, stacked from bottom to top, a large disk, a medium disk and a small disk. The goal is to reach a final state where the disks are placed in this same order on the rightmost location, while respecting three rules. The first is that only one disk can be moved at a time, the second is that if multiple disks are on the same pole, only the smallest can be moved, and the third is that a disk cannot be placed on top of a smaller one. The isomorph versions, whose resolution is compared in the studies, involve different surface features, but the number of steps needed to reach the goal is kept equal (Clément 2009).

For example, two versions of the problem described three acrobats of different sizes who had to jump on each other’s shoulders in order to reach a certain spatial configuration (Kotovsky et al. 1985). The only difference between the two versions resides in the way in which one of the rules was described in the statement. In one case, it was mentioned that an acrobat could not jump on the shoulders of a smaller acrobat, in line with the participants’ knowledge of human physical strength. In the second case, an acrobat could not jump on the shoulders of a taller acrobat. Hence, they had to be placed on a smaller acrobat, which is inconsistent with what is known about the usual relations between humans of different sizes. The results showed that problems were solved more quickly when the rule was compatible with everyday knowledge. Thus, participants rely on the familiar knowledge evoked by a problem statement to guide their understanding of its structure.

Similarly, it was shown that the representations abstracted from isomorph problems are different depending on the presence of ordinal or cardinal quantities, which influences the solving algorithm that is selected (Popov et al. 2017). Here, two problem statements are mentioned with either a cardinal or an ordinal variable.

– Statement with a cardinal variable (number of elements): in the Richards family, there are five people. When the Richards go on vacation with the Roberts, there are nine people at the hotel. In the Dumas family, there are three people less than in the Richards family. The Roberts go on vacation with the Dumas. How many will they be at the hotel?

– Statement with an ordinal variable (ages): Antoine attended painting classes at an art school for 8 years and stopped when he was 17 years old. Jean began at the same age as Antoine and attended the course for 2 years less. At what age did Jean stop attending the classes?

These problems share the same formal structure as both respect the principle: if two sets have a common part, the difference between their wholes is equal to the difference between their complementary parts (Sander 2008). Both can be solved using a one-step direct comparison strategy, which consists of computing 9 – 3 = 6 for the first problem, and 17 – 2 = 15 for the second one. Both can also be solved using a three-step complement strategy, which amounts to calculating: 5 – 3 = 2 and 9 – 5 = 4, then 2 + 4 = 6 for the first problem, and 17 – 8 = 9 and 8 – 2 = 6, then 9 + 6 = 15 for the second one. However, the cardinal quantities, as they evoke knowledge related to unordered elements that we usually put together or separate (e.g. families that meet and leave each other), would induce the extraction of a part-whole schema leading to the three-step complementation strategy being chosen and the values of the different parts being found. With regard to ordinal quantities, they are usually represented on an axis following a defined order (e.g. ages on a time line). This would lead to extracting a comparison schema guiding toward the one-step comparison strategy, where the difference between the wholes is directly sought. Once again, it appears that the representations associated with the elements described in the problem constrain how its structure is encoded. Together, these studies show that structural features which align with participants’ prior knowledge are encoded, whereas those that do not echo familiar concepts are generally bypassed. This process bears important implication regarding the implementation of analogies, since comparisons are generally based on the features that have been encoded.

7.2.3. Prior knowledge and analogical retrieval

The link between encoding and analogical retrieval has been highlighted by many researchers (Gick and Holyoak 1983; Dunbar, 2001; Popov et al. 2017). It is well illustrated by studies showing the consequences of experimental settings aimed at promoting or hindering the processing of a deep encoding on the rates of structurally based retrieval (Catrambone and Holyoak 1989; Hammond et al. 1991; Gentner et al. 2003). The basic idea is that the structure of the analogs must be encoded so as to be able to use it as a retrieval cue. For instance, Sander and Richard (2005) analyzed what it takes to implement an analogy between the sour grapes fable and the story of Harry (Wharton et al. 1994). The fable depicts a fox who, after failing to reach grapes, declares that they looked sour anyway, and the analogous story is about Harry, who was unsuccessful in getting a new job and finally tells his wife that it would have been boring anyhow. The authors noted that the analogy between the two situations is made possible by their encoding through the common concept of the situations where someone denigrates what they cannot obtain.

In the previous section, it was shown that participants’ prior knowledge highly constrains encoding. Hence, we argue that structurally based retrieval is achieved when the common structure aligns with familiar concepts, but that failures occur when the shared structure is more abstract than the initial encoding. The ability to base retrieval on familiar structures has been documented in a series of story-recall studies (Wharton et al. 1996; Catrambone 2002; Raynal et al. 2020; Trench et al. 2020), where participants first read a set of source stories, and after a delay, read a set of target stories sharing different types of similarities with the former. The task of the participants is to indicate whether each target story reminds them of a source story. Notably, Raynal et al. (2020) presented participants with stories about daily-life events that shared either a surface similarity (e.g. stories about pizzaiolos), or a structural similarity involving a familiar concept (e.g. competition resolved amicably). The results demonstrated that a great proportion of the participants recalled the structurally similar stories, whereas only a minority reported the superficially similar ones. We also found that this pattern held during the retrieval of autobiographic experiences encoded in naturalistic settings (Raynal et al. 2018). In this study, we presented participants with short anecdotes referring to familiar abstract categories. For instance, one of them described someone who had to buy a lightbulb but postponed several times before resigning themselves to do it at last (concept of procrastination). Participants were invited to report a maximum of memories that the target would remind them of. The analysis of the participants’ retrievals revealed that most of them were superficially dissimilar analogs. In the procrastination example, participants frequently recalled an experience of postponing the completion of homework several times before getting down to it at the last moment. Such remindings suggest that retrievals based on structural similarity occur readily when the common structure aligns with familiar concepts.

Other studies have reported massive failures to base retrieval on a structural similarity. For example, Trench and Minervino (2015) presented analog targets with or without surface similarity to test the retrieval of popular movies such as Jurassic Park. In this movie, a millionaire clones dinosaurs from DNA found in fossilized mosquitoes to exhibit the creatures in a public park, but the dinosaurs end up attacking the visitors. The superficially similar target was about a businessman who replicated mammoths with the intention of showing them in a zoo open to the public. The superficially dissimilar target was about an astrophysicist who had replicated Martian storms by using images he had taken with a space probe, with the intention of giving his colleagues access to the experimental area to study the phenomenon. The results showed that superficially dissimilar targets rarely triggered the tested retrieval, whereas superficially similar targets were more effective in doing so. This finding can be interpreted as evidence that the superficially dissimilar target did not fall within the scope of the familiar concepts used to encode the movies. As an illustration, the initial encoding of Jurassic Park may involve the concept of reproduction of extinct species presenting a threat to humans, which elicits its retrieval when faced with a target situation preserving this concept (e.g. the mammoth scenario), but not when encountering a target situation sharing similarities at a higher level of abstraction (e.g. the Martian storm scenario and Jurassic Park share the highly abstract concept of reproduction of a phenomenon presenting a threat to humans) (Raynal et al. 2020).

Research investigating spontaneous transfer between superficially dissimilar analog problems has documented other failures to base retrieval on structural similarity. In a series of studies, the general problem was presented as a source analog (Gick and Holyoak 1980, 1983; Keane 1987; Catrambone and Holyoak 1989). In this problem, a General aims to capture a fortress controlled by a dictator. His army is large enough to achieve his goal, but the roads leading to the target are mined, so that only small groups of men can get through without blowing them up. The General reached his goal by using a strategy that consisted of sending small troops of soldiers along the different roads simultaneously in order to converge on the fortress. After a delay, the participants had to solve the radiation problem borrowed from Duncker (1945). This problem describes a doctor who seeks to operate on a tumor in the stomach of a patient. It is stated that the doctor has a beam which is powerful enough to destroy the tumor, but using it at full intensity would also harm the healthy tissues surrounding the tumor. It can be noticed that a common abstract schema implicitly underlies both analogs. It is composed of a goal (using a force to overcome a central target), resources (a sufficient force) and a constraint (the inability to use the force at full intensity along a single pathway). In these experiments, only a minority of participants spontaneously transfer the solution from the General problem to the radiation problem (Gick and Holyoak 1980). Once again, these retrieval failures suggest that participants’ initial encoding was not as abstract as the common schema preserved by the analogs.

Research on experts has suggested that the reason why this schema could not be encoded is that participants lack the knowledge required to encode such an abstract structure. This explanation is supported by research showing that experts, who have developed abstract categories within their field, are able to classify problems according to their deep structural commonalities (Chi et al. 1981). Novick (1988) further demonstrated that experts have better performances than novices at spontaneously transferring solutions toward superficially dissimilar analog problems. Beyond problem solving, experts’ proficiency at basing retrieval on structural similarity has been documented in several studies (Dunbar and Blanchette 2001; Christensen and Schunn 2007; Kretz and Krawczyk 2014). More specifically, Jamrozik and Gentner (2020) tested the possibility that experts’ ability to retrieve structural matches relies on their use of labels referring to relational categories. In their story-recall experiments, source and target stories sharing either a surface similarity (e.g. the medical semantic domain) or a structural similarity (e.g. positive feedback loop) were presented to novice participants. Crucially, in one condition, stories were accompanied by the relational label describing their structure (e.g. “positive feedback loop”), whereas in another condition, they were not. The authors found that the introduction of the relational labels increased the probability of recalling structural rather than surface matches. The results support the possibility that experts superficially encode different examples through similar abstract concepts. Thus, studies on experts highlight that novices’ failure to retrieve based on structural similarity is due to their lack of knowledge preventing them from integrating the critical abstract structures in their initial encodings.

Overall, research on analogical retrieval reports both successes and failures to base retrieval on structural similarity. Importantly, it appears that participants’ initial encoding is focused on familiar concepts from daily-life that allow them to retrieve analogs within the scope of these concepts (Wharton et al. 1996; Raynal et al. 2020), but that prevent them from accessing analogs sharing a more abstract similarity (Gick and Holyoak 1980; Trench and Minervino 2015).

7.2.4. Prior knowledge and mapping

Research has revealed that participants are generally better able to detect structural similarities when mapping two superficially dissimilar analogs presented jointly than when they have to retrieve the source analog from memory (Holyoak 2012). Although participants often fail to spontaneously transfer a solution from a source to a target problem sharing only a structural similarity, they are generally able to transfer the solution after the introduction of a hint inviting them to rely on the source problem to find the solution of the target (Gick and Holyoak 1980). Nevertheless, difficulties to map superficially dissimilar analogs have been documented when the common structure refers to particularly unfamiliar concepts.

In Bassok et al.’s (1995) study, participants were trained to use a formula to solve a source problem, and then they had to instantiate this same formula to solve a target problem. As the formula was presented while solving the target problem, the task involved mapping and transfer, but not spontaneous retrieval. The problems involved a distributive structure (i.e. permutation problems); the participants had to calculate the probability that, following random draws, some elements of one set would end up being assigned to some elements of another set. The surface features of the problems were manipulated. In one condition, the source problem consisted of determining the probability that some computers would be distributed to some secretaries, and the target problem consisted of computing the probability that some prizes would be distributed to some students. According to the researchers, the same familiar concept x obtain y can be used by the participants to interpret the structure of both problems and guide the analogical mapping of the obtained elements from the two problems, that is, between computers and prizes. In another condition, participants were trained on the same source problem, but then had to solve a target problem where children from one school were assigned to children from another school. The prediction was that the structure of this target problem would be encoded through a different concept (i.e. x work with/play with y) than the one used to encode the source problem, which should hinder transfer. More precisely, the role attributed to the computers from the source problem was different from that of the children from the target problem (no obtained items), and thus it could not guide the establishment of the correspondences. In line with the hypotheses, transfer was significantly greater when the structure of both analogs could be interpreted through a similar familiar concept. Further experiments showed that the similarity between the corresponding objects’ role (e.g. computers and prizes are obtained items) was more determinant than their surface similarity (e.g. computers and prizes are inanimate items) for the success of the transfer. These data illustrate that the effect of prior knowledge on encoding has repercussions on the mapping process. Again, it appears that the familiar concept involved in the initial encoding of a situation makes it easier to draw comparisons with analogs instantiating this concept than with other analogs sharing similarity at a more abstract level (e.g. distributive structure).

Developmental studies have also reflected the determinant role of prior knowledge in mapping abilities. The idea is that children are generally able to map a common relation across superficially dissimilar examples when it echoes a familiar concept (Vosniadou 1989). Using a four-term analogy task (A: B: C: ?), Goswami and Brown (1990) tested children’s ability to base mapping on familiar relations (e.g. lives in). Participants were first presented with a source objects’ pair (e.g. bird: nest), then with the first object of the second pair (e.g. dog: ?), and they had to select the fourth term among several options (see Figure 7.1). In the selected example, the possible matches were a doghouse (relational similarity), a bone (thematic association), a cat (categorical association) and a dog (surface similarity).

The results showed that as early as age 4, children were generally able to select the relational match. These results converge with those showing that very young children are able to transfer a solution (e.g. using a tool to pull an object) between problems sharing only low levels of surface similarity, as long as the common relation refers to a familiar concept (e.g. pulling, Brown (1989)). Preschoolers’ ability to draw analogies also appears through their semantic approximations, where they interchange two words sharing an analogical similarity (Gaume et al. 2002; Bowerman 2005). Using the ERP method with 4 year-olds, Raynal et al. (2021) investigated the N400 effect, a component associated with incongruity detection (Kutas and Federmeier 2011), during the simultaneous presentation of either approximative (e.g. “she is undressing the orange”) or invented verbs (“she is rauging the orange”) and pictures of actions (e.g. someone peeling an orange). The results demonstrated that the N400 elicited by approximative verbs was smaller than the one evoked by invented verbs, suggesting that preschoolers encode the abstract category shared by the action and the approximative verb (e.g. the taking of an envelope).

Children’s failures to map analogs when lacking familiarity with the common relations have been documented by Gentner and co-workers. They have shown that with insufficient knowledge, young children are highly influenced by surface similarities during mapping (Gentner and Toupin 1986). This was particularly well illustrated in a study on the interpretation of metaphors by 5- to 6-year-olds, 9- to 10-year-olds and adults. Participants were presented with metaphors based on common object attributes (e.g. “pancakes are nickels”), common relations (e.g. “a tire is a shoe”) or whose interpretation could be based on either object attributes or common relations (called double metaphors, e.g. “plant stems are drinking straws”). The results showed that participants’ tendency to provide a relational interpretation for relational metaphors (e.g. both are used by moving figures as points of contact with the ground) and for double metaphors (e.g. both are used to bring liquids from below to nourish a living thing) increased with age. While 9- to 10-year-olds and adults favored relational interpretations for the double metaphors, the 4- to 5-year-olds did not show such a preference over object attributes-based interpretations (e.g. both are long and cylindrical). Although research has highlighted several factors involved in the development of analogical reasoning (Richland et al. 2006), Gentner has proposed that analogical abilities critically depend on the accretion of conceptual knowledge (Rattermann and Gentner 1998; Gentner and Smith 2013). Thus, developmental studies suggest that young children are able to process analogies when they involve a familiar concept. They support the idea that prior knowledge constrains the initial encoding and influences the type of analogies that can be processed.

As a whole, research on analogy has converged on the determinant role of prior knowledge in encoding (Kotovsky et al. 1985; Clément and Richard 1997; Gvozdic and Sander 2019; Gros et al. 2021), analogical retrieval (Novick 1988; Jamrozik and Gentner 2020; Raynal et al. 2020) and analogical mapping (Bassok et al. 1995). We have argued that the level of abstraction of the initial encoding is tightly constrained by familiar concepts, and that in turn, this encoding influences the analogies that can or cannot be implemented. Analogies appear to be readily processed when the similarity corresponds to the initial encoding (see, for example, Goswami and Brown (1990); Raynal et al. (2020)), but they are more difficult to detect when they rely on a more abstract concept (Gick and Holyoak 1980; Gentner 1988; Bassok et al. 1995). Identifying the processes allowing one to overcome this limitation and implement analogies that were initially out of reach thus constitute a major issue. The proposal is that cognitive flexibility, defined as a change in point of view about a situation, is a key mechanism for detecting analogies that initially went unnoticed (Chalmers et al. 1992; Sander and Richard 1998; Clément 2009).

7.3.1. The abstract recoding process

When solving a problem, the initial encoding induced by the statement does not always find the adequate solution. Some authors have proposed that a process of abstract recoding is necessary to modify the initial encoding and adopt a new, more abstract representation that would better support the discovery of the solution (Sander and Richard 1997, 2005; Clément 2009; Gamo et al. 2010; Gvozdic and Sander 2019). Notably, they have analyzed what it takes to solve a problem requiring the consideration of an unusual function of an object. In the candle problem (Duncker 1945), for example, participants must devise a solution for fixing a candle to a wall. They are given matches, a box of matches and thumbtacks. The solution is to fix the box on the wall with the thumbtacks and to use the box as a support for the candle. Solving this problem involves cognitive flexibility as participants must abandon an initial representation focusing on the usual function of the box (e.g. being a container) to reconceptualize it as a support.

Solving Maier’s (1931) two-rope problem may rely on a similar process (Chalmers et al. 1992). In this problem, two ropes hanging from the ceiling must be attached, but they are too far apart to be caught simultaneously. Participants are given a chair and pliers to complete the task. To find the solution, participants must ignore the usual function of the pliers, in order to think of the tool as a weight to be attached to one rope, allowing them to swing it and grasp the two ropes.

The literature documenting content effects has put forward a similar analysis by proposing that the more difficult problems are those whose surface features induce an initial encoding that must be reconsidered to adopt a more abstract representation (Clément and Richard 1997). Specifically, the study investigated differences in difficulty between isomorphs of the Tower of Hanoi problem, involving changes of state that were presented as either changes of size or changes of place. For example, actions could be described as moving cubes or changing the size of the cubes. In change of size problems, the first rule is that only one cube can be changed in size at a time, the second is that when two cubes are the same size, only the size of the leftmost cube can be changed, and the third is that it is impossible to assign to a cube the size of a cube that is further to the left. The rules are described differently in the change-of-place problems. The first rule is that only one cube can be moved at a time. The second is that if there are several cubes in the same location, only the smallest cube can be moved. The third rule is that you cannot place a cube on top of a smaller cube.

Although both problems can be solved in five steps, participants have more difficulty reaching the desired final state when the change of state is described as a change of size rather than as a change of place. These differences can be accounted for by the fact that a change of state (involving a change of size or place) can be perceived in two ways. It can be interpreted from the point of view of a continuous transformation process, by analogy with biological growth processes, leading from one state to another through all the intermediate states. Alternatively, it can be viewed as the result of the action, where the change is percieved as moving from one state into another state. This latter conception leads to bypassing the intermediate steps leading from one state to another. The result point of view is more adequate to solve the problem because, by decomposing the change in two steps (moving out of a state and moving into a new one), it makes it possible to associate the second rule with moving out of a state, and the third rule with moving into a new one. Such associations are not favored by the description of a change of size, since it is apprehended through the model of biological growth, a continuous process which is less readily decomposed into two steps. Crucially, the authors propose that the initial encoding referring to a transformation process must be abandoned in order to privilege a more abstract perspective, that is, the result of the action. Hence, it appears that solving certain problems relies on a representational change, whereby a familiar concept activated by the statement must give way to a new, more abstract conceptualization (Clément forthcoming).

A convergent analysis is proposed by Gros et al. (2020), who developed a theoretical model (SECO, for semantic congruence) of the semantic recoding process that must be implemented when the initial encoding does not perfectly match the formal structure of the problem. This process would consist of reinterpreting the problem so as to move from an initial encoding based on daily-life knowledge evoked by the statement to a representation, which is closer to the formal structure. In line, discovering the one-step solution in cardinal problems inducing a three-step strategy would depend on an abstract recoding of the problem’s structure. In other words, the participant has to give up a representation based on the cardinality of the variables, in order to build a representation that is closer to the critical mathematical principle. Thus, problem-solving studies have highlighted an abstract recoding process allowing one to give up an initial encoding and consider a more appropriate perspective to find a solution. It will be proposed that the flexible reinterpretation of situations is precisely what is required to process challenging analogies.

7.3.2. Abstract recoding and the implementation of challenging analogies

The ability to reinterpret a situation in a flexible way is central to perceiving certain analogies (Hofstadter and Mitchel 1994; Hofstadter and Sander 2013). Indeed, the knowledge involved in the initial encoding of two analogs does not always allow one to grasp their shared structure. The implementation of certain analogies depends on abandoning an inadequate initial concept to reconsider the analogs through an alternative concept that reveals the common structure (Sander and Richard 2005). Sander and Richard (1997) have proposed that reinterpreting the situations through a more abstract concept can be the key to detecting previously unnoticed structural similarities. In this view, the recoding of the royal crown and his own body through the abstract category of concrete object may have been the process allowing Archimedes to implement his insightful analogy.

When the initial encoding of a target situation does not provide a match with source analogs stored in memory, analogical retrieval may critically depend on an abstract recoding process. This idea is supported by work on re-representation, which refers to the process of altering the representation of one or both analogs, in order to improve the match (Gentner and Forbus 2011). Indeed, Clement et al. (1994) provided empirical evidence showing that better analogical retrieval performances are obtained when the domain-specific terms of the analog stories are replaced by domain-general words, which apply to the domains of both analogs. These findings suggest that certain retrievals depend on the ability to alter the representation of the analogs to adopt a more abstract one that creates a better match. Moreover, results from studies reporting failures to retrieve superficially dissimilar analogs (Gick and Holyoak 1980; Trench and Minervino 2015) illustrate that abstract recoding is not systematically implemented spontaneously to afford retrieval. One possible explanation for this is that access would be a relatively passive process, depending on how the analogs had been initially encoded (Clement et al. 1994; Popov et al. 2017).

However, recoding can occur in a way that promotes spontaneous transfer under specific conditions, even in young children (Brown 1989). In a study reported by the author, the participants were familiarized with a source problem where a genie rolled up his magic carpet into the shape of a tube in order to use it to transfer jewels from one bottle to another (Holyoak et al. 1984). Then, they had to solve a target problem describing a farmer who was trying to transfer cherries across a large fallen log that was blocking a path. The child was given various objects, including a sheet of paper. The question of interest was whether the rolling-up solution would be transferred to propose rolling up the sheet to get the cherries from one side of the tree trunk to the other. It can be noted that finding this solution requires the child to think of an unusual use for the sheet of paper. Different groups of children were formed and trained in different ways before being exposed to the genie source problem. One group of children was encouraged to draw three pictures on sheets of paper, another group was asked to use the paper for three different purposes (e.g. building a tent, an airplane and a house), and a last group performed an unrelated activity. The results showed that the training that consisted of considering different uses of a sheet of paper improved transfer, whereas the training during which only one use (e.g. drawing) was considered had a negative effect on transfer. They confirm that the ability to represent an item through different concepts is critically involved in analogical retrieval, and show that this process can be implemented from an early age.

Further, Sander and Richard (1997) have gathered empirical evidence demonstrating the spontaneous implementation of successive recodings following increasing levels of abstraction in adults. In their experiment, the participants were introduced to text-editing and had to perform various tasks (e.g. deleting, moving, duplicating elements, and so on). Analysis of the participants’ activity revealed that they initially completed the tasks by borrowing procedures from the familiar source conception of the typewriter (e.g. typing from an insertion point). Then, when these procedures proved inadequate to accomplish a goal, participants referred to the more abstract conception of writing in general, including both handwriting and typography (e.g. replacing one word with another). Finally, when this last conception was in turn inadequate, they borrowed procedures relating to the superordinate conception of object manipulation (e.g. moving a set of elements). These results suggest, on the one hand, that a situation can be spontaneously recoded through concepts of increasing levels of abstraction, and, on the other hand, that recoding through a very abstract concept can be costly.

This leads us to the possibility that the failures to retrieve superficially dissimilar analogs (Gick and Holyoak 1980; Trench and Minervino 2015) are not only due to an insufficiently abstract initial encoding, but also to the necessity to spontaneously recode the analogs through very abstract concepts. This hypothesis can be illustrated by analyzing what it takes to solve the tumor problem. Gick and Holyoak (1980) proposed that the convergence solution is difficult to find because the divisibility property is not initially associated with the concept of a laser beam. In fact, attributing this property to lasers requires us to reconsider it through the very abstract concept of concrete objects, since the part-whole relation associated with them induces the possibility to separate and group them (Sander and Richard 1997). Hence, the challenge to recode an item through a very abstract concept may explain the massive failures to spontaneously transfer the onvergence solution. Trench and Minervino’s (2015) results may be explained in a similar way. The Martian storm in the superficially dissimilar target scenario and the dinosaurs in the source movie Jurassic Park must be recoded through the very abstract concept of phenomena in order to constitute a retrieval cue. Overall, it appears that when the initial encoding does not afford retrieval cues to a source analog, a recoding is required. This recoding can occur spontaneously, but the necessity to recode through highly abstract concepts induces retrieval failures.

Works on re-representation have illustrated how mapping depends on altering the initial encoding of the analogs (Gentner and Forbus 2011). Research suggests that recoding more readily occurs during mapping than during analogical access (Clement et al. 1994), possibly because both analogs are maintained in working memory, which is prone to elicit explicit comparison (Holyoak 2012). The intervention of re-representation during mapping has been highlighted by Day and Asmuth (2017), who tasked participants with comparing a source analog and a first target analog. Then, participants had to assess the similarity between the same source analog and a second target analog. In one condition, the first and the second analogy were based on the same structural similarity. Thus, it was predicted that both analogies would lead to interpreting the source in a similar way. In another condition, the two analogies were based on a different structural similarity and thus relied on different interpretations of the source. The results highlighted that participants judged the source and the second target to be more similar when both analogies were based on the same structural similarity, than when the second analogy relied on a different common structure. These results suggest that the representation of the source had been altered during the comparison with the first target, so as to provide a better match.

In a similar way, research has shown that participants who had to judge the similarity between a pen and a bottle were more likely than participants who did not complete such a task, to accept that a pen is a container (Kurtz 2005). The interpretation of the author was that the comparison induced a change in the set of semantic elements that were activated to represent a pen. The possibility that mapping induces a flexible adaptation of the concept used to represent an analog appears most clearly in a study by Oberholzer et al. (2018). The study provides evidence that when comparing two analogs, the target can be re-represented through the concept of which the source is a prototypical member. For example, participants will be more likely to describe the target “lighting a candle in a basement” (a marginal exemplar of the superstitious behavior category) as an act of superstition after having compared it to the source “hanging garlic on the door” (a prototypical exemplar of the superstitious behavior category), than after comparing it to an unrelated situation.

Hence, research shows that a recoding process can generally take place so as to map non-identical elements of the analogs. However, the failures to map analogs, which have been documented in developmental studies (Gentner and Toupin 1986; Gentner 1988; Ratterman and Gentner 1998) and to a lesser extent in studies on mature reasoning (Bassok et al. 1995) reveal that this recoding can be costly, even during mapping. In Bassok’s experiments, for example, participants may hardly give up their initial representation (e.g. x obtain y) to adopt a new one (e.g. x are assigned to y) that would allow them to map any analog (Gros et al. 2020). The general picture that emerges from the literature is that abstract recoding is a key mechanism in analogy-making, and that the difficulty to recode through highly abstract concepts can lead to bypassing relevant analogies, especially when they have to be retrieved from memory. These missed opportunities make it necessary to design devices promoting the implementation of flexibility.

7.3.3. Comparison to promote flexibility and analogy

A prominent line of research has focused on analogical comparison as a means to induce a more abstract representation than the initial encoding (Doumas et al. 2008; Holyoak 2012; Gentner and Hoyos 2017). The idea is that comparing superficially dissimilar analogs would make their structure more salient, leading to the extraction of an abstract schema and the specific examples being re-represented (Gentner and Smith 2013). Schematic representations would be readily retrieved when faced with a superficially dissimilar analog because it has few surface differences and maximizes the structural similarity (Gick and Holyoak 1983; Gentner et al. 2009). For example, it has been shown that participants were better at transferring the convergence solution to the radiation problem when they were previously prompted to compare the source problem of the General with another analogous source problem (e.g. a fire chief manages to extinguish a blaze by ordering the firefighters to encircle the fire and simultaneously throw buckets of water) (Gick and Holyoak 1983; Catrambone and Holyoak 1989). The improvement in transfer performance was related to the quality of the extracted schema, as measured by subjects’ descriptions of the similarities of the two analogs (see also Kubricht et al. 2017). The benefits of analogical encoding have also been shown in transferring knowledge from a paper-andpencil task to a dynamic real-time situation (Gentner et al. 2003). Furthermore, this type of encoding promotes transfer when comparison takes place at recall time, during the encoding of the target situation. Indeed, comparing superficially dissimilar target analogs allows us to better retrieve superficially dissimilar source analogs from memory (Kurtz and Loewenstein 2007; Gentner et al. 2009).

Developmental studies have shown that comparison enhances children’s performances at establishing relational mappings. When children focus their initial encoding on surface features, analogical comparison allows them to reorient their representation on relational features (Doumas et al. 2008; Gentner and Hoyos 2017). This process would induce a reinterpretation from a domain-specific encoding to a more generalizable representation (Kotovsky and Gentner 1996). In a study led by Christie and Gentner (2010), novel spatial patterns (e.g. symmetry) were presented to children with an invented label (e.g. “this is a toma”). Then, children had to extend the label to one of two alternatives: a relational match (i.e. preserving the relational pattern) or an object match (i.e. presenting a similar animal). In one condition, children were prompted to compare two exemplars before being asked to select a match, whereas in another condition, they only received one exemplar. It appeared that children who compared two analogs were more likely than those who did not, to select the relational match. Other studies have further shown that language (e.g. introducing a relational label) also promotes the processing of a relational encoding (Gentner et al. 2011; Son et al. 2012). Kotovsky and Gentner (1996) highlighted the contribution of progressive alignment, whereby comparing examples sharing a structural similarity at a moderate level of abstraction (e.g. symmetry in object size) facilitates the establishment of analogies with new exemplars sharing a more abstract structural similarity (e.g. symmetry in object color). The underlying process would consist of abandoning an initial domain-specific representation of the relation to adopt a more general representation.

The efficiency of analogical comparison has not only been documented in the lab, but also tested in educational contexts (Gvozdic and Sander 2019). For example, Gamo et al. (2010) trained 10- to 11-year-old students to solve analog problems with cardinal or ordinal variables that can be solved using either a one-step or a three-step strategy. The aim of the comparison activity was to induce a more abstract encoding than that induced by the statement, in order to be able to apply the one-step algorithm to any analog problem. In order to highlight the presence of a common part and its link with the one-step strategy, the class was encouraged to compare the two possible strategies for the same problem and describe the similarities between different problems. The results supported the effectiveness of the training, since the pupils who followed the training were more likely to use the one-step strategy for cardinal problems, when compared to a control group. These results highlight the additive value of using comparison to emphasize the essential conceptual aspects of the situations at hand.

Overall, research has shown that one of the keys to allow participants to implement challenging analogies is to induce a representational change leading to a more abstract point of view than the one initially adopted. A great deal of experimental results has revealed that analogical comparison is an efficient way to initiate this flexible recoding process.