Which of the following is the reasoning process in which two similar cases are compared?

Abstract

Analogical reasoning is a kind of reasoning that is based on finding a common relational system between two situations, exemplars, or domains. When such a common system can be found, then what is known about one situation can be used to infer new information about the other. The basic intuition behind analogical reasoning is that when there are substantial similarities between situations, there are likely to be further similarities. This article describes the processes involved in analogical reasoning, reviews seminal research and recent developments in the field, and proposes new avenues of investigation.

Abstract

Analogical reasoning is a fundamental aspect of human cognition: People, even young children, engage spontaneously in analogical reasoning to make sense of unfamiliar situations. However, people often fail to use analogies productively, when the analogies are generated by someone other than themselves (e.g., a teacher). In this chapter I will discuss the challenges of using analogies in mathematics instruction, using as an example the analogy “numbers are points on the line.” This analogy is the product of a long-term comparison between numbers and the Euclidean line, and underlies a common representation of numbers, namely, the number line. Drawing on empirical evidence, I will illustrate the affordances of this analogy along with the challenges that it presents for students. Finally, I will discuss the conditions under which the use of analogies can be fruitful in mathematics instruction, elaborating on the features of successful interventions, particularly on the “bridging analogies” teaching strategy.

Analogical Reasoning

Analogical reasoning theory provides a foundation for metaphor-enhanced design of multisensory representations. Analogists have established that people learn from mapping relational structure from one domain to another (Gentner, 1983; Gentner & Markman, 1997; Holyoak, Gentner, & Kokinov, 2001). Figure 6.2 displays the CyGaMEs approach (Reese, 2007, 2009) to the application of cognitive science analogical reasoning theory in the design of digital game instructional systems. The model begins with the rectangle on the left representing the real-world relational structure for some targeted knowledge domain. The ovals signify objects (e.g., people, places, events, things) and the directional arcs represent the relations connecting the objects. When people encounter new, unfamiliar, or abstract domains, people make inferences (learn) by mapping relational structure from the known domain (left) to the unknown domain (the right-hand rectangle). Immediate, pragmatic goal structures shape the characteristics of those mappings (Gentner & Holyoak, 1997; Holyoak, 1985, 2012; Holyoak & Thagard, 1989; Spellman & Holyoak, 1996).

In metaphor-enhanced design approaches, the designer uses a specification (knowledge map) of the to-be-learned knowledge domain (the right-hand rectangle) to constrain, guide, and evaluate a game world system for alignment with that knowledge domain (left-hand rectangle). The game world models the relational structure of targeted domain knowledge, and the game goals (analogs of targeted learning goals) guide and reinforce the player to discover and apply targeted new knowledge. Thus, the learner constructs a mental model (right-hand rectangle) that represents targeted, real-world knowledge (e.g., scientific phenomena, the left-hand rectangle).

Procedurally, engineering multisensory metaphor enhanced game-based learning environments begins with domain specification. In collaboration with subject matter experts, a learning engineer who specializes in this technique (metaphorist) conducts a cognitive task analysis (Chipman, Schraagen, & Shalin, 2000; Clark, Feldon, Merriënboer, Yates, & Early, 2008) to discover and specify the relational structure of the to-be-learned domain. Then multisensory human-computer interface designers, game designers, and game developers invent (define) and realize each sensory representation as an integrated and integral component of the embodied learning system (game world: game goals, game rules, mechanics, feedback, etc.). In other words, the game design team translates targeted knowledge-domain relationships into player transactions with the game world. In a multisensory learning game, these transactions might derive from haptic or auditory sensory representations in addition to or in place of the more typical visual stimuli.

Transactions may also take advantage of innate sensory-specific processing mechanisms to facilitate the mappings, at least initially. This is because perceptual factors such as Gestalt principles provide a grounding that influences symbolic reasoning (Goldstone et al., 2010). For example, Goldstone et al. studied the influences of perceptual grouping on symbolic reasoning in arithmetic. According to conventions for order of precedence of operations in arithmetic calculations, multiplication should occur before calculations using addition. Goldstone et al. applied the Gestalt principle of visual proximity to facilitate students’ application of the order of operations. When multiplication symbols and numbers were placed closer together and those for addition were placed farther apart, the students processed equations involving order of operations more quickly. When spatial relationships were reversed, the processing of equations occurred more slowly and the error rate increased (six times as many errors when spacing was inconsistent with the operator convention). Even though participants received immediate feedback after each trial, behaviors were robust. Goldstone et al. concluded that “perhaps all cognition may intrinsically involve perceptually grounded processes” (p. 31) and “grounding cannot be ignored for any cognitive task” (p. 27). Scaffolds using perceptual grounding, like proximity, can be faded and removed after learning. Indeed, Goldstone et al. suggest fading will promote transfer from a specific instructional context to more general application.

Analogical Reasoning

Several studies have found that analogical reasoning engages frontopolar cortex. Furthermore, different components of analogical reasoning appear to differentially engage frontopolar versus dorsolateral prefrontal areas. Whereas dorsolateral areas are recruited for processing externally generated information (e.g., the monitoring and manipulation of presented facts), frontopolar areas are recruited additionally for the evaluation and manipulation of internally generated information.

One study assessed the neural systems that support analogical reasoning in the Raven’s Progressive Matrices task. Study participants received a 3 × 3 matrix of figures with the bottom right figure missing, and had to infer the missing figure by selecting one of four possible alternatives. Participants received three types of problems that differed in their degree of relational complexity (0-relational, 1-relational, 2-relational). The 0-relational problems involved no relation of change and thus required no relational processing. The 1-relational problems involved one relation of change in either the horizontal or vertical dimension, and thus required relational reasoning. Finally, the 2-relational problems involved two relations of change, in both the horizontal and vertical directions, and thus required even more relational reasoning.

A region-of-interest analysis was performed to assess the role of prefrontal cortex in processing multiple relations simultaneously (i.e., relational integration). This analysis produced two main findings. First, activation occurred in frontopolar prefrontal cortex (BA 10), reflecting the internal generation of relations required to form complex analogies. Interestingly, this activation only occurred for 2-relational problems, not for 1-relational problems, suggesting that frontopolar cortex is important for processing complex relational structures. Second, activations also occurred in right DLPFC (BA 46), which reflected greater manipulation of externally presented information in more complex problems. Other studies using the Raven’s Progressive Matrices task have found similar results, and have also reported bilateral posterior parietal activations (BA 7).

In another study, individuals received a source picture of colored geometric shapes, followed by a target picture of colored geometric shapes. Pictures that did not share similar geometric shapes but that did share the same system of abstract visuospatial relations were also presented. Individuals judged whether each source-target pairing was analogous (analogy condition) or identical (literal condition). Analogical reasoning (analogy minus literal) recruited the dorsomedial frontal cortex (BA 8) and left-hemisphere regions, including frontopolar (BA 10), inferior frontal (BA 44, BA 45, BA 46, and BA 47), and middle frontal (BA 6) cortices, and also inferior parietal cortex (BA 40). These findings suggest that analogical reasoning is mediated by a predominantly left-hemisphere frontoparietal system.

Another study systematically evaluated the component processes of analogical reasoning. Specifically, this study assessed the neural systems that underlie (1) the storage of abstract relations in working memory and (2) the process of integrating abstract relations to form analogies. These researchers also found that analogical reasoning activated a left frontoparietal system, with some regions of this circuit mediating working memory processes and others mediating abstract relational integration. In particular, left frontopolar regions (BA 9/10) were again central to the processing of relations that underlie analogical reasoning.

In summary, the observed findings for analogical reasoning are consistent with theories that advocate distributed rather than localized representations (e.g., dual-process, dual-code, embodied theories). As we saw, distributed frontal and posterior systems appear to play a wide variety of roles as representations are retrieved, stored, and integrated. Findings on analogical reasoning further support the cognitive demand and task specificity hypotheses (e.g., dual-process, dual-code, embodied theories). Different kinds of information recruited different prefrontal areas (e.g., external vs. internal information), and the harder the reasoning, the more areas recruited (e.g., 1-relation vs. 2-relation problems). The general importance of frontopolar cortex for generating relations internally suggests that this area is especially important for processing complex analogies.

Analogical (Inductive) Reasoning

In broad terms, analogical reasoning involves a classic analogy in the form of A:B:C:? and problem solving by transferring strategies from analogous problems (Gentner and Markman, 1997). Over the past two decades, findings from several studies have challenged previous reports that young children are incapable of analogical reasoning. Children may have difficulty in solving classic analogy tasks because they are unfamiliar with the tasks, but they often are able to identify structural relationships and map analogous relations through classic analogies presented in familiar, child-appropriate ways. Goswami and Brown (1990) administered a very simple analogy task to 3- and 4-year-old children. The context of this task was familiar to the children as it presented problems such as “bird:nest:dog:?” Children were required to choose a picture from four potential items that completed the analogy. Possible choices included the item that correctly completed the analogy (in this example, a doghouse); an item that was associated with the analogy (a bone); and an item that was part of the same category and was semantically related. Preschoolers were capable of accurately mapping the relations and completing the analogies, and therefore demonstrated early competences in solving classic analogy problems.

The basic paradigm for testing young children's analogical problem solving or transfer involves presenting children with source stories that describe problems and solutions, and then observing whether and how the children solve isomorphic problems using the source strategies. Studies (e.g., Brown, 1990) have furnished substantial evidence that even preschoolers exhibit analogical transfer based on structural similarity. For example, Holyoak et al. (1984) found that 4- and 5-year-olds who were exposed to a story (a genie using a magic staff to pull one bottle closer to the other in order to transfer his jewels) were able to solve a target problem by selecting a walking cane from among other items to transfer gumballs.

Different paradigms and age-appropriate tasks have been used to explore early competencies in solving analogical problems (e.g., Chen et al., 1997). Ten- to 12-month-olds first observed a parent solving a problem: obtaining an out-of-reach toy by removing a barrier, pulling a piece of cloth to reach a string, and pulling the string to bring the attached target toy within reach. The children then transferred the strategy to new problems that shared the same goal structure but differed in superficial features from the original task. In another study (Chen and Siegler, 2000), 18- to 35-month-olds demonstrated effective transfer of a tool-use strategy across a series of isomorphic problems. Three-year-old children, but not 1.5-year-olds, transferred the learned strategies across the analogous tasks more readily and effectively, revealing the developmental differences in analogical problem solving. These studies demonstrate that, from infancy, children possess a rudimentary ability to analogize in problem solving. Despite early competencies, younger children's acquired strategies tend to be more closely welded to specific learning situations, thus impeding the flexibility needed for cross-task transfer. Children's ability to perceive and utilize structural similarity in solving analogy problems becomes more sophisticated with age, from infancy to toddlerhood, to preschool, and through elementary school and beyond.

1 The Closing of Gaps in Human Problem Solving

Deduction, induction, and analogical reasoning, the three basic forms of logical thought, were derived from natural or everyday situations, where they were originally embedded in commonplace forms of interaction and communication. The differences are explained in the respective paragraphs.

Depending on the context, the German word ‘schließen’ (‘conclusion’) may mean to ‘close,’ ‘conclude,’ ‘reason,’ or ‘infer.’ Generally speaking, the term implies the bridging or closing of a gap, be this a literal gap between two entities or, in the metaphorical sense, a logical gap between a problem and its solution. The closing of such metaphorical gaps by means of reasoning or inference usually entails a series of cognitive ‘steps.’

Problems can be defined in terms of three characteristic stages: (a) an initial state (e.g., the arrangement of the figures on a chess board, an algebra or geometry exercise, or a balance of trade), (b) an end state (e.g., putting the opponent's king in checkmate, solving the maths exercise, or finding ways of improving the balance of trade), and (c) a well-defined series of cognitive ‘steps’ by which the initial state is transformed to the end state: the solution to the problem (or one of a number of solutions).

They are still to be found in their original, undifferentiated form as IF-THEN relations: in barter, trade, the manufacture of implements, and the planning of collective action, for example. Once raised to the metalevel of pure thought, these forms of reasoning became a subject of philosophical inquiry in the fields of mathematical logic, linguistics, and psychology. Over the course of history they even became independent areas of scientific systematics. It was only at a very late stage, in the time of the great Greek philosophers, that the three types were differentiated, defined, and each traced back to its logical structure. This represented the first step on the path to European or Western thought and the scientific advances that were entailed. And the first to tread this path were the great trio of Ancient Greek thought—Socrates, Plato, and Aristotle.

8.9 Analogy in Law

In the Anglo-American common law tradition, analogical reasoning is fundamentally a matter of taking due notice of legal precedents9 The doctrine of precedent is called stare decisis. In English law, which embodies the strictest version of the doctrine, precedent is governed by three main conditions [Cross and Harris, 1991, p. 5].

Decisions of any superior court must be respected.

Any decision of such a court constituents a binding precedent for that court and for any lower court.

Any decision of such a court constitutes a persuasive precedent for higher courts.

A precedent is binding when it requires a judge in a given case to decide it in the same way as the previous case irrespective of the merits of the case presently before him. A precedent is pervasive when a judge must honour it in the present case, unless he has sufficient reason not to.

Deciding a case in the same way as an earlier case involves the application of the ratio decidendi of the prior case to the present one. A ratio decidendi is a general principle of law, or set thereof, on the basis of which the judge reaches his decision. In actual juridical practice, discerning such principles is often far from easy and, in any event, always contextually influenced by the particular facts of the case in question [Cross and Harris, 1991; Levi, 1949]. There is a famous case in which the House of Lords found for the plaintive in an action brought against a producer of ginger beer. The charge was that the manufacturer was liable for the plaintiff’s illness upon consuming the beverage from a bottle containing a dead beetle. The ratio decidendi of the finding included the determination that

… a manufacturer of products, which he sells in such a form as to show that intends them to reach the ultimate consumer in the form in which they left him with no reasonable possibility of intermediate examination, and with the knowledge that the absence of reasonable care in the preparation or putting up of the products will result in an injury to the consumer’s life or property, owes a duty to the consumer to take that reasonable care. (Donoghue v. Stevenson 1932 AC 599.)

In a case from a court of equal or higher jurisdiction, the judge in a present case must apply the reasoning of the previous case unless there exist relevant differences between the two. Should the present judge determine that relevant differences exist, he must weigh the possibility that the previous ratio generalizes in such a way as to override these differences. We note in passing the adaptability of the doctrine of stare decisis to the MATAA model of analogical argument. In the preceding case we find two elements of this model. The judge’s ratio decidendi, which corresponds to MATAA’s Generalization Argument, and the judge’s finding, which corresponds to MATAA’s instantiation of the Generalization Argument (which corresponds to The Violinist in our reconstruction of Thompson’s example). The third component of the MATAA model is embedded in the subsequent trial. It is the court’s finding in that case (corresponding, again, to The Pregnancy), drawn by instantiation from the previous ratio (corresponding to The Generalization). The law’s unanalyzed notion of relevant difference is easily handled in the MATAA model. There is no relevant difference between the present and prior cases just when they each instantiate a common ratio in a P-preserving way. P-preservation here answers to the legal notion of reasoning adequate for the determination of a legal fact.

Where a judge finds relevant difference to exist between his present and some earlier cases, the duty to ascertain whether the ratio of the preceding case can be generalized to the present case may strike two different forms, both of which can be accommodated in the MATAA model. In the first instance the judge attempts to determine whether the prior ratio generalizes in a P-preserving way to a more general form of reasoning from which a like finding in the present case could be seen as a P-preserving instantiation. In the second instance, the present judge leaves the generality of the preceding ratio untampered with, and tries instead to determine whether the facts of the present case constitute an approximately P-preserving instantiation of the previous ratio. As we saw earlier in this chapter if the judge proceeds in the first way, his finding in the present case satisfies precedent but is not abductive, since P-preservation conflicts with the ignorance condition. If the judge proceeds in the second way, he allows himself a presumptive finding for the facts of his present case from the prior ratio. But this too is not abduction. In finding that the facts of the present case approximate to an instantiation of a prior ratio, it is not in general a requirement that his estimate of the similarities be conjectural.

The authority of legal precedent is subject to loose and strict interpretations [Llewellyn, 1930]. Precedents are usually interpreted strictly when they are considered as defective in some way. Strict interpretations limit the harm done by bad juridical determinations. In contrast, precedents are interpreted loosely when they are thought good enough instances of legal reasoning to justify their widest possible application. A strict interpretation of a precedent often involves the judge in finding a relevant difference in the present case which almost certainly would be regarded as nonexistent under a loose interpretation. This is natural occasion for a foolish or inexperienced judge to make mistakes of relevance. Thus in English legal practice strict interpretations of precedent tend to be reserved for highly experienced judges.

The contrast between relevant and irrelevant similarities works in tandem with the contrast between strict and loose interpretations of precedent. We have indicated how the first contrast is captured by the MATAA distinction between P-preserving instantiation and P-approximating instantiation. Likewise, a judiciously made strict interpretation will require P-preservation for strict values of P. Loose interpretations, in turn would apply to those more open to P-approximation for less strict values of P. Accordingly

5.4 Inference and Analogy

A rapidly growing body of research has begun to illuminate a deep relationship between thematic thinking and analogical reasoning. In addition, although relatively little research has examined the role of thematic relations in more basic inferential reasoning, preliminary results suggest that thematic thinking also supports some inferences.

Inference is typically based on taxonomic knowledge. For instance, the taxonomic knowledge that cricket is a sport allows one to validly infer, even if one is unfamiliar with the sport, that it requires physical effort and (often) results in a winner. However, thematic knowledge can also support inference. The thematic knowledge that cricket involves a ball and a bat allows one to infer that there must an athlete who delivers the ball (i.e., a bowler) and another who attempts to hit it (i.e., a batsman). Lin and Murphy (2001) tested for thematic inference by presenting scenarios in which a base animal is related to another animal either thematically (i.e., they interact) or taxonomically. Critically, the base animal was described as having a particular bacterium, and participants judged whether the two other animals were likely to also have the same bacterium. People were more likely to infer that two animals have the same bacterium if those animals were thematically related than if they were taxonomically related (see also Saalbach & Imai, 2007, described in Section 6.2). This makes sense because bacteria are transmitted by proximity, and animals that interact with one another will have opportunities for contact. Chaigneau, Barsalou, and Zamani (2009) found similarly that the accuracy of inferences is improved when participants have knowledge about the events and situations in which objects are used. For example, participants more accurately inferred the function of a novel object when it was presented with other objects used in the same event (e.g., a projectile to be used in a catapult) than when presented in isolation.

Thematic relations also support analogical inference. Comprehending an analogy requires one to recognize the relation between two source concepts (e.g., pen : write) and infer that same relation between two target concepts (e.g., scissors : cut). Indeed, the relationship between thematic thinking and analogical reasoning appears to be strong and interactive. Doumas, Hummel, and Sandhofer (2008) developed a powerful computational model in which relational concepts (including thematic relations) themselves are abstracted from experience with multiple instances of analogous relations. With sufficient exposure to various causal relations, for instance, one develops cause and effect role concepts, which are then used to more efficiently detect and represent new causal relationships. And conversely, because many analogies involve thematic relations, Leech, Mareschal, and Cooper (2008) developed another powerful computational model in which analogical reasoning develops from the more basic process of relation priming (see Section 4.4). Essentially, they argue that analogical inference is bootstrapped from our natural propensity for apprehending thematic relations. Thus, analogies appear to enable the development of relational themes, and those thematic relations subsequently sustain more advanced analogical inference.

Several studies support this presumed link between thematic thinking and analogical reasoning. Understanding an analogy activates the relation between terms (Green, Fugelsang, & Dunbar, 2006), and that relation can be recognized even after the terms have been forgotten (Kostic, Cleary, Severin, & Miller, 2010). Apprehension of the relation between source items not only facilitates relational transfer to a target pair (e.g., Bendig & Holyoak, 2009) but also facilitates retrieval of previously experienced, relationally similar examples (Gentner, Loewenstein, Thompson, & Forbus, 2009; Markman, Taylor, & Gentner, 2007). And just as literal similarity can either help or hinder analogical reasoning (Gentner & Colhoun, 2010), thematic relations can also facilitate a correct response or distract from it (Thibaut, French, & Vezneva, 2010). For example, the highly accessible thematic relation between train and track induces the correct analogical inference in car : road :: train : ??, but it decreases accuracy in car : petroleum :: train : ??. In addition to the relation itself, the relational roles are also an important factor in analogical reasoning (Estes & Jones, 2008; Hummel & Holyoak, 1997, 2003; Morrison et al., 2004). Faced with the analogy wind : erosion :: smoke : ??, people tend to incorrectly complete the analogy with the highly accessible and thematically related fire. However, given that the direction of the causal relation in the source pair is cause → effect, a more appropriate response would be suffocation. Together, these studies suggest that thematic thinking underlies analogical inference and may also influence more basic inferences.

1 Devising Negative Analogies

Chen and Daehler (1989) investigated the effects of positive and negative analogies on young children’s analogical reasoning in a problem analogy paradigm that required 6-year-old children to extract a bead from a narrow glass cylinder without inverting it. The cylinder was 12” deep and 2” wide, and the bead was floating in a small amount of water at the bottom. Various tools were provided, including a cup of water, scissors, a hammer, two sticks, and a short spoon. Two different solutions were possible with these tools. One involved adding more water to the cylinder until the bead floated to the top and could be manually removed, and the other required connecting the spoon to one of the sticks to make a tool long enough to reach the bead.

Two groups of children were informed about one or other of these solutions via analogy stories. One analogy story was about someone retrieving a Ping-Pong ball from a hole by pouring water into the hole, and the other was about a monkey that retrieved food placed outside its cage by joining two sticks together so that it could reach the food. The valency of these analogies to the target problem was manipulated by varying the adequacy of the solution tools provided. For one subgroup of children who had heard the Ping-Pong ball story, sufficient water was provided in the cup to bring the bead to the top of the cylinder (positive analogy group), and for a second subgroup (negative analogy group) the water was insufficient. Similarly, for one subgroup of children who had heard the monkey story, the spoon could be attached to the stick that was provided (positive analogy group), and for a second subgroup it could not (negative analogy group).

Chen and Daehler predicted that hearing one solution when only the other was practicable would impede solution of the target problem. This prediction was confirmed. Sixty percent of the children who received a positive analogy solved the bead extraction problem, compared to 29% of children in a control group who heard a neutral story, and only 8% of the children who received a negative analogy solved the problem. Positive analogies significantly enhanced the children’s performance, and negative analogies significantly impeded it.

6.2 Legal Reasoning and Transfer

The use of single-instance reasoning has implications well beyond its use by the layperson. A variant on single-instance and analogical reasoning is legal reasoning. The finding of similarity is the key ingredient in legal reasoning. Thus, much of legal reasoning is based on transfer. As the history of legal reasoning shows, it is based largely on reasoning from a single or a few instances or cases. Attorneys and judges often argue whether some previous case is applicable to a current case. Even reasoning on the basis of legal principle involves transfer reasoning; judges often decide cases “on principle,” demanding that like cases should be treated alike.