CLASSIC ANALOGY AND ANALOGICAL PROBLEM SOLVING

Analogical thinking lies at the core of human cognition and is a key component for a multitude of functions such as problem solving, reasoning, and discovery and learning. It has been argued that the very act of forming an analogy requires a kind of “mental leap,” inasmuch as it necessitates seeing one thing as if it were another (Holy-oak & Thagard, 1995). Many scientific discoveries frequently rely on these mental leaps, and analogy forms the basis for our everyday problem solving, from the simplest instance to the most sophisticated reasoning strategy. Analogical reasoning has been classified into two common types: classic analogy and analogical problem solving. Both types of analogical reasoning involve analogical mapping, a central process of discovering which elements in the target correspond to specific elements in the source and aligning them together.

## CLASSIC ANALOGY AND ANALOGICAL PROBLEM SOLVING

Classic analogy usually applies knowledge from a set of familiar elements (the relation of A to B) to relations about yet unknown elements (the relation of C to D) (Sternberg, 1977). Classic analogical problems are states A:B::C:? For example, dog is to puppy as cat is to ___? The answer is kitten. By knowing the relation between the first two elements (a puppy is a baby dog), one can use that knowledge to complete the analogy for a new item (cat). Because analogies are based on similarities, one must understand the relational similarity between dogs and cats and puppies and kittens in order to solve the analogical problem above.

Analogical problem solving, the second type of analogical reasoning, consists of a source case (which is generally already understood to some extent) and a target case (about which new knowledge is desired), and a relation that maps these elements from one case to the other. The analogical problem is considered solved when individuals can successfully transfer their knowledge from the source set and apply it correctly to the target set. A classic study by Gick and Holyoak had the participants attempting to solve the “tumor” problem. They were presented with a situation in which a doctor was trying to save his patient's life by eradicating a stomach tumor. The participants needed to advise the doctor on the kind of ray that at sufficiently high intensity can destroy the tumor but not harm the surrounding healthy tissue. The solution was to aim multiple low-intensity rays at the tumor from many angles; the rays would “meet” at the site of the tumor and their “sum” would equal the full strength of the ray. Gick and Holyoak found that the problem was quite difficult to solve. To determine whether participants could transfer a solution from an analogous problem to the tumor problem, they had participants first read an analogous story about the “general and a fortress” in which the general had to divide his troops into many small groups and attack the fortress from different directions. Having an analogous story in structure (source set) prior to reading the target problem allowed the participants to transfer their previous knowledge and correctly solve the analogy. However, most participants required a hint before they noticed that the solution “general and a fortress” problem could be applied to the “tumor” problem.

The analogy helps learners make connections between the pre-existing source knowledge and the new target. It has also been suggested that analogies facilitate abstraction from individual cases to general schemata (Gick & Holyoak, 1983) or help the learner generate new inferences about the target. Studies have shown that analogies can support productive conceptual change when neither the target nor the source are well understood; knowledge of each can be enriched through a process of “bootstrapping” (Kurtz, Miao, & Gentner, 2001). Investigations have demonstrated how analogies are used in non-experimental settings such as political debates (Blanchette & Dunbar, 2001), as well as in more naturalistic settings, such as scientific laboratory meetings. Although analogical transfer is an important component in human thinking and reasoning, lab studies do not always demonstrate effective use of analogies in solving problems. Some studies indicate that in lab scenarios in which participants typically use source information to solve problems superficially similar to the source, scientists often use structural features and higher-order relations in making the analogy during the discovery process.

## COMPONENTS AND PROCESSES INVOLVED IN ANALOGICAL REASONING

Analogical problem solving can be functionally divided into multiple processes. The first component involves representing the source information. Studies have demonstrated that individuals' ways of encoding source analogues and the characteristics and quality of the representations influence subsequent transfer. For example, children who formed a goal structure of a source problem (e.g., goal of the main character; obstacle; and action to overcome obstacle and achieve goal) transferred more readily than those who encoded only specific details (e.g., Brown, 1989). Differences in breadth of learning are related to differences in the depth of initial learning, and this more “robust learning” partially explains the better retention and wider generalization evident in subsequent transfer.

The second component is perceiving the analogical relationship, and a major obstacle to transfer is the failure to access spontaneously a source analogue. Informing problem solvers about the potential usefulness of source analogues, increasing the superficial similarities between tasks, and encouraging solvers to extract a more abstract principle from the source analogues have proved useful in improving the accessing process (Gentner & Rattermann, 1991).

Once a source analogue has been retrieved, the correspondences between the problems' key elements need to be mapped. The mapping process is guided by the common relational structures between analogous tasks, and perceiving these underlying structures presents a great challenge to successful transfer, often not only for young children but also for some adults. It is important to note that the difficulty in analogical transfer is not simply a developmental problem, but in various contexts may pose difficulty for all learners. Accessing an analogue and mapping relations do not ensure successful implementation of a solution. The last component involves executing an acquired strategy or principle. When the source and target problems share only a solution principle but differ in specific procedures, children might experience an obstacle in executing a source solution, despite the availability of the strategy (e.g., Ross, 1989). Experiencing diverse procedures illustrating a solution principle facilitates the extraction of a strategy, which is not embedded in a specific procedure, and its flexible and effective generalization to relevant problems.

The conceptual model concerning transfer processes and the age differences associated with each component addresses the observation that children in particular often experience difficulty with analogical transfer. This model also addresses questions of why some analogues are more difficult to use in problem solving than others and helps pinpoint how people at different age levels differ in solving analogous tasks, and how various factors influence transfer performance.

## THEORIES OF ANALOGICAL REASONING

While it is well documented that analogical reasoning is a sophisticated conceptual process that is central to everyday thinking and learning, many underlying mechanisms that support the development of analogical reasoning are not yet well understood. Previously established theoretical accounts of analogy have two important processes: relational processes, which establish correspondences based on similar relations among the objects in the two problems, and object-matching processes, which create correspondences based on the similarity of an object in the target problem to one in the source problem. However, research has shown existing evidence of consistent developmental differences in drawing analogies, and efficiency in analogical problem solving has been offered as a sensitive index of age-related differences. Younger children are more prejudiced by surface features than by structural or deeper causal properties and they are more dependent on hints pointing to the relations between problems. With age and experience, an expanding knowledge base, and increasing mental resources, children become more effective in perceiving deep relations or causal structures. Theorists have put forth three major hypotheses to account for age-related differences in analogical reasoning.

The first hypothesis is increased domain knowledge. In the late 1970s Piagetian studies suggested that young children are unable to reason analogically prior to achieving formal operations at approximately 13 or 14 years of age. Goswami and Brown have greatly disagreed with that claim and proposed in their 1989 study a “relational primacy hypothesis,” arguing that analogical reasoning is fundamentally available as a capacity from early infancy but that children's analogical performance increases with age due to the accumulation of knowledge about relevant relations. While Piaget's tasks frequently involved uncommon tasks (e.g., a “steering mechanism”) which were likely unfamiliar to youngsters, Goswami presented analogical reasoning tasks that were more relevant to children as young as 3 years old, who in turn, demonstrated knowledge about those relations. That said, having knowledge about relevant relations still cannot fully account for age-related effects in young children's performance on analogical reasoning tasks since children still seem to fail on analogies in systematic ways even when they possess relational knowledge relevant to the task.

The second hypothesis, an alternative explanation of young children's observed age-related increase in analogical reasoning performance, is relational shift. In a 1991 study, Gentner and Rattermann suggested that children primarily begin to attend to feature similarity between objects and will reason on the basis of perceptual features rather than on the basis of relational similarity. Following a relational shift, or a fundamental maturational change in their thinking, children can and will reason on the basis of relational features. Developmental literature has shown support for this hypothesis, demonstrating the interrelations between young children's processing of object similarity and their processing of relational similarity.

A third explanation for developmental changes in analogical reasoning shows the limits on children's working memory capacity that affect their ability to process multiple relations simultaneously. In a 2002 study Andrews and Halford defined relational complexity in terms of the number of sources of variation that are related and must be processed in parallel. In a binary relation in which “a dog chases a cat,” the chase itself is a single relation, while the interaction between the dog and the cat become the arguments or the second relation. A child would need to hold both arguments, dog and cat, as well as the relevant relation in mind to reason on the basis of this relation. Andrews and Halford argued that for a developmental continuum in children's working memory capacity, children can process binary relations (a relation between two objects) after 2 years of age and can go on to process ternary relations (a relation between three objects) after 5 years of age. Age deepens children's development of executive functions, particularly the ability to reflect on the relation between two rules. Cognitive maturation also allows children to efficiently solve analogical problems with higher levels of complexity once they have an increased control over their thoughts and actions.

## THE ROLE OF ANALOGY IN COMPREHENSION, TRANSFER, AND DEVELOPMENT

Whether and how children retrieve relevant information and use examples acquired in the past to solve analogous problems is a fundamental issue in the study of cognitive development. Experimental investigations of analogical transfer can be traced back to the beginning of the 20th century (e.g., Thorndike & Woodworth, 1901); however, the centrality of transfer to children's thinking and learning still remains a rather peripheral aspect in developmental psychology.

In broad terms, analogical thinking involves metaphor comprehension (e.g., Gentner & Markman, 1997), similarity and relational mapping (Markman & Gentner, 2002), classical analogy (Goswami & Brown, 1989), and problem solving by analogy (Brown, 1989). The transfer of strategies from familiar situations to novel problems reflects how deeply children acquire strategies, how broadly they generalize them to different situations, and how flexibly they think and reason. Strategy generalization is one of five critical dimensions of strategic change in children's thinking (Siegler, 2000) and is an ultimate measure of children's learning.

While traditional Piagetian accounts of analogical reasoning suggested that it is an ability that develops later in life, early competencies in solving analogical problems were demonstrated with infants as young as 10 to 12 months. The infants first observed their parent solving a problem and then were able to transfer the strategy themselves to a new problem, which shared the same underlying structure but differed in superficial features.

Substantial evidence further demonstrates that preschool children who observed an experimenter demonstrating with an appropriate tool (a rake that was both long enough to reach the toy and had a functional head with which to pull the toy closer) transferred more effectively than did those who received only a hint about using the tool. Specifically, these children were increasingly likely to choose a tool that was similar in causal function to the source tool (e.g., a long cane) instead of tools sharing only perceptual similarities (e.g., a rake head without a handle or a handle without an effective head). Receiving a hint still proved somewhat helpful; children offered a hint outperformed those who received neither demonstration nor hint. These findings reveal that from a very early age children not only have a rudimentary ability to analogize in problem solving but that help provided by the adult allows the children to move beyond simply understanding analogies based on perceptual similarity and conceptualize them based on their causal function.

With the emergence of evidence that outlines the consistent developmental differences, it is well known that younger children are more prejudiced by surface features than by structural or deeper causal properties, and they are more dependent on hints pointing to the relations between problems. By providing direct instruction in the original learning situation teachers facilitate an effective way of subsequent transfer of strategies. Children's strategy transfer also benefits from specific probing questions that encourage self-explanations that give the children the opportunity to explain both the reasoning behind their strategy choices as well as the conclusions they drew from their design (Siegler, 2000). Children's superior performance in learning and transferring in such a probe condition suggests that direct instruction and asking children to generate self-explanations enhances learning and transfer.

## INSTRUCTIONAL USES OF ANALOGIES

There are numerous research findings in support of analogy-enhanced teaching and learning. The widespread use of analogies as explanatory tools in introducing new concepts occurs across various disciplines in middle schools, high schools, and colleges. This has proved a reliable method of teaching in many countries worldwide.

Analogies, models, and modelling have been recognized as key tools for scientists, science teachers, and science learners. The familiar situation base or source analogue provides a kind of model for making inferences about the unfamiliar situation or the target analogue. By finding a structural alignment between a novel situation and the familiar situation learners are able to bring new ideas or concepts closer to their understanding and achieve eventual mastery.

Classroom-based research demonstrates that the use of models and analogies within the pedagogy of science education may provide an effective understanding of the nature of science. Studies show that in order successfully to develop conceptual understanding in science, learners need to be able to reflect on and discuss their understanding of scientific concepts as they are developing them. Furthermore, it has been found that pedagogies that involve various types of modelling are most effective when students are able to construct and critique their own and scientists' models. Research also suggests that group work and peer discussion are important ways of enhancing students' cognitive and metacognitive thinking skills. Lastly, understanding of science models, analogies, and the modelling process enables students to develop an awareness of scientific knowledge, as well as providing the tools to reflect on their own scientific understanding.

A vast body of research shows the importance of analogy use in math learning. Novice learners have great difficulty spontaneously noticing the similarity between two problems or instances that embody the same principle but have a different form (e.g., Gick & Holyoak, 1983). For example, a novice learner who correctly solves one math problem will have difficulty noticing that a second problem has the same conceptual structure if its form is changed into a word problem, if it is written in an alternate way, or if semantic cues in the problem invoke different background knowledge. Since analogical reasoning follows a series of specific steps that can lead to deep processing of novel information, yielding the capacity to learn and transfer knowledge to unfamiliar targets, the application to learning mathematics is especially relevant.

Mathematics is characterized by abstract structure in which underlying relationships remain the same but the object slots can be filled in various ways. Noticing higherorder similarity relationships between such instances of structural similarity is at the core of complex mathematical thinking. For instance, students may not at first notice the similarity between addition of numbers and addition of variables because variables are different at the surface level. However, with assistance or an increase of expertise, they could generate an analogy between these objects and use what they know about addition of numbers to inform their reasoning about addition involving variables. This simple case illustrates the power of analogy in enabling reasoners to apply information from one mathematical topic to other topics that are structurally similar.

In a 2004 study conducted in eighth-grade mathematics classrooms, Richland, Holyoak, and Stigler discovered that teachers used verbal analogies on a regular basis. When explaining a problem, non-math sources were used almost exclusively when instructors were developing analogies to teach mathematical concepts and when using analogies for socialization purposes. Conversely, when explaining mathematical procedures, teachers were most likely to use non-contextualized math problems. These findings suggest that teachers were explicitly or implicitly tailoring their analogy production to the cognitive needs of students. When students were showing difficulty, teachers generated sources and targets with higher surface similarity, thus making analogy more transparent for learners.

In summary, the use of analogies for instruction, whether initiated by text, by the teacher, or by the students themselves, has been shown to improve conceptual learning. Studies show that analogies presented to students in a school setting, usually via instructional materials or teachers' spontaneous use, promote flexible conceptual learning and problem solving. Analogy allows students to use commonalities of one subject to help understand novel problems or concepts either within a single domain (e.g., physics concept of water pressure confined to a tower) or apply it to a vastly different area of knowledge across domains (e.g. using the water-tower analogy for understanding the cardiovascular system), thereby contributing to integral components of overall cognitive proficiency. Scientific research has increasingly placed analogical reasoning in the foreground, giving further support to notion that reasoning by analogy may indeed be the main engine of inventive thinking.

### BIBLIOGRAPHY

Andrews, G., & Halford, G. S. (2002). A cognitive complexity metric applied to cognitive development. Cognitive Psychology, 45, 153–219.

Blanchette, I., & Dunbar, K. (2001). Analogy use in naturalistic settings: The influence of audience, emotion, and goals. Memory & Cognition, 29, 730–735.

Brown, A. L. (1989). Analogical learning and transfer: What develops? In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 369–412). London: Cambridge University Press.

Chen, Z., Sanchez, R. P., & Campbell, T. (1997). From beyond to within their grasp: The rudiments of analogical problem solving in 10- and 13-month-olds. Developmental Psychology, 33, 790–801.

Gentner, D., & Markman, A. B. (1997). Structure mapping in analogy and similarity. American Psychologist, 52, 45–56.

Gentner, D., & Rattermann, M. J. (1991). Language and the career of similarity. In S. A. Gelamn & J. P. Byrnes (Eds.), Perspectives on thought and language: Interrelations in development (pp. 225–277). London: Cambridge University Press.

Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Science, 15, 1–38.

Goswami, U., & Brown, A. L. (1989). Melting chocolate and melting snowmen: Analogical reasoning and causal relations. Cognition, 35, 69–95.

Holyoak, K. L., & Thagard, P. (1995). Mental leaps: Analogy in creative thought. Cambridge, MA: MIT Press.

Kurtz, K., Miao, C. H., & Gentner, D. (2001). Learning by analogical bootstrapping. Journal of the Learning Sciences, 10, 417–446.

Richland, L. E., Holyoak, K. J., & Stigler, J. W. (2004). Analogy use in eighth-grade mathematics classrooms. Cognition and Instruction, 22(1), 37–60.

Ross, B. H. (1989). Distinguishing types of superficial similarities: Different effects on access and use of earlier problems. Journal of Experimental Psychology: Learning, Memory, & Cognition, 15, 456–468.

Siegler, R. S. (2000). The rebirth of children's learning. Child Development, 71, 26–35.

Sternberg, R. J. (1977). A component process in analogical reasoning. Psychological Review, 84(4), 353–378.

Thorndike, E. L., & Woodworth, R. S. (1901). The influence of improvement in one mental function upon the efficiency of other functions. II. The estimation of magnitudes. Psychological Review, 8, 384–395.