Analogical reasoning refers to reasoning on specific exemplars or cases. In this process, knowledge about one exemplar is used to infer knowledge about another exemplar (Gentner, 2003; Holyoak, 2005). Only parts of AR research are directly relevant to the present purpose. We focus on analogies in problem solving, largely neglecting other analogies (e.g., between stories or personal relations). We have also left out important subfields such as computational modeling work (Holyoak, 2005, 2012).
2.3.1. Relevance and effectiveness of examples
Some researchers regard AR (i.e., the reliance on already-familiar examples or cases) as fundamental to achievements such as scientific discovery, problem solving, and identifying causal relations (e.g., Holyoak, 2005; Holyoak & Cheng, 2011). Some authors even regard AR as the core of human cognitive capabilities and, in particular, of effective learning (e.g., Gentner, 2010; Gust & Kühnberger, 2006). AR is also considered as one solution to the learning paradox (Bereiter, 1985): If a learner does not understand something, then she cannot learn it because she does not know enough to even begin. Analogy from a different domain might provide a schema as an initial template that can boost understanding (Chi & Ohlsson, 2005). Holyoak (2012) postulates that analogy is a “strong ‘weak’ (i.e., domain-general) method” that is very powerful if knowledge about an analog is available.
Much of the analogy literature has analyzed basic questions of cognition and learning. There is hardly any research comparing instruction based on AR to other approaches (for an exception, see Nokes-Malach et al., 2012). However, there is indirect evidence: In Hong Kong and Japan—countries with high levels of mathematics achievement—teachers provide much more AR support (e.g., a familiar source problem to be compared to a target analog) as compared to the U.S. “standard instruction,” in which average mathematics achievement is relatively poor (Richland, Zur, & Holyoak, 2007).
AR research provides explanations as to why example-based learning is effective. Transfer is achieved when both the abstract principles and exemplars are provided, and when learners relate the exemplars to abstract principles (e.g., Fong & Nisbett, 1991; Ross & Kilbane, 1997). In other words, transfer requires that both abstract and concrete knowledge be encoded and interconnected (Reeves & Weisberg, 1994). Example-based learning is thus effective because it provides affordances for interrelating abstract and concrete knowledge.
2.3.2. Phases of skill acquisition
Analogical reasoning research has not provided a stage model for an entire skill-acquisition process. However, it has provided a very fine-grained account of different stages within the phases of skill acquisition as described in WE and OL research (cf. the analogy phase by Anderson et al., 1997; intermediate phase, VanLehn, 1996). Typically, four core stages are postulated (Holyoak, 2005, 2012; see also Gentner, 2003). First, the examples—in some cases together with the underlying principles—are presented as sources of transfer. They are encoded in memory. A schema including an abstract principle might already be constructed (stage 1). When encountering a transfer problem, potentially relevant analogical examples—encoded in stage 1—are activated and selected (stage 2). The transfer problem is mapped to the analog; this mapping process can be regarded as the core process of AR (stage 3). Finally (stage 4), induction of an abstract schema can arise from this mapping process (or a schema modification provided it had been constructed in stage 1).
2.3.3. Learning processes
As just mentioned, in the first stage, the examples presented as transfer sources are encoded, and an abstract schema, including the solution principle, might already be constructed. There are two main instructional options in this stage: providing examples from which the principle must be induced (embedded principle method; Ross & Kilbane, 1997) or providing the principle up-front (abstract principle method), the former method being more typical in AR. In the embedded-principles approach, learners do not automatically go beyond the given surface features of the source examples (e.g., cover stories) and induce schemas (Reeves & Weisberg, 1994). Abstraction is fostered when providing multiple examples, particularly when the learners are simultaneously instructed to compare these examples (Holyoak, 2005; Reeves & Weisberg, 1994). The more specific such instructions are, the more likely schemas will be abstracted (e.g., Gentner, Loewenstein, & Thompson, 2003). In the abstract principle approach, the principles and examples must be actively interrelated (e.g., Reed, 1989, 1993; Ross, 1987, 1989). It makes sense to integrate the principle's presentation and an initial example so that learners establish tight connections between them (Ross & Kilbane, 1997). Even if learners encode examples by referring to abstract principles, concrete problem features are usually stored as well (Reeves & Weisberg, 1994; Ross, 1989). Learners also represent contextual features of the learning situation in their memories. For example, a similar context (source example and target problem presented by the same person) facilitates transfer better than a dissimilar context (different persons; Spencer & Weisberg, 1986).
In the second stage, when a transfer problem is to be solved, potentially relevant analogs are (a) activated and (b) selected. These processes are especially difficult for learners (Holyoak, 2005). Either they fail to notice that a known example solution applies to a problem, or they select an incorrect example. The latter problem arises from the fact that learners primarily use their knowledge traces of surface features (e.g., cover stories) to retrieve analogs (Ross, 1987, 1989), which can lead to the retrieval of analogs that do not fit. Note that even when an abstract schema was formed, learners still have the concrete examples encoded in long-term memory (e.g., Reeves & Weisberg, 1994) and they are reminded of related problems by superficial similarities (Ross, 1987, 1989). Such reminders occur even when learners have encoded an example several days before (e.g., Fong & Nisbett, 1991; Keane, 1987). Knowledge traces of concrete examples seem to be relatively stable. Even advanced learners “first” rely on superficial features (Novick, 1992). However, compared to less advanced learners, they are better able to decide whether the retrieved example actually fits or whether they better seek another example. In any case, reliance on surface features can provide useful cues for analogs and corresponding principles because structural and surface features are often correlated in real life (Bassok, 1996; Blessing & Ross, 1996; Novick, 1992). Hence, surface similarity often paves the way for principle-based analogy. For example, the verb “share” is typically a valid cue that division must be applied in a mathematics word problem, even if this is not necessarily true (the word “share” does not correspond to a structural feature; see About.com, 2013).
In the third stage, the problem to be solved is mapped to the analog. The learners determine the correspondences between the features of the known example(s) and those of the problem at hand. Note that for productive AR, not just surface features are considered. The structure mapping theory by Gentner and colleagues (e.g., Gentner & Markman, 1997) emphasizes that AR involves mapping on the basis of structural features that consist of the relations between elements and not (necessarily) on the basis of the elements. Although the relations and not the objects are crucial for successful mapping, learners also rely on surface features when trying to map two problems. For example, when the roles of objects (e.g., cars) and persons (e.g., mechanics) were reversed in source examples and transfer problems, learners often mapped erroneously the corresponding objects to each other, leading to incorrect solutions (e.g., Ross, 1987, 1989). Nevertheless, as mentioned, relying on surface features may also facilitate structural mapping (Bassok, 1996; Reeves & Weisberg, 1994). Moreover, if there are multiple ways to map an example and a problem, the multi-constraint theory (Holyoak, 2012; Holyoak & Thagard, 1997) postulates that a good mapping procedure (i.e., helping in the pragmatic contexts) maximizes correspondences between (surface) elements.
In the final fourth stage, some schematic abstraction might arise from a mapping process. Such a first-schema abstraction may not necessarily lead to a highly generalized representation enabling transfer to any problem with the same deep structure; learners tend to be conservative in this respect (Reeves & Weisberg, 1994). For further abstraction, a multitude of examples and deliberate comparison processes are necessary. If a first schema has already been constructed in stage 1, it may be modified and made more abstract in this stage. Sometimes, however, learners may follow previous examples “verbatim” (e.g., VanLehn, 1998) without relying on abstract principles. In these cases, learners do not induce a schema. They need to engage in elaboration and analogical mapping during problem solving so that abstractness is added to the concrete representations. Overall, AR research assumes that there are two “opportunities” for schema abstraction processes. First, an initial schema might be constructed when (multiple) examples are encoded (stage 1); second, when the knowledge gained from studying examples is applied to a new problem, further abstraction can take place (stage 4).
Interestingly, working memory resources are crucial in AR (cf. cognitive load theory). The comparison and mapping of two analogical problems induce heavy demands on working memory resources, especially when the problems are complex (Holyoak, 2005, 2012). Hence, sufficient working memory resources are crucial for effective analogical transfer. Restricted capacity leads to reliance on surface features rather than structural features (e.g., Waltz, Lau, Grewal, & Holyoak, 2000). Relying on “directly” perceivable surface feature requires less capacity than focusing on relations that must typically be inferred (Holyoak, 2012). Working memory constraints come into play especially when (a) there are multiple relations (instead of just one) to be mapped (complex comparison and mapping demands) and (b) when “seductive” features (e.g., misleading surface similarity) must be inhibited, which is usually attributed to the executive function of working memory (e.g., Cho, Holyoak, & Cannon, 2007; Viskontas, Morrison, Holyoak, Hummel, & Knowlton, 2004). In addition, the performance of learners with restricted working memory capacity, that is, older (Viskontas et al., 2004), distracted (e.g., Waltz et al., 2000), or young learners whose executive functions are not fully developed yet (Richland, Morrison, & Holyoak, 2006) are particularly hampered by multiple relations and by correspondences to be inhibited. Reducing working memory demands should therefore foster the acquisition of effective problem-solving schemas and, thereby, transfer (Richland, Stigler, & Holyoak, 2012). At first glance, these assumptions on the importance of working memory capacity seem to parallel cognitive load theory. However, there are important differences as well that we discuss later on.
In summary, as the WE and OL approaches, AR research regards examples as an important source of the construction of abstract problem-solving schemas. All three approaches also assume that encountering examples is insufficient for effective learning. Instead, learners must actively compare and map examples to each other and to principles. An obvious difference refers to grain size. Whereas research on WEs and on OL focuses the acquisition of cognitive skills on a relatively coarse grain size, AR research analyzes the use of examples in great detail.