Novick, L. R. & Holyoak, K. J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory and Cognition, 17, 338-415.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....will be the representations encountered earlier. Previous research into when transfer occurs shows that transfer can happen between exercises taking place in different representations, through mechanisms such as analogy, and that transfer can occur between similar processes (Novick 1988, Novick and Holyoak 1991, Singley and Anderson 1989) Hence, we seek to find out if and how these processes extend to the very first stages of learning how to use and generate a representation. We are interested both in positive transfer, and in overgeneralization, where knowledge is transferred inappropriately. ....

....both appropriately and inappropriately. We do not yet know which common features between representations are essential to this process certainly it seems that surface features are more important than deeper features, a finding compatible with those in analogical transfer (cf. Novick 1988; Novick and Holyoak 1991) but which surface features are most salient is an important question in itself. For example, stem and leaf plots have three large differences from histograms: flipped axes, the need to remove the tens digit, and the need to count up values. Determining which of them is most important will have ....

Novick, L.R. and Holyoak, K.J. (1991) Mathematical Problem Solving by Analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition. 17 (3), 398-415.

....model integrating problem solving and learning) 1 Introduction Analogical problem solving is commonly described by the component processes retrieval, mapping, and transfer. We will focus on analogical transfer. Transfer can be faulty or incomplete, even if retrieval and mapping were successful (Novick Holyoak, 1991). We are especially interested in transfer of non isomorphic sources that is, problems which are not structurally identical to a target problem. Several studies (cf. Reed, Ackinclose, Voss, 1990; Spellman Holyoak, 1996) show that subjects can transfer non isomorphic sources successfully ....

Novick, L. R., & Holyoak, K. J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition, 17(3), 398-415.

....In the analogy approach a new problem is solved by adapting the solution of an already solved, structurally similar, problem. Learning is described as generalization over the structures of the newly solved problem and the analog example problem retrieved from memory (e.f. AT89 ] Chap. 4, NH91 ] Schmid ( Sch94 ] proposed a hierarchical memory structure with RPS as basic cognitive units. This memory structure is acquired bottom up: When a new problem is solved by adapting a stored example solution, both problems are represented as predecessors of the infered generalized RPS. If ....

L.R. Novick and K.J. Holyoak. Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition, 14:510--520, 1991.

....worthwhile to further investigate structural properties: It is evident that in realistic settings source problems are usually not isomorphical to target problems. And there are experimental data which show that succesful mapping between source and target is not sufficient for succesful adaptation (Novick Holyoak, 1991). But the question which kind of structural properties are necessary and sufficient for succesful adaptation is seldom addressed in psychological experiments (see Hummel Holyoak, 1997 for an overview of recent work) and there are no computational models dealing with structure mapping and ....

Novick, L. R., & Holyoak, K. J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition, 17 (3), 398-415.

....model integrating problem solving and learning) 1 Introduction Analogical problem solving is commonly described by the component processes retrieval, mapping, and transfer. We will focus on analogical transfer. Transfer can be faulty or incomplete, even if retrieval and mapping were successful (Novick Holyoak, 1991). We are especially interested in transfer of non isomorphic sources that is, problems which are not structurally identical to a target problem. Several studies (cf. Reed, Ackinclose, Voss, 1990; Spellman Holyoak, 1996) show that subjects can transfer non isomorphic sources successfully ....

Novick, L. R., & Holyoak, K. J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition, 17(3), 398-415.

....we store it in memory. These RPSs enlarge the set of predefined functions available to IPAL and they can be used for analogical programming. Analogical programming can be seen as a special case of analogical problem solving which is described by four subprocesses in cognitive science literature [15]: 1) Retrieval of an example problem already solved which is structural similar to the current problem; 2) mapping the structures of example and goal problem; 3) adapt the example solution to the goal problem; and (4) generalize over the structure of example and goal problem. In the case of ....

L. R. Novick and K. J. Holyoak. Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition, 14:510--520, 1991.

....other areas of analogy and problem solving research. Salvucci Anderson, Integrating Mapping and Problem Solving 3 Introduction Analogy the process of finding and using correspondences between concepts plays a fundamental and ubiquitous role in human cognition. From mathematical problem solving (Novick Holyoak, 1991) to computer programming (Anderson Thompson, 1989) to creative discovery (Holyoak Thagard, 1995) analogy facilitates better understanding of old knowledge and the formation and inference of new knowledge. Given the importance of analogy to cognition, it has become clear that any general ....

....For instance, in Dunckers (1945) tumor ray problem, a doctor removing a tumor can be mapped to a general attacking a fortress. While other subprocesses play a role in the analogy process (e.g. access: Ross, 1987, 1989; induction: Gick Holyoak, 1983; transfer: Catrambone, 1994; adaptation: Novick Holyoak, 1991), analogical mapping has stood out as a fundamental yet complex component of analogy that warrants detailed study. This paper presents the path mapping theory, an extension of the ideas in Salvucci and Anderson (1998a) in which analogical mapping is incorporated into a general problem solving ....

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Novick, L. R., & Holyoak, K. J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition, 17, 398-415.

....execution of several operations in an unconstrained order. For example, geometric analogies are solved by applying transformations such as rotate , change size , and add a part . The relationship is only stated; Novick and Tversky do not make any follow up connections between the similarity. Novick and Holyoak, 91] have also commented on the general idea that mathematical problems can be solved by analogy. MacDonald, 84] has shown that rudimentary mathematics can be taught effectively via analogical reasoning. MacDonald s experiments did not involve algebra; he concentrated on basic mathematical ....

Novick, L.R., and K.J. Holyoak. "Mathematical problem solving by analogy." Journal of Experimental Psychology : Learning, Memory, and Cognition Vol: 17 No.: 3, (1991), 398-415.

.... of source and target problem, and (3) transferring the source solution to the target (cf. Falkenhainer, Forbus, Gentner, 1989; Anderson Thompson, 1989; Hummel Holyoak, 1997) Constructing a systematic and consistent mapping between source and target is a necessary prerequisite for transfer (Novick Holyoak, 1991). For domains such as algebra word problems (Reed, Ackinclose, Voss, 1990; Novick Holyoak, 1991) constructing proofs or programs (Anderson Thompson, 1989) or generating plans (operation sequences to fulfill a given goal; Carbonell, 1983) analogical mapping and transfer are typically ....

.... Forbus, Gentner, 1989; Anderson Thompson, 1989; Hummel Holyoak, 1997) Constructing a systematic and consistent mapping between source and target is a necessary prerequisite for transfer (Novick Holyoak, 1991) For domains such as algebra word problems (Reed, Ackinclose, Voss, 1990; Novick Holyoak, 1991), constructing proofs or programs (Anderson Thompson, 1989) or generating plans (operation sequences to fulfill a given goal; Carbonell, 1983) analogical mapping and transfer are typically described in the following way: mapping is performed on the initial problem descriptions represented ....

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Novick, L. R., & Holyoak, K. J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition, 17(3), 398-415.

.... (e.g. Novick, 1988) analogue retrieval (e.g. Holyoak Koh, 1987; Keane, 1987; Wharton et al., 1994) analogical mapping (e.g. Clement Gentner, 1991; Gentner Toupin, 1986; Keane, 1985, 1988, 1995; Spellman Holyoak, 1992) adaptation (Keane, 1994; Holyoak, Novick Melz, 1994; Novick Holyoak, 1991) and induction (Gick Holyoak, 1983; Novick Holyoak, 1991) Furthermore, many of these sub processes have been modelled computationally and specific tests of these models have been carried out (see e.g. Keane Transfer Between Analogies 4 Falkenhainer, Forbus Gentner, 1989; Holyoak ....

.... Holyoak Koh, 1987; Keane, 1987; Wharton et al., 1994) analogical mapping (e.g. Clement Gentner, 1991; Gentner Toupin, 1986; Keane, 1985, 1988, 1995; Spellman Holyoak, 1992) adaptation (Keane, 1994; Holyoak, Novick Melz, 1994; Novick Holyoak, 1991) and induction (Gick Holyoak, 1983; Novick Holyoak, 1991). Furthermore, many of these sub processes have been modelled computationally and specific tests of these models have been carried out (see e.g. Keane Transfer Between Analogies 4 Falkenhainer, Forbus Gentner, 1989; Holyoak Thagard, 1989; Keane Brayshaw, 1988; Keane, Ledgeway Duff, ....

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Novick, L.R., & Holyoak, K.J. (1991). Mathematical problem solving by analogy.

.... In complex problems, transfer is very hard to induce, unless a number of examples are presented, the shared schema is emphasized, and salient structural features are highlighted (Catrambone Holyoak, 1989; Cooper Sweller, 1987; Holyoak Koh, 1987; Klauer, 1989; Larkin, 1989; Novick, 1988; Novick Holyoak, 1991; Reed, et al. 1990) Transfer can also be induced when the underlying principle is explicitly taught (Perry, 1991) or when the subject is encouraged to explain the relation between the surface and structural features (Brown Kane, 1988; Lewis, 1988) However, even with some form of hint to ....

....could not solve distant variants of the problem using the explanations they were given. One of the main reasons was that the the explanations still left the subjects with a large number of inferences to make (Robertson Kahney, 1993) The relation between example and test problems was examined by Novick Holyoak (1991). Using algebra word problems they looked at the effects of giving subjects specific numerical mappings for transfer problems. For example, in the number mapping hint condition, subjects were presented with hints such as: the 12, 8, and 3 in the band problem are like the 10, 4, and 5 in the ....

Novick, L. R., & Holyoak, K. J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition, 17(3), 398-415.

....underlying the phenomenon; that is, representation, retrieval, mapping, adaptation and induction. Many empirical studies now substantiate these theories (see e.g. Clement Gentner, 1991; Gentner Toupin, 1985; Gick Holyoak, 1980; Holyoak Koh, 1987; Keane, 1985, 1987, 1988, 1994; Novick Holyoak, 1991). The distinctive, core sub process in analogy is analogical mapping; it establishes the correspondences between the concepts in a base domain of knowledge and a target domain, performing any analogical inferences that follow from these correspondences. For example, in drawing an analogy between ....

Novick, L.R. & Holyoak, K.J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory and Cognition, 17, 398-415.

....are semantically distant to one another (e.g. Gentner Landers, 1985; Keane, 1987) Analogical mapping is the core process during which the analogy is drawn between the two domains. After an analogy is mapped it may have to be adapted in order to make the solution fit the target problem (see Novick Holyoak, 1991; and the Artificial Intelligence literature on case based reasoning, e.g. Kolodner, 1993; Smyth Keane, 1993) With respect to induction, Gick Holyoak (1983) have shown that subjects can induce a generalisation from the correspondences between two analogous stories which facilitates ....

....it has to be verified and may have to be adapted to fit the target situation. Intuitively, the above causal, inference seems reasonable but experimentation could reveal that the dependency is only one of enablement, in which case the inference would have to be adapted (see Falkenhainer, 1987) Novick Holyoak (1991) have demonstrated that adaptation occurs in solving mathematics problems by analogy to stories. But, very little is known about the nature of adaptation and how it influences the course of analogical problem solving. The Theoretical Issue One of the basic issues in analogical mapping is how one ....

Novick, L.R. & Holyoak, K.J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory and Cognition, 17, 398415.

....and showed the differences in experts and novices use of analogy. Subprocesses The earliest psychological models for analogy as well as more recent ones echo psychological theories stemming from research on analogy comprehension and problem solving, e.g. in [77, 132] Several theories, e.g. [52, 71, 107], suggest that the following subprocesses constitute analogical reasoning: ffl Representation of the source and the target, ffl Retrieval of a potentially useful source, ffl (Partial) mapping between the source and the target, ffl Elaboration of the mapping, ffl Evaluation and adaptation of ....

L.R. Novick and K.J. Holyoak. Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory and Cognition, 14:398--415, 1991.

....analog can at times be very difficult. Some researchers have named a fourth essential component of analogy, that of schema induction. Anderson (1993) and Anderson and Thompson (1989) have used analogical reasoning in the creation of new production rules that act as schemata for similar problems. Novick and Holyoak (1991) provided evidence that schema induction is a natural consequence of successful analogical transfer. Ross and Kennedy (1990) found that the cueing of memorized sample problems can facilitate generalization of already known formulas. The literature is somewhat undecided on whether schema induction ....

Novick, L. R., & Holyoak, K.J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition, 17, 398-415.

....copying during component adaptation is a key problem to be avoided during software reuse, otherwise cognitive short cuts are likely to result in inferior or incorrect software designs. Studies have also revealed other difficulties during adaptation of analogical solutions. Novick Holyoak [49] investigated mathematical problem solving by analogy and observed clear distinctions between understanding and adapting old solutions. Surprisingly, good analogical understanding did not lead to successful transfer. Novick Holyoak also reported induction of key mental abstractions during ....

....which were similar or prominent in the reusable specification. On the other hand, expert software engineers avoided copying and exhibited strategies which maximised transfer of the specification and exploited all reusable components. They tended to understand the specification while adapting it [49]. However, understanding the reusable specification was difficult despite the best efforts of the experts. They also only understood components which were similar and prominent in the reusable specification. Study conclusions: these two studies revealed that complex component understanding and ....

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Novick L.R. & Holyoak K.J., 1991, 'Mathematical Problem Solving by Analogy', Journal of Experimental Psychology: Learning, Memory, and Cognition 17(3), 398-415.

....solving experiences. That is, he she commits the new program into memory. Additionally, he she can profit from the identified structural similarities between problems. An abstract scheme generalizing over the common structure of both programs is generated and also stored in memory for further use [3, 16]. If a current problem cannot be solved by analogical transfer, it has to be solved by general mechanisms of inference (as incorporated in IPAL) This is typically a more time consuming process. An inferred program is also stored in memory. While there is a large and still growing number of work ....

L. R. Novick and K. J. Holyoak. Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition, 17(3):398-- 415, 1991.

....directed to do so. In analogical mapping, the core process in analogical thinking, the analogy is drawn between the two domains (e.g. between the story and the problem) After an analogy is mapped it may have to be adapted to make the analogous solution fit the target problem (see Keane, 1990; Novick Holyoak, 1991). With respect to induction, Gick Holyoak (1983) have shown that subjects can induce a generalisation from the correspondences between two analogous stories that facilitates subsequent problem solving. In this study, we will be concerned with factors that impact the mapping and adaptation ....

....solving [In Artificial Intelligence, adaptation is a key stage of case based reasoning (see e.g. Kolodner, 1993; Riesbeck Schank, 1989; Smyth Keane, 1994, 1995) In this paper, I test the novel hypothesis that adaptability factors can also affect the interpretation selected for an analogy. Novick Holyoak (1991) have demonstrated the importance of adaptation in solving mathematics problems by analogy to stories. They showed that even when subjects know the correspondences between two domains, they had difficulty applying the solution plan in the base story (the mathematical procedure of finding the ....

Novick, L.R., & Holyoak, K.J. (1991). Mathematical problem solving by analogy.

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Novick, L. R. & Holyoak, K. J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory and Cognition, 17, 338-415.

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