in the Solution of Mathematical Word Problems

Researchers typically have assumed that analogy is a "weak method" most often used by novices.
Weak methods are general-purpose methods that can be used to solve many different types of
problems in a variety of content domains. They tend to be used by less skilled people in a domain,
because those solvers lack the specialized knowledge needed to use more powerful domain-specific
methods. Analogy -- that is, finding the correspondences between two similar situations and using
them to transfer information from the better known to the less well known situation -- would seem to
fit the definition of a weak method because it can be applied to problems in almost any domain.
Contrary to this conception of analogy as a weak method primarily used by novices, the experiments
reported in Novick (1988a) showed that problem solvers who were more expert (i.e., more skilled)
in the domains of arithmetic and algebra were more likely to transfer a solution procedure across
superficially dissimilar complex arithmetic story problems (i.e., from a problem about a vegetable
garden to a structurally-similar problem about a marching band).
Although analogy is a general-purpose solution method, and less skilled solvers in a domain do try to use it, its successful use
depends on domain-specific knowledge (e.g., concerning the types of problem features that are
relevant to the solution), which more skilled solvers are more likely to have.
**Developing a More Precise Model of Procedural Transfer**

A major goal of my research in this area was to more precisely delineate the processes involved in transferring a solution procedure from a worked-out example problem to a similar test problem. Through about the mid-1980s, researchers distinguished two component processes of transfer: retrieving a relevant example and determining the corresponding elements in the example and test situations (i.e., mapping). Researchers tended to assume that once the mapping had been determined, it was a relatively straightforward matter to use the correspondences as the basis for transferring knowledge from one situation to the other. Thus, using the mapping to guide problem solving was not distinguished from constructing the mapping in the first place.

A primary focus of three of my papers was to show that it is important to distinguish the
mapping process from two other processes that occur subsequently. In the Novick and Holyoak (1991) and Novick (1995) articles, I showed that there is a crucial *adaptation* process that solvers must execute in order to modify the example problem's solution procedure to fit the requirements of the test problem. In the Holyoak, Novick, and Melz (1994) chapter, we proposed
that the adaptation process needs to be subdivided by distinguishing *pattern completion* from adaptation. The pattern completion process involves making inferences about the test problem, such as inferring appropriate solution operators to use, by directly carrying over information from the example. The adaptation process is called into play subsequently if these "direct" inferences need to be modified to take into account structural constraints that are unique to the test problem. In Novick (1995), I provided the first empirical evidence for the hypothesized pattern completion process.

The main focus of the Novick (1995) paper, however, was adaptation. I proposed two factors that may be important for distinguishing types of adaptations for mathematical word problems. This analysis, which refined and extended that given by Novick and Holyoak (1991), applies to problems involving arithmetic and algebra, but it may also be applicable more broadly in the domain of mathematics. The adaptation process is particularly important because successful adaptation of a solution procedure to account for slightly altered mathematical structures is what will allow problem solvers to develop abstract schemas for types of solution procedures that can be flexibly applied when appropriate (see the next section on schema induction). In particular, knowledge about when a procedure needs to be modified and how to modify it is what will provide solvers with crucial information about the "conditions of applicability" of the procedure. A common complaint of educators is that students often know how to solve problems if they are told what procedure to use, but they have difficulty determining on their own the best procedure to apply. This is because their knowledge is inert -- i.e., it does not include information about the conditions under which the procedure can be applied.
**Schema Induction as a Consequence of Procedural Transfer**

Much of the early work on analogical transfer concerned factors that facilitate its occurrence. For example, Keith Holyoak and his colleagues found that having an abstract schema for the type of problem and its solution makes it easier for solvers to transfer that solution procedure to new problems for which it is relevant. In the early 1990s, researchers began to consider whether there are any long-term benefits of successful transfer beyond the effects on solving a specific test problem. In particular, they hypothesized that solving a test problem by analogy to an example problem would help solvers learn about a general category of similar problems about which they were previously unfamiliar. Such "schema induction" would be an important consequence of procedural transfer because research on expertise has shown that highly skilled problem solvers tend to rely on abstract schemas for types of problems to guide their problem solving. In collaboration with Keith Holyoak, I provided some of the first direct evidence for the link between transfer and schema induction (Novick & Holyoak, 1991).
**Individual Differences**

The results of most of my research on the contributions of algebraic expertise and general ability to solving mathematical word problems by analogy are summarized in Novick (1992). I found that the contribution of expertise depends on the transfer component being considered. More skilled algebraists have no advantage over their less skilled peers in executing the pattern completion process or in inducing an abstract schema from their successful transfer attempt(s). However, they do have an advantage in retrieving an appropriate example problem, mapping the elements in the example and test problems, and adapting the example solution procedure to fit an altered mathematical structure. These are the processes that are most highly dependent on being able to appropriately analyze the underlying structure of a problem. Extensive research on expertise indicates that this is a prime area in which more expert solvers have a distinct advantage. Thus, my results suggest that instruction aimed at teaching students how to analyze mathematical structures is likely to ameliorate most of the difficulties less expert solvers experience in transferring a learned procedure to a new problem.

Induction involves the development of general principles from one or more specific examples and the generalization of those principles to new cases. Analogies of the form "A is to B as C is to D" are the most common measure of inductive reasoning. For analogical transfer, the example problem and solution correspond to the A and B terms of the analogy, and the test problem and solution correspond to the C and D terms. Thus, it is reasonable to expect inductive reasoning ability to be highly related to success at analogical transfer. In contrast, I found no relation between these two constructs in two studies that used different measures of inductive reasoning ability and that assessed transfer using different types of problems (Novick & Holyoak, 1991; Novick, 1995).

In the later study (Novick, 1995), I predicted that higher levels of deductive reasoning ability would facilitate execution of the mapping process of analogical transfer, because the existence of certain correspondences between the example and test problems implies the existence of other, related correspondences. These implications can be deduced logically. The results of my 1995 study supported this prediction. Thus, in contrast to previous theorizing, the core process of analogical transfer seems to be more highly related to deductive than inductive reasoning.

Novick, L. R. (1998, April). Highly skilled problem solvers use example-based reasoning to support their superior performance. In E. R. Rothkopf (Chair), What do experts know: General principles or specific problems? Symposium conducted at the annual meeting of the American Educational Research Association, San Diego, CA.

Novick, L. R. (1995). Some Determinants of Successful Analogical Transfer in the Solution of Algebra Word Problems. Thinking and Reasoning, 1, 5-30.

Holyoak, K. J., Novick, L. R., & Melz, E. (1994). Component processes in analogical transfer: Mapping, pattern completion, and adaptation. In K. J. Holyoak & J. A. Barnden (Eds.), Advances in connectionist and neural computation theory, Vol. 2: Analogical connections (pp. 113-180). Norwood, NJ: Ablex.

Novick, L. R. (1992). The role of expertise in solving arithmetic and algebra word problems by analogy. In J. I. D. Campbell (Ed.), The nature and origins of mathematical skills (pp. 155-188). Amsterdam: Elsevier.

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

Novick, L. R. (1991, April). The role of expertise in analogical problem solving. In J. Gallini (Chair), Analogical problem solving: The mechanisms underlying what develops. Symposium conducted at the Annual Meeting of the Society for Research in Child Development, Seattle, WA.

Novick, L. R. (1988a). Analogical transfer, problem similarity, and expertise. Journal of Experimental Psychology: Learning, Memory, and Cognition, 14, 510-520.

Novick, L. R. (1988b). Analogical transfer: Processes and individual differences. In D. H. Helman (Ed.), Analogical reasoning (pp. 125-145). Dordrecht, The Netherlands: Kluwer Academic Publishers.

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