Memory & Cognition 2001, 29 (3), 462-477 The representations of the arithmetic operations include functional relationships JAMES A. DIXON College of William and Mary, Williamsburg, Virginia JULIE K. DEETS Trinity University, San Antonio, Texas and ASHLEY BANGERT College of William and Mary, Williamsburg, Virginia Current theories of mathematical problem solving propose that people select a mathematical oper- ation as the solution to a problem on the basis of a structure mapping between their problem repre- sentation and the representationof the mathematical operations. The structure-mapping hypothesis re- quires that the problem and the mathematical representations contain analogous relations. Past research has demonstrated that the problem representation consists of functional relationships, or principles. The present study tested whether people represent analogous principles for each arith- metic operation (i.e., addition, subtraction, multiplication, and division). For each operation, college (Experiments 1 and 2) and 8th grade (Experiment 2) participants were asked to rate the degree to which a set of completed problems was a good attempt at the operation. The pattern of presented an- swers either violated one of four principles or did not violate any principles. The distance of the pre- sented answers from the correct answers was independently manipulated. Consistent with the hy- pothesis that people represent the principles, (1) violations of the principles were rated as poorer attempts at the operation, (2) operations that are learned first (e.g., addition) had more extensive prin- ciple representationsthan did operations learnedlater (multiplication),and (3) principles that are more frequently in evidence developed more quickly. In Experiment 3, college participants rated the degree to which statements were indicative of each operation. The statements were either consistent or in- consistent with one of two principles. The participants’ ratings showed that operations with longer de- velopmental histories had strong principle representations.The implications for a structure-mapping ap- proach to mathematical problem solving are discussed. The ability to apply mathematics to real-world do- 1991). Three major lines of research have demonstrated mains is an important skill and an important issue for the- a strong relationship between people’sintuitiverepresen- ories of problem solving.How children and adults gener- tations and their mathematical solutions. First, research ate mathematical solutions to novel problems is not well on comprehending word problems has shown that how understood. Problems, of course, do not arrive labeled, people interpret and represent the problem affects their so- and although cue words embedded in the problem may lution strategies (see, e.g., Kintsch & Greeno, 1985). Sec- help, people apply mathematics to problems that have ond, research comparing experts and novices has shown unique phrasing (Geary, 1994). One important factor in that the problem representation of experts is superior to generating mathematical solutions is the person’s con- that of novices (e.g., Schneider, Gruber, Gold, & Opwis, ceptual or intuitiverepresentationof the problem. The in- 1993). Differences in the problem representation explain tuitiverepresentationof the problem is a qualitative,non- many of the differences in the formal solutions used to formal representation of the relationships between the solve the problem (e.g., Hardiman, Dufresne, & Mestre, objects in the problem (e.g., Moore, Dixon, & Haines, 1989). Third, research on analogy has shown that being able to analogicallymap the structure of a new problem to a better understood problem improves performance with Preparation of this article was supported in part by National Science formal strategies(e.g., Novick & Holyoak, 1991; Ross & Foundation Grant BCS-9996353.Weare grateful to Jennifer Abernathy Kilbane, 1997). and Justin Green for help with data collection and to Jamie Campbell Although a person’s intuitive representation of the and David Geary for comments on an earlier version of the article. We problem is strongly related to the mathematical solution also thank the participating schoolsand students. Correspondencecon- cerning this article should be addressed to J. A. Dixon, Department of he or she generates, past work has shown that the rela- Psychology, College of William and Mary, P. O. Box 8795, Williams- tionship is not simple. Having a good intuitive representa- burg, VA 23187-8795(e-mail: [email protected]). tion of a problem is necessary,but not sufficient, to arrive Copyright 2001 Psychonomic Society, Inc. 462 ARITHMETIC OPERATIONS 463 at an appropriate mathematicalsolution(Dixon & Moore, cess. Because the basic operations are initially learned 1996, 1997). One implication of this relationship is that with positive integers, we focused on principles for this the intuitive representation does not automaticallycue the class of number. A description of each principle for each appropriate mathematics. That is, knowing how a physi- operation can be seen in Figure 1. cal system works (e.g., how temperatures combine)does The monotonicity(Mono) principle specifies whether, not give direct access to the mathematics that instanti- as the operands change in a single direction,the answers ates those relationships. Rather, intuitiverepresentations also change in a single direction.The basic arithmeticop- and mathematical solutions appear to be related through erations all have the same value for the Mono principle. a more complex process. Dixon and Moore (1996, 1997) That is, this principle,unlike the others considered below, and Bassok, Chase, and Martin (1998) have independently is constant across all the operationschildren learn in grade proposed that this process is similar to structure mapping school. Therefore, we predict that this principle will de- in analogy (Gentner, 1983; Gentner & Markman, 1997). velop early in a child’s representation of each operation. Under this hypothesis, people select a mathematical op- To illustrate the Mono principle,considera pattern of an- eration on the basis of its structural match to their intu- swers that violates it: 14 + 15 = 35, 14 + 17 = 37, or 14 itive representation. + 19 = 29. Increasing the second operand (i.e., from 15 A central but untested prediction of this hypothesis is to 17, from 17 to 19) does not always result in the same that people represent relational information about each direction of change in the answers; 35 to 37 is an increase, mathematicaloperationthat is analogousto the relational 37 to 29 is a decrease. Therefore, the pattern of answers information in their intuitive representation. That is, just is not monotonic. as the intuitive representation contains information The relationship-to-operands (RO) principle specifies about the relationships between objects in the physical the relationship between the answer and the operands. system, the representation of each mathematicaloperation This principle takes on different values for the different should contain information about the relationships be- operations.For addition and multiplication, the RO prin- tween objects in the mathematical system (i.e., operands ciple specifies that the answer must be larger than either and answers). The purposeof this article is to propose and of the two operands.For subtractionand division,the RO provide evidence for such a representation of the arith- principle specifies that the answer must be smaller than metic operations of addition,subtraction,multiplication, the minuend or the dividend, respectively.This principle and division. can be evaluated for any of the operationsby examininga single problem. That is, children are exposed to this prin- Principle Representation ciple for a particular operation each time they correctly of Mathematical Operations solve a problem with that operation.Therefore, we expect Although we often think about the arithmetic opera- this principle to develop fairly quickly.As an example of tions in terms of producing particular answers, each op- this principle,consider a single addition problem that vi- eration also produces a signature pattern. This invariant olates the RO principle, 8 + 6 = 4. Obviously, this prob- pattern emerges when the values of the operands are al- lem does not have the correct relationshipbetween the an- lowed to vary within a particular class of number (e.g., swer and the operands; the answer is smaller, rather than positive integers). Figure 1 illustrates the signature pat- larger, than the operands. terns for the arithmeticoperationsfor the class of positive The direction-of-effect (DE) principlespecifies whether integers. For example, addition shows a pattern in which changes in one operand result in increases or decreases the answer increases as either operand increases. The pat- in the answers. This principle also takes on different val- tern is also completely parallel; all the curves have the ues for the different operations. For addition and multi- same slope. Subtraction also shows a parallel pattern, but plication, increases in either operand result in increases in the effect of increasing the operands is quitedifferent; in- the size of the answer. For subtraction and division, in- creasing one operand decreases the answer, whereas in- creasing the minuend or the dividend increases the an- creasing the other operand increases the answer. We pro- swers; increasing the subtrahendor the divisordecreases pose that as children and adults work