Knowledge or attention -- 1 Running Head: REPRESENTATION OF PRINCIPLES REPRESENTATION OF ARITHMETIC PRINCIPLES BY NOVICES: KNOWLEDGE OR ATTENTION?

Four experiments examined the nature of novice and expert representations for both surface features and principled (relational) properties of arithmetic equations. Experiment 1 used a forced-choice categorization task, in which surface features of equations (e.g., numbers) competed with principled properties of mathematics (associativity and commutativity). It was found that experts were more likely to focus on principles in their judgments than were novices, who focused more often on surface features. Experiment 2, using a similar task, introduced trials in which only principled properties varied. Novices were able to focus on principled properties in this case, but failed to transfer these representations when surface features were reintroduced. These findings indicate that novices had knowledge of the principles, but that they did not attend to them when competing surface features were present. Experiment 3 used a recognition task to determine whether or not novices encode deep principled properties, whereas Experiment 4 used the recognition task to determine the manner in which features and principles are encoded. A process model describing probable mechanisms for novice representations of principled (relational) properties is discussed. Knowledge or attention -3 REPRESENTATION OF ARITHMETIC PRINCIPLES BY NOVICES: KNOWLEDGE OR ATTENTION? It has been well established that in different knowledge domains (e.g., physics, mathematics, or chess), experts approach problems in a manner different from that of novices (Chase & Simon, 1973; Chi, Feltovich, & Glaser, 1981; Larkin, 1983; Simon & Simon, 1978; Reed, Ackinclose, & Voss, 1990). In particular, experts and novices often diverge as to what components of a problem they represent. While experts are more likely to focus on hidden relational properties of a problem, such as equations that can be used to solve the problem, novices are more likely to focus on less important surface features of a problem, such as the problem's content. Note that although the term hidden relational property refers to properties that can not be directly observed, the term has been used in at least three ways. First, hidden (or “deep”) properties are those that require knowledge, such as taxonomic or functional properties (e.g., the fact that a whale is a mammal is hidden). Second, hidden properties are relational as opposed to object properties. For example, given two pictoral dyads, (1) daughter-mother and (2) mothergrandmother, the mother in the first picture corresponds to the mother in the second picture in terms of object properties (it is the same object), whereas it corresponds to the grandmother in terms of hidden relational properties (it is the same relation: the mother of X). Finally, hidden properties constitute important principles in a knowledge domain (e.g., Boyle's law) as opposed to surface or accidental properties (e.g., the story line of a particular problem). It seems that the three meanings are interdependent as they jointly emphasize that hidden properties are (1) knowledge-dependent, (2) not directly perceptible, and (3) relational. In this paper, we will use Knowledge or attention -4 the terms deep relational features or deep principles as referring to those hidden properties that have (1), (2), and (3). The term surface feature also affords several meanings. For example, clinical cases could be grouped on the bases of medical symptoms (e.g., high fever vs. headache), demographic variables (e.g., age and gender), or patients' hair colors. All of these would be surface features as opposed to deep relational properties, such as clinical syndromes (e.g., bronchitis or depression) relied upon by experts (see Custers, Boshuizen, & Schmidt, 1998, for a discussion). However, medical symptoms and demographic variables could be "meaningful" surface features, because they could be predictive of a clinical syndrome, whereas patent's hair color is a superfluous surface feature. Although the distinction between meaningful and superfluous surface features may appear arbitrary, it has proved useful in several knowledge domains (see Reeves & Weisberg, 1994 for a discussion). In the current studies, we used those surface features that were superfluous with respect to the deep relational properties. There is a large body of literature indicating that in problem solving, reasoning, learning and transfer, and problem categorization, novices tend to focus on surface features rather than on deep relational properties. These effects have been demonstrated in a variety of knowledge domains. These domains include chess (Chase & Simon, 1973), mathematics (Blessing & Ross, 1996; Hinsley, Hayes, & Simon, 1977; Schoenfeld & Herman, 1982; Bassok, 1996, 1997; Novick, 1988; Reed, et al, 1990; Ross & Kilbane, 1997; Silver, 1981), physics (Chi, et al 1981; Simon & Simon, 1978; Larkin, 1983; Larkin, McDermot, Simon, & Simon, 1980), and computer programming (Adelson, 1984). Similar effects have been observed in a variety of knowledgelean domains, such as deductive and inductive inference. When presented with deduction problems, untrained reasoners often tended to ignore the argument's logic (i.e., its deep structure) Knowledge or attention -5 while relying on the argument's surface features, such as content and believability (Cheng & Holyoak, 1985; Evans, Newstead, & Byrne, 1993; Johnson-Laird & Byrne, 1991; Byrne, 1989). When presented with induction and analogy problems, novices often ignored deep relational structure while relying on the surface features (Gentner, 1989; Gentner & Toupin, 1986; Kotovsky & Gentner, 1996; Holyoak & Koh, 1987). While there is little disagreement that novices focus on surface features, several issues remain unclear. Why do novices tend to represent surface features and not deep relational properties? And why does instruction often fail in helping novices represent deep relational properties, such as rules of logic (Nisbett, Fong, Lehman, & Cheng, 1987) or algebraic equations (Reed, et al 1990; although see Schoenfeld & Herrmann, 1982, for diverging results). In addressing these issues, we will also touch upon another issue of whether superfluous surface features constitute an impediment or aid in extracting the deep relational structure of a task. One possible explanation of novices' tendency to focus on surface features is that novices merely have little knowledge of deep structural relations. However, while this possibility is capable of explaining expert-novice differences in extremely knowledge-demanding domains, such as medical diagnostics, chess, or advanced physics, it falls short of explaining these differences in fairly simple domains, such as elementary mathematics and physics. For example, researchers examining novices' representations in mathematics and physics often drew examples from students' textbooks, thus reasonably assuming that students should be familiar with the deep structure underlying these problems (Chi, et al, 1981; Larkin, 1983; Novick, 1988). The credibility of the lack of knowledge explanation is further undermined by findings that even those novices who receive instruction in a domain often continue to focus on surface features rather than the deep structure of a problem. These has been demonstrated across a variety of Knowledge or attention -6 knowledge domains, including mathematics (Morris & Sloutsky, 1998), logic (Nisbett, et al. 1987), and physics (Kaiser, McCloskey, & Proffitt, 1986; McClosskey, 1983). Finally, the fact that findings on novices' representations in knowledge-lean domains are compatible with those in knowledge-rich domains (Cheng & Holyoak, 1985; Evans, Newstead, & Byrne, 1993; JohnsonLaird & Byrne, 1991; Byrne, 1989, Gentner, 1989; Gentner & Toupin, 1986; Kotovsky & Gentner, 1996; Holyoak & Koh, 1987) makes the low knowledge hypothesis even less plausible. Another possibility is attentional: even when novices know about deep relations and are capable of extracting these relations, they still fail to represent these relations because surface features are more prominently present in the problem. In failing to represent relational properties, they may either fail to encode these relations, or they may initially encode these relational properties, but discard them in favor of more salient surface features. It is also possible that novices use surface features more frequently than relational properties, and, as a result, when surface features are well-known, they are more accessible than deep relational properties (cf. Anderson, 1990). The salience of surface features could be demonstrated using task presented in Figure 1. If one considers similarity of objects, then Test 2 is more similar to the Target (both comprise circles), whereas if one considers similarity of relations, then Test 1 is more similar to the Target (both comprise a monotonic increase). Note that it is easier to overlook relational similarity than to overlook object similarity: while the latter is directly perceptible, the former is not. These intuitions have been corroborated in a series of studies where participants often failed to represent relational similarity in tasks similar to those depicted in Figure 1 (Gentner & Toupin, 1986; Kotovsky & Gentner, 1986; Gentner & Markman, 1997). As mentioned above, the attentional explanation of the tendency to represent surface rather than deep relational properties Knowledge or attention -7 comprises several options: novices may fail to encode deep relational properties, or they may initially encode both surface and deep relational properties but discard the latter in favor of more salient surface features. The first option, of course, makes surface features more benign than the second, because these features are often predictive of the deep relational structure (e.g., Bassok, 1996; 1997; Blessing & Ross, 1996). Therefore, if novices would fail to encode deep relational properties anyway, attention to the surface structure may facilitate novices' learning, transfer, and problem solving (see Bassok, 1997, for a discussion). The second option suggests that surface features, by their virtue of being salient, interfere with the process of representation of deep relational properties (Bassok, 1997; Gentner, 1989; Gentner & Toupin, 1986). Note that because in the current experiments, we did not use surface features that were predictive of deep relational properties, answers provided by the current research would be limited only to superfluous surface features. To distinguish among the above-mentioned explanations (i.e., knowledge vs. attention), we constructed tasks where surface features were completely superfluous and were not predictive of deep relational properties. Do novices know the deep relational properties? Do they represent both surface features and deep relational properties? And do they encode both surface features and deep relational features? The present research addresses these questions. Of course, it is also possible that experts and novices differ dramatically with respect to their general intelligence levels, with experts being a self-selected group, while novices represent more general segments of the population. At the same time, different intelligence theorists agree that the ability to detect relations is indicative of general ability (Horn & Noll, 1994; Kyllonen, 1994; Sternberg, 1985). While this possibility is not likely to tell the whole story (e.g., Knowledge or attention -8 Schoenfeld & Herrmann, 1982, demonstrated that novices can improve their problem representation), it might tell a part of the story. Therefore, we deemed it necessary to control for overall ability levels when comparing experts and novices. The goal of the current studies is to provide a detailed examination of problem representation in novices along the lines of the questions outlined above. To achieve this goal, we deemed it necessary to control for the intelligence and knowledge factors, while manipulating representational factors. In controlling for knowledge factors, we (a) used simplified tasks and (2) selected only those deep relational properties (or principles) that were well familiar to novices in mathematics. In particular, we selected the commutative and associative properties of arithmetic, because these principles are likely to be familiar to the majority of college undergraduates (Everyday Mathematics: Teacher's Reference Manual, 1998). In this paper, we present four experiments. In Experiment 1, experts and novices in mathematics were asked to group arithmetic equations. These groupings could be based either on the commonality of surface features (e.g., numbers used, the number of constituent elements in the equations) or on the commonality of a deep mathematical relation (principles of commutativity or associativity). It has been argued that researchers often arbitrarily categorize certain relations as "deep and structural" (Brown, 1989), taking their intuitions as face validity. To counter this argument, we deemed it important to ascertain that associativity and commutativity are treated as important mathematical relations not only by math textbooks, but also by mathematics experts. In Experiment 2, we introduced a two-phase grouping task. During the first phase, deep relations were "unmasked," such that surface features were not varied among the compared equations. During the second phase, the deep relations were "masked" again by reintroducing Knowledge or attention -9 competing surface features. In this experiment, we were particularly interested in performance in the "unmasked" phase and in transfer from the "unmasked" to the "masked" phase. In Experiment 3, equations similar to those used in Experiments 1 and 2 were presented in a “new/old” recognition paradigm. The goal of this experiment was to examine whether or not novices encode deep relational properties. This experiment was designed to help distinguish between the two options mentioned above: (a) that novices fail to encode deep relational properties, or (2) they initially encode both surface features and relational properties, but discard the latter in favor of more salient surface features. Experiment 4 used the same methodological paradigm as Experiment 3, but decreased the encoding time in which participants were exposed to the equations in the study phase. The goal of Experiment 4 was to examine the manner in which surface and relational properties are encoded. For example, if surface and relational properties are encoded in a parallel manner, then shortening the encoding time should affect recognition on both of these features. However, if the are encoded in a serial manner, then it is more likely that recognition for only one type of feature will be effected. EXPERIMENT 1 Method