Ail existing accounts of similarity assume that it is a function of matching and mismatching attributes between mental representations. However, Bassok and Medin (19%) found that the judged similarity of sentences does not necessarily reflect the degree of overlap between the properties of paired stimuli. Rather, similarity judgments are often mediated by a process of thematic integration and reflect the degree to which stimuli can be integrated into a common thematic scenario. We present results of a study which extend this surprising flnding by showing that it also applies to similarity ratings of objects and occurs whether or not subjects explain their judgments. Also, consistent with the Bassok and Medin findings, the tendency towards thematic integration was more pronounced when the paired stimulus shared few attributes--but was still an important factor in similarity judgments between objects which shared many attributes. We discuss the implications of these findings for models of cognitive processes which use similarity as an explanatory construct.

Explanations of many cognitive processes, including probability judgment, rely on the construct of similarity. The present paper is concerned with the similarity-based explanation of reasoning in the conjunction task. Although high positive correlations have been found between similarity and probability judgments in this task, these alone cannot validate the assumption that similarity is judged by a process of feature comparison or that similarity judgment is an explanation of probability judgment. Preliminary results from a study in which we collected written justifications from subjects who made both types of judgment suggest that these assumptions are not tenable. Subjects cited considerations of causality and statistics ~ not just feature overlap -- when judging both similarity and probability, indicating that (1) feature comparison is only one way in which people judge similarity and (2) similarity judgment can involve processes usually associated with probability judgment. These findings suggest that the role of similarity in explaining other cognitive processes needs to be revised. It is proposed that the power of similarity and probability to predict one another can be exploited for the purpose of making either type of judgment.

e examined similarity judgments between simple Noun-Verb?Noun statements that were matched either in their verbs or nouns (separate matches) and made either analogous or non?analogous assertions (combined matches). An analysis of written justifications that accompanied subjects' similarity judgments revealed that matching verbs and matching nouns lead to two qualitatively different types of alignments. Matching verbs (e.g., "The carpenter fixed the chair" and "The plumber fixed the radio") led subjects to construct structural alignments and evaluate the quality of the resulting analogies (e.g., "Not analogous because plumbers don't fix radios as part of their job"). By contrast, and contrary to any traditional account of similarity as a process of comparison, matching nouns (e.g., "The carpenter fixed the chair" and "The carpenter sat on the chair") led subjects to construct thematic ahgnments and evaluate similarity based on the plausibility of the resulting causal or temporal scenarios (e.g., "He sat on the chair to see whether he fixed it well").

The standard number system includes several distinct types of notations, which differ conceptually and afford different procedures. Among notations for rational numbers, the bipartite format of fractions (a/b) enables them to represent 2-dimensional relations between sets of discrete (i.e., countable) elements (e.g., red marbles/all marbles). In contrast, the format of decimals is inherently 1-dimensional, expressing a continuous-valued magnitude (i.e., proportion) but not a 2-dimensional relation between sets of countable elements. Experiment 1 showed that college students indeed view these 2-number notations as conceptually distinct. In a task that did not involve mathematical calculations, participants showed a strong preference to represent partitioned displays of discrete objects with fractions and partitioned displays of continuous masses with decimals. Experiment 2 provided evidence that people are better able to identify and evaluate ratio relationships using fractions than decimals, especially for discrete (or discretized) quantities. Experiments 3 and 4 found a similar pattern of performance for a more complex analogical reasoning task. When solving relational reasoning problems based on discrete or discretized quantities, fractions yielded greater accuracy than decimals; in contrast, when quantities were continuous, accuracy was lower for both symbolic notations. Whereas previous research has established that decimals are more effective than fractions in supporting magnitude comparisons, the present study reveals that fractions are relatively advantageous in supporting relational reasoning with discrete (or discretized) concepts. These findings provide an explanation for the effectiveness of natural frequency formats in supporting some types of reasoning, and have implications for teaching of rational numbers.

To understand the development of mathematical cognition and to improve instructional practices, it is critical to identify early predictors of difficulty in learning complex mathematical topics such as algebra. Recent work has shown that performance with fractions on a number line estimation task predicts algebra performance, whereas performance with whole numbers on similar estimation tasks does not. We sought to distinguish more specific precursors to algebra by measuring multiple aspects of knowledge about rational numbers. Because fractions are the first numbers that are relational expressions to which students are exposed, we investigated how understanding the relational bipartite format (a/b) of fractions might connect to later algebra performance. We presented middle school students with a battery of tests designed to measure relational understanding of fractions, procedural knowledge of fractions, and placement of fractions, decimals, and whole numbers onto number lines as well as algebra performance. Multiple regression analyses revealed that the best predictors of algebra performance were measures of relational fraction knowledge and ability to place decimals (not fractions or whole numbers) onto number lines. These findings suggest that at least two specific components of knowledge about rational numbers--relational understanding (best captured by fractions) and grasp of unidimensional magnitude (best captured by decimals)--can be linked to early success with algebraic expressions.

To understand the development of mathematical cognition and to improve instructional practices, it is critical to identify early predictors of difficulty in learning complex mathematical topics such as algebra. Recent work has shown that performance with fractions on a number line estimation task predicts algebra performance, whereas performance with whole numbers on similar estimation tasks does not. We sought to distinguish more specific precursors to algebra by measuring multiple aspects of knowledge about rational numbers. Because fractions are the first numbers that are relational expressions to which students are exposed, we investigated how understanding the relational bipartite format (a/. b) of fractions might connect to later algebra performance. We presented middle school students with a battery of tests designed to measure relational understanding of fractions, procedural knowledge of fractions, and placement of fractions, decimals, and whole numbers onto number lines as well as algebra performance. Multiple regression analyses revealed that the best predictors of algebra performance were measures of relational fraction knowledge and ability to place decimals (not fractions or whole numbers) onto number lines. These findings suggest that at least two specific components of knowledge about rational numbers-relational understanding (best captured by fractions) and grasp of unidimensional magnitude (best captured by decimals)-can be linked to early success with algebraic expressions.

Previous research on multiplicative reasoning has shown that for whole numbers, understanding of division is intimately linked to multiplication, as retrieval of division facts is often accomplished through reverse multiplication. We recently extended this research to rational numbers, and found that inverse multiplication problems can serve as primes for one another (e.g., a √ó b/a = a primes b √ó a/b = b) when the second multiplier is expressed as a fraction, but not when it is expressed as a decimal. In the current paper we propose a process model of how such relational priming takes place, and report two experiments that test the limits of this priming effect. The first varies the format of the equations as fractions or a total division equation, and shows that priming is only observed using the fraction format; the second varies the multiplicative complexity of the factors in the equations, and shows that priming requires a common factor linking the successive problems.