Reasoning strategies with rational numbers revealed by eye tracking.

Recent research has begun to investigate the impact of different formats for rational numbers on the processes by which people make relational judgments about quantitative relations. DeWolf, Bassok, and Holyoak (Journal of Experimental Psychology: General, 144(1), 127-150, 2015) found that accuracy on a relation identification task was highest when fractions were presented with countable sets, whereas accuracy was relatively low for all conditions where decimals were presented. However, it is unclear what processing strategies underlie these disparities in accuracy. We report an experiment that used eye-tracking methods to externalize the strategies that are evoked by different types of rational numbers for different types of quantities (discrete vs. continuous). Results showed that eye-movement behavior during the task was jointly determined by image and number format. Discrete images elicited a counting strategy for both fractions and decimals, but this strategy led to higher accuracy only for fractions. Continuous images encouraged magnitude estimation and comparison, but to a greater degree for decimals than fractions. This strategy led to decreased accuracy for both number formats. By analyzing participants' eye movements when they viewed a relational context and made decisions, we were able to obtain an externalized representation of the strategic choices evoked by different ontological types of entities and different types of rational numbers. Our findings using eye-tracking measures enable us to go beyond previous studies based on accuracy data alone, demonstrating that quantitative properties of images and the different formats for rational numbers jointly influence strategies that generate eye-movement behavior.

Relational Priming Based on a Multiplicative Schema for Whole Numbers and Fractions.

Why might it be (at least sometimes) beneficial for adults to process fractions componentially? Recent research has shown that college-educated adults can capitalize on the bipartite structure of the fraction notation, performing more successfully with fractions than with decimals in relational tasks, notably analogical reasoning. This study examined patterns of relational priming for problems with fractions in a task that required arithmetic computations. College students were asked to judge whether or not multiplication equations involving fractions were correct. Some equations served as structurally inverse primes for the equation that immediately followed it (e.g., 4 × 3/4 = 3 followed by 3 × 8/6 = 4). Students with relatively high math ability showed relational priming (speeded solution times to the second of two successive relationally related fraction equations) both with and without high perceptual similarity (Experiment 2). Students with relatively low math ability also showed priming, but only when the structurally inverse equation pairs were supported by high perceptual similarity between numbers (e.g., 4 × 3/4 = 3 followed by 3 × 4/3 = 4). Several additional experiments established boundary conditions on relational priming with fractions. These findings are interpreted in terms of componential processing of fractions in a relational multiplication context that takes advantage of their inherent connections to a multiplicative schema for whole numbers.

A set for relational reasoning: Facilitation of algebraic modeling by a fraction task.

Recent work has identified correlations between early mastery of fractions and later math achievement, especially in algebra. However, causal connections between aspects of reasoning with fractions and improved algebra performance have yet to be established. The current study investigated whether relational reasoning with fractions facilitates subsequent algebraic reasoning using both pre-algebra students and adult college students. Participants were first given either a relational reasoning fractions task or a fraction algebra procedures control task. Then, all participants solved word problems and constructed algebraic equations in either multiplication or division format. The word problems and the equation construction tasks involved simple multiplicative comparison statements such as "There are 4 times as many students as teachers in a classroom." Performance on the algebraic equation construction task was enhanced for participants who had previously completed the relational fractions task compared with those who completed the fraction algebra procedures task. This finding suggests that relational reasoning with fractions can establish a relational set that promotes students' tendency to model relations using algebraic expressions.

Neural representations of magnitude for natural and rational numbers.

Humans have developed multiple symbolic representations for numbers, including natural numbers (positive integers) as well as rational numbers (both fractions and decimals). Despite a considerable body of behavioral and neuroimaging research, it is currently unknown whether different notations map onto a single, fully abstract, magnitude code, or whether separate representations exist for specific number types (e.g., natural versus rational) or number representations (e.g., base-10 versus fractions). We address this question by comparing brain metabolic response during a magnitude comparison task involving (on different trials) integers, decimals, and fractions. Univariate and multivariate analyses revealed that the strength and pattern of activation for fractions differed systematically, within the intraparietal sulcus, from that of both decimals and integers, while the latter two number representations appeared virtually indistinguishable. These results demonstrate that the two major notations formats for rational numbers, fractions and decimals, evoke distinct neural representations of magnitude, with decimals representations being more closely linked to those of integers than to those of magnitude-equivalent fractions. Our findings thus suggest that number representation (base-10 versus fractions) is an important organizational principle for the neural substrate underlying mathematical cognition.

Conceptual and procedural distinctions between fractions and decimals: A cross-national comparison.

Previous work has shown that adults in the United States process fractions and decimals in distinctly different ways, both in tasks requiring magnitude judgments and in tasks requiring mathematical reasoning. In particular, fractions and decimals are preferentially used to model discrete and continuous entities, respectively. The current study tested whether similar alignments between the format of rational numbers and quantitative ontology hold for Korean college students, who differ from American students in educational background, overall mathematical proficiency, language, and measurement conventions. A textbook analysis and the results of five experiments revealed that the alignments found in the United States were replicated in South Korea. The present study provides strong evidence for the existence of a natural alignment between entity type and the format of rational numbers. This alignment, and other processing differences between fractions and decimals, cannot be attributed to the specifics of education, language, and measurement units, which differ greatly between the United States and South Korea.

Conceptual Integration of Arithmetic Operations With Real-World Knowledge: Evidence From Event-Related Potentials.

Research on language processing has shown that the disruption of conceptual integration gives rise to specific patterns of event-related brain potentials (ERPs)-N400 and P600 effects. Here, we report similar ERP effects when adults performed cross-domain conceptual integration of analogous semantic and mathematical relations. In a problem-solving task, when participants generated labeled answers to semantically aligned and misaligned arithmetic problems (e.g., 6 roses + 2 tulips = ? vs. 6 roses + 2 vases = ?), the second object label in misaligned problems yielded an N400 effect for addition (but not division) problems. In a verification task, when participants judged arithmetically correct but semantically misaligned problem sentences to be "unacceptable," the second object label in misaligned sentences elicited a P600 effect. Thus, depending on task constraints, misaligned problems can show either of two ERP signatures of conceptual disruption. These results show that well-educated adults can integrate mathematical and semantic relations on the rapid timescale of within-domain ERP effects by a process akin to analogical mapping.

From rational numbers to algebra: separable contributions of decimal magnitude and relational understanding of fractions.

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.

Modeling discrete and continuous entities with fractions and decimals.

When people use mathematics to model real-life situations, their use of mathematical expressions is often mediated by semantic alignment (Bassok, Chase, & Martin, 1998): The entities in a problem situation evoke semantic relations (e.g., tulips and vases evoke the functionally asymmetric "contain" relation), which people align with analogous mathematical relations (e.g., the noncommutative division operation, tulips/vases). Here we investigate the possibility that semantic alignment is also involved in the comprehension and use of rational numbers (fractions and decimals). A textbook analysis and results from two experiments revealed that both mathematic educators and college students tend to align the discreteness versus continuity of the entities in word problems (e.g., marbles vs. distance) with distinct symbolic representations of rational numbers--fractions versus decimals, respectively. In addition, fractions and decimals tend to be used with nonmetric units and metric units, respectively. We discuss the importance of the ontological distinction between continuous and discrete entities to mathematical cognition, the role of symbolic notations, and possible implications of our findings for the teaching of rational numbers.

Conceptual structure and the procedural affordances of rational numbers: relational reasoning with fractions and decimals.

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.

Magnitude comparison with different types of rational numbers.

An important issue in understanding mathematical cognition involves the similarities and differences between the magnitude representations associated with various types of rational numbers. For single-digit integers, evidence indicates that magnitudes are represented as analog values on a mental number line, such that magnitude comparisons are made more quickly and accurately as the numerical distance between numbers increases (the distance effect). Evidence concerning a distance effect for compositional numbers (e.g., multidigit whole numbers, fractions and decimals) is mixed. We compared the patterns of response times and errors for college students in magnitude comparison tasks across closely matched sets of rational numbers (e.g., 22/37, 0.595, 595). In Experiment 1, a distance effect was found for both fractions and decimals, but response times were dramatically slower for fractions than for decimals. Experiments 2 and 3 compared performance across fractions, decimals, and 3-digit integers. Response patterns for decimals and integers were extremely similar but, as in Experiment 1, magnitude comparisons based on fractions were dramatically slower, even when the decimals varied in precision (i.e., number of place digits) and could not be compared in the same way as multidigit integers (Experiment 3). Our findings indicate that comparisons of all three types of numbers exhibit a distance effect, but that processing often involves strategic focus on components of numbers. Fractions impose an especially high processing burden due to their bipartite (a/b) structure. In contrast to the other number types, the magnitude values associated with fractions appear to be less precise, and more dependent on explicit calculation.

Introduction to the special section on the neural substrate of analogical reasoning and metaphor comprehension.

The special section on the neural substrate of relational reasoning includes 4 articles that address the processes and brain regions involved in analogical reasoning (Green, Kraemer, Fugelsang, Gray, & Dunbar, 2011; Maguire, McClelland, Donovan, Tillman, & Krawczyk, 2011) and in metaphor comprehension (Chettih, Durgin, & Grodner, 2011; Prat, Mason, & Just, 2011). We see this work as an example of how neuroscience approaches to cognition can lead to increased understanding of cognitive processes. In this brief introduction, we first situate the 4 articles in the context of prior cognitive neuroscience work on relational reasoning. We then highlight the main issues explored in these articles: different sources of complexity and difficulty in relational processing, potential differences between the roles of the 2 hemispheres, and the impact of individual differences in various cognitive abilities. The 4 articles illustrate a range of methodologies, including functional magnetic resonance imaging (fMRI; Green et al., 2011; Prat et al., 2011), event-related potentials (ERPs; Maguire et al., 2011), and different types of semantic priming (Chettih et al., 2011; Prat et al., 2011). They highlight the connections between research on analogy and on metaphor comprehension and suggest, collectively, that a cognitive neuroscience approach to relational reasoning can lead to converging conclusions.

Following the standard form: effects of equation format on algebraic modeling.

When people construct algebraic equations to represent quantitative relations, they often reverse the roles of the variables (6S = P instead of 6P = S). Results from three experiments show that a major reason for such reversal errors is people's adherence to, and interpretation of, the Standard Form of algebraic models. College students constructed, selected, and interpreted algebraic models that either had the standard multiplication format (MF: ax = y), or the mathematically equivalent division format (DF: y/a = x). A large minority of participants constructed reversed MF models, whereas most participants who were instructed to use DF equations constructed correct models (Experiment 1). Most participants understood that MF and DF models of the same relation should be mathematically equivalent (Experiment 2), but they drew a conceptual distinction between these models (Experiment 3). The authors discuss the impact of notational conventions on people's interpretation and use of representational tools.

Priming addition facts with semantic relations.

Results from 2 relational-priming experiments suggest the existence of an automatic analogical coordination between semantic and arithmetic relations. Word pairs denoting object sets served as primes in a task that elicits "obligatory" activation of addition facts (5 + 3 activates 8; J. LeFevre, J. Bisanz, & L. Mrkonjic, 1988). Semantic relations between the priming words were either aligned or misaligned with the structure of addition (M. Bassok, V. M. Chase, & S. A. Martin, 1998). Obligatory activation of addition facts occurred when the digits were primed by categorically related words (tulips-daisies), which are aligned with addition, but did not occur when the digits were primed by unrelated words (hens-radios, Experiment 1) or by functionally related words (records-songs, Experiment 2), which are misaligned with addition. These findings lend support to the viability of automatic analogical priming (B. A. Spellman, K. J. Holyoak, & R. G. Morrison, 2001) and highlight the relevance of arithmetic applications to theoretical accounts of mental arithmetic.

Effects of semantic cues on mathematical modeling: evidence from word-problem solving and equation construction tasks.

Mathematical solutions to textbook word problems are correlated with semantic relations between the objects described in the problem texts. In particular, division problems usually involve functionally related objects (e.g., tulips-vases) and rarely involve categorically related objects (e.g., tulips-daisies). We examined whether middle school, high school, and college students use object relations when they solve division word problems (WP) or perform the less familiar task of representing verbal statements with algebraic equations (EQ). Both tasks involved multiplicative comparison statements with either categorically or functionally related objects (e.g., "four times as many cupcakes [commuters] as brownies [automobiles]"). Object relations affected the frequency of correct solutions in the WP task but not in the EQ task. In the latter task, object relations did affect the structure of nonalgebraic equation errors. We argue that students use object relations as "semantic cues" when they engage in the sense-making activity of mathematical modeling.