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

DeWolf, Melissa; Bassok, Miriam; Holyoak, Keith J.

doi:10.1016/j.jecp.2015.01.013

Cited by 85 publications

(106 citation statements)

References 44 publications

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“…The results of the current study strongly support the basic hypothesis first proposed by DeWolf et al (2014DeWolf et al ( , 2015aDeWolf et al ( , 2015b: fractions have a unique advantage over other rational number types, such as decimals, when they are used to represent relations between countable sets (e.g., the ratio of boys to girls in a classroom). Conversely, decimals (which correspond to onedimensional magnitude representations) are very strongly linked to continuous quantities, which also serve as representations of one-dimensional magnitudes (e.g., the volume of water in a beaker).…”

Section: Discussionsupporting

confidence: 86%

“…Fractions, with their bipartite structure, are literally and mentally used to represent relations, whereas decimals are better-suited to represent magnitudes. DeWolf et al (2015b) found evidence of that these two aspects of mathematical understanding make differential contributions to acquisition of higher-level mathematical concepts, such as those involved in algebra. The present set of studies suggest that representational differences between types of rational numbers are not specific to cultural or educational variations.…”

Section: Discussionmentioning

confidence: 98%

“…Studies have shown that magnitude comparisons can be made much more quickly and accurately with decimals than with fractions (DeWolf, Grounds, Bassok, & Holyoak, 2014;Iuculano & Butterworth, 2011), but that fractions are more effective than decimals in tasks such as relation identification or analogical reasoning, for which relational information is paramount (DeWolf, Bassok, & Holyoak, 2015a). Importantly, various aspects of performance with both fractions and decimals predict subsequent success with more advanced mathematical topics, such as algebra (Booth, Newton, & Twiss-Garrity, 2014;DeWolf, Bassok, & Holyoak, 2015b;Siegler et al, 2011Siegler et al, , 2012Siegler et al, , 2013.…”

Section: Conceptual and Processing Differences Between Fractions And mentioning

confidence: 99%

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

et al. 2016

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a b s t r a c tPrevious 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.

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Section: Discussionsupporting

confidence: 86%

Section: Discussionmentioning

confidence: 98%

Section: Conceptual and Processing Differences Between Fractions And mentioning

confidence: 99%

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

et al. 2016

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“…Because of their bipartite structure, fractions have a much more natural correspondence to relational concepts based on ratios of countable sets (DeWolf et al, 2015a). The relational aspects of fraction representations appear to make fraction understanding a critical bridge to learning algebra (DeWolf et al, 2015b), which depends critically on grasping the concept of a variable (understood to represent a quantity of unknown magnitude). The "isolation" technique introduced in the present paper (imaging activity evoked by an individual number as the participant prepares for a specific mathematical task performed immediately afterwards) might usefully be extended to compare the neural patterns evoked by the same symbol (e.g., a fraction) in preparation for tasks that require different types of information (e.g., magnitudes or relational concepts).…”

Section: Future Directionsmentioning

confidence: 99%

Neural representations of magnitude for natural and rational numbers

et al. 2016

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a b s t r a c t a r t i c l e i n f oArticle history: Received 12 January 2016 Accepted 26 July 2016 Available online 26 July 2016 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.

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“…A 27-question paper-and-pencil assessment provided a baseline measure of participants’ algebra understanding (DeWolf, Son, Bassok, & Holyoak, 2015; adapted from DeWolf et al, 2015b). This assessment included algebra problems that were either taken from the California State Standards for Grade 8 or adapted from Booth, Newton, and Twiss-Garrity (2014).…”

Section: Methodsmentioning

confidence: 99%

The role of visual representations in college students’ understanding of mathematical notation.

Atagi

DeWolf

Stigler

et al. 2016

Journal of Experimental Psychology: Applied

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Developing understanding of fractions involves connections between non-symbolic visual representations and symbolic representations. Initially, teachers introduce fraction concepts with visual representations before moving to symbolic representations. Once the focus is shifted to symbolic representations, the connections between visual representations and symbolic notation are considered to be less useful, and students are rarely asked to connect symbolic notation back to visual representations. In two experiments, we ask whether visual representations affect understanding of symbolic notation for adults who understand symbolic notation. In a conceptual fraction comparison task (e.g., Which is larger, 5a or 8a?), participants were given comparisons paired with accurate, helpful visual representations, misleading visual representations, or no visual representations. The results show that even college students perform significantly better when accurate visuals are provided over misleading or no visuals. Further, eye-tracking data suggest that these visual representations may affect performance even when only briefly looked at. Implications for theories of fraction understanding and education are discussed.

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

Cited by 85 publications

References 44 publications

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

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

Neural representations of magnitude for natural and rational numbers

The role of visual representations in college students’ understanding of mathematical notation.

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