Conceptual distinctions and preferential alignment across rational number representations
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Conceptual distinctions and preferential alignment across rational number representations Kailun Qiu 1 & Yunqi Wang 1 Received: 28 August 2019 / Revised: 27 June 2020 / Accepted: 31 August 2020 # Instituto Universitário de Ciências Psicológicas, Sociais e da Vida 2020
Abstract
Rational numbers can be represented in multiple formats (e.g., fractions, decimals, and percentages), and a rational number notation can be used to express different concepts in different contexts. The present study investigated the distribution of the multiple concepts expressed by these different rational number notations in real-world contexts as well as the semantic alignment between entity type (discrete vs. continuous) and rational number format (decimal vs. fraction) in the Chinese context. Textbook analysis and two paper-and-pencil experiments yielded the following four major findings: (1) Decimals were more likely used to represent numerical magnitudes, while fractions were more likely used to represent relations between two numerical magnitudes. (2) Decimals were more often used to represent continuous entities while fractions were preferred to represent discrete entities. (3) The strength of the association between different formats of rational numbers and their preferred conceptual meanings seemed more pronounced than the semantic alignment between number type and entity type. (4) Percentages were used in a way more similar to fractions than decimals in terms of the concepts they express. These findings indicate that different formats of rational numbers differ dramatically in their use in real-world contexts both in terms of the conceptual meanings they express and the entities they model. Educational implications of this study are discussed. Keywords Rational numbers . Decimals . Fractions . Percentages . Semantic alignment . Magnitude . Relations
Introduction Unlike whole numbers, rational numbers can be represented in multiple forms like fractions, decimals, and percentages. Rational numbers are also multifaceted and can be interpreted in
* Yunqi Wang [email protected]
1
Department of Linguistics, School of International Studies, Zhejiang University, No. 866 Yuhangtang Road, Hangzhou 310058, People’s Republic of China
K. Qiu, Y. Wang
such different ways as measure, part-whole relation, and ratio (Kieren 1976, 1980). Therefore, the use of rational numbers in real-world contexts is a complex process, involving the choice of both the proper representation and the proper type of conceptual interpretation. The present study examined the factors supposed to constrain people’s preference in the use of different rational number forms in specific real-world contexts. The two factors considered in the present study are the entities to be modeled and the conceptual meanings intended to express.
Semantic alignment People’s interpretation and use of arithmetic operations are mediated by the alignment between the real-world situations and the analogous mathematical relations (Bassok et al. 1998, 2008; Guthormsen et al. 2016; Mar
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