Analogical Reasoning with Rational Numbers: Semantic Alignment Based on Discrete Versus Continuous Quantities


Non-integer rational numbers, such as fractions and decimals, pose challenges for learners, both in conceptual understanding and in performing mathematical operations. Previous studies have focused on tasks involving access and comparison of integrated magnitude representations, showing that adults have less precise magnitude representations for fractions than decimals. Here we show the relative effectiveness of fractions over decimals in reasoning about relations between quantities. We constructed analogical reasoning problems that required mapping rational numbers (fractions or decimals) onto pictures depicting either part-whole or ratio relations between two quantities. We also varied the ontological nature of the depicted quantities, which could be discrete, continuous, or continuous but parsed into discrete components. Fractions were more effective than decimals for reasoning about discrete and continuous-parsed (i.e., discretized) quantities, whereas neither number type was particularly effective in reasoning about continuous quantities. Our findings show that, when numbers serve as models of quantitative relations, the ease of relational mapping depends on the analogical correspondence between the format of rational numbers and the quantity it models.

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