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

- Melissa DeWolf,
*University of California, Los Angeles, Los Angeles, California, United States*
- Miriam Bassok,
*University of Washington*
- Keith Holyoak,
*University of California, Los Angeles*

## Abstract

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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