Relational Reasoning with Rational Numbers
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 two-dimensional relations between sets of discrete (i.e., countable) elements (e.g., red marbles/all marbles). In contrast, the format of decimals is inherently one-dimensional, expressing a continuous-valued magnitude (i.e., proportion) but not a two-dimensional relation between sets of countable elements. These differences in format and conceptual structure are reflected in both behavioral and neural patterns associated with different types of rational numbers. Decimals naturally align with continuous quantities, whereas fractions align with discrete quantities. Magnitude comparisons are made more quickly with decimals than fractions, but fractions are advantageous for relational reasoning with discrete (or discretized) quantities. Processing a fraction evokes greater neural activity (in the intraparietal sulcus and elsewhere) than is triggered by decimals or whole numbers. Individual differences both in relational knowledge of fractions and in magnitude processing with decimals predict degree of early success in algebra. Implications for quantitative reasoning and education will be discussed.