Semantic Alignment of Fractions and Decimals with Discrete Versus Continuous Entities: A Textbook Analysis

When people use mathematics to model real-life situations, their modeling is often mediated by semantic alignment (Bassok, Chase, & Martin, 1998): The entities in a problem situation evoke semantic relations (e.g., tulips and vases evoke the functionally asymmetric “contain” relation), which people align with analogous mathematical relations (e.g., the noncommutative division operation, tulips/vases). Here, we applied the semantic-alignment framework to understand how people use rational numbers as models of discrete and continuous entities. A textbook analysis revealed that mathematics educators tend to align the discreteness vs. continuity of the entities in word problems (e.g., marbles vs. distance) with distinct symbolic representations of rational numbers—fractions vs. decimals, respectively. We discuss the importance of the ontological distinction between continuous and discrete entities to mathematical cognition, the role of symbolic notations, and possible implications of our findings for the teaching of rational numbers.