Relational Reasoning with Rational Numbers: Developmental and Neuroimaging Approaches

The study of how children and adults learn mathematics has given rise to a rich set of psychological phenomena involving mental representation, conceptual understanding, working memory, relational reasoning and problem solving. The subfield of understanding rational number processing and reasoning focuses on mental representation and conceptual understanding of rational numbers, and in particular fractions. Fractions differ from other number types, such as whole numbers, both conceptually and in format. Previous research has highlighted the extent to which fractions and other rational numbers pose challenges for children and adults with respect to magnitude estimation and misconceptions. The goal of this dissertation is to highlight the distinct differences in reasoning with different types of rational numbers. First, a neuroimaging study provides evidence that fractions yield a distinct pattern of neural activation during magnitude estimation that differs from both decimals and integers (Chapter 2). Second, a set of behavioral studies with adults highlights the affordances of the bipartite format of fractions for relational reasoning tasks (Chapter 3). Finally, a developmental study with pre-algebra students provides evidence for a significant relationship between relational understanding of fractions and algebra performance, and specifically algebraic modeling. This work is presented in the context of viewing mathematical notation as a type of conceptual modeling. In particular, decimals have advantages in measurement and representing magnitude. Fractions, on the other hand, have advantages in relational contexts, due to the fact that fractions, with their bipartite (a/b) format, inherently specify a relation between the cardinalities of two sets. When mathematics is viewed as a type of relational modeling, rational expressions provide a gateway to more complex mathematical notations and concepts, such as those in algebra