Flexibility in Abstract Thought: Context-Dependent Representations of Integer Magnitude

Negative numbers are challenging because they lack direct perceptual referents and require reasoning about magnitude in a relational, bidirectional space. This review examines how people represent and use integer magnitude on the mental number line, a cognitive tool that learners initially use for positive numbers, focusing on the flexibility with which this tool is deployed across contexts. Synthesizing evidence from cognitive psychology, developmental science, and mathematics education, we evaluate three broad representational accounts: componential accounts, which treat sign and magnitude as separable symbolic elements; extension accounts, which posit a continuous magnitude representation extending across zero; and reflection accounts, which propose a symmetry-based organization of magnitude around zero. Across experimental paradigms, task formats, and age groups, findings reveal substantial variability in behavioral signatures of integer processing. We argue that this variability reflects context-dependent recruitment of multiple coexisting representations rather than a single, fixed structure or a stage-like developmental progression. On this view, the mental number line functions as a dynamic cognitive tool that can be reparametrized in the moment to support rule-based, analog, or relational reasoning. Understanding how integer magnitude is flexibly represented and used provides insight into the mechanisms of abstract thought, coordination of representations, and conceptual change.