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Category Theory, a branch of mathematics, has shown promise as a modeling framework for higher-level cognition. We introduce an algebraic model for analogy that uses the language of category theory to explore analogy-related cognitive phenomena. To illustrate the potential of this approach, we use this model to explore three objects of study in cognitive literature. First, (a) we use commutative diagrams to analyze an effect of playing particular educational board games on the learning of numbers. Second, (b) we employ a notion called coequalizer as a formal model of re-representation that explains a property of computational models of analogy called “flexibility” whereby non-similar representational elements are considered matches and placed in structural correspondence. Finally, (c) we build a formal learning model which shows that re-representation, language processing and analogy making can explain the acquisition of knowledge of rational numbers. These objects of study provide a picture of acquisition of numerical knowledge that is compatible with empirical evidence and offers insights on possible connections between notions such as relational knowledge, analogy, learning, conceptual knowledge, re-representation and procedural knowledge. This suggests that the approach presented here facilitates mathematical modeling of cognition and provides novel ways to think about analogy-related cognitive phenomena.

Educational board games are a promising teaching method due to their low cost, playful, exploratory, and engaging nature. By drawing on analogical research, we created a game whose structure of spatial relationships mirrored the structure of rational numbers. We expected that children playing this game would improve their knowledge of fractions. We conducted a school intervention with an active control group and pretest‐posttest assessments to evaluate our board game. Playing this game promoted the learning of fractions, even after controlling for nonverbal cognitive abilities. This low‐cost educational game might help reduce the knowledge gap that separates less and more affluent children.

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