Mathematics is critical for making sense of the world. Yet, little is known about how people evaluate mathematical explanations. Here, we use an explanatory reasoning task to investigate the intuitive structure of mathematics. We show that people evaluate arithmetic explanations by building mental proofs over the conceptual structure of intuitive arithmetic, evaluating those proofs using criteria similar to those of professional mathematicians. Specifically, we find that people prefer explanations consistent with the conceptual order of the operations (“9÷3=3 because 3×3=9” rather than “3×3=9 because 9÷3=3”), and corresponding to simpler proofs (“9÷3=3 because 3×3=9” rather than “9÷3=3 because 3+3+3=9”). Implications for mathematics cognition and education are discussed.