Children exhibit remarkable variability in their mathematical problem-solving abilities, yet the cognitive, metacognitive and affective mechanisms underlying these individual differences remain poorly understood. We developed a novel Bayesian model of arithmetic problem-solving (BMAPS) to uncover the latent processes governing children's arithmetic strategy choice and efficiency. BMAPS inferred cognitive parameters related to strategy execution and metacognitive parameters related to strategy selection, revealing key mechanisms of adaptive problem solving. BMAPS parameters collectively explained individual differences in problem-solving performance, predicted longitudinal gains in arithmetic fluency and mathematical reasoning, and mediated the effects of anxiety and attitudes on performance. Clustering analyses using BMAPS parameters revealed distinct profiles of strategy use, metacognitive efficiency, and developmental change. By quantifying the fine-grained dynamics of strategy selection and execution and their relation to affective factors and academic outcomes, BMAPS provides new insights into the cognitive and metacognitive underpinnings of children's mathematical learning. This work advances powerful computational methods for uncovering latent mechanisms of complex cognition in children.