Background: The sophistication of young children's arithmetic problem-solving strategies can be influenced through experience and instructional intervention. One potential pathway is through encountering story problems where the location of the unknown quantity varies. Aims: The goal of the present study is to characterize how arithmetic problem-solving strategy sophistication can evolve through opportunities to solve story problems. Sample: We used microgenetic principles to guide the coding of arithmetic problem-solving behavior (8843 attempts) across three timescales (time within-session, attempt to solve, and between sessions) for nine story problem structures (N = 40, 19 girls). Data come from a teaching experiment conducted in a Mountain West US state in Spring 2018. Methods: We employed a Bayesian hierarchical ordinal regression with a nine-level response variable. The model contained fixed effects for session, attempt, story problem structure; a smooth time within session effect; and random effects for student, instructor, and equation. Results: Our analysis indicates which transitions from less to more sophisticated strategies are better supported by additional attempts to solve the same problem vs. additional instructional sessions. Strategy sophistication also varied by the location of the unknown quantity (result unknown, find difference, start unknown), but not operation (join, separate, part-whole). Conclusions: If confirmed by other studies, including experiments, what teachers offer children in terms of learning opportunities (more attempts within the same problem or more problems across work sessions) should vary based on the transition they are making.