Teaching and assessing conceptual/mathematical coherence in introductory physics problem solving

Research on physics problem solving has demonstrated the importance of explicitly teaching students to connect conceptual and mathematical reasoning. For instance, the most well-studied class of effective problem-solving instruction teaches students to begin their problem-solving process with a conceptual analysis that drives subsequent mathematical analysis. Current research in physics education is continuing to build out more sophisticated models of conceptual/mathematical coherence in physics teaching and learning. I will discuss other ways in which conceptual/mathematical coherence (or the lack thereof) connects to known issues in physics education and how a focus on coherence can drive the search for effective problem-solving instruction and assessment principles. I will share the results of a teaching comparison demonstrating how an instructional approach emphasizing conceptual/mathematical coherence can produce positive outcomes in (i) student reasoning – including detecting errors, finding conceptual insights, and spontaneously using calculations on qualitative questions – and (ii) student beliefs about problem solving. Finally, I will discuss the open questions raised by these results and the current directions of our research program.