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Mathematical Sensemaking as Seeking Coherence between Calculations and Concepts: Instruction and Assessments for Introductory Physics

What kind of problem-solving instruction can help students apply what they have learned to solve the new and unfamiliar problems they will encounter in the future? We propose that mathematical sensemaking, the practice of seeking coherence between formal mathematics and conceptual understanding, is a key target of successful physics problem-solving instruction. However, typical assessments tend to measure understanding in more disjoint ways. To capture coherence-seeking practices in student problem solving, we introduce an assessment framework that highlights opportunities to use these problem-solving approaches more flexibly. Three assessment items embodying this calculation-concept crossover framework illustrate how coherence can drive flexible problem-solving approaches that may be more efficient, insightful, and accurate. These three assessment items were used to evaluate the efficacy of an instructional approach focused on developing mathematical-sensemaking skills. In a quasi-experimental study, three parallel lecture sections of first-semester, introductory physics were compared: two mathematical sensemaking sections, with one having an experienced instructor (MS) and one a novice instructor (MS-nov), and a traditionally-taught section acted as a control group (CTRL). On the three crossover assessment items, mathematical sensemaking students used calculation-concept crossover approaches more and generated more correct solutions than CTRL students. Student surveyed epistemological views toward problem-solving coherence at the end of the course predicted their crossover approach use but did not fully account for the differences in crossover approach use between the MS and CTRL groups. These results illustrate new instructional and assessment frameworks for research on mathematical sensemaking and adaptive problem-solving expertise.