  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.
