[Paper Review] Mathematical Sensemaking as Seeking Coherence between Calculations and Concepts: Instruction and Assessments for Introductory Physics

Eric Kuo, Michael M. Hull+2 more|arXiv (Cornell University)|Mar 13, 2019

Science Education and Pedagogy75 references5 citations

TL;DR

This paper proposes mathematical sensemaking—seeking coherence between calculations and conceptual understanding—as a core instructional goal in introductory physics. Using a novel assessment framework and quasi-experimental study, it demonstrates that students taught with this approach use calculation-concept crossover strategies more effectively and solve problems more accurately than those in traditional instruction, especially when guided by experienced instructors.

ABSTRACT

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.

Motivation & Objective

• To address the gap in physics education where students struggle to transfer learning to novel problems.
• To develop and validate an assessment framework that captures students' use of coherence-seeking problem-solving strategies.
• To investigate whether instruction emphasizing mathematical sensemaking improves student performance on unfamiliar problems.
• To examine how instructor experience influences the implementation and outcomes of sensemaking-focused instruction.
• To explore the role of students' epistemological views in predicting their use of calculation-concept crossover approaches.

Proposed method

• Designing three assessment items that require students to integrate calculations and conceptual reasoning, creating 'calculation-concept crossover' tasks.
• Implementing a quasi-experimental study with three parallel sections: two focused on mathematical sensemaking (one with experienced instructor, one with novice), and one control group using traditional instruction.
• Using a framework to score student responses based on the use of coherence-seeking strategies, particularly the integration of formal math with conceptual understanding.
• Analyzing student performance on the three assessment items to compare correct solution rates and strategy use across groups.
• Administering surveys to assess students' epistemological views on problem-solving coherence and correlating these with strategy use.
• Applying statistical analysis to evaluate differences in performance and strategy use between the MS, MS-nov, and CTRL groups.

Experimental results

Research questions

• RQ1 How does instruction emphasizing mathematical sensemaking affect students' use of calculation-concept crossover strategies in problem solving?
• RQ2 To what extent do students in sensemaking-focused courses outperform those in traditional instruction on assessments requiring conceptual and mathematical coherence?
• RQ3 Does instructor experience moderate the effectiveness of mathematical sensemaking instruction in promoting coherent problem-solving strategies?
• RQ4 To what extent do students' epistemological views about problem-solving coherence predict their use of calculation-concept crossover approaches?
• RQ5 Can an assessment framework effectively capture and measure students' coherence-seeking practices in physics problem solving?

Key findings

• Students in mathematical sensemaking sections used calculation-concept crossover approaches significantly more often than those in the control group.
• Mathematical sensemaking students generated more correct solutions on the three crossover assessment items compared to the control group.
• The experienced instructor (MS) group showed higher use of crossover strategies than the novice-instructor (MS-nov) group, indicating that instructor expertise influences implementation.
• Students' epistemological views toward problem-solving coherence predicted their use of crossover strategies, but did not fully explain the performance gap between the MS and control groups.
• The assessment framework successfully identified and quantified coherence-seeking practices, demonstrating its utility in measuring adaptive problem-solving expertise.
• The results suggest that mathematical sensemaking instruction enhances both strategic flexibility and solution accuracy in introductory physics problem solving.