Assessing mathematical sensemaking in physics through calculation-concept crossover

Andrew Elby,Michael M. Hull,Eric Kuo,Ayush Gupta

doi:10.1103/physrevphyseducres.16.020109

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.

Highlights

When solving problems, the work of professional physicists and engineers relies on reasoning that leverages coherence between formal mathematics and conceptual understanding [1,2,3,4,5,6]—a form of reasoning which has been described as mathematical sense making [7,8,9,10,11]
We propose a novel assessment paradigm of calculation-concept crossover that highlights one dimension of mathematical sense making identified in the literature and distinguishes it from common physics education research (PER) assessment approaches
The MS group used more crossover approaches than the CTRL instructional group on all three crossover assessments: qualitative judgment, χ2ð1;N 1⁄4 206Þ 1⁄4 7.25, p 1⁄4 0.007 isomorphic calculation, χ2ð1;N 1⁄4 206Þ 1⁄4 16.5, p < 0.001, and cued symbolic evaluation χ2ð1; N 1⁄4 206Þ 1⁄4 12.8, p < 0.001. This confirmed our main prediction for all three crossover assessments: MS instruction better supported the calculation-concept crossover when solving physics problems compared to CTRL instruction

Summary

Introduction

The work of professional physicists and engineers relies on reasoning that leverages coherence between formal mathematics and conceptual understanding [1,2,3,4,5,6]—a form of reasoning which has been described as mathematical sense making [7,8,9,10,11]. We propose a novel assessment paradigm of calculation-concept crossover that highlights one dimension of mathematical sense making identified in the literature and distinguishes it from common physics education research (PER) assessment approaches. We instantiate this paradigm through three assessment questions, each one probing this dimension of mathematical sense making in students’ problem-solving practice in a different way. The results show that an instructional approach designed to foster mathematical sense making—in conjunction with other PER-based active learning strategies—can produce problem-solving benefits detectable by targeted assessments

Objectives

Results

Conclusion