Assessing mathematical sensemaking in physics through calculation-concept crossover

Abstract

Professional problem-solving practice in physics and engineering relies on mathematical sense making-reasoning that leverages coherence between formal mathematics and conceptual understanding. A key question for physics education is how well current instructional approaches develop students' mathematical sense making. We introduce an assessment paradigm that operationalizes a typically unmeasured dimension of mathematical sense making: use of calculations on qualitative problems and use of conceptual arguments on quantitative problems. Three assessment items embodying this calculationconcept crossover assessment paradigm illustrate how mathematical sense making can positively benefit students' problem solving, leading to more efficient, insightful, and accurate solutions. These three assessment items were used to evaluate the efficacy of an instructional approach focused on developing students' mathematical sense making skills. In a quasi-experimental study, three parallel lecture sections of first-semester, introductory physics were compared: two mathematical sense making sections, one with an experienced instructor and one with a novice instructor, and a traditionally taught section, as a control group. Compared to the control group, mathematical sense making groups used calculation-concept crossover approaches more often and gave more correct answers on the crossover assessment items, but they did not give more correct answers to associated standard problems. In addition, although students' postcourse epistemological views on problem-solving coherence were associated with their crossover use, they did not fully account for the differences in crossover approach use between the mathematical sense making and control groups. These results demonstrate a new assessment paradigm for detecting a typically unmeasured dimension of mathematical sense making and provide evidence that a targeted instructional approach can enhance engagement with mathematical sense making in introductory physics.

Seeking physical/mathematical coherence by recruiting and reconciling reasoning: A case study in E&M

“…Given these assumptions, Blake notes that these two opposing variables make it impossible to qualitatively reason out which one has more charge using their current line of reasoning. The idea of two opposing influences has been identified previously as a "conceptual schema" [8], which physics students can use to conclude incorrectly that these two opposing influences exactly compensate for one another [11][12][13][14][15]. However, here Blake explicitly notes that a definite determination cannot be made.…”

How conceptual blends support sensemaking: A case study from introductory physics

“…Kapon (2017) argues that we assess the value of scientific explanations based on their consistency with how the world works and their ability to explain a range of phenomena based on the fewest possible principles. Sikorski and Hammer (2017) and Kuo et al (2020) propose that sensemaking is a process of coherence-seeking, and so the explanation that provides resolution is one which achieves coherence between disparate pieces of knowledge.…”

“…Returning to my second research question, what affordances do conceptual blends provide for sensemaking, I argue that conceptual blends can facilitate resolution in at least two ways. First, they allow students to create conceptual frameworks that connect and incorporate disparate knowledge elements into a coherent whole, essentially defragmenting one's knowledge space and achieving the desired feeling of coherence discussed by Sikorski and Hammer (2017) and Kuo et al (2020). The case study I have presented illustrates this process: Jake and Liam initially expressed a great deal of confusion about voltage and related concepts, and although they knew many of the "rules" and features of these concepts, they had difficulty connecting these pieces of information together.…”

Categorical framework for mathematical sense making in physics

“…While in specific instances a particular mode may be preferred, expert reasoning likely involves fluidity with all of them and an understanding of which mode may be most productive in a given situation. For example, the two mixed reasoning modes (Psm-M and Msm-P) are likely at play during instances of what Kuo et al have called "calculation-concept crossover" [22] where conceptual reasoning is used to answer a quantitative question or calculations are used to answer a qualitative question.…”

“…Mathematical problem solving is a critical component of the physical sciences, and so a topic of great focus in science learning. As such, there has been a substantial amount of work in physics problem solving [1][2][3], mathematical problem solving in physics [4][5][6][7][8][9][10], the coordination of mathematics with other representational forms [11][12][13][14][15], scientific sense making [16,17], and mathematical sense making in physics [18][19][20][21][22]. This large body of work has identified both student-centric problem solving strategies aimed at helping students engage with mathematical formalisms more like experts [1][2][3] and researcher-centric frameworks for considering student use of mathematics in physics [4][5][6][7][8]12].…”