How students blend conceptual and formal mathematical reasoning in solving physics problems

TL;DR

This paper investigates how physics students integrate conceptual understanding with formal mathematical reasoning during problem solving, emphasizing the importance of opportunistic blending for developing expertise.

Contribution

It introduces the concept of opportunistic blending of reasoning types during manipulation of equations, supported by case studies and analysis of student interviews.

Findings

Students often blend reasoning during equation manipulation.

Blending reflects a deeper conceptual understanding of equations.

Opportunistic blending can be targeted in instruction.

Abstract

Current conceptions of expert problem solving depict physical/conceptual reasoning and formal mathematical reasoning as separate steps: a good problem solver first translates a physical Current conceptions of quantitative problem-solving expertise in physics incorporate conceptual reasoning in two ways: for selecting relevant equations (before manipulating them), and for checking whether a given quantitative solution is reasonable (after manipulating the equations). We make the case that problem-solving expertise should include opportunistically blending conceptual and formal mathematical reasoning even while manipulating equations. We present analysis of interviews with two students, Alex and Pat. Interviewed students were asked to explain a particular equation and solve a problem using that equation. Alex used and described the equation as a computational tool. By contrast, Pat found a shortcut to solve the problem. His shortcut blended mathematical operations with conceptual reasoning about physical processes, reflecting a view - expressed earlier in his explanation of the equation - that equations can express an overarching conceptual meaning. Using case studies of Alex and Pat, we argue that this opportunistic blending of conceptual and formal mathematical reasoning (i) is a part of problem-solving expertise, (ii) can be described in terms of cognitive elements called symbolic forms (Sherin, 2001), and (iii) is a feasible instructional target.