Contract grant sponsor: National Science Foundation.
How students blend conceptual and formal mathematical reasoning in solving physics problems
Article first published online: 18 DEC 2012
© 2012 Wiley Periodicals, Inc.
Volume 97, Issue 1, pages 32–57, January 2013
How to Cite
KUO, E., HULL, M. M., GUPTA, A. and ELBY, A. (2013), How students blend conceptual and formal mathematical reasoning in solving physics problems. Sci. Ed., 97: 32–57. doi: 10.1002/sce.21043
Contract grant number: EEC-0835880.
The views expressed here are those of the authors and do not necessarily reflect those of the National Science Foundation.
- Issue published online: 18 DEC 2012
- Article first published online: 18 DEC 2012
- Manuscript Accepted: 28 SEP 2012
- Manuscript Received: 9 NOV 2011
- National Science Foundation. Grant Number: EEC-0835880
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 of 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.