How students blend conceptual and formal mathematical reasoning in solving physics problems
How students blend conceptual and formal mathematical reasoning in solving physics problems
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学生如何在解决物理问题时融合概念推理和形式数学推理
DOI：
10.1002/sce.21043
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发表时间：
2011
影响因子：
4.3
通讯作者：
Andrew Elby
中科院分区：
文献类型：
--
作者：
E. Kuo;M. M. Hull;A. Gupta;Andrew Elby
物理学中的定量问题解决专家的当前概念以两种方式结合了概念推理：用于选择相关方程(在操纵它们之前)和用于检查给定的定量解是否合理(在操纵方程之后)。我们提出，即使在处理方程的同时，解决问题的专业知识也应该包括概念推理和形式数学推理的机会主义融合。我们提供了对亚历克斯和帕特两名学生的采访分析。受访学生被要求解释一个特定的方程式，并用这个方程式解决一道题。亚历克斯使用并描述了这个方程作为一种计算工具。相比之下，帕特找到了一条解决问题的捷径。他的捷径将数学运算与关于物理过程的概念推理结合在一起，反映了他早先在解释方程式时表达的一个观点，即方程式可以表达总体的概念意义。利用Alex和Pat的案例研究，我们认为这种概念推理和形式数学推理的机会主义融合(I)是问题解决专业知识的一部分，(Ii)可以用称为符号形式的认知元素来描述(Sherin，2001)，(Iii)是一个可行的教学目标。
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.