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 calculation-concept 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.

中文翻译：

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通过计算概念交叉评估物理中的数学意义

物理学和工程学中的专业问题解决实践依赖于数学意义上的理解-这种推理利用了形式数学与概念理解之间的连贯性。物理教育的一个关键问题是当前的教学方法如何很好地发展学生的数学意识。我们介绍了一种评估范式，该范式可实现数学意义上通常无法衡量的维度：使用关于定性问题的计算以及使用关于定量问题的概念性论证。体现此计算概念交叉评估范式的三个评估项目说明了数学意义制造如何可以积极地帮助学生解决问题，从而产生更有效，更有见识和更准确的解决方案。这三个评估项目用于评估侧重于发展学生的数学理解能力的教学方法的有效性。在一个准实验研究中，比较了上学期入门物理课的三个平行授课部分：两个数学意义形成部分，一个由经验丰富的讲师指导，一个由新手指导，一个传统授课部分作为对照组。与对照组相比，数学意义上的小组更频繁地使用计算概念交叉方法，并在交叉评估项目上给出了更正确的答案，但对相关的标准问题却没有给出更正确的答案。此外，尽管学生对解决问题的连贯性的事后认识论观点与他们的交叉使用有关，他们没有完全考虑数学意义和对照组之间在交叉方法上的差异。这些结果证明了一种新的评估范式，用于检测通常无法测量的数学意义感知的维度，并提供证据表明定向教学方法可以增强对入门物理学中数学意义感知的参与。