Researchers generally agree that physics experts use mathematics in a way that blends mathematical knowledge with physics intuition. However, the use of mathematics in physics education has traditionally tended to focus more on the computational aspect (manipulating mathematical operations to get numerical solutions) to the detriment of building conceptual understanding and physics intuition. Several solutions to this problem have been suggested; some authors have suggested building conceptual understanding before mathematics is introduced, while others have argued for the inseparability of the two, claiming instead that mathematics and conceptual physics need to be taught simultaneously. Although there is a body of work looking into how students employ mathematical reasoning when working with equations, the specifics of how physics experts use mathematics blended with physics intuition remain relatively underexplored. In this paper, we describe some components of this blending, by analyzing how physicists perform the rearrangement of a specific equation in cosmology. Our data consist of five consecutive forms of rearrangement of the equation, as observed in three separate higher education cosmology courses. This rearrangement was analyzed from a conceptual reasoning perspective using Sherin’s framework of symbolic forms. Our analysis clearly demonstrates how the number of potential symbolic forms associated with each subsequent rearrangement of the equation decreases as we move from line to line. Drawing on this result, we suggest an underlying mechanism for how physicists reason with equations. This mechanism seems to consist of three components: narrowing down meaning potential, moving aspects between the background and the foreground and purposefully transforming the equation according to the discipline’s questions of interest. In the discussion section we highlight the potential that our work has for generalizability and how being aware of the components of this underlying mechanism can potentially affect physics teachers’ practice when using mathematics in the physics classroom.

中文翻译：

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重新排列方程以发展物理推理

研究人员普遍认为，物理学专家以将数学知识与物理学直觉相结合的方式使用数学。然而，数学在物理教育中的使用传统上倾向于更多地关注计算方面（操纵数学运算以获得数值解），这不利于建立概念理解和物理直觉。针对这个问题已经提出了几种解决方案；一些作者建议在引入数学之前建立概念理解，而另一些作者则认为两者是不可分割的，并声称数学和概念物理需要同时教授。尽管有大量工作研究学生在处理方程时如何运用数学推理，但物理专家如何将数学与物理直觉相结合的具体细节仍然相对未得到充分探索。在本文中，我们通过分析物理学家如何重新排列宇宙学中的特定方程来描述这种混合的一些组成部分。我们的数据由方程的五种连续形式的重新排列组成，正如在三个独立的高等教育宇宙学课程中观察到的那样。使用 Sherin 的框架从概念推理的角度分析了这种重新排列象征形式。我们的分析清楚地表明，随着我们从一行移动到另一行，与方程的每次重新排列相关的潜在符号形式的数量如何减少。根据这一结果，我们提出了物理学家如何用方程进行推理的基本机制。这种机制似乎由三个组成部分组成：缩小意义潜力、在背景和前景之间移动方面以及根据学科感兴趣的问题有目的地改变方程。在讨论部分，我们强调了我们的工作具有普遍性的潜力，以及了解这一潜在机制的组成部分如何可能影响物理教师在物理课堂上使用数学时的实践。