Much of physics involves the construction and interpretation of equations. Research on physics students’ understanding and application of mathematics has employed Sherin’s symbolic forms or Fauconnier and Turner’s conceptual blending as analytical frameworks. However, previous symbolic forms analyses have commonly treated students’ in-context understanding as their conceptual schema, which was designed to represent the acontextual, mathematical justification of the symbol template (structure of the expression). Furthermore, most conceptual blending analyses in this area have not included a generic space to specify the underlying structure of a math-physics blend. We describe a conceptual blending model for equation construction and interpretation, which we call symbolic blending, that incorporates the components of symbolic forms with the conceptual schema as the generic space that structures the blend of a symbol template space with a contextual input space. This combination complements symbolic forms analysis with contextual meaning and provides an underlying structure for the analysis of student understanding of equations as a conceptual blend. We present this model in the context of student construction of non-Cartesian differential length vectors. We illustrate the affordances of such a model within this context and expand this approach to other contexts within our research. The model further allows us to reinterpret and extend literature that has used either symbolic forms or conceptual blending.

中文翻译：

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在方程构造和解释中使上下文变得明确：符号混合

许多物理学涉及方程的构造和解释。对物理学生对数学的理解和应用的研究采用了谢林的符号形式或福康尼尔和特纳的概念混合作为分析框架。然而，以前的符号形式分析通常将学生的上下文理解视为他们的概念图式，旨在表示符号模板（表达式的结构）的上下文数学论证。此外，该领域的大多数概念混合分析都没有包含一个通用空间来指定数学物理混合的基础结构。我们描述了一种用于方程构造和解释的概念混合模型，我们称之为符号混合，它将符号形式的组件与概念模式结合起来，作为构建符号模板空间与上下文输入空间的混合的通用空间。这种组合补充了具有上下文含义的符号形式分析，并为分析学生对方程作为概念混合的理解进行分析提供了基础结构。我们在学生构造非笛卡尔微分长度向量的背景下提出了这个模型。我们在这种背景下说明了这种模型的可供性，并将这种方法扩展到我们研究中的其他背景。该模型进一步允许我们重新解释和扩展使用符号形式或概念混合的文献。