We engaged five mathematicians who conduct research in the domain of complex analysis or use significant tools from complex analysis in their research in interviews about basic concepts of differentiation and integration of complex functions. We placed a variety of constructivist, social-constructivist, and embodied theories in mathematics education in conversation with one another to explore the development of the expert participants’ construction of mathematical meanings while moving between varying levels of abstraction from embodied concepts and real-world contexts to symbolic manipulation and formal theories. The mathematicians relied heavily on direct application of concepts and analogies from differentiation of real-valued functions and employed rotation and dilation as a local linear description of the action of a complex differentiable function with attendant repeated mental imagery and physical gestures. They also employed reasoning about real-valued line integrals to interpret contour integrals but acknowledged significant struggle to conceptually interpret what was analogously accumulated in the complex case. Instead, they all developed more personal meanings through a process of reconciling various aspects across their concrete to formal domains of reasoning. Much of the observed construction of meaning was manifested through contextualizing well-understood aspects of formal mathematical theory. We consequently explore implications for characterizing mathematical conceptual development as an interplay between concrete and formal reasoning rather than a development from one to another.

中文翻译：

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专家对复值函数的导数和积分的数学意义的构​​造

我们采访了五位在复杂分析领域中进行研究或在研究中使用复杂分析中的重要工具的数学家，他们对复杂功能的微分和集成的基本概念进行了采访。我们互相交谈，将各种建构主义，社会建构主义和具体化理论放置在数学教育中，以探讨专家参与者数学意义建构的发展，同时在从具体化概念和现实环境中抽象化的不同层次之间移动象征性的操纵和形式理论。数学家在很大程度上依赖于从实值函数的微分中直接应用概念和类比，并采用旋转和扩张作为复杂的可微函数的作用的局部线性描述，并伴随着反复的心理意象和肢体手势。他们还采用了关于实值线积分的推理来解释轮廓积分，但承认在概念上解释复杂情况下类似积累的内容方面付出了巨大的努力。取而代之的是，他们都通过协调其具体方面与正式领域之间的各个方面，发展出更多的个人意义。所观察到的意义建构中的许多内容是通过将形式化数学理论中易于理解的方面进行情境化而体现的。