Abstract Counting problems offer opportunities for rich mathematical thinking, and yet there is evidence that students struggle to solve counting problems correctly. There is a need to identify useful approaches and thought processes that can help students be successful in their combinatorial activity. In this paper, we propose a characterization of an equivalence way of thinking, we discuss examples of how it arises mathematically in a variety of combinatorial concepts, and we offer episodes from a paired teaching experiment with undergraduate students that demonstrate useful ways in which students developed and leverage this way of thinking. Ultimately, we argue that this way of thinking can apply to a variety of combinatorial situations, and we make the case that it is a valuable way of thinking that should be prioritized for students learning combinatorics.

中文翻译：

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在枚举组合中定义和证明等效的思维方式

摘要 计数问题为丰富的数学思维提供了机会，但有证据表明学生很难正确解决计数问题。需要确定有用的方法和思维过程，以帮助学生在他们的组合活动中取得成功。在本文中，我们提出了对等思维方式的表征，我们讨论了它如何在各种组合概念中以数学方式出现的例子，我们提供了来自本科生的配对教学实验的片段，展示了学生发展的有用方式并利用这种思维方式。最终，我们认为这种思维方式可以适用于各种组合情况，