Abstract

In this paper, we present data from two iterative teaching experiments involving students’ constructions of four basic counting problems. The teaching experiments were designed to leverage the generalizing activities of relating and extending to provide students with opportunities to reflect on initial combinatorial activity when constructing these formulas. We discuss three combinatorial ways of thinking that emerged from their work on this task and provide commentary on the interplay between the students’ generalizing activities and engagement in the emergent ways of thinking. We address the practical goals of identifying ways that students productively reason in combinatorics, and we provide a theoretical commentary on students’ reflection on activity as it occurs in the context of particular generalizing actions.

摘要

在本文中，我们展示了来自两个迭代教学实验的数据，这些实验涉及学生对四个基本计数问题的构建。教学实验旨在利用关联和扩展的概括活动，为学生提供机会在构建这些公式时反思最初的组合活动。我们讨论了他们在这项任务的工作中出现的三种组合思维方式，并对学生的概括活动与新兴思维方式的参与之间的相互作用进行了评论。我们解决了确定学生在组合数学中有效推理的方法的实际目标，并且我们对学生在特定概括行动的背景下对活动的反思提供了理论评论。