Abstract Introducing the slope of a curve as the limit of the slope of secant lines is a well-known challenge in mathematics education. As an alternative, three other approaches can be recognized, based on linear approximation, based on multiplicities, or based on transition points. In this study we investigated which of these approaches fits students most by analyzing students’ inventions during a lesson scenario revolving around a design problem. The problem is set in a context that is meaningful to students and invites them to invent methods to construct a tangent line to a curve: an implementation of the guided reinvention principle from Realistic Mathematics Education (RME). The teaching scenario is based on the phased lesson structure of the Theory of Didactical Situations (TDS). The scenario was tested with 44 groups of three students in six grade 9 or 10 classrooms. We classified the strategies used by students and, using the emergent models-principle from RME, investigated to which of the four approaches the student strategies connect best. The results show that the groups produced a variety of strategies in each classroom and these strategies contributed to a meaningful institutionalization of the notion of slope of a curve.

中文翻译：

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用于学习曲线斜率的再发明活动中的紧急模型

摘要 引入曲线的斜率作为割线斜率的极限是数学教育中众所周知的挑战。作为替代，可以识别三种其他方法，基于线性近似、基于多重性或基于过渡点。在这项研究中，我们通过在围绕设计问题的课程场景中分析学生的发明来调查这些方法中哪些最适合学生。问题设置在对学生有意义的环境中，并邀请他们发明构建曲线切线的方法：实现现实数学教育 (RME) 的引导式再发明原则。教学场景基于教学情境理论 (TDS) 的分阶段课程结构。该场景在 6 个 9 年级或 10 年级的教室中对 44 组三名学生进行了测试。我们对学生使用的策略进行了分类，并使用来自 RME 的涌现模型原理，调查了学生策略与四种方法中的哪一种最相关。结果表明，这些小组在每个课堂上制定了各种策略，这些策略有助于将曲线斜率概念有意义地制度化。