ABSTRACT

We present a model for describing the growth of students’ understandings when reading a proof. The model is composed of two main paths. One is focused on becoming aware of the deductive structure of the proof, in other words, understanding the proof at a semantic level. Generalization, abstraction, and formalization are the most important transitions in this path. The other path focuses on the surface-level form of the proof, and the use of symbolic representations. At the end of this path, students understand how and why symbolic computations formally establish a claim, at a syntactic level. We make distinctions between states in the model and illustrate them with examples from early secondary students’ mathematical activity. We then apply the model to one student’s developing understanding in order to show how the model works in practice. We close with some suggestions for further research.

摘要

我们提供了一个模型，用于描述学生阅读证明时理解力的增长。该模型由两条主要路径组成。一个重点是了解证明的演绎结构，换句话说，就是从语义层面理解证明。泛化，抽象和形式化是此路径中最重要的过渡。另一条路径侧重于证明的表面形式以及符号表示的使用。在这条道路的尽头，学生将了解符号计算如何以及为什么在语法层面上正式确立主张。我们对模型中的状态进行区分，并以初中生的数学活动为例进行说明。然后，我们将模型应用于一个学生不断发展的理解中，以展示该模型如何在实践中发挥作用。