Students’ ways of understanding a proof

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.

KEYWORDS:

• Generalization
• abstraction
• formalization
• proof comprehension
• students’ understanding

Acknowledgments

We wish to thank, gratefully, Professor John Mason and other readers and reviewers for useful comments on earlier versions of this article.

Notes

1. We use the word ‘argument’ to refer to any text or utterance that could potentially be convincing, and restrict the word ‘proof’ to those arguments that satisfy the standards of a mathematical community.

2. Fatemeh Ahmadpour (FA) conducted the interviews. In our transcripts slash (/) indicates that the speaker changes her words in the middle of a sentence before completing it. Ellipses (…) indicate omitted speech. Brackets ([]) contain some explanatory notes about the nonverbal events. Star (*) means that the conversation has been paused and the student is thinking about 1 s. Three question marks (???) indicate an inaudible speech or a guess at partly audible speech.