Abstract
Proving and refuting are fundamental aspects of mathematical practice that are intertwined in mathematical activity in which conjectures and proofs are often produced and improved through the back-and-forth transition between attempts to prove and disprove. One aspect underexplored in the education literature is the connection between this activity and the construction by students of knowledge, such as mathematical concepts and theorems, that is new to them. This issue is significant to seeking a better integration of mathematical practice and content, emphasised in curricula in several countries. In this paper, we address this issue by exploring how students generate mathematical knowledge through discovering and handling refutations. We first explicate a model depicting the generation of mathematical knowledge through heuristic refutation (revising conjectures/proofs through discovering and addressing counterexamples) and draw on a model representing different types of abductive reasoning. We employed both models, together with the literature on the teachers’ role in orchestrating whole-class discussion, to analyse a series of classroom lessons involving secondary school students (aged 14–15 years, Grade 9). Our analysis uncovers the process by which the students discovered a counterexample invalidating their proof and then worked via creative abduction where a certain theorem was produced to cope with the counterexample. The paper highlights the roles played by the teacher in supporting the students’ work and the importance of careful task design. One implication is better insight into the form of activity in which students learn mathematical content while engaging in mathematical practice.
Highlights
Proving and refuting are fundamental aspects of mathematical practice (Lakatos, 1976)
Whereas it can be said that mathematical knowledge is generated even during heuristic refutation, in that improved conjectures are produced, in this paper we extend the model of heuristic refutation and propose the framework illustrated by Fig. 1, which adds generation of mathematical knowledge as an element
The analysis presented in this paper shows that mathematical activity in the classroom can be encapsulated by combining our model representing the generation of mathematical knowledge through heuristic refutation (Fig. 1) and the model of the kinds of abduction (Fig. 2)
Summary
Proving and refuting are fundamental aspects of mathematical practice (Lakatos, 1976). Both processes are closely inter-related, and conjectures and proofs are often produced and improved through the back-and-forth transition between attempts to prove and disprove. Such mathematical activity leads to the development of mathematical knowledge and theory. Other studies have explicitly addressed the inter-related activities of conjecturing, proving, and refuting such as how students modify their conjectures, sometimes by capitalising on their proofs, when faced with counterexamples (Balacheff, 1991; Larsen & Zandieh, 2008)
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