A Study on Students’ Explanations by Counter-examples in Elementary School Mathematics: Characterising Students’ Responses Based on a Survey Using “Number Pyramid” as a Task

Abstract

In this study, we aim to characterize students’ uses of counter-examples and their explanations by counter-examples, through a qualitative analysis based on a questionnaire survey targeting sixth graders in elementary school. For attaining this, we first reviewed the literature related to the notion of generic examples, and the role of counter-examples for explanation in mathematics education research. Then, an analytical framework, consisting of two basic aspects, “generic use of counter-examples” and “scope of explanatory power by counter-examples”, was constructed. To illustrate this framework, we conducted a questionnaire survey using a proving task (called “number pyramid”), analyzed the students’ writings, and classified them into each aspect of the framework. As a result, students’ different responses to the tasks are characterized by the aspects of the explanatory power of the counter-examples as well as the aspects of generic use of counter-examples. This study implies that the students have more difficulties with increasing explanatory power rather than increasing the genericity of use of counter-examples. Some implications for teaching and learning in lower secondary schools are also discussed.