# Virtual Manipulatives and Students’ Counterexamples During Proving

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## Abstract

Counterexamples play a crucial role in the disciplinary practice of mathematics, and experiences with their use can enhance students’ engagement with mathematical practice and their learning of mathematical content. Although it is known that students encounter difficulties in producing and addressing counterexamples, and while there is evidence that incorporating appropriate computer technology into classroom tasks can improve the student experience, task design focusing on counterexamples is scarce in mathematics education research. In this chapter, we address such matters by employing a framework for research on computer-based virtual manipulatives to re-examine tasks that were designed using a set of design principles that included a key role for a dynamic geometry environment. Our analysis shows that the tasks, which were originally based on our theory-informed design principles, are further supported with the conceptual framework on virtual manipulatives manipulatives. We also provide an empirical illustration of the affordances of the tasks with a task-based interview where undergraduate students successfully discovered and addressed counterexamples. An implication for mathematics education research is the importance of ensuring that task design is considered from various theoretical and conceptual perspectives in order to strengthen the theoretical underpinnings of the task design.

## Notes

### Acknowledgements

We wish to express our thanks to the reviewer for providing helpful comments on the earlier version of this chapter. This study is supported by the Japan Society for the Promotion of Science (Nos. 15H05402, 16H02068, and 18K18636).

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