Task Design Principles for Heuristic Refutation in Dynamic Geometry Environments

Abstract

Task design is increasingly recognised as crucial for enhancing student learning of mathematics. Even so, and despite the significance of mathematical activity related to proofs and counterexamples in school mathematics, the task design principles underpinning students' success in proof-related activity remains under-explored in mathematics education research. What is more, although the affordances of using dynamic geometry environments (DGEs) have been established in the literature, task design in DGEs remains somewhat under-studied. In this paper, we address both these issues by theoretically developing, and empirically testing, task design principles for supporting students' heuristic refutation (revising conjectures/proofs through addressing counterexamples) in DGEs. We use the existing literature to elaborate three design principles: using tasks whose conditions are purposefully implicit, providing tools that enhance the production of counterexamples, and increasing students' recognition of contradictions. To test these principles empirically, we analyse two sets of task-based interviews, one with a triad of secondary school students and the other with a pair of undergraduates, where tasks designed according to the principles were implemented. Our analysis shows that using the tasks enabled the students to engage successfully in heuristic refutation. The matter of examining mathematical definitions when handling counterexamples emerged as an issue for further research.

Interplay between Paper-and-Pencil Activity and Dynamic-Geometry-Environment Use during Generalisation and Proving

“…Existing studies have tended to emphasise the relative affordances of DGEs in comparison with those of paper-and-pencil environments. The dragging function of DGEs, leading to a variety of diagrams, enables students to engage in dynamic mathematical activity where they can produce conjectures by themselves, discover counterexamples and come up with ideas regarding how to prove conjectures (Baccaglini-Frank and Mariotti 2010; Baccaglini-Frank et al 2018; Komatsu and Jones 2019). This is partly the case with the lessons described in this article, as the students discovered the existence of different cases through dragging and investigated a conjecture generalising the original statement.…”

“…It is within this type of generalisation that DGEs can play a key role, because the dragging function of DGEs allows for relatively easy access to multiple diagrams, while keeping certain the geometrical relationships imposed on the diagrams. The successive transformation of diagrams can highlight continuity and discontinuity between different configurations within an identical statement (Komatsu and Jones 2019).…”

Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities

“…Researchers have argued that dynamic technologies have to be a tool used to develop mathematical thinking rather than simply a tool that executes mathematical operations (Richard et al, 2019). By using dynamic technologies as ways to build knowledge inductively, students can be supported to focus on the more general aspects of the cases they examine and move to thinking more deductively (Baccaglini-Frank, 2019; Komatsu & Jones, 2019;Lachmy & Koichu, 2014).…”

Online assessment of students’ reasoning when solving example-eliciting tasks: using conjunction and disjunction to increase the power of examples

“…The uses of example for demonstrating, confirming, or refuting existential statements are often found to be non-trivial. Komatsu and Jones (2018) reported that students often encounter difficulties in formulating counter-example diagrams. They found that DGEs were highly useful in overcoming these difficulties and helping students produce counter-example diagrams.…”