Using variation tasks to investigate learners’ attention to structure in the area of proving

Thesis

Proving is fundamental to the learning of mathematics because the ability to prove involves understanding, identifying, and expressing generalities. Working with generalities, i.e., generalising, lies at the heart of learning, and hence teaching, mathematics. Diversity across the designs and demands of proving tasks makes identifying cross-task structural and algorithmic generalities difficult. Consequently, no overall learning or teaching approach can be adopted. This pedagogical challenge might explain why, in secondary schools, proof is often taught in isolation to other mathematical activities, i.e., on a proof- or task-specific basis. These isolated teaching approaches can provide poor preparation for proof production because they do not first address the kinds of mathematical actions that proving can involve.

In this study I posit ways to alleviate two key problems that can arise from the challenges of proof pedagogy: (1) insufficient scaffolding, and (2) indistinct characterisation of learners’ proving actions. To address insufficient scaffolding, I designed sequences of slightly varied (consecutive number) proof arguments and accompanying prompts, which I call proof variation tasks. I used the variation tasks to encourage learners to perform the kinds of mathematical actions that can underpin the construction of proofs. I designed an isomorphic pair of variation tasks in each of two distinct domains: number and geometry. Each domain-pair of variation tasks comprised an initial task and a conceptually similar but more challenging transfer task. To address the indistinct characterisation of learners’ proving actions, I formulated the GOLDEN framework. The acronym GOLDEN captures the six broad actions that learners similarly performed on each domain-pair of variation tasks: Generalising, Organising, Localising, Deconstructing, Extending and Networking.

I gave the variation tasks to three Year 8 learners and three Year 10 learners (12- to 13-year-olds and 14- to 15-year-olds, respectively) all of whom were new to any formal notion of proof and argumentation. The learners engaged with the variation tasks in videoed semi-structured interviews that I conducted in the dual role of teacher/researcher. I used the GOLDEN framework to describe and compare learners’ actions on the initial and transfer variation tasks within and across each of the two domains. 

I provide evidence that consecutive-number variation tasks which feature structurally similar arguments can be used to encourage learners to perform a range of proving actions, thereby supporting their development in the area of proof. I also introduce the GOLDEN framework as a task-generic analytical tool for describing and identifying learners’ proving actions. Suggestions for future research, and implications for advancing pedagogy are discussed.

Authors: Bentley, J

• DOI: 10.5287/ora-0zmedrnq7
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: argumentation; learning transfer; proof; proving; structure; variation theory; mathematical actions
• Subjects: learning mathematics; teaching mathematics; mathematics education; proof; proving