An Algebraic Opportunity to Develop Proving Ability

Set-based reasoning and conditional language are two critical components of deductive argumentation and facility with proof. The purpose of this qualitative study was to describe the role of truth value and the solution set in supporting the development of the ability to reason about classes of objects and use conditional language. This study first examined proof schemes – how students convince themselves and persuade others – of Algebra I students when justifying solutions to routine and non-routine equations. After identifying how participants learned to use set-based reasoning and conditional language in the context of solving equations, the study then determined if participants would employ similar reasoning in a geometrical context. As a whole, the study endeavored to describe a possible trajectory for students to transition from non-deductive justifications in an algebraic context to argumentation that supports proof writing. First, task-based interviews elicited how participants became absolutely certain about solutions to equations. Next, a teaching experiment was completed to identify how participants who previously accepted empirical arguments as proof shifted to making deductive arguments. Last, additional task-based interviews in which participants reasoned about the relationship between Varignon Parallelograms and Varignon Rectangles were conducted. The first set of task-based interviews found that a majority of participants displayed ritualistic proof schemes – they viewed equations as prompts to execute processes and solutions as results, or “answers.” Approximately half of participants employed empirical proof schemes; they described convincing themselves or others using a range of arguments that do not constitute valid proof. One particularly noteworthy finding was that no participants initially used deductive justifications to reach absolute certainty. Participants successfully adopted set-based reasoning and learned to use conditional language by progressively accommodating a series of understandings. They later utilized their new ways of reasoning in the geometrical context. Participants employed the implication structure, discriminated between necessary and sufficient conditions, and maintained a disposition of doubt toward empirical evidence. Finally, implications of these findings for pedagogues and researchers, as well as future directions for research, are discussed