Developing Secondary Students’ Understanding of Proof through Constructing, Critiquing, and Revising Arguments

ABSTRACT

We examined high school geometry students’ written work on four proof tasks where they posed a conjecture, drafted an argument, provided written critiques, then revised their argument based on peer feedback. Students’ written work across the tasks was analyzed to determine whether the instructional sequence supported them in improving their arguments and attending to key aspects of proof (justifications, generality, clarity, structure). Claim-level analysis for each of the key aspects revealed minor changes between students’ draft and revised arguments with results varying by task. That said, students attended to the key aspects of proof through the critiques they provided each other with most critiques, if appropriately addressed, having the potential to help improve the draft argument. Students’ reflections also showed this process helped them think about the clarity and level of detail in their arguments. Implications for this study include the benefits of providing proof tasks that offer fewer supports for students, alongside multi-faceted analysis of their written arguments, in terms of providing insights into students’ current understanding of proof.

KEYWORDS:

• Reasoning-and-proving
• justification
• critiquing arguments
• geometry

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

¹ The sum of the angles in a triangle had been previously established as true in the classroom, so we did not want students to prove this conjecture. As a result, there was only one conjecture related to the provided image that could be proven.

² We also coded students’ arguments holistically for their mathematical structure. Since 65 of the 68 arguments contained structural gaps, we chose to exclude this analysis from our paper.

³ While diagrams are not intended to be used as justification, prior studies have found students use aspects of diagrams as justification when verbally explaining a proof (Karaman, 2017) and view measurement of examples as sufficient to determine the validity of a conjecture (e.g., Chazan, 1993).