Mathematics teachers are expected to engage their students in critiquing and constructing viable arguments. These classroom expectations are necessary, given that proof is a central mathematical activity. However, mathematics teachers have been provided limited opportunities as learners to construct arguments and critique the reasoning of others, and hence have developed perceptions of proof as an object that must follow a strict format. In this article, we describe a four-part instructional sequence designed to broaden and deepen teachers' perception of the nature of proof. We analyzed participants' reflections on the instructional sequence in order to gain insight into (a) the differences between this instructional sequence and participants' previous proof learning opportunities and (b) the ways this activity was influential in transforming participants' perceptions of proof. Participants' previous learning experiences were focused on memorizing and reproducing textbook or instructor proofs, and our sequence was different because it actively and collaboratively engaged participants in constructing their own arguments, critiquing others' reasoning, and creating criteria for what counts as proof. Participants found these activities transformative as they became more clear about what counts as proof, began to view proof as socially negotiated, and expanded their conception of proof beyond a rigid structure or format.
* In accordance with MTE policy regarding conflicts of interest with the editor, the review process for this manuscript was handled by Melissa D. Boston, Duquesne University. This article was submitted and accepted under the editorship of Margaret Smith.
Justin D. Boyle, Department of Curriculum and Instruction, 230D Graves Hall, University of Alabama, Tuscaloosa, AL 35487; firstname.lastname@example.org
Sarah K. Bleiler, Department of Mathematical Sciences, 263 Kirksey Old Main (KOM), Middle Tennessee State University, Murfreesboro, TN 37132; email@example.com
Sean P. Yee, Department of Mathematics & Instruction and Teacher Education, LeConte College 317K and Wardlaw 257, University of South Carolina, Columbia, SC 29208; firstname.lastname@example.org
Yi-Yin (Winnie) Ko, Department of Math and Computer Science, Root Hall A-140E, Indiana State University, Terre Haute, Indiana, 47809; email@example.com