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.

### Justin D. Boyle, Sarah K. Bleiler, Sean P. Yee and Yi-Yin (Winnie) Ko

### Eric J. Knuth, Jeffrey M. Choppin and Kristen N. Bieda

Asking middle school students to verify the math they do requires them to think about proof. By doing so, students construct arguments in the middle school and are more ready for proof in high school.

### Charles K. Assuah and Lynda R. Wiest

Can middle-grades students determine which of two rectangular prisms has a larger volume? Can they do so without using a formula? Geometry, and particularly the concept of volume, is important in many subjects, such as physics and chemistry. Carpenter and Romberg (2004) note that students greatly enhance their mathematics knowledge when they make generalizations and construct arguments to justify their methods through problem solving. To realize this goal, teachers should encourage and accept students' self-initiative, allow students' responses to help shape lessons, and inquire about students' understanding of concepts before sharing their own (Brooks and Brooks 1993). We describe a task that requires students to use their own nonformulaic methods and reasoning to investigate volume.

### Karen F. Hollebrands, AnnaMarie Conner and Ryan C. Smith

Prior research on students' uses of technology in the context of Euclidean geometry has suggested it can be used to support students' development of formal justifications and proofs. This study examined the ways in which students used a dynamic geometry tool, NonEuclid, as they constructed arguments about geometric objects and relationships in hyperbolic geometry. Eight students enrolled in a college geometry course participated in a task-based interview that was focused on examining properties of quadrilaterals in the Poincaré disk model. Toulmin's argumentation model was used to analyze the nature of the arguments students provided when they had access to technology while solving the problems. Three themes related to the structure of students' arguments were identified. These involved the explicitness of warrants provided, uses of technology, and types of tasks.

### Sean P. Yee, George J. Roy and LuAnn Graul

statements to one another and constructed arguments to justify their statements, for which the black and white pegs helped support or challenge their conditional statements. We wanted pairs of students to engage with the task in a coordinated way to solve