Reasoning and proving is a difficult topic for teachers. Research has shown that teachers often favor empirical arguments over proofs and believe that even after proving a statement, one might still be able to find a counterexample to mollify the proof (Knuth, 2002a). In addition, many teachers hold a limited view of the utility of proof in the secondary school curriculum (Knuth, 2002b).  In particular, secondary mathematics teachers tend to view proof largely as a specific topic of study rather than as a tool for doing mathematics or as a stance towards mathematics in general. This view stands in sharp contrast with the practice of mathematicians, where proof is used to justify new results and verify the results of others (Hanna, 1995), and is the culmination of a series of activities (e.g., Lakatos, 1976).

Teachers’ current understanding of proof and its value in the secondary curriculum are of particular concern in light of the growing consensus that high school mathematics programs need to include a greater emphasis on reasoning and proof.  For example, The Common Core State Standards Initiative (2010) identifies reasoning abstractly and quantitatively and constructing viable arguments as key mathematical practices that students need to develop across a breadth of content areas. In Focus in High School Mathematics: Reasoning and Sense Making (NCTM, 2009), the authors argue that reasoning and sense making “should occur in every mathematics classroom everyday” (p.5).

Making reasoning and proving a central feature of classroom instruction will require helping teachers understand the role these mathematical processes can and should play in secondary mathematics as well as approaches for developing their students’ abilities to engage in a broad range of reasoning-and-proving activities (i.e., searching for patterns, forming generalizations, making conjectures, providing arguments that demonstrate the viability of the conjecture). In this session, participants will engage in a discussion and analysis of curricular materials for teacher professional development designed to meet this challenge and of the data collected from three pilot studies in which preservice teachers secondary teachers engaged with these materials.

The practice-based materials we have developed focus teachers on learning about reasoning and proving and are organized around three guiding questions: 1) What is reasoning and proving; 2) How do secondary students benefit from engaging in reasoning and proving activity? and 3) How can secondary mathematics teachers support students’ capacities to reason and prove?  The content of this working session focuses on Question 3 -- How can secondary mathematics teacher support students’ capacities to reason and prove?

Supporting students’ capacities to reason and prove will require transforming classrooms from places where the dominant interactions involve teachers demonstrating procedures followed by students practicing and applying learned procedures with no emphasis on reasoning, proving, and sense-making to communities of learners where justifying conclusions and debating with peers are hallmark practices. We argue that three key dimensions of classrooms are critical to this transformation: (a) the tasks or activities in which students engage should provide opportunities for students to look for patterns, make conjectures and develop arguments; (b) tools should be available to support students’ reasoning and sense-making as they engage with challenging tasks; and (c) productive classroom talk must support mathematical discourse and enable students to share and refine their ideas. While tasks, tools and talk have been identified as critical dimensions of classrooms that promote understanding (Carpenter & Lehrer, 1998; Hiebert, et al, 1997), we argue that they are of paramount importance in classrooms that promote reasoning and proving. Tasks, Tools, and Talk work in concert to create an environment that supports students’ growth and development as mathematical thinkers and learners.

In this session, participants will engage in a discussion and analysis of activities drawn from this practice-based curricula and of data from pilot studies aimed at exploring the utility of tasks, tools, and talk as a framework for shifting teachers knowledge and practice related to reasoning and proving in the mathematics classroom. The session will unfold in the following sequence: 1) describe the curricular materials and their purpose (15 minutes); 2) engage participants in analyzing a subset of activities from the materials intended to develop teachers understanding of tasks, tools and talk (30 minutes); 3) have participants review data on teacher learning from three pilot studies (30 minutes); and 4) discuss how to best capture data that allows researchers to investigate questions such as What is learned by teachers, through engagement with these materials, about instruction that supports students’ capacities to reason and prove? (15 minutes). The following five questions will focus the discussion between participants and the research team:

• To what extent do the materials focus on critical aspects of what teachers need to know and be able to develop their students’ capacity to reason and prove?
• To what extent is the task, tools, and talk framework helpful in supporting teachers learning of classroom practices that support the development of reasoning and proving?
• To what extent do the data collected to date provide evidence of teacher learning?
• What additional sources of data or measures of learning might provide more compelling evidence?
• What related work in the field might serve to inform or strengthen the development of these materials and the related research?

The session will address professional learning through the examination of materials designed to support teacher learning and analysis of data regarding what teachers appeared to learn from their experiences. The first and third questions address professional learning directly.