One significant call for change in NCTM’s (1989, 2000) Standards documents was the recommendation that students of all ages should have experiences that facilitate an understanding of reasoning and proof as fundamental aspects of mathematics. In NCTM’s Reasoning and Sense-Making (2009) document, formal reasoning (i.e., proof) was situated as the final of three stages in the reasoning progression required for increasing levels of understanding in the high school mathematics classroom. The authors pointed out that the effort to help students progress from less formal to more formal reasoning requires that “teachers play an essential role in encouraging students to explore more sophisticated levels of reasoning and sense making” (p. 11). As teachers work to adopt the Common Core State Standards, they will undoubtedly continue to play an important role in fostering an understanding of reasoning, particularly as they work on this Standard of Mathematical Practice: “Construct viable arguments and critique the reasoning of others” (p. 6). The objective that, by high school, students will be able to engage with more formal proofs (e.g., of geometric theorems and trigonometric identities) will best be realized if students have had opportunities to engage in mathematical reasoning, argumentation, and justification throughout the grade levels.

Despite such calls for change, however, recent research indicates that the goal of elevating the importance of reasoning and proving in the curriculum and classroom has not yet been realized. For example, Bieda (2010) found that even experienced middle school teachers using the Connected Mathematics curriculum did not provide enough feedback about their students' conjectures and justifications to establish standards of proof in their mathematics classrooms. Other research indicates that, even in the geometry curriculum where proof is typically taught, changes proposed by NCTM publications related to the teaching of proof have not occurred (see, e.g., Herbst, et al., 2009). Some research (Knuth, 2002) has suggested that perhaps the reason that teachers have not moved their students beyond the more traditional approaches to proof is related to teachers’ beliefs about the purpose of proof and their students’ abilities to complete a proof. Additionally, teachers may not have had opportunities to consider alternative ways of teaching proof that fall outside of the “apprenticeship of observation” (Lortie, 1975) experienced in their own mathematics backgrounds. For example, Cirillo (2011) found that some teachers believe that the only way to teach proof is for the teacher to model doing proofs while the students look on.

In this symposium, researchers representing five different research teams provide overviews of their projects and look across these ongoing studies to discuss how, through research and professional development, they are working with teachers to influence their notions of and practices around reasoning and proving. Participants of the session will be asked to participate in this discussion at several key points in time.

Project 1

This project introduces a new strand of research into the ongoing work on proof by embedding proof activities in a context of quantitative reasoning. We worked with middle-school students who engaged in proving when reasoning with functional relationships between quantities. Results indicated that a focus on quantities supported reasoning that was flexible, general, and deductive. We are now preparing to share the student results with practicing teachers in order to foster teachers’ abilities to promote their students’ proof competencies through quantitative reasoning.

Project 2

Project 2 is a research and development project focused on justification in middle grades classrooms. In collaboration with 12 middle grades teachers over two years, we have examined the role of justification in the classroom, analyzed how teachers develop their understanding of justification and proof as content knowledge, and documented teachers' advancement of a pedagogy of justification as they implement proof-relevant tasks. We are currently analyzing data to coordinate across teachers' conceptions and pedagogy and looking at change over time.

 

Project 3

To learn more about and support high school geometry teachers as they engage their students in proving, this project is focused on developing a community of five teachers working together through professional development to improve the ways in which they teach proof. Results from Year 1 of this two-year study indicate that these project teachers frequently wrestled with the challenges of scaffolding proof in meaningful ways and dilemmas related to the amount of rigor they should impose on students.

 

Project 4

We present the results of a pilot study of secondary mathematics teachers who were asked to consider classroom scenarios where a geometry teacher assigned students a proof task without providing the givens or the conclusion to prove, and instead charged students with doing so. We examine the teachers’ reactions to these scenarios and classify the justifications and indictments they provided of what the teacher did in each scenario.    

Project 5

The purpose of this project is to develop a practice-based curriculum for the professional education of preservice and practicing secondary mathematics teachers that: 1) focuses on reasoning and proving; 2) has narrative cases as a central component; and 3) supports the development of knowledge of mathematics needed for teaching. After piloting the materials  with several groups of teachers over the past two years, preliminary results suggest that teachers improved their capacity to write proofs, developed more appropriate criteria for determining the validity of a proof, and were able to select appropriate tasks for engaging their students in reasoning and proving.

Session Organization

 

Activity	Description	Time (minutes)
Introduction	Theoretical Background and Goals of Session	5
Project Overviews	Each researcher will give a 10-minute project overview	50
Small Group Discussions	In groups, participants will be asked to consider the following questions: What working definitions of proof resonated with you and why? How can we use methods and ideas from the various projects to scale up teacher preparation in the area of reasoning and proof? What can be learned from these projects to help support teachers as we look towards CCSSM implementation? What still needs to be done?	20
Whole Group Discussion	Participants will come back together to raise questions or highlight key points from the discussions.	15

 

 

 

Bieda, K. N. (2010). Enacting Proof-Related Tasks in Middle School Mathematics: Challenges and Opportunities. Journal for Research in Mathematics Education, 41(4), 351-382.

Cirillo, M. (2011). "I'm like the Sherpa guide": On learning to teach proof in school mathematics. Paper presented at the 35th Conference of the International Group for the Psychology of Mathematics Education (Ubuz, B., editor), Ankara, Turkey: PME.

Herbst, P. G., Chen, C., Weiss, M., Gonzalez, G., Nachieli, T., Hamlin, M., et al. (2009). "Doing proofs" in geometry classrooms. In D. A. Stylianou, M. L. Blanton & E. J. Knuth (Eds.), The teaching and learning of proof across the grades (pp. 250-268). New York: Routledge.

Knuth, E. (2002). Secondary school mathematics teachers' conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405.

Lortie, D. C. (1975). Schoolteacher: A sociological study Chicago: University of Chicago Press.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: Author.