The National Council for Teachers of Mathematics (2000) states that reasoning and proof should be a standard in every classroom. Similarly, the Common Core State Standards for Mathematics (CCSSM, 2010) emphasizes justification as one of its key mathematical practices. Teaching with justification, however, is challenging. Previous studies have documented low levels of student justifications in U.S. classrooms (e.g., Jacobs et al., 2006) and specific challenges of enactment (Bieda, 2010). The lack of positive examples makes it imperative that we better understand pedagogical moves that do support student justification. Therefore, this study was guided by the question: ÒWhat teacher practices are associated with high levels of mathematical justification?Ó

Theoretical Perspective

Teaching to support studentsÕ justification is a complex and multifaceted activity. In seeking to understand teachersÕ pedagogy of justification, we draw on sociocultural theories as presented by Vygotsky (2002), Wertsch (1998), and others, and focus on teachersÕ discursive acts as a mediating tool and critical resource that teachers use as they make efforts to support, prompt and shape studentsÕ engagement with mathematics. This perspective attunes us to the moment-to-moment interactions, as these specific discursive moves can create space for studentsÕ justification or can close down productive discussions.

Methodology

This study is a collective longitudinal case study; each of the seven selected teachers operated as an intrinsic case study. We analyzed the implementation of the Number Trick task, a task requiring the use of the distributive property or equivalent reasoning. Data included video transcriptions, teacher reflections and student work.

Before coding, transcripts were split into episodes which included all discussion centered on a single question. Then, each student utterance and the episodes as a whole were coded, using two different coding schemes which emphasized the level of justification present. Interrater reliability was 93% when agreement was counted as two out of three researchers agreeing.

Summary of Findings

We will offer preliminary findings by reporting three factors that were identified as supporting studentsÕ use of justification: (1) attending to the mathematics of the students, (2) supporting students in articulating their mathematical ideas, and (3) creating a critical classroom culture where students support each otherÕs justifications. Only the third factor is addressed here. All three factors, and others identified through our continuing analyses, will be reported in the session.

Supporting a Critical Classroom Culture

A critical classroom culture is one where students question each other about their ideas. The excerpt in Figure 1 shows how students negotiated how to represent the first part of the task with an algebraic expression.

Figure 1. An example of a supporting a critical classroom culture.

In turn 1, Dan suggested that the first part of the number trick could be represented by n+5+9. This works when n=5, but not when n is any other number. The students negotiated this, with strategic guidance from the teacher, eventually concluding that this proposed representation of the trick will not work for all values of n.

Though we examined each factor in isolation, and can see how they are distinct, in reality a teacher uses more than one practice in a given moment. This study provides evidence that particular teaching practices may support studentsÕ use of mathematical justification. Further analyses will refine additional pedagogical strategies that are associated with high levels of student justification and focus on those pedagogical moves that support student justification, and not only student participation in classroom discourse.

References

Bieda, K. (2010). Enacting proof-related tasks in middle school mathematics: Challenges and opportunities. Journal for Research in Mathematics Education, 41, 351-382.

Council of Chief State School Officers & National Governors Association Center for Best Practices. (2010). Common core state standards. District of Columbia: Council of Chief State School Officers & National Governors Association Center for Best Practices.

Jacobs, J., Hiebert, J., Givvin, K., Hollingsworth, H., Garnier, H. & Wearne, D. (2006). Does eighth-grade mathematics teaching in the United States align with the NCTM Standards? Results from the TIMSS 1995 and 1999 video studies. Journal for Research in Mathematics Education, 36, 5-32.

National Council of Teachers of Mathematics (2000). Principles and standards of schools mathematics. Reston, VA: NCTM.

Vygotsky, L. S. (2002). Thought and language (13 ed.). Cambridge, MA: MIT Press.

Wertsch, J. V. (1998). Mind as action. New York: Oxford University Press.