The Common Core State Standards for Mathematics (CCSS-M) state there is a “need to connect the mathematical practices to mathematical content in mathematics instruction” (National Governors Association and Council of Chief State School Officers [NGA/CCSSO], 2010, p. 8).  However, for many teachers, enacting the mathematical practices will require changes in their current practice. Similarly, students must rethink what it means to do mathematics in classrooms.   

This symposium will present how one Common Core Mathematical Practice, constructing arguments and critiquing the reasoning of others, is demonstrated across five different classroom settings. The settings vary from an elementary classroom to an undergraduate college calculus course, in which students, to varying degrees, “listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments” (NGA/CCSSO, 2010, p. 7). Together the five studies provide insights on how classroom culture and communication can be enhanced through this mathematical practice. The primary objective of this proposed session is to characterize how this mathematical practice may play out in actual classrooms and the barriers teachers may encounter as they incorporate it into their instruction. Our symposium is designed to help researchers consider how their work can guide classroom practice and to allow practitioners consider how the results of research can be valuable as they learn to teach in ways consistent with CCSS-M. We hope providing an open dialogue on these issues will help to connect the research and practitioner communities.

In the past 15 years, mathematics educators have commonly drawn on Toulmin’s (1958) work in rhetoric to analyze argumentation in mathematics classrooms. Our intention is to collectively extend such work to examine the factors that enhance or impede students’ argumentation in the mathematics classroom. For example, Mueller (2009) identified several factors that influence classroom cultures focusing on reasoning and justification, including a collaborative environment, task design, an emphasis on sense making, and interventions designed to shift authority from the instructor to peers.

The symposium will be organized as follows:

1. Overview of the symposium (5 minutes)
2. Individual research presentations (10 minutes each)
3. Remarks from discussant (5 minutes)
4. Audience discussion (30 minutes)

The audience discussion will be framed by the following questions:

1. What implications do these studies provide for teachers preparing to implement the CCSS-M?
2. How can research inform the implementation of the CCSS-M mathematical practices and what further research is necessary to ensure CCSS-M is enacted with fidelity?
3. What are the similarities and differences among these five research studies regarding communication and argumentation in different classroom contexts?
4. How do teacher developers improve teachers’ ability to promote the mathematical practices in their teaching?

Presentation 1: The first presenter examined a fifth grade classroom’s argumentation of zero being even or odd.  Grounded theory techniques were employed on video recordings of classroom discussions.  The dialogic interactions afforded particular insights about zero and students’ understanding of this number. Closely examining the dynamic discussions and the supports the classroom teacher employed for them shed light on how teacher educators can support students in the Common Core Mathematical Practice of constructing arguments and critiquing the reasoning of others.  Implications of what it means to implement this mathematical practice with fidelity as opposed to integrity will be explored.

Presentation 2: The second presenter will discuss how students in four middle-grade classrooms engaged in a project-based statistics unit. Using techniques from grounded theory to analyze video recordings and transcripts of students’ small group discussions, the presenter examined how students in small groups constructed statistical arguments. Findings include many characteristics of argumentation that are unique to statistics, such as students often 1) use oppositional platforms to construct statistical arguments; 2) fail to connect various aspects of the statistical investigative cycle; and 3) initially focus on individual data points, rather than the aggregate data set. Implications for teaching moves will also be shared.

Presentation 3: The next project examined communication within pairs of Algebra I students collaborating on rigorous tasks requiring understanding of representations of functions during a unit of study on linear functions. This presentation will focus on a task in which students were asked to apply and extend their understanding of linear functions to model a non-linear relationship using the various representations. The rigor of this task created a need for students to communicate about novel mathematics. The analysis reveals strengths and weaknesses in terms of students constructing arguments and critiquing others’ reasoning. The results will be discussed in terms of task design and classroom practice. 

Presentation 4: The fourth presenter examined how two high school teachers supported argumentation during implementation of a process-standards based state curriculum. Each teacher completed a survey, was interviewed three times, and had their classes observed and audio recorded; specific instances of their support for productive argumentation with their tenth graders were noted. These behaviors were attributed to their beliefs about teaching and learning and their opportunities to learn what it means to teach using the process standards. Although originally focused on process standards, examining the data through the lens of Mathematical Practice 3 reveals that teacher developers must help teachers understand what it means to teach content standards using process standards, and, soon, the mathematical practices.

Presentation 5: The final presenter investigated the prominence of Mathematical Practice 3 in a college classroom setting. Videotapes of classroom discourse were analyzed in order to determine the instructor’s effectiveness in fostering Mathematical Practice 3 among students in a second-semester calculus course. In addition, videotapes of students’ work outside of class were analyzed in order to characterize the translation of in-class argumentation to out-of-class interactions with peers. Results suggest that establishing an environment in which critical argumentation takes place regularly has a pronounced effect on students’ discourse outside of class, as well as on their conceptual understanding of calculus topics.

A nationally recognized mathematics educator informed on the CCSS-M will serve as a discussant. This symposium can (1) inform the preparation of teachers to implement the CCSS-M and (2) provide specific ideas for the implementing the mathematical practice of constructing viable arguments and critiquing the reasoning of others.

References

Mueller, M. F. (2009). The co-construction of arguments by middle-school students. Journal of Mathematical Behavior, 28, 138-149.

National Governors Association and Council of Chief State School Officers. (2010). Common core state standards initiative: Mathematics. Retrieved from http://www.corestandards.org/the-standards/mathematics

Toulmin, E. S. (1958). The uses of argument. London: Cambridge University Press.