Mathematical Representations: Instructional Challenges and Insights

Purpose

            While there is a long history in considering the ways in which particular representations can support and constrain mathematical understanding as well as a history of professional development documenting teachers' changing beliefs and practices, little attention has been paid to developing teachers' use and understanding of mathematical representations. The goal of this paper is to describe the ways in which elementary teachers, engaged in a professional development focused on representation, consider the affordances of representation in mathematics teaching.

Research Questions

1.     What challenges do teachers encounter when considering the role of representation in mathematics teaching and learning of rational number?

2.     What patterns of teacher thinking are apparent in the way they negotiate each of these challenges?

Perspectives

            Research has documented the important role of representation in mathematics teaching and learning. Affordances and constraints of fractions (e.g. area models, number lines, and discrete models), has received a considerable amount of attention related to mathematical understanding (Behr, Lesh, Post, & Silver, 1983; Carpenter & Fennema, 1992; Saxe, Taylor, McIntosh, & Gearhart, 2005). 

            Professional development is often considered an important, if not essential, tool for improving mathematics instruction in classrooms, and research has shown it can impact teachers' classroom practice (Franke et al., 1998, 2001). Although there is evidence of successful professional development about many areas of mathematics, even professional development that was motivated through research on learning (Carpenter et al., 1989, 1996), there is little research on professional development specifically focused on considering the use of representations in mathematics teaching and learning. This work seeks to provide some guidance by examining teachers' thinking during professional development designed to further teachers' understanding of the role of representation in mathematics instruction.

Methods and Data Sources

            We analyzed data from three professional development sessions focused on supporting teachers' consideration of the affordances of mathematical representations. Fourteen female elementary school teachers from 6 schools participated in monthly sessions during the school year.  The sessions were structured by the Reflection Connection Cycle (see Taylor, 2011). First teachers read articles, taught new lessons, and wrote reflections.  During the sessions, teachers watched a sample lesson, had large and small group discussions, and planned lessons in grade-level groups.

            We used an open coding process with teachers' monthly reflections and transcripts of audio recordings of small and whole group discussion to identify potential insights and challenges teachers were encountering. After initial coding scheme was developed, we developed a set of focused codes. Three major themes were identified: negotiating students' mathematical autonomy, supporting students' mathematical competence, and balancing instructional goals for incorporating representations.

Results

            Data indicate that many teachers expressed difficulties negotiating students' mathematical autonomy when using multiple representations. In many cases, teachers reported their students' resistant to use of unfamiliar representations, and a few teachers noted that often the highest achieving students were most resistant.  They often had to consider whether they should allow students to choose the representation they preferred, or expose them to many different representations.  One teacher explained this dilemma as, "What proved difficult to me was deciding how much I would orchestrate the student's use of forms. That is, what balance could I strike between requiring the use of certain forms and allowing the students to choose?" Some teachers addressed this dilemma by devising pedagogical strategies that encouraged students to use more representations, while allowing students mathematical autonomy.   

            While teachers noticed that their students had difficulties using new representations, students' struggles did not align with teachers' perception of their mathematical competence.  Almost half of the teachers noticed that students who struggled to use multiple representations were often the strongest mathematics students or that lower-performing students saw more success than usual.  One teacher commented on the reluctance of her best students, "the most reluctant to try the suggested strategy are often the 'top' performing students who pride themselves in their procedural knowledge and adeptness." The teachers' reflections signal that in many classrooms, using different representations may be a way for teachers to differentiate instruction such that successful students experience challenges and lower-performing students experience success. Curiously, these observations did not lead any teachers to question the mathematical proficiency of the "best" students.

            Teachers' reflections showed an extremely wide range of instructional goals, including supporting specific mathematical content, development of important mathematical practices, and supporting their own pedagogical goals related to using multiple representations in mathematics lessons.  Most teachers reported using different representations in their classrooms to further the mathematical understanding of their students (e.g. unitizing or equivalence in fractions). Almost all of the teachers used multiple representations to support valued mathematical practices in their classrooms, (e.g. facility and efficiency in problem solving and flexibility in representation use). For example, one teacher felt strongly about her use of the empty number line, writing, "I believe the empty number line to be a powerful tool to promote development of a 'mental number line' that encourages strategic flexibility, perhaps warding off the rigidity that some students seem to develop as they learn about, then increasingly adhere to, an algorithm for solving problems." Finally, teachers also incorporated multiple representations for pedagogical purposes, such as accessing student thinking or understanding about mathematics. This range in goals shows the complexity of incorporating multiple representations into teaching practice.

Education/Scientific Importance

            Although research has made advances in understanding representation in mathematics teaching and learning, this study has highlighted the major challenges teachers faced when trying to incorporate multiple representations into mathematics instruction. Mathematics educators may benefit from understanding the key challenges teachers face as they negotiate representation use.  In particular, mathematics educators should consider how to guide teachers in exposing students to multiple representations of mathematical concepts without sacrificing students' mathematical autonomy and how teachers can successfully attend to multiple instructional goals for incorporating different representations. Finally, mathematics educators might be able to encourage teachers to reexamine their labeling of students as mathematically proficient because, as this study suggests, it was often students who have success in traditional mathematics that had the greatest challenge using different representations.

References

Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes, (pp. 91–126). New York: Academic Press.

Carpenter, T. P., Fennema, E. (1992). Cognitively guided instruction: Building on the knowledge of students and teachers. In W. Secada (Ed.) Researching educational reform: the case of school mathematics in the United States [Special Issue]. International Journal of Educational Research, 17, 457-470.

Carpenter, T. P., Fennema, E., & Franke, Megan L. (1996). Cognitively Guided Instruction: A Knowledge Base for Reform in Primary Mathematics Instruction. The Elementary School Journal, 97(1), 3-20.

Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C.-P., & Franke, M. L.. (1989). Using Knowledge of Children's Mathematics Thinking in Classroom Teaching: An Experimental Study. American Educational Research Journal, 26(4), 499-531.

Franke, M. L., Carpenter, T. P., Fennema, E., Ansell, E., & Behrend, J. (1998). Understanding teachers' self-sustaining, generative change in the context of professional development. Teaching and Teacher Education, 14(1), 67–80.

Franke, M. L., Carpenter, T. P., Levi, L., & Fennema, E. (2001). Capturing teachers' generative change: A follow-up study of professional development in mathematics. American Educational Research Journal, 38(3), 653-689.

Taylor, E. V. (2011). Supporting children's mathematical understanding: professional development focused on out-of-school practices. Journal of Mathematics Teacher Education.