Researchers (e.g., Hill, Ball, & Schilling, 2008) have posited frameworks and constructs designed to illuminate the nature of elementary teachers’ mathematical knowledge and its role in the teaching of mathematics. Recent policy discussions have assumed the impact of teachers’ mathematical knowledge on classroom instruction, and beginning teachers are an interesting population for this discussion given the recency of their collegiate mathematics experience (e.g., Heid, Karunakaran, Kinol, & Grady, 2010). We focus on secondary mathematics teachers and choose to explicitly define mathematical knowledge as that which is evidenced by participation in mathematical processes and work with the products of those processes (Zbiek, Peters, & Conner, 2008). We take as mathematical knowledge four mathematical processes and the products of those processes (justifying and justification, defining and definition, representing and representation, and generalizing and generalization), each of which has been the focus of mathematics education research and theory. Our study focused on the use of these processes and products in one beginning teacher’s personal mathematics and in her classroom mathematics.

Research Question

What factors influence the use of mathematical processes in classroom teaching by a beginning secondary mathematics teacher who ably used mathematical processes in her personal mathematics?

Methods, Data Sources, and Analyses

Fiona is a secondary mathematics teacher whom we observed teaching algebra and geometry during her first two years of teaching. In each of the five 1-2 hour task-based interviews with Fiona during the last two years of her teacher preparation program, she engaged ably in mathematical processes, but these processes did not play a prominent role in our week-long observations each year of her teaching or in her reflections on those lessons in follow-up interviews.

Line-by-line, the research team coded and annotated transcripts of the task-based interviews for Fiona’s use of mathematical processes and products. Pairs of researchers coded transcripts of teaching observations and follow-up interviews for use of mathematical processes and products as well as other mathematical activity and pedagogical choices and reflections. Disagreements were resolved by review of the data by the entire team. The research team then examined the coding for emergent themes that would help to illuminate Fiona’s use of processes in her classroom mathematics, modifying and adding themes as they emerged.

Emerging Themes

Several themes characterize Fiona’s classroom teaching: thorough planning and organization, use of tasks that reflect the sequence of topics in the textbook rather than the learning goals, precise control over the implementation of the selected tasks, constant student involvement in an activity related to the mathematical topic, and a particular perception of teaching and learning. Fiona pre-plans every part of her lessons, starting the year with a binder that contains each activity she will use in each class in the order in which she will use them. Once tasks have been selected and planned for, Fiona seems to rely on the task itself to carry the mathematical content of the lesson.  She orchestrates her lessons by shepherding her students to produce the exact answers she has identified in the extensive answer keys in her binders. There is no “down time” in Fiona’s classroom. She focuses on students being busy at all times performing tasks that are directed/initiated by her, continually interacting with students as she presents new material, directing student work on sample exercises, leading them in hands-on activities, or engaging them in informal discussions on non-mathematical topics. Finally, Fiona’s apparent view of teaching and learning, that everything should be presented to students and that there should be no complications or problematic discussions in class, influences the classroom mathematics provided for her students. It is unclear whether this tendency is a result of Fiona believing that students are unable to cope with complexity or whether it is a result of a view of teaching and learning in which the job of the teacher and task is to make learning easy for students.

Effect of Emerging Themes on Classroom Mathematics

The emergent themes influence the use of mathematical processes in Fiona’s classroom.  In Fiona’s classroom mathematics, we see some use of definition but very limited engagement with defining.  As a representative example, when Fiona wants students to have the definition of prism to write on their notesheets, she asks a series of questions and uses words and phrases from student responses to shape the definition she has predetermined.  Fiona wants the class to verbalize a final definition that matches the one she identifies because she already has the definition written on a poster and her view of teaching and learning seems to require that students have the exact wording presented to them.  This representative episode serves Fiona’s purpose of having students constantly engaged and having her students exposed to a correct definition.  Fiona exerts such strict control over the discussion that students’ opportunity to engage in defining is very limited.  

Summary

Fiona ably uses mathematical processes in her personal mathematics.  The fact that processes do not play a prominent role in Fiona’s classroom mathematics cannot be attributed solely to her personal mathematics.  Other emerging themes related to Fiona’s role as a teacher and her view of teaching and learning had major influence on Fiona’s classroom mathematics.  Although recent research and policy discussions have focused on the strong impact of teachers’ mathematical knowledge on classroom instruction, it is important to consider other factors that may also be influencing classroom instruction.

Time Use

After presenting a 15-minute overview of the study, we will use the round table sessions to generate discussion of possible factors influencing teachers’ use of mathematical processes in classroom mathematics.

 

References

Heid, M. K., Karunakaran, S., Kinol, D., & Grady, M. (2010). Interplay of Beliefs and Processes in the Classroom Mathematics of a Novice Secondary Mathematics Teacher. In P. Brosnan, D. B. Erchick, & L. Flevares. (Eds.) Proceedings of the 32nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1021-1029). Columbus, OH: The Ohio State University.

Hill, H.C., Ball, D. L., & Schilling, S. (2008). Unpacking “pedagogical content knowledge”: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372-400.

Zbiek, R. M., Peters, S., & Conner, A. (2008). A mathematical processes approach for examining teachers’ mathematical understandings. Unpublished manuscript.