The field of mathematics education research lacks a clear understanding of exactly what mathematical knowledge teachers need in order to teach well (Sowder, 2007).  Part of the problem may be a lack of clear specifications of what teacher mathematical knowledge is (Shulman, 1986).  The Learning Mathematics for Teaching (LMT) research group has developed a theoretical model for Mathematical Knowledge for Teaching (MKT) at the elementary level, making a strong case that this MKT is multi-dimensional, informing the structure of elementary teacher certification programs, and providing a basis on which to build meaningful tests of teacher knowledge (See, for example,  Ball, Thames, & Phelps, 2008; Ball, Lubienski, & Mewborn, 2001; Ball & Bass, 2000). The nature of teacher mathematical knowledge at the secondary level is less well explored, and research has raised the question of whether MKT constructs are materially different at the secondary level.  Speer & King (2009) suggest that what should be considered specialized might be different in post-secondary contexts.  Hill et al. (2007) express concern that the theoretical mapping of the domain is incomplete, and that the proliferation of different frameworks for secondary MKT may undermine coherence in the field.  A strong theoretical framework for MKT at the secondary level would ground teacher education and teacher-knowledge assessments as it did at the elementary level.  This study is a small-scale qualitative study, modeled on the proven methodology associated with the LMT group’s foundational work in MKT for elementary teaching, and is envisioned as an initial step in developing such a framework.

Research Question

How can we characterize the mathematical knowledge needed for a secondary teacher to teach exponents?

Research Design

The study employs case study methodology and uses a cyclical analysis model (also called a constant comparative model). The National Research Council (NRC) describes such descriptive studies as appropriate for the development of new theory or conjecture.  Rigor in such studies depends on purposeful sampling, addressing bias, collecting multiple types of data and working with other researchers throughout the analysis process (Committee on Scientific Principles for Education, 2002).   This study design borrows from the established methodology used by Ball and other LMT researchers in their initial work on MKT for elementary teachers, although the methodology was refined significantly to accommodate the smaller scope of the study.

Data Collection

Data were collected during an instructional unit spanning approximately 3.5 weeks of time, in a classroom selected because it met the criteria specified by Thames (2009) for opportunity to observe meaningful interaction around mathematics.  Data collected included lesson plans for each observed class session, video-recordings of each observed class, audio-recordings of daily video-prompted interviews with the teacher, and researcher notes from the classroom observations and from the teacher interviews.

Data Analysis

The focus of the analysis was on the mathematics demanded of the teacher by the work of teaching the instructional unit on exponents.  The first analytical step was performed in tandem with data collection as the researcher scanned each video recording to identify potential events around which to focus the teacher interviews.  The video-record was broken into analytic units called moments of mathematical intervention (MMI), which represented the teacher’s decision points.  Such moments included tasks that the teacher selected or teacher-presented mathematical ideas, but also included interventions such as choosing to focus students on a particular idea that had been raised or to reframe it in different words, choosing to stay with an idea or to move away from it, and adopting or disregarding student suggestions about how to proceed mathematically.

The remaining analytic steps consisted of line-by-line analysis of the transcript and associated video for each of the identified MMI.  Each cyclical pass through the data focused on a slightly more general analysis question as listed below:

1. What mathematical knowledge did this teacher use in this particular situation?
2. What mathematical knowledge might a teacher require to respond to such a situation?
3. How would the LMT model characterize such knowledge and how else might it be productively characterized?

Results

Results of the study include a detailed set of mathematical ideas around exponents, accompanied by links between these ideas and descriptions of how these ideas might be characterized.  Specific mathematical knowledge that emerged during analysis included a thorough understanding of various definitions of exponents (positive, negative, whole number and rational), how these definitions fit together and can be thought of as extensions over various domains, where they cannot be extended and the reasons and consequences.  Also included was knowledge of the exponent rules, when they are domain dependent, how can it be determined whether a rule is true, and how to manipulate exponential expressions.  More general emergent mathematical themes included what constitutes a “good” definition, what it means for something to be undefined and how this differs from the notion of indeterminate, how the justification for a definition differs from that of proof, and what constitutes proof in a mathematical community.  Many of these ideas can be characterized as knowledge about mathematics, disciplinary or socio-mathematical norms, or mathematical practices, which are dimensions that are included but not well characterized by the LMT’s elementary MKT model, and which some might not describe as content knowledge at all.  While a single case study is insufficient for generalization, these characterizations provide clues about the shape of a framework for secondary MKT, suggest that such a framework will not be identical to the LMT’s elementary MKT framework, and suggest that, especially at the secondary level, a clearer understanding of how disciplinary knowledge affects teaching and how teachers come to acquire it is essential.

Presentation

This proposal is submitted for an interactive paper session.  As required by the format, a 15-minute overview of the study will be given which will provide an overview of results and will touch on one or two examples.  For the roundtable discussions, one or two more detailed mathematical examples from the results will be provided for discussion among participants.

References

Ball, D., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching & learning (pp. 83-104). Westport, CT: Greenwood Publishing Group.

Ball, D., Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers' mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (4th ed., pp. 433-456). Washington, D.C.: American Educational Research Association.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.

Committee on Scientific Principles for Education. (2002). In Shavelson R., Towne L. (Eds.), Scientific research in education Washington, D.C.:National Academy Press

Hill, H., Sleep, L., Lewis, J., & Ball, D. (2007). Assessing teachers' mathematical knowledge. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 111). Charlotte, NC: Information Age Publishing.

Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4

Sowder, J. (2007). The mathematical education and development of teachers. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 157). Charlotte, NC: Information Age Publishing.

Speer, N., & King, K. D. (2009). Examining mathematical knowledge for teaching in secondary and post-secondary contexts. Proceedings for the Twelfth Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education, Raleigh, North Carolina.

Thames, M. H. (2009). Coordinating mathematical and pedagogical perspectives in practice-based and discipline- 
grounded approaches to studying mathematical knowledge for teaching (K-8) 
. Unpublished PhD, University of Michigan, Ann Arbor, MI.