Teachers’ Knowledge of Mathematical Definitions

• In Event: Defining and Measuring What Math and Science Teachers Need to Know: Implications for Professional Development

Abstract

The purpose of this study is to examine teachers’ perceptions on what they need to know about mathematical definitions and what is actually required for them to be effective. Teachers’ knowledge of mathematical definitions and their concept images affect their instructional decisions, the explanations they provide in the classroom, the way they guide their students, and how they conduct mathematical discussions (e.g., Zazkis & Leikin, 2008). However, their knowledge of equivalent definitions of mathematical concepts or their perceptions of the knowledge of definitions may not match with the actual knowledge that supports their flexibility and fluency in a mathematics classroom. Furthermore, much remains unknown about the specific knowledge required to do the mathematical task of teaching involving definitions such as to inspect definitions given by textbooks, to establish the equivalence of alternative definitions for a given concept, and to choose and develop useable definitions (Ball, Thames, & Phelps, 2008).
The Mathematical Knowledge for Teaching (MKT) framework (Ball et al., 2008; Hill, Rowan, & Ball, 2005) categorizes teachers’ content knowledge into three sub-domains: Common Content Knowledge (CCK), Specialized Content Knowledge (SCK), and Horizon Content Knowledge (HCK). We focus on the knowledge of mathematical definitions that is held by teachers. This goes beyond the ability to recite the actual definitions and into the area of understanding variations of definitions (SCK), whether congruent or non-congruent (Usiskin & Griffith, 2008), and understanding mathematically accurate yet useful definitions and its trajectory (HCK).
To gauge teachers’ perceptions and understanding of mathematical definitions and their role in teaching and learning, this paper draws from research studies conducted in Indonesia (Author2, 2011) and Norway (Author1, in press; Authors1, 2011; Authors2, 2009) that examined adaptability of the U.S. developed MKT measures. Qualitative analyses of focus group interviews with teachers in these countries and quantitative results based on psychometric analyses of teachers’ responses on the MKT measures were conducted to examine their perceptions and understanding of mathematical definitions.
Results from this study show that there is a gap in teachers’ knowledge of what constitute mathematical definitions; teachers confuse between formulas, rules, and definitions. Teachers in this study do not perceive the need of remembering definitions or to appraise different definitions or dispute the authority of textbooks. They may not even be aware that multiple mathematical definitions exist. In general, they do not foresee the role of mathematical definitions in teaching or to understand multiple definitions and their connections to students’ learning trajectory.
Leikin and Zazkis (2010) found that prospective mathematics teachers’ knowledge of definitions is situated in content domain of mathematics, and reflects the nature of school mathematics textbooks and of the school curriculum. They also found a gap between the mathematics learned in university courses and school mathematics. Therefore, there is a need for teacher educators to understand the gap between how teachers perceive what knowledge of mathematical definitions they need to know and what they actually need to know to be effective in their teaching in order to support preservice and inservice teachers’ learning.

Authors

• Reidar Mosvold, University of Stavanger

• Dicky Ng, Utah State University

• Arne Jakobsen, University of Stavanger

• Janne Fauskanger, University of Stavanger

• Raymond Bjuland, University of Stavanger