Reconsidering the Organization of Mathematical Knowledge for Teaching

• In Event: 21.050 - Developing Measures of Mathematical Knowledge for Teaching for Secondary Teachers: Current Research and Open Questions

Abstract

We describe efforts to understand the organization of mathematical knowledge of secondary mathematics teachers through explorations of the dimensionality of sets of items designed to measure MKT for high school geometry and algebra content. On the one hand, the conceptualization by Author and colleagues (2008) distinguishes between mathematical knowledge used (a) only by teachers and (b) by teachers and non-teachers alike. This conceptualization has helped researchers develop a reliable measure of MKT for geometry (Author and colleague, 2014). On the other hand, Speer, King, & Howell (2014) have questioned whether the definitions for the MKT domains provided by Author et al. (2008) fit the knowledge of secondary teachers, and researchers have struggled to empirically show CCK items and SCK items as measuring distinct latent traits. In this paper, we present an alternative way of organizing secondary teachers’ mathematical knowledge for teaching high school geometry and algebra that preserves the fundamental insight of Author et al.’s (2008) identification of SCK while also being sufficiently nuanced to account for differences among secondary mathematics teachers.
Because MKT describes the mathematical knowledge used in teaching, our items are posed in scenarios that describe a teacher engaged in the work of teaching mathematics. Hence, we organize MKT items along two categories appearing in those scenarios: the task of teaching that the teacher is doing and the instructional situation that the teacher is managing or overseeing. By task of teaching we mean a general description of what the teacher is doing: creating problems for students to solve, observing students’ work on problems, responding to students’ work on problems, explaining new ideas, etc. By instructional situation (Author, 2006) we mean the types of mathematical work that students do in particular courses of study: In algebra, these include solving, graphing, simplifying, calculating; in geometry, these include constructing, doing proofs, calculating, making conjectures.
Our research on items classified according to two tasks of teaching and seven instructional situations shows promise for measuring teachers’ distinguishable knowledge domains. The two organizers can be used to detect dimensionality among a set of items. A structural analysis of these items shows that a five-factor model (where all the factors reflect different categories of instructional situations) is no worse than a four-factor model (where two of the factors are combined), and is better than models with less factors. Moreover, factors associated with situations in the same course of studies (such as within high school geometry; e.g., geometric calculation, making conjectures) are more highly correlated (.83) than items across courses of study (e.g., geometric calculation, solving equations; .68). Together, these findings suggest that items developed around this organizational scheme identify distinguishable dimensions of teachers’ MKT.
Overall, our approach suggests that it might be possible to operationalize in different ways the definitions of domains of MKT so as to account for both the variation among the general population in what is considered to be CCK and the variation among teachers with different experience in what is considered to be SCK and PCK.

Authors

• Patricio G. Herbst, University of Michigan - Ann Arbor

• Inah Ko, University of Michigan - Ann Arbor