Investigating the Dimensionality of Teachers' Mathematical Knowledge for Teaching Secondary Mathematics Using Item Factor Analyses and Diagnostic Classification Models

Abstract

This study proposes a way of organizing mathematical knowledge for teaching that permits to reveal its multidimensionality. Scholars concerned with teachers’ mathematical knowledge have traditionally distinguished knowledge dimensions by knowledge types, such as mathematical content knowledge or pedagogical content knowledge (e.g., the MKT framework). This approach has been widely adopted in studies that measure teachers’ knowledge using assessment items. But it remains an open question whether these conceptualizations can lead to precise measures of the different domains, as it is highly likely teachers simultaneously use multiple knowledge types when teaching mathematics. This creates challenges in measuring only mathematical content knowledge not mixed with any pedagogical aspects but still used in the work of teaching. While this way of conceptualizing knowledge dimensions has allowed researchers to develop measures that reflect professional knowledge, it has been less adept to documenting whether and how the knowledge varies depending on the specific teaching assignments teachers have experience with. The challenge in developing distinct measures has motivated me to propose a new way to organize assessment items. I describe this new way in terms of an item blueprint that specifies the correspondence between the organization of the items and the dimensions of the knowledge purported to be measured by the items. The proposed item blueprint is then evaluated regarding its purposes: 1) to capture multiple aspects of teachers’ mathematical knowledge used in teaching; 2) to develop precise multiple measures reflecting the identified dimensions of knowledge. Ultimately, the developed measures were designed to allow a fine-grained description of the knowledge used in the work of teaching secondary mathematics. The proposed item blueprint uses two organizers: task of teaching and instructional situation. Task of teaching alludes to each of the activities that comprise the practice of a mathematics teacher (e.g., understanding students’ work). Instructional situation alludes to each of the types of mathematical work students are assigned within a course of study (e.g., doing proofs in geometry). Following the blueprint, I assigned each set of items to measure one knowledge dimension associated with one task of teaching and one instructional situation. By organizing the knowledge using these two organizers, the item blueprint allows a description of teachers’ knowledge with respect to the characteristics of the components of the work of teaching. With this conceptual rationale, the methodological feasibility of the item blueprint was evaluated by fitting item-factor models to the item responses collected from a nationally distributed sample of 602 U.S. practicing mathematics teachers. The distinctions among the factors were examined using model-comparison tests conducted under three different measurement models: structural equation modeling, item response theory, and diagnostic classification models. The results consistently showed that the majority of the hypothesized dimensions are statistically distinguishable by either or both of the organizers within and across both geometry and algebra courses of study. This distinction was further supported by different relationships with teachers’ educational background and teaching experience across the identified knowledge dimensions. By presenting an innovative item blueprint that is theoretically warranted and methodologically feasible, this study shows great promise for measuring multiple dimensions of teachers’ mathematical knowledge used in the work of teaching. It contributes to developing theory of mathematics teaching and to future item development for measuring knowledge used in professional tasks and instructional situations.