Abstract:

This exploratory three-group non-experimental study used multiple regression to explain the relationships among three depth-of-mathematics-knowledge measures with pedagogical content knowledge (PCK). More specifically, 1589 elementary teachers’ depth of content knowledge (DOK) and PCK scores were analyzed to determine the extent to which three mathematics DOK measures—(a) memorization, (b) understanding, and (c) problem solving or reasoning—explained variance in whole number and rational number content PCK. Prior to this study, researchers and theorists had described content knowledge in terms of depth or cognitive complexity or they categorized PCK. None had quantitatively measured relationships among teachers’ mathematics knowledge and PCK or assessed PCK in terms of levels of depth of mathematics knowledge. This study, which indicates that conceptual understanding predicts almost 60% of PCK, offers an important connection between content knowledge and PCK. Other findings included (a) teachers’ understanding of number concepts and algorithms was the best predictor of PCK for both whole and rational numbers; (b) whole number content knowledge was not a predictor of either rational number content knowledge or rational number PCK even though whole number PCK was correlated with rational number PCK; and (c) PCK can be described and understood in terms of depth of knowledge. The findings add to the current teachers’ mathematics knowledge. Not only can items be analyzed for cognitive complexity in terms of how they are written, but also in terms of how they are answered. Teachers’ responses can be analyzed in terms of depth of knowledge displayed. Did the teacher correct a student’s problem or a student’s misconception of the problem? The former demonstrates less cognitive complexity than the latter. Is the response rote, algorithmic, or conceptual? Because research (a) suggests that teachers not only need to be able to teach mathematics content, but do so at a high conceptual level and (b) promotes PCK as a means to develop effective teachers, these results have clear implications for teacher educators, professional development providers, and teachers as they seek to increase student achievement in mathematics.

Keywords:

depth of knowledge, pedagogical content knowledge, mathematics knowledge, teacher.