• Vol. 45, No. 5, November 2014

    Elise Lockwood
    In many STEM-related fields, graduating doctoral students are often expected to assume a postdoctoral position as a prerequisite to a faculty position, yet there is no such expectation in mathematics education. This phenomenon is likely due in large part to an abundance of faculty positions; however, it may also result from the field’s perspective on postdoctoral positions. In this commentary, we call on the mathematics education research community to consider the importance of postdoctoral fellows, and we make the case that prioritizing postdoctoral positions could afford mutual benefits to the postdocs, to faculty mentors, and to the field at large.
    Natalie E. Selinski
    The central goals of most introductory linear algebra courses are to develop students’ proficiency with matrix techniques, to promote their understanding of key concepts, and to increase their ability to make connections between concepts. In this article, we present an innovative method using adjacency matrices to analyze students’ interpretation of and connections between concepts. Three cases provide examples that illustrate the usefulness of this approach for comparing differences in the structure of the connections, as exhibited in what we refer to as dense, sparse, and hub adjacency matrices. We also make use of mathematical constructs from digraph theory, such as walks and being strongly connected, to indicate possible chains of connections and flexibility in making connections within and between concepts. We posit that this method is useful for characterizing student connections in other content areas and grade levels.
    Anne Garrison Wilhelm
    This study sought to understand how aspects of middle school mathematics teachers’ knowledge and conceptions are related to their enactment of cognitively demanding tasks. I defined the enactment of cognitively demanding tasks to involve task selection and maintenance of the cognitive demand of high-level tasks and examined those two dimensions of enactment separately. I used multilevel logistic regression models to investigate how mathematical knowledge for teaching and conceptions of teaching and learning mathematics for 213 middle school mathematics teachers were related to their enactment of cognitively demanding tasks. I found that teachers’ mathematical knowledge for teaching and conceptions of teaching and learning mathematics were contingent on one another and significantly related to teachers’ enactment of cognitively demanding tasks.
    Charles Munter
    This article introduces an interview-based instrument that was created for the purposes of characterizing the visions of high-quality mathematics instruction of teachers, principals, mathematics coaches, and district leaders and tracking changes in those visions over time. The instrument models trajectories of perceptions of high-quality instruction along what have been identified in the literature as critical dimensions of mathematics classroom practice. Included are a description of the methods by which an analysis of interview data was integrated with previous findings from the research literature in order to develop leveled rubrics for assessing visions of high-quality mathematics instruction, a report of the results of using the instrument to code more than 900 interviews, and a discussion of the possible applications and benefits of such a methodological approach.