Megan Wawro

When the Classroom Floor Becomes the Complex Plane: Addition and Multiplication as Ways of Bodily Navigation

In this article we contribute a perspective on mathematical embodied cognition consistent with a phenomenological understanding of perception and body motion. It is based on the analysis of 4 selected episodes in 1 session of an undergraduate mathematics class. The theme of this particular class session was the geometric interpretation of the addition and multiplication of complex numbers. On the basis of these episodes, the article examines 2 conjectures: (a) The mathematical insights developed by an individual or a group are expressed in and constituted by perceptuo-motor activity, and (b) the learning of mathematical ideas is shaped in nondeterministic ways by the setting or learning environment.

Reasoning Using Particulate Nature of Matter: An Example of a Sociochemical Norm in a University-Level Physical Chemistry Class.

In college level chemistry courses, reasoning using molecular and particulate descriptions of matter becomes central to understanding physical and chemical properties. In this study, we used a qualitative approach to analyzing classroom discourse derived from Toulmins model of argumentation in order to describe the ways in which students develop particulate-level justifications for claims about thermodynamic properties. Our analysis extends the construct of sociomathematical norms to a chemistry context in order to describe disciplinary criteria for reasoning and justification, which we refer to as sociochemical norms. By examining how whole class and small group discussions shape norms related to reasoning, we provide suggestions for teaching practices in inquiry-oriented settings.

An Example of Inquiry in Linear Algebra: The Roles of Symbolizing and Brokering

Abstract In this paper we address practical questions such as: How do symbols appear and evolve in an inquiry-oriented classroom? How can an instructor connect students with traditional notation and vocabulary without undermining their sense of ownership of the material? We tender an example from linear algebra that highlights the roles of the instructor as a broker, and the ways in which students participate in the practice of symbolizing as they reinvent the diagonalization equation A = PDP−1.

A hypothetical learning trajectory for conceptualizing matrices as linear transformations

ABSTRACT In this paper, we present a hypothetical learning trajectory (HLT) aimed at supporting students in developing flexible ways of reasoning about matrices as linear transformations in the context of introductory linear algebra. In our HLT, we highlight the integral role of the instructor in this development. Our HLT is based on the ‘Italicizing N’ task sequence, in which students work to generate, compose, and invert matrices that correspond to geometric transformations specified within the problem context. In particular, we describe the ways in which the students develop local transformation views of matrix multiplication (focused on individual mappings of input vectors to output vectors) and extend these local views to more global views in which matrices are conceptualized in terms of how they transform a space in a coordinated way.

Stretch Directions and Stretch Factors: A Sequence Intended to Support Guided Reinvention of Eigenvector and Eigenvalue

In this chapter, we document the reasoning students exhibited when engaged in an instructional sequence designed to support student development of notions of eigenvectors, eigenvalues, and the characteristic polynomial. Rooted in the curriculum design theory of Realistic Mathematics Education (RME; Gravemeijer, 1999), the sequence builds on student solution strategies from each problem to the next. Students’ used their knowledge of how matrix multiplication transforms space to engage in problems involving stretch factors and stretch directions. In working through these problems students reinvented general strategies for determining eigenvectors, eigenvalues, and the characteristic polynomial.

Finding Synergy Among Research, Teaching, and Service: An Example from Mathematics Education Research

Being a faculty member in higher education involves the balance and integration of various roles and demands. In this chapter I present my own story, as a mathematics education researcher in the teaching and learning of undergraduate mathematics focusing on linear algebra. Using my experience as an example, I describe how synergy among research, teaching, and service can impact career goals and institutional needs.