Mathematical actions as procedural resources: An example from the separation of variables

Abstract

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Students learning to separate variables in order to solve a differential equation have multiple ways of correctly doing so. The procedures involved in separation include division or multiplication after properly grouping terms in an equation, moving terms (again, at times grouped) from one location on the page to another, or simply carrying out separation as a single act without showing any steps. We describe student use of these procedures in terms of Hammer's resources, showing that each of the previously listed procedures is its own "piece" of a larger problem solving activity. Our data come from group examinations of students separating variables while solving an air resistance problem in an intermediate mechanics class. Through detailed analysis of four groups of students, we motivate that the mathematical procedures are resources and show the issues that students must resolve in order to successfully separate variables. We use this analysis to suggest ways in which new resources (such as separation) come to be.

Using Framing as a Lens to Understand Context Effects on Expert Reasoning

“…The framing and resources theoretical framework has been used extensively in physics education research. Studies using this framework span physics and science instruction across K-12 (e.g., Elby and Hammer, 2010;Berland and Hammer, 2012) and at the university level (e.g., Hammer, 2000;Wittmann and Black, 2015). When attempting to name resources, investigations most often involve qualitative case studies of single individuals (e.g., Lising and Elby, 2005;Farlow et al, 2019) or small groups of subjects (e.g., Bing and Redish, 2008;Loverude, 2015), and rarely from free-response data (e.g., Goodhew et al, 2019).…”

Mapping activation of resources among upper division physics students in non-Cartesian coordinate systems: A case study

“…To more clearly communicate the flow of Mark's thinking and introduce our new resource nomenclature, we developed two toy models. The first consists of resource graphs [20] organized by the context of each question-one each for unit vectors, position vectors, and velocity vectors (Fig. 10).…”

Students' context-sensitive use of two kinds of conceptual resources for mechanical wave reflection

“…"Context," in this case, may include features of the question at hand and/or the social setting in which the question is asked [4,5,7,8]. Thinking involves activating a variety of resources of different kinds (e.g., conceptual, epistemological, and procedural) and the activation of these resources is intertwined [6,7,9]. Thus, the activation of different epistemological resources may, for example, cue the activation of different conceptual resources.…”