Students’ use of symbolic forms when constructing differential length elements

Abstract

As part of an effort to examine students' understanding of the structure of non-Cartesian coordinate systems and the differential vector elements associated with these systems, students in junior-level electricity and magnetism (E&M) were interviewed in pairs. Students constructed differential length and volume elements for an unconventional spherical coordinate system. A symbolic forms analysis found that students invoked known as well as novel symbolic forms when building these vector expressions. Further analysis suggests that student difficulties were primarily conceptual rather than symbolic.

Chemistry and Mathematics: Research and Frameworks To Explore Student Reasoning

“…As stated by Cooper and Stowe, “...there is little to be gained by simply cataloging misconceptions without paying heed to the mechanisms of their emergence, their organization, and their character.” From a theoretical perspective, the resources framework provides the language and the lens to make the transition from focusing on what students do not know to considering how students can use what they do know productively. Recent work situated using the resources perspective has yielded rich data across topics in chemistry ,,,− and more broadly in physics and mathematics. − In the sections that follow, the resources perspective is used to frame the remainder of the discussion, in which each of the analytic frameworks can be viewed as tools to characterize mathematical resources (e.g., symbolic forms) or the interaction between resources or groups of resources (e.g., blended processing). To illustrate this idea, a resource graph is provided below in Figure , which illustrates the connections among resources; , for examples of how resource graphs have been used as a data visualization tool, see recent work by the authors. , As shown in Figure , symbolic forms, graphical forms, and covariational reasoning can be framed as examples of mathematical resources, and when these resources are combined with resources related to chemistry, conceptual integration (blended processing and engaging in mathematical narratives) occurs between the chemistry and mathematics domains.…”

“…On the basis of the nature of the problem-solving scenario, most of the symbolic forms identified by Sherin were related to algebraic manipulations (e.g., prop- , the meaning encoded in a reciprocal relationship) or were implicitly more relevant for physics contexts (e.g., balancing , terms or groups of terms on opposite sides of the equation are equivalent and reflect a balance of opposing influences). However, more recently, the symbolic forms framework has been used in many contexts that move beyond classical mechanics, describing a range of reasoning, including advanced mathematical ideas such as the meaning encoded in integrals or vector notation. ,,− ,,, …”

“…However, more recently, the symbolic forms framework has been used in many contexts that move beyond classical mechanics, describing a range of reasoning, including advanced mathematical ideas such as the meaning encoded in integrals or vector notation. 10,17,[64][65][66][67][68][69][70]80,87,88 Building on the idea of symbolic forms, graphical forms is a logical extension that provides a lens to characterize ideas associated with patterns in graphs. In his original work and in a subsequent paper, Sherin 86,89 suggested the existence of graphical reasoning analogous to symbolic forms; however, this idea was not developed further or explored in the literature.…”