Making context explicit in equation construction and interpretation: Symbolic blending

Much of physics involves the construction and interpretation of equations. Research on physics students’ understanding and application of mathematics has employed Sherin’s symbolic forms or Fauconnier and Turner’s conceptual blending as analytical frameworks. However, previous symbolic forms analyses have commonly treated students’ in-context understanding as their conceptual schema, which was designed to represent the acontextual, mathematical justification of the symbol template (structure of the expression). Furthermore, most conceptual blending analyses in this area have not included a generic space to specify the underlying structure of a math-physics blend. We describe a conceptual blending model for equation construction and interpretation, which we call symbolic blending, that incorporates the components of symbolic forms with the conceptual schema as the generic space that structures the blend of a symbol template space with a contextual input space. This combination complements symbolic forms analysis with contextual meaning and provides an underlying structure for the analysis of student understanding of equations as a conceptual blend. We present this model in the context of student construction of non-Cartesian differential length vectors. We illustrate the affordances of such a model within this context and expand this approach to other contexts within our research. The model further allows us to reinterpret and extend literature that has used either symbolic forms or conceptual blending.

Figure 1

(a) Conventional spherical coordinate system; (b) the “schmerical coordinate” system, an unconventional spherical coordinate system with radial component $M$ and angles $\alpha$ and $\beta$, that was presented to students in the task discussed. The correct differential length vectors for each system are shown in (c) and (d), respectively. Image from Ref. [9].

Figure 2

Adam added “$M$” in his second term to include units of length when constructing the differential length vector. We identify this as the use of the coefficient symbolic form.

Figure 3

Basic diagrams depicting conceptual blending. (a) Generic model of a conceptual blend. Image reproduced from Ref. [11]. (b) Model for the boxing CEO blend. Adapted from Ref. [11].

Figure 4

Symbolic blending model for student equation construction and interpretation that incorporates symbolic forms elements. The contextual knowledge to be expressed as an equation forms one input space. The symbol template is a second input space, while the conceptual schema is the underlying generic space. The resulting blended space is the represented equation.

Figure 5

Symbolic blend for Elliot and Frank. They first constructed the individual components using magnitude-direction, then added each component together, since the differential length vector included multiple terms for each dimension.

Figure 6

Symbolic blending diagram for Carol and Dan as they began construction of the differential length vector. They started with a combined parts of a whole and magnitude-direction template. Next, they used the idea of length to fill in the magnitude term before they represented it as a differential.

Figure 7

Symbolic blending diagram for Elliot for the differential template with varied conceptual understanding. Here Elliot used the differential as a small quantity rather than a change in a quantity.

Figure 8

Symbolic blending diagram for Adam and Bart including the no dependence symbolic form, with which they acknowledged the alpha component does not include a beta term.

Figure 9

Interpreting symbolic forms for integration using conceptual blending, based on Ref. [7]. Depending on the conceptualization of integration by the student pair as either (a) adding up pieces or (b) area and perimeter, students attached different meaning to each part of the integral template.

Figure 10

An example of symbolic blending from Adam and Bart involving recall and backward projection during their pair interview. In place of conceptual understanding (right), Adam and Bart connected the coordinate system to spherical coordinates and attempted to deconstruct the spherical differential length vector (left) but did not know why the sine function was included. Without the underlying understanding, they pulled the sine function into their schmerical construction because of how the differential length is written in spherical coordinates (right).

Figure 11

Figure provided for the charged sheet task [47] in which students were asked to determine the electric field at a distance $x$ from the center of a uniformly charged sheet. The task required students to construct a differential area to describe the surface.

Figure 12

Symbolic blending for charged sheet task where Jake added differential length terms instead of multiplying the terms. We classify this as Jake invoking the incorrect template.

Figure 13

Comparison of symbolic blending diagrams for Molly (a) and Lenny (b) for the student’s invocation of different differential length elements. Molly correctly invoked the magnitude-direction form while Lenny focused on $\theta$ having a direction and used the vector symbol which is nested within the differential symbolic form $d\square$.