Categorical framework for mathematical sense making in physics

We present a framework designed to help categorize various sense making moves, allowing for greater specificity in describing and understanding student reasoning and also in the development of curriculum to support this reasoning. The framework disaggregates between the mechanisms of student reasoning (the cognitive tool that they are employing) and what they are reasoning about (the object). Noting that either the tool or object could be mathematical or physical, the framework includes four basic sense making modes: Use of a mathematical tool to understand a mathematical object, use of a mathematical tool to understand a physical object, use of a physical tool to understand a mathematical object, and use of a physical tool to understand a physical object. We identify three fundamental processes by which these modes may be combined (translation, chaining, and coordination) and present a visual representation that captures both the individual reasoning modes and the processes by which they are combined. The utility of the framework as a tool for describing student reasoning is demonstrated through the analysis of two extended reasoning episodes. Finally, implications of this framework for curricular design are discussed.

Figure 1

A representation of the structure of activity, as described by activity theory. The upper (solid) triangle represents the mediated—by a tool—and direct pathways by which a subject interacts with an object, and is the unit of mediated cognition discussed by Vygotsky and others [24, 26]. The other nodes in the triangle show the larger activity system, highlighting the connections between individual cognition and the larger sociocultural history in which the current activity is embedded.

Figure 2

The four reasoning structures (modes) of the categorical sense making framework. These modes differ based on the object of the sense making and the tool employed in the sense making.

Figure 3

Left: a circuit diagram used as an evaluation of the conceptual circuit model developed by Shaffer and McDermott [31]. Right: a homework question from the Tutorials in Introductory Physics. Students are asked to rank the resistances of the boxed circuit elements and are explicitly told to not use math in their explanations. [33].

Figure 4

Left: a diagram of the angled-ground projectile problem, drawn here for clarity. Right: the solutions to part f) of the angled-ground projectile problem, as presented by Hahn [19].

Figure 5

Student work from Kuo et al. [21] exemplifying a “plug and chug” type of solution.

Figure 6

An example of a sense making diagram. These diagrams show the individual atoms of reasoning as well as the processes by which they come together to form a molecule of sense making. Each individual shape (triangle, triangle with node, square, etc.) represents a unique reasoning structure, which can be either a local instance of a single mode, or be built up of many modes through the processes of translation, chaining, and coordination.

Figure 7

Qualitatively accurate plots of KE vs $f$ (left) and KE vs $\lambda$ (right).

Figure 8

Student work by Amanda. The leftmost picture is a scan of her actual work, showing her initial answer (the crossed-out plots) and her final answer. For clarity, we have included plots separating her initial sketch (center) and the group’s final answer (right).

Figure 9

A sense making diagram for the analysis of the three scenes of case study I.

Figure 10

A plot of the double finite square well potential. The regions were indicated by Cam, and were not present in the initial diagram. Sketches of the ground state and first excited state wave functions (the answer to the protocol question) are shown on top of the potential.

Figure 11

Student work by Penny and Morgan. The letters show the order in which this work was written and are used for reference in the text.

Figure 12

A sense making diagram for the analysis of the three scenes, prologue, and epilogue of case study II.

Figure 13

Student work (written by Morgan) summarizing the possible solutions to the Schrödinger equation for constant potentials.