Analogical scaffolding and the learning of abstract ideas in physics: An example from electromagnetic waves

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29. We note that this objectlike reasoning may also apply when the sign is in the form of words, and e.g., interpreting the phrase “step function” as describing a stair-step-like object. See, for example, D. T. Brookes and E. Etkina, Do our words really matter? Case Studies From Quantum Mechanics, Proceedings of the 2005 Physics Education Research Conference (AIP, Melville, NY, 2006).
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36. Alternatively, one might hypothesize that this approach would lead to more confusion for students. The string and sound analogies may interfere in problematic ways, or there may be simply too much information for students to learn at once.
37. N. Podolefsky and N. Finkelstein, http://per.colorado.edu/analogy/index.htm.
38. In common physics parlance, sign and representation share the same meaning. In other sciences, representation often refers to internal or mental representations, along the lines of mental models or schemata.
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40. This was the state of scientific knowledge prior to Rutherford.
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42. In the language of blending theory, this is referred to as a “single scope” blend, where the organizing frame of one mental space (solar system) is applied to the elements of another (atom). For more, see Ref. [12].
43. One might ask why an external representation is contained within a mental space. We side with the view that artifacts of the environment, such as pictures on paper, are key components of cognition, and hence mental spaces. See M. Wilson, Six views of embodied cognition, Psychon. Bull. Rev. 9, 625 (2002).
44. Each of these sign-schema-referent triangles may be considered a blend, whereby the sign, referent, and schema are connected by vital relations. In the case of the solar system, a picture of the solar system (sign) may be connected to the real solar system (referent) via the vital relation representation. In the Rutherford atom blend, the size of both the solar system and atom referents are scaled to a human scale (the actual size of the sign) via the vital relation space. Role and value vital relations may be involved, whereby the role central object takes the values sun or nucleus, and the role outer object takes the values planet or electron.
45. In this case, the historical progression happens to match a pedagogy that may be productive. We do not mean to suggest, however, that pedagogy should, in general, follow historical accounts of discovery.
46. These changes to sign-schema-referent relations may be along the lines of changes to Wittmann’s resource graphs. See M. C. Wittmann, Using resource graphs to represent conceptual change, Phys. Rev. ST Phys. Educ. Res. 2, 020105 (2006).
47. It might be more accurate to say Planck’s quantization of energy, which Bohr blended with Rutherford’s model.
48. The delineation of abstract and concrete may depend on the level of expertise. To an expert physicist, an electron is a particular type of particle, and thus the electron is more concrete. To a student, to whom an electron may be an unfamiliar idea, “particle” may be more concrete in the sense of being connected to a real object, like a dust particle. In this sense, students’ prior knowledge plays a role in our model to the extent that we can determine which ideas are already concrete for students, and which remain abstract. For a detailed analysis of levels of abstraction, see S. I. Hayakawa, Language in Thought and Action, 3rd ed. (Harcourt Brace Jovanovich, New York, 1972).
49. Such a notion of abstraction being a series of blends is consistent with Lakoff and Nunez’s notion of layering (Ref. [16]). The level of perceived abstraction may depend on the student, in that as students become increasingly familiar with abstract sign-schema relations, these relations may become increasingly treated as concrete. In this case, nodes may become so tightly coupled for an expert that the sign-schema link is compressed and the nodes disappear. For instance, to the expert physicist, the notions of “light” and “wave” are not separate ideas—to this expert, light is a wave.
50. Saalih Allie (private communication).