Analogical scaffolding and the learning of abstract ideas in physics: Empirical studies

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29. Developing well honed curricular materials would require an iterative cycle of design, testing with students, and modification. Analogical scaffolding serves as a cognitive model on which to base curricular materials and may be considered a guide in this iterative development process.
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36. Unless otherwise stated, all statistics are based on a two-tailed z test.
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38. These interviews were conducted as part of the modifications of the materials used in these studies. The standard EM-wave representations often used in textbooks includes crossed E and B fields represented by superimposed vectors and sine waves. This standard representation is problematic for several reasons which were revealed in student interviews. In these interviews, we found that students often did not distinguish between the electric and magnetic fields, resulting in false positives on question 2. This is because with the B field shown, students might answer P=R since both of these points lie near a wave peak, (the E field for P and the B field for R). We also removed the vectors from these representations in Fig. 2 in order to examine how students make sense of this “stripped down” and more abstract representation.
39. The idea that, for a traveling wave, a point where the wave crosses the x axis at one instant in time will be nonzero as the wave propagates was covered explicitly for a wave on a string. We expected that, combined with the hint on question 2 in Fig. 2 , students would be able to reason productively about the “time averaged” signal of an EM wave. Our classroom observations indicate that students make sense of traveling waves in the context of a wave on a string, but do not, in general, apply this idea when answering question 2 in Fig. 2 . We have some evidence that addressing the term “time averaged signal” explicitly helps students make this connection.
40. Supplementary materials are available at http://per.colorado.edu/analogy/index.htm.
41. We note that since a surface-level interpretation of a sine wave results in productive ideas for a wave-on-a-string, but not for sound or EM waves, a sine wave can be considered an abstract representation of a sound or EM wave, but relatively concrete for a wave-on-a-string. We purposefully designed this particular “abstract” sound wave representation in order to complement the EM wave representation, (e.g., Fig. 2 ) and promote student’s use of the sound wave analogy in a manner that will facilitate understanding of this EM representation. Additionally, what is labeled as abstract or concrete will depend upon the individual using these representations. See the discussion of abstraction in Ref. [5] and notes (Refs. [48, 49]) therein.
42. It may be noted that the PER community recognizes the need for a more refined approach to student thinking than a conceptions or misconceptions approach. However, researchers still argue for the existence of large-scale, stable, consistently activated sets of resources (Ref. [56]). For an in depth examination of conceptions, see Elby (Ref. [33]).
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45. The notion that misconceptions are relatively stable across contexts is testable. For more see Ref. [4].
46. Along the lines of diSessa’s p -prims (Ref. [16]).
47. This might be an example of WYSIWYG type reasoning (Ref. [33]).
48. Note that nearly all of the off-diagonal elements, 35 of 36 cells, are zero, (not including the category Other). In this case a χ² test is invalid. However, because of the nearly perfect diagonalization, we may conclude a strong association between answer choice and stated reasoning.
49. We do not have a compelling explanation for the unexpectedly large number of students in the abstract group answering partially correct on the pre-test. Since the majority of these students answered differently, and incorrectly, on the post-test, we consider this result curious, but insignificant to our broader findings.
50. We observe 25% differences between concrete and blend groups on both questions 1, (correct answer) and 2, (partially correct answer) with p<0.1 . In this case, since we expect high performers to be less susceptible to representational effects than low performers, a one-tailed z test may be appropriate which would result in p<0.05 . Note that dividing the student populations into high and low performers reduces N by half in statistical tests, resulting in reduced significance levels even on observed differences of 25% between treatments.
51. In this lab activity, sound was consistently represented by a sine wave, with one pictorial representation of air particles along the lines of the concrete representation used. No blended representations were used.
52. Z. Hrepic, D. Zollman, and S. Robello, Proceedings of the NARST 2005 Annual Meeting, 2005.
53. According to the model (Ref. [5]), the concrete, (air particles) sign is privileged over the abstract, (sine wave) sign for making meaning of sound. Thus the sine wave inherits the 3D schema from the air particles picture, (and not the other way around).
54. Note that both transverse and longitudinal waves can be generated on a stretched slinky. Most of the students who choose the slinky analogy indicated in their open response that their choice was associated with a longitudinal wave.
55. Here, we observe signs driving schemata. Note that schemata may also drive the meaning, (or creation) of signs. We might consider this latter directionality an indicator of expert reasoning.
56. E. F. Redish, Teaching Physics with the Physics Suite (Wiley, New York, 2003).