An analysis is presented of data on students’ problem-solving performance on similar problems posed in diverse representations. Five years of classroom data on 400 students collected in a second-semester algebra-based general physics course are presented. Two very similar Newton’s third-law questions, one posed in a verbal representation and one in a diagrammatic representation using vector diagrams, were given to students at the beginning of the course. The proportion of correct responses on the verbal question was consistently higher than on the diagrammatic question, and the pattern of incorrect responses on the two questions also differed consistently. Two additional four-question quizzes were given to students during the semester; each quiz had four very similar questions posed in the four representations: verbal, diagrammatic, mathematical/symbolic, and graphical. In general, the error rates for the four representations were very similar, but there was substantial evidence that females had a slightly higher error rate on the graphical questions relative to the other representations, whereas the evidence for male students was more ambiguous. There also was evidence that females had higher error rates on circuit-diagram problems in comparison with males, although both males and females had received identical instruction .
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46.
The “dominance principle” (a term used by Halloun and Hestenes) refers to students’ tendency to attribute larger-magnitude forces to one or the other object in an interacting pair, based on an ostensibly “dominant” property such as greater mass, velocity, or charge.
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This result suggests that some students’ expertise in using vector representations may have increased faster than did their understanding of Newton’s third law, because response B is an accurate representation of an answer based on the dominance principle.
48.
Question #2 in this set was designed by Leith Allen, private communication (2002).
49.
J. P. Guilford, Fundamental Statistics in Psychology and Education, 4th ed. (McGraw-Hill, New York, 1965), p. 184. This test considers each pair of values to be an independent measurement of the difference between the paired quantities. It is the appropriate test here because there are manyyear-to-year variations (in student demographics, course logistics, etc.) but in each individual year, there is no a priori reason to expect differences between the paired quantities.
50.
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51.
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52.
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53.
We have tried to further test this interpretation with interview data [Leith Allen and Larry Engelhardt, private communication (2003)]. Approximately 15 students were interviewed in all; they had volunteered in response to a general solicitation. None of the students interviewed showed any clear evidence of the representation-related difficulties identified in this paper. Our experience (and that of others) has been that most students who volunteer for interviews are well above the average in terms of course performance. It seems that the relatively simple nature of the questions used in this investigation (indicated by the low error rates) was an inadequate challenge for the interview volunteers. It will probably be necessary to target potential interviewees in the future, soliciting students who have already shown (on quizzes or exams) evidence of the learning difficulties being investigated.
54.
Lei
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and Edward F.
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55.
Although the and versions of the gravitation question (and related Coulomb’s law question) include similar options regarding force magnitudes, the version obviously portrays directional information as well. This directional information is an additional bit of complexity which probably contributes to overall confusion, although it is not clear how (or whether) it might make it more difficult for a student to pick out an “equal magnitudes” option.
56.
This convention—that the tail of the arrow representing a force exerted on an object is attached to the object—is certainly not universal. However, in the context of question 8, the attractive nature of the gravitational force guarantees that the force exerted on an object must point toward the other object in the interacting pair. This fact makes the assignment of force vector arrows in question 8 unambiguous; regardless of the convention for locating the tails of the arrows, the arrow corresponding to the force exerted on the moon must point toward the earth. Therefore, it is not merely a confusion about notation or vector conventions that leads to the error identified here. [It is notable that not a single student chose either response or on the electrostatic final-exam question (Fig. 2); these responses would be acceptable representations of a dominance-principle answer, or the correct answer, respectively, if one ignored tail location.] This observation leaves open the question of whether the students’ confusion was primarily with the tail location, the meaning of the arrow direction itself, the meaning of “attractive force,” or some amalgam of these (and possibly other) issues.
57.
Most gender-related differences in this study seem to be smaller than the differences documented to exist between traditional instruction and interactive-engagement instruction;
see, for instance,
Richard R.
Hake
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66
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(1998
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Jill
Marshall
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(2004
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However, one must also consider the possibility that specific differences in the way the questions were worded also may have contributed significantly to the discrepancies in responses that were observed.
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