Students in introductory calculus-based physics were asked about the angular momentum of a particle traveling in a straight line. The tendency to state that the angular momentum is identically zero was widespread, and few students applied l = r × p correctly. The common errors reflect a tendency to conflate angular momentum with angular velocity or with linear momentum. Many students assume that linear and angular momentum are jointly conserved, an error that appears to be linked to their thinking about energy. A tutorial was developed to help students recognize that linear momentum and angular momentum are separately conserved. The results suggest that helping students understand why angular momentum is attributed to a particle moving in a straight line may be more effective in helping them to apply the concept than instructing them only on its correct use. In addition to providing insights into student learning of the concept of angular momentum, we illustrate how students’ own ideas can be the basis for more effective instruction.
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Students are given 15 min to complete the pretest, which they can begin at any time during a period of at least 48 h. They are given credit for completing the pretest whether or not their answers are correct. They are free to consult each other, their textbook, or the internet while they take the pretest. A student response to each question consists of a choice from a pull-down menu and a brief free-form explanation. The choices for the pull-down menus are intended to exhaust all reasonable possibilities that we have observed in student responses. Statistics from the pull-down menu choices are computed, whereas statistics on the explanations themselves are not; the explanations are used to guide our interpretation of students’ selections of various menu choices. For written exam questions, questions were posed to the students in the same manner: a fairly closed question with limited choices followed by an open-ended explanation. Constructing exam questions in this form leads to very few student responses with ambiguous choices.
11.
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The percentage of students answering correctly in each class was within 5%–10% of the average taken over all sections. No systematic variation that could be attributed to the prior instruction was observed.
13.
As we use the calculated statistics to iteratively improve instruction, we have tended to take seriously only differences of 15% or more. Although the data are presented here to 1%, we do not claim that the instruction has improved overall in any case where the increase in the student success rate on relevant questions is less than 15%, which for our measurements, exceeds the usual standards for statistical significance.
14.
This 40% figure is similar in magnitude to the result quoted in Ref. 8 that 35% of students answered the “clay ball” problem correctly under similar instructional circumstances. Both problems may be solved by using the same property of the vector cross product; namely, if the vectors in the cross product are unchanged except for the component of one (for example, r) that is parallel to the other vector (p or F), the cross product itself (L or τ) remains unchanged. We have not yet studied how the same students would respond to both questions.
15.
The initial version of the tutorial is unpublished.
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In this and other cases we have not found systematic variation in results that could be explained by whether or not a grade was awarded for the correct answer. See also
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We have observed that, when told that the thermal energy increases as a result of the collisions and asked which experiment would yield a greater increase in thermal energy, most students choose experiment 2, even though it is more in experiment 1. Further probing of their thinking reveals that they emphasize the process of moving after the collision (though they were told that the environment is frictionless), rather than the collision itself, as the main mechanism for the transformation of energy from kinetic to thermal.
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In the center-of-mass reference frame the kinetic energy in experiment 1 (see Fig. 2) is reduced to zero during the collision. It is conceivable that students recognize this result and use it to infer that the kinetic energy in the original reference frame must be significantly and suddenly reduced, because the change in kinetic energy as a result of the collision is frame-invariant. We expect very few students to change to alternative reference frames, because most students treat one reference frame as showing the “real motion,” which has real causes, while others show only appearances of motion. See, for example,
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Experiments 1 and 2 were performed with the same puck and rod, of approximately equal mass, on a low-friction air table. Each series of images was derived from a video by showing every third frame. Between frames 1 and 2, the puck can be seen to have moved through the same displacement in the same time before the collision, thus confirming that the puck did have the same initial velocity before the collisions. In frame 10, the center of mass of the puck-rod combination (approximately halfway in between the individual centers of mass of the puck and rod) can be seen to have traveled the same displacement across the air table in the same time in experiments 1 and 2. Therefore, though the puck-rod combination in experiment 2 rotated more than 90° during the same time interval, the center-of-mass velocity was the same for each puck-disk combination after the collision.
22.
Throughout the course of this study, we provided all students (enrolled in any of up to 20–25 tutorial sections run concurrently) with a single version of the tutorial and homework exercises, which represented what we thought was the most beneficial instruction at the time. Under this constraint, two groups of students would receive different instructions only if they took the course during different academic quarters. Thus, students who worked through the “revised tutorial” took the course in a later academic quarter than students who worked through the “initial tutorial.” We did not subject pretest and post-test questions to the same constraint, so student responses to a given question could, in principle, come from any academic quarter.
© 2011 American Association of Physics Teachers.
2011
American Association of Physics Teachers
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