This article reports on an investigation of student understanding of the concept of time in special relativity. A series of research tasks are discussed that illustrate, step-by-step, how student reasoning of fundamental concepts of relativity was probed. The results indicate that after standard instruction students at all academic levels have serious difficulties with the relativity of simultaneity and with the role of observers in inertial reference frames. Evidence is presented that suggests many students construct a conceptual framework in which the ideas of absolute simultaneity and the relativity of simultaneity harmoniously co-exist.

1.
For an extensive bibliography, see the relevant sections in
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2.
For an article that illustrates how research can guide the development of curriculum, see
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What we teach and what is learned—Closing the gap
,”
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3.
For examples of curriculum that have been developed through an iterative process of research, curriculum development, and instruction, see L. C. McDermott, P. S. Shaffer, and the Physics Education Group at the University of Washington, Tutorials in Introductory Physics, Preliminary Edition (Prentice Hall, Upper Saddle River, NJ, 1998);
L. C. McDermott and the Physics Education Group at the University of Washington, Physics by Inquiry (Wiley, New York, NY, 1996), Vols. I and II.
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9.
T. E. O’Brien Pride, “An investigation of student difficulties with two dimensions, two-body systems, and relativity in introductory mechanics,” Ph.D. dissertation, Department of Physics, University of Washington, 1997 (unpublished).
10.
That these are concepts that require definition is itself a new idea to many students. For insightful discussions of these definitions and the pedagogical concerns that they raise, see, for instance, P. W. Bridgman, A Sophisticate’s Primer of Relativity (Wesleyan U. P., Middletown, CT, 1962) and A. B. Arons, A Guide to Introductory Physics Teaching (Wiley, New York, NY, 1990).
11.
The definition of a global coordinate system breaks down in noninertial frames and in general relativity. The restriction of an inertial frame to a finite extent in both space and time is a refinement not usually encountered in courses in special relativity.
12.
One method of synchronizing clocks in special relativity includes sending the reading on one clock to a clock at another location by means of some signal. The second clock is synchronized with the first by setting it to read the time sent from the first clock plus the signal travel time. The use of light signals for the synchronization of clocks is customary but not necessary. See, for instance, the first book in Ref. 10.
13.
Although it is possible to define the time of an event as the time at which an observer sees the event, the time then depends on observer location. Einstein, for instance, considered and rejected such a definition. See A. Einstein, “On the electrodynamics of moving bodies,” in The Principle of Relativity: A Collection of Original Papers on the Special and General Theory of Relativity (Dover, New York, NY, 1952).
14.
Some authors define the term “observer” to indicate the full set of measuring devices and procedures that comprise a reference frame. For an example of this approach, see E. F. Taylor and J. A. Wheeler, Spacetime Physics (Freeman, New York, NY, 1992), p. 39.
15.
The process by which “time is spread over space” is described meticulously in the first book in Ref. 10.
16.
For a description of the course for high school teachers, see
L. C.
McDermott
, “
A perspective on teacher preparation in physics and other sciences: The need for special courses for teachers
,”
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58
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734
742
(
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).
17.
The results from the various classes at the University of Washington were consistent within statistical fluctuations. The results from the other universities were within the same range. For the purposes of this article, the results from corresponding classes have been combined.
18.
The problems are “ungraded” in the sense that they are not graded for correctness. Rather, students receive credit for giving responses that reflect an attempt to answer the questions. Student performance on the examination questions and the ungraded problems was very similar. This is consistent with the results of other investigations conducted by our group, in which we have found that students take the ungraded questions seriously.
19.
Each research question was posed in at least two ways, with different context and slight changes in wording. In general, we found that such changes had little effect on student performance. Therefore, in this paper we present only a representative description or example of each question.
20.
The fact that the curved surface of a rotating Earth is not an inertial frame did not elicit student concern. In one version of the question, students were told to neglect noninertial effects. In another, the context was set in deep space. Neither statement seemed to change student responses.
21.
In keeping with standard practice in one-dimensional problems on special relativity, students were told in all questions that all motions were to be considered as occurring along a single line in space. No student seemed to have had difficulty neglecting the vertical dimension.
22.
The prototype of such a qualitative analysis is that of the classic train paradox often used to develop the relativity of simultaneity. See, for instance, P. A. Tipler and R. A. Llewellyn, Modern Physics (Freeman, New York, NY, 1999).
23.
We use the notation δt instead of Δt, etc., to try to minimize confusion between the difference between two quantities and a change in a quantity. We are grateful to Eric Mazur for a discussion on this point.
24.
Since the events are separated by a spacelike interval (c2δt2−δx2<0), students can predict the existence of a frame in which the events are simultaneous without use of the Lorentz transformations.
25.
We have evidence that the students in this class performed as well as they did on the directed version of the Spacecraft question as a result of special instruction that they had received. On the basis of the research described in this paper, we have been developing instructional materials to address the specific difficulties that we have identified. This class had used preliminary versions of these materials. As is demonstrated in Sec. VI, other classes after standard instruction did not do as well on similar questions.
26.
This finding is consistent with our experience that the study of advanced material does not necessarily deepen conceptual understanding.
See, for example,
S.
Vokos
,
P. S.
Shaffer
,
B. S.
Ambrose
, and
L. C.
McDermott
, “
Student understanding of the wave nature of matter: Diffraction and interference of particles
,”
Phys. Educ. Res., Am. J. Phys., Suppl.
68
,
S42
S51
(July
2000
);
B. S.
Ambrose
,
P. S.
Shaffer
,
R. N.
Steinberg
, and
L. C.
McDermott
, “
An investigation of student understanding of single-slit diffraction and double-slit interference
,”
Am. J. Phys.
67
,
146
155
(
1999
);
K.
Wosilait
,
P. R. L.
Heron
,
P. S.
Shaffer
, and
L. C.
McDermott
, “
Development of a research-based tutorial on light and shadow
,”
Am. J. Phys.
66
,
906
913
(
1999
);
T. O’Brien
Pride
,
S.
Vokos
, and
L. C.
McDermott
, “
The challenge of matching learning assessments to teaching goals: An example from the work-energy and impulse-momentum theorems
,”
Am. J. Phys.
66
,
147
157
(
1998
);
L. C.
McDermott
,
P. S.
Shaffer
, and
M. D.
Somers
, “
Research as a guide for teaching introductory mechanics: An illustration in the context of the Atwood’s machine
,”
Am. J. Phys.
62
,
46
55
(
1994
).
27.
The belief in absolute simultaneity may be related to the strong belief of students in a preferred reference frame, which has been documented in the context of Galilean relativity. See Refs. 5, 6, and 9.
28.
This term is related to the amount of time that two specific synchronized clocks in the spacecraft frame are measured to be out of synchronization by observers in the ground frame.
29.
See, for instance, D. Griffiths, Introduction to Electrodynamics (Prentice Hall, Upper Saddle River, NJ, 1989), p. 452. This widely used text states explicitly that the relativity of simultaneity is “a genuine discrepancy between measurements made by competent observers in relative motion, not a simple mistake arising from a failure to account for the travel time of light signals.”
30.
The belief that each observer constitutes a distinct reference frame is similar to the belief documented in Galilean contexts in Ref. 4. The authors describe a tendency of students to treat the extent of an observer’s reference frame as limited to the physical object on which the observer is located (e.g., the deck of a boat).
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