This paper reports on a study of student understanding of the wave nature of matter in the context of the pattern produced by the diffraction and interference of particles. Students in first-year, second-year, and third-year physics courses were asked to predict and explain how a single change in an experimental setup would affect the pattern produced when electrons or other particles are incident on a single slit, double slit, or crystal lattice. The errors made by students after standard instruction indicated the presence of similar conceptual and reasoning difficulties at all levels. Among the most serious was an inability to interpret diffraction and interference in terms of a basic wave model. Other errors revealed a lack of a functional understanding of the de Broglie wavelength. Students often treated it as a fixed property of a particle, not as a function of the momentum. An important goal of this investigation was to provide a research base for the design of instruction to help students develop and apply a basic wave model for matter.

1.
F. M.
Goldberg
and
L. C.
McDermott
, “
Student difficulties in understanding image formation by a plane mirror
,”
Phys. Teach.
24
,
472
480
(
1986
);
F. M.
Goldberg
and
L. C.
McDermott
, “
An investigation of student understanding of the real image formed by a converging lens or concave mirror
,”
Am. J. Phys.
55
,
108
119
(
1987
).
2.
K.
Wosilait
,
P. R. L.
Heron
,
P. S.
Shaffer
, and
L. C.
McDermott
, “
Development and assessment of a research-based tutorial on light and shadow
,”
Am. J. Phys.
66
,
906
913
(
1999
).
3.
P. R. L.
Heron
and
L. C.
McDermott
, “
Bridging the gap between teaching and learning in geometrical optics: The role of research
,”
Opt. Photonics News
9
(
9
),
30
36
(
1998
).
4.
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
).
5.
B. S.
Ambrose
,
P. R. L.
Heron
,
S.
Vokos
, and
L. C.
McDermott
, “
An investigation of student understanding of light as an electromagnetic wave: Relating the formalism to physical phenomena
,”
Am. J. Phys.
67
,
407
415
(
1999
).
6.
L. C. McDermott, P. S. Shaffer, and the Physics Education Group at the University of Washington, Tutorials in Introductory Physics (Prentice–Hall, Upper Saddle River, NJ, 1998), Preliminary ed.
7.
For an article that illustrates the development and assessment of a tutorial in geometrical optics, see Ref. 2.
For an example in physical optics, see
K.
Wosilait
,
P. R. L.
Heron
,
P. S.
Shaffer
, and
L. C.
McDermott
, “
Addressing student difficulties in applying a wave model to the interference and diffraction of light
,”
Phys. Educ. Res., Am. J. Phys. Suppl.
67
,
S5
S15
(July
1999
).
8.
Members of the Physics Education Group have previously investigated student understanding of the particle-like behavior of light. See
R. N.
Steinberg
,
G. E.
Oberem
, and
L. C.
McDermott
, “
Development of a computer-based tutorial on the photoelectric effect
,”
Am. J. Phys.
64
,
1370
1379
(
1996
).
9.
See, for example,
A. P.
French
and
E. F.
Taylor
, “
Qualitative plots of bound state wave functions
,”
Am. J. Phys.
39
,
961
962
(
1971
);
An Introduction to Quantum Mechanics, M.I.T. Introductory Physics Series (Norton, New York, 1978);
D. Zollman, “Hands-on quantum mechanics,” Proceedings of Hands-on Experiments in Physics Education, GIREP, 1998 (to be published).
For a different way of arriving at the qualitative shape of wave functions, see
E. F.
Taylor
,
S.
Vokos
,
J. M.
O’Meara
, and
N. S.
Thornber
, “
Teaching Feynman’s sum-over-paths quantum theory
,”
Comput. Phys.
12
,
190
199
(
1998
).
10.
For some papers that include findings relevant to the research reported in this paper, see
I. D.
Johnston
,
K.
Crawford
, and
P. R.
Fletcher
, “
Student difficulties in learning quantum mechanics
,”
Int. J. Sci. Educ.
20
,
427
446
(
1998
);
H. Fischler and M. Lichtfeldt, “Learning quantum mechanics,” Proceedings of an International Workshop on Research in Physics Learning: Theoretical Issues and Empirical Studies, edited by R. Duit, F. Goldberg, and H. Niedderer (IPN, Kiel, 1992), pp. 240–258;
H. Niedderer, “Alternative frameworks of students in mechanics and atomic physics—Methods of research and results,” Proceedings of the Second International Seminar in Misconceptions and Educational Strategies in Science and Mathematics, edited by J. D. Novak (Cornell U.P., Ithaca, NY, 1987), pp. 335–348.
11.
L. C.
McDermott
and
E. F.
Redish
, “
Resource Letter: PER-1: Physics Education Research
,”
Am. J. Phys.
67
,
755
767
(
1999
).
12.
B. S. Ambrose, “Investigation of student understanding of the wave-like properties of light and matter,” Ph.D. dissertation, Department of Physics, University of Washington, 1999 (unpublished).
13.
See, for example, Ref. 2.
14.
This series includes tutorials on two-source interference, double-slit interference, multiple-slit interference, and single-slit diffraction. See Ref. 6.
15.
Some of the students in the advanced courses had used the tutorials on interference and diffraction of light in their introductory classes. However, only a few had worked through the same revised version that the students in the calculus-based course had used. Although those students seemed to do better than the students who had not worked through that version or who had not used the tutorials at all, the number of students was too small to draw definitive conclusions about significant long-term retention.
16.
Just before publication of this article, type P questions were administered in two calculus-based courses before lecture instruction on the de Broglie wavelength. (The students had recently completed the tutorial sequence on physical optics.) The percentage correct and the percentage with correct reasoning were within 5% of the results for calculus-based students after lecture instruction. (See Tables II(a) and III(a).
17.
In addition to Refs. 2, 4, 5, and 7, see
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
);
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
).
18.
For a description of the tutorial system at the University of Washington, see the articles in Ref. 17.
19.
For a discussion of this instructional strategy, see
L. C.
McDermott
, Millikan Award Lecture: “
What we teach and what is learned—Closing the gap
,”
Am. J. Phys.
59
,
301
315
(
1991
).
20.
We consider a tutorial successful when the post-test performance of introductory students matches or surpasses that of tutorial instructors (graduate students and advanced undergraduates) on the corresponding pretests. Type P questions were given to more than 25 TA’s enrolled in a graduate teaching seminar. About 80% correctly predicted the effect of changing the speed of the electrons (70% with correct reasoning). When the electrons were replaced with other particles of the same kinetic energy, about 50% gave the correct response (40% with correct reasoning). The post-test performance of students at all levels matched or surpassed that of the TA’s on the pretest. (See Tables II(b) and III(b).)
21.
See, for example, the last article in Ref. 17 and P. S. Shaffer, “Research as a guide for improving instruction in introductory physics,” Ph.D. dissertation, Department of Physics, University of Washington, 1993 (unpublished).
22.
See for example, Ref. 4 and the first paper in Ref. 17.
23.
Tutorials in Introductory Physics is being pilot-tested in situations in which the tutorials replace standard problem-solving sections. Typically, we have found that the tutorial students not only perform much better on qualitative questions but as well, or better, on related quantitative questions. The result is consistent with those obtained by G. E. Gladding, University of Illinois, D. Elmore, Purdue University, and E. Mazur, Harvard University (private communications).
24.
For an example of a study in which the factor of time spent on task was explicitly controlled, see
E. F.
Redish
,
J. M.
Saul
, and
R. N.
Steinberg
, “
On the effectiveness of active-engagement microcomputer-based laboratories
,”
Am. J. Phys.
65
,
45
54
(
1997
).
25.
See Ref. 10.
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