We describe an investigation of student learning of the concept of pressure in a static liquid. We document patterns of student answers and explanations in response to written and interview questions. We find that many undergraduate students fail to develop a correct understanding of pressure in this context. Many students have difficulty identifying the forces that act on a liquid and relating these forces to pressure. We describe the development and assessment of research-based instructional materials designed to address student difficulties with pressure and provide evidence that these materials can improve student understanding.

1.
A more complete description of early portions of this work can be found in
M. E.
Loverude
, “
Investigation of student understanding of hydrostatics and thermal physics and of the underlying concepts from mechanics
,” Ph.D. dissertation, Department of Physics,
University of Washington
,
1999
,
and in
C. H.
Kautz
, “
Investigation of student understanding of the ideal gas law
,” Ph.D. dissertation, Department of Physics,
University of Washington
,
1999
.
See also Ref. 7, and
M. E.
Loverude
,
C. H.
Kautz
, and
P. R. L.
Heron
, “
Helping students develop an understanding of Archimedes’ Principle. Part I: Research on student understanding
,”
Am. J. Phys.
71
,
1178
1187
(
2003
).
2.
E.
Engel Clough
and
R.
Driver
, “
What do children understand about pressure in fluids?
,”
Res. Sci. Technol. Educ.
3
,
133
143
(
1985
).
3.
P. A.
Giese
, “
Misconceptions about water pressure
,”
Proceedings of the Second International Seminar: Misconceptions and Educational Strategies in Science and Mathematics
, edited by
J. D.
Novak
(
Cornell University
,
Ithaca, NY
,
1987
), pp.
141
148
,
and
M.
Mayer
, “
Common sense knowledge versus scientific knowledge: The case of pressure, weight, and gravity
,”
Proceedings of the Second International Seminar: Misconceptions and Educational Strategies in Science and Mathematics
, edited by
J. D.
Novak
(
Cornell University
,
Ithaca, NY
,
1987
), pp.
298
310
.
4.
P.
Kariotoglou
and
D.
Psillos
, “
Pupils’ pressure models and their implications for instruction
,”
Res. Sci. Technol. Educ.
11
,
95
108
(
1993
).
5.
See
D.
Psillos
and
P.
Kariotoglou
, “
Teaching fluids: Intended knowledge and students’ actual conceptual evolution
,”
Int. J. Sci. Educ.
21
,
17
38
(
1999
).
6.
See
S.
Rozier
and
L.
Viennot
, “
Students’ reasonings in thermodynamics
,”
Int. J. Sci. Educ.
13
,
159
170
(
1991
),
and
M.
Mayer
, “
Common sense knowledge versus scientific knowledge: The case of pressure, weight, and gravity
,”
Proceedings of the Second International Seminar: Misconceptions and Educational Strategies in Science and Mathematics
, edited by
J. D.
Novak
(
Cornell University
,
Ithaca, NY
,
1987
), pp.
298
310
.
7.
C. H.
Kautz
,
P. R. L.
Heron
,
M. E.
Loverude
, and
L. C.
McDermott
, “
Student understanding of the ideal gas law. Part I: A macroscopic perspective
,”
Am. J. Phys.
73
,
1056
1063
(
2005
).
8.
F. F.
Camacho
and
L. G.
Cazares
, “
Partial possible models: An approach to interpret students’ physical representation
,”
Sci. Educ.
82
,
15
29
(
1998
) (The language is that of the study authors; some of the other studies referenced above also refer to pressure as being “exerted” on an object or surface).
9.
See, for example,
A.
Wilson
, “
The hydrostatic paradox
,”
Phys. Teach.
33
,
538
539
(
1995
), and references therein.
Another such example in which a liquid mixture separates is in
A.
Arons
,
A Guide to Introductory Physics Teaching
(
Wiley
,
New York
,
1995
), p.
327ff
.
10.
In early versions of this problem, we asked students explicitly to explain the effect of the vapor pressure of the liquid in the barometer on the pressure measurements. The results suggest that very few students had a working knowledge of the term “vapor pressure.” For example, very few were able to predict whether the effects of the nonzero vapor pressure of mercury would tend to increase or decrease the height of a mercury column for a given value of atmospheric pressure.
11.
Because the results of this experiment depend on forces of adhesion and cohesion, the result is sensitive to the size of the straw. This effect was not the focus of the interviews. All students had experience with the phenomenon and readily predicted the correct result.
12.
See, for example,
J.
Minstrell
and
P.
Kraus
, “
Guided inquiry in the science classroom
,” in
How Students Learn: History, Mathematics, and Science in the Classroom
, edited by
M. Suzanne
Donovan
and
John D.
Bransford
(
National Academies Press
,
Washington, D.C.
,
2005
), pp.
475
514
.
13.
Some students who are unfamiliar with the Greek alphabet do not distinguish the Greek letter ρ from the roman letter p. Other students support this response by incorrectly invoking Pascal’s principle, as described in the first citation in Ref. 1.
14.
For example, consider the points just under the stoppers in the tubes in Fig. 3. The pressures at points X, Y, and Z are equal. The pressure just under the stopper is less by the amount ρgh. Because the density of oil is less than that of water, the pressure change is smaller in tube C, so the pressure just under the stopper is greater in that tube. Students tend to predict that the pressure is less because the density of oil is less.
15.
Our intuition might have been different had we seen some of the relevant literature in psychology. Several studies have found that many adults predict that the surface of a static liquid in a tilted container is not horizontal. Performance was worse among people (like restaurant wait staff) with real-life experience with the phenomenon. See, for example,
H.
Hecht
and
D. R.
Proffitt
, “
The price of expertise: Effects of experience on the water-level task
,”
Psychol. Sci.
6
,
90
95
(
1995
).
16.
If the volume was the same on both sides, the height ratio should be the inverse of the cross-sectional area rather than the diameter. Student responses may reflect difficulties with area and volume in addition to those with pressure.
17.
A handful of students (at most 1% of those in the entire study) argued that the water level will be slightly different in the two tubes due to capillary action. These students are included in the correct category.
18.
The lecture demonstration involves four tubes of different shapes that are connected to a common reservoir of liquid. The tubes contain a colored liquid that is readily visible. Students are expected to observe that the free surfaces in the four tubes are all at the same level. The demonstration had just been shown and was in fact still visible in the front of the room as students answered the different-diameters U-tube problem. Kraus has cited the students’ failure as evidence that many students have difficulty in correctly making and interpreting observations of lecture demonstrations,
P.
Kraus
, Ph.D. dissertation, Department of Physics,
University of Washington
,
1997
.
19.
One recent suggestion that students are better able to conceptualize pressure when it is taught as an energy density was offered by
C. J.
DeLeone
,
W. H.
Potter
, and
L. B.
Coleman
, “
Model-centered curriculum: Non-traditional approaches to fluid and flow phenomena
,”
AAPT Announcer
30
(
4
),
122
(
2000
).
20.
One student said that the pressure at point X would be equal to the pressure at point Y, but that the pressure at point Z would be less. All of the others gave the ranking PX>PY>PZ.
21.
The differences in student performance between introductory and second-year courses were more pronounced on more difficult questions involving multiple liquids, curved tubes, or tubes with different cross-sections. These situations were the only ones in which success rates fell below 50%. For more details, see Ref. 1.
22.
For example, one instructor in a section of the second-year course discussed “pressure contours” (contours consisting of points with a given value of pressure) after seeing the N-tube question and our research results. Approximately 75% of the students in this section answered correctly, as compared to 65% in other sections, with approximately 10% giving answers apparently based in part on the weight of the material above a point. On the other hand, another instructor who was familiar with this work took great pains to call attention to the proper use of the formula but obtained results no different than in sections in which no special effort was made.
23.
About 5% of the students gave the opposite answer, for example, PX>PW>PY>PZ in the capped U-tube. Some of these students indicated that the atmosphere was the source of the pressure and the pressure decreased with increasing distance from the source.
24.
L. C.
McDermott
,
P. S.
Shaffer
, and
M.
Somers
, “
Research as a guide for curriculum development: An illustration in the context of the Atwood's machine
,”
Am. J. Phys.
62
(
1
),
46
55
(
1994
).
25.
See for example, the discussion in
D.
Halliday
,
R.
Resnick
, and
J.
Walker
,
Fundamentals of Physics
(
Wiley
,
Hoboken, NJ
,
2005
), p.
366
.
26.
About 20% of the students answered both questions correctly. Another 5% answered both problems in a manner consistent with the pressure being related solely to the height of liquid directly above the point. All but one of the students who answered correctly on the curved tube also correctly ranked pressure in the square tube. Based on the relative success rates of the two tasks, we believe that the pressure ranking on the curved tube is a more rigorous test of functional understanding.
27.
Such a connection is also consistent with the experience of many instructors with the student difficulties with the hydrostatic paradox. The paradox is resolved by considering force on a liquid by container walls. See Ref. 9.
28.
One student who mentioned suction explicitly related it to a difference in the pressure inside and outside the straw, but most stated that the air above the water would pull the water upward.
29.
Surface tension does play a role in this situation. This student attributed the upward force solely to surface tension and failed to identify the air below the straw as important.
30.
See
J.
Minstrell
, “
Explaining the ‘at rest’ condition of an object
,”
Phys. Teach.
20
,
10
14
(
1982
) or Ref. 12.
31.
Others have suggested alternative approaches to the teaching of pressure. For example, Karitoglou and Psillos (Refs. 4 and 5) suggested divorcing the teaching of pressure from the notion of force. See also Ref. 19.
32.
Though most of these students ranked the pressures PX>PZ>PY, a few switched the rankings for points X and Z based on the mistaken assumption that oil is more dense than water.
33.
The existence of a force on the top of the oil by the stopper depends on there being a nonzero pressure at the top of the oil. The fact that the leftmost tube is filled with water to the top suggests that there is a nonzero pressure at the top of the oil. The point just below the stopper in the leftmost tube is less than the pressure at the point just below the stopper in the rightmost tube. Therefore, even if the pressure just below the leftmost stopper is zero, the pressure just beneath the rightmost stopper cannot be.
34.
The students who drew correct free-body diagrams were not typically the best students in the class, as measured by course grade. The students who neglected to include the force by the stopper in their free-body diagrams had an average course grade that was above the mean for the course, suggesting that their incorrect responses cannot be explained by poor overall academic performance. The average grade in the course for those students who drew correct free-body diagrams was 2.7, compared to 3.2 for those who neglected the force exerted by the stopper on the oil and 3.0 for those who made errors based on difficulties with Newton’s third law. The overall course mean grade was 2.81 and the standard deviation was 0.82.
35.
See, for example,
P. S.
Shaffer
, “
Research as a guide for improving instruction in introductory physics
,” Ph.D. dissertation, Department of Physics,
University of Washington
,
1993
. Even graduate students in physics made this error.
36.
See Ref. 30 and
J.
Clement
, “
Using bridging analogies and anchoring intuitions to deal with students’ preconceptions in physics
,”
J. Res. Sci. Teach.
30
,
1241
1257
(
1993
).
37.
Although the majority of students takes the three parts of the course in consecutive quarters, some students had a gap of one or more quarters between mechanics and the second quarter of the sequence.
38.
In addition, the students in these classes performed similarly on other problems designed to test their knowledge of Newtonian mechanics. See the first and last citations in Ref. 1.
39.
To determine whether the students who answered the mechanics problem correctly were better students, we compared the mean of the course grades of the students who answered the mechanics problem correctly to that of those who did not. The mean grades of students in the two categories were very similar. These results suggest that the students that make errors in labeling the weight force on the mechanics problem are not the weaker students academically. For example, in one section, the students whose free-body diagrams showed the weight of the upper book as a force acting on the lower book had an average grade of 2.86 compared to the course mean of 2.84.
40.
The Physics Education Group at the University of Washington
,
L. C.
McDermott
, and
P. S.
Shaffer
,
Tutorials in Introductory Physics
(
Prentice-Hall
,
Upper Saddle River, NJ
,
2002
).
41.
For a more detailed description of an interactive tutorial lecture, see Ref. 1 or
P. R. L.
Heron
,
M. E.
Loverude
, and
P. S.
Shaffer
, “
Helping students develop an understanding of Archimedes’ principle. Part II: Development of research-based instructional materials
,”
Am. J. Phys.
71
,
1188
1195
(
2003
).
42.
See, for example, Refs. 24 and 35 and
E. F.
Redish
and
R. N.
Steinberg
, “
Teaching physics: Figuring out what works
,”
Phys. Today
52
(
1
),
24
31
(
1999
).
43.
Although there is no time to dwell on the difference between gauge pressure and absolute pressure, students are asked how their free-body diagrams would differ if they were to consider forces exerted by the atmosphere.
44.
For an example in the context of physics, see
M. T. H.
Chi
,
P. J.
Feltovich
, and
R.
Glaser
, “
Categorization and representation of physics problems by experts and novices
,”
Cogn. Sci.
5
(
2
),
121
152
(
1981
).
45.
L. G.
Ortiz
,
P. R. L.
Heron
, and
P. S.
Shaffer
, “
Student understanding of static equilibrium: Predicting and accounting for balancing
,”
Am. J. Phys.
73
(
6
),
545
553
(
2005
).
46.
A. A.
DiSessa
, “
Toward an epistemology of physics
,”
Cogn. Instruct.
10
(
2&3
),
105
226
(
1993
).
47.
E. F.
Redish
, “
A theoretical framework for physics education research: Modeling student thinking
,” in
Proceedings of the International School of Physics, “Enrico Fermi” Course CLVI
, edited by
E. F.
Redish
and
M.
Vicentini
(
IOS
,
Amsterdam
,
2004
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.