Helping students develop an understanding of Archimedes’ principle. I. Research on student understanding

This research is described in greater detail 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.

L. C. McDermott and the Physics Education Group at the University of Washington, Physics by Inquiry (Wiley, New York, 1996).

These materials are discussed in the companion article:

P. R. L.

Heron

,

M. E.

Loverude

,

P. S.

Shaffer

, and

L. C.

McDermott

, “,” , – (), following paper.

See Ref. 3.

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, 2002).

Although the tutorials are used in algebra-based courses in several colleges and universities, the algebra-based course at the University of Washington does not have small sections at present.

A Cartesian diver is an object of variable density that can be made to sink or float in response to pressure changes in a fluid.

The diver used in the interviews is made from the transparent bulb of a plastic dropping pipette. A weight is attached to its open end. The bulb is placed with its neck down in a two-liter plastic soda bottle nearly filled with water. The bulb contains water with a bubble of air above the water. The amount of air is adjusted so that the diver initially barely floats. When the soda bottle is squeezed, the increase in pressure leads to a decrease in the volume of the air bubble inside the bulb. Water enters through the neck of the diver, thus increasing its density so that the diver sinks. (The students had seen a lecture demonstration involving a Cartesian diver in class, but the apparatus differed significantly in appearance from that used in the interviews.)

Evidence that students confuse the concepts of pressure and weight can be found in Ref. 1, as well as in

E. Engel

 

Clough

and

R.

 

Driver

, “,”  , – () and in

P.

Kariotoglou

and

D.

Psillos

, “,” , – ().

In most cases, we have found that the results for a given problem are essentially the same whether administered on a non-graded quiz or on a graded course examination. For related results, see

C.

Henderson

, “,” , – ().

In our experience, the variation in results from class to class is usually very small. In particular, for large classes $(N∼100$ or more) we have found that the success rates of individual classes usually fall within about 5% of the mean taken over several classes. Therefore, we have reported all percentages rounded to the nearest five percent. In the second-year course the number of students is typically much smaller and we collected data from fewer sections. Our experience has been that the variation is comparable to that in the case of larger classes. However, the primary function of the data from the second-year course is to illustrate the extent to which similar conceptual issues surface among students who typically have a stronger mathematical background and who have had more instruction on the topic.

In many of the written questions discussed in this article, blocks are depicted as floating with surfaces parallel to the water surface. In practice these blocks would float with one corner up. We have not found this to be a source of confusion for students nor did we wish to explore the issue of stability with these questions. We have revised the questions so that current versions refer to cylinders instead of blocks.

Only classes in which we can be certain that none of the students had participated in the relevant lab are included. In one case the students were not asked to write their names and so the lab students cannot be separated from the rest of the class. Therefore we have excluded these results. Also excluded are results from a class in which about half the students had participated in the lab, and there was evidence that a number of students changed their answers while taking the ungraded quiz (an unusual occurrence). Results obtained after the lab are discussed in detail in Ref. 4.

For instances in which results have been similar before and after instruction, see, for example,

L. C.

 

McDermott

, “,”  , – () and

L. C.

McDermott

, “,” , – ().

Compressibility had not been an issue in our investigation of student understanding of hydrostatic pressure. In that study, students attributed pressure gradients to a number of different variables. Density, however, was not one of them.

The prevalence of the descending line response was essentially the same as on versions in which no free-body diagrams were required. It seems that additional questions did not help students arrive at correct answers.

After collecting the students’ responses to the five-block problem, the instructor in one section of the introductory course conducted a class discussion in which he explained the correct answer and discussed the common incorrect descending line answer. This discussion may have had an effect on the students’ subsequent ability to answer other questions.

For other examples in which students apply formulae without understanding the terms, see

B. S.

 

Ambrose

,

P. S.

 

Shaffer

,

R. N.

 

Steinberg

, and

L. C.

 

McDermott

, “,”  , – ();

L. C.

 

McDermott

and

P. S.

 

Shaffer

, “,”  , – ();

L. C.

McDermott

and

P. S.

Shaffer

, , (E) ().

In this case, the students had also performed a laboratory experiment on buoyancy in which they measured the buoyant force on the same block suspended in two different liquids.

See, for example,

R. A.

 

Lawson

and

L. C.

 

McDermott

, “,”  , – () and

M. E.

Loverude

,

C. H.

Kautz

, and

P. R. L.

Heron

, “,” , – ().

Difficulties in distinguishing mass and volume are described in M. L. Rosenquist, “Improving preparation for college physics of minority students aspiring to science-related careers,” Ph.D. dissertation, Department of Physics, University of Washington, 1982, p. 74.

For related results, see

J.

 

McKinnon

and

J. W.

 

Renner

, “,”  , – () and

J.

McKinnon

, “,” , – ().

We note that the mechanics courses did not include relevant tutorials from Ref. 5. However, even if they had, it may be unrealistic to assume that students would gain enough facility during their initial study of dynamics to apply their knowledge of forces in the more complicated environment of a liquid.

For examples in mechanics in which students inappropriately refer to equilibrium positions, see

R. F.

 

Gunstone

, “,”  , – () and

R. F.

Gunstone

and

R. T.

White

, “,” , ().

Similar difficulties with the concept of equilibrium have been documented in the context of thermal physics and rigid-body dynamics. See C. Kautz, “Investigation of student understanding of the macroscopic and microscopic behavior of an ideal gas,” Ph.D. dissertation, Department of Physics, University of Washington, 1999 and L. G. Ortiz, “Identifying and Addressing Student Difficulties with Rotational Dynamics,” Ph.D. dissertation, Department of Physics, University of Washington, 2001.

On a similar question asked on an end-of-high-school examination in Australia, Gunstone reported that about 45% of the students answered correctly, with about 35% giving the answer that the blocks would return to the same level (see Ref. 24).

In some cases, students were instructed to draw free-body diagrams for each block. The results were apparently unaffected.

In addition to the references already cited, see

L. C.

 

McDermott

, “,”  , – ();

D.

 

Hestenes

,

M.

 

Wells

, and

G.

 

Swackhamer

, “,”  , – ();

R. R.

 

Hake

, “,”  , – ().

Besides this representative sample, an extensive bibliography can be found in

L. C.

McDermott

and

E. F.

Redish

, “,” , – (). The emphasis in this resource letter is on university students and on references readily available to physicists.

See, for example,

L.

 

Viennot

, “,”  , – ();

A.

 

Champagne

,

L.

 

Klopfer

, and

J.

 

Anderson

, “,”  , – () and

J.

Clement

, “,” , – ().