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Why do students have such trouble with quantifers?

During a course I taught this summer in Non-Euclidean Geometry for middle school and high school teachers I used Joel Castellanos' excellent NonEuclid program. I assigned as homework a few problems from the list of activities that accompanies the program. Only one of the 13 students in the class answered the following question correctly:

In Euclidean geometry, any polygon can be completely enclosed in some sufficiently large triangle. This is so obvious a statement that I have never even seen it written as a theorem. In, hyperbolic geometry, this is not an obvious statement. Is it a true statement?

The correct answer is that the statement is not true. Counterexamples are easy to come by. For example, consider a regular hexagon, whose center coincides with the center of the Poincare Disk. If the vertices of the hexagon lie on a sufficiently large circle, as in Figure 1, a little experimentation should convince the student that it will be impossible to enclose the hexagon in a triangle. A simple proof (not required for the homework) is based on the fact that with proper normalization the area of a hyperbolic triangle is equal to the defect = (pi - sum of the angles). Thus, no triangle can have an area greater than pi. However, the hexagon can be decomposed into six triangles, each of which has defect = 2*pi/3 - eps, where eps > 0 can be made as small as desired by increasing the radius of the circle. Thus the area of the hexagon = 4*pi - 6*eps can be made significantly larger than pi. Since the part cannot be greater than the whole, no triangle can enclose such a hexagon.

Twelve of 13 students showed, in effect, that given a triangle, they could find a regular hexagon inside the triangle. One student even wrote that her initial attempt didn't work because her hexagon was too big, so she had to use a smaller hexagon. Clearly, her problem was in the interpretation of the question. I am convinced that that was the problem of the other students as well, since almost all of them had previously constructed "large" regular hexagons. See Figure 2 for a typical student production.

Figure 1:
Read this document on Scribd: LargeRegularHexagon

Figure 2:
Read this document on Scribd: LargeTriangleWithHexagon

This can be viewed as a problem with understanding the difference between existential and universal quantifiers that bedevils college students, as anyone who has taught beginning calculus knows. I think that there is a psychological component as well. Most students originally go into mathematics because they are good at following directions. For example, they are asked to multiply two polynomials, and they are rewarded when they can do so. The idea of discovering that something is impossible rubs the wrong way. It is satisfying to be able to create a regular triangle inside a given triangle. It is disturbing to have to conclude that there is a hexagon which cannot be enclosed in any triangle.

If our teachers think that solving a problem in mathematics consists of following some procedure to produce a positive result, how are students going to view mathematics as a search for truth, whether the result be positive or negative?

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