How to teach students the value of concrete counterexamples?

Question

I teach exercise sessions for a Linear Algebra course for 1st semester students in Europe. Students have to prepare some exercises at home. In class, I call on students to present their solutions.

One big problem that I see is that in exercises of the form "prove or disprove", students' disproofs usually ends with an equation with variables instead of a concrete counterexample. For example, to prove that a certain statement does not hold true, a student proof may go like this:

If the statement does hold, then [...] And in the end, we find that, for all real numbers $x,y$, that $x=y$. This is clearly not true, so the original statement is false.

When I tell my students, "This proof is okay, but a concrete counterexample would be nicer", they usually say, "But from this equation I can see it just as clearly".

I see three main reasons to advocate for concrete counterexamples instead:

1. They force you to actually think about the existence of a counterexample, or to recognize that the resulting equation is "vacuous". (However, with some easy Linear Algebra examples, this does not seem to be an issue.)
2. The proof looks simpler and more convincing for the reader. (But admittedly, not so for my students. It seems they would they have to derive the equation, find a concrete counterexample, and then write the disproof again with their newly discovered counterexample.)
3. It is the usual, accepted style.

However, I tell my students these reasons repeatedly and it seems to have no effect. A related issue is that they often "prove" a universally quantified statement via an example.

My questions:

1. Are there any easy examples where you could easily find an empty equation and no counterexample which I could show to my students?
2. Has there been any reasons about why thinking in counterexamples is hard for fresh(wo)men?
3. What can I do to improve the situation?

Answer 1

Perhaps "thinking in counterexamples is hard" because that requires thorough understanding, a grasp of the boundaries where a claim holds, and where it does not. For example, is matrix multiplication commutative, i.e., does $AB = BA$? This hypothesis is best refuted by a counterexample, say, 3D rotations. But then it raises the question of what conditions ensure that matrix multiplication is commutative. $\textrm{SO}(2)$ is commutative, as is the subset of $\textrm{SO}(3)$ representing rotations about a common axis. Exploring the boundary between truth and falsity leads to robust understanding of the involved concepts. For matrix commutativity, one might be led to simultaneous diagonalization.

Answer 2

If you're speaking about counterexamples in general, the following aphorism really struck me:

In order to understand what something is, you must also understand what it is not.

One of the main values for knowing lots of counterexamples is that they give examples of what something is not.

Answer 3

Students are not accustomed to speaking/writing precisely. When presented with an affirmation, such as a homogeneous linear system of equations has always a nontrivial solution a student may realize that the statement is false for some particular overdetermined system without realizing that the existence of a particular counterexample invalidates the affirmation. This occurs because in colloquial speech one customarily qualifies affirmations, so it is habitual to speak of something as true when it is generally true. In the opposite direction, a student might consider the affirmation true because it is true for all underdetermined systems and systems ... We have to teach students the difference between mathematical (or, more generally, careful, precise, technical) speech and colloquial speech. This is perhaps the most important reason we teach linear algebra to so many students who will not later use its results in an instrumental way.

The only way to teach this, so far as I can see, is to force students to explain carefully the reasoning in a number of illustrative examples. One way to do it is to make an affirmation and ask the student to decide whether it is true or false, providing justification in the case of truth, and a counterexample in the case of falsity. Often it helps to put side by side superficially (that is, verbally) similar problems. For example consider the following affirmations:

1. A matrix is orthogonally diagonalizable if and only if it is symmetric.

2. A matrix is diagonalizable if and only if it is symmetric.

Many students will not distinguish between "diagonalizable" and "orthogonally diagonalizable" so will not see any difference between the affirmations. Putting the two affirmations side by side signals to a mildly attentive student that there is a difference between them. This will be true even for students who have already seen in class an example of a $2 \times 2$ diagonalizable matrix that is not symmetric, and had it pointed out to them explicitly that this matrix is diagonalizable but not orthogonally diagonalizable. So one has to explain the reasoning in detail.

At the bottom is the (quite general) confusion between necessary and sufficient conditions. Most people do not distinguish well between the two until taught to do so. This has to do with what Polya called plausible inference. While it is true mathematically that just because A implies B it need not be the case that B implies A, in scientific, and less technical contexts, the truth of B is evidence in favor of A. This mode of thinking is natural to thoughtful and observant people, and useful in many contexts, although it is not correct in purely mathematical contexts. It is the math teacher's job to indicate how mathematicians use language differently from others, and why it is useful to do this. This is part of why we fuss so much about questions such as whether every conservative vector field has a potential.

This is my sense of what V. I. Arnol'd meant when he wrote (in his usual intentionally tendentious style) on p. 740 of this article (although I do not share Arnol'd's antagonism to Bourbaki):

Now it became possible to apply the techniques developed in the problem of adiabatic invariants. As soon as I accomplished that, Kolmogorov suggested that I should submit the paper on perpetual adiabatic invariance to ZHETF, the main physical journal in the USSR.

A few weeks later, M. A. Leontovich (who was, as far as I remember, a deputy to the editor-in-chief of ZHETF) invited me to his home (near the Atomic Energy Institute of the USSR Academy of Sciences) to discuss the manuscript. Having fed me, as usual, by boiled buckwheat and calling me, as usual, “Dimka” (M. A. called me in such a way until his death), Mikhail Aleksandrovich explained to me that the paper could not be published in ZhETF due to the following reasons.

1. The manuscript contained the words “theorem” and “proof” forbidden in ZhETF.

2. The manuscript claimed that “A implies B” while every physicist knew examples showing that B does not imply A.

3. The manuscript used the unintelligible terms “Lebesgue measure”, “invariant tori”, “Diophantine conditions”.

Mikhail Aleksandrovich therefore proposed that I should rewrite the paper.

Now I realize how right he was in defending a physical journal from the Bourbaki-like mathematical jargon.

For instance, indeed, while claiming that “A implies B” the author must point out explicitly whether the converse holds, otherwise any reader not spoiled by the mathematical slang would understand the claim as “A is equivalent to B”.