Bad mathematics good problems

The precision at the heart of our discipline serves us poorly when we try to ease away from sterile, mechanical problems. In an introductory course, well posed questions are often boring; they invite mechanical responses from students, who simply search for an isomorph among the examples in the chapter under study. Assigning too many well posed problems teaches trivial pattern recognition, not analysis or problem solving or any other skill our discipline demands.

Good problems contain too much or too little or contradictory information. Good problems may not have unique solutions, and they do not define the desired solution. They often require that students reformulate a better posed problem from that which is first put to them.

For example, a carefully posed problem is:

Find the unique solution of the initial value problem y' = y, y(0) = 1.

Some more interesting variants that omit or add information or contradict themselves are:

Find the unique solution of y' = y, y(0) = 1, y(1) = 0.

Find the unique solution of y' = y.

Find the general solution of y' = y, y(0) = 1.

Find a nontrivial solution of y' = y, y(0) = 0.

We should always ask the well posed question first to build confidence. We should not stop there, though. Students will better grasp concepts if they see them in different - even wrong - settings. And they will develop critical judgment and problem solving skills if they are forced to formulate and answer a better question than the one they were asked. (There is considerable anecdotal evidence that this process of finding the right problem among a myriad of questions is important in the practice of mathematics in industry [3].)

From a student's perspective, problems that involve parameters instead of numbers are almost as ambiguous as if they were ill posed. As their maturity increases, students realize that the vague ambiguity of a poorly posed problem is much different from the clearly limited range of options introduced by the variability of a parameter. Developing a sense of that distinction is another argument for posing problems that include parameters.

For example,

Find the unique solution of y' = y, y(0) = a.

is no harder than if the initial condition were y(0) = 1. But the parameter is a small additional challenge.

The parameter can be explicit or implicit:

Do solutions of y' = y, y(0) = a, increase or decrease?

Do solutions of y' = y increase or decrease?

Of course, the first question would be more straightforward if it were worded, ``Does the solution of ...''.

Students should have to make decisions about what values of the parameter need to be considered and about the effect of parameter values on both processes and outcomes. Compare these three problems:

Solve .

Solve .

Solve .

The first is straightforward, the second asks only that the parameter a be dragged along, with perhaps a brief look at a = 0, and the last demands separate consideration of a = 1.