Ask about, not for

We too often ask for programmed responses. We seldom ask for judgments. Asking about processes, methods, and results, instead of asking specifically for a given process, method, or result gives students the freedom to break out of the Pavlovian mode.

A typical exercise that asks for the application of a specific process is:

Find the inverse of the Laplace transform .

A problem that asks about the process is:

The Laplace transform of a function is . Does the function grow, decay, or oscillate?

Students could choose to invert the transform, or they could simply observe that the form of the denominator forces an exponential in time.

Of course, that last exercise is not perfectly posed, for the adjectives grow, decay, and oscillate hardly exhaust the set of possible behaviors. To add to the ambiguity, give the transform of a function that does not fall cleanly into one of these three classes:

The Laplace transform of a function is . Does the function grow, decay, or oscillate?

Are grow, decay, and oscillate mutually exclusive choices?

In the same spirit, an instructor can ask about an outcome rather than for a specific outcome. Instead of

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

ask

Does the solution of y' = y, y(0) = 1, increase or decrease?

Leave to the student the choice of the route to an answer. The outcome might be a solution formula, or it might be some alternate analysis of solution behavior.

The fog of ambiguity can be made successively denser by asking:

Do the solutions of y' = y increase or decrease?

Do the solutions of y'' = -y increase or decrease?

Do the solutions of y'' = y increase or decrease?

Ask about methods rather than for their specific application. Don't place in front of a list of equations an instruction like:

Solve the following equations using the methods of this chapter.

Instead, try:

For each of the following equations, list every method you know that can be used to solve it. Justify each claim you make. Use one of those methods to solve each equation.

Include in the list some equations to which no method applies and others which hark back to chapters past; e.g., nonlinear equations when the methods are linear or first order equations in a chapter on second order methods.

Both of the preceding problems exercise the solution processes, but the second requires judgments about which methods are appropriate and why. The art of making such judgments is innately more important than temporary memorized mastery of the steps in a solution process.