Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

When solving assigned problems, do you often think " What does he want us to do?"?

I'm asked this question a lot: "What do you want us to do?" or "I didn't know what you wanted us to do! " after the fact. My standard answer is "I want you to think!". Questions on a test or an assignment should be designed to expand your understanding, i.e., they should take you a little beyond what you have understood before.

Sometimes they are also designed simply to assess how much you have learned. Those questions aren't nearly as interesting to do or to pose, but they may be necessary. We grade because we are required to, not because we love to! As you go on in your studies emphasis should gradually shift from assessment type questions to horizon expanding questions. When you are a graduate student your instructor may ask you questions that nobody in the world knows the answer to - without telling you that fact - just to see how you react to the question.

You should ask yourself: How is this related to what we did before (inside and outside of class)? What is the problem going to teach me? What do I expect the answer to look like? Have I seen this kind of problem before? Can I solve it similarly? How does it relate to other problems? Is the solution obvious? (It often is!) In any reasonably conducted class you should be able to go beyond the immediate problem if you see an intriguing aspect of a problem. I get very excited when I see a student do this, the more excited the more sophisticated the student's action (and I'll let the student or the entire class know).

Don't ask "what does he want us to do?"! I'm not your boss. I'm your guide and servant, and hopefully I can inspire you to get excited about what we are doing in class. I want you to think, to get curious, to ask questions yourself, to be critical, to ask me questions!

Fine print, your comments, more links, Peter Alfeld, PA1UM