I remember one occasion, in my first year of graduate school, when a classmate asked a question in class. I have no memory of what the question was, but I recall the professor’s answer vividly. He told her "that question cannot be asked." Apparently, she had asked a question for which there was no answer. I know that there are such questions in math, as just because a question can be written down does not mean that it can be solved. I believe that this relates closely to the challenges of parenting, where we can sometimes encounter what seem to be impossible situations.
There are several series of books on parenting that encourage parents to let their children make their own mistakes. While I think this is useful advice now that my daughter is getting older, I still have doubts as to whether this is good advice for a parent of a young child. After all, isn’t one of our jobs as parents to protect our children from the dangers of the world? I especially found myself objecting to this approach when I first read these books while she was in kindergarten.
I often think of this approach to parenting when I encourage my students to work on math problems that are particularly difficult. Because I see the value of this approach to learning, my "lectures" often turn into seminars, with students working problems or proofs together, often jumping up to illustrate an idea on the board. Luckily, I am not threatened by sharing board space with them, or of going (slightly) off track. This approach was one that was reflected in a comment I read in Inside Higher Ed recently.
In a discussion on how to encourage students to study and succeed in the STEM disciplines (Science, Technology, Engineering and Math and/or Medicine), one commenter suggested that teachers ask students to work a smaller set of problems, and then to create similar problems which would be presented to the class. As the commenter noted, it is entirely possible that the new problem would be very difficult or even impossible to solve, but the process of encountering a difficult or impossible problem is useful in itself. The parenting advisors believe that making choices and living with the consequences will help children grown in maturity. I believe that, in a similar way, struggling with difficult problems, making mistakes in solving them, and then taking new approaches help a student gain a deeper understanding of math.
I am curious as to whether any of my readers have used such an approach in teaching their classes. If so, how did you manage to keep your students from inventing impossible problems (or did you?) What did you do if those students presented a steady stream of impossible problems to the class? Did your students appreciate this approach, and did you find it enjoyable? I would love to hear from other professors who may have tried it. I think I may try to use it, if not now, in the end of Calculus I, at least as I begin to teach Calculus II in a few months.
P.S. In response to the commenter from last week who asked why we make such an issue out of turning 21; that is exactly the point. Ursuline College tried to come up with a safe, healthy and classy way to celebrate a milestone that is often marked with the excessive consumption of liquor.