# Fear, courage, and place in problem solving

Sorry for the slowdown in posting. It’s been tremendously busy here lately with hosting our annual high school math competition this past weekend and then digging out from midterms.

Today in Modern Algebra, we continued working on proving a theorem that says that if $a$ is a group element and the order of $a$ is $n$, then $a^i = a^j$ if and only if $i \equiv j \ \mathrm{mod} \ n$. In fact, this was the third day we’d spent on this theorem. So far, we had written down the hypothesis and several equivalent forms of the conclusion and I had asked the students what they should do next. Silence. More silence. Finally, I told them to pair off, and please exit the room. Find a quiet spot somewhere else in the building and tell me where you’ll be. Work on the proof for ten minutes and then come back.

As I wandered around from pair to pair I was very surprised to find animated conversations taking place about the proof. It wasn’t because of the time constraint — they’d been at this for three days now. For whatever reason, they were suddenly into it. One pair was practically arguing with each other over the right approach to take. By the end of the 10 minutes, two of the groups had come up with novel and mathematically watertight arguments. Between the two, and with a little bit of patching and a lemma that needs to be proven still, they generated the proof.

One student made the remark that she had been thinking of these ideas all along, but she didn’t feel like it was OK to say anything. This is a very verbal, conversational class done in Moore method style, so I can only interpret that comment to mean that she didn’t feel free enough, or bold enough, to say what she was thinking. The right proof was just bottled up in her mind all this time.

There’s something about our physical surroundings which figures in significantly to our effectiveness as problem solvers. Getting out of the classroom, for this one student at least, was tantamount to giving her permission to have the correct thoughts she was already having and to express them in a proof. I think our problem solving skills are highly inhibited by fear — fear that we will be wrong. And it takes a tremendous amount of confidence and/or courage as a problem solver to overcome that fear.

When you feel that fear in a classroom, it becomes compounded by the dread of looking like an idiot. Changing the surroundings — making things a little less cozy, a little more unusual and uncertain — doesn’t seem to make the fear go away as much as it helps us feel like that fear is perfectly normal and manageable, if not less fearful.

### 3 Responses to Fear, courage, and place in problem solving

1. eclexia

I think you’re right–a lot of times we try to battle fear by talking people out of their fears. Personally, I have found it more helpful to discover contexts where the reality of my fear doesn’t become the issue, but where I’m able to go ahead and do whatever, even with the fear. I guess it’s like rather than fear being conquered, the fear is disarmed enough to stop being such a huge obstacle. Thanks for stirring my thinking on this.

2. Makes me want to go back to school and take a full set of math classes, mostly again. I loved the stuff, but never pulled it together well enough (and in a compact enough period of time).

But look how you combine the good math with the good teacher stuff! I could do that (or at least some of that) as a student… Even with dry lecture and “the proof is left as an exercise for the student” classes, I could be core to killer study groups.

Not going to happen. Oh well. But still loads of fun to read. Sounds like a great class.

Jonathan