Problem Solving: Part 2 of 2. Reflection

I just finished posting my solution to the tangent circle to three circles problem.  As I worked through the problem I tried to pay attention to what I was thinking and feeling.  I’ve never been great at reflection or journaling, either one.  So this is a new process for me.

But I was able to identify a number of things:

  • I loved it.  I love the challenge.  The mystery.  Mostly I strive for the feeling that I get at the “Aha!” moments.
  • I puttered at it.  I’d gnaw at it a while, then leave it to soak.  I did not work at it constantly until it was done.  I have discerned over the years that this is my problem-solving style.  I think that’s okay.  I wonder how many of my students might have the same problem-solving style, but get trampled in the current structure of mathematics curriculum.
  • I went through probably 10 GeoGebra windows, some of them quite messy.  It’s a messy process.  Trial and error.  Travel down one path to see if it might lead somewhere.  It might.  It might not.  Experimentation/play/sandbox.  We don’t teach math this way.  Or at least I should say I don’t teach math this way.  I’m trying to change but it’s hard, and I need more problems centered around what I’m trying to teach, and more flexibility about covering a long list of content.
  • I tried to forget about it, but would find myself thinking about it at strange times. Why do these things “stick” in my head?  Why don’t they “stick” in my students’ heads?  Most of them well and truly forget about it.  Heck, most of them come in the next day and can’t even remember if they had an assignment, let alone what was on it.  Why?  What’s different about me?  Is it just that I know I’ll probably solve it?  Have they never had the little rush that comes with figuring out a problem on their own?  Have they just never actually figured out a problem on their own?
  • I wasn’t sure I could solve it.  I wasn’t positive that I could come up with a solution, and there was no answer key in the back of the book or the teacher’s edition.  I love climbing without a net, but I’m not sure my students do.
  • I felt smarter than hell when I did come up with a solution.  There’s probably some perfectly rational neuro-chemical explanation for how I felt when I came up with the answer, or when I realized that the extra intersection point was the center for the internally tangent circle to all three circles.  But I don’t care what it is.  It just feels, well, like a natural high of some kind.  It’s a rush.  “Schwing!”
  • I would be quite happy if someone would pay me to do this full time.  Really.
  • It took me 3 or 4 days.  If this had been a homework assignment problem that was due the next day, I probably wouldn’t have had it done.  Given my personality, if this were a one-time shot, I might have tried to stay up all night to solve it.  But if problems that did not have immediately obvious solutions to me were placed before me day after day after day with strict deadlines, I would probably have a tendency to quit.  I have renewed sympathy for my students.
  • I relied on a variety of background knowledge to solve the problem.  In addition to my GeoGebra construction skills (which are probably only average for someone who’s been at it for as long as I have), I needed fluent knowledge about circles, tangents, midpoints and hyperbolas to be able to solve this particular problem.  The lack of knowledge in one of these areas might have created an impenetrable roadblock for me.  As much as I dislike the laundry list of seemingly random and disconnected knowledge we are asking kids to learn, they are the necessary ingredients to being able to solve problems.
  • I relied on some problem-solving strategies to solve the problem.  In addition to the background skills, I had some idea how to navigate through the process.  Solving a simpler problem and just messing around with some drawings are time-honored favorites.  Still, I had some idea in which direction to start travelling.  Where this mental compass comes from, I don’t know.  How do you teach it?  Anyhow, both parts are necessary:  problem-solving ability and factual knowledge.  Neither one is of any use without the other.
  • None of the above bullets can be measured with a standardized test.