Saturday, February 28, 2009

Teaching Problem Solving To Kids

If I had to pick one skill that I found sorely lacking in today's mathematics students, it would be problem solving skills. And it's not hard to figure out why. To demonstrate, let me tell a story.

I used to have a pretty busy tutoring practice, and once I was hired to teach geometry to two girls who were part homeschooled, part enrolled at their local junior high. The school had provided a copy of their chosen textbook, so I went through it to make lesson plans, and I discovered something extremely odd. Right smack in the middle of each chapter was a section covering some topic in number theory, such as modular arithmetic or Mersenne primes. Those topics are cool, and the material presented was accessible and age-appropriate, but I couldn't figure out what the heck they were doing in the middle of a geometry textbook, especially since no effort was made at all to tie them into the subject matter in the rest of the chapter. Finally it dawned on me what was happening. I punched up the NCTM Standards and sure enough, the book's entire structure had been shamelessly cribbed from the Standards. And what standard corresponded to the number theory sections?

"Problem Solving In A Geometry Context."

Evidently the usual definitions of these words had been stretched to mean "problems to solve, surrounded on left and right by geometry." I just about laughed out loud.

Real problem solving is learned the exact same way as any other skill, like playing a piano or tying a Windsor knot. You watch it, you try it, you screw it up, you try it again and do better; lather, rinse, repeat. It is NOT learned by having unusual problems foisted upon you in a confusing manner and being taught specific "problem solving strategies" bald of any context in which they might be applied.

So here is Wacky Hermit's Patented Problem-Solving Teaching Method (OK, OK, it's just a list of tips.)
  1. Give the child a problem that you believe he can solve with the knowledge he's got. Affirm that it might be hard for him, but you know he can do it.
  2. When the child solves the problem, allow him to claim credit for his success.
  3. When the child asks for help, DO NOT GIVE HIM THE ANSWERS. Give him guidance and engage him in what will later become his "internal dialogue," giving him credit at the end. Example: "Mom, how do you solve this word problem?" "Mmmm, that's a tough one, son. Did you have any ideas?" "No, I don't know where to start." "OK, well there's some numbers here... what do you think we might do with these numbers?" "Add them?" "Why?" "Because it says 'find the total'?" "Very good, son! See, you figured it out!"
  4. Let the child catch you solving problems. If you solve a problem in front of the child, do it out loud. Example: "I wonder which ketchup is actually cheaper, the small bottles on sale or the large bottles that are regular price."
  5. Keep giving the child problems of gradually increasing difficulty, always validating him if he says the problems are difficult, but affirming your confidence in him that he can solve the problems even though they're hard. IMPORTANT: do not wimp out and continue giving the child easy problems. Children are smart and will eventually see that the problems do not get any harder and will figure out if you're lying about the difficulty level.
  6. Tell your child about famous problems, like the Traveling Salesman Problem, that do not have known solutions. Impress upon your child that solving one of these problems would make him rich and famous. (It's actually true-- if you came up with a solution to any of the NP-complete problems, Bill Gates would likely give you his billions for it, if the government didn't get to you first.)
  7. Allow your child to fail at problem solving and to come back later and try again to solve a difficult problem after engaging in an unrelated activity. If the child then solves the problem, see #2.
That's it! It really is that simple to teach a child true problem solving skills. You don't have to have a fancy degree in education to do it (ask me how I know!) and you don't need an NCTM Standards-aligned textbook either. Too bad people with both of those don't seem to be able to do it...