I read a the book Five Elements of Effective Thinking over the summer, and while lots of things in it struck me as stuff I need to remember to mention to my students, one thing has so universally prompted a “wow, that’s so obvious, how did I miss it” response that it’s quickly becoming something that I mention in pretty much every conversation I have about problem solving.
So here’s the question: which one is easier, writing a first draft or editing a first draft? Without exception, everyone has responded with “editing” as being the easier task. Editing whatever is there, discerning what is good and what is not, what works and what doesn't, has been seen as the easier thing to do. This leads my follow-up question: if editing is easier, then why not just get the first draft out of the way, regardless of how bad it may be. That way you can get to the task of editing, keeping the stuff that was good in the brainstorm first draft, and working with the stuff that wasn't to make it better.
Relating this to a math class: when it comes to problem solving, why not take the same approach? When tackling a problem, try something…anything. Get your thoughts down on paper, and then start sifting through what’s there to see what is worth keeping and what needs to be “edited”. Just brainstorm some ideas about how to attack the problem, not worrying about forcing any “algebra” or “geometry” into the process, but just working through how to solve the exercise. Once the idea about how to solve the problem comes into view, then put the equations and/or the pictures into the solution to communicate your ideas to others in the common languages of algebra and geometry.
For so long we have shown the kids how to solve the problems that they often don’t even consider brainstorming ideas about how to do it, and instead they go looking for “the formula” of “the example” that relieves them of any real thinking, which is a shame. The analog to this would be to have a kid writing an essay for an English class to forget about writing any sort of a rough draft, and instead asking them to simply use a template with lots of almost complete sentences that have a few blanks to fill in. That’s not how we teach kids to write an essay, and it shouldn't be how we teach kids how to problem solve. They learn to write by writing, discussing, and editing. The same holds true when they learn to problem solve.
Just one more reason to run a discussion-based classroom.