When you decide to explain some math to a kid, how should you explain it? Step-by-step, or all at once?
There’s an issue with step-by-step explanations: kids have to remember what you’ve already said in order to understand what comes next. This means that there’s often a lot to hold in their head!
There’s an issue with fully worked-out examples: by not developing math slowly, in full view of the student, you make it seem as if the solution was dropped out of the sky. It can present a false picture of math: as constituted entirely of an encyclopedia of procedures that mathematicians memorize, look-up, and employ on canned problems.
I was thinking, today, about whether there is a way to get the best of both worlds. My mind wandered towards my 8th Grade class. We’re studying slope, so I launched class by putting two right triangles under the document camera. Which is steeper?
Students debated, thought some more, offered good and better approaches. I kept a record of their ideas, which I scanned after class:
I’m wondering, what if I started tomorrow’s class by projecting this image back on the board? I’d say, Here’s what we figured out yesterday. I’d like to give you two new triangles to look at today. I’m going to keep yesterday’s work up, in case it’s helpful.
Would that give a distorted view of mathematics and its development? Would that give students the benefits of worked-out examples?
Teaching With Problems 