Case in point: Yesterday, Eva was working to solve this problem:
Show how the wall might look.
[NOTE: there was a rectangle drawn below the problem.]
She'd already solved the previous problem (painting the wall half red & half blue) correctly and was working on this one quite dilligently. She had a blue colored pencil and had colored in about one fifth of the rectangle (vertically) and was looking at it thoughtfully.
Then, she looked around and saw that some other children were sectioning off their rectangles using pencils, so she took a pencil and looked back at the rectangle. After a moment or two, she started drawing lines to make different sections.
At about that moment, one of the other students needed my attention, so I called them over and spoke to them about their work, then turned back to watch Eva.
"Miz F," she said, her eyebrows wrinkling, "if I make another line there, it will be five."
"Show me what you mean," I said.
She showed me how she'd drawn another two lines to divide the rectangle into four parts with a larger space left over. So, if she drew another one (so they were all equidistant and had equal spaces), there would be five sections of the wall instead of four.
"It's too much spaces," she said.
"I wonder if you could make the blue area bigger -- would that help?"
"I'll see," she said.
Eva then proceeded to extend the blue section, draw two more lines to make four equal sections, then colored them in.
When she was done, she grinned at me and pointed to her work. "I have four equal parts," she said proudly.
The reason this has been on my mind for the past day is simply this: I have to make the time to listen to the children as they're working. I have to keep my own mouth shut more often and let their own thinking be the guide.
The thing is, Eva has a tough time learning. She has a lot stacked against her in life right now, and it's pretty easy for her to get frustrated. This time, though, she didn't get frustrated. She didn't just attack the problem, she thought ahead to what needed to happen next and realized that it wasn't going to be correct. Sure, she used a push from me to get back on track, but who is to say that she wouldn't have found that push on her own?
And you know what? Now we get to think back on that moment. We get to use Eva as a model for thinking ahead when you're working (and not just in math!) and thinking critically to make sure that you're solving the problem that you're trying to solve.
Pretty powerful as far as I'm concerned.