Ha ha. J/k. There isn't a picture for polynomials. That's insane. The question about polynomials comes up, though, especially when we give into the fiction that students can't enjoy math for its own sake.
Let me highlight two positive externalities of WCYDWT, which is to say, benefits of WCYDWT that don't limit themselves to the time that we are actually WCYDWT-ing:
- The class understands that non-standard approaches are awesome.
- The class understands that failure is useful, not shameful.
You can capture those benefits using traditional curriculum but you have to work a lot harder at it and if you stop working harder, you capture the negative externalities: students come to understand that math is a right or wrong endeavor in which "wrong" is an destination unto itself rather than just another waystation to "right."
The last two years of my career I facilitated classes that were often fearless and creative. That meant this: if they were really confident with trinomials like x2 + 7x + 6, I didn't have to lecture. I'd just write on the board: 2x2 + 7x + 6.
Which would offend them. You know, like, "how dare you bring that weak stuff in here, Meyer? You didn't see what we just did to the last trinomials?"
Because they were creative and because failure had little stigma attached to it, students would start putting answers down. They'd experiment. In math. Worst case, maybe one of them would throw down (2x + 7)(x + 6) — just banging the numbers from the question together, hoping to see some sparks. She'd call me over and ask if it was correct. I'd tell her to check it. "You know how to multiply binomials."
She'd see she missed it — 2x2 + 19x + 42 — but we'd notice she nailed the 2x2 — "keep that!" — and ask her to experiment some more. My role in class was to help condense and summarize the findings of student experimentation.
This is how you maintain the spirit of WCYDWT even for concepts that seem to defy the spirit of WCYDWT.