Today, one of my student teachers wrote and taught a pretty solid lesson. We were introducing the standard form of the linear equation (Ax + By = C). The notes asked students to determine the combinations of chairs ($35/each) and/or pillows ($5/each) someone could buy with $70. This starting problem treated students like sense-makers in that it was largely accessible and students could go about solving in any way they pleased. Plus, while the problem is slightly contrived, it is closely related to actual situations (as opposed to, for example, problems where students have to figure out the price of something when obviously if they were at the store, they’d just look up the price).
Why is it illustrative? Because unfortunately, our schedule is such that we “have to” finish standard form by next Tuesday. And therefore, after students found all the combinations, we had to finagle our way into the equation (which a few students generated, while most hadn’t yet abstracted that far). And then we moved on to a bunch more scenarios (aka practice).
So when I talk about traditional math taught more creatively, this is what I’m talking about. That was a pretty creative lesson, I think, where students were respected and engaged. What do I envision for next year? Off the top of my head, for standard form equations, I could imagine students entering the room on the first day and finding various pairs of manipulatives with constraints at different tables. Students would then spend awhile finding ALL the possible combinations. Teachers would push students to represent those combinations in different ways — pictorially, tabularly, graphically, realizing the combinations were linear, and eventually representing them algebraically. Groups would present their representations and as a class, come up with a general form of a standard equation. Soonafter, we could apply standard form equations to systems, where they are more useful.
This is just one idea, but I hope serves to portray the goal for next year versus my current best efforts.