The problem with asking your students to tell you what they know is that then you have to deal with it and the fact that what you thought was a bang up lesson really wasn’t.
Case in point, my “fabulous” concept attainment lesson on rational numbers. I love concept attainment a lot, because I love watching the kids get excited when they think they have it figured out, and the continued cycles of this as I add more complicated examples that smash their ideas so they continue to refine their definitions. The kids really did a bang up job, and as a class did a really good job of defining rational numbers, so good that the formal definition just fell right out into our laps. They were also able to generate lots of examples of things that were not rational numbers, which is just as important. High fives all around right?
Sigh, wrong. Two days later I had them fill out an exit slip that had their own definition of “rational number” as well as two examples of rational numbers and two examples of non-rational numbers. Somewhere around two-thirds of the class gave me a bang up definition of irrational numbers instead… and the examples/counter examples were all over the place.
So it would appear we have some work to do!