When they produce work without really absorbing it deeply, just getting the steps done without reflecting on them, I wish they had stopped me and said “We really don’t get this!” But when they don’t produce anything and tell me “We really don’t get this!”, I wish they would not give up, keep looking, keep going. It seems I may be trapping them in some conflicting expectations.
What’s in between being stopped cold, and getting someone (me, another student) to show you a procedure you can carry out mindlessly? Prior to having any understanding, what’s in between is to “try something”. What does that mean? When I solve math problems, it means executing one or another procedure on paper and then looking at the result to see if it seems to move me toward a solution.
Miguel Cura, one of my coaches (I am lucky to have a number of great coaches this year), pointed out that many of my students don’t know how to represent their guesses, conjectures, and trials-and-errors on paper. They know how to do guess-and-check by substituting values and simplifying, but rewriting expressions, substituting variables, factoring expressions and other techniques are all things they’ve learned to do when told, but not things that occur to them to try, just to see if the result looks like something they can move forward from.
Ben’s right: the problems are too hard at this stage. But not grasping the nature of the puzzle being posed is only half the reason. The other half is, I want them to keep looking, but “looking” at this stage usually means rewriting things many different ways without knowing in advance how it’s going to work out, and they are unfamiliar with that whole mode of exploration.
All the modeling that I’ve done, that other students have done, none of it is showing up on the tests (even from the students who did the modeling!) because they are still trying to memorize routes from A to B rather than wandering around, getting oriented to landmarks, so when they have to get from C to B or from B back to A they can do it because they know the neighborhood.
It’s not the curriculum: CPM is good about this, even explicit about it. It’s just taken me this long to figure out that this is what I need to use these materials to teach them how to do.