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.

Have you tried “Keep staring at it until you get an idea?” (I think I stole that from George Polya.) It sounds stupid but it conveys the message: “I’m not going to tell you steps 1, 2, and 3, and I don’t expect you to instantly know what to do, and I have faith in your capability to at least make progress.”

Polya’s “How to Solve It” is a great resource for showing others how to go about tackling a problem. One key idea is to work backwards: what would I need to have in order to do the problem easily? This can give you an intermediate, possibly simpler goal to aim for.

I haven’t read Polya for years – worth going back, I think!

Kate: I do that with my Calc students; I wonder why I haven’t tried it with Algebra 2? Thanks!

I am using Cuoco et al’s and Avery Pickford’s ideas about using routine pattern-seeking as a forum for teaching my Algebra 2 students some mathematical habits of mind, particularly how to tinker. In my classroom we call this process “wallowing.”

I’ve come to believe there are two kinds of people in this world — those who are natural tinkerers and those who (like me) were clueless about the idea of mathematical tinkering until we watched somebody with more confidence model the tinkering process for us and with us. For example, just today, in working with my Algebra 2 students on dealing with logarithmic and exponential functions, where you have a base raised to the power of an exponent that is itself a log with the same base it’s being raised to, I suggested that we turn the impossible-looking problem into a much less-impossible problem they already know how to solve — in this case, by substituting some arbitrary variable like “w” for the insane-looking logarithmic exponent.

Suddenly 2-to-the-log-base-2-of-8-equals-3 becomes a much more manageable problem (2-to-the-w-equals-3) and as a result, the transitive property starts to have actual value as a tool in their eyes. Students gasped and asked me, wait– you can DO that? To which I replied, Why not? It’s YOUR thinking process!

From then on, the light bulbs started going off all over the room.

I can’t help but think of a quote from Richard Feynman’s Nobel speech in which he talks about how misleading it can be in publication (or I would substitute “direct instruction”) when we try to make the finished work look as straightforward as possible by erasing and overwriting all the false starts, blind alleys, and wrong approaches we chase after before we stumble upon an approach that will take us somewhere that is actually meaningful.

Sometimes the most pedagogically meaningful thing we can model for students is the process of taking a wrong turn! Then they get to see how we backtrack and find our way to a better direction.

– Elizabeth (aka @cheesemonkeysf on Twitter)

Thanks, Elizabeth – there’s been a lot of talk this week here at school about modeling what attempts look like, and I’m thinking a lot about that this week! Very helpful and timely!