Two weeks ago I went to an AIM Math Teachers’ Immersion workshop. Did some math. Met some nice people. Ate some good (free) food. Had myself a good-old time playing around with some new, some familiar mathematical concepts. In the morning we did some problems given to us by Alan Schoenfeld, who was leading the pre-lunch part of the day. He did my favorite kind of leading–giving us some problems and then letting us get to them.
My table group spent most of the time working on a problem about finding a construction for a square whose corners touched three sides of the triangle. We worked mostly individually, and then checked in with one another periodically, making some headway, but not quite figuring out how to construct the square in the time we had. Towards the end of the morning workshop, I decided to look at a few other problems before Alan did the wrap-up. So, I began working on a problem about magic squares.
I was happily playing around with the numbers in my as-yet non-magical square and seeing what patterns I could come up with, what I could deduce, when Alan walked by and saw me “struggling” on a problem that (as quoted from the handout) “wasn’t much of a challenge.” [I had seen this line previously, and had decided that the best means of dealing with this line was to pretend that it hadn't been written.] Alan intervened with me in order to help me move past this initial step. He did this very kindly and with some great teaching strategies. He asked me “what’s the most important piece?” and “what one piece of information would you need to help you make progress?” These were great questions to push me past my so-far fruitless wanderings in the problem towards a more productive path–and indeed, shortly thereafter I solved this problem with minimal effort.
Unfortunately, I didn’t want this help. And having succeeded in solving the problem after getting this help was really unsatisfying for me.
To his credit, Alan had no way of knowing that I had only been working on this particular problem for about five minutes and not for what could potentially have been much longer. I was still in the exploration phase of the problem; “making progress” towards the solution was not my focus at this point. I was looking for patterns, and I was thinking about this as collecting data. I was just playing around.
This interaction got me thinking about the choice we make all of the time as teachers about when to intervene in students’ work. This is a tricky road to navigate, as evidenced by my own experience as a learner who didn’t need and didn’t want intervention–at least not at that point in my work. How do you decide when to intervene? Is this a function of what students need/want, or is it due to other factors such as the fact that class will be over in ten minutes? Should intervention in student thinking only be a matter of what students want, or should the teacher prioritize the needs of the class and curriculum at times? How do you figure out whether or not someone is frustrated by not making progress or if they are just playing around?
I think I have an answer for the last question. My strategy for determining whether or not a student wants to keep playing or wants a nudge in the “right” direction is simple: ask them “What have you figured out so far?” The answer to this question can tell you an amazing amount of information and can serve as a window into their state of mind.
I’ll leave the other questions for better minds than my own to grapple with.