The third chapter of The Art of Problem Posing begins with an anecdote of a speaker relating the following observation to a group of math teachers:
There are nine Supreme Court Justices. Each year, in an act of cordiality the Supreme Court session begins with each judge shaking hands with every other judge. (p. 19)He then asks what the question was for this setup.
What would you ask?
If you're like me you asked the, "How many handshakes..." question. The book states that the question was so obvious the speaker treated it as rhetorical, stated the question himself, and moved on with his talk. The authors then drop kick me in the head with this:
Many of us are blinded to alternative questions we might ask about any phenomenon because we impose a context on the situation, a context that frequently limits the direction of our thinking. We are influenced by our own experiences and frequently are guided by specific goals (e.g., to teach something about permutations and combinations), even if we may not be aware of having such goals.
The ability to shift context and to challenge what we have taken for granted is as valuable a human experience as creating a context in the first place. (emphasis added because it's so awesome)
I had a strong reaction to reading Ashli's experience working with Dan Meyer's Boat in the River problem. When I sat in to watch Dan present this video (IIRC, there were about 20 of us and presumably, most of us were teachers), the "right"question was asked by nearly all of us. I recall being shocked that Ashli's students were so all over the place and only one group got the "right" question. In her shoes, I can imagine myself being mildly irritated at what appears to be kids shouting out random questions in attempts to be, ummm, kids.
Look at this question - "Why does he put the headphones in?"
I would definitely get this question. I would get it from the kid who is always trying to not-so-smoothly listen to his ipod by resting his head on his hand and running an earbud through his sleeve. He'd probably follow up the question with a statement about how Dan's ipod is old and the newer gen is much better. I'd be dismissive and move on.
Except.... I'm the one at fault here. It's actually quite interesting why he needs headphones and is certainly worth investigation. Discovering that Dan needs to maintain a constant speed only enriches the problem that I had intended for us to investigate originally. If you want to get crazy, the followup question, "What if he didn't maintain a constant speed?" opens up a whole new, and perhaps more interesting, investigation.
My tunnel vision is only exacerbated when I've taken the time to develop and create the problem myself. It's fairly easy for me to divorce myself from a lab I found on google. On the other hand, when I've taken hours to create a lab or demo in order to launch a specific investigation, I find myself invested in the idea that this demo so wonderfully elicits the question I'm hoping my students will ask so I can teach the thing I wanted to teach.
Stepping back and allowing the process of problem solving and problem posing to grow organically is something I struggle with daily.
It hurts my brain to think about how often I've limited questions because I've imposed my own context.
One of the most valuable practices I did as a student teacher was to sit down with three other teachers on a regular basis and go through the work we were going to do with kids and try to find all of the possible misconceptions students might have when working on the problems. There were more often than not ones that we missed on our first go-around, since we couldn't anticipate when and where our students would take the material somewhere different than anticipated.
ReplyDeleteGreat post. It reminds me of some great parenting advice I got from my sister-in-law (who is also awesome). She says that she responds to the kid's emotion first, and then to the content. So if they spill something that's not really a big deal but are crying about it, she empathizes first with "oh no, that sucks!" before "it's no big deal, here I'll help you clean it up."
ReplyDeleteThe context issues you raise here remind me of that because if you can match a student where they're at with their passion and curiosity then you've made it. The rest is just gravy.
I sure I'm terrible at this but you've inspired me to be on the watch for opportunities.
Wonderful. Boy, do I ever not have answers either. I'm not sure I'd call the issue "context" though; to me, it's more like not going with the archetypal question for a given situation.
ReplyDeleteHere's something this made me wonder:
People shaking hands—we immediately go to "how many handshakes," but why? I guess because we've seen it a lot and it's a clean situation, easy to abstract, that gets at several important math ideas, ideas you need to come back to over and over. In your math life, you have to sum the integers up to n, so it helps, when you're trying to remember if it's n+1 or n–1 in the numerator, to have some situation in your head you can return to. Or if you come across some other problem that's isomorphic to this one, you can just say, "oh, this is a handshake problem…" and continue.
As math teachers, we've learned a slew of these. Recognizing these problems in all their guises is part of our problem-solving armamentarium, so it's no wonder we so easily fall into them. And the kids, well, they don't have them yet—maybe all problems have the same weight, in a way—so these archetypes have less gravitational pull.
I have often thought that we might do well to teach students these archetypes intentionally. This requires at least two things:
(1) We identify them as archetypes to students and give them prominence.
(a) We figure out what the archetypes are.
I'm even trying that a bit in my stats class right now; there's a particular problem (the "Aunt Belinda" problem) that I'm telling them they really have to memorize in all its detail because it's a model for a gazillion other problems.
But your post suggests that this whole idea may be wrong-headed, chaining our free-thinking students to an unnecessary orthodoxy!
I suppose, as with all things, there is a balance. I love it when I stumble across a question that flouts (instead of flaunts) orthodoxy. But I'm also curious about (a) above. What are the archetypal math-class problems? Certainly, boats in a river. Painting the fence. Later on, the Königsberg bridges. Cutting the opposite corners out of a chessboard. Let's Make a Deal. AIDS testing.
Hmmm. It all gets me thinking. Thanks!
While this has never been a strength of mine, I actually think I was better at this as a freshly minted teacher than I am now, simply because I had more of an "outsider" perspective and wasn't yet familiar with all the archetypes. The catch, however, was that I was scared and had no confidence in my own understanding-- I assumed that if I hadn't seen it in a book, it probably was somehow wrong or pedagogically counterproductive (probably indicative of my own haphazard math education).
ReplyDeleteTwo other random thoughts:
1) I find myself being intellectually lazy about these sorts of things much more frequently than I care to admit. I see a problem and think "oh nice" because it looks familiar enough to make sense and doesn't create enough cognitive dissonance. I wonder what it would take to make myself pause long enough to examine it critically and/or train myself to be jarred by bad problems.
2) I'm going to narrow the scope and talk about procedural problems here-- the kind that aren't nearly as interesting but (imo) still necessary for building a baseline automaticity or fluency-- I've been wrestling with how to help new teachers vary the procedural problems they give rather than issue worksheets with 20 problems and the numbers switched out, and suspect that part of the problem is similar to the one you and commenters have described: an inability to be truly generative and envision alternatives. Will write more about this on my own blog sometime, but got excited about the potential connection :)
Nice post! BTW, this is an example of a problem where the teacher is really interested in n handshakes and the kids don't really give a hoot. Pseudo-context, at best. Funny how the archetype problems can be.
ReplyDelete