I just recently learned of an organization in NYC called the Grassroots Education Movement, which last Thursday premiered a documentary film with the awesome title The Inconvenient Truth Behind Waiting for Superman. They will apparently send you a copy for free; I just ordered mine.
Meanwhile, the city of New York continues to besiege its own public schools with budget cuts, looming layoffs, and a multi-year hiring freeze. (Having spent the year training 12 new teachers, let me not even get started on the hiring freeze.) Another thing that happened on Thursday was that East Side Community High School, a wonderful school on the Lower East Side where I used to teach and where the math teaching is strong enough that we placed four student teachers there this quarter, had its first fund-raiser. Like, big event, speakers, performances by students, pay to get in, as though it were a non-profit, carrying out its own civic mission and in need of private funding to do it, rather than a public school, charged with a civic mission by the state, which no longer sees fit to pay for it.
I missed both the documentary premiere and ESCHS’s fund-raiser because I was teaching the final class of a 3-session minicourse at Math for America on the fundamental theorem of arithmetic. Let me do a little reflecting on the execution:
At the end of the 2nd session, I gave participants about a half-hour to try to figure out something quite difficult. I attempted to scaffold this with some unobtrusive PCMI-style tricks in a previous problem set: sequences of problems with the same answer for a mathematically significant reason. It turned out not to be enough. There was high engagement the whole time, but no one seemed to be headed in my intended direction after that half-hour. On the other hand, that half-hour had made the group into a legit mathematical research community. What was afoot was a live process of trying things out, questioning, pressing on others’ logic, and generally behaving like research mathematicians. I was left with a dilemma. I had one session remaining. I wanted to protect that process, meaning I did not want to steal from them any of the deliciousness (or pain – also delicious) of the process they were in the middle of by offering them too much direction. But at the same time I felt I needed to guarantee that we would reach resolution. (Storytelling purposes.)
The solution I went with: I had them pick up in the final session where they left off, but I brought in a sequence of hints on little cut-up slips of paper. I tried to call them “idea-starters” as opposed to “hints” to emphasize that the game was you thinking on your own, and this is just to get you moving if you’re stuck, rather than I have a particular idea and I want you to figure out what it is, but I don’t think I was consistent with this, and I think they pretty much all called them “hints,” and I don’t think it really mattered. They were in an order from least-obtrusive to most-directive. None of them were very directive. Most importantly, I told the participants that if they wanted to get one, they needed to decide this as a table. (There were 6 tables with 3-4 folks each.)
How this went: a) it preserved the sense of mathematical community. I do not think there was much of a cost to participant ownership of what they found out. b) People were actually pretty hesitant to use the “idea-starters.” Most of them went untouched. This would probably be different with a different audience. (High schoolers instead of teachers?) c) The “idea-starters” worked great, but very slowly. I planned to spend 45 min letting them work in this arrangement, but after 45 min, most of the groups were still deep in the middle of something. After over an hour, I asked two groups to present what they had, however incomplete, for the sake of a change of pace and the opportunity for cross-pollination of ideas between the tables. I had actually meant to do a lot more of this but had forgot to mention it at the beginning. I let everybody work for another 10-15 min while these groups laid out their presentations. By the time they presented, I realized that there wasn’t enough time left for everyone to really get back to work afterward, but in any case their ideas had gotten more fully developed in that 10 min. so they actually had pretty much figured out everything I had wanted them to. I presented the final link in the logical chain, just to fill in the picture, in the last 5 minutes. It was pretty satisfying to me to watch the presentations, except that it happened so late in the session. This for two reasons. One was that I would have ideally liked to have time to encourage the participants to interrogate the presenters more, but there wasn’t time for that. The other was that I had intended to spend the last half-hour with the participants consolidating their understanding of the argument by applying it to a new situation in which they didn’t know the outcome and it would tell them; but we didn’t have time for that either. I really feel a loss about that.
If I were to repeat it I think I would interrupt much earlier to have people present partial work. The cross-pollination of ideas might or might not accelerate the figuring-out process. Either way I think the change of pace would have been good for concentration. Also, I could have put some of the questions I used as “idea-starters” into the Session 2 problem sets, trying to move some of the combustion I got in session 3 into session 2. But these would both be experiments as well. I hope I get a chance to try them.
Research in Practice 
Ben:
Thanks for being a voice for reason when it comes to public school education. I am attaching a reply that I wrote to the NYTimes yesterday in response to a front page article about Bill Gates and the out-sized influence he has had as a behind-the-scenes lobbyist for educational privatization and reform.
One of the many problems resulting from the ever growing disparity of income in our country is the fact that a few individuals with out-sized resources have the capacity to impose their often uninformed wills on so many others. Bill Gates, certainly a successful businessman, but nevertheless a college dropout, has had more influence on educational reform in recent years than anyone else in our country. Despite this, one has to doubt whether he will ever be held accountable for what, has so far, been the failure of most of the reforms that he has financed. With local economies struggling and taxpayers always looking for relief taking Gates’ money even with the strings attached seems like a no-brainer.
Whether his motives be charitable or simply service for the rich in reducing the power of unions is still unclear, but the education of millions of children hangs in the balance.
Sounds like a fun time with the MfA teachers. I feel there’s always a tough balance between goals and timing when you’re exploring like this, and one thing you might try for next time is setting several goals you’d be happy with as an “ender”. Then, when you get pressed for time, you can change the target instead of having to change the way the class works.
At PCMI we have more flexibility and opportunity since there are 14 or 15 sessions instead of 3; but in cases like this where you need to get somewhere specific TODAY we are more likely to stop-and-go. Instead of 45 minutes of group time, we might do 20, a 5-to-10 minute discussion, then another 15-20 minutes of group time. The second half of that time usually ends up being more directed toward the goal, for better or worse.
Hopefully you’ll get to do it again, it sounds like it went well.
Ah – thank you for elucidating the tension between non-stealing and storytelling. I’ve been feeling it without knowing what I was feeling. Of course when all goes
wellas planned, there’s not a distinction–they’re writing the story. When not, I’ve been tending to lose the story for fear of stealing. Love the idea of putting the hints/helps on separate slips.@ Howard – right on!
@ Bowen – yeah I totally meant to do the stop-and-go (in the sense I described above – groups presenting ideas for cross-pollination) and forgot to set it up at the beginning so I just let them work; but it would have been better I think. But the real something-for-nothing move would be to have had the idea-starters ready the previous session. But of course I didn’t know I needed them yet.
@ Dan – I took a course 2 falls ago with a great NYC teacher named Larry Zimmerman. He began every session of the class with his thoughts on the central tensions of teaching. Intuition vs. rigor, and theme vs. variations, were the two I remember. We can add non-theft vs. story. This one is recent language for me, though of course it’s been at play all along, and it feels closely related to one I’ve been thinking about explicitly for at least 3 years: freedom vs. orientedness. I have also been thinking a lot (and have written about, and intend to write more on) concreteness vs. abstraction – what Bob & Ellen Kaplan poetically call Ramanujan vs. Hilbert.
Wow that’s helpful. I could poetically call intuition vs. rigor “Period 8 vs. Period 2.” It’s funny; I don’t see all these as being completely orthogonal. I might put concreteness, intuition and freedom together against abstraction, rigor and orientedness. But I may be conflating things simply because particular students embody them in those groups. (And I am going to Bob and Ellen’s summer institute; I am so excited I can hardly sit still.)
Not entirely orthogonal but I think it’s worth taking the case that they’re closer to orthogonal than they sometimes seem. I am speaking as somebody who came of age in a department riven by the math wars, i.e. surrounded by a tendency of people to perceive all dichotomies in math pedagogy as essentially the same. This headspace didn’t get anyone anywhere.
For more concreteness ;), note that concrete contexts are orienting, so it’s never gonna be possible to make concreteness vs. abstraction line up completely with freedom vs. orientation.
I found my notes from Larry’s class, so here are the other two tensions he mentioned:
technique vs. theory
application vs. pure math
let me add ours, just to have them all listed together!
non-theft vs. story
freedom vs. orientedness
(these two really do feel related to me)
and Larry’s other ones:
intuition vs. rigor
theme vs. variations
I should probably put all these in its own blog post…
Oops I left out
concreteness vs. abstraction
Ok, so if I’m counting right, we’re in 7-space. Fine. I confess I don’t really know what these 14 directions (2 in each dimension) mean, or better: I know what I think they mean, but I don’t know what you or Larry think they mean.
For a dimension to be a tension I’ve got to feel a pull in both directions at the same time. I know how I experience that tension 15 minutes before the end of the period for non-theft vs. story. When do I feel the others, in your view?
I’d support a separate post (or this discussion is going to get annoyingly skinny).