Some Followup on “A Note to My Fellow White People” Sunday, Mar 17 2013 

If you were interested, challenged or otherwise engaged by my Note to My Fellow White People, I have come across a bunch of other things recently you will be interested in:

Here is the other video he refers to in the video:

Also a propos is this recent opinion piece in the NYT by Ta-Nehisi Coates.

I was talking in general about white people receiving feedback about race, but several people who commented took it (very reasonably) in the direction of how to have conversations about race in the classroom. In which case I have the following strong book recommendation:

High Schools, Race, and America’s Future: What Students Can Teach Us About Morality, Community, and Diversity

I am cross-posting my review of this book on goodreads.com:

Full disclosure: the author of the book is my dad. The high school featured in the book is the one I both attended and taught at.

THAT SAID.

This is a beautiful book. The author is a (white, Jewish) professor of philosophy at a university. The book chronicles his venture into teaching a class about race and racism at his local racially diverse public high school. It offers a model of what a functioning, productive cross-race conversation about race and racism can look like, in an era where (depressingly) this is still a rarity. It makes a case for the civic value of integrated public education in an era where we seem to be forgetting that education even has a civic purpose.

It belongs broadly to the genre of teaching memoirs, along with books like Holler if You Hear Me. But two related features distinguish it in this genre:

(1) The author is a serious scholar. Unsurprisingly, then, the content of the course he taught features heavily in the book. So this teaching memoir also functions, with no cost to readability, as a scholarly book about race. (As an aside, I am very proud of him on the readability front. It was a real stretch for him to write a book whose style didn’t place a technical burden on the reader, and it took a lot of rewrites, and help from his editor, but he totally pulled it off!)

(2) The genre is characterized by taking students seriously as moral and psychological beings. That’s one of its strengths as a genre as a whole. But this is the first book I’ve read that takes students equally seriously as intellects. The author often writes with plain admiration for his students’ ideas. This may be my favorite feature of all. Developing students as minds is, after all, the point of education. So it strikes me as surprising that it’s so rare for a memoir about the lived experience of teaching to give such loving attention to what those minds produce.

A Note to My Fellow White People Friday, Jan 18 2013 

I haven’t talked openly about race or racial difference on this blog before, but I actually think about it a lot. One of the most damning legacies of our racist history has been systematic libel against the minds of black and brown children (and adults for that matter). Meanwhile, in our culture, math is the ultimate signifier of intelligence. So the math classroom has heightened power, both to inflict injustice and to rectify it. Given this, plus the diversity of teachers and students, a comfortable cross-race conversation about racial matters is a must for the profession. In the spirit of contributing to that conversation, I offer

A Note to My Fellow White People

Guys, we have to chill out a little. It has to be possible for somebody to say to you, “that was ignorant,” or “that was racially offensive,” or even “that was racist,” without you flipping out, getting offended or defensive, or needing to be reassured you are not a horrible person. It’s not a good look, on any level: it’s not dignified, and it makes it impossible to have a productive conversation about race across racial lines.

I was at a cafe a couple months back trying to get some schoolwork done when I found myself distracted by a profoundly uncomfortable conversation at the next table. There was a white man in his early 50s and two black women, one close to his age and one closer to mine. They seemed to be sharing a familiar and friendly meal. Things started to go south when the man admitted to being afraid of a young black man on the street. The younger of the women said something to the effect of, “you might have work to do on that.”

Her tone was warm: she wasn’t being accusatory but rather seemed to be offering her words in the spirit of holding her friend to a high standard. But the man immediately became anxious, although his face and words were all smiles and jokes. His first response was that white people make him more uncomfortable than black people, as though he could re-establish his lost racial coolness with sufficiently loud declamations of prejudice against white people.

The women weren’t having it. “You’re being ignorant against white people now.” I interpreted their response as saying, “you can’t get off the hook with this diversionary tactic.” But he kept trying. His anxiety was as audible to me as a fire alarm, even when he wasn’t talking. I tried to concentrate on my math but I couldn’t get anything done.

Things stayed in this state, a tense, anxious impasse overlaid by a thin layer of too-eager conviviality and jokes, for about 20 minutes, till they got up to leave, no noticeable progress having been made in the conversation. At this point the man, in that same overly-eager joking tone, almost-but-not-quite-explicitly asked for reassurance that everybody was still his friend. They gave him the reassurance. On their way out, the younger woman leaned over to my table and apologized for her “ignorant friend.”

I’m not telling you this story to put the man down or call him ignorant. I don’t remember the context of the conversation and I don’t have my own opinion about it. Also, I think in all likelihood he’s a completely nice and decent person, and so are the women.

The point of the story is the man’s intense anxiety at being put on the spot racially, and the way that anxiety dominated both the conversation and its goals (so that what started as an attempt to raise consciousness was aborted, and turned into a reassurance fest), and the social and public space (so that the younger woman felt the need to apologize to a neighboring table).

Now I don’t fail to have empathy for him. If you are a white person with a modicum of sense and decency, you know that you are the beneficiary of an unjust history. (Shout out to Louis CK.) Just knowing that you’re benefiting is already a little uncomfortable to begin with. Feeling like you might be participating in that injustice can make the discomfort acute. I’ve been there many times.

But, guys, we’ve got to get it together! It is necessary to learn how to be with that discomfort and still function. First of all, the story I just told you is about a grown-a** man! Trying to prove how un-racist you are, and then needing to be coddled and preened so that you know the trouble is past, is unbefitting of the dignity of an adult. So is any other response aimed at removing the source of your discomfort rather than tolerating it – throwing a fit, acting defensive or offended, etc. Shouldn’t we aspire to some grace here?

Secondly, it makes it impossible for the conversation to advance! If we want to avoid participating in injustice we have to be willing to tolerate the possibility that we already are participating. Otherwise how will we learn what to avoid? In the anecdote I’ve recounted here, the man’s anxiety shut down the ability of the conversation to make any progress. He was blessed with friends who were willing to hold him to a higher standard and he was too busy freaking out to get the benefit of that! The bottom line question is, would you rather spend your time and energy proving how un-racist you are, or would you actually like to learn how to make the world better?

All of this puts me in mind of a much more public incident. In 2009, Attorney General Eric Holder gave a speech at the Dept. of Justice Black History Month program in which he said that Americans are afraid to talk about race and called upon us to do better. Multiple commentators immediately jumped down his throat.

Thereby proving his point.

The Attorney General made an effort to hold the nation to a higher standard. At the time, we didn’t react with grace or manifest any interest in growing.

How about now?

Featured comment

Aiza:

IMO the best thing white teachers, or any teachers who find themselves teaching classes of black/brown students can do is to constantly hold their students to the same high standards they would hold their own biological children to. Giving these kids a high standard education is one of the few ways to equip these kids to deal with racism.

Wherein This Blog Serves Its Original Function Wednesday, Nov 21 2012 

The original inspiration for starting this blog was the following:

I read research articles and other writing on math education (and education more generally) when I can. I had been fantasizing (back in fall 2009) about keeping an annotated bibliography of articles I read, to defeat the feeling that I couldn’t remember what was in them a few months later. However, this is one of those virtuous side projects that I never seemed to get to. I had also met Kate Nowak and Jesse Johnson at a conference that summer, and due to Kate’s inspiration, Jesse had started blogging. The two ideas came together and clicked: I could keep my annotated bibliography as a blog, and then it would be more exciting and motivating.

That’s how I started, but while I’ve occasionally engaged in lengthy explication and analysis of a single piece of writing, this blog has never really been an annotated bibliography. EXCEPT FOR RIGHT THIS VERY SECOND. HA! Take THAT, Mr. Things-Never-Go-According-To-Plan Monster!

“Opportunities to Learn Reasoning and Proof in High School Mathematics Textbooks”, by Denisse R. Thompson, Sharon L. Senk, and Gwendolyn J. Johnson, published in the Journal for Research in Mathematics Education, Vol. 43 No. 3, May 2012, pp. 253-295

The authors looked at HS level textbooks from six series (Key Curriculum Press; Core Plus; UCSMP; and divisions of the major publishers Holt, Glencoe, and Prentice-Hall) and analyzed the lessons and problem sets from the point of view of “what are the opportunities to learn about proof?” To keep the project manageable they just looked at Alg. 1, Alg. 2 and Precalc books and focused on the lessons on exponents, logarithms and polynomials.

They cast the net wide, looking for any “proof-related reasoning,” not just actual proofs. For lessons, they were looking for any justification of stated results: either an actual proof, or a specific example that illustrated the method of the general argument, or an opportunity for students to fill in the argument. For exercise sets, they looked at problems that asked students to make or investigate a conjecture or evaluate an argument or find a mistake in an argument in addition to asking students to actually develop an argument.

In spite of this wide net, they found that:

* In the exposition, proof-related reasoning is common but lack of justification is equally common: across the textbook series, 40% of the mathematical assertions about the chosen topics were made without any form of justification;

* In the exercises, proof-related reasoning was exceedingly rare: across the textbook series, less than 6% of exercises involved any proof-related reasoning. Only 3% involved actually making or evaluating an argument.

* Core Plus had the greatest percentage of exercises with opportunities for students to develop an argument (7.5%), and also to engage in proof-related reasoning more generally (14.7%). Glencoe had the least (1.7% and 3.5% respectively). Key Curriculum Press had the greatest percentage of exercises with opportunities for students to make a conjecture (6.0%). Holt had the least (1.2%).

The authors conclude that mainstream curricular materials do not reflect the pride of place given to reasoning and proof in the education research literature and in curricular mandates.

“Expert and Novice Approaches to Reading Mathematical Proofs”, by Matthew Inglis and Lara Alcock, published in the Journal for Research in Mathematics Education, Vol. 43 No. 4, July 2012, pp. 358-390

The authors had groups of undergraduates and research mathematicians read several short, student-work-typed proofs of elementary theorems, and decide if the proofs were valid. They taped the participants’ eye movements to see where their attention was directed.

They found:

* The mathematicians did not have uniform agreement on the validity of the proofs. Some of the proofs had a clear mistake and then the mathematicians did agree, but others were more ambiguous. (The proofs that were used are in an appendix in the article so you can have a look for yourself if you have JSTOR or whatever.) The authors are interested in using this result to challenge the conventional wisdom that mathematicians have a strong shared standard for judging proofs. I am sympathetic to the project of recognizing the way that proof reading depends on context, but found this argument a little irritating. The proofs used by the authors look like student work: the sequence of ideas isn’t being communicated clearly. So it wasn’t the validity of a sequence of ideas that the participants evaluated, it was also the success of an imperfect attempt to communicate that sequence. Maybe this distinction is ultimately unsupportable, but I think it has to be acknowledged in order to give the idea that mathematicians have high levels of agreement about proofs its due. Nobody who espouses this really thinks that mathematicians are likely to agree on what counts as clear communication. Somehow the sequence of ideas has to be separated from the attempt to communicate it if this idea is to be legitimately tested.

* The undergraduates spent a higher percentage of the time looking at the formulas in the proofs and a lower percentage of time looking at the text, as compared with the mathematicians. The authors argue that this is not fully explained by the hypothesis that the students had more trouble processing the formulas, since the undergrads spent only slightly more time total on them. The mathematicians spent substantially more time on the text. The authors speculate that the students were not paying as much attention to the logic of the arguments, and that this pattern accounts for some of the notorious difficulty that students have in determining the validity of proofs.

* The mathematicians moved their focus back and forth between consecutive lines of the proofs more frequently than the undergrads did. The authors suggest that the mathematicians were doing this to try to infer the “implicit warrant” that justified the 2nd line from the 1st.

The authors are also interested in arguing that mathematicians’ introspective descriptions of their proof-validation behavior are not reliable. Their evidence is that previous research (Weber, 2008: “How mathematicians determine if an argument is a valid proof”, JRME 39, pp. 431-459) based on introspective descriptions of mathematicians found that mathematicians begin by reading quickly through a proof to get the overall structure, before going into the details; however, none of the mathematicians in the present study did this according to their eye data. One of them stated that she does this in her informal debrief after the study, but her eye data didn’t indicate that she did it here. Again I’m sympathetic to the project of shaking up conventional wisdom, and there is lots of research in other fields to suggest that experts are not generally expert at describing their expert behavior, and I think it’s great when we (mathematicians or anyone else) have it pointed out to us that we aren’t right about everything. But I don’t feel the authors have quite got the smoking gun they claim to have. As they acknowledge in the study, the proofs they used are all really short. These aren’t the proofs to test the quick-read-thru hypothesis on.

The authors conclude by suggesting that when attempting to teach students how to read proofs, it might be useful to explicitly teach them to mimic the major difference found between novices and experts in the study: in particular, the idea is to teach them to ask themselves if a “warrant” is required to get from one line to the next, to try to come up with one if it is, and then to evaluate it. This idea seems interesting to me, especially in any class where students are expected to read a text containing proofs. (The authors are also calling for research that tests the efficacy of this idea.)

The authors also suggest ways that proof-writing could be changed to make it easier for non-experts to determine validity. They suggest (a) reducing the amount of symbolism to prevent students being distracted by it, and (b) making the between-line warrants more explicit. These ideas strike me as ridiculous. Texts already differ dramatically with respect to (a) and (b), there is no systemic platform from which to influence proof-writing anyway, and in any case as the authors rightly note, there are also costs to both, so the sweet spot in terms of text / symbolism balance isn’t at all clear and neither is the implicit / explicit balance. Maybe I’m being mean.

Math Is Not a Spectator Sport Monday, Nov 5 2012 

Neither is democracy.

Everybody in the USA better vote tomorrow.

Take the Effing Tests Guys Friday, Jun 1 2012 

In December, Kate had a great idea, which I seconded.

We just got backed by Diane Ravitch.

I’d say it’s time to take this one to the streets.

Justin Lanier Thursday, Apr 26 2012 

How did I miss that Justin Lanier started blogging (finally!) last August?

His blog is called I Choose Math. Keep your eye on this one, he’s the real deal.

P.s. Japheth Wood Is My Dude Friday, Mar 2 2012 

One reason why.

Dispatches from the Learning Lab: Yup, Time Pressure Sucks Friday, Mar 2 2012 

Continuing the series I began here and here, about snippets of new-feeling insight about the learning process coming from my new role on the student side of the desk…

This one is funny, because I knew it, I mean I knew it in my bones, from a decade working with students; but yet it’s totally different to learn it from the student side. I’m a little late to the blogosphere with this insight; I’ve been thinking about it since December, because it kind of freaked me out. Even though, like I keep saying, I already knew it.

Learning math under time pressure sucks. It sucks.

It sucks so much that I ACTUALLY STOPPED LIKING MATH for about 5 days in December.

I didn’t know this was possible, and I don’t think anyone who’s ever worked closely with me in a mathematical context (neither my students, colleagues, or teachers) will really believe it. But it’s true. It was utterly, completely unfun. There was too much of it and too little time. It was like stuffing a really delicious meal down your throat too quickly to chew, or running up the Grand Canyon so fast you puke. Beautiful ideas were everywhere around me and I was pushing them in, or pushing past them, so hard I couldn’t enjoy them; instead they turned my stomach, and I had the feeling that the ones I pushed past in a hurry were gone forever, and the ones I shoved in weren’t going to stay down.

I had some independent study projects to work on during winter break, and what was incredible was the way the day after my last final exam, math suddenly became delicious again. Engaging on my own time and on my own terms, that familiar sense of wonder was back instantly. All I had to do was not be required to understand any specific thing by any specific date, and I was a delighted, voracious learner again.

Now part of the significance of this story for me is just the personal challenge: most of the grad students I know are stressed out, and I entered grad school with the intention of not being like them in this respect. I was confident that, having handled adult responsibilities for a decade (including the motherf*cking classroom, thank you), I would be able to engage grad school without allowing it to stress me out too much. So the point of this part of the story is just, “okay Grad Program, I see you, I won’t take you for granted, you are capable of stressing me out if I let you.” And then regroup, figure out how to adjust my approach, and see how the new approach plays out in the spring semester.

But the part of the story I want to highlight is the opposite part, the policy implication. Look, I frickin love math. If you’ve ever read this blog before, you know this. I love it so much that most of my close friends sort of don’t feel that they understand me completely. So if piling on too much of it too quickly, with some big tests bearing down, gets me to dislike math, if only for 5 days, then the last decade of public education policy initiatives – i.e. more math, higher stakes – is nothing if not a recipe for EVERYONE TO HATE IT.

And, not learn it. Instead, disgorge it like a meal they didn’t know was delicious because it was shoved down their throat too fast.

In short. The idea of strict, ambitious, tested benchmarks in math to which all students are subject is crazy. It’s CRAZY. The more required math there is, and the stricter the timeline, the crazier. I mean, I already knew this ish was crazy, I’ve been saying this for years, but in light of my recent experience I’m beside myself. If you actually care about math, if you have ever had the profound pleasure of watching a child or an adult think for herself in a numerical, spatial or otherwise abstract or structural context, you know this but I have to say it: the test pressure is killing the thing you love. Its only function is to murder something beautiful.

If you teach, but especially if you are a school leader, and especially if you are involved in policy, I beg you: defend the space in which students can learn at their own pace. Fight for that space.

Take the Tests, Decisionmakers! Thursday, Dec 8 2011 

In case any of you missed this at f(t):

A school board member in Orange County, FL had the guts to sit for his state’s high-stakes test, the type of test a lot of decisionmakers are all in such a rush to have students’ futures and teachers’ livelihoods resting on.

Kate is asking her readers to call on NY Governor Cuomo to do the same thing.

This is effing brilliant. I say we take it up a notch. If you live in the US, pick an elected or appointed government official or purveyor of “education reform” who is rushing to rest more and more human futures on the results of a test, and call on them to take the test. I am not trying to be an organizer right now; I suppose it would be smart to make some strategic choices about whom to contact and via what medium (Kate: Cuomo / Twitter), but that’s not my style. I do have some nominations:

Arne Duncan
Bill Gates

Because these folks are operating at the national level, it’s not obvious which test to tell them to take. I want to say all of them, but maybe that’s just cuz I’m pissed off. Abnegating my role as organizer I’ll let you call it. Here’s one that’s easy:

NYC Mayor Michael Bloomberg

Take the NY Regents, mayor, and make the results public. I don’t care how you do, but I want you to know what you’re talking about when you make policy, and I want you to be willing to be scrutinized as you are insisting that students, teachers, and schools be.

*****

Here’s what I love about this.

The last few years have felt to me like American schools are riding on top of a malfunctioning robot that is careening inexorably toward more and more insane school policy. The robot is being driven by an inflated sense of the importance and automatic legitimacy of numerical data. For a decade, a chorus of voices (many of people directly involved in the practice of education) have been crying out that this is madness,[1] but the robot has only sped up.

<Interlude>

During the same decade, and especially in the last few years before this fall, the language used by national political figures advocating for justice and progressive change has felt more and more tepid to me. The clearest instance of this is the way that Democrats and even some progressive advocacy groups have latched onto the phrase “middle class.” Y’all are giving up the fight, guys. If you feel you are not allowed to advocate for working people or (God forbid) poor people, that in order to be a legitimate public interest your cause has to be sanded down and shellacked with a patina of educated white-collarness, then the folks who are only looking out for the interests of rich people have already prevailed.

My mood in relation to this language was not unlike my mood when beholding current debates about education: the feeling that justice and sanity are speaking, but being ignored; and they cannot find the language that will make the powerful listen.

So, imagine my thrill when this fall a new language took over: the 99%. Whatever you think of the Occupy Wall Street movement, you have to give it credit for a complete reshaping of the vocabulary available to discuss economic inequality. It seemed like everywhere I went this fall, somebody was talking about either “the 99%” or “the 1%” or both. This is just what I was missing: a way of talking about economic justice that feels powerful and relevant. That interrupts the inexorable slide into tepid lameness that characterized the national discourse till now.

</Interlude>

What I’m getting at here is that we need ideas to interrupt the inexorable careening of the malfunctioning education reform robot, and Kate may just have found one. In the words of Rick Roach, the Orange County school board member who took the Florida tests,

“I can’t escape the conclusion that decisions about the FCAT in particular and standardized tests in general are being made by individuals who lack perspective and aren’t really accountable.”

You know I don’t love the word “accountable,” the way it is thrown around these days. But these are the folks who do love it. So if they love it so much, let’s make them accountable. What I really mean is this: the public defamation of public schools and teachers, and the concomitant policy initiatives, have been based on numerical data from tests whose contents are public, but this is the only public thing about them. Most critically, their development is opaque, the way the data is used is opaque, and the way that decisions get made about how the data is used is therefore not subject to legitimate public scrutiny, or even, in all probability, based on any real understanding of the tests. The decisionmakers don’t even know what taking the tests is like!

So, decisionmakers, take the tests! You are willing to force students to take them, to scrutinize the results, and to make important decisions about students, teachers, and schools on their basis. Finding out what you’re actually forcing on them, and opening yourself up to the same scrutiny, is the least you could do.

[1]One of these voices was the television show The Wire, which aired well before the latest and most intense phase of this insanity, but which in spite of this develops a beautifully articulated critique of numbers-driven accountability in municipal institutions. Schools are included, but the brunt of the criticism is aimed at the police department and the city government. However, the essential problem is the same in all cases: when you demand numbers from people who are supposed to be doing a job requiring creative problem-solving and perseverance, you divert their attention from their actual work to the problem of giving you what you’re asking for. If you’ve never seen the show, you can get the whole thing from Netflix. You won’t be sorry. If you think I shouldn’t be citing a fictional television show regarding public policy, let me quote Mathnet: the names are made up but the problems are real. Not convinced? Read this.

The Inconvenient Truth Behind Waiting for Superman and other stories Monday, May 23 2011 

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.

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