# Teaching proof writing

I’m at BEAM 7 (formerly SPMPS) right now. I just taught a week-long, 18 hour course on number theory to 11 awesome middle schoolers. I’ve done this twice before, in 2013 and 2014. (Back then it was 20 hrs, and I totally sorely missed those last two!) The main objective of the course is some version of Euclid’s proof of the infinitude of the primes. In the past, what I’ve gotten them to do is to find the proof and convince themselves of its soundness in a classroom conversation. I actually wrote a post 4 years ago in which I recounted how (part of) the climactic conversation went.

This year, about halfway through, I found myself with an additional goal: I wanted them to write down proofs of the main result and the needed lemmas, in their own words, in a way a mathematician would recognize as complete and correct.

I think this happened halfway through the week because until then I had never allowed myself to fully acknowledge how separate a skill this is from constructing a proof and defending its soundness in a classroom conversation.

At any rate, this was my first exercise in teaching students how to workshop a written proof since the days before I really understood what I was about as an educator, and I found a structure that worked on this occasion, so I wanted to share it.

Let me begin with a sample of final product. This particular proof is for the critical lemma that natural numbers have prime factors.

Theorem: All natural numbers greater than 1 have at least one prime factor.

Proof: Let $N$ be any natural number $> 1$. The factors of $N$ will continue descending as you keep factoring non-trivially. Therefore, the factoring of the natural number will stop at some point, since the number is finite.

If the reader believes that the factoring will stop, it has to stop at a prime number since the factoring cannot stop at a composite because a composite will break into more factors.

Since the factors of $N$ factorize down to prime numbers, that prime is also a factor of $N$ because if $N$ has factor $Y$ and $Y$ has a prime factor, that prime factor is also a factor of $N$. (If $a\mid b$ and $b\mid c$ then $a\mid c$.)

There was a lot of back and forth between them, and between me and them, to produce this, but all the language came from them, except for three suggestions I made, quite late in the game:

1) I suggested the “Let $N$ be…” sentence.
2) I suggested the “Therefore” in the first paragraph.
3) I suggested the “because” in the last paragraph. (Priorly, it was 2 separate sentences.)

Here’s how this was done.

First, they had to have the conversation where the proof itself was developed. This post isn’t especially about that part, so I’ll be brief. I asked them if a number could be so big none of its factors were prime. They said, no, this can’t happen. I asked them how they knew. They took a few minutes to hash it out for themselves and their argument basically amounted to, “well, even if you factor it into composite numbers, these themselves will have prime factors, so QED.” I then expressed that because of my training, I was aware of some possibilities they might not have considered, so I planned on honoring my dissatisfaction until they had convinced me they were right. I proceeded to press them on how they knew they would eventually find prime factors. It took a long time but they eventually generated the substance of the proof above. (More on how I structure this kind of conversation in a future post.)

I asked them to write it down and they essentially produced only the following two sentences:

1. The factoring of the natural number will stop at a certain point, since the number is finite.
2. If $X$ (natural) has a factor $Y$, and $Y$ has a prime factor, that prime factor is also a factor of $X$.

This was the end product of a class period. Between this one and the next was when it clicked for me that I wanted proof writing to be a significant goal. It was clear that they had all the parts of the argument in mind, at least collectively if not individually. But many of the ideas and all of the connective tissue were missing from their class-wide written attempt. On the one hand, given how much work they had already put in, I felt I owed it to them to help them produce a complete, written proof that would stand up to time and be legible to people outside the class. On the other, I was wary to insert myself too much into the process lest I steal any of their sense of ownership over the finished product. How to scaffold the next steps in a way that gave them a way forward, and led to something that would pass muster outside the class, but left ownership in their hands?

Here’s what I tried, which at least on this occasion totally worked. (Quite slowly, fyi.)

I began with a little inspirational speech about proof writing:

Proof writing is the power to force somebody to believe you, who doesn’t want to.

The point of this speech was to introduce a character into the story: The Reader. The important facts about The Reader are:

(1) They are ornery and skeptical. They do not want to believe you. They will take any excuse you give them to stop listening to you and dismiss what you are saying.

(2) If you are writing something down that you talked about earlier, your reader was not in the room when you talked about it.

Having introduced this character, I reread their proof to them and exposed what The Reader would be thinking. I also wrote it down on the board for them to refer to:

1. The factoring of the natural number will stop at a certain point, since the number is finite.

(a) What does finiteness of the number have to do with the conclusion that the factoring will stop? (b) Why do you believe the numbers at which the factoring stops will be prime?

2. If $X$ (natural) has a factor $Y$, and $Y$ has a prime factor, that prime factor is also a factor of $X$.

What does this have to do with anything?

(I don’t have a photo of the board at this stage. I did do The Reader’s voice in a different color.)

Then I let them work as a whole class. I had the students run the conversation completely and decide when they were ready to present their work to The Reader again. In one or two more iterations of this, they came up with all of the sentences in the proof quoted above except for “Let $N$ be…” and minus the “Therefore” and “because” mentioned before. They started to work on deciding an order for the sentences. At this point it seemed clear to me they knew the proof was theirs, so I told them I (not as The Reader but as myself) had a suggestion and asked if I could make it. They said yes, and I suggested which sentence to put first. I also suggested the connecting words and gave my thinking about them. They liked all the suggestions.

This is how it was done. From the first time I gave the reader’s feedback to the complete proof was about 2 hours of hard work.

Let me highlight what for me was the key innovation:

It’s that the feedback was not in the teacher’s (my) voice, but instead in the voice of a character we were all imagining, which acted according to well-defined rules. (Don’t believe the proof unless forced to; and don’t consider any information about what the students are trying to communicate that is not found in the written proof itself.) This meant that at some point I could start to ask, “what do you think The Reader is going to say?” I was trying to avoid the sense that I was lifting the work of writing the proof from them with my feedback, and this mode of feedback seemed to support making progress with the proof while avoiding this outcome.

Postscript:

As you may have guessed, the opening phrase of the sentence “If the reader believes…” in the final proof is an artifact of the framing in terms of The Reader. Actually, at the end, the kids had an impulse to remove this phrase in order to professionalize the sentence. I encouraged them to keep it because I think it frames the logical context of the sentence so beautifully. (I also think it is adorable.)

# Sue’s Book Is Ready for Press and Needs Crowdfunding!

Hey y’all, I am incredibly excited about Sue’s book, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. If you have been around the math education blogosphere for more than a short time, you probably are too.

It needs crowdfunding to cover publication costs. I am about to help out and I invite you to do so too!

# Kids Summarizing

Back in the spring, I resolved to make a practice of having students summarize each others’ thoughts whenever I have classroom opportunities. This summer, I got the opportunity to give this technique a sustained go, when I taught at SPMPS (which was completely awesome btw). And:

It is an effing game-changer.

This summer, when I or a student put forth an idea, I regularly followed it with, “who can summarize what so-and-so said?” Or (even better), “so-and-so, can you summarize what so-and-so just said?” Following the models of Lucy West and Deborah Ball, I carefully distinguished summary from evaluation. “Not whether you buy it, just the idea itself.” When dipsticking the room on an idea, I would also make this distinction. “Raise your hand if you feel that you understand what was just said; not that you buy it, just that you understand what they’re trying to say.” Then, “leave your hand up if you also buy it.”

These moves completely transformed the way whole-class conversation felt to me:

* Students were perceptibly more engaged with each others’ ideas.
* The ideas felt more like community products.
* Students who were shy to venture an idea in the first place nonetheless played key roles as translators of others’ ideas.

Furthermore, for the first time I felt I had a reliable way past the impasse that happens when somebody is saying something rich and other people are not fully engaged. More generally, past the impasse that happens when somebody says something awesome and there are others for whom it doesn’t quite land. (Whether they were engaged or not.)

A snippet of remembered classroom dialogue to illustrate:

Me: The question before us is, do the primes end, or do they go on forever? At this point, does anybody think they know?

(Aside: This was after a day of work on the subject. Most kids didn’t see the whole picture at this point, but one did:)

[J raises his hand.]

J: They don’t end. If they ended, you’d have a list. You could multiply everything on the list and add 1 and you would get a big number N. Either N is prime or it’s composite. If it is prime, you can add it to the list. If it is composite, it has at least one prime factor. Its factor can’t be on the list because all the numbers on the list when divided [into] N have a remainder of 1. So you can add its factor to the list. You can keep doing this forever so they don’t end.

Me: Raise your hand to summarize J’s thought.

(Aside: although J has just basically given a complete version of Euclid’s proof of the infinitude of the primes, and although I am ecstatic about this, I can’t admit any of this because the burden of thought needs to stay with the kids. J is just about done with the question, but this is just the right thing, said once: the class as a whole is nowhere near done. This is one of the situations in which asking for summaries is so perfect.)

[Several kids raise their hands. I call on T.]

T: J is saying that the primes don’t end. He says this because if you have a list of all the primes, you can multiply them and add one, giving you a big number N. If N is prime, you can add it to the list. If N is not prime, and its prime factors are not on the list, you can add them.

Me: J, is that what you were trying to say?

J: Yes.

(Notice that a key point in J’s argument, that the factors of N cannot already be on the list, was not dealt with by T, and J did not catch this when asked if T had summarized his point. This is totally typical. Most kids in the room have not seen why this point is important. Some kids have probably not seen why J’s argument even relates to the question of whether the primes end. All this has to be given more engaged airtime.)

Me: raise your hand if you feel that you understand the idea that J put forth that T is summarizing.

[About 2/3 of the room raises hands. I raise mine too.]

Me: Leave your hand up if you also find the idea convincing and you now believe the primes don’t end.

[A few kids put their hands down. I put mine down too.]

N [to me]: Why did you put your hand down?

Me [to class]: Who else wants to know?

[At least half the class raises hands.]

Me [to T]: Here’s what’s bugging me. You said that if N is not prime and its prime factors are not on the list, I can add them. But what if N is not prime and its prime factors are already on the list?

T [thinks for a minute]: I don’t know, I’ll have to think more about that.

[J’s hand shoots up]

Me [to T]: Do you want to see what J has to say about that or do you want to think more about it first?

[T calls on J to speak]

J: Can’t happen. All the numbers on the list were multiplied together and added 1 to get N. So when N is divided by 2, 3, 5, and so on, it has a remainder of 1. So N’s factor can’t be 2, 3, 5, and so on.

T: Oh, yeah, he’s right.

Me: Can you summarize his whole thought?

[T explains the whole thing start to finish.]

T: Yes.

Me: Who else wants to summarize the idea that J put forth and T summarized?

Unexpectedly, this technique speaks to a question I was mulling over a year and a half ago, about how to encourage question-asking. How can the design of the classroom experience structurally (as opposed to culturally) encourage people to ask questions and seek clarification when they need it? The answer I half-proposed back then was to choose certain moments in the lesson and make student questions the desired product in those moments. (“Okay everyone, pair up and generate a question about the definition we just put up” or whatever.) At the time I didn’t feel like this really addressed the need I was articulating because it had to be planned. Kate rightly pressed me on this because actually it’s awesome to do that. But I was hungering for something more ongoingly part of the texture of class, not something to build into a lesson at specific points. And as it turns out, student summaries are just what I was looking for! The questions and requests for clarification are forced out by putting students on the spot to summarize.

A last thought. Learning this new trick has been for me a testament to teaching’s infinitude as a craft. Facilitating rich and thought-provoking classroom discussions was already something I’d given a lot of thought and conscious work to; perhaps more than to any other part of teaching, at least in recent years. I.e. this is an area where I already saw myself as pretty accomplished (and, hopefully with due modesty, I still stand by that). And yet I could still learn something so basic as “so-and-so, can you summarize what so-and-so said?” and have it make a huge difference. What an amazing enterprise to always be able to grow so much.

# Deborah Ball and Lucy West are F*cking Masters

I recently saw some video from Deborah Ball’s Elementary Mathematics Laboratory. I actually didn’t know what she looked like so I didn’t find out till afterward that the teacher in the video was, y’know, THE Deborah Ball, but already from watching, I was thinking,

THAT IS A F*CKING MASTER. I F*CKING LOVE HER.

It put me in mind of a professional development workshop I attended 2 years ago which was run by Lucy West. Both Ball and West displayed a level of adeptness at getting students to engage with one another’s reasoning that blew me away.

One trick both of them used was to consistently ask students to summarize one another’s train of thought. This set up a classroom norm that you are expected to follow and be able to recapitulate the last thoughts that were said, no matter who they are coming from. Both Ball and West explicitly articulated this norm as well as implicitly backing it up by asking students (or in West’s case, teachers in a professional development setting) to do it all the time. In both cases, the effect was immediate and powerful: everybody was paying attention to everybody else.

The benefit wasn’t just from a management standpoint. There’s something both very democratic and very mathematically sound about this. In the first place, it says that everybody’s thoughts matter. In the second, it says that reasoning is the heart of what we’re doing here.

I resolve to start employing this technique whenever I have classroom opportunities. I know that it’ll come out choppy at first, but I’ve seen the payoff and it’s worth it.

A nuance of the technique is to distinguish summarizing from evaluating. In the Ball video, the first student to summarize what another student said also wanted to say why he thought it was wrong; Ball intercepted this and kept him focused on articulating the reasoning, saving the evaluation step until after the original train of thought had been clearly explicated. Which brings me to a second beautiful thing she did.

Here was the problem:

The first student to speak argued that the blue triangle represents half because there are two equal wholes in the little rectangle at the top right.

He is, of course, wrong.

On the other hand, he is also, of course, onto something.

It was with breathtaking deftness that Deborah Ball proceeded to facilitate a conversation that both

(a) clearly acknowledged the sound reasoning behind his answer

and

(b) clarified that he missed something key.

It went something like this. I’m reconstructing this from memory so of course it’s wrong in the details, but in overall outline this is what happened –

Ball: Who can summarize what [Kid A] said?

Kid B: He said it’s half, but he’s just looking at the, he’s just…

Ball: It’s not time to say what you think of his reasoning yet, first we have to understand what he said.

Kid B: Oh.

Kid C: He’s saying that the little rectangle has 2 equal parts and the blue is one of them.

Ball [to Kid A]: Is that what you’re saying?

Kid A: Yeah.

Ball: So, what was the whole you were looking at?

Kid A [points to the smaller rectangle in the upper right hand corner]

Ball: And what were the two parts?

Kid A [points to the blue triangle and its complement in the smaller rectangle]

Ball: And are they equal?

Kid A: Yes.

Ball [to the rest of the class]: So if this is the whole [pointing at the smaller rectangle Kid A highlighted], is he right that it’s 1/2?

Many students: Yes.

Ball: The question was asking something a little different from that. Who can say what the whole in the question was?

Kid D [comes to the board and outlines the large rectangle with her finger]

Kid A: Oh.

I loved this. This is how you do it! Right reasoning has been brought to the fore, wrong reasoning has been brought to the fore, nobody feels dumb, and the class stays focused on trying to understand, which is what matters anyway.

# A Note to My Fellow White People

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.

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.

# What She Said

Three weeks ago Sue VanHattum and Kate Nowak recommended Bob and Ellen Kaplan’s Math Circle Training Institute. If you are looking for a PD opportunity this summer and you are interested in cultivating students thinking for themselves, I strongly second their recommendation.

This is a weeklong training on the campus of Notre Dame in South Bend, Indiana where you learn how to run a math circle in the spirit of the Kaplans. What that means is that you ask thought-provoking questions and you facilitate students discussing them. Heaven, right? The setup is that in the morning, the Kaplans run a math circle on you, and in the afternoon they bus in local kiddies for you to try out your thought-provoking questions on, and watch others do it, and give and receive feedback. At lunch and at night you hang out with like minded educators talking about math and education. The \$850 includes room and board for the whole week.

I did this training in the summer of 2009 and it was a key step on my path to being the educator I am now. In 2007-8 I had come to the realization that my most central, pressing goal as an educator was to empower students to find their own mathematical curiosity, and I started stretching my pedagogical boundaries to find out what it would look and feel like to teach with this as the only goal. But I felt like I was reinventing the wheel. Reading the Kaplans’ book Out of the Labyrinth, I felt like I had found my comrades. Going to the Summer Institute, I felt like I had met them.

For example, Sue and Alex, and my fairy blogfamily Kate and Jesse Johnson. See what I mean?

Tangential to the math PD but also a wonderful benefit was the opportunity to spend a week on the Notre Dame campus. As a Jew I did not go into the experience expecting to be so moved by the shrines and sanctuaries of this Catholic institution, but I was. After my first experience with a labyrinth (the meditative kind), Alex McFerron said to me, “the Catholics really ace those sacred spaces.” True that.

# Still Here, Still Learning

I last posted in October. I wrote a review of Waiting for Superman that generated more traffic than I’d ever seen before on this blog. Since I had been intending to continue my series on the idea of mathematical talent since the summer, I decided not to post again until I was done with the next installment of that series. But because it involves some research, and I care about it a lot and want to get it just right and tend to get kind of obsessive about things like that, and because there’s been a lot of other stuff going on so I haven’t been working on it consistently, this has kept me from posting anything at all for 4.5 months. So maybe it was time to revisit that agreement with myself?

And a few days ago, JD2718 wrote me an email to the effect of, “yo, what happened to you?”

So, here’s a partial answer –

a) I learned a lot about leadership. One of my jobs this year has been to facilitate the weekly math department meeting at a high school, and plan the agenda for this meeting. This has gotten me involved with the communication channel between the department and the principal. I feel really grateful to have had the opportunity to do this. It has caused me to start to develop a completely different skill set than I’ve ever had to use before. (To give you a whiff of what I mean, it inspired the following facebook status: “Ben Blum-Smith thinks it is important to be a straight-shooter and a diplomat, and that you do each better by doing the other one.”)

b) I learned a lot about training new teachers. Another of my jobs this year has been as a faculty member of an MAT program. In the fall, my colleague Japheth Wood and I taught a “math teaching 101” typed course for our cohort of 12 preservice folks; this winter we taught the “math teaching 102” installment. They’ve been in apprenticeships for 9 weeks and we’ve just gone through observing them actually teach a few times, so now on my mind is – what am I happy with in their teaching? What’s missing? And what implications does all that have for our fall and winter courses?

c) I’ve continued to design and implement a graduate course on algebra and analysis for the faculty of a high school. This has been both awesome and very challenging. We chose to organize the course to culminate with the Fundamental Theorem of Algebra. At the beginning of the year I thought this was a reasonable goal and the course would not feel hurried. Now, 2/3 of the way in, somehow I’ve found myself feeling pressure to go through significant chunks of material at breakneck speed. That tension is of course absolutely part of the lives of all the participants in their own classrooms, so in a way it’s cool that this is parallel; but still. I am implicitly making a case with this course for the principles of math teaching I believe in, so I’d better be living those principles in my teaching of it. A few of them I feel like I’ve been 100% consistent with:

* Every day I will bring you questions that are worth your time, questions that even I think are exciting to think about even though I already know the content.
* A math course should have a plot, with beginning, middle, end, dramatic tension, resolution. (Math teaching as storytelling.)
* Central to learning math is the interplay between formal/rigorous thoughts, definitions etc. and intuitive notions. I will always stress the connections between the two.

Other principles I feel like I’ve nailed some of the time and totally let slip away other times in my concern to make sure we get to the content:

* (Closely related) The most powerful certification of new knowledge is consensus of the learning community, the same way new knowledge is certified in the research community.

3 classes ago I had them prove the irrationality of $\sqrt{2}$, spent the whole period on it, left them all the heavy lifting, noticed and brought out points that were bothering people, and generally aced these last two principles. The last two classes have felt the opposite way. I think I was talking 80% of the time in the most recent class. Lots of questions never got answered because they never got aired; lots of productive thoughts never got formed because they never had time to. Anyway, getting this course right will continue to be an engaging challenge.

d) I applied to doctoral programs in math. Now I need to decide where to go. The choices are NYU, CUNY and Rutgers. I feel very excited and torn.

e) If anybody remembers the ellipse problem that Sam Shah brought back from PCMI, and which I wrote about back in August… Japheth and I have completely solved it. I am going to tease you with this tidbit and not the solution itself because we wrote a manuscript on it which we hope to get published.

f) Okay this doesn’t fit under the rubric of “what happened to me” but here are some links you might enjoy:

* A Teacher Story by Anna Mudd. Anna’s blog, Drawmedy, is a beautiful kind of writing which I won’t try to describe. It’s not an education-themed blog so I was delighted to see her take on her experience as a teacher.

* This gem from Vi Hart: Wind and Mr. Ug

* Taylor Mali’s What Teachers Make. This poem is definitely amazing, and if you’ve never seen it, I think you won’t be sorry if you watch it before reading the next sentence. <pause>Pause while you watch the video.</pause> It brings up some ambivalent feelings in me too – these are a story for another time, but here’s the short version: It’s related to the tone of the current national conversation about education, which is all about how the incompetent slovenly dumb*sses in front of our children are f*cking everything up. In this context, Mali’s piece is an eloquent testament to the value of our work, but it also makes me uncomfortable. Mali appears to have been amazingly happy with the job he was doing as a teacher when he wrote and performed this. But I don’t think that (especially in light of the current climate of the conversation) feeling like you’re doing an amazing job should be in any way a requirement for testifying to the value of your work; especially since most of us do not feel that way, most of the time.

* Speaking of the current national conversation about education, a new study by the National Education Policy Center came out on New York City’s charter schools, which are often touted as models for the nation.

* It’s weird to experience yourself as an unwitting participant in a historical zeitgeisty trend, but I do. I have the strong feeling that the traditional distance between the mathematics education community and the mathematics research community is closing, and I, a classroom teacher and teacher trainer entering into a math PhD program, am like completely an example of that. Another is the latest issue of the Notices of the American Mathematical Society, which is the research community’s professional association. It is devoted to education. You can download it for free.

(Thanks, JD2718, for making me write all this.)

# Talking Openly about How to Do It Better

Last week I somewhat impulsively picked up and read cover-to-cover the new book of an important mentor of mine.

Linda is the principal of the Boston Arts Academy, where I did my student teaching a decade ago, in what I believe was the school’s 3rd year of existence. The book is largely a collection of vignettes from the BAA’s 12-year history. The vignettes have a theme:

Education involves facing difficult dilemmas. The thing that needs to be done is to bring together the people involved, open up the lines of communication, and try to figure out jointly what to do.

Some of these dilemmas are pedagogical, some pragmatic, some political, and some interpersonal. Some are a combination. The community of people involved may be administration, teachers, students, parents, or a combination. But however configured, Nathan is saying this process is at the heart of education: put the hard choice to the community, and keep everyone engaged with each other as you undertake to work it out.

This book was a very refreshing read for me. We are deep in the days of Arne Duncan, Michelle Rhee, Race to the Top, the Common Core Standards, and the tendency among journalists1 to regard the KIPP schools as the greatest thing that have ever happened anywhere in the universe because they have high test scores. Now I have some nice things to say about some of these things. The Common Core Standards in 6th to 8th grade math are an order of magnitude better (i.e. shorter and less concerned with trivia) than the New York State standards have been, and while I have no firsthand knowledge of KIPP schools, I’ve been curious about them in a good way since my student teaching year at BAA, when a fellow student teacher came back from a visit to a KIPP school very excited about SLANT. But what this list is meant to capture is that I can’t escape the feeling that the highest-profile conversations about education in this country, in their frenzy regarding accountability and competition, have totally lost sight of the following facts:

a) Students are people and they have cares and values.
b) Teachers are people and they have cares and values.
c) Everybody involved has cares and values.
d) Education takes place in a community. (Corollary: improving education involves improving community.)

Reading The Hardest Questions… felt like walking into a room full of people who had never lost touch with any of this. Nathan is talking about thinking through educational dilemmas with her staff and students and being guided by what all the people involved value. Stating and working for what matters to her, and asking her teachers and her students what matters to them. It’s absurd that this should feel like a refreshing notion, but to me right now, it does. The Race to the Top funding criteria include a lot about assessments and data that will be used to measure teacher and principal effectiveness, and no encouragement whatsoever for students, teachers, principals or even state superintendents to reflect on what they value.

Another refreshing aspect of The Hardest Questions… is that it doesn’t uniformly make Linda or the BAA look good. (Often – and from firsthand experience they are good – but not uniformly.) The book narrates some play-by-play encounters with some difficult conundrums that don’t have clear resolutions, so it airs some missteps. (Different readers will probably count different moves as missteps.) One of the most pernicious elements of the accountability-and-results orientation in the national conversation about education is that it gives everybody (states, schools, teachers and students) a great reason to hide every mistake. You can’t learn math while you’re trying to hide your mistakes and you can’t learn to teach that way and you can’t learn to run a school that way. You can’t learn that way, period.2

Some specific themes and highlights:

* Schools need to develop a “unifying framework” – what the school stands for educationally. This is not a mission statement that collects dust in an administrative folder but a vision articulated frequently to students of the most important themes in their education. The faculty needs to be involved in developing it. The administration needs to be willing to commit to it in a long-term way. The school community needs to periodically revisit whether and how the school is implementing this shared educational vision. At the BAA, the unifying framework the faculty eventually came to, after 2 years of discussion and debate, is a list of four “habits of the graduate” – refine, invent, connect, own. The idea is that these words are the faculty’s answer to the question, “what we are committed to cultivating in every student?” and that this goal defines the school. Nathan makes a point that she initially tried to have faculty sign on to other lists of words (that to an outsider now don’t look so different), but it turned out to be necessary for the faculty to go through the intense and time-expensive process of answering this question for themselves.

I am suspicious of statements that begin with “All schools should…” But this is one I truly stand behind: all schools should develop and use a unifying framework. The “new initiative every year” model doesn’t work. Teachers need to be involved in articulating the framework, and a school must be willing to commit to the implementation of the framework over the long haul. Finally, I would argue that schools without a unifying framework still have an unspoken one – a de facto assumption of what this school is about. If it were expressed in posters on the wall, these frameworks might be “We Are Failing: Who Should We Blame?” or “High Scores and College Admissions – Everything Else Be Damned!” To honestly answer the question “What does your school stand for?” takes a willingness to ask again and again how your practices are improving, what students know and can do, and how day-to-day realities in the classroom match the ideals you have articulated. pp. 30-1

* Developing a school’s commitment to social and moral values also takes a community-wide process, and this one has to go beyond the faculty to the students. And it needs to be continually recreated, because new kids come every year. Chapter 2 of Nathan’s book describes how the BAA faculty first articulated a group of “Shared Values” in response to a community crisis (a “white power” graffito in the bathroom), and then slowly learned more and more, over the course of a series of other community crises (involving theft, homophobia, alcohol…), about what it would take to make these shared values a part of student culture. Some highlights:

As Shared Values became a way to talk about what was important in our community, and even the way to address some of our rules, a few students suggested that we change our quarterly honor-roll assemblies to be called Honor Roll/Shared Values assemblies. They wanted the school to recognize students when they were “Caught in the Act of Shared Values,” a phrase they coined. Students or faculty could nominate students who had done something to exemplify a shared value. The action wouldn’t have to be a big deal, but it had to be something that everyone could applaud. We have, for instance, acknowledged students “caught in the act” of putting up posters that someone had ripped down, staying behind to help clean up a classroom, bringing in doughnuts for everyone in the class after a strenuous day of testing… pp. 38-9

In the spring of 2005, some BAA music students performed at a local music club… It was a wonderful concert; the house was packed… However, the next day the owner of the club called to report that alcohol had been stolen from her establishment.

Ms. Torres [the assistant principal] gathered all the musicians together, and initially had an awful time getting any of the students to say they had seen anything. Finally, one of the young musicians, Martin, a leader in the band, said to the whole group, “Hey, listen, someone saw something. It will be terrible for our school and our reputation if we don’t figure out who did it and make sure it doesn’t happen again….” Martin spoke fervently, but still nobody talked, not for another few days. During these days, the entire school was buzzing with talk about expulsions and rumors that the music department would never be able to perform outside of the school again. In the meantime, Ms. Torres and security personnel managed to uncover the truth: which students had actually stolen the alcohol, which had looked away but knew what was going on… They were all suspended and the ringleader… was expelled…

Even though this incident only directly involved one group of students, so many students were talking about it that Ms. Torres decided to hold another whole school assembly. She also decided to have students talk to students rather than… expect administrators to chastise everyone. Ms. Torres asked Martin if he would address the student body and explain why this was such a big deal… Ms. Torres explained, “I need you to talk about the larger issues, Martin…” He agreed.

At the assembly, Martin got out of his seat, twirling his drumsticks in one hand. “We all know this school is pretty amazing,” he began. “Sure, we’ve got beefs and there are things that we all think are stupid and try to change. Sometimes we do. I know all you freshmen want to have lunch off campus, for example. Well, maybe you can change that. But, you know, one thing that keeps us together is that we have these Shared Values. Sure, some of us might laugh when Ms. Torres gets on the intercom every morning and tells us to live one of the Shared Values, but it’s cool. We do believe in diversity with respect. Just look around at how many different kinds of people are in here. And passion with – ” And then he held his mic out to the audience like a DJ as they responded, “Balance!”

“Yeah, that’s right,” Martin continued. “And we believe in community with – ” And again the audience responded, “Responsibility.”

“So, like you’ve heard from Ms. Torres, they’re dealing with the students who did this, but I just think we all have got to think about what this means for our whole community and our reputation out there. We live by our reputation as artists, and if it gets tight out there for us, we won’t be performing…”

We didn’t want students to dismiss the incident as “just something that happened to the music majors.” Dumb, destructive behavior like this is common among adolescents… As sad as I was that BAA students had stolen alcohol, and as disappointed as I was that other students hadn’t turned them in, I was proud of our school’s overall response to the incident. Martin’s leadership meant so much to me. It established a norm that respected student leaders could support school values publicly… pp. 48-51

* Great teachers are empowered to be great by the community they’re a part of. The principal needs to work for the creation and maintenance of this community in order to empower teachers to be great. Building a great school involves “transforming a faculty into a professional learning community.”

Success truly begets success… This plays out in Ms. Chan’s [dance] class, but we see it even more clearly in Mr. Ali’s [humanities class], where students are not all here by choice. Mr. Ali can build on Aleysha’s engaged identity as an artist to encourage in her an engaged identity as a scholar. He has listened to her concerts over the years, and he knows she has a gift and love for music. It is his challenge to create the same set of expectations and joys in his own humanities classroom. p. 78

Teaching at BAA is decidedly not a solitary activity. While I have very little influence on what goes on moment-to-moment in Ms. Chan’s or Mr. Ali’s classroom, I can, and do, work on the schedule (the skeletal system of a school) to ensure that teachers help each other, that worries and questions are shared among team members and the entire faculty. Mr. Ali meets weekly with academic and arts colleagues to discuss students and to develop curriculum. At the end of the year, he will spend two days with his team reviewing and critiquing each other’s units and lessons, and creating notebooks on the year’s courses so that they continue to build a collective archive of work.

Mr. Ali and Ms. Chan are not “one-offs” or “the exceptions” at BAA. I tell their stories here as representative of the ways in which our teachers can be successful. As a leader, it is my job to build a school in which all teachers work in teams, and have the time built into their schedules to talk, to visit each other’s classrooms, and to create curricula as carefully and self-critically as artists create their pieces. pp. 80-1

* A school that wants to make progress on the achievement gap needs to have frank and potentially uncomfortable conversations with faculty and students about race.

There are a lot of really compelling passages to quote on this one but it’s already several hours past the time I told myself I would have finished this post. Read the book.