## Teaching proof writing Friday, Jul 28 2017

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.)

## Hamilton: Visibility/Invisibility of Brown Brilliance, part II Thursday, Jun 29 2017

It was written in the books of Europeans we were savage
That our history was insignificant and minds below average
But how can one diminish the worth
Of the most imitated culture on this Earth? – Akrobatik, Black Dialogue

TL;DR

Rap music is extremely brainy. This is objectively obvious, but American culture outside of hip-hop culture has been systematically ignoring it for decades and insisting that rappers are dumb.

Enter Lin-Manuel Miranda and Hamilton. The show has other aims too, but one of them is to explicitly combat this by connecting hip-hop’s linguistic creativity and power – “words’ ability to make a difference” – to the intellectual seriousness and historical import of the writing at the heart of the nation’s founding.

We have responded by enthusiastically missing the point: elevating LMM as a genius, and Hamilton as a work of genius, while continuing to ignore the brilliance of the rap luminaries he is explicitly crediting.

At length

This post was inspired by the following episode:

I was having lunch with a charming gentleman who happens to be the executive director of a media institution. (I will call him “Ed.”) We got on the topic of Hamilton, and Ed gushed:

“The words! The words are so brilliant! It never occurred to me that those words rhymed!”

Ed is not alone in this particular enthusiasm. There are a lot of things about Hamilton that have resonated with audiences, but this type of response (“The words! The WOOORRRDDDS!! Lin-Manuel’s genius words!!!”) is a throughline. (Here is a print example.) And, I appreciate the enthusiasm. The words are frickin awesome.

I am the A-L E-X A-N D
E-R we are meant to be
A colony that runs independently
Meanwhile Britain keeps sh*ttin on us endlessly
Essentially, they tax us relentlessly
Then King George turns around and runs a spendin spree
He ain’t never gonna set his descendants free
So there will be a revolution in this century! – Lin-Manuel Miranda as Alexander Hamilton, Hamilton

Nonetheless, I left lunch secretly pissed. Not at Ed, but at our culture as a whole. Here is the rant that my brain generated:

<rant>

The lyrical pyrotechnics on display in Hamilton are nothing more (nor less) than competent rapping.

There are a lot of things to love about Hamilton, but what you’re talking about — the rhymes — Lin-Manuel Miranda didn’t invent that. He learned it from Biggie, Mobb Deep, Busta. So on. I believe he’d be the first to tell you this.

It’s like it took a highly celebrated Broadway musical lionizing a Founding Father to get you, finally, at long last, to listen intently to rap lyrics, for the first time ever. But then, having listened, and having of course been blown away, you proceeded to tell yourself that you were listening to the unique genius of LMM, rather than acknowledge the radiance hip-hop put before you this whole time.

What’s ironic is that calling your attention to this brilliance is one of the clear intentions of the show. Ever since the first public appearance of the opening number, LMM has made it clear that he sees Alexander Hamilton’s story (“impoverished Carribean immigrant uses his way with words to rise to fame and influence”) as a hip-hop story.

Hamilton literally wrote a verse to get him off an island — that’s the most hip ­hop shit ever. He transcends the struggle, and if you look at your favorite rapper, that’s most likely what they did. – LMM

In this way the play asserts that hip-hop (not Hamilton but hip-hop) is continuous with no less an intellectual and political achievement than the nation’s founding.

So you have a lot of d*mn nerve gushing over this very play while simultaneously acting like you have never heard rhymes like this before.

</rant>

I want to make it very clear that I am not trying to throw Ed under the bus. He’s a great guy. And while his comment did lead me to the thoughts above, it was only because his point of view seemed emblematic of something much bigger. It’s actually part of the lore around Hamilton that both Stephen Sondheim, the legendary musical composer and wordsmith, and Ron Chernow, the biographer who wrote the book on Alexander Hamilton that inspired LMM to create the show, were initially skeptical about hip-hop’s power to tell such a nuanced story.

My point is their skepticism. If American culture had been able to be objective about what hip-hop was doing this whole time, it wouldn’t have taken Stephen Sondheim till this late date to see its sharpness and sophistication. Ample evidence has been all over the place for decades. Sondheim of all people would have recognized kindred spirits.

So while you fumin I’m consumin mango juice under Polaris
Ya just embarrassed, cuz it’s ya last tango in Paris – Lauryn Hill, Zealots

I’m submitting that rap music is not just one of America’s great artistic achievements but one of its great intellectual achievements.

I bomb atomically. Socrates’ philosophies and hypotheses
Can’t define how I be droppin these mockeries – Inspectah Deck, Triumph

But it’s high time for me to hop off any kind of soapbox I may have just climbed, because this is supposed to be dental hygiene: I am not trying to act pure. The truth is that while I beat Ed, Stephen Sondheim, and Ron Chernow by 20 years, I too was late when it came to hearing rap clearly.

I grew up a very nerdy, meticulously well-behaved, and somewhat delicate white kid in an East Coast city. I got to junior high in the early 90’s, by which time rap was beginning to be everywhere, but I didn’t connect to much of it right away. At the time I was listening to folky stuff continuous with what I was raised on. It was serious between me, Paul Simon, and Tracy Chapman. The defiant attitude, crime metaphors, and drug talk of NWA and Cypress Hill didn’t make me feel safe or welcome.

They weren’t supposed to, of course.

What I had trouble seeing at the time — what our country’s ambient but hidden racial ideology meticulously trains us not to see — is that this is completely independent of whether NWA or Cypress Hill were being smart, or interesting, or substantive. Of course they were.

Nonetheless, I resolved the discomfort of my unease with the music’s content by feeling intellectually superior. This is retrospectively preposterous, since it is no longer possible for me to hear these same artists without being amazed by their cleverness. But I was not the first nor the last person (white or otherwise) to make this move. My purpose in telling you this story is the hope that some readers might recognize themselves in it. Maybe distinguishing between “dumb” and “not designed to make you feel at home” is a lot to ask of a white 12-year-old in a country like ours; honestly, I don’t really think it is, but what do I know?

Well, I at least know that when I got to high school, and Biggie dropped, I assumed, entirely incorrectly, that he was just some *sshole who didn’t have anything to offer me. I wasn’t listening.

We used to fuss when the landlord dissed us
No heat, wonder why Christmas missed us
Birthdays was the worst days
Now we sip champagne when we thirstay – Notorious B.I.G., Juicy

It’s not a coincidence that the first rap album I ever bought (in 1994 or so) was the resolutely nonthreatening Tribe Called Quest album containing Can I Kick It?, and the album that fully converted me was The Fugees’ The Score, which, among many other (retrospectively much more amazing) things, name-checks Tracy Chapman in the opening track.

This is a confession of sorts, because although I don’t think it’s unreasonable that I didn’t really get into rap before I felt invited into it in this way, I do think it’s emblematic of a bigger problem that it took this type of invitation for me to even perceive it clearly, as I’ve been at pains to show. But I’m also pointing, in a morally neutral way, to the present moment and Hamilton. My story is that once I felt invited, I started listening to rap (not just hearing but listening), and increasingly on its own terms as time passed. As I did this, the distortion in my young perception fell away. Well, as of Hamilton, we have all, even the squarest of us (Ron Chernow? Ed?), officially been invited.

So, as with any time one receives an invitation, we now have a choice about whether to accept it. We can insist that Hamilton‘s lyricism is the product of Lin-Manuel Miranda’s unique genius, or we can start listening.

LMM himself could not have made it clearer which one he wants us to do. In addition to releasing a Spotify playlist back in 2015 of songs that inspired him while writing the show, he produced an album, The Hamilton Mixtape, handing the show back to other artists to remix. Some of them are clearly people LMM has admired for a very long time. All of them are, equally clearly, people he thinks we should all be listening to.

The most iconic song from the show is My Shot (“hey yo I’m just like my country / I’m young, scrappy and hungry…”), and The Hamilton Mixtape gives a certain pride of place to Busta Rhymes by giving him the dominant verse on that remix. There’s something so fitting about this. When Busta’s amazing gravely voice enters it’s like a homecoming: like meeting the teacher after spending a long time with the disciple. There’s no other way for me to close this post than with that verse.

Throughout my travels and journeys through life I been searchin
And been learnin to be the type of person
To display how determined I get when I’m certain
Inside I feel like fire that’s burnin
Like a knife that is turnin, I fight while I’m hurtin
Sometimes they’re right ’cause life is a burden
Like the pain from a bite that’ll worsen
Tryna stifle the light that’ll shine on me first and
Before I ride in a hearse and…

## The dental hygiene mode of thinking and talking about race Friday, Mar 24 2017

So back in January I promised a pair of posts entitled “Visibility/Invisibility of Brown Brilliance.” Part I went up almost right away, but Part II has proven to be a lot of work. I tried to bang it out a couple times but got stuck in questions of exactly how personal I wanted to get. So I shelved it until after I finished and defended my PhD thesis.

Which, by the way: defended! You may now address me as “Motherf*cking Doctor.”

Also, I’m on twitter now, and plan on actually using it.

So, anyway, looking forward to finishing Part II. But I realized it might help to more explicitly create the frame for the type of conversation I want to have. I got added impetus by reading Yen Duong’s sweet and brave post the other day, entitled Am I Racist?

In it, Yen describes going to a football game with her spouse, and noticing that she perceives the white players as younger than the black players. She connects this with a 2014 study showing that white male police officers and white female undergraduates tend to overestimate the ages, and underestimate the innocence, of black boys aged 10 and up. She asks her spouse if she is being racist. He recoils and insists she’s not.

What came up for me was the critical, critical importance of being able to talk about the way that living in this world and this country, with all their glorious and sordid history, distorts our perceptions of each other based on race, without getting sidetracked by a conversation about whether or not we are good people.

I think something really beautiful and important was said about this some years ago by Jay Smooth. I’ve linked the below video twice before, but let me make it the focus this time.

The main idea:

Being a good person, with respect to race (and more generally), is like being a clean person. It’s not something you are or not, it’s a practice. Like dental hygiene.

The world we have inherited has racial “dirt” everywhere — tendencies to misperceive each other accrete in our minds, like plaque on our teeth, daily, just from living life in this world. The root causes of this fact were in place long before anyone alive today was born. So when we notice one of these accretions in ourselves, or have it pointed out to us, the question of whether that makes us a bad person is a red herring. It doesn’t: these accretions are inevitable, for everyone. The right question is how to train ourselves to perceive each other more clearly.

The video:

Watch this right now. I’ll wait.

Two things.

1) In the video, Jay says, “There are many things in our day-to-day lives that lead us toward developing little pockets of prejudice.”

I think one aspect of the racial “dental hygiene” he’s calling for is the search for awareness and understanding of these processes. My major purpose in writing the Visibility/Invisibility of Brown Brilliance posts is to call attention to the subtlety and effectiveness with which our media and cultural environment, whether by design or not, programs us to underestimate the minds of the black and brown Americans among us. (How could I not have noticed, before Queen of Katwe and Hidden Figures were announced, that I’d practically never seen a movie centered on the brainy pursuits of a brainy black woman, despite the many brainy black women in my life?)

But for the benefit of those reading who are unsure what is being referred to, here is a very concretely documented example:

Here is a twitter user comparing Google image searches of the phrases ‘three black teenagers’ vs. ‘three white teenagers’, turning up mugshots in the former case and cutesy, wholesome stock photos in the latter.

This is the “dirt.” It is going to get on us, every day. The question is what to do with it.

2) I love Yen for her reflectiveness about the football players and the study. This is what the “dental hygiene” looks like — this is how you do it.

I also relate to her spouse. If somebody (even your partner) is calling your partner a bad name, you defend! BUT, I have the feeling that trying to reassure Yen she wasn’t being racist was pulling them both away from the good stuff. Look, a study of hundreds of cops and college kids found that on average they tended to overestimate black boys’ ages a dramatic amount. Presumably, lots and lots of people do this. I bet I do it. What are we then going to do? Take note, and look for ways to do a better job? Or, waste energy trying to prove the improbability that we’re somehow immune from this poison?

Again, I feel him. And I don’t blame him. The issue is that our cultural understanding of how to be a good person is so limited. An alien watching video of lots of Americans talking publicly about race would surely conclude that we believe that good people are never prejudiced and if you ever have a prejudiced thought, you’re bad. In the language of the video, the “tonsils paradigm of race discourse” — “I can’t be prejudiced, I had my prejudice removed in 2005!” We would all grant that this is absurd, abstractly, and yet we have an anxiety meltdown, or get angry and defensive, at the slightest suggestion of prejudice — what other conclusion could our hypothetical alien come to?

This limited frame makes it impossible to attend to a racially problematic habit of thought without implying that you’re a bad person. This forces us to hide the dirt. Then we just get dirtier and dirtier and keep hiding it.

I’m offering Jay’s video as an alternative frame. What if instead of hiding our racial dirt we were trading ideas about how to deal with it? Working on better and better “toothbrushes” for our stereotypes?

On that note — above I mentioned Google image searches as a quick and concrete measure of the “dirtiness” of our environment of racial images — here is a “toothbrush” that was designed in response. A photo / video / poetry art piece by 19-yr-old Myles Loftin, addressing these images. Enjoy!

## BEAM in the NYT! Saturday, Feb 18 2017

A paragraph I was not expecting to read in the NYT today:

Even as movie audiences celebrate “Hidden Figures,” the story of black women who overcame legally sanctioned discrimination to perform critical calculations in the race to put a man on the moon, educators say that new, subtler obstacles to higher-level math education have arisen. These have had an outsize influence on racial prejudice, they contend, because math prowess factors so heavily in the popular conception of intelligence.

Another one:

“Fundamentally, this is a question about power in society,” said Daniel Zaharopol, BEAM’s director. “Not just financial power, but who is respected, whose views are listened to, who is assumed to be what kind of person.”

Anyway, big ups to Amy Harmon and the NYT for this beautiful article about Bridge to Enter Advanced Mathematics, which is one of my all-time favorite places to teach.

## Hidden Figures: Visibility / Invisibility of Brown Brilliance, Part I Sunday, Jan 22 2017

Has everybody seen Hidden Figures yet?

It’s delightful: a tight, well-acted, gripping drama, based on a true story about an exciting chapter in national history. You can just go to have a good time. You don’t need to feel like you are going to some kind of Important Movie About Race or whatever. It is totally kid friendly, and as long as they know the most basic facts about the history of racial discrimination, it doesn’t force you to have any kind of conversation you aren’t up for / have every day and don’t need another… / etc. Just go and enjoy yourself.

THAT SAID.

Everybody, parents especially, and white parents especially, please go see this film and take your kids.

I was actually fighting back tears inside of 5 minutes.

Long-time readers of this blog know that I am strongly critical of the widespread notion of innate mathematical talent. I’ve written about this before, and plan on doing a great deal more of this writing in the future. The TL;DR version is that I think our cultural consensus, only recently beginning to be challenged, that the capacity for mathematical accomplishment is predestined, is both factually false and toxic. My views on the subject can make me a bit of a wet blanket when it comes to the representation of mathematical achievement in film – the Hollywood formula for communicating to the audience that “this one is a special one” usually feels to me like it’s feeding the monster, and that can get between me and an otherwise totally lovely film experience.

In spite of all of this, when Hidden Figures opened by giving the full Hollywood math genius treatment to little Katherine Johnson (nee Coleman), kicking a stone through the woods while she counted “fourteen, fifteen, sixteen, prime, eighteen, prime, twenty, twenty-one, …,” I choked up. I had never seen this before. The full Good Will Hunting / Little Man Tate / Beautiful Mind / Searching for Bobby Fischer / Imitation Game / etc. child-genius set of signifiers, except for a black girl!

What hit me so hard was that it hit me so hard. For all the brilliant minds we as a society have imagined over the years, how could we never have imagined this one before now? And she’s not even imaginary, she’s real! And not only real, but has been real for ninety-eight years! And yet this is something that, as measured by mainstream film, we haven’t even been able to imagine.

You’ll do with this what you will, but for me it’s an object-lesson in the depth and power of our racial cultural programming, as well as a step toward the light. I am a white person who has had intellectually powerful black women around me, whom I greatly admired, my whole life, starting with my preschool and kindergarten teachers, and including close friends and members of my own family, as well of course as many of my students. And yet the type of representation that opened Hidden Figures is something that only fairly recently did it begin to dawn on me how starkly it was missing.

So, go see this movie! Take your kids to see it! Let them grow up easily imagining something that the American collective consciousness has hidden from itself for so long.

## This blog and the nation Sunday, Jan 22 2017

I have been relatively inactive on this blog for a while now. This has been due 100% to the necessity to focus on my schoolwork and other offline pursuits, and will continue to be true for a few more months at least. (Btw, I’m on twitter now! But won’t be using it much for the same few months.)

Also, the scope of this blog, while broad (I think) within the general umbrella of math and education, has never ventured out from beneath this umbrella.

But the sea change in our national political context is on all of our minds, certainly on mine, and there are a number of themes and ideas that I want to explore with you here, relating to the state of our union and our democracy. Some of them are related to math and education directly; others more obliquely.

Much of the writing I intend to do will have to wait at least the above-referenced few months. But I am going to commence a pair of hopefully pretty short blog posts now, entitled Visibility / Invisibility of Brown Brilliance, concerning the way that some recent exciting pop-cultural events have thrown into really stark relief for me the doggedness and obstinacy of our refusal, as a culture as a whole, to acknowledge the power of our black and brown citizens’ intellectual contributions to our nation.

I hope the relevance to the political moment is felt, but I don’t want to draw explicit connections here because I don’t want what I’m going for to get drowned out by partisanship, mine or anyone else’s. I hope to steer clear of self-righteousness (and please let me know if I’m unsuccessful). These posts are intended to invite introspection — I’m aspiring to the dental hygiene paradigm of race discourse. When I talk about our refusal as a culture as a whole to acknowledge brown brilliance, I mean all of us – me and you and all of us. Not “the bad guys” / “the others”.

Anyway. Look for a pair of posts on this theme in the next few days. I hope you’ll find them useful.

## Education and Markets (reblog) Thursday, Oct 13 2016

Ben Orlin kills it on the complete incoherence of the notion that public education can only be improved by increased exposure to market forces. This is something I’ve been brewing thoughts on for years, and Ben says pretty much everything I would want to say, except with his signature drawings and economical word use in place of my epic and probably gratuitous verbosity.

While everything he says is gold, I will pull out one point I want to amplify:

The difference between businesses and schools is that nobody cares if most businesses fail.

## Think of a Brainy Black Woman in a Hollywood Film Sunday, Sep 25 2016

So I’m psyched about Queen of Katwe (Disney), starring Lupita Nyong’o and David Oyelowo, based on the true story of young Ugandan chess champion Phiona Mutesi, which just came out. I’m definitely gonna see it this week.

I am also looking forward to the release this winter of Hidden Figures (20th Century Fox), starring Taraji P. Henson, Octavia Spencer, and Janelle Monae, based on the true story of Dorothy Vaughan, Mary Jackson, and Katherine Johnson, and their foundational mathematical contributions to the US space program. I have never ever ever seen a black female mathematician in a major film before.

This got me thinking: in my entire life up til now, have I ever seen a film released by a major Hollywood studio that centered on a brainy black woman and her brainy pursuits? I’ve been musing on this for about 24 hours now. I thought of exactly one: Akeelah and the Bee.

Can you think of any others?

Update 9/29: I thought of two more candidates. They don’t have that same “this woman is taking over the world with her mind alone” quality as all of the above, but they do have something:

Home (20th Century Fox, 2015): it’s not a major theme of the film, but the generally resourceful and awesome main character, voiced by Rihanna, does at a key point figure out the mechanism of a piece of alien technology while boasting of her “A in geometry”…

A Raisin in the Sun (Columbia Pictures, 1961): Beneatha’s intellectual pretensions don’t exactly drive the plot, but they are pretty central to her character. If you want to see what I mean and are up for being made a little upset, click here (the “in my mother’s house…” scene if you know it).

Update 1/7/17: Having sat on this blog post for a few months now, I feel that the previous update dilutes the point a bit. Akeelah and the Bee, Queen of Katwe, and Hidden Figures, are the only movies of their kind I can think of. (Per the description above: produced by Hollywood, centered on a brainy black woman and her brainy pursuits.) I earnestly want to know if more exist. I am very excited there have been 2 inside of 6 months.

If I ask for “that kind of movie” only without the requirement that the lead be black and female, then we are swimming in them: Good Will Hunting, Theory of Everything, Imitation Game, Beautiful Mind, The Man Who Knew Infinity, Little Man Tate, Searching for Bobby Fischer, … shall I keep going?

For a quick and dirty numerical sample of the status quo: here is a list, compiled by a random IMDB user, of “movies about geniuses.” I found it among the first few hits upon googling “movies about smart people.” On this list I see 35 distinct titles. (The list says 42 but I see 7 repeats.) Of these, by my count the “geniuses” include 32 white boys/men, 1 black man, 1 East Asian man, and 1 white woman.

The fact that I managed, scraping my memory, to find a movie (Home) centered on a black girl who at some point in the film does something cool with her brain, is irrelevant to this stark picture. (This is not a knock on Home, which I loved.) If we want to bring it into the conversation, then we should put it in the context of every movie centered on a character that at some point does something cool with their brain. This is a lot of movies, way too many to make any kind of list.

If I allow the character in question not to be the main character (as in Raisin in the Sun; and if I allow us to leave Hollywood, 2012’s Brooklyn Castle and 2002’s Spellbound come to mind), then we are talking about every movie containing a character with plausible intellectual aspirations. Again, way too many to start listing.

The upshot: representations of brainy black women in (Hollywood) film have been exceedingly, shockingly rare. If you have taught in any place that has black people, you know that brainy black women are not rare in real life. Our national culture has had a very limited imagination in this regard. So let’s all effing go see Hidden Figures as soon as we possibly can. Independent of all this, I’ve heard it’s very good.

## What It Comes Down To Monday, Jun 13 2016

I was just reading A. K. Whitney’s piece in The Atlantic about the Hacker-Tanton debate. She gets to the heart of the matter.

Actually, not just the heart of the matter of the Hacker-Tanton debate, but, like, The Heart of The Matter in math education.

Is math for everybody?

I have come to feel like I can hear this question somewhere in the background of almost every debate about math education and math education policy that I encounter.

Almost everyone will say “yes.” But do they mean it? Or more precisely, what do they mean?

Is ‘rithmetic for everybody but that abstract stuff is just for eggheads? Is being put through the paces of the corpus of school math for everybody but enjoying it is just for dorks or smartypants? Is having to take tests for everybody but math as a tool to exercise agency is just for white and Asian men?

Or is all of it for everybody?

## The History of Calculus / Honor Your Dissatisfaction Saturday, May 21 2016

I was just rereading an email exchange with a friend (actually the O of this post), and found that I had summarized the history of calculus from the 17th to 20th centuries, up through and including Abraham Robinson’s invention of nonstandard analysis, in the form of a short play! I’m sharing it with you.

Mainly this is for fun, but it’s also part of my ongoing campaign promoting the value of honoring your dissatisfaction. The dialectic between honoring our impulse to invent ideas to understand the world better and honoring our dissatisfaction with these ideas is where mathematics comes from.

Here’s the play!

# The History of Calculus, in 4 Extremely Short Acts

Featuring a lot of oversimplification and a certain amount of harmless cursing

Act I

Late 17th century

Leibniz, Newton: Look everybody, we can calculate instantaeous speed!

Everybody: How??

Leibniz: well, you consider the distance traveled during an infinitesimal interval of time, and you divide distance/time.

Everybody: Leibniz, what do you mean, “infinitesimal”? Like, a millisecond?

Leibniz: No, way smaller than that.

Everybody: A nanosecond?

Leibniz: Nah, dude, you’re missing the point. Smaller than any finite amount.

Everybody: So, zero time?

Leibniz: No, bigger than that.

Some people: Oh, cool! Look we can use this idea to accurately calculate planetary motion and stuff!

Other people: WTF are you talking about Leibniz? That makes no effing sense.

Act II

18th century

Bernoullis, Euler, Lagrange, Laplace, and everybody else: Whee, look at everything we can calculate with Newton and Leibniz’s crazy infinitesimals! This is awesome!

Bishop George Berkeley: But nobody answered the question of WTF they are even talking about. “What are these [infinitesimals]? May we not call them the ghosts of departed quantities?”

Lagrange: Hold on, let me try to rebuild this theory from scratch, I will make no mention of spooky infinitesimals, and will do the whole thing using the algebra of power series.

Everybody: Cool, good luck with that.

Act III

19th century

Cauchy: Lagrange, homie, it’s not gonna work. $e^{-1/x^2}$ doesn’t match its power series at zero.

Lagrange: Sh*t.

Everybody: I think we don’t actually understand this as well as we thought we did.

Ghost of departed Bishop Berkeley: OMG I HAVE BEEN TRYING TO TELL YOU THIS.

Cauchy: How about we forget the whole “infinitesimal” thing and just say that the average speeds are approaching a certain limit to whatever desired degree of accuracy. As long as we can identify the limit and prove that it gets as close as we want it to, we can call that limit the “instantaneous speed” without ever trying to divide some spooky infinitesimals by each other.

Everybody: Awesome.

Weierstrass: I have an even better idea. Let’s formalize Cauchy’s thinking into some tight symbols and quantifiers. “Let us say that the limit of a function $f(x)$ at $c$ is a number $L$ if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that whenever $0 <|x-c|<\delta$, it follows that $|f(x)-L|<\varepsilon$…”

All the mathematicians: AWESOME. Down with spooky infinitesimals! Calculus can be built soundly on the firm footing of “for any $\varepsilon>0$ there exists a $\delta>0$ such that…” and you never have to talk about any spooky sh*t!

All the mathematicians, in private: … but thinking about infinitesimals sure streamlines some of these calculations…

[Meanwhile all the physicists and engineers miss this whole episode and continue blithely using infinitesimals.]

Act IV

20th century

Scene i

Mathematicians: Infinitesimals are satanic voodoo!

Mathematicians: Whatever dude, don’t you know about Weierstrass and $\varepsilon$ and $\delta$?

Physicists and engineers: Um, no, and I don’t care either! What’s the point when everything already works fine?

Mathematicians, in public: No, dude, there are all these tricky convergence issues and you will F*CK UP EVERYTHING IF YOU’RE NOT CAREFUL!

Mathematicians, in private: … but those infinitesimals are indispensible as a heuristic guide…

Scene ii

Abraham Robinson: Um, whatever happened to infinitesimals?

Mathematicians: I mean we rejected them as satanic voodoo because nobody was ever able to tell us WTF THEY ARE.

Robinson: I have a proposal. How about we consider them to be [fancy-*ss definition based on formal logic and other fancy sh*t]. Would you say that constitutes an answer to “wtf they are?”

Mathematicians: … why, yes!

Some mathematicians: omg awesome I can now RESPECTABLY use infinitesimals in calculations, I don’t have to hide anymore!

Other mathematicians: Whatever, I have no need to do the work to master this fancy sh*t. It doesn’t do anything good ole’ Weierstrass $\varepsilon$ and $\delta$ couldn’t do.

Physicists and engineers: wow, you guys are way over-concerned with the little stuff. Literally.

# End

(Long-time readers of this blog will recognize the bit of dialogue with Leibniz from something I shared long ago.)

The point is that the whole episode is driven by uncertainty about what is even being discussed. The early developers of calculus shared the conviction that there was something there when they talked about “infinitesimals”, but none of them (not even Euler) gave a definition that was satisfying to everybody at the time (let alone to a modern audience). But this encounter, between the intuition that there’s something there and the insistence of the world to honor its dissatisfaction until a really satisfying account was given, was a generative encounter, resulting in several hundred years’ worth of powerful math progress.