BEAM in the NYT!

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

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.


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

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)

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

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).

I want more! Please help!

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

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

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.


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!

Physicists and engineers: What are you talking about, what about CALCULUS?

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.


(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.

So. Honor your dissatisfaction.