Math is Democracy II: Math is Democracy!

I announced a series on math and democracy back in October.

It will deal with a lot of concrete areas. Last time I talked about a case that is before the Supreme Court and will influence voting law throughout the land. In the future I’ll be talking about voting, political participation, technology and who has a say over its development, and of course the classroom.

But I want to properly kick things off with a post that is essentially philosophical. I am here to assert the following proposition:

Math is democracy!

What do I mean?

Democracy — from Greek — literally, “rule by the people.” I am referring to the ideal itself, not any particular system of government. Throughout the world we have various systems attempting to implement this ideal. One can ask questions about the degree of success of these attempts, but that’s not what this post is about. I’m just isolating the ideal — democracy — rule by the people.

Mathematics — from Greek — literally, “learning.” Of all the domains of human inquiry, math occupies a privileged place in terms of our confidence in its conclusions. It is the only field where practitioners regularly express unqualified certainty about its results. We sometimes discuss the wisdom it gives us as some sort of celestial gift (as in Wigner’s classic essay on its applicability to the sciences).

I am about to draw a connection. I expect it is still opaque at this point, but hang on.

If math is a miracle, then there is a second miracle: the divine gift was implanted in each of us, since it springs solely from the universal human capacity for rational thought. The wisdom of mathematics was not given us by way of Mt. Sinai, handed down from on high by somebody with privileged access to The Boss. Although many people think back to childhood and recall inscrutable formulas dispensed by a teacher who mysteriously knew the answer (how did they know??), this memory conceals the real truth, which is that the only place mathematical knowledge comes from is a community of peers reaching some kind of consensus after a period of engaged discussion. Furthermore, at least in principle (if not always in practice), anybody in this community has the right at any time to raise good-faith questions about the logic underlying any of our mathematical knowledge, and the matter is not really settled unless these questions have a good answer.

Thus, the only true source of mathematical authority is the consensus of a community of equals.

The principle of democracy is that this is also the only true source of legitimate political authority.

Broadening further, I offer that the principle of democracy holds that the only source of authority (of any kind) over a community is consensus of that community. So math is literally democracy.

Addendum 3/29/18:

This is edited from the version I posted yesterday, where I used the phrase “functional consensus” instead of “consensus.” This was to acknowledge that in a large-scale community such as a nation, or the international community of mathematics researchers, true consensus is not a viable goal. That said, the “functional” didn’t sit well with me overnight, because I thought it could be taken to suggest some sort of majoritarian principle. To me, majoritarianism is a fatal compromise of the principle of democracy articulated here, and it defeats the purpose of the analogy with math.

The thing about math is that, in principle, if an objection is raised to what is regarded as established fact, then that objection needs to be dealt with. Maybe something was overlooked! In actual practice, it may or may not be, because the question of whether you can get people to pay attention to your objection depends on things like if you’re famous, if you’re well-connected, how much work other people have to do to understand it, etc. But mathematicians’ collective understanding of what we’re doing holds that if somebody raises a new objection to something thought to be well-established, we have to answer it, not ignore it, in order to hold onto the established knowledge. This ideal isn’t attained, but it is still how we think about it.

By the same token, it seems to me that the democratic ideal insists that a minority view has the right to be processed rigorously by the community. I am making a high-level analogy so I’m not getting into what that processing might look like. But the failure of a community to take into account minority constituencies in some way is a failure of democracy.

Addendum 3/31/18:

I want to acknowledge some intellectual debt!

In 2008, I went to the Creating Balance in an Unjust World conference and saw a presentation by Sarah Bertucci, Jason Cushner, and several of their current and former students, entitled Consensus is the Answer Key: Empowerment in the Math Classroom. The presentation was on using consensus as the source of mathematical knowledge in the classroom. Later (in 2009?), I visited the school in Vermont where Jason and Sarah were then teaching, and saw Jason’s class. (Random aside: I also met Jasmine Walker!) The ideas have shaped how I saw both mathematics and the classroom ever since. You can see their clear imprint above (and in many of the things I’ve written on this blog over the years).

In about 2010, I was having a conversation with Jay Gillen of the Baltimore Algebra Project. At the time, I was preparing to apply to graduate school in math. Jay asked me many questions about how I thought about the math classroom and the subject itself. At some point he paused and said, “Everything you love about math is what free people love about democracy.” This comment has been continuously blowing my mind for 8 years, and again you can see its clear imprint in the above.

Advertisements

“Gifted” Is a Theological Word

Quite the juicy convo on Twitter:

Hard not to reply with every thought I have, but I want to keep the scope limited. One idea at a time.

In some sense, I work in “gifted education.” Big ups to BEAM, my favorite place to teach. This is a program that is addressing the intellectual hunger of students who are ready to go far beyond what they are doing in school. I have profound conviction that we are doing something worthwhile and important. (NB: to my knowledge, BEAM does not use the word “gifted” in any official materials, and most BEAM personnel do not use it with our kids out of growth mindset concerns.)

It is also true that I myself had a very different profile of needs from my peers at school as a young student of math. I taught myself basic calculus in 5th grade from an old textbook. I read math books voraciously through middle school, and in class just worked self-directedly on my own projects because I already knew what we were supposed to be learning. I am not mad that I didn’t have more mathematical mentorship back then — my teachers did their best to find challenges for me, I appreciated them both for that and for the latitude to follow my own interests, and in any case things have worked out perfectly — but looking back, at least from a strictly mathematical point of view, I definitely could have benefited from more tailored guidance in navigating my interests.

In this context I want to open an inquiry into the word “gifted” as it is used in education.

I hope the above makes clear that this inquiry is not about whether different students have different needs. That is a settled matter; a plain fact.

The subject of my inquiry is how we conceive of those differences. What images, narratives, stories, assumptions, etc., are implicit in how we describe them. In particular, what images, narratives, stories, and assumptions are carried by the word “gifted”?

This question is too big a topic for today. Today, I just want to make one mild offer to that inquiry, intended only to bring out that there is a real question here — that “gifted” is not a bare, aseptic descriptor of a material state of affairs, but something much more pregnant — containing multitudes. It is this:

“Gifted” is a theological word.

What do I mean?

A gift is something that is given; bestowed. My nephews recently bestowed on me a set of Hogwarts pajamas, fine, ok, but when we speak of “giftedness,” you know we are not discussing anything that was bestowed by any human.

By whom, then, is it supposed to have been bestowed?

You know the answer — by God. Or if not God, then by “Nature,” the Enlightenment’s way of saying God without saying God.

When we say a child is “gifted,” we are declaring them to have been selected as the recipient of a divine endowment. Each of these words carries a whole lot of meaning extrinsic to scientific description of the situation — selected; recipient; divine; endowment.

When we use this word in contemporary educational discourse, we usually aren’t consciously evoking any of this. Nothing stops a committed atheist from saying a kid is gifted. Nonetheless, I don’t think it can really be avoided.

Why I say this is how easily and quickly the full story — selected, recipient, divine endowment — becomes part of the logic of how people reason about what to do with a student so labeled. To illustrate with a contemporary slice of pop culture, the 2017 film Gifted, starring McKenna Grace, Chris Evans, Lindsay Duncan and Jenny Slate, hinges on the question of what is a family’s obligation to its child’s gift? How can a bare material state of affairs create a moral obligation? — but being chosen as the custodian of a divine spark on the other hand, it’s easy to see how to get from that to something somebody owes.

So, this is my initial offer. I’m not saying anything about what to do with this. For example, I am not evaluating Michael’s assertion that “giftedness is true.” I’m just trying to flesh out what that assertion means — to call attention to the sea of cultural worldview supporting the vessel of that little word.

Math is Democracy I: The Citizen and the Gerrymander

I am intending a series of my typically long, elaborate blog posts entitled Math is Democracy. The ideas have been brewing for years although they have been rapidly expanding and taking on new urgency since January. I alluded to this intention previously.

I wasn’t ready to start it yet, but I feel I must. I was reading the oral arguments in Gill v Whitford, the Wisconsin partisan gerrymandering case currently before the Supreme Court. I had to stop and have a moment when I read this:

CHIEF JUSTICE ROBERTS: Mr. Smith, I’m going to follow an example of one of my colleagues and lay out for you as concisely as I can what — what is the main problem for me and give you an opportunity to address it.

I would think if these — if the claim is allowed to proceed, there will naturally be a lot of these claims raised around the country. Politics is a very important driving force and those claims will be raised.

And every one of them will come here for a decision on the merits. These cases are not within our discretionary jurisdiction. They’re the mandatory jurisdiction. We will have to decide in every case whether the Democrats win or the Republicans win. So it’s going to be a problem here across the board.

And if you’re the intelligent man on the street and the Court issues a decision, and let’s say, okay, the Democrats win, and that person will say: “Well, why did the Democrats win?” And the answer is going to be because EG was greater than 7 percent, where EG is the sigma of party X wasted votes minus the sigma of party Y wasted votes over the sigma of party X votes plus party Y votes.

And the intelligent man on the street is going to say that’s a bunch of baloney. It must be because the Supreme Court preferred the Democrats over the Republicans. And that’s going to come out one case after another as these cases are brought in every state.

And that is going to cause very serious harm to the status and integrity of the decisions of this Court in the eyes of the country.

Now, there’s a lot here one could react to.[1] But the main thing I reacted to was this:

The Chief Justice of the highest court in the land thinks Americans don’t feel empowered to judge an argument on the merits if there’s math involved.

You know what? He’s probably right about that.

But this situation is very, very wrong.

Math is being used increasingly to make decisions governing our lives, for good or ill. Increasingly sophisticated math.[2] The instance most familiar to readers of this blog is probably teacher value-added scores, but the many various uses share this: they are not accountable to the public.

One reason the Wisconsin case is so hot is because the process that led to the map currently being challenged included a lot of fancy mathematical modeling intended to make the Republican legislative majority as bomb-proof as possible — an effort that appears to have worked really well. That the map was drawn with this goal and these tools is not a controversial point in the case. This was a use of math by legislators aimed at becoming less accountable to the public.

What I’m getting at: math is a species of power, and it’s a species that multiple antidemocratic forces are using, very effectively. And it’s a kind of power that citizens, by and large, totally lack.

So, the game is unfair. We the People are supposed to be able to participate in public decision-making. That’s the heart of democracy. But math is increasingly becoming a kind of secret key to power that, if the Chief Justice is right, We the People mostly don’t have. As soon as there’s math involved, we can’t even participate in debates about the very consequential choices that are being made. In which case, nobody who wants to use the power of math (for good or ill!) needs to be accountable to us.

I mean, this was true before the explosion of data-science driven business and governmental practices Cathy writes about, or the computer-assisted 2010-11 legislative redistricting.[3] But now it is more intensely true than ever.

What this leaves me with is that doing our jobs well as math educators is completely urgent for democracy. Every kid we leave traumatized and alienated from formulas and data analysis is a citizen that doesn’t have a voice.

Don’t let anybody tell you it doesn’t matter.

Notes:

[1] For example: This author at ThinkProgress thinks Roberts has a lot of nerve claiming to be concerned with the perception that the court is partisan when he has so consistently voted along partisan lines in landmark cases. This author at WaPo thinks it’s not legitimate for the Court to be considering its public perception in the first place. I am personally inclined to believe that Roberts is earnestly concerned about the court’s reputation and that his question was earnest (mostly because of his surprising and apparently similarly-motivated vote in NFIB v Sebelius), although I do think that the fact that he doesn’t appear to be equally concerned with the perception of partisanship if the court does not “allow the claim to proceed” reflects a rather striking partisan limitation in his image of the “intelligent man on the street.” I know plenty of intelligent men, and women, who would be inclined to conclude that he himself is a partisan hack on the basis of the above quotation alone.

[2] Shout out to Cathy O’Neil.

[3] This seems like a good moment to acknowledge the deep debt of my thinking here to Bob Moses, who has been on this tip for a long time. Also, there is some relationship to the work of math educators in the Freirean tradition such as Marilyn Frankenstein and Rico Gutstein, though I can’t take the time now to figure out exactly what it is.

My Favorite Nerds on Television

Speaking as somebody who has been a nerd since long before that was a thing, these last 30 years have really been a trip as far as the way the word “nerd” has changed in the public sphere. I was a kid in the ’80s. Back then, nerds in pop culture meant short goofy men, usually named Louis, who couldn’t get it together under any circumstances. Now we have Zac Efron, Chris Hemsworth, Mila Kunis, Karlie Kloss, Michael Fassbender, and Selena Gomez all identifying as nerds on the record.

This is a real shift. It’s a juicy sociological question why and how. I don’t think anybody doubts that the ascendancy of Silicon Valley, e.g. the kingmaking of Mark Zuckerberg, had something to do with it. I’m inclined to believe that the internet had a more democratic role to play as well: the birth of virality allowed us, the people, at least briefly, to start declaring what was awesome without corporate mediation. Suddenly everybody’s private nerdiness had a mechanism to go public, and when it did, we crowned things that the arbiters of the pre-Youtube media landscape would have dismissed instantly, if they had even noticed them. Remember Chewbacca Mom? How about Chocolate Rain? Nerdiness has been validated by visible numerical strength. Well, anyway, I’m not trying to do sociology here, I’m just speculating. But something has really changed.

But it also hasn’t. But it has, but it hasn’t, but it has, but it hasn’t. The highest-rated non-sports TV show of the 2016-2017 season was The Big Bang Theory, which this fall will enter its 11th season. (I’m not presuming Nielsen ratings are still definitive of anything, but clearly it’s at least a big deal.) I feel like I’m supposed to like this show, but it’s always rubbed me wrong. It’s 2017 and “nerd” still means overgrown child? Female nerdiness is still essentially secondary and nonwhite nerdiness essentially tokenistic? Brainy people can’t aspire to social maturity and socially mature people can’t aspire to braininess? Maybe I’m being unfair to the show but that’s how it makes me feel.

Nonetheless, the more democratic side of nerd ascendancy has furnished us with a wider variety of screen representations than I could have imagined back then. So I want to take a moment to give some props to three + two of my very favorites.

Quick disclaimers: (1) I do not watch a ton of television. I’m sure there are a bunch of awesome nerds I don’t know anything about. (2) Spoiler alert! Information about these characters is freely discussed. You’ve been warned.

Ok, without further ado, and in no particular order,

My favorite nerds on television!

Willow Rosenberg, Buffy the Vampire Slayer

willow

C’mon, y’all, of course! Buffy’s shy, self-effacing, brainiac-hacker-turned-sorceress bestie is the first time I think I saw a nerd on TV get to be a whole person. This show was written into nerd canon the moment in the very first episode when Buffy, courted by mean-girl Cordelia, decisively sides with Willow instead —

and its place was sealed in episode 2 when Willow quietly sticks up for Buffy, and then for herself —

But Willow wouldn’t have been part of the inspiration for this post if things had stayed where they were early in season 1. The thing I love about the portrayal of Willow was that she got to be a multidimensional, changing human. I’ve seen seasons 1-5 and part of 6, and over the course of that time Willow investigates many different sides of herself and ways of being — group belonging vs. autonomy; sexuality and partnership; power, creation and destruction; selflessness vs. ego. A really wide range of self-experience is part of being human, but they never used to write nerds this way.

Case in point: when an ’80s / ’90s nerd obtains some swagger, it’s usually due to some sort of magical or science-fictional intervention, cf. Stefan Urquelle. (Drugs and alcohol can serve the magical purpose as well, cf. Poindexter.) The entertainment value is the contrast between the magic/science/psychotropics-enhanced version of the character and the swaggerless everyday version. Buffy plays with that trope too — in a classic episode in season 3, an evil vampire version of Willow shows up in town, rocking leather and taking absolutely no sh*t from anyone.

latest

But in the Buffyverse, this is an opportunity for the character to grow. A plot device occasions the real Willow to have to impersonate her evil vampire twin, and she’s forced to try on some unaccustomed ways of being — assertive; fear-inspiring; fearless; sexually confident. They feel weird and uncomfortable to her in the moment, but they also resonate — indeed, it was a shy but defiant experiment in power and danger by real Willow herself that (accidentally) brought evil twin Willow to town in the first place. And without doubt, the whole experience opens up new avenues of selfhood for Willow to explore.

Seymour Birkhoff, Nikita

Birkhoff-seymour-birkhoff-nikita-cw-32081184-500-281

I don’t know why the CW’s reboot of La Femme Nikita wasn’t more of a thing. A and I were totally obsessed with it. And one of the (many) reasons was Seymour Birkhoff, the Star-Wars-Lord-of-the-Rings-quoting black-ops technology specialist.

In a lesser show, Birkhoff would have been a purely instrumental character, there to solve plot problems. “We need to hack into this network — where’s Birkhoff?” In this show, he’s a principal, and his relationship with the other leads, especially Michael and Nikita, are at the heart of the whole thing.

(Spoiler warning if you’re not in season 2 yet!)

Like Willow, over the course of the show’s 4 seasons, Birkhoff gets to be a whole person. Fearful, brave, valorous; selfish, loyal; supportive, needy; a truthteller and a deceiver. Powerful and vulnerable.

900a4e83ac7c13586efec46dc1158bdf

Like Willow, this range of experience never compromises the legit nerdiness. It’s a different flavor than hers: a familiar awkward cockiness coupled with a constant stream of references to canonical nerd material, from the aforementioned Lord of the Rings and Star Wars to Harry Potter and X-Men. Including, at the risk of a spoiler, literally my favorite use of “may the force be with you” in all of film, including the OT.[1] At one point he almost gets himself killed with a poorly chosen Mr. Miyagi quote, but it’s not a joke at his expense. He reads to me as a “for us, by us” representation — if the writers and/or the actor don’t identify as nerds, somebody is really convincingly faking it.

Jane Gloriana Villanueva, Jane the Virgin

JanetheVirgin

I conceived of this post when I was still in season 1 of Jane the Virgin. Even though I relate to Jane as a fellow nerd, I wasn’t completely sure it was right to claim her this way publicly. Whereas Birkhoff and Willow are clearly delineated by the scripts as their respective shows’ Designated Nerds — Birkhoff is literally nicknamed “Nerd” by Nikita — Jane is not explicitly so constructed. Was I “calling her a nerd,” then? (This used to be rude.)

image

Season 2 fully cleared that up as all the relevant features of Jane’s personality came into clearer focus. Between her late-night informational internet binges, her anxiety around school success (she’s working on a creative writing degree), her urgent need to get everything right, her tendency to overthink things, and her not even playing it a little bit cool around her father’s celebrity friends (see below), it was settled. And then, oh, right, she’s a virgin, deep into her twenties.

All of these are important aspects of Jane’s story and/or personality, but none of them pigeonhole her.

I think that’s the unifying theme of this blog post. Being a nerd is not a limitation on what’s possible in terms of the range of human experience. Nerds are not a homogeneous bunch — we are not even homogeneous internally as individuals. TV doesn’t always recognize this, but when it does, it’s glorious.

Runners up!

Cosima Niehaus, Orphan Black

While for me Cosima doesn’t quite meet the “for us, by us” standard set by Birkhoff, it still feels worth celebrating that we now have an earnestly-geeked-out-on-science character who is also “the hot one”.

Brian Krakow, My So-Called Life

My So-Called Life is a classic show for a reason. Every one of the characters had an interior life that was more richly and empathetically rendered than any prior teen show that I know of. From Angela Chase (to this date, Claire Danes’ greatest work imho) to Rickie Vasquez to Rayanne Graff, Jordan Catalano, Sharon Cherski, and the resident nerdy neighbor Brian Krakow, nobody was denied a point of view.

It’s not possible to overstate how much I identified with Brian when I was 18. I kind of felt like he was literally based on me. I’m putting him here in the “runners up” only because I’ve changed so much, and my historical identification with Brian reflects limitations in how I saw myself.

I guess that’s the point of all of this. Nerdy or not, humans are infinite. May TV reflect this infinitude.

Notes

[1] (a) Do not look this up on Youtube! It needs to be appreciated in context. If you’re curious, watch the entirety of season 2. (b) I suspect there are those who would question my nerd cred for suggesting that my favorite use of MtFBWY occurs elsewhere than the OT. Now, I forcefully reject the notion of “nerd cred.” An exclusionary posture about nerddom is both limiting (cf. the rest of this blog post) and a singularly bad look on people who have ever felt excluded. Nonetheless, I am happy to establish mine. Saying your favorite MtFBWY occurs outside the OT is kind of like saying that your favorite lightsaber fight is RvD2. You say it in the full acknowlegement that whatever you’re naming as your favorite owes its whole existence to the OT. Happy now? 😉

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.)

Hamilton: Visibility/Invisibility of Brown Brilliance, part II

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:

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.

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? (“If you feel the urge to freak, do the jitterbug / Come and spread your arms if you really need a hug”), 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. These songs made me feel explicitly invited.

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

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 “MF 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.

What I want to add:

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!