Math Is Democracy III: A Short Rant about Voting Theory

I have been full tilt exploring the relationship between mathematics and democracy for the last at least year and a half. E.g. check out my TED talk.

And I’ve been thinking about this relationship, in some form, for the last 8 or 9 years, ever since Jay Gillen said to me, “Everything you love about math is what free people love about democracy.” (See this previous post.)

Given this, it may be surprising that until fairly recently, the well-established mathematical subfield that explicitly addresses democracy, voting theory (aka social choice theory), never grabbed my interest. FWIW, this has changed, but I think my journey around it is indicative of something worth keeping in mind for mathematical people who want to use math to think about how society works. (E.g., me, and hopefully you.)

Voting theory asks: given a large number of people forming a collective (e.g. a nation or state!), and a choice the collective is faced with (e.g. electing a political leader!), what are the possible ways one could aggregate individual preferences into a collective decision? (E.g. everybody vote for one candidate, and the candidate with the most votes is elected, aka plurality vote, aka how most political elections work in practice; but other possibilities too, like instant runoff, the Borda count, approval voting, and score voting.) And what are their properties? (E.g. in plurality voting, there is a potential spoiler effect when similar candidates split the vote of the majority, which is mitigated in approval and score voting since they do not force voters to choose only one candidate to support.)

It’s sort of a puzzle: given how much I’ve always cared about math, and how much I’ve always cared about democracy, why wouldn’t I want to understand this developed and beautiful theory exploring this most fundamental democratic operation — how does a collective make a decision?

Well, I’ll tell you.

The first and loudest thing I heard about voting theory, beyond just what it’s about, was Arrow’s impossibility theorem. This is a “foundational” result in the theory. It is often glossed as asserting that “there is no perfect voting system.” (Do not take this interpretation at face value. More below.)

From this, I got the impression that the point of voting theory was for us to brain out about the possibilities, and then conclude (with mathematical certainty!) that “we the people” can’t win no matter what. I’m supposed to be excited about this?

Was the whole thing just an intellectual exercise? Many years ago, I listened to a mathematician excitedly describe the Borda count to some graduate students. I remember thinking, with some confusion, “Where is the excitement coming from?” It seemed to me that the mathematician was excited only intellectually, about the mathematical properties of this system, and was totally disengaged from the question of whether anybody had used it or would use it or should use it to make any collective decision.

In this context, voting theory seemed almost grotesque to me. I love math for math’s sake — my PhD work is in pure algebra. But taking the urgent moral and political problem of building a system of government that adequately reflects the will of the people, and using it as loose inspiration for some math for math’s sake — ick.

Now, I take full responsibility for this impression. Many practitioners of voting theory believe the system can be radically improved, and are much more engaged with the realities of elections than I was giving them credit for. Steven Brams, now a mentor of mine, was involved in the adoption of approval voting for leadership elections in several professional societies. (Steve’s attempts to push something similar with political elections have mostly been frustrated so far, although advocacy by the Center for Election Science, an organization Steve has advised, played a role in the recent adoption of approval voting by Fargo, ND.) And Warren Smith has been developing a website to spread information about score voting since 2005. Just as two examples.

Now that I understand that voting theory is meaningfully engaged with actual elections, I’ve done a 180, and I’ve been reading up on both the foundations and recent research, and am looking for ways to contribute myself.

That said, I retain the feeling (with hopefully due humility that I could be wrong again) that the field would benefit from greater accountability to the problems it’s trying to speak to. The imperatives of mathematical exploration are different than the imperative of improving the mechanisms by which the will of the people expresses itself, and these differing imperatives can be in tension.

I’m reading one of Donald Saari’s books on voting theory. Saari has developed a beautiful, geometric way of looking at election systems that offers great clarity in making sense of some of the counterintuitive things that can happen. This is beautiful and important math. He also tends to portray the Borda count as kind of the best voting system, and I cannot shake the feeling that this is more because of the mathematical beauty of its properties, when viewed through a particular mathematical lens, than a sober assessment of its value to society.

So. I want to see and be part of a voting theory that is responsive to the problem it involves itself in. We can talk about voting systems using the tools of academic discourse, but we must remember the stakes. This is not an academic exercise.

Appendix

I promised above a little explanation of my comment that Arrow’s theorem should not be taken as stating that “there is no perfect voting system”. Lemme get into that for a sec. Arrow’s theorem says a voting system of a very specific type cannot obey a certain very specific list of prima facie desirable properties all at once. My basic objection is that the theorem is too specific for this broad an interpretation. My more fine-grained caveat is that there is nothing God-given about the particular set of desirable features Arrow chose to analyze, so deviation from them is just one definition of “imperfection”, and there are plenty of others.

In either case, I don’t blame Arrow for this, but I definitely thought I was being told that making the system better isn’t on the table. My real motivation here is to disabuse you of that misimpression if you had it too.

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Measuring Gerrymandering

I gave a talk at MoMath back in early May, about mathematical efforts to measure gerrymandering, culminating with recent very exciting efforts such as those of the Duke Quantifying Gerrymandering group.

It’s online! Check it out!

Acknowledgement: this talk owes a lot to the conferences organized this past academic year by the Tufts Metric Geometry and Gerrymandering Group, and especially to Mira Bernstein.

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.

“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.

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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.)