The Irrelevant Innumeracy of the Swarmed GOP Town Halls

Surely by now you've seen the stories about GOP congressmen, in deep red districts, being absolutely swarmed by angry constituents furious that they're not standing up to Trump, Musk, DOGE, and the buzzsaw attack on hardworking federal employees.

My thesis about this will be twofold. First, there's objectively less to these events than meets the eye. And second, it doesn't matter that there's objectively less to these events than meets the eye, and we should all behave like they're exactly what they appear to be.

Start with the first. The excitement over these protests relates to the sense that anger and outrage over Trump has expanded beyond the blue bubbles and is penetrating even dark red terrain. But the mistake here is something I alluded to in my How To Tokenize with Proportions post. A congressional district where a Republican won by, say, a 66-33 margin is by any measure a dark-red district. But it also is a district where one in three voters voted Democrat. One in three is a lot of people! In a congressional district of 750,000, it's 250,000 people! It is not hard to fill a high school auditorium, particularly if that 250,000 is feeling especially angry and activated.

It's an issue of framing, if you're generous, or innumeracy, if you're not: 33% doesn't feel very common, 1 in 3 feels very common. Politically speaking, the former is closer to accurate, which is why congressional districts where one sees 66/33 margins aren't typically treated as competitive.

But what innumeracy taketh away, innumeracy also giveth. The fact is that most people see a crowd of angry constituents filling an auditorium in a deep-red district and don't start doing math about how easy or hard it is to fill up the space given the baseline number of Democrats around. They just see the crowd. Politics is often a game of perception and of momentum -- people see others in their community and in their spaces expressing anger and fear and frustration, and it validates their own nascent feelings of anger, fear, and frustration. It makes them feel like they're not alone. It encourages them that these sentiments are common in their community, and that they're not weird or outcasts or loners if they feel them too. All of that starts to build a narrative conducive to resistance. And even if it doesn't mean the deepest-red congressional districts will flip blue in 2026, it gets that permission structure going that will make life very difficult for Republicans in more vulnerable seats.

So keep swarming. Keep yelling. Keep sharing those vids. Build up that narrative that people everywhere are mad as hell, and they're ready to fight. In politics, image becomes reality before you know it.

The "Rigorous" Case Against Teaching More Students More Math

California is in the process of updating its statewide framework for Mathematics education, and unsurprisingly there is controversy -- controversy that will strike many observers as having striking similarities to the recent flare-ups about "ethnic studies" in the state. Monica Osborne in Newsweek has a concise summary of the position of opponents (largely summarizing the objections made in this open letter). There are the de rigueur drive-bys against the curriculum becoming too "political" and non-"neutral", to the point at which it is allegedly not even "mathematics" anymore -- I actually addressed some of those objections in a prior post. But the main objection is claiming that the new math framework abolishes (or at least sharply discourages) "accelerated" or "advanced" math classes, doing disservice to students who genuinely are "gifted" in a misguided nod towards "equity".

One thing I will say is that the open letter (perhaps learning from the Ethnic Studies debacle, where a refusal to do basic citation led to outright fabrications being bandied about by curriculum opponents) does everyone the helpful service of citing its sources so one can see where, exactly, the Mathematics Framework allegedly does say the things its critics contend it is saying. It isn't perfect -- for example, the letter claims that the Framework "[e]ncourages keeping all students together in the same math program until the 11th grade and argues that offering differentiated programs causes student 'fragility' and racial animosity (Ch. 1, p. 15)"; but I wasn't able to find either the "keeping all students together" claim or the "racial animosity" claim on that page (as for the "fragility" point, which was mentioned, it is interesting to note that -- perhaps contrary to one's intuition reading that word out of context -- the claimed "fragility" is that of the gifted students who come to "fear times of struggle in case they lose the label"). Nonetheless, it is tremendously helpful in assessing these claims that they are consciously linked to particular portions of the draft framework. Kudos for that!

Having said that, what do we make of how the new framework addresses issues like "tracking" and "giftedness" and other like practices? To some extent, there seems to be the usual "talking past one another" that one finds in controversies such as this; the State Board explicitly denies that the new framework removes programming to serve "gifted" students, or eliminates "accelerated" classes. But clearly they are attempting to make some alterations that are designed -- I think it is fair to say -- to de-emphasize the degree to which math education is sharply divided between the naturally-gifted "haves" and the hapless "have-nots". Are those modifications salutary? Do they underserve brilliant minds? Or are they increasing access to rigorous mathematical education for all students?

I want to digress for a moment to talk about my own journey through mathematics education. I am not, to be clear, a math teacher (though I am a teacher, and so to that extent have practitioner-level experience with pedagogy in general). I also never considered myself, and to this day do not consider myself, "good at math" (a label which to some extent makes this debate more interesting to me). This is so even though under any objective metric I was actually very good at math -- I got a 720 on my math SAT and a 4 on my AP Calculus exam. That's quite good, especially for someone who always considered himself "bad at math"!

My recollection of how math education occurred at my (high-performing, suburban public) school is that between sixth and seventh grade students made a single, fateful choice. They could enroll in advanced math -- that would put them on track to take advanced calculus (multi-variable calculus) by twelfth grade. They could enroll in honors math -- that would put them on track to take regular calculus in twelfth grade. Or they could enroll in some form of "basic" or "remedial" math, in which case they would not be on track to take calculus at all.

Notice that this single choice at age 12 basically laid out the entirety of one's mathematical education for the next six years. There was essentially no moving up the ladder once one made the choice (there was theoretically the possibility of falling down the ladder, although that was a terrifying prospect that kept many a "gifted" student awake cramming at night). For my part, I chose the middle ("honors") route -- I didn't think of myself as "good at math", so the top path wasn't for me, but I was a conscientious student, so I wasn't going to go the bottom route either. And from that point forward, my path was set; and to a large extent so was my self-identity vis-a-vis math. Despite the fact that I objectively was perfectly capable of doing well at math (again, check those SAT and AP scores), I always viewed myself as not a math guy -- the people who were taking the advanced classes; they were the ones good at math. As someone who tends to think very verbally, I never connected with math, never fully saw its usefulness to someone like me and with my interests.

This persisted even after I graduated high school. Carleton had a math/science distribution requirement, which I skirted as best I could with classes like "Science and Society" or "Conservation Biology" (which basically was a semester of taking pleasant hikes through nature). The political science department did require one to take a statistical methods sequence, which I satisfied by taking (1) the lowest level stats class the school offered and then (2) a "methods" class where I resolutely avoided doing any of the hard statistical analysis by deliberately choosing a research question where I found no correlation between my variables (so no need for robustness checks). When I got to graduate school, there was no way in hell I was going to touch the methods classes (which were not required for political theorists) -- that was way to advanced for little ol' anti-math me. I can't quite say I "regret" not taking them -- even now, the thought of it fills me with dread -- but I can say that lacking the ability to conduct independent empirical research is probably the biggest gap in my scholarly toolset dividing the sorts of scholarship I'm interested in doing from that which I'm capable of doing, and I do regret that.

Thinking back on it, it is simply insane that this entire mathematical-education arc was more or less set in stone via a single decision made when I was twelve. That's nuts! All the more nuts since it ended up becoming quite obvious that I was perfectly capable of learning advanced math; the choice made then did not map onto any "natural" capacity I did or did not have. I can fully accept that some other students at twelve might have had more fire in the belly for math than I did at the time, and that in turn could suggest different styles of teaching math to them than would have been appropriate for me. But to lock either of us into a particular rigid track at that age, telling us "you will go exactly this high, and no further" (and "you can go lower, but only if you cop to being a disgraceful failure") seems absolutely absurd.

It was remembering that personal trajectory that informed my read on what California is proposing. The core of their argument is, more or less, "all or nearly all students are capable of learning high-level math, and so making decisions in middle school that lock students in or out of taking high-level math classes at the end of high school is both unnecessary and foolish". Instead, we should restructure math education so that there are many different pathways that offer the opportunity for more students to end up enrolled in high-level math courses -- the choice you make at twelve shouldn't be your destiny. And likewise, we should recognize that "calculus" is not the only example of "high-level" math -- it is one, but not the only one. The current system is defined by the "race to calculus", where success is defined solely by whether and how fast one gets to the calculus class of the math sequence. This was certainly my experience -- the advanced students got there in 11th grade (then took an extra class of advanced calculus senior year), the middle group I was in get there in 12th grade, and the bottom group doesn't get there at all and is thus looked down upon. We actually had an AP Statistics class offered as well, but I honestly do not recall who even took it -- it was definitely seen as the "lesser" math class, even though in retrospect a strong knowledge of statistics would have been far more useful to me than the calculus class I did take. But it did not lie on the rigid trajectory of the math sequence, which meant it was implicitly downgraded as the off-ramp for the failures.

Math education, even more so than other disciplines, is addicted to the "cult of the genius", where we can identify relatively early on certain students as naturally gifted and others as hapless drones, and sort them into appropriate tracks based on those assessments. The California Framework suggests that this cult, like many cults, is not backed by empirical evidence -- one needn't be a gifted genius to learn high-level math, and working on the assumption that one does need to be such a genius means providing decidedly suboptimal math education to the median California student. We could be successfully teaching more students more math, but we don't because we somehow decided that we can sort the geniuses from the worker bees at the tweenage years and after that we just let nature take its course. It is unsurprising that these gut-check instincts about who is a "genius" track the usual lines of social hierarchy and stereotyping, but that is just the tip of the problematic iceberg.

The reformulation proposed by the California framework is not to prohibit strong, enthusiastic math students from being able to pursue their interests. For those students, it suggests that optimal math education is not defined by racing them to calculus as soon as possible, but may entail giving them more opportunities to explore more elements of math in greater depth or rigor. Moreover, by broadening the pathway that leads to high-level math, we reduce the sometimes overwhelming pressure that falls upon these student where if they struggle anywhere they are a failure and have sacrificed their future. It stands to reason that there will be plenty of "gifted" students for whom algebra clicks easily and for whom geometry is a nightmare. The status quo assumes that anyone who happens to be good at that type of math that is prevalent in grade seven will likewise sail through all the other math concepts at a similar pace. It would be better if we recognized that all students likely will have some parts of math they find easy or which naturally connects (or which they just enjoy more), and other areas they find difficult and need more time on (or just find profoundly uninteresting), and that acknowledging the latter does not come at the cost of giving up one's chance to take upper level courses. The rigidity of the current framework doesn't serve "advanced" students even on its own terms. It locks them into a rat race with only one destination and exceptionally high stakes for failure. It is not a good thing that strong math students at my high school essentially could not take AP Statistics even if that interested them more than AP Calculus.

But the bigger reform being pitched is that, for the average California student, the math sequence being offered isn't rigorous enough. It knocks too many students off the path of taking advanced math classes at far too early a point. Pluralizing the sequence of math classes and making it so that many students -- not just those identified as "stars" as tweens -- are in position to take advanced coursework in the field is a step towards stronger math education. It is step towards teaching more students more math at a higher level -- an unambiguously good thing, as far as I'm concerned.

In this, Osborne's reference to the opportunities wealthier families have to place their children in bespoke math "enrichment" programs actually proves the opposite of her point. It is unlikely, after all, that all or even most of the students who enroll in such programs are "geniuses". But that's the point -- they don't have to be: give them access to advanced programming, and they're perfectly capable of learning the material, because this education is not the province of geniuses but of ordinary students. The lesson California draws from those programs is that most of their students, "genius"-labeled or no, can and should have access to strong, rigorous mathematical education. So they should reorient their math education away from rigid, imprecise sorting at young ages, and instead think in terms of providing a plurality of pathways to get as many students as possible to advanced classes. Rich parents get this already; they can provide their kids with advanced instruction regardless of how they're labeled as tweens. This is powerful evidence that the rest of the public school system should follow suit.

What may be true is that California is trying to at least diminish the obvious hierarchy within math, where we can say the students who took multi-variable calculus their senior year are better than the those who only took AB calculus are better than those who didn't take calculus at all (recall my high school, where taking AP Statistics was a sure mark of being a lesser student). Reorienting the math curriculum so that many different end points (calculus, statistics, formal modeling, and so on) are all rigorous and robust means consciously trying to blur hierarchical lines that say one of these ends is "higher" than the others. If the most important part of math education is providing for status-differentiation -- being crystal clear about who are the elite students and who are the normies -- then this is a loss. It is, we should be clear, a very different loss from depriving "gifted" students of the opportunity to learn advanced math -- they're still learning it, they if anything have more choices on exactly what arena of advanced math they want to concentrate on, but they are not simply by virtue of staying on the top rung of the ladder they climbed at age twelve automatically viewed as superior math students to their peers. There is good ground to believe that it is that loss --the status loss, not the educational loss -- which is what's motivating much of the anger at the reform (how can I guarantee I'll get into Stanford unless everyone knows I'm better than my classmates who weren't imprecisely identified as math geniuses as a tween?). 

But California is making a conscious decision to trade having clear ordinal hierarchical ranking for giving more students more rigorous math instruction than they have now. That, to me, sounds like a good trade. And to the extent it isn't -- the problem isn't that the new curriculum lacks rigor or ambition, or tries to bring everyone down to the lowest common denominator. It is because it is sincerely committed to doing more for more students that the framework is generating this animosity.

ACES Wild

Last we encountered the "Alliance for Constructive Ethnic Studies" (ACES), they were pushing fabricated evidence and wild screeds against "critical race theory" in a failed attempt to derail the California Ethnic Studies Model Curriculum after it was reformed in accord with tremendous efforts by a range of California Jewish (and non-Jewish) organizations.

Now they're back in action, and this time their target is California's new draft Mathematical Framework. What horrors are contained inside? Let's look!

The first draft of the California Mathematics Framework is out for review, and it includes as a resource "A Pathway to Equitable Math Instruction," a guide that labels teaching practices like "addressing mistakes" and  "focus on the right answer" as "white supremacy culture."

This is critical race theory.

This is discrimination. 

(Is this "critical race theory"? Nope, not going to get sucked into that).

Unfortunately, as was the case in the ESMC debacle, we are given only the thinnest possible citations to the primary sources for the alleged offending content. The link to the CMF draft goes to a website offering a thirteen chapter document, all in separate files, comprised of hundreds of page, with no indication of where in the morass the "Pathway" document is included. The link to the Pathway itself, for its part, goes to a site that contains five separate documents, again totaling hundreds of pages, with nary a clue as to where this language about "addressing mistakes" might be found. All of this, I suppose, is left as an exercise for the reader.

Well, I may not be a math expert, but I have gotten familiar enough with the strategies of ACES and its friends to know better than to accept what they say on faith. So I went in search of this resource and this language, to see if it is as scary and offensive as they say.

I want to begin with some good news: unlike the Ethnic Studies case, ACES and its allies do not appear to have completely fabricated the inclusion of the putatively offensive material. Congratulations, ACES! This is a big step forward for you as an organization, and you should give yourself a hearty pat on the back.

Alas, if we ask for more than "not fabricated" and stretch all the way out to "not abjectly misleading", things get dimmer.

Start with the CMF draft. From what I can tell, the section they refer to (where the Pathway document is "included as a resource") is on page 44 of chapter two (lines 1010-13). Here, in its totality, is what's included:

Other resources for teaching mathematics with a social justice perspective include... The five strides of Equitable Math.org: https://equitablemath.org/

That's it (The website "Equitablemath.org" is titled "A Pathway to Equitable Math Instruction"). It is mentioned, unadorned, in the "other resources" conclusion -- and as far as I can tell, nowhere else. Wowzers. I can feel the racial divisiveness cracking up from here.

One thing I'll observe on this is that often times one hears critics of "critical race theory" (or whatever random buzzword they're using today to connote "scary left-wing idea with a vaguely identity-politics kick") say that their problem isn't that the idea is included, but only that it's indoctrinated -- it's not one perspective of many, it's the only perspective on offer. This protestation was always rather thin -- the many many bills banning "critical race theory" are decidedly not about ensuring viewpoint diversity -- and one sees just how hollow it is here. The raw, unadorned inclusion of the Equitable Math resource -- as part of a broader whole, not even quoted from directly -- is too much for these people to tolerate. This is not about ideological heterodoxy. This is about censoring ideas, full stop.

But maybe Equitable Math is such an awful or inane document that it would be wrong to include it, even as one resource among many. The way it's described, after all, makes it sound like Equitable Math is a group of hippies saying "2+2 = 4 is the white man's answer, man! Fight the power!" Is that what's happening? Is this a fever dream of post-modernism where nothing is true and everything is permitted?

Once again, I had to dig for myself to figure out where this content was so I could see it in context. The answer appears to be the first document on the site, titled "Dismantling Racism in Mathematics", on pages 65-68. Do they deny that there are such things as "right" answers in math? No: "Of course, most math problems have correct answers," but there are math problems (particularly word problems, but also data analysis) that can be interpreted in different ways that yield different "right" conclusions, and students and teachers should be attentive to that possibility. Do they say one should never "address mistakes"? No again, but mistakes should not simply be called out flatly but rather used as "opportunities for learning" with an emphasis on building on what the student does understand to lead them to recognize what they misapprehend.

I don't teach math, obviously, but there are many occasions where I'll say "such-and-such is the doctrinally correct answer -- but if we look at the problem from this other vantage, doesn't this other position become more plausible?" So when the Equitable Math site suggests, as an alternative to obsessive focus on the one correct answer, classroom activities like " Using a set of data, analyze it in multiple ways to draw different conclusions" -- well, that doesn't seem weird to me. Certainly, as someone who is also trained as a social scientist, I can say confidently that it's quite valuable to anyone who has seen how the same dataset can be deployed by different people with different priors to support different agendas.

Even more than that, the suggestions around "addressing mistakes" resonate with how I try to teach in my classrooms. Sometimes my students say something wrong. When they do so, for the most part I don't say "bzzzt" and move on; instead I try to guide them to the correct answer by having them unpack their own thinking. There's a lot of "I see what you mean by [X], but suppose ..." and ask questions which hone in on the problems or misunderstandings latent in what they're saying. And eventually they get there, hopefully without feeling like they've just been put inside an Iron Maiden for daring speak up. 

Admittedly, I've never thought of what I'm doing as "dismantling White supremacy" -- I just viewed it as good pedagogy. But then again, that's kind of what I've always thought when asked about such subjects -- we act as if there's this deep magic to fostering equity and inclusion in the classroom, when really it's employing the basic strategies of being a good teacher, one of which is declining to engage in a measuring contest where you prove you know more than the student does. Obviously I know more than the student does. I don't need to prove anything. So if they say something wrong, I do not gleefully pounce on them for it, I do my best to build on what they do know to get them to a position of right. Is that so outrageous?

Finally, ACES in its tweet identifies one other area of crazy-lefty-craziness in this resource: "the incorporation of 'Ethnomathematics'". What does that mean? They don't say, correctly surmising that fevered imaginations will produce something far worse than anything they might quote. So I'll do the quoting for them (this comes from page 8):

Center Ethnomathematics: 

• Recognize the ways that communities of color engage in mathematics and problem solving in their everyday lives. 

• Teach that mathematics can help solve problems affecting students’ communities. Model the use of math as a solution to their immediate problems, needs, or desires. 

• Identify and challenge the ways that math is used to uphold capitalist, imperialist, and racist views. 

• Teach the value of math as both an abstract concept and as a useful everyday tool. 

• Expose students to examples of people who have used math as resistance. Provide learning opportunities that use math as resistance.

I know, I know -- we're all going to pitch a fit about challenging "capitalist views". But apart from that, this seems ... very normal? We all know, to the point of cliche, that a barrier to getting kids interested in math is that they fail to see how it's useful to them or "in the real world". So they advise that math be taught in a way that resonates with real world experience. And likewise, sometimes, for some people "in the real world", math can feel like an enemy (think "am I just a statistic to you?"). So figure out ways to name that and challenge that. For the most part, "ethnomathematics" just reads as a particular social justice gloss on "being a good teacher", as applied to teaching in diverse communities.

Now perhaps one disagrees with these concepts as pedagogical best practice. I'm not a math teacher, I'm not going to claim direct experience here. But that goes back to the intensity of the backlash -- that these ideas need to be banned, that they are outright dangerous and unacceptable and neo-racism. Can that be right? Surely, these ideas are not so outlandish that we should pitch a fit about their being (deep breath) single elements of an 80 page document which is itself part of a five part series being incorporated as a single "see also" bullet point in the second chapter of a thirteen chapter model state framework. Seriously? That's where we're landing? That's what's going to drive us into a valley of racial division and despair?

It's wild. The people engaged in this obsessive crusade to make Everest size mountains over backyard anthills are nothing short of wild.

The Ring Announcer's Dilemma

The below feels like one of those math puzzles, though I don't actually know enough about math puzzles to know if it really is one. It is something I've genuinely noticed and wondered about when watching boxing.

In boxing, there are functionally six types of decisions: A unanimous decision, a split decision, a majority decision, as well as a unanimous draw, a split draw, and a majority draw.

• In a unanimous decision, all three judges agree in scoring the bout in favor of one fighter.
• In a split decision, two judges score the bout in favor of one fighter, while the third scores it favor of the other.
• In a majority decision, two judges score the bout in favor of one fighter, while the third scores it draw.
• In a unanimous draw, all three judges score the bout a draw.
• In a split draw, one judge votes for fighter A, one for fighter B, and the third scores it a draw.
• In a majority draw, one judge votes for one fighter, but the other two judges score it a draw.

When a ring announcer gets set to tell the audience the judges' decision, there are several pieces of information he needs to communicate. By the end of his announcement, the audience should know how each judge scored the fight, and for whom, and of course they need to know the actual result of the fight (who won, or that it was a draw). In general, however, he can announce the three judges' decision in any order he likes. Being a good performer, the announcer would like there to be as much suspense as possible. In practice that means he wants to the last piece of information he reveals to be the result.

The dilemma is as follows: how, if it all, can an announcer accomplish that goal in the case of a majority draw?

Start with a unanimous decision. A bad announcer might deliver the decision this way:

"Judge A had the bout 77-75 for Doe, Judge B had the bout 78-75 for Doe, and Judge C had the bout 78-74 for the winner, John Doe!"

Notice how once the crowd knows both A and B voted for Doe, they know the result even before it is announced. John Doe has at least a majority of the judges, so he won. In order to achieve the result of not tipping off the audience until the very end, a better announcement might go as follows:

"Judge A had the bout 77-75, Judge B had the bout 78-75, and Judge C had the bout 78-74, all for the winner, by unanimous decision, John Doe!"

Notice how by the end everyone knows how each judge voted, and for whom, but the last piece of information they got was the result. Until "John Doe" was said, the crowd didn't know the result of the match.

One can do this for most types of decision. A split decision can be announced like this:

Judge A had the bout 77-75 for Doe. Judge B had the bout 77-75 for Smith. And Judge C had the fight 77-75 for the winner, by split decision, John Doe!"

This works so long as the announcer is permitted to choose what order he delivers the judges' verdicts (i.e., he can make sure the one card for Smith is announced either first or second).

Here's a split draw:

Judge A had the bout 77-75 for Doe. Judge B had the bout 77-75 for Smith. And Judge C had the fight 76-76 -- this fight is a draw!

Here's a unanimous draw: 

All three judges scored the fight the same, 76-76 -- this bout is a draw!

And here's a majority decision:

Judge A scored the bout 76-76. Overruled by Judge B, who scored the bout 77-75, and Judge C, who scored the bout 78-74, for the winner, by majority decision, John Doe!

The majority decision is the toughest one so far -- the alert listener knows once the announcer says the word "overruled" that a majority decision is coming*, but still doesn't know who won.

But what of a majority draw? Consider the following:

Judge A scores the fight 77-75 for Doe. Overruled by judges B and C, who each score the bout even -- this fight is a majority draw!

Here one knows the result of the fight -- that it's a majority draw -- the moment the announcer says "overruled". Why? Well, there are only two ways that A's score for Doe could be overruled -- either B and C voted for Doe's opponent, or they voted for a draw. If it was the former, though, this would be a terrible way to announce it, as the audience would know who won as soon as just one of Judge B or C's card was announced for the other fighter. There'd be no suspense. Given that, we all know that the card was overruled by two judges voting for a draw even before we actually hear it.

What happens if you announce the cards in a different order?

Judge B and C score the fight 76-76, a draw, overruling Judge A, who scored the fight 77-75 Doe.

Nope -- that gives away the result before we ever hear Judge A's card. Similar problems emerge if you try to do something like going B (draw), A (Doe), C (draw) -- once you've revealed that B voted draw and A voted for Doe, then you know that if anyone won it has to be Doe (by majority decision), which means that if Doe did win you'd know as soon as the announcer gave a non-draw score even before they told you who the judge voted for -- and knowing that the announcer wouldn't do that, you know that C's score is going to be a draw and that the fight will be a majority draw. 

*Deep breath*

So ... is there a resolution to this? Is there a way for a ring announcer to announce a majority draw without sapping it of all the drama? I don't know. And I don't know if this "dilemma" reveals anything interesting. But I have noticed it, and haven't been able to solve it (if it can be solved).

* How does he know? Because Judge A's decision to score the fight a draw could only be "overruled" if the two other judges did not have it a draw and did vote for the same person to win. If one or both scored it a draw, then the fight would be either a majority or unanimous draw. And if they didn't score the fight even but voted for different fighters to win, then the result would be a split draw.

If You're One in a Million...

Many of you are familiar with the saying "If you're one in a million, there are a thousand people just like you in China alone."

It helps illustrate that while one in a million is certainly very rare, on another way of looking it at it's also quite common. A thousand people! You could fill a high school gymnasium with that!

Push the proportion down a bit and things get even more stark. Imagine a political view held by only 1% of the population. That's pretty fringe, right (for reference, 33% of Americans believe that alien UFOs have visited Earth)? But it's also one in a hundred -- in America, that translates to well over three million people. That's a lot! (We explored this dynamic previously in my "how to tokenize with proportions" post.)

One thing I often think about is how modernity and modern technology, in conjunction with our decidedly pre-modern lizard-brains, don't always mesh well. We know, for example, that fat tastes delicious because in the primordial environment it was rare and vital, and thus highly desirable to consume -- unfortunately, this doesn't translate well to a contemporary context where calories and fat are plentiful and we can easily over-saturate ourselves.

I suspect there's something similar going on with political opinions. One of the oft-proclaimed virtues of the internet is it allows you to find communities of like-minded persons no matter how obscure or random the interest. Obsessed with underwater basketweaving? You can find dozens of people who share that passion with minimal effort!

What does it mean when the same is true for political opinions? I suspect our brains have a rough heuristic at the ready that correlates how difficult it is to find holders of a given opinion with how uncommon it is in society. If one struggles to come across individuals who believe ideology X, one assumes that X is rarely believed in a given society. If one comes across X-believers without too much trouble, one infers that X is a common ideology. If 1% of Americans hold a particular political stance, that may be three million people -- but (at least until recently) they're not going to be easy to find via the normal modes of political engagement. If you just read newspaper columns, chatted with your neighbors, watched TV pundits, and so forth, you'd probably come across it rarely, if ever. If one really wanted to find a sizable chunk of Americans who believe this 1% view, one would have to expend considerably more effort.

Now to be clear: what I'm describing is only a heuristic, which means it's imperfect -- there are all sorts of reasons why, for example, a rare opinion might nonetheless be easy to spot "in the wild" (it's favored among extroverts or celebrities, e.g.) or a common one might be rarely seen (it's embarrassing). But it has some logic as a rough-and-ready way of telling us which views are common in our social circle and which aren't. It's not quite the same as the availability heuristic, but it is similar. Call it the search heuristic. Something easy to find upon commencing a search for it is common; something hard to find even when searching for it is rare.

The problem is that if modern technology makes pretty much any opinion with even a speck of public salience "easy to find", that hijacks our heuristic circuitry to make all of these opinions register in our minds as "commonplace". What is the result of that?

One potentially positive result is that it might offset some mechanisms that serve to silence dissident views via the so-called "spiral of silence" -- they learn that they're not alone, and so they're more willing to air their dissident views knowing that there are peers who share their perspective.

But there are also some potential upshots that I'm more ambivalent about. One thing that we might experience is the erosion of perceived consensus -- a sense of widespread opinion balkanization and a corresponding vertiginous inability to tell when there is an opinion that carries significant social agreement. There's a push/pull on this -- sometimes, a feeling of "consensus" is dependent on wrongly not perceiving the existence of dissent, and so the elevation of dissident voices corrects a widespread social misperception. But, assuming "consensus" does not require universal agreement, sometimes, a feeling of dissensus is falsely inspired by the presence of high-profile but ultimately negligible dissenters. To the extent that modern technology makes very small ideological minorities loom larger, we might believe ourselves to be far more disunited than we actually are. And if the search heuristic causes a wide range of opinions (many mutually incompatible with one another) to register as "common", we may have trouble grasping onto distinctions between actually common versus fringe outlooks.

In a similar vein, it is at least plausible that in a democracy there is a prima facie obligation to consider and give airing to certain viewpoints simply by virtue of the fact that they're common. This wouldn't necessarily mean that uncommon views can be automatically rejected, only that they must "earn" their space on the democratic agenda by means other than "because many people believe it". If this is so, then the perception that more views are "common" mean that more views can claim access to this prima facie obligation of consideration. Perhaps that doesn't strike you as a bad thing -- but consider it in the case of, say, openly avowed racism or extremism -- views which might objectively be as rare as ever, but perhaps feel more common than they've been in recent memory.

There are also risks latent even for the holders of the dissident opinions themselves, for they as much as anyone might be mislead into thinking their views are more widely shared than they are. If someone holds a view they know is rare but wish was widely shared, they must endeavor to persuade others to adopt it. If they then, say, run for office on its platform whose tenets are held by only 10% of the population, if (or when) they lose they probably won't be happy but they at least probably won't be confused. Unpopular opinions don't win elections.

But things are different if the search heuristic misfires and makes the dissident believe they are actually expressing a very common view. If they nonetheless persistently lose in the democratic arena, they might suspect bias, corruption, institutional barriers, or other forms of foul play are obstructing them. To be clear: there are many cases where such things are at work; I'm not saying that everyone who believes their views are not carrying the democratic day because of various social biases is simply misleading themselves. But sometimes a democratic spade really is just a spade; and there is at least the potential for this sort of self-deception to accelerate -- the result being greater mistrust and resentment of social institutions.

It's worth noting that there isn't an "objective" way of declaring whether a view is "rare" or not. Much of it already lies in framing: "held by 1% of the population" sounds uncommon, "held by three million Americans" sounds reasonably common. So we can't quite say that, even if the search heuristic is misfiring, it is objectively causing us to label "uncommon opinions" as "common". But I do suspect that our wider net of appraisals around how we relate to an opinion based on its perceived "commonality" are tied to the same set of assumptions under which the search heuristic should function at least roughly well -- meaning that if we no longer exist in that social world, the whole edifice comes under serious strain (if it doesn't collapse outright).

These are preliminary thoughts; they are not wholly hashed out in my mind yet, and I'm curious to hear others' views. Here's the tl;dr

1. The search heuristic tells us that, roughly speaking, a view that is hard to find upon searching for it is rare, and a view that is easy to find upon searching for it is commonplace.
2. The social media revolution has drastically reduced the search costs required to find large absolute numbers of persons who hold any particular view, even when they are actually relatively uncommon.
3. Together, (1) & (2) cause us to mentally code many viewpoints which we'd perceive as uncommon as quite common (since we are able to find examples of them with little effort).
4. The effects of this are unclear, but may include (a) increased willingness to air dissident views; (b) decreased sense of social consensus; (c) decreased ability to distinguish relatively common versus uncommon views; (d) decreased trust that formal mechanisms for measuring public opinion reliably track actual public viewpoints (even when they are in fact doing so reasonably well).

Not My President's Day Roundup

Our apartment's water heater is being replaced tomorrow. That means my one true joy in life -- long, languid, hot showers -- will also have to go for the day. It will be terrible.

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China's crackdown on religious liberty threatens the tiny but ancient Kaifeng Jewish community.

One of the few Black mathematicians in American academia recounts the microaggressions and subtle racism which alienated him from his own discipline.

I think the tone of this column is a little off, but the broad point -- that leftist anti-Zionists have no friend more highly placed in Israel than Netanyahu himself -- is on the mark, and it's important that someone like Eric Yoffie is saying it.

Alabama newspaper editor urges the return of the KKK in order lynch Democrats (and insufficiently conservative Republicans). Yes, really.

Apparently, Louisiana has a bad habit of not releasing prisoners after they've finished serving their sentences.

"As a Jew, I’m either furious or eating. Sometimes both."

For all the talk of "creeping Sharia", the fact is that the American Muslim community is actually experiencing something very different: creeping liberalism. For a community that, for much of recent electoral history, at least leaned Republican (especially on social issues), the rapid embrace of feminism, gay equality, and sexual liberation among the younger generation is coming as a bit of a shock to the more conservative old-guard.

How To Tokenize with Proportions

13% of American Muslims voted for Donald Trump.

That's a minuscule proportion. It is around half the proportion that Hillary Clinton got in Idaho. It is fair to say that Muslims overwhelmingly voted against Trump, just like it is fair to say that Idahoans voted overwhelmingly against Clinton.

13% also translates, roughly, into "1 in 8". And when you think of it that way, it shouldn't be that hard to find a Muslim Trump supporter. Statistically, all you'd need to do is know eight American Muslims, and one of them is probably a Trump voter. And across a population of roughly 3.3 million Muslims, that means there are roughly 412,500 American Muslims who support Trump -- a lot of people! Yet it would be clearly, obviously wrong to use those "lot of people" to try and argue against the above conclusion that "Muslims overwhelmingly voted against Trump."

In short, it is simultaneously true that "Muslims overwhelmingly dislike Trump" and "it is not hard to find Muslims who do like Trump." Likewise, we can simultaneously know that Idaho is exceptionally conservative and know that finding liberal Idahoans doesn't take any herculean effort.

When one doesn't keep those two thoughts in mind, it is very easy to mislead oneself. I've noted that 13% is also the percentage of UK Jews who planned to vote Labour last election, but that still means it should not be remotely hard to find Jews -- quite a few Jews -- who are loud-and-proud for Jeremy Corbyn. If one is a Corbyn fan, one can (accurately!) think "look at all the Jews I know who support Corbyn" and then (inaccurately) conclude that the stories of widespread Jewish consternation over Corbyn are ginned-up nonsense. Same with Black Republicans -- they're simultaneously rare and not that difficult to find, and so it is easy for conservatives to dupe themselves into thinking they have no race problem by pointing out all the Black Republicans out there.

Ditto when one sees big crowds of angry constituents in a deeply conservative or liberal representative's town hall meeting. One can see those and think "wow -- even here people are turning against [Insert Party]!" But even in the most electorally lopsided districts, there are still going to be quite a few members of the other side -- certainly enough to pack an auditorium, if they're feeling motivated.

Or take this article, "To Understand White Liberal Racism, Read These Emails." It is about angry emails sent to school administrators regarding the decision by Seattle school teachers to wear "Black Lives Matter" t-shirts. The article observes that these emails came from one of the whitest, most affluent" and "staunchly liberal neighborhoods" in the city, places "dotted with rainbow yard signs that say 'All are welcome.'"

Applying the "staunchly liberal" label to these neighborhoods is entirely justified. The (Democratic) state senator in this part of Seattle was last re-elected with 80% of the vote. That's a crushing margin! But it still means that 1 in 5 voters in the district cast their ballot for Republicans. On the one hand, that's not a lot of people. On the other, that's a lot of people! Certainly, if 1-in-5 school parents have retrogressively conservative views on  race, that'd be enough to make their voices known in a letter-writing campaign.

Now, to be clear, it is entirely possible -- plausible even -- that these emails didn't come from the 1-in-5 Republicans but from the 4-in-5 Democrats. "Democrats" are a wide tent, and there are, indeed, plenty of putative progressives who are on a hair-trigger about race issues and would be prime candidates to send out letters like these. I'm not saying that because these emails were racist, they couldn't have come from liberals. They very much could have.

What I am saying is that we can't say "because this neighborhood is staunchly liberal, these emails must have come from liberals." That's because that conclusion entails a shift from the accurate observation that this part of Seattle is overwhelmingly liberal, to the inaccurate observation that any political or social activity substantial enough to make it onto the social radar screen must be emerging from liberals. It's quite possible for conservatives in a place like Northeast Seattle to be simultaneously a marginal presence and a visible one, under the right circumstances. Ditto liberals in a place like Idaho.

More broadly, this is just a particular example of an obvious point: words with the same meaning can nonetheless communicate very different messages. When we want to erase the minority presence, we talk in percentages (20% is teensy-tiny!). When we want to elevate it, we talk in ratios (1:5 is really common!). Both are right, and in fact both connotations are right: a minority of 20% is a very small minority (as against an 80% majority), but 1:5 people is very common. Keeping both connotations in mind is good deliberative practice. Jumping from one to the other as argumentatively-necessary is very bad practice.

Return on Investment

Tom Perkins (whom you may remember for suggesting that the rich were at risk of being victims of Nazi-style genocide) suggests that we apportion votes to tax dollars:

"The Tom Perkins system is: You don't get to vote unless you pay a dollar of taxes," Perkins said.

"But what I really think is, it should be like a corporation. You pay a million dollars in taxes, you get a million votes. How's that?"

Just like a corporation! Because the idea behind government is that you should get a cut of the proceeds in proportion to your investment! No, wait, in the public sphere that's known as "corruption". My bad.

Also, if I were more of a mathematician, I'd be (even more) offended by this too:

Pressed for examples of how the rich were being demonized, Perkins said that he feared higher taxes.

"The fear is wealth tax, higher taxes, higher death taxes -- just more taxes until there is no more 1%. And that that will creep down to the 5% and then the 10%," he said.

Yeah, not really sure that's how percentages work.

Clarification: We're Dumb

I love it when newspapers issue "clarifications" regarding statements that were actually just flagrantly false. Here's USA Today, "clarifying" that a pay raise "could very well bump you into the next tax bracket, possibly leaving you with less money." As Jon Chait notes, there is no situation where a pay raise could leave you with less money due to increased taxes, because our tax system is graduated. Worse yet, the column was entitled "math tips". Come on, people -- even I can get this.

Non-SOTU Roundup

'Cause that would be cliched.

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Pennsylvania Superior Court overturns a slew of anti-gay child custody decisions from the 1980s.

Indian roads create new opportunities.

The impact of Obama's school speech, one year later.

People are casting this post as Ta-Nehisi Coates pasting Chris Matthews, but I think the important points are far broader than anything Matthews-specific. That said, it's a fabulous post.

A U.Chicago study finds that female math teachers who are anxious about their own skills transmute that anxiety onto female pupils, resulting in reduced performance.

Bill Clinton hails Israeli mission to Haiti.

A touching post by Al Brophy on a friend of his who recently passed away.

As a pinball fan, I actually knew of the "call the shot" story which got pinball legalized in New York -- but that doesn't mean I won't share it.

Missing Women in Science

Carleton has a nearly legendary reputation for the relative gender-balance in our math and science programs. Our active, tenure or tenure-track Chemistry department is more than half female (five to four men), which is virtually unheard of. The Geology department is three men and two women. Physics is five/three, Biology is seven/three, and Math is six/four.

Most places are not like Carleton in this respect. Indeed, normally in professional math, science, and engineering settings, women are outnumbered by men by a 3:1 ratio. And studies cited in a recent Scientific Daily article show how this imbalance can seriously effect the performance of women who are so marginalized in these environments.

Murphy and colleagues showed a group of advanced MSE undergraduates a gender balanced or unbalanced video depicting a potential MSE summer leadership conference. To assess identity threat, the researchers measured the participant’s physiological arousal during the video, cognitive vigilance, sense of belonging and desire to participate in the conference.

The results are telling. The women who watched the gender unbalanced video- where women were outnumbered by men in a 3 to 1 ratio- experienced faster heart rates, higher skin conductance (sweating), and reported a lower sense of belonging and less desire to participate in the conference.

They also found that women were more vigilant to their physical environment when they watched the video in which women were outnumbered. Throughout the testing room, Murphy planted cues related to Math, Science, and Engineering such as magazines like Science, Scientific American, and Nature on the coffee table and a portrait of Einstein and the periodic table on the walls. Women were able to recall more details about the video and the test room, indicating that they paid more attention to the identity-relevant items in order to assess the likelihood of encountering identity threat. “It would not be surprising if the general cognitive functioning of women in the threatening setting was inhibited because of this allocation of attention toward MSE-related cues,” write the authors. Thus, it is likely that this kind of attention allocation would interfere with performance and might help explain the performance gap between men and women in these fields.

While men, in either condition, showed no significant difference in physiological arousal, cognitive vigilance, or sense of belonging, both men and women expressed more desire to attend the conference when the ratio of men to women was balanced. Murphy says that while it’s interesting that both men and women want to be where the women are, the motivations of men and women for wanting to be there are probably quite different. “Women probably feel more identity-safe in the environment where there are more women- they feel that they really could belong there- while men might simply be attracted by the unusual number of women in these settings. Men just aren’t used to seeing that many women in these settings, because the numbers in real Math, Science, and Engineering settings are so unbalanced.”

I think it's a little too cheap to say the only reason men prefer gender-balanced conferences is because of the novelty of it (this seems to play more than a little on "geek guys can't get girls" stereotypes), though I don't think it's wrong to discount it either. But regardless, the point is that everybody seems to be happier, and a significant chunk of the population more comfortable (and thus presumably more productive) when there isn't a notable, gaping imbalance among the genders at these conferences. Carleton is a great pipeline for female science graduates in part simply because women here are very visible in these departments. It isn't weird, it isn't out of the ordinary, there's very little sense that one is "trailblazing" or entering a terrain dominated by men (though I hasten to add that I imagine female math and science majors are certainly aware of those tropes in broader society, and will still have to deal with them there). This indicates that simply having a few familiar and friendly faces can do wonders for expanding the pool of qualified, competent, and comfortable science graduates, and provides yet more reason for colleges and universities across the country to try and adjust their departments accordingly.

Hat tip to the lovely Kawaii Kid, whose scientific exploits (Physics, I believe?) I have not been particularly supportive of (but only because I believe it's a waste of her debating talent!).