Nature on “decolonizing” mathematics

February 2, 2023 • 12:30 pm

The latest issue of Nature, one of the world’s most prestigious scientific journals, has a long (4-page) feature about the “decolonization” of mathematics. As we’ve learned to expect from this kind of article, it points out gender and ethnic inequities among mathematicians, ascribes them to structural racism existing today, and seen as ubiquitous in math, and and then proposes untested ways to achieve equity in math (proportional representation of groups) by infusing the teaching of math with aspects of local culture.

The problem with this paper, like similar “decolonization” screeds, is that while it certainly means well (I agree that everyone should have the chance to learn math), and is sensitive to differences among cultures, it gives no evidence that “decolonizing” mathematics (that is, removing its “whiteness” and “Westernness”, and using as math subjects features of the local culture) actually works. It’s a gift package of suggestions and assertions wrapped around, well, nothing.  This doesn’t meant that the suggestions are not worthwhile, but there’s nothing to be gained by blaming inequities, which could be due to a number of factors, to existing bigotry and racism in math, for which it offers no evidence.  More important, the “course” they chart has to be shown to actually lead to more understanding than alternatives.

Here’s where blame is affixed. You don’t have to be a rocket scientist to know that it adheres to white men.

Maths is built on a modern history of elevating the achievements of one group of people: white men. “Theorems or techniques have names associated to them and most of the time, those names are of nineteenth-century French or German men,” such as Georg Cantor, Henri Poincaré and Carl Friedrich Gauss, all of whom were white, says John Parker, head of the mathematical sciences department at Durham University, UK. This means that the accomplishments of people of other genders and races have often been pushed aside, preventing maths from being a level playing field. It has also squelched wider access to rich mathematical ideas developed by people of different backgrounds — such as Chike Obi, James Ezeilo and Adegoke Olubummo, a trio credited by the website Mathematicians of the African Diaspora with having pioneered modern maths research in Nigeria. Another example is Mary Golda Ross, a Cherokee mathematician and engineer who was a founding member of ‘Skunk Works’, a secretive division of the US aerospace manufacturer Lockheed. There, she developed early designs for space travel and satellites, among other things.

Where is the evidence that high quality and non-white mathematicians, of which until recently there were very few, are now being pushed aside by racism? I don’t doubt that there was discrimination in the past against women and minorities, but even then I keep thinking of the Indian Srinivasa Ramanujan, an immensely talented autodidact from Tamil Nadu who in 1913 sent a bunch of his theorems and proofs to G. H. Hardy at Cambridge, who instantly recognized the man’s talent and arranged for him to study at Cambridge. I can’t imagine anyone more “minoritized” in the UK than Ramanujan, dark of skin, poor, and humble of origin. And yet people helped him, and he’s still regarded as a giant in the field. Would people push him aside today—or anyone like him? I doubt it, just as I doubt that mathematics is presently rife with structural racism—that the playing field is still “far from level”. If “level” means “equal opportunity”, then I’d say we’re pretty close. If it means “equal outcoms”, I’d say, yes, it’s not level. But that’s not what a tilted playing field means: it means that right now there is not equal opportunity. Yes, the pipeline needs to fill up after a past of sexism and bigotry, but the article gives evidence for “structural bias” or “system bias” at the pipeline’s distal end.

Here’s what the advocates of decolonization advocate to replace the kind of math teaching we have today:

Edward Doolittle, a mathematician at First Nations University of Canada in Regina, contrasts Indigenous mathematics with the mainstream, global way of teaching maths, in which instructors essentially present the same content regardless of where they’re teaching.

Doolittle, who’s also a Mohawk person from Six Nations in southern Ontario, says that calculus courses are structured so similarly that he could teach the subject “anywhere the students speak English”, and even take over teaching a course midstream.

By contrast, he says that Indigenous mathematics involves getting inside a culture and examining the mathematical thinking in it. He draws a further distinction between Indigenous mathematics and the practice of what he calls “indigenizing mathematics”, which, he says, involves searching for cultural examples to use in courses taught in the global version of mathematics.

Indigenizing mathematics tweaks the curriculum when it isn’t feasible to fully immerse students in ideas from an Indigenous culture, Doolittle says. “It’s very hard, if not impossible, to break out of” the global mathematics system, he notes. By indigenizing mathematics, instructors can stay within the parameters of what they’re required to cover while broadening the cultural scope of their curriculum.

Using that approach, “we have respected the knowledge of Indigenous people and are furthering our ties with Indigenous people” while still teaching students core topics, he says. For example, when teaching statistics courses, Doolittle has discussed a simplified version of the Peach Stone Game, which is based on making wagers and is played in his community. “You can analyse this in terms of a binomial probability distribution,” or the chances of two outcomes over time, he says.

“I would like to encourage many of my colleagues to engage in indigenization efforts, and hopefully to turn up interesting examples from their local area,” Doolittle says.

As for how to “indigenize” math, the article gives a couple of examples beyond the Peach Stone Game: teaching about Polynesian navigation to Hawaiians  in Hawaii and using aspects of local culture to teach math in five African countries (“the next Einstein will be African” is the motto of this five-nation consortium, the African Institute for Mathematical Sciences, or AIMS). And that’s about it.  There is a lot of noise, but, as of yet, little to show that this kind of training produces results better than “non-indigenous” training. If it does work, more power to them. So far, most of the “indigenizing” appears to be mainly trying to increase the diversity of people going into math. That’s great, too, but it’s not a revolution in teaching math.

And even some of these endeavors involve bringing in mathematicians who aren’t indigenous. Here’s what AIMS does:

Faculty members at the centres are hired from African countries, often through partnerships with local universities. AIMS also hosts visiting lecturers from outside Africa who teach courses that range from a few weeks to two months in length. Bringing in outside researchers exposes students to top talent while they continue to expand their roots in Africa’s mathematical communities.

But isn’t it counterproductive to bring in “top talent”, probably white people, who undoubtedly teach math in decidedly non-Indigenous ways?

It’s clear that while I have no strong beef against using local culture or examples to teach math—or any form of science—this will go only so far (what happens when you get to really high-level math?), and if you’re going to do something like this, it’s better to start by showing in pilot projects that it really works.  Blaming whiteness or the West on holding down math education in places like Africa (where whites are actually a minority), is no longer tenable, and even counterproductive.

But here’s the part I most object to. Durham University in the UK is itself mounting a decolonization effort that involves Ric Crossman, a statistician, and John Parker, head of Durham’s maths department. Here’s their philosophy of education:

Durham’s senior mathematicians felt that their curriculum-reform process had to be led by the students, because otherwise “we’re in the awful situation of deciding for ourselves what’s best for them”, Crossman says. That, Parker adds, would be at odds with the concept of decolonization, because colonization “was some group of people thinking they knew best for some other group of people”.

What an AWFUL situation!  It’s certainly feasible for some students to tell you the best ways they can absorb mathematics, but this will certainly differ among students, and not every student knows. But to put the curriculum and all the teaching methods in the hands of the students, ignoring the experience of teachers who have spent years finding out which forms of pedagogy work in general, is a recipe for disaster. It’s simply invidious to denigrate the expertise of teachers by comparing it to “colonizers.” But such are the rhetorical tactics that progressives have learned to use.

h/t: Carl

Video: Alternative math takes over

January 18, 2023 • 12:45 pm

This video, “Alternative Math,” has been around for six years, and has won 15 awards for short features and funny videos. The sad thing is that while it’s funny, it’s also true: truer now than it was when it was made. It documents the “2 + 2 = 5” alternative-truth mentality that is represented by “other ways of knowing.” But it also has a funny ending, so be sure to watch the whole thing (it’s nine minutes long).

The IMDb summary (which also has info about the film and the cast) is this: “A well meaning math teacher finds herself trumped by a post-fact America.”


Four awarded the Fields Medal for mathematics, including only the second woman to get it (and she’s Ukrainian)

July 5, 2022 • 9:00 am

Every four years since 1936, the prestigious Fields Medal is awarded to a maximum of four young mathematicians (under 40) for outstanding accomplishments. The awarding organization is the International Congress of the International Mathematical Union.

It’s seen as the “Nobel Prize in Mathematics,” even though it isn’t formally a Nobel. But in one way it’s better: it’s awarded every four years instead of yearly. Since the Nobel Prize in any area can be given to up to three people, the maximum number of Nobelists in four years is twelve—compared to four for the Fields.

The down side, if there is one given the immense prestige the Fields confers, is that it doesn’t come with a lot of dosh—about $15,000 Canadian. In contrast, a Nobel Prize comes with a sum of 10 million Swedish kroner—almost exactly one million U.S. dollars. (If there are two winners it’s split evenly, if three the division is decided by the Swedes.) $15,000 won’t enable you to buy a beach house, as Feynman did with his Nobel money. But money seems of much smaller consequence than the fact that winners are topped for the life with the halo “Fields Medal Winner.” (See the movie “Good Will Hunting.”)

The Fields was just awarded to four people, including only the second woman ever to win. And she’s from Ukraine!

Click to read:

The details and accomplishments of the four are in the article, but here are their names and institutions:

Hugo Duminil-Copin; Institut des Hautes Études Scientifiques, France andUniversity of Geneva, Switzerland

June Huh Princeton University, US

James Maynard;  Oxford University, UK

Maryna Viazovska;  École Polytechnique Fédérale de Lausanne, Switzerland

Viazovska is the only woman to win the prize besides Maryam Mirzakhani of Stanford, who won in 2014 and is of Iranian descent.

I’ll highlight Maryna Viazovska to applaud not only the advance of women in math, but as a boon to the much-beleaguered Ukraine. Here’s what the NYT says about her in a summary by Kenneth Chang:

Maryna Viazovska, a Ukrainian who is now a professor at the Swiss Federal Institute of Technology in Lausanne, is known for proofs for higher-dimensional equivalents of the stacking of equal- sized spheres. She is also only the second woman ever to win the Fields Medal.

Of the 60 mathematicians who won Fields Medals before this year, 59 were men. In 2014, a Stanford mathematician, Maryam Mirzakhani, was the first and, until now, the only woman to receive one.

“I feel sad that I’m only the second woman,” Dr. Viazovska said. “But why is that? I don’t know. I hope it will change in the future.”

Dr. Viazovska’s work is a variation of a conjecture by Johannes Kepler more than 400 years ago. Kepler is best known for realizing that the planets move around the sun in elliptical orbits, but he also considered the stacking of cannonballs, asserting that the usual pyramid stacking was the densest way that they could arranged, filling up just over 75 percent of the available space.

Kepler could not prove that statement, however. Neither could anyone else until Thomas Hales, then at the University of Michigan, succeeded in 1998 with a 250-page proof and, controversially, the help of a computer program.

Proving something similar for the packing of equal-size spheres in dimensions higher than three has been impossible so far — with a couple of exceptions.

In 2016, Dr. Viazovska found the answer in eight dimensions, showing that a particularly symmetric packing structure known as E8 was the best possible, filling about one-quarter of the volume. Within a week, she and four other mathematicians showed that a different arrangement known as the Leech lattice was the best possible packing in 24 dimensions. In high dimensions, the filled volume is not very full, with the Leech lattice of 24-dimensional spheres occupying about 0.2 percent of the volume.

What’s so special about eight and 24 dimensions?

“I think that’s a mystery,” Dr. Viazovska said. “It’s just in these dimensions, certain things happen which don’t happen in other dimensions.”

She said that a method that generally gives an upper bound on the packing density turns out to be the exact solution in these cases.

High-dimensional sphere packings are related to the error-correcting techniques used to fix garbles in the transmission of information.

She said that the Russian invasion of Ukraine had taken its toll on her family. “It’s very difficult,” she said.

Her parents still live near Kyiv, Dr. Viazovska said, while her sisters, nephew and niece left and joined her in Switzerland.

Here’s the Fields Medal (caption from Wikipedia, the Latin translation is “Rise above oneself and grasp the world”), and a photo of Viazovska:

Photo of the obverse of a Fields Medal made by Stefan Zachow for the International Mathematical Union (IMU), showing a bas relief of Archimedes (as identified by the Greek text). The Latin phrase states: Transire suum pectus mundoque potiri

Maryna Viazovska, from the Guardian:

h/t: Tom

The intellectual vacuity of mathematical arguments against evolution

June 2, 2022 • 12:00 pm

UPDATE: Somehow I missed that Jason has a new book that expands on this problem (I didn’t see it on the Amazon site). Here’s the cover, and click on it to go to the site:


Jason Rosenhouse is a professor of math at James Madison University in Virginia and also a friend. Besides teaching and researching in his field, he’s also written a lot about applying math to popular culture, including books on Sudoku and the perplexing Monty Hall Problem. But to me his biggest contribution has been his series of books and writings about creationism. Jason has not only immersed himself in creationist culture, attending lots of meetings to suss out the psyche of anti-evolutionists, but also written about it in both books and articles (see his 2012 book Among the Creationists: Dispatches from the Anti-Evolutionist Front Line).

He’s just come out with an article in the Skeptical Inquirer (see below) in which he summarizes how Intelligent Design creationists use mathematical arguments to show that evolution is impossible, and then Rosenhouse debunks the tactics they use. Jason writes very well and very clearly, so this article is accessible to the layperson. It’ll give you a strarter background on the creationists’ arguments (yes, IDers are creationists), and why those arguments re misguided.

Click to read (it’s free).

Jason explicates and then demolishes two ID arguments against evolution. Quotes from Jason are indented, my own prose is flush left.

1.) The probability of evolution producing complex features, like bacterial flagella, is almost nil. 

The ID argument rests on the idea that if the probability of an amino acid in a protein, say tyrosine, being in a specific position is small, then the probability of getting a protein of 100 amino acids with tyrosine in the right position and the other 19 amino acids in the other right positions is effectively zero. (They simply multiply probabilities for each site together.) But, as Jason shows, that’s not the way that evolution works. Proteins are built up step by step, with each step adopted only if it incrementally improves fitness. The probability-multiplying argument is so transparently false that I’m surprised people believe it, but of course most people don’t have a decent understanding of probability.


However, this argument is premised on the notion that genes and proteins evolve through a process analogous to tossing a coin multiple times. This is untrue because there is nothing analogous to natural selection when you are tossing coins. Natural selection is a non-random process, and this fundamentally affects the probability of evolving a particular gene.

To see why, suppose we toss 100 coins in the hopes of obtaining 100 heads. One approach is to throw all 100 coins at once, repeatedly, until all 100 happen to land heads at the same time. Of course, this is exceedingly unlikely to occur. An alternative approach is to flip all 100 coins, leave the ones that landed heads as they are, and then toss again only those that landed tails. We continue in this manner until all 100 coins show heads, which, under this procedure, will happen before too long. The creationist argument assumes that evolution must proceed in a manner comparable to the first approach, when really it has far more in common with the second.

That’s a very good explanation.

IDers, however, have made the argument a bit more sophisticated:

Let us return to coin-tossing. Suppose we toss a coin 100 times, thereby producing a chaotic jumble of heads and tails. It was very unlikely that just that sequence would appear, but we do not suspect trickery. After all, something had to happen. But now suppose we obtained 100 Hs or a perfect alternation of Hs and Ts. Now we probably would suspect trickery of some kind. Such sequences are not only improbable but also match a recognizable pattern. ID proponents argue that it is the combination of improbability and matching a pattern that makes them suspect that something other than chance or purely natural processes are at work. They use the phrase “complex, specified information” to capture this idea. In this context, “complex” just means “improbable,” and “specified” means “matches a pattern.”

As applied to biology, the argument goes like this: Consider a complex, biological adaptation such as the flagellum used by some bacteria to propel themselves through liquid. The flagellum is a machine constructed from numerous individual proteins working in concert. Finding this exact functional arrangement of proteins is extremely unlikely to happen by chance. Moreover, they continue, the structure of the flagellum is strongly analogous to the sort of outboard motor we might use to propel a boat. Therefore, the flagellum exhibits both complexity and specificity, and it therefore must be the product of intelligent design.

That is, natural selection, say critics like William Dembski, can’t create “complex specified design”. But we have no idea what organismal features would imply intelligent design (“specificity”) rather than selection. Further, as for “complexity”, Jason says this:

The argument likewise founders on the question of complexity. According to ID proponents, establishing complexity requires carrying out a probability calculation, but we have no means for carrying out such a computation in this context. The evolutionary process is affected by so many variables that there is no hope of quantifying them for the purposes of evaluating such a probability.

In summary, any anti-evolutionist argument based on probability theory can simply be dismissed out of hand. There is no way to carry out a meaningful calculation, and adding “specificity” to the mix does nothing to improve the argument.

2.) Because mutations are degrading processes, much more likely to make DNA coding for a protein less adapted to the environment than more adaptive, there is no way that new genetic information can be created. Ergo, complexity, much less adaptation, can’t increase. rgo God—the Creator of Complexity. In some ways this resembles the old Second Law of Thermodynamics argument against evolution: entropy must increase, and evolution appears to violate entropy by making matter less random.  Thus we need God to get the entropy down.

The problem with that is, of course, the Second Law holds only in a closed system, but evolution occurs in an open system: the Earth in its surrounding universe. Evolution is fueled by radiation from the Sun, which involves an increase in entropy, and any decrease in entropy produced by evolution is more than compensated for by the increased entropy produced by generating evolution’s fuel: solar energy.  Ergo, in the whole system, the Second Law is obeyed.

There’s already one way known whereby new genetic “information” can increase: gene duplication.  Sometimes due to errors in replication, a gene is duplicated, and we have two copies instead of one (there are always two copies in a diploid genome, but I’m talking about what happens when a gene on one chromosome duplicates in addition. When this happens, there is an opportunity for that new copy of the gene to diverge in function from the old one, for the old one’s still around doing its thing. The new copy can do a new thing. Ergo, new information.  This in fact has happened a gazillion times in evolution: all of our globins, for instance (alpha, beta, fetal hemoglobin, and myoglobin) were produced by gene duplication and subsequent divergence. In Antarctic fish, an enzyme used to digest food has, after duplication, evolved into a blood antifreeze protein to allow them to inhabit waters below the freezing point.

Jason mentions gene duplication (I’m just giving examples), and then goes into the “No Free Lunch” ideas of Dembski and others, showing that these ideas irrelevant to the possibility of evolution.  I’ll let you read that part for yourself (read the whole thing!), and will just give two more quotes from Jason:

Even if we accept everything Dembski and his coauthors are saying about these theorems, this whole line of attack simply amounts to nothing. Most of us did not need difficult mathematical theorems to understand that Darwinian evolution can work only if nature has certain properties. The search problem confronted by evolution arises ultimately from the laws of physics, but it is well outside biology’s domain to wonder why those laws are as they are. Dembski and his cohorts argue that the fundamental constants of the universe encode information of a sort that can arise only from an intelligent source, but they have no more basis for this claim than they did for the comparable claim about genetic sequences.

He finishes like this:

Everyone agrees that complex adaptations require a special sort of explanation. Scientists argue that actual biological systems show copious evidence of having resulted through evolution by natural selection. Anti-evolutionists reject this claim, but the ensuing debate, such as it is, has nothing to do with mathematics. This makes you wonder why anti-evolutionists insist on padding their work with so much irrelevant and erroneous mathematical formalism. The answer is that their literature has far more to do with propaganda than it does with serious argument. Mathematics is unique in its ability to bamboozle a lay audience, making it well suited to their purposes. But for all its superficial sophistication, anti-evolutionary mathematics is not even successful at raising interesting questions about evolution.

Jason knows whereof he speaks, as he knows both math and evolution.

More bias in Scientific American, this time in a “news” article

January 15, 2022 • 1:00 pm

Scientific American has tendered a news piece in their “Mathematics” section, reporting on a schism in the math community. I’ve followed this schism for a while but haven’t written about it. As I understand it, what happened is that last October the Association for Mathematical Research (AMR) was formed, breaking away from the two older associations, the Mathematical Association of America (MAA) and the American Mathematical Society (AMS), primarily because the latter two societies were becoming too woke, trying to dilute the mathematical goals of their organization with social-justice considerations, considerations favoring the performative and “progressive” ideology we know too well.

While the article starts off okay, giving the facts above, it quickly devolves into somewhat of a hit piece on the new AMR for being racist and sexist. This is in line with the total lack of objectivity of Scientific American, which, as we all know, has diverted much of its mission to teach science so that it can further social justice, though in a misguided and ineffective way. In this piece, the bias of Sci. Am. is reflected in both the imbalance of quotations from pro- and anti-AMR people (much more from the latter) and in its own commentary and slant.

Now I tend to be opposed to the new direction Sci Am is taking, so I may be biased, but I don’t think I am: I think this article is what’s slanted, not me. But read it for yourself by clicking on the screenshot.  The dissing of the AMR starts with the subheading, where critics get their say without any mention of why the AMR was formed.

This bit is pretty accurate, as far as I know, though you can see a bit of pro-woke bias nosing in:

A new organization called the Association for Mathematical Research (AMR) has ignited fierce debates in the math research and education communities since it was launched last October. Its stated mission is “to support mathematical research and scholarship”—a goal similar to that proclaimed by two long-standing groups: the American Mathematical Society (AMS) and the Mathematical Association of America (MAA). In recent years the latter two have initiated projects to address racial, gender and other inequities within the field. The AMR claims to have no position on social justice issues, and critics see its silence on those topics as part of a backlash against inclusivity efforts. Some of the new group’s leaders have also spoken out in the past against certain endeavors to diversify mathematics. The controversy reflects a growing division between researchers who want to keep scientific and mathematical pursuits separate from social issues that they see as irrelevant to research and those who say even pure mathematics cannot be considered separately from the racism and sexism in its culture.

Then, throwing off the mantle of objectivity, the author goes full steam ahead. All quotes from the piece are indented:

Criticism of the AMR (selected bits)

With bias, harassment and exclusion widely acknowledged to exist within the mathematics community, many find it dubious that a professional organization could take no stance on inequity while purporting to serve the needs of mathematicians from all backgrounds. “It’s a hard time to be a mathematician,” says Piper H, a mathematician at the University of Toronto. In 2019 less than 1 percent of doctorates were awarded to Black mathematicians, and just 29 percent were awarded to women.

. . .Louigi Addario-Berry, a mathematician at McGill University in Montreal, wrote about the AMR on his blog. He told Scientific American he is speaking up because “I think this is an organization whose existence, development and flourishing will hurt a lot of members of the mathematical community who I respect. It is being founded by people who have publicly stated views I find harmful—both hurtful to me as an individual and detrimental to the creation of an inclusive and welcoming mathematical community.”

Hass responded in a statement to Scientific American: “The focus of the AMR is on supporting mathematical research and this goal benefits all members of the mathematics community.” But Addario-Berry questions how the AMR can be neutral on social justice issues when some of its leaders have previously taken strong public stances on some of these topics.

This is very strange. It’s like saying that the University of Chicago cannot be organizationally neutral on social-justice issues when many of its faculty have taken strong stands one way or the other. Can the author not conceive of an organization being officially politically neutral even though its members may have strong views? This isn’t rocket science. It’s just the University of Chicago.

There’s some discussion both ways about UC Davis math professor Abigail Thompson’s criticism of requiring diversity statements for faculty jobs (see my post here), and a note that Thompson is secretary of the new AMR. But that’s seems like an attempt to tarnish the AMR by picking out members who themselves opposed wokeness. It says nothing about the organization’s own stance, which is indeed neutrality. Thompson is also listed as one of the “current vice presidents of the American Mathematical Society” in her Wikipedia bio. But none of this really has to do with the issue at hand, except to try to cast aspersions on Thompson and, by extension, the AMR. But wait! There’s more!

Another AMR founding member and a member of its board of directors, Robion “Rob” Kirby, is a mathematician at the University of California, Berkeley. In a post entitled “Sexism in Mathematics???” on his Web site, he wrote, “People who say that women can’t do math as well as men are often called sexist, but it is worth remembering that some evidence exists and the topic is a legimate [sic] one, although Miss Manners might not endorse it.”

In fact, I don’t think that Kirby is right; as far as I knew, men and women in secondary school perform equally well in math, but the women excel in reading. Women like reading more than math, and thus they tend to go on more often to the humanities. Whatever is responsible for inequity between men and women, it’s not skill.

Or course conservatives are going to leave an organization disproportionately if it becomes too woke, for wokeness is the purview of the Left, not the Right. You don’t have to be a conservative to try to keep your discipline pure, but if you’re a liberal like me who doesn’t like performative wokeness, you’re going to have to live being associated with some politically inconvenient bedfellows. At any rate, the statement above doesn’t represent someone supporting the new AMR, it’s Sci Am’s attempt to denigrate it.

Then there’s this:

The AMS and the MAA have publicly acknowledged the need to work toward a more inclusive mathematical community. Last year an AMS task force released a 68-page report that, in the organization’s words, details “the historical role of the AMS in racial discrimination; and recommends actions for the AMS to take to rectify systemic inequities in the mathematics community.” In 2020 an MAA committee stated that the mathematics community must “actively work to become anti-racist” and “hold ourselves and our academic institutions accountable for the continued oppression of Black students, staff, and faculty.” It also addressed Black mathematicians specifically, saying, “We are actively failing you at every turn as a society and as a mathematics community. We kneel together with you. #BlackLivesMatter.”

In contrast, the AMR has not released any official statements about injustice.

Okay, that’s pretty snarky, but is followed by something even snarkier:

“I am supposed to believe, in the year 2021, that this omission is not itself an act of racism?” asks Piper H, who spoke to Scientific American late last year. “How am I, as a 40-year-old Black American mathematician, parent, and person who has paid a bit of attention to American history and American present, supposed to believe that AMR’s refusal to address the actual obstacles that real mathematicians face to doing mathematical research and scholarship is anything other than an insult and a mockery?”

This is pure Kendian mishigass: if your organization doesn’t make an explicitly anti-racist statement, then your organization is racist. Note that they add that Hass denies that the AMR’s silence on diversity is a message (see below).

. . . It’s not just a coincidence that the AMR was founded on the heels of a greater push for diversity within the AMS,” wrote Lee Melvin Peralta, a mathematics education graduate student at Michigan State University, in the November 16, 2021, newsletter of the Global Math Department, an organization of math educators. The AMR, Peralta added, “seems more like a separatist organization for those people who are striving for some kind of ‘purity’ within mathematics away from ‘impure’ considerations of race, gender, class, ability, sexual orientation, and socioeconomic status (among others).”

And, at the end of the article, there’s this parting shot:

Some of the AMR’s founding members have left the organization amid the controversy. “To create an organization to do something positive requires the trust and goodwill of the community that it wants to affect. And this is something that the AMR does not have at this point,” wrote Daniel Krashen, a mathematician at the University of Pennsylvania, in a November 14, 2021, Twitter thread. “I have no desire to negatively impact the mathematical community by my actions and words. I see that some people feel less safe and less heard by my actions, and for this I apologize. I have decided to withdraw my membership.”

Less safe? How has Krashen made anybody less safe or less heard? For crying out loud, this whole article is a megaphone handed to the critics of the AMR! Nobody has been silenced and the only harm has come to people’s feelings. (That said, I of course oppose those social conditions that have denied women or minorities entry into the math “pipeline.”)

Defense of the AMR:

Joel Hass, a mathematician at the University of California, Davis, and current president of the AMR, describes the group as “definitely focused on being inclusive.” He adds that the AMR “welcomes all to join us in supporting mathematical research and scholarship. In early 2022 we plan to open membership to anyone in the world who wishes to join us. There will be no fees or dues. By removing financial barriers to entry, we will make it easier to have participation from anyone across the world. Mathematical research is a truly global endeavor that transcends nation, creed and culture.”

. . . Hass denies that the AMR’s founding had anything to do with the antiracism push at the AMS or the MAA. The changes in the research environment caused by the COVID pandemic revealed new opportunities for the development and communication of mathematical research, allowing for incorporation of new technologies and international activities,” he says. “We felt there was room for a new organization that would explore these.” Hass adds that “the AMS and MAA are wonderful organizations that we hope to work with, along with other organizations such as SIAM [Society for Industrial and Applied Mathematics], ACM [Association for Computing Machinery] and many non-U.S.-based groups.”

I think Hass is being disingenuous here, for what I’ve heard is that the AMR is a reaction to the wokeness of the other two organizations. I don’t see that as a sign of racism; I see it as a sign of trying to keep an objective discipline from being diverted into political pursuits.*********

So there we have it:  four mathematicians criticizing the AMR for racism/sexism or “harm”, and one defending its mission. That’s not to mention the way that Scientific American has structured the article, providing a critical sub-header for the title and ending with a critical slam.

I’m not by any means a fan of the views of all AMR members: in fact I’ve just criticized two statements of their members. But with this article, Sci. Am. is casting its lot in with the woke, as it always does. There is no rationale, they’re saying, for a mathematics organization that is not explicitly devoted to achieving Social Justice.

This is my view, which of course might be conditioned by my extreme dislike for the direction that Sci. Am. is taking. So read for yourself and let me know if the piece seems objective to you.

More than half of Americans oppose the use of Arabic numerals!

December 29, 2021 • 1:30 pm

Just a bit of fun, but the headline below is true. The survey on which it’s based is reported in this article in from the Independent, which you can see by clicking on the screenshot:(you can register for free with email and a password if it’s blocked; there’s no paywall)

So, here are some results given in the article:

More than half of Americans believe “Arabic numerals” – the standard symbols used across much of the world to denote numbers – should not be taught in school, according to a survey.

Fifty-six per cent of people say the numerals should not be part of the curriculum for US pupils, according to research designed to explore the bias and prejudice of poll respondents.

The digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are referred to as Arabic numerals. The system was first developed by Indian mathematicians before spreading through the Arab world to Europe and becoming popularised around the globe.

A survey by Civic Science, an American market research company, asked 3,624 respondents: “Should schools in America teach Arabic numerals as part of their curriculum?” The poll did not explain what the term “Arabic numerals” meant.

Some 2,020 people answered “no”. Twenty-nine per cent of respondents said the numerals should be taught in US schools, and 15 per cent had no opinion.

John Dick, who happens to be the head of Civic Science, issued this tweet with the data in graphic form, which I’ve put below as well:

Now Dick thinks this is an example of bigotry—”Islamophobia,” I suppose. I’m not so sure. Although I am sure that many of us know that Arabic numerals are the numerals we use every day, some people don’t, and, this being America, it’s possible that nobody has told children that they are learning “Arabic numerals.” The 56% figure could thus represent ignorance rather than bigotry, although both could play a role.  But Dick seems wedded to the latter explanation. Regardless, if it is ignorance, it’s pretty appalling. After all, everyone knows what Roman numerals are!

But wait! There’s more. There was so much doubt about this survey’s results that Snopes had to investigate it.

In its headline Snopes says “It’s difficult to answer survey questions if you don’t fully understand the meaning.” I’m pretty sure, from following them, that Snopes is woke,but their assumption that there’s no anti-Arabic bigotry involved is just a guess.

You can read their analysis, in which they reluctantly admit that the claim is true, by clicking on the screenshot below.

But wait! There’s still more! You get this special grapefruit-cutting knife if you read on—for free!


Those were the results of a real survey question posed by the polling company Civic Science. John Dick, the Twitter user who originally posted a screenshot of the survey question, is the CEO of Civic Science.

The full survey doesn’t appear to be available at this time (we reached out to Civic Science for more information), but Dick has posted a few other questions from the poll, as well as some information regarding the purpose of the survey.

Dick, who said that the “goal in this experiment was to tease out prejudice among those who didn’t understand the question,” shared another survey question about what should or shouldn’t be taught in American schools. This time, the survey found that 53% of respondents (and 73% of Democrats) thought that schools in America shouldn’t teach the “creation theory of Catholic priest Georges Lemaitre” as part of their science curriculum. Here are the results:

33% of Republicans, a whopping 73% of Democrats, and 52% of independents thought that Lemaître’s theory should NOT be taught.

Now this question is more unfair, because, really, how many Americans know what the “creation theory of Georges Lemaître” was? If you read about science and religion, or have followed this site for a while, you’ll know that, although he was a Catholic priest, Lemaître held pretty much the modern theory of the Big Bang and the expanding Universe. As Wikipedia notes:

Lemaître was the first to theorize that the recession of nearby galaxies can be explained by an expanding universe, which was observationally confirmed soon afterwards by Edwin Hubble. He first derived “Hubble’s law”, now called the Hubble–Lemaître law by the IAU, and published the first estimation of the Hubble constant in 1927, two years before Hubble’s article. Lemaître also proposed the “Big Bang theory” of the origin of the universe, calling it the “hypothesis of the primeval atom”, and later calling it “the beginning of the world”.

Yes, and Lemaitre did other science, including analyzing cosmology using Einstein’s theories of relativity. He was a smart dude, and should have gone into physics instead of the priesthood. There’s a photo of him with Einstein below.

Why did so many people answer that Lemaître’s theory, which is, as I said, is pretty much the current theory of the Universe’s origin, NOT be taught? Surely it’s because the question identified Lemaître as a “Catholic priest”. That means that people probably thought his “theory” was the one expounded in Genesis chapters 1 and 2—God’s creation. So they didn’t want a religious theory taught in school.

Two points: most Republicans didn’t mind as much as Democrats of Independents, and that may be because more Republicans are creationists than are Democrats. But why did so many Democrats not want Lemaître’s theory taught? Are they that much less creationist than are Republicans? Perhaps that’s one answer. Another is that they are more anti-Catholic, but that seems less likely. But underlying these data—as perhaps underlying much of the data about Arabic numerals—is simple ignorance. I, for one, wouldn’t expect the average Joe or Jill (oops!) to know what Lemaître said.

One final remark: Accommodationists sometimes use the fact that Lemaître got it right as evidence that there’s no conflict between science and religion. I’m not sure if Lemaître thought God created the Universe, but if he did, he might have thought that the Big Bang was God’s way of doing it. (He was surely NOT a Biblical literalist). So yes, religious people can and have made big contributions to science. But that doesn’t mean that religion and science are compatible—any more than Francis Collins’s biological work shows that science and Evangelical Christianity are compatible. I’ve explained what I mean by “compatible” before, and it’s NOT that religious people can’t do science.

In the case of Lemaître, Francis Collins, or other religious scientists, they are victims of a form of unconscious cognitive dissonance: accepting some truth statements based on the toolkit of science, and other truth statements based on the inferior “way of knowing” of faith. And that is the true incompatibility: the different ways that we determine scientific truth as opposed to religious “truth.”

But I digress, and so shall stop.

George Lemaître (1894-1966), photo taken in 1930:

From Wikipedia:

(From Wikipedia): Millikan, Lemaître and Einstein after Lemaître’s lecture at the California Institute of Technology in January 1933.

h/t: Phil D.

“Everybody has won and all must have prizes”: The drive to end merit-based schooling

November 9, 2021 • 12:15 pm

There are two articles you can read that show how quickly merit-based educational assessment is vanishing in the U.S. The first, from the New York Times, discusses California’s downgrading of math instruction, turning it as well into an instrument for teaching social justice. The second, from the Los Angeles Times, describes the move to eliminate grading, or at least the lower grades of D and F so that everyone must have the prize of a “C” (required to get into the Cal State system of colleges).

Click on the screenshot to read the pieces. I’ll give a few quotes from each (indented):


If everything had gone according to plan, California would have approved new guidelines this month for math education in public schools.

But ever since a draft was opened for public comment in February, the recommendations have set off a fierce debate over not only how to teach math, but also how to solve a problem more intractable than Fermat’s last theorem: closing the racial and socioeconomic disparities in achievement that persist at every level of math education.

The California guidelines, which are not binding, could overhaul the way many school districts approach math instruction. The draft rejected the idea of naturally gifted children, recommended against shifting certain students into accelerated courses in middle school and tried to promote high-level math courses that could serve as alternatives to calculus, like data science or statistics.

The draft also suggested that math should not be colorblind and that teachers could use lessons to explore social justice — for example, by looking out for gender stereotypes in word problems, or applying math concepts to topics like immigration or inequality.

No matter how good the intentions, math—indeed, even secondary school itself—is no place to propagandize students with debatable contentions about social justice. The motivation for this, of course, is to achieve “equity” of achievement among races, since blacks and Hispanics are lagging behind in math. (Indeed, as the article notes, “According to data from the Education Department, calculus is not even offered in most schools that serve a large number of Black and Latino students.”)

Everything is up for grabs in California given the number of irate people on both sides. Some claim that school data already show that the “new math” leads to more students and more diverse students taking high-level math courses, while other say the data are cherry-picked. I have no idea.

Complicating matters is that even if the draft becomes policy, school districts can opt out of the state’s recommendations. And they undoubtedly will in areas of affluence or with a high percentage of Asian students, who excel in math. This is not a path to equal opportunity, but a form of creating equity in which everybody is proportionately represented on some low level of grades. I wish all the schools would opt out! There has to be a way to give every kid equal opportunity to learn at their own levels without holding back those who are terrific at math. I don’t know the answer, but the U.S. is already way behind other First World countries in math achievement. This will put us even farther down.

From the L. A. Times:


This issue is a real conundrum, more so than the above, because it’s not as easy to evaluate.  Here are a few suggestions of what teachers are doing to change the grading system—the reason, of course, is racial inequity in grades that must be fixed.

 A few years ago, high school teacher Joshua Moreno got fed up with his grading system, which had become a points game.

Some students accumulated so many points early on that by the end of the term they knew they didn’t need to do more work and could still get an A. Others — often those who had to work or care for family members after school — would fail to turn in their homework and fall so far behind that they would just stop trying.

“It was literally inequitable,” he said. “As a teacher you get frustrated because what you signed up for was for students to learn. And it just ended up being a conversation about points all the time.”

These days, the Alhambra High School English teacher has done away with points entirely. He no longer gives students homework and gives them multiple opportunities to improve essays and classwork. The goal is to base grades on what students are learning, and remove behavior, deadlines and how much work they do from the equation.

But I had always assumed that grades were based on what students were learning: that’s what tests do. You ask students questions based on what you’ve taught them and what they’ve read, and then see if they’ve absorbed the material.  I have no objection at all to basing grades on “what students are learning” so long as you don’t grade them on the basis tht you have different expectations of what different students can learn. (In fact, as you see below, that may be the case.)

As for behavior, well, you have to conduct yourself in a non-disruptive manner in class; and as far as deadlines and quality of papers and work, those are life lessons that carry over into the real world. You don’t get breaks from your employer if you finish a project late.  I always gave students breaks if they had good excuses, or seemed to be trying really hard, but can you give a really good student a lower grade because she’s learning the material with much less effort than others? Truly, I don’t understand how this is supposed to work.

There is also much talk about “equity” in grading, and I don’t know what that means except either “everyone gets the same grade”, which is untenable, or “the proportion of grades among people of different races must be equal”, which, given the disparity in existing grades between whites and Asians on one hand and blacks and Hispanics on the others, means race-based grading. That, too, seems untenable.  But of course this doesn’t negate my own approval of some forms of affirmative action as reparations to groups treated unfairly in the past. Nobody wants a school that is all Asian and white, and nobody wants a school that is all black or all Hispanic.

Again, I don’t know the solution except to improve teaching while allowing everyone to learn to the best of their ability. And that means effort must be judged as well as achievement. Here’s a statement from L.A. Unified’s chief academic officer:

“Just because I did not answer a test question correctly today doesn’t mean I don’t have the capacity to learn it tomorrow and retake a test,” Yoshimoto-Towery said. “Equitable grading practices align with the understanding that as people we learn at different rates and in different ways and we need multiple opportunities to do so.”

Somehow I get the feeling that this refers not to different individuals‘ capacity to learn, but on assumptions about the capacity of members of different races to learn—assumptions that are both racist and patronizing. This is supported by the fact that San Diego’s school board said this:

“Our goal should not simply be to re-create the system in place before March 13, 2020. Rather, we should seek to reopen as a better system, one focused on rooting out systemic racism in our society,” the board declared last summer.

Similar to Los Angeles, the San Diego changes include giving students opportunities to revise work and re-do tests. Teachers are to remove factors such as behavior, punctuality, effort and work habits from academic grades and shift them to a student’s “citizenship” grade, which is often factored into sports and extra-curricular eligibility, said Nicole DeWitt, executive director in the district’s office of leadership and learning.

It seems to me that you can’t solve the problem of unequal achievement by adjusting grades based on race. In the long term, that accomplishes very little. You solve the problem by giving everybody equal opportunities in life from the very beginning of life. Since minorities don’t have that, we should be investing a lot of time and money in providing those opportunities. In the meantime, some affirmative action is necessary to allow more opportunity than before, and because we owe it to people who have been discriminated against and haven’t had equal opportunity.

The new math in Toronto

November 5, 2021 • 1:30 pm

A graduate of the University of Toronto called my attention to this mathematics course as a harbinger of the decline of that great university. I have no idea what “liberated” mathematics is, and can’t find out anything about it, or the course, on the Internet. Several other people have tweeted this course, and I note that the only requirement for it is “high school level algebra.” I gather, therefore, that this course, taught by the mathematics department, is more about ideology than math.

I’ve put the transcript of the course description below, or you can click on the screenshot.

Currently, mathematics is at a crossroads between tradition and progress. Progress has been led in large part by women mathematicians, in particular Black women, Indigenous women, and women from visible minorities. Intertwined in their studies of mathematics is a daring critique of traditional mathematics, re-imagining of mathematics culture, and more. This course will compare and contrast new forms of accessible mathematics with standard sources that draw dominantly on the experiences and narratives of men. Restricted to first-year students. Not eligible for CR/NCR option.

The solution to this morning’s puzzle

September 20, 2021 • 2:39 pm

I haven’t yet looked at the comments about the puzzle I reported this morning that had been proposed by Russian Prime Minister Mikhail Mishustin when visiting a science-oriented “sixth form” class.

Here’s the puzzle again:

Construct a perpendicular from the (red) point on the circle to the diameter, without using any measuring devices.

In other words, given a circle with a diameter marked on it, and a point on the circle, can you find a way to draw a line from the point that hits the diameter at a right angle. (As marked in green above.)

The beauty of this question is the seemingly outrageous restriction not to allow measuring devices, which means that you cannot use a compass or a marked ruler. All you are allowed is an unmarked ruler to draw straight lines.


The Guardian has now published the four-step solution (there may be others); go to the preceding link to see it. Below you can see the PM drawing the solution (he must know his math).

A geometry puzzle from Russia’s prime minister

September 20, 2021 • 8:00 am

In lieu of Readers’ Wildlife today, we have a puzzle, one posted (as the Guardian reports) by Russian Prime Minister, Mikhail Mishustin when he was visiting a science-oriented “sixth form” (what age of kids are these?) school. Matthew sent me a link.

Here’s the problem, and there’s a clue in both the photo below and in the Guardian article:

Construct a perpendicular from the (red) point on the circle to the diameter, without using any measuring devices.

In other words, given a circle with a diameter marked on it, and a point on the circle, can you find a way to draw a line from the point that hits the diameter at a right angle. (As marked in green above.)

The beauty of this question is the seemingly outrageous restriction not to allow measuring devices, which means that you cannot use a compass or a marked ruler. All you are allowed is an unmarked ruler to draw straight lines.

Matthew couldn’t solve it and, as I haven’t had my coffee, I’m not even going to try.

Here’s, a picture of Mishustin posing the problem (and giving a bit of a solution):

Russia’s Prime Minister Mikhail Mishustin draws on a chalkboard while visiting the Kapitsa Physics and Technology Lyceum in the town of Dolgoprudny earlier this month. Photograph: Dmitry Astakhov/TASS