Category: mathematics

Indigenous mathematics: smoke and mirrors

April 18, 2024 • 11:30 am

I used to think that the “decolonization” of STEM was strongest in New Zealand and South Africa, which of course is a movement to dethrone so-called “Western” science in favor of indigenous science. But now I’m beginning to wonder if the “indigenization/decolonization of science” isn’t making its way deep into Australia as well.  I have followed developments in New Zealand far more closely than these other places, because I hear often from Kiwi scientists who beef about the dethroning of modern science (which hasn’t been “Western” in a while) in favor of Mātauranga Māori (MM), the “way of knowing” of the indigenous Māori people. Also, I have visited New Zealand, love the place, and would be devastated if science were watered down with superstition, myth, legend, and morality.

And that’s the first issue with “decolonizing” science. Usually those movements intend to either defenestrate modern science or at least teach “indigenous science” alongside it as an equally valid “way of knowing”. Yet indigenous science, like MM, is a grab-bag of empirical knowledge based on trial and error (the premier example is the navigation of Polynesians, the ancestors of Māori; another is how to catch eels), but is also imbued with superstitions, myths, legends, word-of-mouth tales, and “rules for living”, including morality. And rarely is indigenous science vetted with the same rigor as is modern science, because modern science has many features missing in indigenous “ways of knowing” (double-blind testing, deliberate replication, hypothesis testing, and so on).  One result is that “indigenous science” can be wrong more often. One example is the insistence of some New Zealand researchers that Polynesians discovered Antarctica in the early seventh century. This is based on oral legend combined with mistranslation; in fact, the Russians were the first to glimpse the continent—in 1820.

Now trial and error methods can indeed produce empirical knowledge in the sense of “justified true belief”, but that is practical knowledge, designed to help people where to find things to eat, how to navigate, how to herd bison, when to plant food, and the like. Its ambit is far narrower than that of modern science, and examples of “indigenous science” that have made valuable contributions to modern science are thin on the ground.

Which brings us to the second issue with indigenous science. Although it’s touted loudly and passionately, examples of indigenous knowledge making substantial contributions to modern science are either scant or missing. Most of the written defenses of enthroning indigenous science I’ve seen are based on a need to pay attention to marginalized people as oppressed victims, whose knowledge must be elevated precisely because they were victims.  But that’s no way to judge science.

And that is precisely the content of this puff piec, coming from Australian National University (ANU) in Canberra, touting the dethroning of “mainstream European-based mathematics” in favor of mathematics produced and used by “Indigenous and First Nations peoples around the world.” The article highlights Professor Rowena Ball of ANU’s College of Science, who lists these as her research interests:

Mathematics Without Borders, Truth-Telling in Mathematics History, Decolonisation of STEM, Indigenous and Non-Western Mathematics, Emergence of life, Nonlinear and complex dynamical systems, Thermochemical instabilities and oscillators, Thermodynamic analysis, Railways and trains, Country pub lunches

What is mathematics? What is included in mathematics? Who gets to say? How and why did Western mathematics exclusively colonise minds and curriculums over the whole world? Should that situation continue unabated?

It will not escape your notice if you read the piece, heavy with quotes from Dr. Ball, that she neglects the contributions of anything other than “mainstream European-based mathematics” to modern mathematics, leaving out the contributions of the Egyptians, Greeks, Arabs, Romans, and Babylonians to modern mathematics. Those people were apparently not “indigenous” and at any rate were not “colonized”. But Ball goes on and on, proffering only one tepid example of how a group of Australian aboriginals in Mithaka Country (an area of east-central Australia) had a form of mathematics that was useful. It turns out that it wasn’t mathematics at all, but practical knowledge that we wouldn’t recognize today as “math” at all.

Click the screenshot to read this short piece (h/t Peter Forsythe):

First I’ll give some of her quotes from the ANU piece (indented) and then her holotype specimen of indigenous math.

What constitutes mathematical knowledge? What is included in mathematics? Who gets to decide? These are some of the questions being asked in a growing decolonisation movement.

“Mathematics is a universal human phenomenon, and students of under-represented and minority groups and colonised peoples are starting to be more critical about accepting unquestioningly the cultural hegemony of mainstream European-based mathematics,” says Professor Rowena Ball from the ANU Mathematical Sciences Institute.

Professor Ball leads a research and teaching initiative called Mathematics Without Borders, aimed at broadening and diversifying the cultural base and content of mathematics.

“Mathematics has been gatekept by the West and defined to exclude entire cultures. Almost all mathematics that students have ever come across is European-based,” she explains. “We would like to enrich the discipline through the inclusion of cross-cultural mathematics.”

“Indigenous and First Nations peoples around the world are standing up and saying: ‘Our knowledge is just as good as anybody else’s − why can’t we teach it to our children in our schools, and in our own way?’

“And this is happening in New Zealand, North and South America, and Africa, and also in a great movement in India to revive traditional Indian mathematics.”

But wait!  There’s more:

. . .“There is a lot of gatekeeping going on,” Professor Ball says of having to justify Indigenous maths. “One effect of colonisation of the curriculum is defensive protection of what is thought to be an exclusively European and British provenance of mathematics.”

“Like most mathematicians I was educated in European and British mathematics,” says Professor Ball, “and it’s fine stuff – I still love my original research field in dynamical systems.” But that mathematics did not develop in isolation, she says, and now there’s even more to learn about how non-Western societies have been seeing the world mathematically that many of us haven’t yet tuned into.

“What the general public think of as mathematics tends to be whatever they learned (or, more likely, did not learn) at school. But in many Indigenous societies, mathematics is lived from when you are born to when you rejoin your ancestors,” Professor Ball says.

Rejoin your ancestors? Does she mean as underground worm food? I don’t think so. But I digress.  Ball argues that indigenous math is largely non-numerical, though in her one unpublished paper that is mentioned in the article (see below), numbers and counting figure largely.

At any rate, here is the single example Ball gives of valuable indigenous mathematics. I am not making this up: it involves the direction of smoke signals.

“One interesting example that we are currently investigating is the use of chiral symmetry to engineer a long-distance smoke signalling technology in real time,” Professor Ball says. “If you light an incense stick you will see the twin counter-rotating vortices that emanate − these are a chiral pair, meaning they are non-superimposable mirror images of each other.”

A memoir by Alice Duncan Kemp, who grew up on a cattle station on Mithaka country in the early 1900s, vividly describes the signalling procedure, in which husband-and-wife expert team Bogie and Mary-Anne selected and pulsed the smoke waves with a left to right curl, to signal “white men”, instead of the more usual right to left spiral.

Mithaka country is southwest Queensland − Kurrawoolben and Kirrenderri (Diamantina) and Nooroondinna (Georgina) river channel country − and for thousands of years this region was a rich, well-populated cultural and trade crossroads of the Australian continent.

To create and understand these signals, you have to be a skilled practical mathematician, Professor Ball says.

“Theory and mathematics in Mithaka society were systematised and taught intergenerationally. You don’t just somehow pop up and suddenly start a chiral signalling technology. It has been taught and developed and practised by many people through the generations.”

At that time in the early twentieth century, British meteorologists were just beginning to understand the essential vortical nature of atmospheric flows.

“Imagine if the existing Indigenous Mithaka knowledge of vorticity had been recognised, nurtured and protected? In what ways may it have fed into the high performance, numerical weather forecasting capabilities that we all rely on now?” she asks.

I don’t find this at all convincing. First, Bogie and Mary-Anne sound like white oppressors to me. But even if they weren’t, is the “reverse curl” something the locals actually used to signal “white people around”? It couldn’t have been going on for thousands of years because the first European people arrived in Australia in the early 17th century. So was there an elaborate system of smoke signals before that? Perhaps, but how are they based on mathematics? Patterns of smoke, like drumbeats, is a kind of language, and how to make the patterns and get them understood correctly is based on trial and error. Where does the math come in?

Further, the claim that the Mithaka knowledge of vorticity—I’m not sure what that knowledge is beyond empirical ways to make smoke signals—would have revolutionized “high performance numerical weather forecasting” long before now is simply risible.

Well, that’s enough. But I’d be remiss not to at least mention a paper by Xu and Ball that defends the thesis above. It’s called “Is the study of Indigenous mathematics ill-directed or beneficial?“, and appears at Arχiv.org, which means it hasn’t been published or peer-reviewed. There are a few examples of indigenous mathematics, which I put below. In some cases you’ll have to look up the references given to check on which people they’re referring to:

Much of ordinary day-to-day arithmetic and geometry performed by ‘illiterate’ women, artisans, carpenters and many other workers are unwritten and even unspoken (Wood, 2000). The apprentice learns by watching carefully then doing the mathematics themselves. The use of tools–an unwritten approach–to support arithmetic has a long history; there are different media for recording and computing with numbers, including stones, twigs, knots and notches (Hansson, 2018). People of many Indigenous Pacific and Australian nations can use parts of the body to count quickly and accurately (Goetzfridt, 2007; Owens & Lean, 2018; Wood, 2000), communicating methods, operations and results through speaking, listening and gesture. Weaving skills were taught unwritten to next generations to construct the numerical relationships that give rise to the desired complex geometrical designs with symmetries (Hansson, 2018). Knotted quipus were used by ‘illiterate’ Inca people of South American
Andes regions to allot land and levy taxes (Ascher & Ascher, 2013). The quipu (Figure 1), with its columns of base-10 numerical data encoded as knots, can be thought of as a spreadsheet, and it seems likely that the Inca knew and applied some array and matrix operations.

Dan, an Indigenous language of central Liberia, is non-written but Dan speakers can carry out arithmetic operations orally, including addition, subtraction and division, play games that require fast counting, tracking and calculating skills, and practice geometric principles in constructing buildings (Sternstein, 2008). Fractal geometry, developed to a high art in Western mathematics from the late 1960s and executed in silico, has non-Western antecedents that were implemented in the built environment in Africa (Eglash, 1998). Chaology and fractal geometry have also been a part of traditional Chinese architectural and garden design for thousands of years (Li & Liao, 1998).

Clearly some indigenous people could count and calculate, though the calculating seems to fall largely to the Chinese, not usually considered indigenous. At any rate, what’s above doesn’t jibe with the claim and quote in the article:

Numbers and arithmetic and accounting often are of secondary importance in Indigenous mathematics.

“In fact, as most mathematicians know, mathematics is primarily the science of patterns and periodicities and symmetries − and recognising and classifying those patterns.”

A lot of the above sounds like counting and accounting to me.  Regardless, it’s clear that some indigenous people could count and figure out patterns that involved counting.  I’m not so sure about the Inca “matrix” operations,  but one can hardly carry out some kind of commerce or taxation without being able to count. At any rate, yes, indigenous people had a form of “counting and pattern mathematics,” but to put them on a par even with what the ancient Egyptians and Greeks accomplished mathematically is to give indigenous people too much credit.

Hamas plays fast and loose with the casualty numbers from Gaza

March 10, 2024 • 11:35 am

This article from Tablet describes “How the Gaza Ministry of Health Fakes Casualty Numbers“, and while I have a few quibbles with it (or rather, alternative but not-so-plausible interpretations), the author’s take seems pretty much on the mark. Abraham Wyner simply gives the daily and cumulative death-toll accounts of Palestinians taken from the Hamas-run Gazan Health Ministry between October 26 and November 10 of last year, and subjects them to graphical and statistical analyses.

The conclusion is that somebody is making these figures up.  They aren’t necessarily inaccurate, but the article makes a strong case that there’s some serious fiddling going on. And the fiddling seems to be, of course, in the direction that Hamas wants.

I’ve put the figures Wyner uses below the fold of this post so you can see them (or analyze them) for yourself. As the author notes, “The data used in the article can be found here, with thanks to Salo Aizenberg who helped check and correct these numbers.”

Click on the link to read.

The data are the daily totals of “women”, “children”, and “men” (men are “implied”, which probably means that Wyner got “men” by subtracting children and women from the “daily totals”). Also given are the cumulative totals in the third column and the daily totals in the last column.

When you look at the data or the analysis, remember three things:

  1. “Children” are defined by Hamas as “people under 18 years old”, which of course could include male terrorists
  2. “Men” include terrorists as well as any civilians killed, and there is no separation, so estimates of terrorists death tolls vary between Hamas and the IDF, with the latter estimating that up to half of deaths of men could be terrorists
  3. A personal note: I find it ironic that Hamas can count the deaths to a person but also say they don’t have any idea of how many hostages they have, or how many are alive.

On to the statistics. I’ll put Wyner’s main findings in bold (my wording), and his own text is indented, while mine is flush left.

The cumulative totals are too regular. If you look at the cumulate death totals over the period, they seem to go up at a very even and smooth rate, as if the daily totals were confected to create that rate. Here’s the graph:

(From author): The graph reveals an extremely regular increase in casualties over the period. Data aggregated by the author and provided by the United Nations Office for the Coordination of Humanitarian Affairs (OCHA), based on Gaza MoH figures.

Cumulative totals will always look smoother than the daily totals, so this may be a bit deceptive to the eye. However, Wyner also deals with the daily totals, which are simply too similar to each other to imply any kind of irregular daily death toll, which one would expect in a war like this.  As he says of the above:

This regularity is almost surely not real. One would expect quite a bit of variation day to day. In fact, the daily reported casualty count over this period averages 270 plus or minus about 15%. This is strikingly little variation. There should be days with twice the average or more and others with half or less. Perhaps what is happening is the Gaza ministry is releasing fake daily numbers that vary too little because they do not have a clear understanding of the behavior of naturally occurring numbers. Unfortunately, verified control data is not available to formally test this conclusion, but the details of the daily counts render the numbers suspicious.

The figures for “children” and “women” should be correlated on a daily basis, but aren’t.  Here’s what Wyner says before he shows the lack of correlation:

Similarly, we should see variation in the number of child casualties that tracks the variation in the number of women. This is because the daily variation in death counts is caused by the variation in the number of strikes on residential buildings and tunnels which should result in considerable variability in the totals but less variation in the percentage of deaths across groups. This is a basic statistical fact about chance variability. Consequently, on the days with many women casualties there should be large numbers of children casualties, and on the days when just a few women are reported to have been killed, just a few children should be reported. This relationship can be measured and quantified by the R-square (R² ) statistic that measures how correlated the daily casualty count for women is with the daily casualty count for children. If the numbers were real, we would expect R² to be substantively larger than 0, tending closer to 1.0. But R² is .017 which is statistically and substantively not different from 0.

This lack of correlation is the second circumstantial piece of evidence suggesting the numbers are not real. But there is more. . .

This seems reasonable to me, although if a large number of “children” are really terrorists fighting the IDF and are not with women, this could weaken the correlation. But given Hamas’s repeated showing of small children in its propaganda, one would indeed expect a pretty strong correlation. In fact, the probability of getting this value of R² (actually, the proportion of the variation in daily women killed explained by the number of men killed) is a high 0.647, which means that if there was no association, you would get an R² this large almost 65% of the time. To be significant the probability should be less than 0.05: less than a 5% probability that the observation association would have happened by chance alone.

(From author): The daily number of children reported to have been killed is totally unrelated to the number of women reported. The R² is .017 and the relationship is statistically and substantively insignificant.

There is a strong negative correlation between the number of men killed and the number of women killed.  The daily data plotted over time shows that this is a very strong relationship: the more women killed on a given day, the fewer men killed on that day.  Below is the plot and what the author says about it.

The daily number of women casualties should be highly correlated with the number of non-women and non-children (i.e., men) reported. Again, this is expected because of the nature of battle. The ebbs and flows of the bombings and attacks by Israel should cause the daily count to move together. But that is not what the data show. Not only is there not a positive correlation, there is a strong negative correlation, which makes no sense at all and establishes the third piece of evidence that the numbers are not real.

The correlation between the daily men and daily women death count is absurdly strong and negative (p-value < .0001).

The figure is indeed strongly negative, and isn’t due to just one or two outliers. The  R value itself (the Pearson correlation coefficient) is a huge -0.914 and what we would call “highly significant”, with a probability that a correlation this large have occurred by chance being less than one in ten thousand. It’s clearly a meaningful relationship.

Is there a genuine explanation for this, one suggesting that the numbers are not made up? I could think of only one: on some days men are being targeted, as in military operations, while on other days both sexes are targeted, as if Israel is bombing both sexes willy-nilly. But that doesn’t make sense, either—not unless the men and women are in separate locations (when a lot of women are killed on a given day, almost no men are killed). Look at the data below the fold, for example: on October 30 no women were reported killed but 171 men were killed.  That could happen only if on that day Israel was targeting only men, which would mean they were going after terrorists. But that’s not Hamas’s interpretation, of course.

Conversely, on the next day 6 men were reported killed and 125 women.  Was the IDF targeting women? None of this makes sense.

There are other anomalies in the data. Here’s one:

. . . . the death count reported on Oct. 29 contradicts the numbers reported on the 28th, insofar as they imply that 26 men came back to life. This can happen because of misattribution or just reporting error.

Indeed, as on October 29 there were 2619 deaths in the cumulative total of men (implied), but on the day before, October 28, there were more: 2645! Take a look at the chart below the fold.

One more anomaly:

There are a few other days where the numbers of men are reported to be near 0. If these were just reporting errors, then on those days where the death count for men appears to be in error, the women’s count should be typical, at least on average. But it turns out that on the three days when the men’s count is near zero, suggesting an error, the women’s count is high. In fact, the three highest daily women casualty count occurs on those three days.

Here’s how the author explains the data:

Taken together, what does this all imply? While the evidence is not dispositive, it is highly suggestive that a process unconnected or loosely connected to reality was used to report the numbers. Most likely, the Hamas ministry settled on a daily total arbitrarily. We know this because the daily totals increase too consistently to be real. Then they assigned about 70% of the total to be women and children, splitting that amount randomly from day to day. Then they in-filled the number of men as set by the predetermined total. This explains all the data observed.

After deciding that we can’t get any numbers other than these, and adding that we can’t differentiate civilians from soldiers, or accidental deaths caused by misfired Gazan rockets, Wyner leave us with this conclusion:

The truth can’t yet be known and probably never will be. The total civilian casualty count is likely to be extremely overstated. Israel estimates that at least 12,000 fighters have been killed. If that number proves to be even reasonably accurate, then the ratio of noncombatant casualties to combatants is remarkably low: at most 1.4 to 1 and perhaps as low as 1 to 1. By historical standards of urban warfare, where combatants are embedded above and below into civilian population centers, this is a remarkable and successful effort to prevent unnecessary loss of life while fighting an implacable enemy that protects itself with civilians.

People tend to forget this ratio, which is stunningly low for fighting a war in close quarters against an enemy that uses human shields. (The link to “historical standards” goes to PBS and an AP report, so it isn’t exactly from Hamas).  Besides showing us that we can’t trust Hamas’s figures, which nevertheless are touted in all the media, it also shows that there is no indication that the Israelis are trying to wipe out the Palestinian people; that is, there is no genocide going on.

But it would be nice, if newer figures were available, to see if these anomalies are still there. This article is from March 6, so it’s pretty new.

Click “continue reading” to see the data

Continue reading “Hamas plays fast and loose with the casualty numbers from Gaza” →

Conor Friedersdorf (and Alexander Barvinok) on ideological coercion in American colleges

December 15, 2023 • 11:00 am

The Atlantic‘s Conor Fridersdorf is a breath of fresh air among liberal woke media.  He is in fact a liberal, but not a “progressive”, and in the new article below he reports one mathematician’s observations of how DEI has insinuated itself into academics, creating not only viewpoint homogeneity, but authoritarianism.

Like my colleague Anna Krylov, Alexander Barvinok began his studies in Russia (math for him, chemistry for Anna), and then moved to the U.S., where, at the university of Michigan, he initially enjoyed the academic freedom he lacked overseas. But then, as time crept towards the present, Barvinok found that math began changing in the direction of a Soviet-style academia, with required conformity and fealty to non-scienctific but ideological ideals. Granted, we don’t kill American academics for bucking conventional ideology, but the parallels between Russia and the U.S. about how the authoritarianism started and advanced are striking. Anna’s story is very similar.  Right now we’re at the “DEI stage” of authoritarianism, but will it stop here? I doubt it.

Click to read, though it’s paywalled. I didn’t find it archived, so a judicious inquiry might yield the piece.

I’ll quote from Friedersdorf’s nice piece, in which Barvinok’s words are given in quotation marks.

“I grew up in the Soviet Union, where people had to affirm their fealty to ideals, and the leaders embodying those ideals, on a daily basis,” he told me. “As years went by, I observed the remarkable ease with which passionate communists turned first into passionate pro-Western liberals and then into passionate nationalists. This lived experience and also common sense convince me that only true conformists excel in this game. Do we really want our math departments to be populated by conformists?”

Barvinok insists that it isn’t diversity to which he objects. Any coerced statement, he says, would trouble him as much. “Even if one is required to say ‘I passionately believe that water would certainly wet us, as fire would certainly burn,’” hewrotein his resignation letter, “the routine affirmation of one’s beliefs as a precondition of making a living constitutes compelled speech and corrupts everyone who participates in the performance.”

It is this compelled speech, which we all know has to adhere to the Kendi-an au courant antiracist doctrine, that makes DEI statements offensive and inimical.  If you diverge from the accepted narrative, your job application might not even get looked at, and you may not be promoted. This is neither inclusive nor promotes diversity, since it leads to everyone having—or at least professing—the same views on diversity.  Those who aren’t “included” keep their mouths shut. And do mathematicians really need to even deal with such issues? I haven’t heard a good argument for it. So long as professors and universities don’t discriminate on the grounds of race, gender, or any other such immutable trait, and concentrate on academic merit and service instead, all will be well.

Barvinok graduated first in his class in St Petersburg, but couldn’t get a good job because he was Jewish. (Yes, Russia was and is still rife with antisemitism.)  After Gorbachev came to power and things loosened up, he got a postdoc in Stockholm, and then moved to Cornell and finally to Michigan in 1994, where he became tenured.

The ideological rot in mathematics (and other sciences, I’m guessing) began soon thereafter, and occurred, Barvinok says, in three stages.

STAGE 1: The NSF kicks off the “broader impact” requirement for scientists

When did he first fret about the political environment in American universities? Looking back, he recalls steady growth in three broad trends that he began to notice sometime in the early 2000s. “Initially, as an American professor, you were in good standing if you taught well, did reasonable research, and were a good colleague, which was demonstrated by your willingness to do committee work,” he recalled. “The first trend I noticed was that to count as a ‘good citizen,’ you were increasingly expected to contribute, so to speak, to the betterment of humanity at large.”

In particular, he recalled the National Science Foundation introducing a requirement starting in 1997 to describe the “broader impacts” of research proposals and how that changed the experience of participating as an expert on panels convened to judge grant applications. In a few cases, applicants he knew as “excellent mathematicians” and “pretty decent individuals” would describe their research objectives with care, knowledge, and enthusiasm, whereas in the mandatory “broader impact” section, they would have “nothing better to say than that they plan to write joint papers with women and supervise female graduate students,” so mething that outraged some members of the panel

“I understood the outrage, of course, but for me it was an indication that this perhaps well-intended requirement was in fact ill-advised, as it pushed otherwise decent people to behave in a silly, sometimes obnoxious manner,” he reflected. “It also appeared that the university and college policies in hiring, and to some extent promotions and merit raises, were increasingly motivated by the desire to effect some positive social changes, in the form of a contribution to DEI.”

STAGE 2:  College administrations grow with new hires involved in DEI

The second trend that Barvinok began to notice in the aughts was the growth of college administrators and the growing coherence of the messages issued by the administrators across institutions. Today, the University of Michigan’s DEI bureaucracy is huge: According to The Chronicle of Higher Education, “Michigan’s DEI structure, with $85 million in initial funding and more than 100 employees contributing at least part time to diversity efforts, is widely considered among the most ambitious and well-funded offices in the nation.” Multiple pages on the University of Michigan website emphasize efforts to infuse DEI values into faculty hiring, research, and more.

Let me emphasize that I’ve been at the University of Chicago since 1986, and although DEI has grown and is more visible, it hasn’t been as noticeable here nor as oppressive as at Michigan, which has a much bigger DEI bureaucracy.

STAGE 3: DEI comes to dominate and invade all academic effort

The third trend he noticed was the changing nature of debate. More and more often, someone would claim that an argument was harming a particular demographic, often without specifying how. As he recalls it, “It was at this ‘harm’ stage that people became afraid to speak their minds.”

This is where the diversity of opinion began diminishing and those who didn’t feel “included” became too intimidated to oppose the trend.

Barvinok sees all these stages as related and thus, perhaps, an “organic progression”:

In Barvinok’s telling, the three trends he described—the institutional requirement to do public good, the growth of the academic bureaucracy, and the accusations of causing harm to silence opponents—are interrelated. “The more social goals one wants to effect, and the more ambitious they are, the more administrators one needs, who in turn may put forth new social goals or make the existing ones more ambitious,” he explained. “If you are convinced that what you do is in the public good, then clearly your opponent is causing harm.” Furthermore, if someone’s actions fail to advance progress (as self-styled reformers see it) quickly and smoothly, or have the opposite effect, there’s a tendency “to search for the enemies within, who hinder the effort, maybe unwittingly.”

As an example of how well the termites have been dining, Barvinok uses the case of UC Davis math professor Abigail Thompson (recounted on this website), who was demonized simply for publishing an article in Notices of the American Mathematical Society favoring DEI but arguing against mandatory DEI statements. That caused a maelstrom (read the link above). It was, for Barvinok, a watershed moment when he realized that, as in Russia, simply expressing a heterodox idea could get you in trouble.

But wait! The rot kept spreading. Here’s what happened after the death of George Floyd, when academic ideology became totally authoritarian:

On June 26, 2020, about a month after the murder of George Floyd by a Minneapolis police officer, an email addressed to the math department arrived in his inbox. In it, the department chairman told everyone that a math-department committee that focuses on the “climate” of the department, with the vetting of the department’s executive committee, had written a statement on the department’s response to racism.

It contained the following lines:

We understand and acknowledge that systemic racism permeates all aspects of our society. We acknowledge that systemic racism permeates the culture of our own institution and department. For this we are deeply sorry; we know that we have work to do.

Once again, Barvinok thought of the society he’d left as a young man.

“In my memory from Brezhnev to more recent Putin times, the slogan ‘We have work to do’ served as an unmistakable indicator that no work would be done, as those who work don’t have an appetite for sloganeering,” he told me. “Still, to gather my wits, I decided to do some work of my own and went to wash the deck of my house.” When he returned, an email to the whole math faculty by a fellow professor was in his inbox. The math department’s email, it read, “essentially states that the majority of my colleagues are racists, a false accusation which I refuse to join.”

Barvinok then responded himself. “I wrote a couple emails to the effect that the department has no business making political (or religious, or artistic, or gastronomical) statements on behalf of its members, and suggested that those who support the statement should just sign it,” he told me. “It became clear then that people were afraid to voice their opinion, unless it was aligned with what was perceived as the dominant narrative. There were many fewer people who expressed their disapproval of the statement publicly than those who did it privately.”

The Michigan fealty statement is very similar to the University of Chicago’s Lab School Science Department statement that I highlighted yesterday, a likely violation of our University’s principle of institutional neutrality. It’s harmful for reasons that Barvinok gives above.  Are we becoming like Russia?

Read and weep (Lab School, please ditch this statement!) The markings are from a parent.

It’s not that different from the Michigan statement, right?

The result is, as you’d expect, a reduction in expressed viewpoint diversity and a “non-inclusion” of those who don’t accept the loyalty oaths.  Friedersdorf issues a call for those who dissent to speak up:

Asurveyof 1,500 faculty in the U.S. conducted by the Foundation for Individual Rights and Expression found that half of respondents considered such statements “an ideological litmus test that violates academic freedom.” If Barvinok’s analysis is correct, that cohort could speak out against DEI statements without significant risk to their career, if only they coordinate and do so in numbers.

Then again, the longer the current system is in place, the harder it may be to reform––if hiring processes are biased in favor of a given ideology, and against those who dissent from it, the faction benefiting from bias will come to dominate the institution over time. As Barvinok put it in his letter to the American Mathematical Society, “I anticipate an argument that the AMS is ‘not involved in politics.’ But this is the kind of ‘politics’ that, rephrasing Pericles, will get involved with you, whether you like it or not, and hence inaction is just as political as action.”

For dissenters, the best time to speak up is now.

And so I have.  Onward and upward!

Here’s Barvinok lecturing at MIT in 2019:

Oh, and here’s an informative tweet about U. Mich. Look at those salaries!

American secondary schools ditch algebra and advanced math requirements in the name of equity

July 23, 2023 • 9:30 am

Here’s a bit of Nellie Bowles’s weekly news summary that I highlighted on Saturday.

→ Make algebra illegal! Progressives have been waging a long battle against accelerated math courses in middle and high school, and they are winning. A lot. First they won San Francisco, where Algebra I was banned in public middle schools. Now this week, they basically got that to be the new California math policy. And it’s been spreading: Cambridge, Massachusetts, and other school districts have followed suit. Basically, white parents are 1) convinced that black kids simply can’t learn algebra and the only possible solution is to ban the class, and 2) alarmed how much better the Asian kids are at this class and worried it might hurt little Miffy’s prospects. For now, just read this great takedown by economics writer Noah Smith: “Refusing to teach kids math will not improve equity.”

Well, of course you have to check the references for yourself, but by and large they do check out. Remember that in America “middle school” is all secondary school from grade 6 up to the beginning of high school, which is grade 9—students from about twelve to fifteen years old.  Nellie’s explanation for the banning of algebra, however, is undoubtedly correct.

First, let’s check out her three claims, which I’ve put in bold below. Two of them are accurate, and one is semi-accurate:

1.) San Francisco bans algebra in public middle schools: This appears to be true: go here or here.

2.) New California math policy bans algebra in middle schools: This appears to be questionable. The source above says this (my emphasis):

Critics, including many parents of high-achieving students, worried that students would be prohibited from taking appropriately challenging courses—and that delaying Algebra until 9th grade wouldn’t leave students enough time to take calculus, generally viewed as a prerequisite for competitive colleges, by their final year in high school.

That language has since been revised. The approved framework still suggests that most students take Algebra I or equivalent courses in 9th grade, through either a traditional pathway or an “integrated” pathway that blends different math topics throughout each year of high school.

But the framework notes that “some students” will be ready to accelerate in 8th grade. It cautions that schools offering Algebra in middle school assess students for readiness and provide options for summer enrichment support that can prepare them to be successful.

This implies that algebra will be optional (as other sources say) in the 8th grade, the last year of “middle school” (“junior high school” as mine was called). It’s possible that some schools won’t offer it, though.

HOWEVER, the new California standards don’t appear to ban algebra, though I haven’t read them carefully. What they seem to offer up to grade 8 is a form of  optional algebra: “algebra lite”. Perhaps that’s why Nellis said “basically” that is the new California math policy.  From a FAQ on the state’s website:

Chapter 8 of the draft Mathematics Framework notes that: “Some students will be ready to accelerate into Algebra I or Mathematics I in eighth grade, and, where they are ready to do so successfully, this can support greater access to a broader range of advanced courses for them.”

The framework also notes that successful acceleration requires a strong mathematical foundation, and that earlier state requirements that all students take eighth grade Algebra I were not implemented in a manner that proved optimal for all students. It cites research about successful middle school acceleration leading to positive outcomes for achievement and mathematics coursetaking, built on an overhaul of the middle school curriculum to prepare students for Mathematics I in eighth grade, teacher professional development and collaborative planning time, and an extra lab class for any students wanting more help.

To support successful acceleration, the framework also urges, in chapter 8: “For schools that offer an eighth grade Algebra course or a Mathematics I course as an option in lieu of Common Core Math 8, both careful plans for instruction that links to students’ prior course taking and an assessment of readiness should be considered. Such an assessment might be coupled with supplementary or summer courses that provide the kind of support for readiness that Bob Moses’ Algebra project has provided for many years for underrepresented students tackling Algebra.”

3.) Cambridge, Massachusetts bans algebra in middle schools. The link above, via the Boston Globe, appears to give an accurate account: algebra is banned until high school:

Cambridge Public Schools no longer offers advanced math in middle school, something that could hinder his son Isaac from reaching more advanced classes, like calculus, in high school. So Udengaard is pulling his child, a rising sixth grader, out of the district, weighing whether to homeschool or send him to private school, where he can take algebra 1 in middle school.

Udengaard is one of dozens of parents who recently have publicly voiced frustration with a years-old decision made by Cambridge to remove advanced math classes in grades six to eight. The district’s aim was to reduce disparities between low-income children of color, who weren’t often represented in such courses, and their more affluent peers. But some families and educators argue the decision has had the opposite effect, limiting advanced math to students whose parents can afford to pay for private lessons, like the popular after-school program Russian Math, or find other options for their kids, like Udengaard is doing.

Now getting rid of the algebra option in middle school, which is where I took it, is about the dumbest thing I can imagine, even if you buy the rationale: to “level the playing field of knowledge” so that the variation in math knowledge is reduced among all students, providing a kind of “knowledge equity”. Because minority students don’t do as well in algebra as white students or especially Asian students, by eliminating algebra you reduce the disparity in achievement among groups.  But preventing advanced students from taking algebra before high school only punishes those students, including minority students, who have the ability and desire to handle algebra. It prevents those students from going on to calculus, and perhaps other advanced math classes, early in high school. The result: a impediment in the way of students who want to and have the ability to go onto STEMM careers. This may be the craziest move I’ve seen done in the name of “equity”: removing the ability of capable students to access classes they want and can handle.

But Noah Smith’s column, cited by Nellie above, gives a much better summary, underlining the sheer lunacy of this policy. Click to read:

An excerpt:

A few days after Armand’s post, the new California Math Framework was adopted. Some of the worst provisions had been thankfully watered down, but the basic strategy of trying to delay the teaching of subjects like algebra remained. It’s a sign that the so-called “progressive” approach to math education championed by people like Stanford’s Jo Boaler has not yet engendered a critical mass of pushback.

And meanwhile, the idea that teaching kids less math will create “equity” has spread far beyond the Golden State. The city of Cambridge, Massachusetts recently removed algebra and all advanced math from its junior high schools, on similar “equity” grounds.

It is difficult to find words to describe how bad this idea is without descending into abject rudeness. The idea that offering children fewer educational resources through the public school system will help the poor kids catch up with rich ones, or help the Black kids catch up with the White and Asian ones, is unsupported by any available evidence of which I am aware. More fundamentally, though, it runs counter to the whole reason that public schools exist in the first place.

The idea behind universal public education is that all children — or almost all, making allowance for those with severe learning disabilities — are fundamentally educable. It is the idea that there is some set of subjects — reading, writing, basic mathematics, etc. — that essentially all children can learn, if sufficient resources are invested in teaching them.

. . . When you ban or discourage the teaching of a subject like algebra in junior high schools, what you are doing is withdrawing state resources from public education. There is a thing you could be teaching kids how to do, but instead you are refusing to teach it. In what way is refusing to use state resources to teach children an important skill “progressive”? How would this further the goal of equity?

. . .Now imagine what will happen if we ban kids from learning algebra in public junior high schools. The kids who have the most family resources — the rich kids, the kids with educated parents, etc. — will be able to use those resources to compensate for the retreat of the state. Either their parents will teach them algebra at home, or hire tutors, or even withdraw them to private schools. Meanwhile, the kids without family resources will be out of luck; since the state was the only actor who could have taught them algebra in junior high, there’s now simply no one to teach them. The rich kids will learn algebra and the poor kids will not.

That will not be an equitable outcome.

In fact, Smith cites a fairly well-known study from Dallas Texas in which students were all put into honors math classes and were forced to opt out instead of opt in. This policy was implemented in 2019-2020, and the result was a dramatic increase in ethnic diversity in honors math classes in the sixth grade (students about 12 years old). The rise is stunning.  This is what we could have if we challenge students rather than accept their deficiencies. But no, that’s not the “progressive” way, which is to dumb down everything to the lowest level.

, , , , How did we end up in a world where “progressive” places like California and Cambridge, Massachusetts believe in teaching children less math via the public school system, while a city in Texas believes in and invests in its disadvantaged kids? What combination of performativity, laziness, and tacit disbelief in human potential made the degradation of public education a “progressive” cause célèbre? I cannot answer this question; all I know is that the “teach less math” approach will work against the cause of equity, while also weakening the human capital of the American workforce in the process.

We created public schools for a reason, and that reason still makes sense. Teach the kids math. They can learn.

I’m not even going to get into the debate about those who suggest that math class could be a way (surprise!) of teaching social justice. That’s also part of the revised California standards, and is summarized in this article by the Sacramento Observer (click to read):

A short excerpt:

The state of California is under scrutiny for its release of a math framework that aims to incorporate “social justice” into mathematics, despite calls from parents for improved education. The California Department of Education (CDE) and the California State Board of Education (SBE) unveiled the instructional guidance for public school teachers last week.

One crucial section of the framework  [JAC: go to chapter 2 of the link] emphasizes teaching “for equity and engagement” and encourages math educators to adopt a perspective of “teaching toward social justice.” The CDE and SBE suggest that cultivating “culturally responsive” lessons, which highlight the contributions of historically marginalized individuals to mathematics, can help accomplish this goal. The guidance further advocates for avoiding a single-minded focus on one way of thinking or one correct answer.

It’s clear from reading the California standards (especially Chapter 2 above) that “equity” means not just equal opportunity, but equal outcomes.  I want to take a second to address that because a few readers have maintained that “equity” simply means “equal opportunity”. If that were the case, we wouldn’t need the word “equity,” would we? No, equity is understood, in all the discussions above, to mean equal outcomes: children of all ethnic groups should be on par in their math learning.

That this is the standard meaning of equity (i.e., “groups should be represented in a discipline exactly in proportion to their presence in a population”) is instantiated in this well known cartoon:

Now this cartoon has a valid point: “equality” means little if groups start out with two strikes against them. But it’s also clear that “equity” means “equal outcomes” (more boxes) not equal opportunity (everybody gets a box).  I’m completely in favor of equality of opportunity for all groups, recognizing at the same time that this is the “hard problem” of society, one that won’t be solved easily. But it has to be solved if you believe in fairness.

I’m not a huge fan of equity, simply because it’s often used as proof of ongoing “systemic racism”, when in fact there are many other causes for unequal representation. Further, it’s the single-minded drive for “equity” that has led to to ridiculous actions like removing algebra from middle school.

Mathematicians warn of ideology polluting their discipline

May 25, 2023 • 11:30 am

It looks as if today will be about ideology infecting science—in this case, mathematics. One would think that math would be relatively impervious to the ideological tides inundating other sciences, but one would be wrong. This article from the Torygraph (click on screenshot, or on the archived version here), discusses nonbinding but injurious ideological guidelines given to college teachers of math in the UK. These guidelines have nothing to do with improving math education, of course, but everything to do with propagandizing students with certain approved political views.

Excerpts are indented:

More than 50 of Britain’s leading mathematicians have accused standards bosses of politicising the curriculum with new diversity guidance.

Academics at top UK universities have signed an open letter criticising guidance on academic standards that states that values of Equality, Diversity and Inclusion (EDI) “should permeate the curriculum and every aspect of the learning experience”.

The guidance was published in March by the Quality Assurance Agency (QAA), an independent body that receives membership fees from more than 300 UK higher education providers and distributes advice on courses.

In an open letter, the mathematicians write: “We reject the QAA’s insistence on politicising the mathematical curriculum.

“We believe the only thing that should permeate the mathematics curriculum is mathematics. Academics should teach from a perspective informed by their academic experience, not from a political perspective determined by the QAA.

“Students should be able to study mathematics without also being required to pay for their own political indoctrination.”

I believe the letter of protest to the QAA guidelines is here, though it may be an earlier version. The link to the guidelines themselves (given in the letter) seems to be gone, but the letter’s signers paraphrase some guidelines:

A particular concern is that the new edition states: “the curriculum should present a multicultural and decolonised view of MSOR, informed by the student voice.”

We abhor racism, but one can abhor racism without subscribing to the theory of decoloniality.

The theory of decoloniality is a postmodernist critique of the “European paradigm of rational knowledge”. We believe that history suggests that mathematics is not a particularly European paradigm. On the contrary there are many examples where the same mathematical ideas have been developed independently across cultures. As just one example, the Japanese mathematician Seki and the Swiss mathematician Bernoulli both studied what are now called Bernoulli numbers. We agree that where practical the mathematical community should use terminology that gives nonWestern mathematicians proper credit, but this is not the meaning of decoloniality.

The QAA suggests promoting a decolonialist perspective as follows:

Students should be made aware of problematic issues in the development of the MSOR content they are being taught, for example some pioneers of statistics supported eugenics, or some mathematicians had connections to the slave trade, racism or Nazism.

The mathematicians are correct; math curricula should be about math alone.  But what is the QAA recommending? This is hard to believe, but seems to be true:

The QAA guidance suggests that professors should note that “some early ideas in statistics were motivated by their proposers’ support for eugenics, some astronomical data were collected on plantations by enslaved people, and, historically, some mathematicians have recorded racist or fascist views or connections to groups such as the Nazis”.

Maths professors said that the agency wanted to teach “a skewed view of the history of mathematics”. They noted that the QAA did not recommend teaching “the universality of mathematical truth, the use of statistics to disprove historical racial theories or about the Jewish mathematicians persecuted by Nazis”.

If you take this tactic, then every single academic subject must devote its time to showing how famous achievers in its area were politically impure. If you want to discuss things like how slaves collected astronomical data, do it in a history or sociology of science class.

But the scariest thing in these guidelines—and I can’t verify this because I can’t find the guidelines themselves—is that the QAA did NOT recommend teaching “the universality of mathematical truth, the use of statistics to disprove historical racial theories or about the Jewish mathematicians persecuted by Nazis”.  Is mathematical truth not universal?  Yes, I know that Euclidean geometry differs from non-Euclidian geometry, but that itself is a universal truth. And they recommending teaching how mathematicians promoted slavery, racism, and Nazism, but, curiously, don’t recommend teaching how slaves enriched astronomy or how Jewish mathematicians were persecuted by Nazis? And, as a secular Jew, I want to know why Jewish persecution get a pass here.

In truth, none of this should be in math class, but I find it deeply weird that of all the philosophies held by some mathematicians, including the morality of slavery and of Nazism, they leave out Jews, who of course were the very victims of Nazi persecution, just as slaves were the victims of racism.

But there’s pushback beyond the letter:

Dr John Armstrong, a reader in financial mathematics at King’s College London, and a signatory of the letter, said: “Education for sustainable development may sound like a positive thing, but when you look into what that is, what they are promoting is encouraging all students to become activists on issues of social justice.

“It’s really quite a remarkable thing to change education from goals such as understanding, learning and appreciating art and shift everything towards consideration of social justice.”

It is simply bizarre that we all sit back and accept this explicit injection of ideology into science, a practice that not only takes time away from science (and, in this case, math), but tries to turn young mathematicians into ideologues. Were I a parent, I’d want my children to decide their views themselves, not have propaganda stuffed down their throats by math teachers.  These are bizarre times we live in, but we can’t let those who are most vocal foist their politics onto children who want to learn math or science (or anything else, for that matter).

Oh, and in light of the letter, the QAA has added this:

A spokesman for the QAA said: “Subject benchmark statements are written by groups of academics from the relevant discipline. Institutional autonomy and academic freedom are crucial principles, and therefore the statements do not mandate academics to teach specific content – they are a reflective tool to support course design and are not compulsory. We agree with the letter’s assertion that course content should be taught by academics in line with their own expertise and academic judgment.”

Indeed. Why, then, did they insist on producing a benchmark statement? And, as one of my friends asked, “How did it all go off the rails?”  It’s almost as if we’re being subject to “extraordinary popular delusions and the madness of crowds,” as the famous book was called.  This foisting of ideology on education is our version of Tulip Mania.

A trigonometric proof of the Pythagorean theorem at last? The news said it was done by two high school students!

March 28, 2023 • 10:30 am

The latest mathematical news involves two high school students from New Orleans who have found a new way to prove the Pythagorean Theorem, which of course states that for a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.  Now of course there are many proofs of this theorem (the link above gives some), but, according to the Guardian article referenced below (click on screenshot), and the POCIT article below that (also click on screenshot), one form of proof is missing: a proof based on trigonometry. The Guardian article reports that this proof was supplied by Calcea Johnson and Ne-Kiya Jackson from St Mary’s Academy.

Here’s a tweet from the American Mathematical Society:

Click to read the Guardian article:

 

Now this is somewhat above my pay grade, but perhaps a mathematician will weigh in. The Guardian notes this:

The 2,000-year-old theorem established that the sum of the squares of a right triangle’s two shorter sides equals the square of the hypotenuse – the third, longest side opposite the shape’s right angle. Legions of schoolchildren have learned the notation summarizing the theorem in their geometry classes: a2+b2=c2.

As mentioned in the abstract of Johnson and Jackson’s 18 March mathematical society presentation, trigonometry – the study of triangles – depends on the theorem. And since that particular field of study was discovered, mathematicians have maintained that any alleged proof of the Pythagorean theorem which uses trigonometry constitutes a logical fallacy known as circular reasoning, a term used when someone tries to validate an idea with the idea itself.

Johnson and Jackson’s abstract adds that the book with the largest known collection of proofs for the theorem – Elisha Loomis’s The Pythagorean Proposition – “flatly states that ‘there are no trigonometric proofs because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean theorem’.”

What is the circularity? To me (and again I may be wrong) it’s instantiated in the section below from the Wikipedia article.  The supposed trigonometric “proofs” of the Theorem depends on an identity:

The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.

The identity is

. . . and that is proven simply from the observation of right triangles and the definition of sines and cosines (see here).  Thus, since this equation is derived from definitions and the use of a right triangle, it cannot itself be used to prove the theorem. Solid proofs have depended on arguments from geometry, rearranging similar triangles, algebra, and differentials. What is said to be new here is the proof of the Theorem using trigonometry, but not in a circular way. Because I haven’t seen the students’ proof, I can’t comment on it, and probably don’t have the chops, either. I’ve put one mathematician’s video interpretation below.

A lot of the press has gotten this mixed up, like this article below (click to read):

It says this:

The Pythagorean theorem is a fundamental theorem in trigonometry that describes the relationship between the three sides of a right-angled triangle. It is expressed with the formula a² + b² = c².

The theorem holds true in every plausible example and has been around since the days of the Ancient Greeks.

American mathematician Elisha Loomis argued that no mathematician has been able to establish its truth without using circular logic i.e. without using Pythagorean Theorem. However, others have argued it can be proved using the notion of similar triangles.

Well, that’s a bit misleading, as you can establish the truth of the Pythagorean theorem without using circular logic—so long as the proof is not a trigonometric one. And that’s what the teens are said to have overcome. However, I noticed this morning that an addendum had been added to the article.

This article was corrected on March 28, 2023. The article previously stated that the Pythagorean theorem had yet to be proven without using circular logic. The article has been amended to acknowledge that this argument was posited by American mathematician Elisha Loomis and is not accepted by all mathematicians.

Even so, the correction (which doesn’t itself invalidate what the students did) still leaves out the fact that even if Loomis was wrong, his claim was not that it couldn’t be proven without using circular logic, for it can be. What he said is that it can’t be proven trigonometrically without using circular logic.

Here’s a video which is one person’s interpretation of Johnson and Jackson’s proof based on the slides he saw of their presentation at the AMS meeting:

 

It’s quite clever, and doesn’t depend on the use of the Pythagorean Identity, merely the definition of “sine”.  Note that the person who made this video isn’t 100% sure that he’s presenting Johnson and Jackson’s proof.

So, were they the first people to prove the Pythagorean theorem without trigonometry? I did some Googling, and found this at the site cut-the-knot.org (click to read), which apparently came out before 2019.

It merely reprises another trigonometric proof of the Pythagorean Theorem, one given in the paper below. This is from the article above:

Now, Jason Zimba has showed that the theorem can be derived from the subtraction formulas for sine and cosine without a recourse to sin²α + cos²α = 1. [JAC: the latter is  of course the Pythagorean Trigonometric Identity.]

The paper it refers to is this:

J. Zimba, On the Possibility of Trigonometric Proofs of the Pythagorean Theorem, Forum Geometricorum, Volume 9 (2009) 275-278

And I found it online. You can read Zimba’s paper by clicking on the screenshot:

Zimba says he can derive the Pythagorean Trigonometric Identity independently of the Pythagorean Theorem, and thus you can use the former to prove the latter without circularity. He defines sine and cosine without using right angles, and then employs a subtraction procedure to come up with the Pythagorean Identity. That identity then establishes the truth of the Pythagorean Theorem. Here’s a bit of the ending, but it’s a short paper.

The author of the cut-the-knot.org piece says that Zimba’s proof is correct. But what do I know? If it is, then Johnson and Jackson were not the first people to produce a non-circular trigonometric proof of the Pythagorean theorem. But even if that’s the case, if the video gives an accurate take of the two students’ proof, what they did is still remarkably clever.

I suppose mathematicians will weigh in on the Internet in the next few days, so stay tuned.

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.”

Enjoy!