Category: mathematics

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!

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:

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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.

Jason:

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.