The new math in Toronto

November 5, 2021 • 1:30 pm

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

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

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

61 thoughts on “The new math in Toronto

    1. Left-handedness, Discordianism, homosexuality, and schizophrenia are invisible minorities. Just looking at someone, I can’t be guaranteed to know someone’s chirality, religion, sexual orientation, or emotional/psychological status, meaning that people in those demographics are safer from at-a-glance discrimination. Certain ethnic backgrounds in certain demographic contexts are visible. Certain physical disabilities are visible. Those constitute visible minorities. ‘Visible minority’ is a buzzword in some jurisdictions.

        1. Contributions from those people are taught, but their background is not visible, with the implication that they are not role models for other people of the same background, for much the same reason that friends from Spain are not considered visibly enough Hispanic. As someone who would be considered a relatively straight-acting queer person, I hate that my actions are judged for what I, as a member of an invisible minority, can /pass/ for. What is worse is that my behaviors would not necessarily have been construed as straight-acting 20 years ago; today, though, I am not considered flamboyant enough to be a member of a visible minority. For anyone who struggles to understand CRT, I strongly recommend you look into the debate as to whether queer culture should be required to be flamboyant, and how ‘passing’/straight-acting members of queer culture are being disenfranchised. S02E11 of Rick & Steve: The Happiest Gay Couple in All the World pretty much sums it up.

            1. And it was aired in 2008! It would be appropriate social commentary today, but it’s fascinating to realize people saw it coming 13 years ago!

    1. “… standard sources that draw dominantly on the experiences and narratives of men.”


      I see what they did there. Of course, Newton had a “lived experience”, … so he is equal to, say, me, who has merely a different “lived experience”. So if I want to claim my own “lived experience” has a lot to say about math, it is unfair – because he was first…. Or something.


    2. One simple problem-writing approach is to use names like Alice Bob Clara, or Ajit Balaji and Chitta. However, some problems incorporate names that start with J, T, M … largely missing the point that the names are merely hinting to simplify the problem into A, B, and C… unless of course there is only one person – then this scheme does not apply.

  1. Utter drivel. Pray, how does mathematics depend on the experiences of the mathematician? Do theorems have sex (or gender)? I suspect Pythagoras’s theorem would have been just the same if Mrs Phythagoras had discovered it. Uh, what’s “CR/NCR”?

    1. CR/NCR I’m guessing would be “credit/no credit”, the equivalent of “pass/fail” except that you can’t fail.
      SO our brave would-be alternative mathematicians must take this course for a letter grade.
      Any guess as to how many actual would-be mathematicians will take it?

    2. According to Bing and a University of Toronto web page:

      Choosing CR / NCR for CMS programs:
      CR courses cannot be used to satisfy program requirements for any UTSC Computer Science, Mathematics or Statistics program, regardless of where the course is taken (UTSC, StGeorge, or UTM).

      CR courses cannot be used to satisfy admission requirements for any UTSC Computer Science, Mathematics or Statistics program, regardless of where the course is taken (UTSC, StGeorge, or UTM)

      So not real maths, then.

    1. Haha so tempted to write an equation only involving X and Y and I then realised that this could one day be accepted as proof by the airheads.

      1. I was going to make a joke about non-Euclidean geometry, and pre-emptively cancelled it (oops!) for much the same reason. It has enough truthiness (there is no objective reason to pick rectilinear/Euclidean geometry instead of rounder/hyperbolic geometry – and they all are useful in specific cases.) While hyperbolic and elliptic geometry are not objectively feminine, I’m sure people would argue that they are.

    1. Oops, I meant to add that as a non-academic I only have access to the opening page of the paper, hence my uncertainty about how relevant it is…

    1. I wonder if the cultural aspect may refer to some indigenous languages. For instance they may have only 2 or 3 words for numbers one, few and many. How hard would it be to use these concepts to engineer real world items? Just a thought bubble anyway

  2. If students can meet a mathematics requirement with this, I suppose it is an attractive option if one dreads sitting down at a table and working problem after problem until understanding kicks in.

  3. I dunno. The history of mathematics is full of diversity of thought. Algebra originated in nonwestern culture as did the concept of zero. The origin of mathematics and even counting is fascinating (think the Lembobo bone). The idea of a course on cultural mathematics is not necessarily objectionable to me. Many advances in mathematics were made by unorthodox thinkers. Although, this course described here may not be that.

  4. Gee, I always thought mathematics was about 2 + 2 = 4, and so on. I had no idea that any “liberation” was needed. “Mathematics culture”? Huh. You learn something new every day.

    1. I believe it has already been suggested, in fiction at least, that ‘2+2=5’ is the simplest example of making people say something they know is untrue, but they say it anyway as you have all the power and they have none….
      Any form of mathematics that even hints at this belongs in Room 101.

  5. Next, U. Toronto activists will demand that Canadian companies switch to liberated mathematics, as well as maintaining proper quotas of approved groups in their workforces. So, perhaps we can look forward to liberated mathematics in the design of snowmobiles, trains, airplanes, and other vehicles at Bombardier, and in the maintenance and operation of airplanes at Air Canada. It will be grand.

    Jokes aside, this course illustrates the way the universities are culpable for unleashing all of this folly on the rest of the world. Curriculum committees would presumably not authorize courses on
    astrology, demonology, witchcraft, ghost studies, alien abduction studies, etc. etc.— but for 40 years they have been authorizing charades like the one under discussion.

  6. I didn’t know what the CR/NCR option was, which this course is not eligible for. If the course is eligible, you can drop the course at any time before the last day of classes and not have your (presumably poor) performance in it affect your GPA. When I was there you had to drop a course by November if you wanted to bail without academic penalty.
    So, now curious, I went to the undergraduate Academic Calendar of the Faculty of Arts and Science and started scrolling.

    The first 10 courses are “ABPxxx”, for the academic bridging program. None are CR/NCR option-eligible. One is entitled Decentring “Canada” (quotation marks in original)
    The first course that is not CR/NCR ineligible is ACT230H1, Mathematics of Finance for Non-actuaries. OK, I think I get it.

    Browsing through the Math program courses I find “Women’s Mathematics” (not eligible, right after Liberating Math), Mathematics of Cryptology (not eligible, requires only high-school algebra and is not eligible for the Math Specialist or Major program), and Calculus with Proofs (eligible!). And the offerings go on for pages. At a glance, many beyond second year appeared to be pitched at a level that I could not pass, as one would expect at a serious university. I am not despairing yet.

    It seems that if you can’t pass a CR/NCR-eligible course, you don’t belong at the U of T. Choose your bird courses carefully. An exception seems to be MAT295H1, an independent reading course only for strong math students and needs approval by the supervisor and the associate chair. After all that investment they aren’t going to let you off the hook if you don’t deliver.

    1. *Sorry, that ought to read, “If you can’t pass a CR/NCR-INeligible course. . .” Apologies…tripped up by the negatives.

  7. Google for “woke math” and you will see that this has become a thing, unfortunately. My favorite article title in the search results:

    Woke Educators Declare Objective Math White Supremacy

    This article examines, and counters, an 83-page article by a group of educators whose cover says:

    A Pathway to Equitable Math Instruction
    Dismantling Racism in Mathematics Instruction
    Exercises for educators to reflect on their own biases to transform their instructional practice


    1. It’s on the official U. Toronto website. What makes you think it’s a hoax?
      Would you like to bet, say, $50? I’m game if you’re “sure”! My bet: the course is real If it’s a hoax, you win $50.

  8. When I was a student there in the mid-80s there was a breadth requirement – science students had to take a certain number of arts courses and arts students had to take a certain number of science courses. A lot of arts students struggled in the science courses (I started out in science then switched to arts so didn’t need to worry about breadth) so I remember around the time I left they started introducing ‘science lite’ courses with titles like “The Way of Physics” (high-level discussion of physics without any math or equations). I suspect this is one of those courses.

    (I should mention that some science students struggled in the arts courses. A friend of mine who stuck with science complained that he had to write *an essay*! in one of his arts courses.)

    1. A Canadian, I did the standard ‘honours’ pure math degree at U. of T. from 1959 to 1963.

      There seems to be no info at all about what is supposed to be the content of the course here. It seems safe to say that the proof of Fermat’s theorem about 25 years ago would have nothing to do with the “progress” referred to in the flabby-minded course description here.

      There is no question that way back then women were often dismissed by academics in mathematics, and that attitude undoubtedly rubbed off on us students.

      But truth versus bullshit is surely so manifest in math, similarly depth versus shallowness, that virtually any male white mathematician who knows ‘anything’ can easily find somebody female, somebody black, etc. etc. who is easily seen to have a far superior mathematics intellect compared to themselves. So that the kind of narrow-minded prejudice which these people desperately seek, in order to justify their nonsensical ideas about different ‘kinds’ of mathematics, has become very much just a small number of nutters with few followers.

      Maybe I’m missing something in my old age, long retired, but I certainly doubt that.

      Things seem to be proceeding very well in (not U. of T.) my own department now. And one major reason of many for that goes back about 15 years or so when, really by a fluke, we happened to hire three female mathematicians in the same year. They are major part of the present day success. I have no hesitation in simply saying that, despite being retired and not being privy to much information that regular tenured people have. But I try to keep my ear to the ground!

      1. It seems safe to say that the proof of Fermat’s theorem about 25 years ago would have nothing to do with the “progress” referred to

        I think, from having seen him presenting on his proof, that Andrew Wiles is approximately white, and within one standard deviation of male.
        Obviously, finding the proof is back up for grabs.

  9. I am willing to bet that there will be little or no time in this class devoted to doing sums, solving equations, or doing any of what most people would consider math.
    I like the phrase “daring critique of traditional mathematics”.
    Also, focusing on math that does not “draw dominantly on the experiences and narratives of men”

    But math does not need anyone’s experience or narrative. There is no subjective element to it, at least at the levels I have studied.

    What would be cool is if they had a class covering the ways that different cultures solve universal math problems using very different numbering systems and languages. The Romans did a lot of complex engineering, and did it with what seems to me to be an awkward system. The Mayans had a really interesting system. Lots of cultures did. So they each had methods to determine the areas of various shapes, and to find the length of the side of a triangle, and so on.
    It would be really interesting to walk through how such calculations were achieved.

    Another interesting thing might be an exploration of the limits of math in cultures that lack a written language, and interesting work-arounds they might have come up with.

    Or maybe looking at relics of extinct systems that can still be found in languages today. Danish is one that comes to mind.

    However, none of these cool things seem to be part of the class under discussion. What matters is being a visible minority, for some reason.

  10. I am not sure what this one is. But social Justice math curricula has been around for a while. At first I thought “oh boy, here we go” when I first saw those things, but from what little I’ve seen they seem like good math problems that are aimed at being more relatable to the lives and problems of under-represented groups.
    Here is an example of an “old” kind of question that is not in the S.J. math curricula:

    “It costs $1.50 to travel each way on the city bus. A transit city “fast pass” costs $65 a month. Which is the more economical way to get to work, the daily fare or the fast pass?”

    This is a good and well-intentioned problem, but critiques say that neither option in the question may be afforded by a poorer family, so in these little and subtle ways, which happen every day to poorer kids in school is that they are reminded that they are not in the loop.

    So the aim of S.J. Math is to write new problems that are more relatable. They would include questions about crime, access to healthy food in the city, the Flint water crisis, and so on. But they are still solid questions.

    1. But the example you give is really something like from grade 5 elementary school, hardly post secondary, as this supposed course in mathematics is puffed up to be at U. of T. If such a problem is considered in typical US universities as appropriate level, perhaps that’s a tiny example of this supposed slide of US into becoming a 2nd world power.

      I’m not so sure it is quite the same difficulty considered by some to need fixing.

      Do you know of any textbooks, say linear algebra or several variables calculus, which has this nervous Nelly worry about being a ‘violence’ threat to students in a university?

      Surely a student coming from such humble origins materially, but now sitting justifiably in a university mathematics course should feel proud, rather than threatened by such a reminder of her humble origins vis a vis material background.

      U.of Toronto is not exactly analogous to the Bobby Jonesy Bible College, preparing students for the job of passing around the collection plate.

    2. Admittedly, it’s many years since I did my maths degree, but there were no questions in it like that. By the time you get to be an undergraduate maths student, it’s assumed you don’t need your maths problems dressed up to pretend they have anything to do with day to day life to give them some sort of faux relevance..

  11. I couldn’t help but hear Tom Lehrer’s song from the 60’s “New Math”…though it has nothing to do with wokeness or this Toronto class in particular. But there might be a correlation somewhere. Anyway, it’s one of my favorites, for your delectation.

      1. Yes, I was aware of his mathematical prowess at Harvard. Thanks for the link, though. I cannot say I have the depth of knowledge of TL I would like to have.

    1. Nobody contributed more to ‘new maths’ than Jacques Lacan, the wackiest of all post-modern wackos:

      “Thus, by calculating that signification according to the algebraic method used here, namely:
      S (signifier)/s(signified) = s (the statement), with S = (-1), produces s = Square root of -1. So the erectile organ is equivalent to square root of -1 of the signification produced above, of the jouissance that it restores by the coefficient of its statement to the function of lack of signifier (-1). ”

      From Richard Dawkins ‘Postmodernism disrobed’. Nature 394:141-143

      1. I’m worried about imaginary penises here.
        Does it translate (roughly) to, “The angle of the dangle is proportional to . . .”

  12. In the NY Times is a relevant/related article: “California Tries to Close the Gap in Math, but Sets Off a Backlash”:

    As of this moment, approx. 2600 comments and still receiving comments.

    Yep, I guess de-emphasizing/dumbing down/not requiring calculus is one way to (attempt to) close the gap.

    “Even in heavily Democratic California . . . the draft guidelinew encountered scathint criticism.” A Stanford U. professor of education: “Even parents who hated maths in school will argue to keep it the same for their kids.”

    What fatuous non sequiturs.

    (I’m reminded of a high school classmate who, reflecting on his daughter getting a full scholarship to major in middle school science and math education, saying, “She didn’t get it from me.” As if an interest in and aptitude for science and math is something one necessarily does or does not get from one’s parent(s). He attributed her success words-to-the-effect to divine intervention. He himself did enough to just get by. I remember him saying, during high school senior year, to the effect that during his senior year he intended to “have a good time” because, after high school graduation, it was “all downhill” after that. I remember specifically thinking, “I don’t agree with that!”)

    Another fatuous reportorial comment: ” . . . the draft of the California guidelines favored a more conceptual approach to learning: more collaborating and problem solving, less memorizing formulas.”

    This tickler file formula-memorizing trope is all too-easily pulled out and uttered by not a few journalists likely challenged to solve two-step algebra problems. Shall a student take time to derive from scratch the quadratic formula during a test (assuming s/he doesn’t know how to “complete the square”)? For time-management purposes it is convenient to memorize a few formulas and the multiplication tables. After all, we memorize the alphabet, and trouble ourselves to correctly spell many if not most words. It strikes me that during the last 20-30 years there has been a substantial emphasis on conceptual thinking in K-12 math.

    1. +1 on memorizing a few basic math “facts”.I was appalled at how badly a number of my Grade 12 and 13 Calculus students screwed up fractions.

      1. When I went to community college, I took some catch-up maths to get me ready for my college maths. I had been out of school for years, and took my classes very seriously. When we were doing something with fractions I realized I didn’t remember the rules. I look up at my professor, raised my hand and asked, “Ms. Harvey, I’m sorry but would you mind giving me the quick and dirty on fractions?”. I was embarrassed that I had to ask, but later she thanked me for it. Because if I was thinking it, other students were as well. Number fluency is important, but not all maths rules are intuitive.

  13. I always knew there was something deeply wrong about partial differential equations. Now I know what it is- they are partial, whereas justice requires that they be impartial. Or rather, partial in the direction of correcting past injustices.

    Maybe I’ve got this all wrong– maybe PDEs are the corrective form of DEs.

    I’m just so confused… 🙁

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