As somebody who is passionate about math education reform, I was excited to hear that Rochelle Gutiérrez, Professor of Curriculum and Instruction at the University of Illinois, was coming to speak about “Rehumanizing Mathematics” at convocation. Most people would agree that the way we teach math in this country is deeply flawed; I would argue that it is, in fact, a critical issue for social justice, the soul of mathematics, and the basic humanity of the schooling experience.

In her talk, Gutiérrez outlined a phenomenal vision for revolutionizing math education, and returning beauty, joy, and humanity to the subject—and on the way, made me question everything I thought I knew about what math is.

On Friday afternoon, as I raved to my friend about rehumanizing mathematics, she asked me: “But isn’t the reason people like math because it’s so inhuman?” The question gave me pause. This is how our society sees math—abstract, separate from human beings, a sign of intelligence—something you’re either good at or you aren’t. At its core, math is far more of an art than a science, and shows itself everywhere in nature—and yet our public schools turn it into children’s daily torment session, where they are forced to spend an hour doing computers’ work, drudging through computations, nomenclature, and meaningless algorithms.

Most people have had to live through more than 13 years of this “slow violence,” as Gutiérrez aptly calls it, to the point where saying “I was never any good at math” is almost cool in a way that “I never got the hang of that whole ‘reading’ thing” never could be. The pent-up trauma and anger at the math education system could be felt bubbling beneath the surface in the chapel, as audience members shared stories of how they had felt dehumanized in math class. They had been told that they were not good at math, and thus not intelligent, and they had had to fight to believe that was not true.

The truth is that we are all born mathematical—in the words of Bob and Ellen Kaplan, founders of the Global Math Circle, we all have the “architectural instinct”—but only those who thrive on the narrow, detached-from-humans slice of it that we teach in school are left liking what we call “mathematics.”

Although this negative experience of math is widespread, it is not distributed equally across the population. The math education system consistently excludes and disadvantages women, people of color, and other minorities, preventing them from full economic access as the highest-paying jobs increasingly require quantitative reasoning skills.

In his book Radical Equations, Robert Moses argues that this is an issue on the same level of importance as the civil rights movement of the 60s. Gutiérrez seeks to combat this injustice by rehumanizing mathematics.

To do so, Gutiérrez challenges the very nature of math; to her, math can be found in musical rhythms, in dance steps; it can be found when deciding on a Tupperware for your leftovers, or when understanding “to what degree the seal nation as a whole is sick” based on the health of an individual seal.

By looking to indigenous perspectives and ethnomathematics from around the world, Gutiérrez paints a picture of math that extends beyond imagination, far past the confines of the arbitrary Algebra-Trigonometry-Calculus sequence that we stick to so religiously, or even the proofs of upper-level math that math majors tend to cite as “real math.”

Mathematics is taught for its supposed utility like no other subject. To Gutiérrez, mathematics should be a creative, artistic process. In her ideal world, the answer to the often-asked question of “when am I ever going to use this?” is: “you might not use it, and that’s okay; that’s not the point.”

I have struggled with this very question—even as a self-proclaimed math lover, I have often questioned the entire of field of mathematics, asking how they can justify exploring abstract concepts which seemingly have no foundation in reality—no “use.”

What Gutiérrez is suggesting is revolutionary. In the post-convocation lunch, she gave the example of an assignment her daughter was given in English class: to write a page of nonsense. Such an assignment has no practical “use,” but is simply done for the artistic value of it, and the building of a craft.

Just as you would create something unrealistic in art class, or do more than just play scales while jamming on an instrument, Gutiérrez wants children to be creative authors of mathematics, rather than consumers. She wants them to break rules; to “come up with a fifth” basic operation, to see what happens when the sum of the angles of a triangle exceed 180 degrees.

Gutiérrez made it clear that she does not advocate for an “anything-goes” sort of mathematics; in her classroom, falsehoods will still not pass for truths. But she sees these rule-breaking pursuits, which can lead to entirely new realms of structure such as non-Euclidian geometry, as inherently valuable and inherently human, and as a critical step in rehumanizing mathematics for everyone.

If a life-long math lover like me finds this idea liberating, I can only imagine what it could do if unleashed in the schools.

Gutiérrez stated clearly that the rehumanization of math is an ongoing process; we are just at the beginning of this revolution, which is vital for social equality, the soul of mathematics, and the wellbeing of our children.

Her talk was full of ideas and suggestions, as she reexamined every angle of math education, but her message rung out loud and clear: math is vast, varied, and human, and we should teach it as the creative, beautiful, joyful thing it is. I highly recommend her talk (a recording can be found on the Carleton Convocations website), and I hope that you will join the movement to rehumanize our children’s math education.

You know, as an engineer, I have had to study a lot of math, and I’m pretty good at math.

I agree that the way we teach mathematics could be greatly improved. I had a very hard time with trigonometry in high school and calculus in college. Differential equations was even worse. My college instructor for differential equations actually felt that everyone in their university career should encounter at least one class that was beyond their capabilities. He accomplished that goal with me, but still gave me a passing grade. To this day I do not understand the wisdom (?) of his approach.

What I don’t agree with Rochelle Gutierrez about is that there is actually any particular “whiteness” or even inherent bias in mathematics as a subject. I mean we use Arabic numerals, right? In my growing up in the ’60’s, it was my experience that girls were inherently better at math than boys.

This article proposes that a triangle’s angles can equal more than 180 degrees, and seems to imply that we should somehow be teaching falsehoods like that, especially to minority students. I don’t believe that Jaime Escalante would stand for such foolishness – though he would likely embrace any approach to teaching math that would be more effective.

I’m also guessing that Rochelle Gutierrez somehow missed the movie Hidden Figures…….

I know I’m probably ignorant about the requirements of an engineering career, but I would guess that the material you learned for your profession should more aptly have been called “modelling and computation” rather than “mathematics”. As I understand it, the author (and Gutiérrez) has a different vision of mathematics, one which is usually advanced by mathematicians working in academia. The view is that the aim of math is not to teach us all of the tricks to find the area under a curve or be able to multiply large numbers with pencil and paper, but to use our creativity and pattern-finding abilities to investigate lofty concepts such as quantity, structure, space, and change (as per Wikipedia).

Saying that having triangle angles add up to more than 180 degrees is a “falsehood” tells me that you, like the vast majority of people in the US, never had the opportunity to see the vast universe of mathematics that exists outside our normal curriculum. Just look up “non-Euclidean geometry” and you will see that this silly-sounding question is related to an issue which baffled mathematicians for almost 2000 years. Instead of dismissing such an outlandish claim about triangles, mathematicians play the “yes, and…” game to see what consequences might come out of such a strange situation. It’s hard to understate how much of the mathematical progress of the last few centuries over the past few centuries hinges on the possibility that triangles could have more (or less) than 180 degrees.

I think academic mathematicians generally want math education to teach students how to freely investigate complex ideas with the aid of a logical structure. Maybe you say this is idealist and useless, that engineers need to know how to do arithmetic and solve differential equations rather than study the peculiarities of some imaginary geometry. I think that’s true, that people still do need hard and tangible quantitative skills. But you can also argue there’s utility in the idealist approach. Knowing how to rigorously analyze and solve complex problems from multiple perspectives is a vital skill in any profession. I think this is part of the reason why so many math PhD students are hired into tech, finance, and government positions, even though the abstract theories they studied for their dissertation almost certainly will never contribute anything to their employer’s profits.

Here’s a much longer piece, “A Mathematician’s Lament” by Paul Lockhart, that goes into more detail on this different ideal of math education:

https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

I assume that Professor Gutiérrez’s convocation address also goes into more detail, though I haven’t watched that. I like these viewpoints because I’m in the academic math world, but I recognize that mathematicians are fighting an uphill battle in this discussion. It’s the same as when people look at me and say “don’t we know all the math already?”. We’re all just coming into this with completely different understandings of what’s even being talked about. The layperson’s “math” is not my “math”, and that’s why a lot of people might scoff at these reformative approaches. My hope would be that the two could be combined, that our schools could teach both “mathematical computation” and “mathematical reasoning” in one class, and students could see how the two feed off of each other. Overhauling an education system is a huge task however, and it will take a lot of work to get to that point.

As to the point about whiteness/bias in math, I think this is part of a larger educational issue: Black Americans and other minorities disproportionately live in poorer areas with poorly-funded schools. Math education just happens to be very critical for job success in this day and age, so the disadvantage plays out in the job market. And maybe your female classmates were better at math in the 60’s, but when I was in college in the 2010’s every single student (and professor) in my theoretical math courses was a white male, despite the university having an equal gender divide and 65% white students. White guys aren’t born to be better at math, so something must be happening to everyone else along the way that’s discouraging them from even trying to pursue the subject further.

J WALTZ, you state that “This article proposes that a triangle’s angles can equal more than 180 degrees, and seems to imply that we should somehow be teaching falsehoods like that, especially to minority students.”

The following lines from the article “She wants them to break rules; to ‘come up with a fifth’ basic operation, to see what happens when the sum of the angles of a triangle exceed 180 degrees […] she sees these rule-breaking pursuits, which can lead to entirely new realms of structure such as non-Euclidean geometry…” make it clear that Gutiérrez is referring to the development of Lobachevskian/hyperbolic geometry via negation of Euclid’s fifth postulate (i.e. the parallel postulate).

Nobody is advocating for the teaching of falsehoods, and the article states that clearly in this line: “Gutiérrez made it clear that she does not advocate for an “anything-goes” sort of mathematics; in her classroom, falsehoods will still not pass for truths.”

Three thoughts:

There are more students studying and more teachers teach mathematics in China than in the USA (and Russia and Europe). The curricula employed in China (as well as in Japan, India and other Asian countries) at the K-12 level is such that students who study collegiate mathematics arrive at college, on the whole, better prepared than their American peers. Should we consider those students and their teachers as indoctrinated in “Whiteness” because their dominant teaching and learning approach to mathematics is similar to that in the USA? I suspect that most people in Asia would be grossly insulted to be told that learning classical mathematics is inauthentic for Asians, and that being successful makes them “White”. The issue is not whiteness per se, the issue is that mathematical success apparently plays a huge role in individual advancement in the dominant culture of modern economies. It just happens that in the USA and in Europe, such people tend to be of European ancestry (and in the USA, of also of Asian, Southeast Asian, Middle Eastern, … ancestry), and in China, they tend to be from the Han ethic group. White, indeed. A quick review of the faculty at the top 100 mathematics departments in the USA would suggest that mathematics is NOT just the pastime of white Europeans, and increasingly, not just the pastime of males.

The kinds of introductory college level skills – algebraic facility, basic understanding of calculus, the ability to work with the simplest mathematical models –that Professor Gutierrez wishes to de-emphasize have consistently those that are necessary for advancement into higher paying jobs (finance, economics, the hard sciences, data science, computer science). The softer skills — working collaboratively, assessments that do not measure acquisition of advanced concepts and tools, drawing on “alternative ways of knowing”, cheerleading your working group — that Professor Gutierrez advances have NOT shown their value in accessing career paths leading to high paying jobs. If we are going to steer oppressed groups (Professor Gutierrez’s preferred term for underrepresented minorities) towards a mathematics that emphasizes the softer skills at the expense of the more classical facts and skills, are we actively consigning those groups to lower earning careers? That sounds … racist.

Professor Gutierrez is an expert on the intersection of K-12 mathematics education and the theory of intersectionalism. She is not, however, a trained mathematician. She has taken almost no masters level courses in mathematics, and it would appear has never studied graduate mathematics. She has published no research in traditional mathematics. I would be very reluctant to take her ideas about what mathematics is and what the joy of doing mathematics might be, as representative of those who actually are mathematicians.

2+2 Never Equals 5

“Freedom is the freedom to say that 2+2=4. If that is granted, all else follows” -George Orwell, Nineteen Eighty-four

>Occasionally in the course of human events it becomes necessary to have to explain something no one would ever have expected to have to defend. In the present moment, we find that circumstance to be the case and that thing to be that two and two do, in fact, make four.

Please read more: https://newdiscourses.com/2020/08/2-plus-2-never-equals-5/