Noble Math

Let's disabuse education.

Right now, the way math is taught is abusive.

I don’t use that term lightly, or as hyperbole. I mean it in a pretty straightforward way.

The start of my Twitter rant on bad math education

It’s a prime example of what I mean by “bad teaching”. Instead of enticing the students with the beauty, value, meaning, and relevance of the subject, the system threatens children into attending required classes. It often adds a layer of gaslighting: if the students feel unmotivated, they’re often told it’s because there’s something wrong with them (“You’re not taking your future seriously”, “You aren’t being a good student”, etc.).

Bad teaching is worse than no teaching. We would be better off if we dropped all required math classes and replaced them with literally nothing at all. At least then we wouldn’t be beating the curiosity out of children. And frankly, hardly anyone needs to know what’s taught in math classes, and what little they do need they’ll tend to pick up as they need it.

Education should not feel like this:

By Halfpoint / shutterstock.com

I lead with all this because I want to illustrate what I think a good future could be. I want to use math education as an example since it’s definitely my area of expertise. But math trauma is so widespread that I basically have to start with some validation:

If you feel unease, nausea, boredom, inadequacy, numbness, hopelessness, cringe, etc. when math is brought up…

…I want you to know that that’s an extremely sane and correct response to abuse. Your reaction makes sense. It’s more than understandable. It’s actually the correct reaction to have when you don’t have better strategies for avoiding an abusive situation.

If you’re willing, I invite you to consider that maybe your reaction is to the abuse, not the math. Something utterly beautiful and precious was probably hidden from you. I want to try to show you a glimpse of the delight lying at the core of the art when we remove the harmful part that doesn’t belong there.

I want to do this for two reasons. The lesser one in this case is that I just really want more people to be aware of what math actually has to offer. I think it’d result in a healthier society in many ways. I also think it’s just fun and meaningful in its own right.

But the main reason is that I want to give an example. I think a way forward to a really good future comes from us cherishing our humanity and helping one another to enrich our lives. I want to see education become more decentralized, more profoundly respectful of all living beings, and built on meeting the deep heartfelt needs and desires of humanity.

I want us aiming for the future we want, rather than just reacting against futures we don’t want.

In my dream future, people study math because it’s interesting and/or relevant to them, and they can tell. There are no required classes. Not even implicitly: you don’t have to have good grades in math classes to enter the vocational training (e.g. college) you want. Instead, if certain math skills are truly needed to enter a given context (e.g. engineering school), they’d test you on those math skills. Because it’s the skills that are relevant, not the obedience training.

But even so, I’d want to see education focusing on deepening our experience of being human rather than on making people employable. This is for several reasons:

• We don’t know what will be employable in ten years. But we do know that delighting in music, falling in love, savoring insight into how the universe works, and enriching our connections to one another will be good no matter what happens.

• Focusing on employability feeds an empty economic engine. It’s why getting our jobs replaced by AI and robotics can be threatening: if they can be more productive than we are, and if our economic value is based on being competitively productive, then we’ll make ourselves obsolete. But we cannot ever become obsolete in an economy based on ennobling the human experience.

• An economy based on solving problems does better if it generates more problems, or if the problems always seem tractable but are never actually solved. That’s part of why war is ongoing in geopolitics, and why dating apps and services are universally terrible, and why there’s planned obsolescence. But an economy based on enriching the human soul does well because it creates good in a way that people can tell is good. Exploitations and manipulations can’t survive in that economic environment. You thrive by helping others thrive.

• It’s just wholesome. Plain and simple.

This might sound utopian. Maybe it is. I’m very open to talking about it, debating it, etc. I think we need to discuss it in order for something like what I’m describing to work.

But I think some steps in this direction are quite practical, honestly. I think we might be able to fix math education for instance. And possibly education in general by similar methods.

Fixing math ed starts by highlighting the connection between math and the human soul. Not to convince, but to offer the truth. That truth as I articulate it won’t speak to everyone, but it’ll speak to some people, and then those people will know what to seek out.

Let me take a stab at showing what that might look like.

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It’s a bit intimidating to try to convey what I really love about math. It feels a bit like trying to tell someone about a person I really like. Sometimes the words just don’t work, and I feel terrible because I’ve given a bad impression about someone I really respect.

So I’m going to say some things, and I hope that if it comes across badly you’ll consider that I’m just awkward at explaining it — either in general, or to you, or just right now.

So with that:

Math occurs to me as the sacred art of relating to truth.

I feel a bit shy using the word “sacred” there. It often puts people off. But anything less feels dishonest.

There’s a truly glorious feeling when I glimpse the truth of a situation. Many mathematicians have long used the word “euphoria” for the experience that accompanies the click of suddenly seeing the truth.

And yet, it’s also humbling. I’ve been very sure of some truths in math, only to discover a counterexample later on. I learn I was cleverly wrong.

And it’s not just me! The history of math is made of cases like this. An entire field of math was built on some people trying to make sense of a weird property of polyhedra. They kept coming up with really compelling arguments for why the property should be a certain way and then they’d find counterexamples. This went back and forth for a long time until someone figured out a way to rephrase the question. The rephrased question gave birth to what’s now known as “topology”.¹

I get the sense that people who haven’t really grappled with math don’t understand this intimate and complex relationship between confidence and humility. What does it mean to really truly know something so solidly that you can do this:

…but still hold your belief so lightly that you’re gladly willing to look for counterexamples and completely change your mind the moment you find them?

(Can you imagine what political discourse would feel like if everyone were approaching those discussions this way? Standing strongly for what they can see clearly, while respectfully taking their opponents’ arguments seriously and truly revising their thinking in contact with those arguments? What would the world look like if voters expected this kind of nobility from people running for office? If arguments went viral because of how gracefully respectful and clear they are instead of how infuriating and unreasonable they make one side of the debate look?)

There is so much nuance to the art of math. I’m quite expert in it, and yet I find it’s almost trivial for me to dive deeper. Reflecting on it helps me notice ways that my thinking in a day-to-day way can be refined.

But it’s also just beautiful in its own right. There’s something profoundly elegant about noticing that two magical numbers — π and e — keep showing up in intimate relationship with one another. One has to do with the shape of circles. The other has to do with exponential growth. Why would those two things be related? Well, their relationship goes through what are called “imaginary numbers”, which have their own profoundly rich structure….

I could go on for a long time like that. It feels like peeking into the deep secrets of the universe. Or like glimpsing the mind of God.

I want to give you a few examples of this. I’m aware I might be making it sound kind of mystical, and I do think there’s something spiritually quite deep about math, but it’s also extremely grounded. It doesn’t have to be weird at all. That’s part of why it’s so marvelous.

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I’m going to lift an example straight from Paul Lockhart’s essay “A Mathematician’s Lament”:

If we draw a triangle inside a box, how much of the box does the triangle take up?

Does it depend on how square the box is? Or where the triangle touches the top of the box? If it’s the same in all those cases… why?

I love the simplicity of this question. And its approachability. You can totally give this to kids as something to tackle. If they’re old enough (and geometrically mature enough) to understand the question, they’ll usually find it pretty engaging.

It so happens that there’s a really elegant move available: what happens if we draw a vertical line through the topmost point?

It’s now quite obvious (I think!) that each half of the triangle cuts its corresponding part of the box in half. Which means that the triangle takes up half the box.

If you read this far without tackling the initial puzzle, in some sense it’s sort of spoiled for you. You don’t get the joy of struggling with the initial question, and trying things on, and running into dead ends, and possibly developing your own explanation for why the triangle takes up half the area. It’s quite normal for students to grapple with a question like this one for a few days.

Although you can get a glimpse of the delightful challenge if you did grapple with the initial puzzle and concluded the triangle takes up something other than half the area of the box. Like, if you were sure it changes, now you can ask: was there something wrong with your argument? If so, what was it? Or is the above argument with the vertical line flawed somehow?

This is the mathematical dance. You look at something you can understand, and you try to make sense of it, and you try on explanations and reasoning. Eventually you come to realize something — and maybe you’re exactly right, and you can tell! But maybe you’re mistaken. This waltz of confidence and humility, curiosity and exploration, articulation and wonder defines the whole art.

Do you see a glimmer of what I mean?

Here, try it on this: notice that if you have the tip of the triangle reaching way way over to one side, the vertical line trick doesn’t work so clearly:

Does the triangle still take up exactly half the area of the box? If so, why? If not, why not, and what proportion does it take up?

Can you see why tackling this is so much more interesting and enriching than is memorizing and “applying” some given formula for the area of a triangle? Does it make sense why figuring out mathematical properties this way might make them a lot easier to remember?

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Here’s another puzzle:

You probably learned that “multiplication is commutative”. Which is to say, a x b = b x a. (Think of commuting to work: it’s saying that the symbols can travel around.)

But that’s a jargon-heavy way of hiding the interesting claim being made here. It’s not a statement about how to rewrite symbols. It’s a rather surprising description of how physical reality actually behaves.

Here’s a use case: if I have four bags of apples, and each bag has seven apples, then I can work out how many apples I have. I just add 7 to itself four times: 7 + 7 + 7 + 7.

Normally we write that as “4 x 7”. So the “4” there is talking about how many groups there are — in this case the number of bags of apples. Whereas the “7” is counting apples, not bags.

The statement “4 x 7 = 7 x 4” is actually pretty bizarre. It’s saying that if we swap it around so that we have seven bags of apples where the bags have four apples each, we will not have changed the total number of apples whatsoever. Even though I get there very differently: 4 + 4 + 4 + 4 + 4 + 4 + 4.

Can you see why that fact might be a bit surprising?

What’s even more surprising is that this holds for any two numbers. You can just swap which number is “number of bags” and which is “number of apples per bag”, with no effect at all on the total, for any pair of numbers.

Now, why would that be? Why can’t there be some numbers where swapping their roles changes the total? Can you really be sure that 102,487 x 982,238 = 982,238 x 102,487? That’s an awful lot of little things to keep track of! What tells you that these two ways of arranging things involve the same total number, no matter what? Can we actually be that confident?

Normally when I give people this puzzle, the scars of bad teaching show up. They’ll often say that they know it works this way because that’s what they’ve been told. It’s deference to authority. Together with the fact that when they plug it into their calculator it works out.

If you’re caught by that way of thinking, I want to encourage you to try looking for an explanation that you find elegant and compelling, that would give you the confidence needed to correct authorities and calculators. Like, what if there were some pair of numbers that didn’t work this way in calculators, and that all mathematicians and teachers said is the one weird exception? Can you land on a way of seeing this commutativity property such that you could stand in your own clear knowing and say “No, you are all wrong”?

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Math is full of puzzles like this. The real art isn’t at all about memorizing methods and solving exercise problems. It’s about reflecting on a curious situation, and trying on explanations, and discovering that you’re wrong, and trying again, and getting glimpses of insight, and sometimes getting an explosion of clarity that makes the whole thing incredibly and gloriously obvious.

It’s the art of clearly understanding truth.

In my dream world, people who are called to it just play with puzzles like these. Those puzzles inevitably lead to more. (What does it mean for multiplication to commute when one term is a fraction? Why would commutativity still work there?) The exploration becomes the whole point! It’s enriching. It’s beautiful.

And people who don’t find that stuff interesting just… don’t study it! And that is totally fine. Hardly anyone needs to know how to solve a quadratic equation, just as hardly anyone needs to know what the circle of fifths is in music.

And then, what is a math teacher for?

Well, to ask really damn good questions.

I know what questions lead somewhere interesting in math. I know that most students aren’t going to struggle with (say) even vs. odd when it comes to understanding commutativity. There’s no reason to distinguish between isosceles and right triangles in the triangle-in-a-box example. But I did name some questions up above that in fact do stimulate students.

A really good math teacher knows where to point. “Look here. What do you see? Can you make it do this other thing?” Their job is rarely to explain things clearly. It’s to guide attention in ways that entangle the student with the topic independent of the teacher. And hopefully lead the student to something they find enriching and fascinating.

One common objection to this direction for teaching math is:

But won’t that take too long? Lots of ideas in math took centuries to develop. Is it really reasonable to expect students to reinvent all of math?

This concern is precisely why teachers are useful. But not by just telling the students a list of results and coercing them into plowing through a bazillion mind-numbing application drills. A huge amount of math’s history is about finding good questions or noticing anomalies. Why not offer good questions up front? When students come up with an argument that ignores an anomaly, a skilled teacher can say “Cool, that seems pretty compelling. It makes me wonder about this strange thing though. What do you think?”

At which point another common objection arises:

But there just aren’t that many skilled teachers! And training teachers this way is impractical. So is this even doable?

I agree, it’s not practical to get most teachers in the current school system to become skilled this way. There were sincere efforts to do exactly that in the USA in the late 20th century, and it badly backfired. It turns out that institutional reform is extremely difficult.

(That’s the main reason I left academia, in fact. I realized that no matter how much research we did on math education and effective interventions, the nature of the political and technological landscape would never allow that research to become relevant at scale. We could help dozens of students, and maybe hundreds, but not even thousands. Barely a blip.)

That said, here are two points I think address the concern:

• No teaching is better than bad teaching. I mean it. I think we should halt math education as it’s currently done even without a replacement. It’s abusive and morally wrong, on top of being ineffective. We’d be better off without it at all.

• Teachers like me are more accessible thanks to the internet. And with some care and attention, we might be able to nurture AI into supporting good teaching this way too. Or at least develop well-known guidelines for using AI in self-enhancing ways!

This last point is something individuals can just do. People like me can just offer good math education. We can lay out the value to the human soul of what we’re doing. If it seems true and good and beautiful, some people will want to study with us. The clearer we all are about how wholesome this kind of transaction is, the less power the coercive systems will have to confuse people. And then more people can check out of the abusive math systems, and can come join more joyful and fulfilling explorations instead.

That dynamic creates an economic incentive for more people to become good math teachers, and/or for AI to become excellent at good math tutoring. Eventually I think this snowballs into a full decentralized replacement for formal math education.

And it never requires institutional reform. We just… replace it. By sincerely caring for the nobility of the human spirit. As individuals.

I wish I had a better answer for what current students in the coercive system could do. I’d love for them to at least have their experience acknowledged (yes it’s abusive, you’re right to hate it, there’s nothing wrong with you) and to give them some immunity to the gaslighting and misdirection that’s so standard. Help them get good at consciously solving the real problem facing them, namely the social problem of having to appease a coercive system in order to get what they want.

But whatever turns out to be the sane, kind answer here, I think we get there by keeping our eye on the future we want.

I think we want a future that’s made of exalting the human spirit. Not just stopping wickedness and wrongdoing, but nurturing our individual nobility and encouraging goodness, wholesomeness, depth, meaning, and compassion in everything we do.

And that vision isn’t just about math. I think it applies to all the academic disciplines. I think we can encourage something really beautiful if we’re on the lookout for what graces the heart and broadens the mind, enriches our lived experience and helps us cherish our connections with one another.

Let’s make what we focus on really, really good.

“Let us have a universal mind that loves and protects all creation and helps all things grow and develop.”

- Koichi Tohei

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I don’t yet know of good widely available material for learning math in what I consider to be a good way. I think lots of people are working on similar things. But right now the available material is (to the best of my knowledge) pretty sparse just yet.

The YouTube channel 3blue1brown has some really wonderful explanations — though I’ll warn that even incredibly elegant explanations won’t bring you into contact with the art of grappling with a question, which I consider way, way more central.

The closest I’ve personally encountered is a physics book: Thinking Physics by Lewis Carroll Epstein. Particularly the first section, on mechanics. It walks the reader through a question-led pathway of deeply understanding ideas of motion, time, energy, momentum, acceleration, etc.

If you’re interested in my skills as a tutor in the deep art of math, shoot me a message:

1. The book Proofs and Refutations by Imre Lakatos tells this story. The property in question is: if you take a given polyhedron and count each of its corners, edges, and faces, then the pattern “#corners – #edges + #faces” will always yield the number 2. Some mathematicians (such as Cauchy) proved this was the case, and then afterwards some others gave examples of polyhedra for which this pattern gave some other number. This threw what “proof” meant into crisis. Eventually Poincaré suggested that we explore what we can deduce about a polyhedron given that we know what this sum-and-difference pattern (called “the Euler characteristic”) is.