As someone who majored in math in college, this essay did an excellent job of communicating ideas that I've struggled to explain to family and friends who practically flinch when I tell them I studied math.
If you connected with this essay (or, perhaps, if you did not), I highly recommend checking out a book called 99 Variations on a Proof, by Philip Ording. It is an enjoyable, easy to understand journey through the art of making a mathematical argument by way of showing the many forms such an argument can take.
Hadn't seen The Cauchy-Schwartz Masterclass, but after a quick glance around, it seems to be very different from 99 Variations on a Proof.
The book I'm recommending is pretty far from a textbook. Rather it is an Exercises in Style type book that explores many different ways to express a mathematical proof of the same simple fact. Some of the proofs are whimsical, and others offer genuine insight. Many cover the ways that such mathematical discussions would have taken place in earlier historical periods.
In other words, 99 Variations is more like a jaunt through mathematics culture rather than a book that focuses on actual mathematical content.
I'll take a look at this book, though, because it looks quite good!
I'm in the reverse position: I have TSCM on my shelves but all I know of 99VoaP is what I can tell from Amazon's look-inside feature. My impression is the same as yours: the two books are very very different. TSCM is very much a book that intends to teach the reader some nontrivial mathematical ideas and techniques. 99VoaP sticks with a single rather elementary proposition and shows lots of ways of looking at it.
The book The Cauchy-Schwartz masterclass was crucial in the first paper I published, introducing the concept of Schur convexity. I am indebted to that great book.
Have you read the book? I came across it for the first time last week, and I’m considering purchasing it. The ebook is more affordable, but I wasn’t sure if it had diagrams, etc that would make me prefer a physical copy.
Though I'm not the same commenter, I did read the book. The book has diagrams, though they are more like sketches you might see on a chalkboard or whiteboard in a lecture (e.g. a ray diagram, several rows of hand-drawn circles that are slightly uneven non-uniform sketches, and other geometric sketches, such as triangles inside of semicircles).
There are some nice abstract artistic illustrations at the start of each "part" of the book (and also the covers), though they are more for aesthetic decorations rather than for communicating mathematical concepts. For example, one of the illustrations is an abstract medieval-like artwork with a sun that has a human-like face, a starry sky, and a person in robes whose head is peeking through the starry sky to look at a world across the boundary. So, you won't lose out on comprehension at all by reading the ebook, but the book does have some minor artwork only tangentially related to the book's content, which might make a physical edition preferable for aesthetic/collector's purposes.
The e-version is perfectly fine, though; as the prologue states, the book was first published as part of an online column on the Mathematical Association of America's website, so the e-version appears close to the original version. The prologue writer who helped publish the book also first received the essay as a 25-page printout from an audience member at a lecture.
I read the first few chapters, then went off and just hacked on some math for the joy of it.
It was incredibly freeing for me at least to hear the message of ‘think deeply about simple things’ with the realisation I don’t care like I used to that there are other people are smarter and faster than me.
Funny, I read this paper for the first time last weekend after a family discussion about public school math sent me searching. I thought about posting it here myself, but I noticed that prior submissions didn’t generate much discussion.
He identifies people that were told they were “‘good at math’ [but] in fact they have no real mathematical talent and are just very good at following directions.” I fit into that group and it served me all right, but I hope I can find ways to give my kids a more intuitive sense and curiosity as he describes
The main counter-argument presented was that if a student makes it all the way to graduate school in mathematics while following instruction, they must have some base level of mathematical talent. Then, continued hard work and persistence can set you up for a strong career in mathematics, without necessarily being brilliant.
The perspectives shared there were quite interesting. There were quite a few users who have worked as professional mathematicians—some very successfully—who talked about how they didn't really worry about whether they had mathematical talent, but just worked hard at their goals to become mathematicians because it's what they wanted to do.
So, in many cases, it's a valid approach to work hard at mathematics without worrying if you have enough mathematical talent, and eventually find challenging and interesting work along the way.
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Separately, there was an interesting Medium post discussed on Mathstodon and r/math a while back, about a father encouraging curiosity for his daughter to learn about the definition of the derivative: https://sunilsingh-42118.medium.com/the-death-of-the-mathema...
I liked the second part of his essay more when he talks about his approach in encouraging intuition. He wrote about creating a sense of mystery by rolling a marble up and down a curve to introduce the idea of a tangent line and the concept of a slope, then got his daughter curious about finding ways to numerically calculate the slope.
I thought that was an interesting approach for kindling a innate motivation to learn mathematics for a person new to mathematics, rather than focusing on physics, engineering, or economics applications like many textbooks too. Though motivating students to learn calculus for applications can also be interesting (especially for different types of students), it's a bit rarer to find ways for educators to motivate students to learn mathematics for the sake of itself, especially for students learning calculus for the first time.
> “‘good at math’ [but] in fact they have no real mathematical talent and are just very good at following directions.”
Ain't that the truth. I was always told the same thing, but I was awful at most math concepts past high school algebra. I didn't realize it until after college, but the misunderstanding seems to stem from the fact that, for most people, being good at math means being fast at arithmetic and remembering the algebra strategies. That's all their experience in math is, so it's all they can comment on.
I think I was good at easy math, and I refused to follow directions. I saw that arithmetic and early algebra had a lot of generally useful principles and I avoided memorizing and would just "think really hard" and I did well enough. I struggled with factoring polynomials (I think it was), and then really started to struggle in Calculus 2. I wish someone had sat me down and said "look, you're not learning universally applicable tools anymore, you just need to memorize these tricks, know that they only work sometimes in the real world, but they will work on the test, just follow the directions!"
Yeah, basically I crashed and burned there and have been stuck ever since. I was studying at a university where CS was treated effectively as a second math department, which is fine, but not when I had a mental block for integration.
Its funny, I was always very good at math in school, that's why I chose not to study it in college! It seemed boring to me, I thought all I'd be doing was pushing numbers around...now I have a humanities degree (I don't regret it one bit, I learned to appreciate everything beautiful in the world), but I'm finding myself growing interested in math once again, but in things like set theory and topology and the areas that seem to ask the questions of representation that I felt what was called "STEM" never addressed.
It's not my fault! The author is right, things like mathematics, physic (in fact, nearly the whole of STEM, at this point) have been subordinated to the logic of "rationality," instead of desire for beauty in the world and human creative effort. What kind of "rationality" builds a society of busybodies and functionaries of power whose only job is to perpetuate a norm of bureaucratic dictatorship, who only have the "freedom" to watch youtube and scroll tiktok after work. True freedom only comes with power, with willful, creative action. And we aren't building a society that teaches kids to do anything but suppress their most natural tendencies for fear of total social alienation.
Buddy is missing the point of school. The most important things is to learn how to pass tests, to hit deadlines, and to find a way to work when you'd rather be doing something else.
Of course he is lamenting, he's a mathematician, so he's made it through all the shit, and his intuition is now a mathematicians intuition. Changing the curriculum will be great for people like him, but terrible for great politicians, who happen to be learning algebra, or chemists, or chefs, or CEO's. Inclusivity is a virtue, so let's not forget the future cannon fodder, the brutes, the blunt instruments, the weak reeds, and the evangelicals.
He envisions an untapped strata of mathematicians. I would say that people are not being undersold on mathematics, and it's elegance. There's just not a lot of people who are mathematicians. Rebranding it as art won't help, even art has the fiddly bits that are the bedrock of life. Cut the construction paper. Gather the glue, don't get caught huffing it, etc.
In fact, if anything they should stop teaching algebra and calculus and start teaching statistical methods and probability, which will expose students to evaluating risk. Give em some dice. ( parents might be ambivalent about this ) Discrete math, where you can do it with marbles and a strip of tape.
This would make intro science more like numerical methods, and then they'd move into continuous math a bit later.
It does not really matter though. Hit deadlines, pass tests, and know how to grind your way through something simple and tedious. Because simple love of the work might not be enough. Maybe you end up having to strip to pay for textbooks. Maybe shit just doesnt work out for you right away. Those rote tasks will keep you moving toward your dreams.
Funnily enough the way he describes musical instruction is actually a good idea in my opinion. I think it’s weird that we don’t teach people to read music effectively as a regular part of musical instruction. Playing music is boring and mundane unless you’re good at it and becoming good first involves imitation. The most effective way at to imitate is to be able to read. I know everyone says there are loads of amazing guitarists who can’t read music but there are zero amazing guitarists who didn’t start by imitating someone else.
Why not give people the best way to imitate as a first step?
Music and maths are very similar in that they are as much a skill as they are an art form. You have to practise both regularly, you can’t cram for maths on the night before an exam and you can’t cram the night before a music recital.
I think music education should be more like mathematics but I also think that both should bring “big ideas” and outcomes into play sooner, with less emphasis on theory and more emphasis on practical implementation. In maths that means project based learning in music it means focusing on reading and practise rather than theory.
You made an interesting observation, but in my personal experience, educators can help students by striking a balance between rigorous fundamentals and getting to improper-but-enjoyable interactions with a new skill, when teaching a new subject—if the student isn't already strongly self-motivated.
My personal experience is with language learning, which might be comparable to focusing on sheet music and scales. When learning languages like Mandarin, Japanese, or Arabic, you can spend a lot of time focusing on fundamentals before getting started with how to say basic useful phrases (e.g. proper tones for Mandarin, and the alphabet system for Japanese and Arabic).
I spent more than a month learning the mundane fundamentals of tones in Mandarin before learning basic useful phrases. I was personally strongly convinced this was useful and was prepared for the mundanity, but I could easily see myself or other people quitting after spending several weeks without knowing more than a few phrases. With Arabic, too, I spent weeks with one teacher just focusing on the alphabet and how the letter shapes transform in certain letter combinations, without learning more than very basic phrases—which I appreciated, as the strong basis paid off down the line. Another teacher, however, spent about two-thirds of the time on phrases with romanization with English letters, and about a third of the time on the alphabet.
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For learners—especially adult students—already convinced about learning a subject, a mundane approach is acceptable because they already have the proper motivation to learn. But especially for younger students, perhaps it's okay to spend a good amount of time having fun learning the subject—but not in a way that's necessarily optimized for time.
For adolescents learning musical instruction, a better approach might be a split between theory and learning to play a simple but nice song without knowing how to read notes. But for already strongly-motivated students and/or adults, a few weeks or even months on more mundane can be even preferable, especially if the student has volunteered to spend their time working through the mundanity.
An important part of designing a curriculum that seems often overlooked is how likely it is that a student will enjoy it, and consequently whether they will even go through with it. Sure, we could speculate at what the optimal training regime would be to force upon a child, but schools unfortunately are not allowed to do that.
Too often, good teachers find themselves working as hype-men rather than educators. I've often said that if a student doesn't want to learn a subject, you basically can't teach them anything about it. And on the flip side, if a student really wants to learn something, you almost can't stop them.
I'm a jazz musician, though I started on classical music. Reading was central to how I was taught to play. Today, I use my reading chops on the bandstand, though I'm also good at picking things up by ear. Most classical students learn very little theory unless they study music in college, which I didn't.
I would say that learning by ear is the other best way to imitate.
One problem with traditional teaching is that it results in a great deal of attrition. In fact, most students give up in frustration. Also, if they choose to take up music on their own, it tends to be at an age where they've had their fill of school, and any kind of formal training is a turn-off. Genres such as folk and rock, and instruments such as the guitar and electric bass, lend themselves to informal self-training and learning material by ear.
Also, if reading isn't used, then writing isn't used. There's a huge written literature for classical music, but virtually none for contemporary popular music. There's a primitive tabulature system, but it doesn't convey much information beyond fingerings.
Reading music is absolutely something that most students of music learn. For stuff like guitar, bass, and drums, there’s tablature instead but for piano you’re definitely gonna learn to read sheet music first thing.
When I was taught piano as a young kid (I stopped after some months or a year) I was taught to start learning how to read sheet music through numbers. Which was a mistake as numbers are important to my brain (OCD counting).
For others (I have an anecdote but it's not mine so I won't share it), only if they can afford private lessons. Which has an impact on opportunities in high school, and consequently on college, if one goes to college. Raw talent only gets you so far if sight reading is expected.
I don't know anyone who wasn't taught how to read music who actually went through an official music program. It was close to the very first thing I was ever taught when learning an instrument.
I think the point still stands though that musical notation isn't the point of music - nobody gets into music because of the notation so if you focus on it too much you'll just ruin something for many people that should ultimately just be about enjoyment.
I'd also contend that reading music just isn't difficult. Interpretation and actual execution are the hard parts.
>> There’s no ulterior practical purpose here. I’m just playing. That’s what math is— wondering, playing, amusing yourself with your imagination
So that's what you do as a working mathematician, a professional mathematician.
And -let me get this right- you want society to keep paying you, so you can play with your imagination?
Uh-huh. I see. The rest of us will sweep streets, build bridges, pop and raise babies, take care of the old and inform, cook, drive, teach, make stuff... but you, we will keep feeding only so you can keep playing with your imagination.
Maybe I'm missing the point, but I think that whole idea would really go down like a led balloon with most people.
A lot of modern technical marvels were built on top of the leisurely work done by people who were just playing.
"I wonder what would happen if the square root of -1 existed"
"I wonder what happens if numbers were cyclic"
"Can you actually cross all the bridges of Köningsberg without walking any of them twice"
The impetus to solve these problems was that they were fun to think about, not that they a century or several centuries later would enable us to do harmonic analysis, invent the basis for cryptosystems or spectral graph theory. Which are billion or even trillion dollar inventions.
I don't think this is accurate. For example square root of -1 was motivated by clear applications in mathematics, it was definitely not just idle leisurely math. Most of important pure math has not been leisurely or recreational in any sense in its origin
Before the 19th and 20th century, much of the basis of modern math was entirely leisurely and recreational. Even today that still rings true. I recommend reading Barry Mazur's "Number Theory as Gadfly" to add some context to that claim.
> And -let me get this right- you want society to keep paying you, so you can play with your imagination?
No, pure mathematics is almost never completely recreational. No serious paper gets published in a decent journal without having some motivation and connection to the rest of mathematics.
It is of course valid to ask why we should fund research in pure mathematics, which is not directly related to any "real" applications.
Same question applies to things like niche art, literary studies, history, etc. One answer is that these things are relatively cheap, and our culture would be poor without these things.
With pure maths the advantage is also that it is cheap, and the payoff in applications (if and when they happen) is high.
Note well that the article makes a strong point that any practical benefits of mathematics are, essentially, a side-effect, and even a distraction:
It would be bad enough if the culture were merely ignorant of mathematics, but what is far
worse is that people actually think they do know what math is about— and are apparently under
the gross misconception that mathematics is somehow useful to society! This is already a huge
difference between mathematics and the other arts. Mathematics is viewed by the culture as
some sort of tool for science and technology. Everyone knows that poetry and music are for pure
enjoyment and for uplifting and ennobling the human spirit (hence their virtual elimination from
the public school curriculum) but no, math is important.
Also see the first two stanzas of the dialogue between "SIMPLICIO" and "SALVIATI" that follow the above paragraph. I think, in other words, that the author would say that the "motivation" in pure mathematics papers is more of a pretext to be able to get published and continue one's work unobstructed, than something that the mathematicians themselves really care about.
It is not always clear what will turn out to be useful or not, even in terms of morale. Witness Richard Feynman’s curiosity about a wobbling plate that someone threw in the air, that he credits with solving his burnout with science: https://pubs.aip.org/aapt/ajp/article/75/3/240/1056339/Feynm...
Riemann was 'just playing' with non-Euclidean geometry. Einstein used it as the basis for the modern theory of Relativity. Of course Galileo had 'played' with Relativity three centuries earlier. Now we have GPS navigation systems that depend utterly on the mathematics that they and an unknown number of other 'playful' mathematicians worked out over the centuries, much of it with no obvious immediate benefit to society.
My career was in practical physics and engineering, much of it with immediate effect on the world. Without all the groundwork done by people without any aim other than working out intellectual problems none of my modest achievements would have been possible.
Or was your point that we should pay people only to work on things that we know we need now?
My own work has been dismissed as too theoretical and impractical. I'm just pointing out the reaction that this language of "playing" can elicit in those strata of society that want to see "results". And, more importantly, to the people who do all the work that is needed to keep people like me alive: those who make the food I eat, who keep the trains running, who keep the sewers unclogged, the streets clean, the houses from falling, etc.
There is a certain arrogance in the article above, and I felt it should be pointed out. When we set out on a grand adventure of the mind, we should not forget who it is that's footing the bill.
One of the most surprising things about mathematics is how useful it can turn out to be even when the mathematicians involved don't think that it will ever apply outside of their obscure domain of e.g. boolean logic.
> I have no objection to a term like “right angle” if it is relevant to the problem and makes it easier to discuss. It’s not terminology itself that I object to, it’s pointless unnecessary terminology.
Yet, what about orthogonal, normal, and perpendicular? How does being [vertically] (up)right relate to the angle of y=-|x|? Why line instead of isodi(a)gon(al)? Why parallel instead of paraisodiagonal? Are there abnormal, prostrate or for that matter left angles? Would not this angle of the normal line come before the definition of an angle formed by two straight lines?
Hard disagree with the sentiment of this author. Maths is cool on an aesthetic level AND maths is a tool that people should learn regardless of their interest in it.
If you want to give people a liberal education in maths, fantastic, do it, but not at the expense of Joe Plumber's ability to quickly estimate a job's costs. It's simply not as important, and this idea that we should preserve the spirit of maths aggravates me because of its entitlement.
You're making assumptions about my motive and experience, and have sadly misinterpreted the point I was attempting to make. I am a little disheartened by your interpretation.
My aggravation is not because I have had a bad experience with maths. I had quite a pleasant one actually - despite elementary and secondary school being rather easy. Even in elementary school, though, there was an obvious tension for teachers. There are students who are excelling and need to be extended, and those who were struggling. The issue is, teacher time is limited, and they need to ensure the entire class is on track. This means there has to be a compromise between the two paces.
If you have taught maths, you will be acutely aware that a students performance depends on a number of things, with one of the most important variables being 'do they do enough questions'. This is the source of the 'volume' of tasks that can seem repetitious.
There is another important variable, which is the appropriateness of the questions that the student is studying. This is discussed in the original essay but in a perspective that I think is unproductive, because it emphasises the issue as student autonomy and not as student comprehension. It can be seen on page 23 within the dialogue, where the student could enjoy the subject even more had they led the investigation.
Importantly, you can reason for yourself that a student who is doing well that is doing lots of problems that are too easy will not improve at the same rate as if they solved suitably difficult problems (though admittedly this can depend on whether boredom inspires creative solutions in that particular student). Similarly, a struggling student will not be able to internalise the meaning of solutions to problems that assume too many structures they have not yet seen (though admittedly this can depend on the tenacity of the student and their motivation to understand structures they haven't learnt).
The idea of managing question appropriateness is then a source of attention tension for the teacher. Do you track the expected progress of the class against the strongest students, and give extra attention to the weaker ones? What if the stronger students notice this and act out due to the lack of contact time?
Or conversely, do you only track against the weakest students, and then provide extension activities for the stronger performers? What if the stronger students notice that they get "more work" for doing well?
These are not the only approaches btw, but it is sufficient to service the point that there is an optimisation problem in student engagement.
So maybe yes, it is possibly true that self led learning would service a students interest in the topic, but it presupposes that the gifted student doing well is of greater value than the struggling student reaching a minimum level to enter a majority of study programs they might be interested in. This assumption is wrong, and I won't debate it - if you disagree there is simply too much different between us to hope to reconcile our views.
Importantly, though, I do think gifted students should be able to ALSO pursue their interests, in a tertiary environment which they can elect to participate in. This is the nature of my "liberal" education comment - I don't refer to the political liberal, but the philosophical one.
What math will Joe Plumber use? What he does might just as well be called "Spreadsheet Skills" instead of math.
The closest thing I can think of to doing math is he might divide hours by number of fittings to see how long it took him to do one joint, then multiply that to find how long he thinks it will take now to do N.
I'd still call that spreadsheet and calculator skills rather than anything that actually requires being able to do math, because someone with zero algebra training can figure out those steps and do them.
Whatever the reason is to learn math, it's gotta be something more than just doing what we can already do with free software and $100 used laptops, unless we expect to have to do without tech.
But in that case, there are probably way more important skills one would need if they expect to survive in a tech free world, and I don't know enough to even start commenting on that.
Arithmetic is separate from mathematics and should be taught as such. The spirit of mathematics should be kept because it is part of the human spirit, not because of any entitlement.
Well, that was refreshing :) I found myself mentally cheering Paul on with a bunch of YYEAAAAH!!! YEAH's!!
I picked up a grade school refresher textbook a few months ago - opened up the geometry refresher summary page...the barrage of symbols, letters, horrid cramped type setting..i just closed the book and walked it back to the library...
For a self learner, can anyone recommend anything in the spirit of Mr Lockhart? (I book marked both his Arithmatic and Measurement books). I currently know of:
> "A piece of mathematics is like a poem, and we can ask if it satisfies our aesthetic criteria: Is this argument sound? Does it make sense? Is it simple and elegant? Does it get me closer to the heart of the matter?"
This reminds me of June Huh, the 2022 Fields Medel winner.
He dropped out of high school to focus on writing poetry after becoming bored and exhausted by the routine of constantly studying during his youth.
If you connected with this essay (or, perhaps, if you did not), I highly recommend checking out a book called 99 Variations on a Proof, by Philip Ording. It is an enjoyable, easy to understand journey through the art of making a mathematical argument by way of showing the many forms such an argument can take.
https://www.amazon.com/99-Variations-Proof-Philip-Ording/dp/...