A Mathematician's Lament (2002) [pdf]

Have you seen The Cauchy-Schwartz Masterclass? Is it similar?

Thanks! Macroexpanded:

A Mathematicians Lament [pdf] - https://news.ycombinator.com/item?id=30829704 - March 2022 (3 comments)

A Mathematician's Lament [pdf] - https://news.ycombinator.com/item?id=15385104 - Oct 2017 (1 comment)

A Mathematician’s Lament (2002) [pdf] - https://news.ycombinator.com/item?id=14331752 - May 2017 (27 comments)

A Mathematician’s Lament [pdf] - https://news.ycombinator.com/item?id=14016877 - April 2017 (1 comment)

A Mathematician’s Lament (2002) [pdf] - https://news.ycombinator.com/item?id=8845507 - Jan 2015 (89 comments)

A Mathematician's Lament [pdf] - https://news.ycombinator.com/item?id=6994939 - Jan 2014 (1 comment)

A Mathematician’s Lament (2002) [pdf] - https://news.ycombinator.com/item?id=6187014 - Aug 2013 (119 comments)

A Mathematician’s Lament - https://news.ycombinator.com/item?id=870885 - Oct 2009 (1 comment)

On Math Teaching: Lockhart's Lament - https://news.ycombinator.com/item?id=256176 - July 2008 (21 comments)

Lockhart's Lament: On Mathematics at School - https://news.ycombinator.com/item?id=130499 - March 2008 (20 comments)

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.

There was an interesting discussion about this quote on r/math about a year back: https://old.reddit.com/r/math/comments/obipos/academics_when...

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.

~~

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!"

https://xkcd.com/2117/

> subordinated to the logic of "rationality," instead of desire for beauty

I don't think that those involved in the Renaissance would agree that rationality and beauty are in opposition.

I think that it is not rationality that is the enemy of beauty but a narrow conception of utility.

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.

~~

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.

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.

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.

> 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.

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?

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.

Why would that "liberal" math education come at the expense of basic arithmetic? You are putting up a straw man.

Your aggravation is likely the result of your experience with the current education system, precisely what the author is up against.

The importance of a subject cannot be judged by people ignorant about it. That's entitlement.

How many plumbers feel Like most of the math they did was soulless and irrelevant, but don’t mind or even like that they got to do some art in school?

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.