6 ms·

A Mathematician's Lament (2002) [pdf]

132 points by lisper 3y ago

nahnahg 3y ago
[dead]
chiggsy 3y ago
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.
an_aparallel 3y ago
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: betterexplained.com No BS guide Anything else in this spirit? Does TAOPS qualify?
mkl 3y ago
(2002) Previous discussions: https://news.ycombinator.com/item?id=6187014 https://news.ycombinator.com/item?id=6187014 2013, 119 comments https://news.ycombinator.com/item?id=8845507 https://news.ycombinator.com/item?id=8845507 2015, 89 comments https://news.ycombinator.com/item?id=14331752 https://news.ycombinator.com/item?id=14331752 2017, 27 comments More: https://hn.algolia.com/?q=mathematician%27s+lament https://hn.algolia.com/?q=mathematician%27s+lament It's an important essay and worth reading. The author went on to write the book Measurement: https://www.hup.harvard.edu/catalog.php?isbn=9780674284388 https://www.hup.harvard.edu/catalog.php?isbn=9780674284388
- ngai_aku 3y ago
  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.
  - nohaydeprobleme 3y ago
    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.
jstrieb 3y ago
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. https://www.amazon.com/99-Variations-Proof-Philip-Ording/dp/0691158835 https://www.amazon.com/99-Variations-Proof-Philip-Ording/dp/...
- syntaxfree 3y ago
  Have you seen The Cauchy-Schwartz Masterclass? Is it similar?
  - jstrieb 3y ago
    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!
    - gjm11 3y ago
      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.
  - 3y ago
ngai_aku 3y ago
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
- xboxnolifes 3y ago
  > “‘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.
- Buttons840 3y ago
  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/ https://xkcd.com/2117/
  - macintux 3y ago
    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.
dools 3y ago
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.
- cammikebrown 3y ago
  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.
  - anonymouskimmer 3y ago
    Most. 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.
- nohaydeprobleme 3y ago
  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.
Jason_Protell 3y ago
When I was in high school, I was lucky to take a Pure Mathematics class taught by Paul Lockhart. What a brilliant teacher...
downvotetruth 3y ago
> 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?
bx376 3y ago
> "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.
DiscourseFan 3y ago
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.
- kwhitefoot 3y ago
  > 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.
auggierose 3y ago
That was made into a full book: https://www.ams.org/notices/201304/rnoti-p461.pdf https://www.ams.org/notices/201304/rnoti-p461.pdf
plaguepilled 3y ago
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.
- leroy-is-here 3y ago
  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.
- eternityforest 3y ago
  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.
- hazbot 3y ago
  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?
- Loquebantur 3y ago
  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.
YeGoblynQueenne 3y ago
>> 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.
- beatley 3y ago
  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.
  - ta123456789 3y ago
    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
    - beatley 3y ago
      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.
- mjklin 3y ago

132 points by lisper 3y ago

[dead]

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.

an_aparallel 3y ago

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: betterexplained.com No BS guide Anything else in this spirit? Does TAOPS qualify?

mkl 3y ago

(2002) Previous discussions: https://news.ycombinator.com/item?id=6187014 https://news.ycombinator.com/item?id=6187014 2013, 119 comments https://news.ycombinator.com/item?id=8845507 https://news.ycombinator.com/item?id=8845507 2015, 89 comments https://news.ycombinator.com/item?id=14331752 https://news.ycombinator.com/item?id=14331752 2017, 27 comments More: https://hn.algolia.com/?q=mathematician%27s+lament https://hn.algolia.com/?q=mathematician%27s+lament It's an important essay and worth reading. The author went on to write the book Measurement: https://www.hup.harvard.edu/catalog.php?isbn=9780674284388 https://www.hup.harvard.edu/catalog.php?isbn=9780674284388

ngai_aku 3y ago
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.
- nohaydeprobleme 3y ago
  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.

jstrieb 3y ago

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. https://www.amazon.com/99-Variations-Proof-Philip-Ording/dp/0691158835 https://www.amazon.com/99-Variations-Proof-Philip-Ording/dp/...

syntaxfree 3y ago
Have you seen The Cauchy-Schwartz Masterclass? Is it similar?
- jstrieb 3y ago
  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!
  - gjm11 3y ago
    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.
- 3y ago

ngai_aku 3y ago

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

xboxnolifes 3y ago
> “‘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.
Buttons840 3y ago
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/ https://xkcd.com/2117/
- macintux 3y ago
  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.

dools 3y ago

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.

cammikebrown 3y ago
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.
- anonymouskimmer 3y ago
  Most. 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.
nohaydeprobleme 3y ago
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.

Jason_Protell 3y ago

When I was in high school, I was lucky to take a Pure Mathematics class taught by Paul Lockhart. What a brilliant teacher...

downvotetruth 3y ago

> 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?

bx376 3y ago

> "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.

DiscourseFan 3y ago

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.

kwhitefoot 3y ago
> 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.

auggierose 3y ago

That was made into a full book: https://www.ams.org/notices/201304/rnoti-p461.pdf https://www.ams.org/notices/201304/rnoti-p461.pdf

plaguepilled 3y ago

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.

leroy-is-here 3y ago
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.
eternityforest 3y ago
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.
hazbot 3y ago
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?
Loquebantur 3y ago
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.

YeGoblynQueenne 3y ago

>> 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.

beatley 3y ago
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.
- ta123456789 3y ago
  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
  - beatley 3y ago
    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.
mjklin 3y ago