Proof you can do hard things
blog.nateliason.com
blog.nateliason.com
The question is never asked about any other school subject and only mathematics has to justify itself that way. I had to learn about categories of plants and animals, interpret 20th century literature, learn about events from a thousand years ago, I did presentations on the demographics of European countries, how certain chemicals react and much, much more. I never used any of that knowledge for anything, certainly not in my career or in university.
But somehow mathematics is the one field which needs to justify its own existence? Mathematics needs to bend itself over and "be relevant" so that people will actually learn about it? Why? Why not ask the same of any other subject.
Justifying mathematics is easy, especially such a universally applicable subject as calculus. But I see no reason why it should have to justify itself in any way.
There is no difference between learning a name in history or in biology, or some workflow.
But math reasoning really filters out people in a way nothing else does.
All other topics just require memory and basic reasoning.
Even physics mostly is hard because of math. And philosophy, mostly because of jargon and references. Otherwise, if you break it down, it's not that complicated.
But you can break down a math problem as much as you want, some of them are beyond what you can do comfortably. And a lot of students reach this limit early in their life.
I see this when I play board games with people: there is a threshold of rules and calculation power above which I lose 90% of the players. They just can't enjoy it, because it requires too much effort to play.
It's similar for me and sport. I've been doing exercise all my life, but my brother will always do more, and harder, because there is a barrier after which it's just too painful for me.
Math and athleticism can be trained, but there is a hard ceiling. And even before you reach that ceiling, closing the gap gets more and more expensive for some people, so much the ROI is difficult to justify.
Yes, math is hard and definitely not for everyone, which I think is perfectly fine. But arguing that something has no use to you, because the first time in your life you actually have to engage in complex logical thinking is just lazy. The value of mathematics is independent of how hard it is.
My guess is that answering "what am I ever going to use this for" is easier for other subjects, sometimes almost intuitive. For Maths, it needs to be defined as soon as it starts getting abstract.
"what am I ever going to use this for" is a perfectly valid question. People who are supposed to answer it should do a better job of doing so, writing good curriculum, and overall communication.
How so? What am I ever going to use my skills in analysing 20th century literature for. What am I ever going to use my knowledge about Roman history for? What am I ever going to use my knowledge about the structure of our government for? Certainly no employeer of mine ever cared, my University didn't care, nobody seems to have ever cared, except my teachers.
The question is bad because:
- People ask it, but only of mathematics
- People answer it, as if mathematics alone has to face that challenge
If school is about job training the answer is simply "because it contains the broadest possible techniques to understand the world and is vital in basically any data-based scientific pursuit and a basic understanding is needed for any trade which uses measurements". Somehow I think ancient history does not have such an answer...
You can use the same skills to spot when a news article is trying to manipulate you.
> What am I ever going to use my knowledge about Roman history for?
You can see how empires fall, and vote against those things.
> What am I ever going to use my knowledge about the structure of our government for?
Understanding why things happen around you is the first thing you need if you ever want to try and change something you don't like.
> Certainly no employeer of mine ever cared, my University didn't care, nobody seems to have ever cared, except my teachers.
You might care, though. Or you might not know what you know now without them.
Thankfully you did all those things, so now you can write a well-written letter to the correct person in government, citing the parallel problems with the current education system and the fall of Rome, and stop children being taught this stuff (-:
>Thankfully you did all those things, so now you can write a well-written letter to the correct person in government, citing the parallel problems with the current education system and the fall of Rome, and stop children being taught this stuff (-:
Okay, that was funny.
I think what you are trying to articulate is education as a hollistic system, not with the goal of expertese, but with the goal of offering some moral foundation and grounding in the world and society. But if you believe in that the utilitarian stance of education immediately becomes irrelevant and "when am I ever going to use this" becomes a non-sensical question, which misses the point.
I said see how they fall. Not every way, and not necessarily why, but how.
> then you could just teach that, the historical context would be almost irrelevant
I don't think you can see how they fall without historical context. That is the how.
> even the premise of a cyclical history, which can be broken by voting, seems completely insane
I didn't mention a cyclical history. Some things repeat, but that doesn't mean in general history is cyclical.
I don't mean to comment in a way that's too much on the nose, but I've just broken out various statements of yours and shown how they didn't follow from what I said. Spending time studying literature in a controlled environment, can help train children to read what's being said, as opposed to what they guess is being said. It's not perfect, but it can help a lot.
That’s why you hear discussions about pushing algebra out, but not about pushing arithmetic out.
That’s why some schools replaced algebra with “data science”.
Abstraction isn’t easy.
But that’s precisely why math, especially algebra and the like, is so essential. It’s the only path to building abstract reasoning among students.
I've often heard it asked of languages, specifically Irish.
The main difference for me is that it's relatively easy to "fill in the gaps" for any other subject we learned from the ages of 12-18. Irish however, is niche enough - and learning resources generally technologically behind - that it's still quite tricky to self-teach or "quickly google" answers to things. That makes it unique from most other primary/secondary school subjects which are more universally taught across the world.
I have known many people, to whom mathematical reasoning comes as easily and naturally as flight does to a swallow, be defeated by literature. Every technical university in the world is brimming with bright young people who could effortlessly describe a Jacobian but would rather gnaw off their own hand than produce a response to a story or sonnet.
> And philosophy, mostly because of jargon and reference. Otherwise, if you break it down, it's not that complicated.
Quite the claim.
It's not a matter of the field being too difficult. It's a matter of personality.
> Quite the claim.
Yes. I've discussed this claim with people defending the opposite idea, and I've yet to find anybody making a good demonstration that philosophy is hard in itself.
They point that there is a lot to know. So reference and memory. Like most expertise in the world.
That texts are hard to read. So style, vocabulary. Well, not different than programming or the law on that point. They are easier than higher level math.
They say few people get novel ideas. Yes, creativity is rare, yet it's not particularly linked to the field. You need creativity to find new theorems too.
So they try to bring complicated topics again and again. Once we are done mapping the references, the vocabulary and cleaned out the awful style a lot of writers use to obfuscate their reasoning, the ideas behind are quite straightforward to grasp.
I've stopped doing it now, it takes a very long time, and it's always disappointing. It feels like debugging a program, only to realize it could have been 10 times shorter and easier to read.
> When a student who likes literature does poorly in mathematics, it is because he is not clever enough.
> When a student who likes mathematics does poorly in literature, it is because it is boring.
If you truly believe this I have nothing more to say to you.
I will not respond to your comments on philosophy, though I think they are wrongheaded, because I think your position to discuss such things is made untenable by the obvious contempt in which you hold the humanities.
Well, I've tried sincerely to find out what was difficult about philosophy for many years. Have you done the same with math?
> because I think your position to discuss such things is made untenable by the obvious contempt in which you hold the humanities.
You assume I was a math head, and that I have disdain from humanities.
That's a total projection which tells me you are not being objective in this discussion. So you are right, it's a good thing we stop there.
But for the record I actually like humanities.
Just because you think a topic is harder than another doesn't mean you prefer one.
I don't even particularly like math, and I'm very bad at it.
I'm probably better than the average Joe, but I'm way better at writing than at calculating.
And I enjoy reading philosophy and sociology.
It's ironic that someone in humanities can be so judgmental of its fellow human, and in a very black and white way.
Disagree. I say: Bad math teachers really filter out people in a way nothing else does.
"Here is a list of rules which can be used in specific circumstances. I am not going to tell you why you would need them and the exam will feature several corner cases. Also, I will not provide any examples or detailed explanations because what would other people think of me, then?"
This is over the top, but boils down my experience starting from highschool. I was in love with math when I was in primary and middle school. I really did.
But Highschool and university killed every last bit of enjoyment I used to have.
Okay, I did really like the first 2 semesters of math. I found an amazing online class which covered the same topics we did. Finally I understood and doing the exercises with such ease felt like cheating.
But even with the best teachers for every field, math or sport will filter out people way more harshly, and sooner, than any other field.
You can smooth things out with quality teaching, but there is a human hardware limit, and you reach it sooner with those two.
Many people don't like to hear it because their ego is bruised or because it goes against the idea that "everybody can do anything".
This is wrong. Birds fly, fishes swim. There is nothing bad about either.
I have a hard time with this mentality. I would have agreed a few years ago perhaps, and do still agree that we all have our own aptitudes, but the level of absolution in a statement like this just isn’t really quite as applicable to humans in my opinion. There are things I never thought I’d be capable of that I’ve recently begun to excel at.
What I thought was a hardware limit turned out to just bug in my psycho-software that needed a patch. Obviously at the extreme end’s of the spectrum, the human hardware limit is very real. But for most (healthy) people, engaging in most activities, I believe that any perceived immutable barriers or obstacles are almost entirely psychological.
Laws of physics are laws of physics. And biology is applied physics.
People in sport have known that for a long time. You can train as much as Usain Bolt, he still has a huge advantage.
Thinking the other human characteristics are not affected by this is weird. You will never be Von Neumann, no matter how much you spend in the class room.
It's not that training has no effect, it does, a lot. Yet this doesn't remove the fact we have a variability between individual.
And there is not a tipping point, it's a spectrum for each characteristic.
It turns out math exhibit some early barrier in that spectrum. There is nothing wrong with that.
But there is some number of seconds which 90% of people can be trained to run 100m below.
And it wouldn’t surprise me if it’s a much lower number than you think.
Likewise not 100% of people can develop the mathematical horsepower of Von Neumann. But 90% of people can attain some level of proficiency in higher mathematical concepts… how high a level would that be? Higher than most high schools aim I would suspect.
The vast majority of people are not biologically incapable of making it through high-school level math, or even becoming a very capable mathematician with enough high-quality teaching and dedication.
The difference is not limited to the extremes. Why would it be?
Hence spectrum.
And math meet people early in the spectrum.
EDIT: thread limit reached, so I can't answer down there.
But in short, it's a HN comment, not a study. But I've been in 11 schools in my life. Some private ones with their own church inside, some for special kids, some for poor kids with a higher degree of violence.
I saw it again and again. The average math teacher is not worse than any other teachers. But the first field kids have a hard time is more often math than other fields. In fact, kids that are good at math usually can do ok in any other, even if not great. The reverse is not necessarily true.
At this point I'm just repeating the argument in the other comments.
I can think of several anecdotal counter-examples... students who seemed to have been falling behind, but with the right teacher were able to make significant improvements and become a top performer.
You can also take a look at many of the studies done on the educational gap, to see that better schooling can have a massive impact on secondary and post-secondary educational outcomes. Surely this wouldn't be the case if poor performers had some "biological limit" that prevented them from succeeding in high school math.
It was hard for me to accept, but I know that after many years of University math education there are things I will never be able to understand. I have had very good grades in my Masters, but I realized that I will never be able to do research in pure mathematics.
Almost every person I know struggled with math far more than I did. How would they become research mathematicians if I couldn't after years of study? The belief that secretly almost all people are better at mathematics than me, despite me trying for many years, seems utterly absurd.
Do you think I really am in the bottom ~50% of the population in terms of mathematical ability? Because I know I will never be a pure mathematician.
I don't know nearly enough about your personal background to make any sort of judgement on to what extent your success up through university came from innate "mathematical ability" versus other factors.
But I do know that making it through a Masters-level program makes you a very capable mathematician, even if were unable to start a career in pure mathematics.
You made it further along in your mathematical education than 99.9%+ of students, and I have a very hard time believing that "biological predisposition" is what's stopping many people from making it past high-school level math.
But even if I were, why did the hundreds of people I studied with in school and university fail far more than I did. No, I didn't work especially hard and often times I was explaining things to other students, it weren't the teachers, we all had the same. It also wasn't motivation, I didn't need all that much to succeed. Was it just luck that I outperformed most people given the opportunity?
Shmaybe. But I doubt many people ever approach it. What they hit is, instead, a "software lockout" - a limit of the kind that some hardware vendors (such as those making oscilloscopes or farming machines) put in their products, where the hardware is already there, but its capability is restricted in software, and you need to pay to have the restrictions lifted.
If you'd seen me as a kid, you'd say I am one of those people with rather low "hardware limit" when it comes to math. But then, when I was around 13, that limit suddenly shot up so high and so fast, it may as well have disappeared. What happened? I started to learn to code, with an intent to make videgames. Eventually I reached the point when I was trying to figure out how to rotate some sprites, and made a mental connection between what I want (understanding how to implement rotating images) and what I was taught at school (trigonometry). Immediately, math switched from being something I had to learn, and became something I wanted to learn. So, what you'd think was a severe hardware limit, was actually a purely software one - and it went away the moment I found the magic unlock code (which, in my case, was "find something that makes math seem relevant and exciting, instead of a bullshit chore").
And yes, I'll admit that I do have a hardware limit relevant here - most kids do well at math without having to find a motivational "cheat code", but people like me, with impaired executive functions, can't naturally power through bullshit, and need workarounds to keep up. But this is a different kind of limit. It's also one we're getting increasingly good at understanding, recognizing and overcoming, and it teaches us that what you see as hardware skill limit, may just be a software limit induced by another problem.
Or, in short, if someone told the twelve year old me the things you wrote here today, it would have derailed my life. Fortunately, nobody did, and this lazy and dumb-ish kid ended up handling undergrad-level math perfectly fine.
If not, what makes you confident that your peers were falling behind due to some sort of innate inability, instead of as a result of compounding poor teaching or other environmental factors?
And I also got into teaching between dev gigs.
Unless you can mass edit all the factors, then it remains that some people simply cannot do math past a certain level.
But for what it’s worth, it was a very small school in a VERY poor area. (Median income probably at or below $20k / year) The teaching was actually above-average for the area, but probably below average over a national rating.
I do think factors matter in my personal anecdote of my particular high school. However my LIFE has reinforced my observations: most people are profoundly bad at math, from a seemingly fundamental level.
I told her to come to office hours and I gave her problem sets to solve, first easy then progressively harder. As she got confidence that she could do this her demeanor in class changed and in the space of a month she went from failing to a B-
After this I started asking anyone who said they had math anxiety or did not like math if they had a bad math teacher they remembered. I have asked close to a hundred people this over the last 30+ years and invariably they not only remembered the name of said bad math teacher but also relived how they were told they were dumb and would never be good at math. They stopped learning at that point.
Nonetheless math is hard because it is not an evolutionary survival skill we developed. I say jokingly that math is an unnatural act. I am not a gifted math student but I like solving problems and I would just solve every problem in the book ploddingly and stubbornly. So I dont hate math and I am not in love with it either. It is a very powerful tool for problem solving and also gives you powerful analytical capability you can use in other places in life.
Maybe that is all it takes to be a "gifted" math student?
Tangental:
I have always hated the term gifted. I will not deny there are some absolute phenoms out there in regards to any domain. Those individuals are few and far between and should be celebrated because of it. Maybe I am being semantical, but being "slightly better" does not make one gifted.
Learning a subject that is being taught to someone quicker than their peers does not make them gifted. Show me what they have created or discovered with their alleged "gift," and then we will see.
Von Neumann is not remembered because of how much quicker he learned math than others or how easy the subjects were for him. He is remembered for what he discovered and what he created -- that is what makes one truly gifted in my opinion.
Short story time: I never could memorize various geometry equations, like surface area or volume of basic shapes. Then one day while bored at my part time job, I found a paper and pencil and decided to use what I'd just learned in calculus to see if I could derive one of those equations, by leaving variables in instead of using concrete numbers.
It totally worked, I reinvented the volume of a sphere equation and ever since that day I've never forgotten it because now every part of the equation has meaning. I know why it is the way it is, and it makes sense.
This means loading the entire problem space in your head.
And this is hard.
Learning Math is just like learning a language, the issue is the script/alphabet/symbols used to communicate, its dictionary, its grammar are all complicated its hard for common people to get a grip on it.
Almost all people can follow logical sequence of arguments provided they understand the language in which they are described.
This language so far is the greatest barrier to bringing Math to public. Probably a new teaching method is needed.
Saying math is about the language is like saying programming is about learning syntax. It is entirely irrelevant, the meaning is what is important.
It indeed is, how do you communicate meaning without words/symbols?
Make an attempt to reply to this very comment by any means apart from using words/symbols.
No.
I know people that have been explained repeatedly with absolute patience and love that correlation was not causality, with simple examples. They agree on the moment, but wait a week and they still mix it up again and again.
That's one of the smallest and easiest logical concept to grasp, and they never integrate it.
Practically, in day to day life, mixing correlation and causality served them very well. It's actually often true. But as soon as it's on the news or a graph on the internet, it's over. Because it's abstract.
And the persons are not dumb either: they are very competent in other fields.
You just proved to them correlation doesn't imply causation in some cases, but no amount of examples will be sufficient enough to prove that is the case in all the cases. This is exactly my point.
If they have seen cases where a cause was strong correlated with some data, so as far as they are concerned, at least in that case they were logically able to conclude there was some connection.
To prove your point, you have to in some way prove a total disconnect between the two(correlation and causation), until then you really haven't proven anything substantial to them. Note you will need some mathematical formulation to do it(Symbol manipulation), which is basically learning a new language from grounds up. It could be impossible. Hence my point.
That said, I lean towards the other option: it's considered a waste of time, so no one puts in the effort, so it seems harder than it really should be.
However, I hated math. HATED it. Did so poorly in it through high school I ended up in college in basic algebra. It wasn't until college I found out I loved math and it was only because I got lucky and had a string of amazing professors.
I suspect this is true for other subjects as well. I also hated exercise until I discovered what I liked doing. PE teachers ruined it for me. Runs were terrible because there was no cadence. Push ups were terrible because no one checked form. The entire exercise of pre-college education is a box checking effort to get you ready for the factory. Perhaps we simply need to stop having the unmotivated teach the motivated. This goes true for research professors forced to teach as well.
Are you kidding? Every tech-bro on the planet makes their disdain for literature, philosophy, etc. extremely well known.
But if it is, then my point is that people absolutely do make the same complaint about any subject they don't like being told to learn.
Tech bros (generally) don't like being told to learn about art. And non-tech bros (generally) don't like being told to learn math.
I just said it can be hard to understand something if you are offended by it. In this case it seemed likely, because the obvious point was that different people have different preferences towards subjects and look down on others. For example, some people in tech have a disdain for literature or other "soft" subjects like the humanities.
When you complain about the stereotype and state that it obscures this point suggests that one could be offended by it and it is clouding their judgement. The fact that the OP simplified their point and you still refuse to understand it because it's a "made-up category" supports this notion.
But hey, maybe you aren't offended by the term "techbro". It's perfectly possible that you lack reading comprehension skills. It's also possible that you are being deliberately obtuse and disingenuous
When you say:
> when someone is more interested in being offended
Because I disagree, that is the stereotype I'm talking about. You weren't making an abstract point; you were stereotyping me based on nothing other than my questioning of paulcole's point.
Technically it's true that you never claimed that everyone who disagrees with you is offended, but that's you getting confused: you were commenting on something I said to disagree with someone who isn't you.
> The fact that the OP simplified their point and you still refuse to understand it because it's a "made-up category" supports this notion.
Please quote where the OP simplified their point and I replied?
You didn't reply directly to that but you responded here: https://news.ycombinator.com/item?id=36674076#36712940
Maybe you disagree with this stereotype, but you didn't counter it. You just said it's too hard to understand. It can be hard to understand things when your emotions are clouding your judgement, but that is just one possibility. I listed a couple other possibilities in my previous comment. I guess you can call it a stereotype, I would call it an inference. You just seem butthurt about a fairly innocuous term, because clearly you have the reasoning skills to understand what is being said. Feigning ignorance is not a good look. Best of luck to you!
You keep guessing emotional reasons: "butthurt", "emotions clouding judgment", "being offended".
If this is the way you view the world, and how you perceive motivations that's fine, but please don't pretend that this constant insinuation is reasonable or anything but lazy, or in any way useful to any discussion.
It's purely subtractive, and the discussion would've been better (other than any benefit you got out of talking to someone this way) if you hadn't made these comments.
The other one comment I meant to paste there was the simplified point he made. You couldn't figure out this is the one I meant to share? He only made two comments, I find it hard to believe you couldn't figure out what I was referencing, seems like you are just being disingenuous again. https://news.ycombinator.com/item?id=36674076#36698701
If you want to add something to the conversation, maybe you can explain why the stereotype is completely ungrounded in reality. You would have a tough time doing so, since in other comments you're talking to other tech-inclined people who are downplaying the utility of learning ancient history and literature.
You claiming you don't understand the point paulcole is making at all is ludicrous. None of your comments have added anything at all of value, whether you had an emotional motive behind them or not. You saying his point is hard to understand is just undermining yourself.
Huh? People around me asked that all the time in high school & beyond.
If you cannot justify to children why you force them to sit in school for hours every day you shouldn't be surprised by the consequences. Though weirdly schools tend to be very surprised in my experience. And yes, this question is absolutely asked in other subjects too, but less often as they're not as abstract and usually related to things people encounter every day.
> Justifying mathematics is easy
And thus left as exercise to the reader?
Nobody told me why I should learn about ancient history or 20th century literature.
>usually related to things people encounter every day.
Basically nothing I did in school had anything at all to do with things I encountered every day. I played video games and hung out with friends. I didn't study European art history in my time away from school.
>And thus left as exercise to the reader?
Yes. I assumed that most people on this website understand the inherent value in abstract thiking and broadly applicable problem solving techniques.
This absolutely doesn't match what is thought in elementary and secondary schools.
You are thought how to apply things and not how to think (in an abstract way or not).
If I need to tell you how one of the most important theories in mathematics, with broad applicability to all data-based academic fields and all careers which have to do with manipulating and investigating data, is important, then the entire school system has failed you. And I am not sure what you are doing on a site like this.
>Oh, you mean the kind of abstract thinking and problem solving skills you develop from reading history and literature and applying them to the events of the day?
lol
Did you ever ask?
I assume not, since you claim nobody ever asks that question about ancient history - so why would anyone have told you? If you did ask, doesn't that rather undermine your claim that nobody ever asks this question about other subjects and people only ask it about mathematics?
No. But I also didn't question its value.
Zing! I feel you'd want to know some of your readers get it :)
Math is a language that is required to describe how the world works. It's the official language of certain spheres of society that seriously impact your life. If you don't speak it, other people will make decisions for you that you can't understand.
It's also really useful if you deal with finance, engineering, computers, statistics etc. If you don't speak math, these doors are closed to you.
This would be a lot more evident if they had us solve real world problems with math.
Hard disagree here. When I was in school, there were constant discussions among the teenagers about "why do I need to learn a dead language (irish)", "why do I need to learn french/german, they can all speak english", "why do I need to study geography, I'm never going to need to be able to categorise volcanic vs fold mountains".
I'm convinced that justification doesn't make students more motivated anyway.
That being said, justification is needed, not for the students, but for people designing syllabus. They require arbitrage because there are so many hours in a day.
But the same does also apply, and get called out just as often, regarding the humanities, especially given 99% of people can fulfil their humanities interests for free and at their leisure and on their phone with Wikipedia.
But i had better things to do. I started programming in grade 7, from a book, with a Apple 2 (circa 1982). There were no forums, no Internet just me and a thin booket that came with the computer, plus later, the odd magazine.
It was never "hard", but it was fun. It taught me how to build things, how to approach problems, how to throw something away and make things better. How to imagine.
Ultimately I would study comp Sci- people would teach me the right way to program, and 40 years later it's still my career. As I predicted at the time (somewhat obnoxiously to my then teachers, sorry Mrs Hodge), I had better things to do than raise my Geography score from a C to an A.
School is important, but finding something you enjoy, where "work" feels like "fun" is the truely "gifted child".
But there are lots of professions that can start as teenage interest. A friend of mine was obsessed with birds, he went into nature conservation, started in a bird reserve, and has had a long career climbing the ladder in that space. And also tour-guides bird watchers.
Sure, not everything works out, but if you can turn your fun into your job, then you're one off the lucky ones.
I really wish the world economy would be structured like this, with smaller entities taking more risk and responsibilities and less enormous corporations filled with people hating their lives.
When it all goes tits-up, you just walk away. No responsibilities, no debt. Polish your resume, blame the result on "bad management" and move on to the next gig. Want to go to another country? - quit your job, and buy a ticket. No responsibilities, no debt, no nothing. Just pack your desk and walk away.
By contrast a small business employer (or a one-man-shop) doesn't get to go on vacation, answers the phone day and night, hustles each month just to keep the lights on, risks all their time and assets on uncertain outcomes. They get to work, look at the parking lot, and take on responsibility to make sure all those employees are paid at the end of the month - their families and their creditors are relying on you.
Even in the good times you're the one making decisions, deciding what risks to take, when to spend, when to save. All the while your employees tell you what you're doing wrong. And complain you're getting rich off their labor. While your bank manager refuses you a loan because you're "self employed". And you get to take personal surety for company debts and leases... Oh, and when cash is tight you're first in line to get short-paid.
Who wouldn't want to be a small business owner? It's a barrel of laughs.
That was me 100%. I passed elementary school with flying colors without ever doing any homework or actively participating in class (I had better things do to: reading comics and building LEGO contraptions) and simply continued this approach in highschool, with much, much less success. But neither did I care nor did I have any time for studying or doing homework - I taught myself programming in Batch and BASIC in the 90ies from the MS-DOS manpages, and later, when we had access to the internet, Delfi, Visual Basic and C++. I had finished a small tool with a few 100 users when I was 13, and I had to take care of them and fix bugs. There simply wasn't time left for any school work. University was just a natural extension of this childhood interest into adulthood. It didn't feel like school at all, and there I quickly discovered that math (well, computer science math at least) is actually pretty easy, and all you need is practice.
If somebody says stuff like this, then they do not understand math, imo.
Im a little bit sad whenever somebody argues for math by using "no phone available at the moment" argument.
Math is insanely powerful world modeling tool.
Starting from calculating right amount of fence for your garden, to estimation of 500km route arrival time while taking traffic statistics into the account, to data science, ML, whatever more complex.
Since math modeling is everywhere in "modeling" industries like engineerings, financial-ish jobs and other
Then you basically not only get better tools to operate (model) in real world, day2day life, but also it opens you doors to highly paid careers.
But the goal is not to have fancy jobs, but being able to do real world modeling.
> Since math modeling is everywhere in "modeling" industries like engineerings, financial-ish jobs and othet
In other words, certain career choices, as the OP says.
(Also, while it might not be the tools needed for the average homeowner, there are plenty of optimisation problems similar to "how much fence do I need" which are most easily solved by solving the Euler-Lagrange equations)
While it's always better to work out the numbers if possible, sometimes it isn't possible and people get trapped trying.
In the long run, the mechanics of math (how to do long division, or differentiate an expression, or expand a geometric series) are important only insofar as to help us model and predict and analyze the world, intuitively. However, students who do not intuitively see the power of mathematical intuition as a tool for understanding and modeling the world better, think that they are taught the mechanics of math just for the sake of "no phone/calculator/computer available" circumstances.
The more abstract it gets, the more it warrants an introduction with "here's how it's used in real life".
I can recall the point when it started becoming mechanical - it was when I started doing Derivatives and Integrations. I was 'just solving puzzles' until I hit a chapter, tucked away way back in the syllabus, almost at the end of the 2 year arc - a chapter about Applied Differential Calculus. I still remember the feeling of "Oh... I get it now" euphoria, with a tinge of sadness - why wasn't this covered early on?
Same thing would have happened in Engineering as well, but at the time my teacher was good. They started by explaining the applications of Fourier Transform before we actually got to learn the mechanics of it.
I have the same thing with learning languages: I despise focusing on the rules of conjugation and word order and rote memorizing rules, exceptions to them and exceptions to exceptions. I much prefer to bootstrap with some vocab and then acquire by immersion, even if that means I get the conjugations wrong in the beginning. But I also find myself in the minority by far, to the extent that pretty much all language teachers I ever had (except my high school French teacher - she was awesome letting me translate MC Solaar lyrics for grades) didn't even really seem to believe that there are other ways of learning other than studying rules.
Mathematics is about examining things and understanding their essence - what statements can we make about all such things? How can we prove that? How can we use that to figure out other things?
But nobody ever tells kids that - they think they’re just in ‘advanced counting’ lessons.
I wonder whether a split in curriculum could help - similar to English language/literature. Mathematics needs a ‘numeracy’ program that starts in kindergarten and covers the mechanical ‘how numbers work’ stuff… then a separate mathematics program that starts in middle school and teaches reasoning and proof. Start with geometry.
I forgot all my calculus after high school, had to relearn it in uni and then I promptly forgot again.
Exactly as the article says, it was more about proving we had the capacity (proxy iq test?) to learn it.
You don't need calculus in real life and I think the focus on calculus is ridiculous when we could explore other more practical areas, like category theory (which only my lucky friends who did advanced math got to play with)
I love the wikipedia intro: Calculus is the mathematical study of continous change, the same way geometry is the study of shape and algebra... that's it perfectly. And the most basic application is in everyone's life and also one of the basic physics thing: The relationship between location, speed and acceleration. I find this very essential, vs category theory at least..
Category theory is about connecting the dots between different areas of maths. The "general application" is to allow you to reason over the structure of a problem you're interested in, while throwing away all the superfluous details. It arose when geometers and topologists realised they were working on the same problems, dressed up in different ways. I think the utility for technical people, from this perspective, is pretty clear.
As for the general working person? I think it's just an exercise in learning to do abstractions correctly, which is valuable in any line of work.
There are actually people who advocate that we should base maths education on category theory much earlier (much as New Math was interested in teaching set theory early on, as a foundational topic). CT is an unreasonably effective tool in a large section of pure maths, so this doesn't sound unreasonable to me; it wouldn't be nearly so scary if it were introduced gently much earlier on (in the same way we start to learn about things like induction in the UK in secondary school, long before formalities like ordinals are introduced at uni). Currently only a very specific, highly-specialised section of the population learn CT, but if something like this were to happen, I'm sure we'd see lots of benefits which are hard to identify at the moment.
I guess to try and mirror your calculus example, I'd try and motivate why someone should care about abstraction itself, perhaps with examples like 'calculating my taxes each year is exactly the same problem, except the raw numbers have changed'.
Alternatively it might go over better to say something like: "Imagine you have a map with a bunch of points, and paths which you can walk between them. CT is the study of the paths themselves, the impact of walking down them in various routes: for mathematicians, this means looking at things like turning sentences such as 'think of a number, add 4 to it then divide by 2 then add 6 then subtract 1' into 'think of a number and add 7'. Once you've spotted this shortcut on this silly toy map, you'll recognise the same paths and the same shortcut when you see on your tax form 'take your income, add £400 to it, divide by 2, add £600 and subtract £100"
I've only read pieces of it, but I think this moves in the right direction towards making category theory useful to day-to-day life in non-trivial ways.
I don't even know what "calculus" is really.
I have had plenty of math classes both in high school and later at university but I don't recall any significant distinction that would leave me with some concrete idea of "algebra" vs. "calculus" vs. "whatever" years later.
It's the continuity/limits/integrals stuff.
Also it was never optional. Either in high school or CS at uni.
"Calculus" is the application of that theory without argument. It's an advanced high school class or an early college one. There you'll integrate or differentiate real valued functions for use in optimisation problems or for determining qualitative features of such a function (e.g. where is it flat, where is it defined, etc).
In the US, you can probably pass calculus without writing a proof, but you can't pass mathematical analysis without at least understanding epsilon/delta proofs.
Hey now that I think of it, "mathematical analysis" had continuity, limits, some integrals. And then every mathematically inclined uni specialization had "integral and differential calculations (let's shorten it to calculus)" which was more advanced use of integrals :)
A rose by any other name would smell as sweet, but it may be called a thorn in a different locale.
In Poland we do those in high school. :)
Yes, calculating the right amount of fence is useful, but not only (as someone pointed out) you don't need a lot of math for that, people just take take a ruler on the plan, or count steps in field, going across the entire length, then estimate, and it works.
> estimation of 500km route arrival time while taking traffic statistics into the account
Who does this? How would you convince a friend to learn math in order to do this? What people do is they remember how long a 200 km route took and so they estimate the 2.5× longer path will take 2.5× more time.
We need examples of real world math applications, because such examples are scarce across the Internet.
Who does it? Navigation app in your phone. They modeled this problem using math. While you could probably model it somehow without math, but why would you want to do it other way, when you have reliable, mature and flexible tool
I didnt write that you shouldnt leverage tech.
Do it, yet be aware how it works and why. You can leverage math in other custom scenarios
Let's be honest: advanced math is useful in programming contexts. Everywhere else, in 2023 you're using computer programs that will do the hard math for you.
Math is not about calculators or computers.
It is about modeling (e.g real world)
It is useful whenever non-standard, already modeled in your calculator problem apprears.
The real problem is you can cannot find a language which you don't speak, useful.
The only reason why things like Addition, Subtraction, Multiplication, Division come easily even to people without any education is because it has to become a part of their everyday vocabulary if you have to as little as start a lemonade stand. If Im not wrong Arabs arrived at algebra trying to workout inheritance laws from the Quran. Want to build a home, you can only brute force that much, eventually you are likely to invent the Pythagoras theorem, measurement techniques, myriad other geometrical methods.
All this is possible because if you have to do a thing, anything at all, well, you have to arrive at the most logically consistent formulation of the problem. That helps you to not only do it well, but do it right.
The real question there fore is introducing Math in a way it becomes a part of people vocabulary and general language.
Perhaps he didn’t and isn’t it a part of the problem to apply math?
>Then you basically not only get better tools to operate (model) in real world, day2day life, but also it opens you doors to highly paid careers.
can you elaborate on how “it opens you doors to highly paid careers”?
Depends on the level of proficiency
Examples can be engineering or fancy financial related companies like Jane Street
> outside of certain career choices.
As someone who did a lot of calculus in university, and definitely hit a eureka moment while integrating over vector fields that helped me conceptualise some general day-to-day stuff better in my head, would most people generally benefit from that same conceptual "grokking": of course. Would it be worth the time & effort investment for them if they're not using it in their career: no.
Abstract thinking is really useful during arguments, even about politics, religion, etc.
The "mental block" you're referring to here is a combination of individual aptitude, capacity, access to educational resources & time-/effort-budgeting over a lifetime. This isn't some free casual object people are refusing to collect on their travels, it requires investment.
I was in a shop this weekend where the price per 250g of coffee was displayed, but the woman in front of me only had a 175g container. Neither her, the person serving her, or the other person working in the café knew how much to charge her. It's 175/250 * £PRICE_PER_250.
In supermarkets, prices of items are displayed beside each other - a sharing bar of chocolate is £x/100g, but the multipack is £y/item, and each item is z grams. Which is better value?
Cooking - I have a recipe in a book that serves 2 people, but I'm cooking for 3. How much X do I need?
Historically many problems have also been hard until people figure them out, and then they stop being considered hard problems. In recent years this has been mostly true of AI-related topics.
A lot of people have achieved mastery over really hard problems and synthesized their learnings over countless hours, making the information much more easily accessible for future generations.
If you keep hearing someone talk about how some field is hard you should take that as an opportunity to challenge yourself rather than shy away from it. One field that has recently interested me is organic chemistry, which I'm interested in learning mostly because of how many people I've heard talking about how it's so challenging. May I find a worthy opponent.
Edit: This is relevant to HN when talking about C and C++. People talk about these languages as if they're some magical beasts, but in reality you can get really far with them by treating it as a serious endeavor. People will talk about how they don't have full mastery over the language, but you don't need anything close to that in order to be effective. If you know how to program in other languages you can pick up C++ just as easily and start being effective very quickly. No mastery required. It's not that hard.
Hard problems or domains are unknowns. Working towards solving hard problems involves thinking through unknowns, which may or may not lead to understanding. An aversion to hard problems is an aversion to the unknown.
Back when I was a high school student math (not just calculus - but the entire subject of mathematics) had this reputation as being a "hard" subject as a result scores of my fellow students just decided math is to difficult I'm not going to engage with this.
I suspect this is related to a fear of failure or kids being afraid of "looking dumb" in front of their friends - There was a definite "if I don't try then it doesn't matter if I can't do it." attitude, so they just switched off in those particular classes.
A lot of these attitudes carry forward into adulthood. I'm almost 40 and amongst my generation programming has a similar reputation. People I grew up with think if you can read or write code you are some kind of mystical wizard with powers beyond the understanding of mere mortals.
I see it today at my day job - I work as an engineer (the non software kind). I've seen my coworkers completely baulk at computer code I hear all the same things I heard back in high school. "This is too hard, I can't learn this stuff, I'm not going to bother attempting to understand it".
Fluid Dynamics was a hard subject (in my opinion), Solid Mechanics was challenging a dozen lines of python code is not on the same level.
To me it seems like a lot of the fault was with the curriculum: basically full steam ahead regardless of the class’ understanding. That’s especially bad in math when each chapter uses what the previous taught.
But the point about adult salaried professionals complaining that they supposedly can’t figure something out is disappointingly relatable. I generally believe that most people are "smart" and just don’t tend to bother using their brain as a muscle and that seems to make it doubly irritating to hear such complaints.
I didn't find out until much later (and it still feels like much too late) that I just had an aptitude for learning the subjects traditionally taught in schools in the ways schools normally teach them. I got lucky. Everybody else was just as capable at learning some things in some ways as I was at learning school things in school ways. They just didn't have the luck that their aptitude lined up with what was measured and lauded in childhood like I did.
That meant they all got to learn how to work hard to learn things outside their areas of aptitude in school. I didn't. I didn't realize there were any such things that might matter someday.
I think learning how to learn things that are hard FOR YOU is quite possibly the most important skill in life. The sooner you master it, the better.
Framing some things as "hard" when everything is hard for some people and easy for other people undercuts the more important lesson.
I’d certainly agree it’s extremely valuable. There are some failure modes from taking it too far (like anything).
People tend to be happier when they’re very good at things. You also contribute more to the world. If you’re always doing things that are hard for you, you won’t do as much or as well, really by definition. It’s okay to do things that are easy for you. There are a lot of upsides!
Even if you choose an easy path, there will always be hard parts. So if you want to get anywhere, it’s essential to have practice navigating that. Just consider if that’s where you want to be all the time. I’ve done it myself and seen it in others, where you think you’re always challenging yourself, but you’ve actually just put yourself into a life that’s a bad fit.
Find a niche, do one thing and do it well. You do you.
Or if you want to be a generalist, that's very valid too, but again, you do you.
The constant glorification of growth and the massive industry built around it doesn't help.
You don't really have to learn how to learn things that are easy for you. That's what gets them classified as "easy." But you'll never accomplish anything unless you learn how to work through those inevitable hard parts, and the sooner you can learn that, the better.
I think it's important to learn early enough that things are usually hard when you get far enough into them, and that it's OK, it's not a brick wall, it just takes some effort to keep advancing.
I've found this professionally useful in combating the seductive idea that "because I'm -really- good at one thing, I'm good at everything. "
I've seen the opposite in customers sometimes. Doctors who are very good doctors, have strong feelings about UI (that are objectively just wrong[1].) But because they operate in a culture which treats their word as law, they find it hard to accept that others may have skills in other areas they lack.
If you are an expert in something I recommend including something else in your life to keep you humble. That humility allows you to be a better spouse, parent, and human being. (And ironically a better expert who's able to recognise and adopt an idea or solution that'd better than yours even in your area of expertise.)
We choose to do these these things, not because they are easy, but because they are hard.
[1] think yellow comic sans italic text on blue background levels of wrong.
[2] golf is fun precisely because it is hard. Things that are easy are not fun. The pleasure of 1 perfect shot out of 100 tries is the dopamine that keeps us coming back.
Fighting games are very good for this except they've always invited a certain mental block in people where they'll declare certain things "cheap" and end up offloading a lot of the humbling experience and not take it to heart. This type of scrub mentality exists everywhere but in fighting games it seems to come extra easy to people, perhaps because they have too many potential things they can blame (tier of their character, match-up supposedly being bad, etc.).
I console myself by believing that Tiger Woods is pretty bad at CSS :)
Sadly no. Playing SC2 online weaned me off online competitive games. Because of all the kids calling me a stupid noob ... when I won. The ones beating me were polite.
- ADHD symptoms being mild,
- Having slept well (ties in with former point), and
- Having no imminent major worries at the moment (family/health/financial).
Any of these can make the difference between casually trying to understand quantum mechanics, and crawling under the table because I just can’t make this one rectangle on the screen do what I want.
The easiness or difficulty of a domain or discipline is always in relation to some individual context; and that context includes variables that the learner controls. To the impatient, disinterested or undisciplined, I imagine calculus, learning the kanji, or playing the oboe all seem hard. But to the extent I can marshal patience, curiosity and discipline, the difficult domain becomes just a series of small steps integrated over time. I’m a musician and when a student complains about how hard a piece is, I ask if they can play the first note, then the second. If so, then it’s not hard. Because the process to acquire the whole thing is right there. Yes there are interpretive elements and techniques to be acquired along the way. But nothing is hard unless you are in a great hurry or you don’t really want to do the thing.
> But nothing is hard unless you are in a great hurry or you don’t really want to do the thing.
I got told this many, many times in my life, and it was incredibly frustrating when it was something I really wanted to do. I discovered after 34 years that I have ADHD, which makes a lot of stuff that can eventually become easy/easier with patience and perseverance to in practice be extremely hard.
I'm bringing this up because a lifetime of guilt and shame for not being able to accomplish something when it was deemed easy, that it "just requires some discipline", said by someone else pushed me away from a lot of things I'm interested in but wasn't able to keep motivated to do them after shame set in. It can spiral if you feel inadequate, and if you live with this you feel inadequate and "catching up" a lot of times.
Specifically, one of those things was music. I tried learning instruments when I was younger but the motivation was not in learning the instrument itself, it was music as a whole. I wanted to understand how it worked and how I could create it, not plow through guitar strumming exercises for months and months, then fingering techniques, then be able to play a few songs, and maybe in some years actually start to create something. To me what worked, in my natural branching way of thinking/learning, was to start producing electronic music some 4 years ago. Just some stupidly cacophonic basic loops in the beginning, which pushed my interest to learn the basics of music theory, learning the basics cleared to me a map I could guide myself through skills I was missing: rhythms, harmony, active listening, etc. After I started understanding what skills I needed to achieve what I wanted then it pushed my motivation to learn an instrument, the piano, and then learning the mechanical skills of the instrument made sense.
I bring this up because since I was diagnosed I had multiple conversations with people that suffered through the same as myself: being called undisciplined, inpatient, disinterested when they couldn't muster the motivation to plow through a structured path when it got boring to them. And that is not under my control, ADHD is much more about lacking motivation control than being hyperactive or actually having an "attention deficit", I get obsessed by things I'm interested in (music is an example), it's just that most of the resources to educate oneself on a discipline/domain is not tailored for people who needs to branch out, find pockets of skills that are interesting and motivating to learn, and putting the puzzle back together after acquiring some skills in a haphazard way than the usual structured learning path.
My daily work is 80% C# and 20% Python (to make internal Blender tools for our artists). And I'm really bad at Python. I don't know any of itertools. I don't know zip() besides its name. I don't even use lambda.
The result? My bad code can be easily understood by some of more tech-savvy artists.
for x,y in zip(['a', 'b', 'c'], ['1', '2', '3']):
print(x + y)
>>'a1'
>>'b2'
>>'c3'
Zip was simpler than I thought when I first saw it.
Example 1: "I learned five notes in shape 1 of the minor pentatonic scale. That took a bit of practice, but now I'm able to play a bunch of cool licks. Neat! If I continue this path, who knows what other cool licks I can pull off!".
Example 2: "I learned how to import libraries. My lesson had me register a twillio account. I imported the twillio library into my python script. And I copied some code that'll instruct the library to send me a text message. I don't quite understand these python concepts, but wow, this is really cool; I just got a text message from my computer program. The fact that libraries can give me abilities like these is neat. I can already imagine how I can build some basic automation to leverage them. Who knows what else I can accomplish if I discover more libraries and understand python better to actually build something automated!"
I’ve used this same idea to dig myself out of ruts. When things are fucked up I’ll start paying attention to small things and deliberately “defer” progress on a few bigger things that are harder to do and more costly to fail. Each small win helps build momentum into the next-biggest challenge.
I’ve found this super useful for avoiding “habit destruction” during major life events/travel/moving.
But it will cost you everything if you don't."
Discipline is a muscle. Go Build it. Key is to understand different activities require different muscles.
Be mindful of picking your activities, but dont keep on waiting.
E.g. I know many people who go through bouts of intense fitness or diet fixations, take a lot of well-deserved pride in their discipline, and then hit a major event that temporarily precludes the fixation. They really struggle to get back on the train.
One major factor, IMO, is that they’re daunted by the intensity of what they achieved before. Obviously in fitness there’s a physical component to this, but there’s a significant mental component as well — especially outside of fitness.
Basically all I’m saying is you can (and should) gradually and deliberately dial up your sense of self-efficacy when it inevitably crosses some local minimum due to events outside of your control. You ought to build a self-image that’s robust to occasional and sometimes significant failures.
Goal Mindset = "reach difficult state of fitness"
Process Mindset = "Time to do my gym routine, just with lighter weights bc I'm temporarily weakened"
So I gave up on that. I go with the flow of my brain. When I can, I build and shore up systems that will survive chaos and reduce my cognitive load. When I can't do plan A, hopefully I can manage plan B, C, or D.
This seems to work pretty well. I'm more reliable long-term when I allow myself to be unreliable day-to-day. :)
It's not about a thing being hard. Walking is hard to a paraplegic. It's about overcoming a thing and feeling good about it (instead of external rewards, like a piece of candy or good grades).
The real problem with school is that it replaces empowerment with gamification externalized rewards. You're not learning calc for the sake of understanding the world, you're doing it for a line item on a checklist. That doesn't come with empowerment.
With the mere framing of "you can do [hard thing] to prove you can do hard things" is a bad framing because it could be anything — from doing calc, to bungee jumping, to drinking a gallon of milk (please don't). This framing doesn't actually lead to empowerment (and then self-improvement).
The reason we learn calculus in high school is because it is foundational for many advanced STEM fields, and we will yield better results during university for the small percentage of students who go into those fields by forcing everyone to learn it in high school. Or, moreso, that's a viable justification for learning it today. Had history taken a different shape maybe we would learn something else, or maybe not. But the point is that calculus is not an arbitrary hard thing we learn for arbitrary reasons.
I fear that you might be right about this
To understand how things actually work, you need math, especially calculus. Deep learning? Calculus. Statistics? Calculus. Finance? Calculus. Physics? Calculus. Mech E, robotics, earth science, econ? Calculus.
Second, calculus, like all math, is easy. Like that’s the point, it’s the science of simple things. That math is competitive and presented as a cryptic challenge is beside the point — it is designed to make it possible for anyone to reason for themselves and solve problems. The sense of impatience and criticism around math is totally unwarranted and isn’t good for anyone.
I get kind of bummed when I see schools spending so much creativity and enthusiasm on art and theater. There really is no reason why science should be thought of as judgmental, difficult and painful, while putting on a play is creative, inviting and fun.
The formulas usually come from what you study in school (e.g. the Redlich-Kwong equation in Physical Chemistry to estimate the properties of real/non-ideal gases).
What's neat is that before computers were popular in the early 20th century, these calculations (mostly empirical equations' iterations) had to be done by hand by engineers. So yeah... you can imagine how tedious it was (especially when you have tiny errors due to humans compounding) yet they were still able to build complex things (like factories!).
R is usually used to do data stuffs in my experience. Like, "take in this data from this CSV, and do these manipulations" which may involve math but not often calculus.
calculus is the most basic of mathematical machinery, that it is essentially a requirement to do anything else.
But I would be very skeptical of a professional data scientist that doesn't understand things like derivative, integral, limit on an intuitive level. I don't know how you would understand distributions without that knowledge
if you make a class without calculus, it is essentially just a bag of tricks and surface level understanding
And engineers should usually be using battle-tested models rather than coming up with their own derivations, so to some extent using the calculus day-to-day shouldn’t be necessary in many cases. But it is necessary in order to understand where the models came from. This is what separates engineers from tech-priests.
That’s just my perspective, though, I have a pretty simplistic, algebraic way of looking at math. I could never be a mathematician or see true beauty in math, it is just a bunch of little rules to me.
IMO calculus class is fundamentally just relatively abstract compared to the stuff before it. But once you’ve finished it, your engineering and science classes should be full of practical, reinforcing applications, right?
Teaching tricks without teaching the concepts not only fails in the way you describe (it prevents building ideas further). It also fails to teach anything useful at all. Because barely anyone outside of STEM wil ever have to solve any kind of integral or derivative in their life.
Learning how to actually solve integrals or differential equations was useless other than it teaching me more about calculus and how it is useful. For calculations in practice I will turn to wolfram alpha, but it has some value to understand what wolfram alpha is doing under the hood.
I think a calculus course whose tests defocusses calculations might be very valuable for practically minded people. Knowing to think about derivatives is much more useful in practice than actually being able to calculate a derivative.
I may not be solving differential equations by hand but I'm using knowledge about calculus everytime I reason about our industrial process.
The excel part is probably referring to solvers - where you plug in boundary conditions and spits out a solution. Edit - and excel or R (or Matlab) is what you use in lieu of needing to solve this stuff by hand.
So . . . logic and number bases? Do tell!
lol, typical HN answer. Mathematics is not "easy", it is a niche; some people are good at, some are average, some are bad. Expecting every person to be able to do math is folly. People will fail. People already fail at memorizing math concepts and literally just applying said concept by plugging the numbers around. Some people just don't "get" it, I know I don't "get" probability, but is pretty good at other branches of math like group theory and calculus. To some people, deriving derivatives is basically black magic, but to me it's pretty intuitive.
Thus, if the world economy relies on people being good at probability then I am screwed. Fortunately for me, the world economy somehow relies on people being good at writing texts on a computer to tell it what to do (programming). I personally think programming is piss easy (its the actual problem being solved that is hard, programming is just knowing how to knock a hammer) However, there are people out there that simply can't program, either because they are not interested or not capable. Perhaps they are good at something else that is not entirely marketable? Is it wrong to be that way?
Being humble is one thing, but not realizing one's gift is another.
"If you do not believe that mathematics is simple, it is only because you do not realize how complicated life is." -von Neumann
Very true, but this doesn't change the fact that math is actually simple, but it is generally taught so badly that most students can't "get" it.
I did a bit of private tutoring back when I was in college (and I'm still doing it for my own children), and every person learns in a different way. It is not always easy to find the right way to convey an idea, but once you find it you can see in the student's eyes how it just clicked.
Totally anecdotal, but I once helped someone who "didn't get how percentages work" get a really high (with respect to her previous attempts) GMAT score in maths.
It's the same with calculus and almost everything you mentioned. People can create algorithms, make statistics calculations and financial predictions, build robots, etc. All without any knowledge of calculus of any kind.
The skills of all of those things are based on calculus like surfing is based on physics. Related, but not in the sense of practical application. Knowledge of the math that underlies the math that underlies the thing is neither required nor sufficient for actually doing the thing.
I've been a professional web developer since 2005 and a development manager (who still codes) since 2017. I don't understand the first thing about Calculus or even logarithms. I'm sure if I did, I'd probably be a better developer. I've had employees try to explain to me fairly basic log notation and my eyes just glaze over. It's never impacted my abilities, nor the respect and admiration I get from them as a well-experienced and knowledgable developer, but I can't help but feel ignorant.
I need to go back to the basics and work my way up; I've lost a lot of it. Where do I start? Kahn Academy?
In your case, follow something like Khan academy through the normal grade school programs to pick up where you left off and work backwards on picking up any concepts you're weak on then pursue whatever threads interest you. Wolfram can also help you look up specific things or find necessary formulas if, e.g. you just need the formula for a cone or to know how to integrate sin().
When we say someone has a 6 figure salary, we are counting how many 0’s (10s) to takes to get there.
For memory, we say something has 32 bits and can have 2^32 possible values. It’s more graspable to talk about the “address size” vs the “number of possible values”, especially for things that grow fast (like storage).
I’d suggest starting with your intuitions and slowly translating them to math.
(Without being a shill, I wrote about real world logs here, it may help: https://betterexplained.com/articles/using-logs-in-the-real-...)
One very good place to start is Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry by George F. Simmons (less than 150 pages!).
You might want to check out my book "No Bullshit Guide to Math & Physics," which starts with a high school math review, and goes up to calculus. It's specifically written for adult learners (self contained + lots of practice exercises).
You can see a PDF preview here https://minireference.com/static/excerpts/noBSmathphys_v5_pr...
The concept map from the book is independently useful to check out: https://minireference.com/static/conceptmaps/math_and_physic... And you should also check out this SymPy tutorial https://minireference.com/static/tutorials/sympy_tutorial.pd... which can help you build a bridge between coding skills and math operations.
Just going to register my dislike of this particular trope. Mostly because "certain career choices" is doing a huge amount of work. "Certain career choices" can give you access to a very high income. They can give you access to a deeper understanding of some pretty interesting stuff and put you in a position to accomplish all sorts of amazing things.
People use calculus in the sciences, of course, but also in business, the arts, music, politics, and beyond.
We learn calculus when we're young as a part of expanding ourselves, casting the widest possible net to find that intersection of what we're good and and what we're excited by. And to fill our quiver with as many arrows as possible for when we hit the adult world. And, of course, to build our self-esteem by showing ourselves we can accomplish hard things.
But I would highly recommend against taking the attitude that "you'll probably never use it." That's counter-productive. And the people who "don't use things" are boring...
While I do like mathematics (thus I clearly have no negative feelings about learning calculus), this argument is dubious: the longer I live, the more I realize that the capability to learn complicated scientific stuff (including mathematics) hardly does transfer to other areas of life.
Just consider this: if such skills transferred so well, one would expect that those people who make a steep career in companies have learned lots of insanely hard stuff, often hard mathematical and scientific stuff. The reality is that the people who have such a steep career are rather great "policitians" (in the negative sense), sycophants and self-promoters. On the other hand, learning hard scientific/mathematical stuff nurtures the personality trait of "no bullshitting" and developing less tolerance for claims that clearly cannot be abided by.
Thus: I really like learning hard things, but this personality trait is in my opinion often a career killer.
It’s a little sad that the best value that the OP can impute for learning calculus is masochism — I cannot imagine saying that for anything in the school curriculum that I actually learned/understood. I wonder (in good faith) whether the OP actually even absorbed calculus at all… (i.e. can they solve calculus problems even today, for example?) — if not, they’re not the person who should be making authoritative comments on the usefulness of calculus.
Calculus (Taylor approximations, perturbation modeling, error propagation, significant figures in measurement precision, gradient descent, etc — just off the top of my head) is so deeply embedded in my thinking, that it strongly shapes how I think and amplifies my effectiveness!
I disagree with the OP’s claim so fundamentally — they might as well claim that schools/education focus on literacy for the same reasons.
It's as you say. I didn't absorb it at all. I stuffed it in short-term memory and passed an exam. But that's all I learned. I learned how to stuff things in short-term memory.
I didn't learn calculus.
I'd argue that as long as someone reaches that attitude toward something that they choose, they have lived a good life. And that something doesn't need to be a high school math class. The hard thing could be trying to become a chef when you're 50, or deciding to write your next app in Assembly knowing none at all, or surviving a month in the woods, or going to a foreign country with the intention of learning the language. It has to be hard to make it worthwhile, but it has to be your own to make it valuable as an accomplishment to you, as opposed to something imposed on you which you merely endured. I think this is why a lot of people come out of incredibly hard ordeals in the military with much less personal sense of self-worth than they were sold they would get going in.
[edit: removed a critique. I had misinterpreted the words "errant period" to allude to something other than punctuation. My mistake.]
>> We’ll cry into TikTok over an errant period at the end of a text message.
My brain read that as, "At the end of a text message, we'll cry into TikTok over an errant period." lol
If somebody had told me that calculus is how you transition between dimensions, or that techniques of integration would enable me to generate 3D shapes from 2D lines, I think I would have been much more motivated to progress rapidly in math, and much less discouraged when I hit the "hard parts." Those are the answers I tend to give today when somebody asks me, "why take calculus?" Demystifying it doesn't even have to be a wholly practical explanation, like deriving acceleration from velocity.
Segregating out the "hard stuff" doesn't even necessarily lead to great learning outcomes, either. At my high school, and it seems many others, the honors kids were put on the track leading to calculus while everyone else ended up in a dedicated statistics class. The honors kids were expected to pick up statistics through supplementary assignments in their laboratory science classes, and this same approach carried over into lower-division undergrad. As an adult, I feel like that approach has only given me cause to go back and seek out a firmer grounding in statistics.
But yeah, I can think of so many examples of things that I would have been way more into in school if I understood how they mapped onto the adult world. Statistics is probably at the top of that list though.
The formulas to do integrals aren't important but the concept of integrals is.
Honestly I think we should try to focus on differential equations instead but maybe it's necessary to do calculus first.
It’s absolutely crucial to economics and business, and it is a travesty that it isn’t a required part of lower division curriculum. You cannot grasp micro/macro/applied/business economics without understanding relationships between changing variables.
But the idea that if you know how the rate of change of a thing changes over time, that that gives you enough information to understand it completely? That’s pretty important and deep.
It also underscored the poor teaching methods at my school. I was somewhat vindicated by being the only person in my class to get a 5 on the AP test. I also ended up in a major where I did almost nothing but calculus for undergrad and grad work.
This essay wouldn't have impressed me in middle school; I don't know why it's on our front page.
> But if we avoid hard things
I don't see how you can justify the former by arguing the latter. These two are orthogonal. If I were that teenager, I think what I really would want to ask is that why it has to be calculus instead of some other things that is also hard but with obvious real world application like writing a small 3D game engine.
And my answer to that question is you probably shouldn't if your were in an ideal education system. You would be taught what interesting interactions you could have with the physical world, and be induced to discover calculus or some other math tools that helps you understand how the interactions really work and demonstrates you really need such tools. You're more likely to grasp them when you're driven by curiosity.
I studied maths in uni and while I've not used it much, even as a programmer, I still enjoy doing it. My dad never had much schooling but now that he's retired he actually picked up a few of my books and slowly worked through high school to now undergraduate courses. He's having a lot of fun with it.
(a) brevity - it's a deep topic so the post length can't do it much justice &
(b) naïvety - their extremely oversimplified explainer for the “C students hire A students” trope is that the C students are actually secret A students in other areas. This assumes a utopian world where academic-style assessment of study/effort maps to career progression. C students do well for a range of reasons including-but-not-limited-to: charisma, nepotism, unconscious-bias-hiring on the basis of race/gender/accent/height/hair/appearance, normative mental health characteristics. It's not because of their secret extracurricular study: that's in direct contradiction of the message behind the trope.
Newton had discovered the law of gravitation, but didn’t publish it for a long time because he couldn’t justify how the gravitational effect of a sphere could be the same as if all its mass was concentrated at a point in the center. He had to invent calculus to prove it. Isn’t that amazing?
I know the author is using calculus as an example of a hard thing, but if someone is genuinely following their curiosity, then things do not seem hard. Personally I wouldn’t want to work on hard things just to prove to myself that I can do hard things. I’m happy being lazy.
Math, and proof based Math such as Number Theory and Analysis is definitely in the same league if not for a career in academics.
Doing hard things proves you can do hard things. Where does calculus enter into it?
>The more hard things you push yourself to do, the more competent you will see yourself to be
"The whole problem with the world is that fools and fanatics are always so certain of themselves, and wiser people so full of doubts."
>Most C students are not doing other hard things instead of school. They’re just goofing off, so they end up working for the A student.
This is ridiculous. Author goes on to imply these ("most C students") are on social media and using drugs. I'd put any money on them being much more likely to have been raised by the type of parent who believes their kid needs a reason to want to learn calculus.
Oh, and what is "hard" by the way? Is it doing something you're not good at until you're good at it? Does it need to be valuable from my perspective or that of my parent? Should it take a certain amount of time to become "good" at it? When am I good at it? Can it be a hobby I enjoy even while it's difficult, or must it be a chore? What if I'm forced to overcome some hurdle in my life for example growing up with a foolish parent with warped worldviews?
This doesn't explain why Calculus over any other number of hard things.
I would also be cautious about setting yourself up for a hard life. The takeaway resonates well with the HN crowd, myself included, because we like challenging ourselves. There's a lot of people out there who are simply looking to satisfice their lives, and you'll need another way to motivate them to learn calculus.
Both changed everything in an epistemologically qualitative manner. Life before calculus/microwave, and life after calculus/microwave.
How did we warm up leftovers before the microwave?
What did the world look like before I learned calculus?
And no I'm not kidding, a couple years ago when I was studying distributed systems, I saw the CAP theorem everywhere. Why isn't distributed systems part of the high school curriculum? It's used in basically ALL computing devices (before cloud computing, distributed system theories were applied in multicore systems...)
In any case, IMHO best advice to young people - try to learn hardest things you can. Later, there will be less time, less energy and more distractions.
> I don’t particularly care what grades my kids get once they start school. But I do care that they consistently prove to themselves they can do hard things. If Calculus is how they want to do it, fine, but there are many, many more options.
Separately, as someone just recently learning calculus as an adult now I'm digging the obliviousness of the "math is just inherently easy" folks. I get where y'all are coming from but there's a deep lack of empathy there. Go teach some people something (anything) and learn that other people's brains don't necessarily contain your knowledge.
Concerning the empathy point: I do believe that to become good in mathematics, you seriously have to believe that everything in mathematics is actually easy, and you either don't work hard enough (so you fail understanding it; solution: work harder) or you are too stupid to understand the material (which means that you should better work even harder to understand the material to compensate for this weakness).
It's a different story for education up to high school. School is pretty much mandatory these days, and basically everyone has to go through it. It's a worthy question to ask why learn X instead of Y, if X has little real application? Why glorify those who can do maths well, as opposed to those who can best remember the names of Pokemon? Or those who can play LoL the best?
Who defines which areas of knowledge counts as "education", and which areas are frivolous trivia?
In the case of maths, the mere fact that it has tonnes of applications in many technical subjects is sufficient justification for me, but if you reject that line of reasoning, I'm not sure you can convince me (or anyone with critical thinking) that it's more worthy to pursue knowledge in maths than knowledge in "Pokemon studies".
The latter is something you see creep in as you get older and it's something that's always attracted me to hobbies that require you to keep your nerve like skiing, scuba diving and flying planes. There's nothing like having to land a plane solo with a bit of a crosswind to give you confidence in yourself.
Some things are vaguely impossible, like finding a solution to fluid mechanics, or finding a theory that combines quantum mechanics and general relativity.
Others are proven to be impossible under some conditions like the halting problem.
Others still require access to some amount of resources, or some amount of leverage, like buying a $100M yacht.
Lots of these hard things are quite unlikely for me to be able to do.
I think most people refer to the resource one? And that some people have more access to resources than they think, and the scales of resources needed are smaller than assumed?
Many people (especially those in a lucky environment) can just try to draw realistically (which is actually not that hard, given a good teacher/instruction), study a “hard” topic in philosophy, get good at playing a “hard” song on an instrument or even learn how to prove mathematical statements or how to model and analyze complex systems. We can just try, there is nothing at stake except our time and energy.
When something is really interesting to me, I try not to stop myself from pursuing it just because it is considered “hard”. I have many mental issues (executive dysfunction, bad memory, etc.) but I am also stubborn and curious enough to try anything, again and again if I have to, which I never regretted. It doesn’t hurt and it can be so much fun with a humble and relaxed attitude and especially if I don’t try to compare myself to others, which might be one of the main things that is holding most of us back.
Heh, I don't know about that.
I got into motorcycles because of how dangerous they are. When I was just learning dirt riding I specifically picked out an old barely used logging road to do so. It's 2 hours from the nearest town, about 40 miles long winding road with a sheer drop on either side.
Getting there is tiring in of itself. Every time I started it, I was already fatigued from the ride out and I just go straight in without a break. But there's very little room of error. There's no cellular service, traffic is almost non existent. I don't tell my family when I got out there or where I am exactly. If I overcook a corner, loose focus for more then a second, fail to spot where the road's been eroded out, make just one little fatigued induced mistake... that's it. There's no getting out. There's no help coming. They won't even have my body to bury.
Finding the strength to crank the throttle wide open and hold it open right until the last millisecond before disaster is hard as hell when every instinct is screaming to slow down.
But in many ways, it's easier then trying to make friends or ask a someone out on a date. If and when I do make that fatal mistake, I don't have live with the knowledge that I screwed up a relationship or have to live with the regret what may or may not have not have been. Or worse; disappointing those who's opinion has weight.
Dying is easy. Living with the consequences is hard.
You will see that math plays a very fundamental role in our reality, and once you start seeing it in such a manner it may begin to interest you.
I entered college without a rock solid foundation in mathematics and it made things much more difficult.
Right. Tell that to my partner, who has been nursing our newborn every two hours for the last 5 months. There is no way to wake up 2-4 times every night for months and not be tired.
Weight lifting would be zero help with that.
But perhaps my FU-mode is stronger than other peoples. Someone on here said I'd never work for a FAANG company, so I guess that's something I'll have to do just because. Not impossible.
Most people pass a class because they have to for their degree.
Most people raise a child while being sleep deprived because they frankly have to.
When people want to do something, they don’t need to prove to themselves that they can do hard things because difficulty hardly matters to one with their mind set on something. For example anyone who decides to run a marathon one day.
Instead for the things you have to do, one could reframe the “have to” with a “get to”. Gratitude is empowering. Not everyone gets to go to college. Not everyone gets the opportunity of being a parent. Etc.
Proving to yourself or anyone else that "you can do hard things" since you did more or less math in school/collage/university will leave you trainwreck at the first real hard thing that bumps your way.
And why some people don't do the math? I guess, because they are told its boring and/or hard, but they should do it anyway. And people don't like to be told what to do.
The harder thing was keeping the pressure up to pass the tests and in total, pass a course, while dealing with the severe discomforting exhaustion of coming in early in the morning for classes, having to understand rather complex abstract notions. I still cannot think when tired or even inebriated. But going through that was really to get that good pencil pushing job reserved for college graduates, no? :D
Instead, in the end, I like what I do and if the spirit moves me or need it for something, I would sharpen those calculus skills.
I suspect I could have passed it with better teaching. I hated that I had to memorize things, which felt tedious, not hard in a good way. If you dont memorize cos,sin,tan stuff you can't take tests fast enough. The class was just how good can you memorize things. I also hated the proofs, pages of proof. No idea why, and the teacher didn't speak english well or communicate well in general
Sure most people can get a long way without understanding anything under the hood but I think I am better developer knowing assembly and architecture. The same for ML, you can get by most of the time. You can also make a whole career not doing hard things. That's not for everyone.
It's all bits and bytes under the hood, which in turn is just electric signals under the hood, and ultimately (hopefully) quantum mechanics under the hood.
Are you an expert in all of these? Why arbitrarily stop at the calculus layer?
The same with the underlying physics. I have never had any use for that and I am not sure why they taught it hat first up. Interesting all the same
Doing random hard things in order to get into college or get a white-collar job is really no different than pointless test preparation that shows you can follow instructions, and it has produced strange hierarchies such as Qing-era mandarins.
Much better to find something that intrinsically motivates you to do hard things that feel less hard to you. It'll take you further. Or you can prove to the world that you are capable of a yearlong mindless grind...
Basically: https://www.smbc-comics.com/comic/why-i-couldn39t-be-a-math-...
I reckon there are plenty of PhDs earning less than $100k around the world, who know calculus and matrix algebra like their ABCs.
The point of the article is that calculus is not taught because it might be incredibly useful for a small percentage of students, but because it measures their ability to pick up a hard subject and ace it.
(As an aside, the idea that calculus is hard is a pedagogical failure. Calculus is easy to learn if it's taught well. Most of our education systems actually make learning much more difficult than it is.)
My current explanation for this is more along the lines of being well-rounded. By being exposed to a number of different skills and learnings throughout your education, your more likely to be able to connect the dots in other areas - and just hold better conversations with people.
Don't need to ace all your classes for that, but put in the effort to try and structurally understand whatever it is you're coming across.
The brain is like a muscle, if you don't train it, it wont grow AND you will basically be STUPID. That is it.
While I do love to solve such problems, in many business areas there are hardly any hard problems to solve as part of your job: either because they don't exist, or because hardly any boss would be willing to let you work focusedly for years to potentially solve one the hard problems that do exist (which is what a PhD in mathematics or physics is about).
Although I'd argue that just having the ability to press on in the face of challenges is a closely related and widely useful skill.
The topic itself really doesn't matter that much -- which is also good, because then you can freely choose it to your liking, if possible.
TL;DR: I think this is very helpful advice to people who question stuff.