I really despise all the "why you should care about math" takes.
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.
> math reasoning really filter out people in a way nothing else does.
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.
> You can smooth things out with quality teaching, but there is a human hardware limit, and you reach it sooner with those two.
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.
> 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.
> 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.
Nobody is claiming that 100% of people can be trained to run a 100m in 9.58s.
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.
There's a big difference between edge cases like Usain Bolt and Von Neumann, and the barriers that most people actually run into.
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 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.
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.
> Do you think I really am in the ~50% of the population in terms of mathematical ability?
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.
I am not a mathematician at all and I have never been. My contributions to mathematics are exactly zero.
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?
Just using the extremum to show there is a difference.
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.
What evidence do you have that the biological limitations for mathematical ability meets people early in the spectrum?
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.
I wasn't the best math student in school and there was a lot I didn't fully understand back then, but I eventually learned it all and more as I got older. I know I'm not the only one. It seems more like a function of time and effort than a "human hardware limit".
Different humans (in the 90th percentile at least, to eliminate the extreme outliers) are not equipped so vastly differently for mathematics or athletic activity as fish and birds. Thinking that some people can fly and some can swim and that’s just that seems very reductive.
Very interesting tgat you mention bad math teachers. I agree. During my teaching assistant stint during graduate studies in applied and pure math, I had one student who was failing. Talking to her I learned that a math teacher in high school had made her feel very stupid - she stopped learning. This happens more often than is recognized especially to girls but not just to girls.
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.
> I like solving problems and I would just solve every problem in the book ploddingly and stubbornly.
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.
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?
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.
Perhaps teaching math in an accessible way is the real challenge. When I was a kid...a looooooong time ago...you were greeted with "Math Problems" on day one. Perhaps a better word than "problem" would attract more devotees.
> math reasoning really filter out people in a way nothing else does.
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.
They are not defeated by literature, they are just not that interested in it, and therefore never practiced, and dislike working on it.
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.
I can second that. I was an pretty good programmer as a kid. I'd go home and write whatever I could imagine on the old 286 box my parents had. I spent time in "gifted" programs at school, etc.
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.
Math as a field of study might be hard but in my experience math pre grad-school is just memorization as much as any other subject. I studied for my exams by simply doing countless exercises and memorizing every problem pattern to the point where I'm just plugging numbers into a series of formulas, even if I don't fully know why I'm doing so.
This is my experience as well. And unfortunately, many of my teachers feed into this. "Ok kids, today we're going to learn this new formula. You put a number in, do some arithmetic, and you get an answer out. I am not going to bother explaining the importance of this formula, how it came into existence, nor its practical application — likely because even I don't know. But what I do know is it's important because everyone says it is and it will be on the exam. Now get cracking." A parody of course, but I say captures the sentiment which completely drained my desire to learn math. I got good grades, but only after I learned to accept that it is futile to learn the importance of what I am learning and instead simply focus on rote memorization of solving the problem, even if I don't know why it works.
> Ok kids, today we're going to learn this new formula. You put a number in, do some arithmetic, and you get an answer out. I am not going to bother explaining the importance of this formula, how it came into existence
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 gets to the heart of what math class needs. Which is a process of learning how to re-invent formulas, constants, and other concepts. Through this, students will come to appreciate why such things exist. And they will appreciate it further if students are first challenged to solve such problems in the absence of it.
There is truth in this, but even then, to apply the formulas to many problems, you need to have seriously strong pattern recognition abilities, and the capacity and flexibility to adapt what you know to the given situation. Then piping the right thing into the other right things requires a deeper understanding, not always of what the thing very nature, but at least what it does, and what's it compatible with.
This means loading the entire problem space in your head.
>>But math reasoning really filters out people in a way nothing else does.
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.
> Almost all people can follow logical sequence of arguments provided they understand the language in which they are described.
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.
>>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.
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.
What I specifically dislike is the question "what am I ever going to use this for" or versions of it. Which I only ever hear about mathematics, but which can be asked about every other subject as well.
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.
I don’t think it’s the abstraction, I think it’s the perceived lack of usefulness. I’ve known many individuals who are bad at math but can understand the abstraction of most programming constructs. I think it’s mainly because the pay off of learning programming has a higher immediate value (it can be used for a variety of tasks and can even help score a high paying job).
I feel your complaint is valid, but I your ire should be directed towards people who are supposed to answer this question, not the people who are asking it.
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.
>My guess is that answering "what am I ever going to use this for" is easier for other subjects, sometimes almost intuitive.
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...
> What am I ever going to use my skills in analysing 20th century literature for.
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 (-:
You are completely missing the point here. If you know "why empires fall" (nobody does) then you could just teach that, the historical context would be almost irrelevant and even the premise of a cyclical history, which can be broken by voting, seems completely insane.
>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.
I always disliked Irish (and languages, in general) in school. However, now that I'm over a decade out of college I find it to be one of the few subjects I wish I'd paid more attention to when I'd had the chance... Sure, it's not particularly useful in adult life, but neither is organic chemistry, for most folks.
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 asked that question in school (strange to look back on, as I'm now fascinated by languages), and the only answer the teacher could give was that doing well in Irish would give extra Leaving Cert points. That struck me at the time as depressing.
There's a major chicken and egg problem here—is math hard and therefore it has to justify itself, or is it hard because people don't think it's relevant and so they care less to try?
Because why put in the effort if it isn't going to bring any benefit? An easy subject can be swallowed even if you don't see the purpose because it doesn't cost you much.
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.
> I see no reason why it should have to justify itself in any way
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.
>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.
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.
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? Yeah, I totally understood that. Still not sure where Calculus fits in, though.
>Yeah, I totally understood that. Still not sure where Calculus fits in, though.
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?
> "Nobody told me why I should learn about ancient history or 20th century literature."
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?
This drove me nuts when I was a student. No one gave me a solid reason, even though it was right there.
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.
Same. For K-12, at least, I believe the reason why no one could give a solid reason is because the teachers who are teaching the subject have literally no professional industrial experience with math. They have nothing to share with no real-world insights. I presume this should come to no surprise. According to GTP, the average salary range of a mathematician in the US is, "$80,000 to $100,000, while more experienced mathematicians can earn up to $150,000 or more." With salaries like these, no one working professionally as a mathematician would choose to downgrade to become a teacher, with teacher pay, lower respect and deal with uninterested, rude, problematic students.
> The question is never asked about any other school subject and only mathematics has to justify itself that way
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".
Another stereotype. I'm not offended by people substituting lazy stereotypes for thinking. I just think it comes out badly when they try and articulate points.
What stereotype do you think I am referencing? Their point was so simple I am honestly baffled at how anyone could possibly be confused or genuinely find it inarticulate.
That's not what a stereotype is. And I didn't even make the claim that everyone who disagrees with me is offended.
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
> And I didn't even make the claim that everyone who disagrees with me is offended
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?
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 pasted the same link twice, so I don't know what you mean. However I can guess that you've realised I didn't respond badly to a clarification, but don't want to retract what you said.
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.
That's a poor guess mate, since the link I posted twice was your own response. I have nothing to retract. I guessed other reasons too in previous comments, like maybe you are being disingenuous.
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.
Teachers of calculus should have to justify why people should learn it, given 99% of them will never use it, and could be taught something else challenging that they will use instead.
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.
> somehow mathematics is the one field which needs to justify its own existence
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.
This resonates a lot with me. At school I coasted, ignored all home work, never studied, and was a consistent C. I was continually told I could fo better by putting in some work (an objective viewpoint I agreed with.)
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".
Yeah, it didn't hurt that my interest co-incided with history.
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.
If you escape the employee mentality you can turn almost anything into a small business which provides value to the world.
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.
The "employee mentality" as you put it has a lot of upsides. You get paid every month. You don't have to worry all the time about making payroll. Your day starts at 9, and ends at 5, what happens outside those hours is not your problem.
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.
Selling is the hard part of this world view. Most people don't have a well-suited psychological framework and it's hard to develop one. It's difficult to tell if it's even possible for so many people to become minimally good at selling to support themselves financially through a small venture.
> At school I coasted, ignored all home work, never studied, and was a consistent C. I was continually told I could fo better by putting in some work (an objective viewpoint I agreed with.)
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.
This reminds me of a powerful post I read a while ago with the charter to "Half-ass [your projects] with everything you've got!". Pre-commit to what level of quality you're looking to achieve and then strive to do just what is necessary to get there: https://mindingourway.com/half-assing-it-with-everything-you...
>You know they will never use it in adulthood, outside of certain career choices.
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.
IME a big part of this misdirection stems from school focusing on the mechanics of math, rather than the intuition of math.
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.
One of the problems I find with mathematics education is that we seamlessly morph the introductory numeracy lessons of ‘counting’ and ‘arithmetic’ into the subject of ‘mathematics’, without ever stopping and telling kids ‘okay, now you can count, we’re going to start doing something different called ‘mathematics’.’
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 can only speak for myself here, but my journey with Maths did start with concrete, practical relationship between concepts and usage. I still remember fascinated by Geometry and Trigonometry, because how it can be used to create graphics and video games is obvious.
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've never taught mathematics but I think it's much harder to teach the intuition, and I also think that more people are capable of learning the mechanics than the intuition. So from the point of view of someone designing a methodology and curriculum - you have to design for scale.
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.
Wholeheartedly agree about bootstrapping with vocab and learning through immersion. Eventually all the nuances with grammar and rules get understood implicitly with enough practice (at least to the extent I'm satisfied with). For me, it makes language learning a lot more fun and palatable
You've given one day-to-day example (garden fence) which requires only extremely elementary math, and then listed a bunch of stuff related to specialised career paths... which was the OP's original qualifier:
> 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.
There is no license required for using math. Yea, if you dont want, then you will not use math. But, if you dont have some mental block, then you may find ways to apply math to your problems.
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.
It does indeed require investment of mental effort. But that's it. I'd argue maths is one of the cheapest monetarily skills it is possible to pick up - there have been several great contributions from people with little or no formal schooling. It's not like it's available only to a select few, I think pretty much everyone that gets deep enough in to be able to apply it continues to do so throughout their life. It can be used to fill in for all sorts of problems. I've been a tutor and ive long been a strong advocate for yes you can do it. I've met so many diverse people with excellent spacial reasoning that for whatever reason decided they weren't smart enough or couldn't understand it, yet when given a zealous advocate to encourage them picked it up just as quickly as any stereotypical maths genius
I'm using math(s) a lot, trigonometry, matrices, algebra… But most of my friend, family, I couldn't convince to learn math just based on the argumentation they can use it in practice.
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 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.
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
So what I'm hearing is that math is only good when there is no GPS / no phone.
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.
Actually everybody needs it, they just outsource it to somebody who can do it better than them.
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.
This makes the same mistake being called out in the comment you're replying to. The point isn't about the mechanics of solving a differential equation, it's about gathering the intuition about a way of approaching problems.
(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)
Interesting I've seen the opposite in highly trained mathematicians - they can really struggle with taking an intuitive leap without the safety net of the working out the numbers.
While it's always better to work out the numbers if possible, sometimes it isn't possible and people get trapped trying.
They're not everyday things for a lot of people though.
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?
I don't get why calculus is always brought as an example.
It wasn't particularly hard, the entire class had to learn it in high school. We all had it (in a slightly harder shape) in every university course (no matter how detached from what we actually needed)
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)
But what is the general application of category theory, outside computer science.. and even there the average programmer who hasn't some type theory experience will stare at you with huge eyes when you mention it..
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.
Don't need to convince me ;), but I mean this is for the very average person who argues like "why do I need more math than adding two numbers"... and even just "allow you to reason over the structure of a problem you're interested in, while throwing away all the superfluous details" and "learning to do abstractions correctly" would not seem more approachable to them than the calculus description I gave above? imo. Is there a simple real-life problem or model everyone should know or has touched in school this can relate to? I'd be curious, because as you frame it I may even need to revisit my faded memory, hm.
I honestly don't know. This might be a skill issue on my part (very much not an educator), but I think of it as a language for thinking about structural abstraction, so to me the question is akin to "is there a real-life problem that German relates to?"; I can certainly think of lots of problems that would be made much much easier by understanding the language (e.g., getting around Germany, i.e. noticing abstractions), but it's tough to point to anything for this explicit question other than "conversing with someone in German".
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 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.
In the US we'd reserve "mathematical analysis" (or more specifically, Real analysis) to the college level classes which involve writing proofs about the continuity of functions between sets of real numbers. You'd probably end up with a lecture on the mean value theorem here, and leave with the ability to prove it, among other things
"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.
I'm pretty sure we did proofs in high school. But that was a while ago, don't know what they do now.
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 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.
I hate the framing of problems or domains as hard, since it kept me from pursuing them further for many years. And the years later when I tried my hand at those problems, I found that it wasn't nearly as hard as it was being made out to be.
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.
I hate the framing of problems and domains as hard because I found it so confusing when they turned out to be easy for me. It gave me a false sense of competence and superiority. Because I found calculus (and everything else in school people said was hard) easy, I thought I was just exceptionally capable and other people were incompetent and/or stupid.
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 think learning how to learn things that are hard FOR YOU is quite possibly the most important skill in life.
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.
I agree that it wouldn't be a good idea to spend your life focused on things that are hard for you, but since, as you said, there will be hard parts no matter what you do, learning how to learn the things that are hard is the critical skill.
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.
Ah, the trap of the gifted student! When everything other students find somehow hard is easy for you, you get a delusion that nothing is hard, and that putting in some effort is not necessary. A rude awakening may come at high school, or at college. Those who kept toiling just keep toiling, and overtake you, because.you're not used to pushing through.
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.
This extends into life as well. I play golf (for fun [2]), which cunningly has a scoring system that tells me I'm objectively rubbish, and getting (mostly) worse.
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.
I've found that one of the best ways to dismantle an overblown ego is to play StarCraft (Brood War or SC2, the result will be the same) or really any other 1v1 game. The direct feeling of being beaten, potentially overrun completely, time and time again, especially after putting in hard work is very humbling.
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've found that one of the best ways to dismantle an overblown ego is to play StarCraft (Brood War or SC2, the result will be the same) or really any other 1v1 game.
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.
Golf might be better because its me against me. No excuses about the other guys ability. But sure, as long as you find your fun, and your humility, it doesn't really matter where it is.
I console myself by believing that Tiger Woods is pretty bad at CSS :)
I was always told that at the next level of school, I would finally meet a challenge. I got through a master's degree without its happening, and unfortunately that just drove the initial point deeper, that I was somehow fundamentally different from other people rather than just very luckily in an environment exceptionally well-suited to my aptitude (even though the rest of the world is not).
For me, being able to do hard things is not something I learned as a kid. It’s a sum of:
- 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.
I've seen the enthusiasm for any subject wither away when a child or teen is told a particular subject is hard, whether that be math, a programming language, or learning a musical instrument. I was this way. In their totality, yes, the subject is hard. But what they aren't taught about any difficult subject matter is that they are achievable by breaking them up into a series of small, easier to understand concepts. Their practical utility grows as the number of these small steps are achieved. And as they are achieved, mini demonstrations of their use should be performed so the student understands the importance and gets exited to continue.
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!"
Rightly on Wrongly some areas of study have the reputation/stigma of being difficult attached to them.
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.
> math (not just calculus - but the entire subject of mathematics) had this reputation as being a "hard" subject
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.
> 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.
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'
Usually you just use it to group two items you're iterating through that are the same length. You CAN do items of different lengths but then when one gets used up the rest of the other get tossed IIRC. Can use it in list comprehensions as well of course.
Zip was simpler than I thought when I first saw it.
> I hate the framing of problems or domains as hard
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.
> 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
> 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.
It is useful to use the words hard and easy. As you mention, changing perspective around these concepts is the crux.
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.
Self-image really is important, especially on the dimension of self-efficacy. There are compounding effects in both the positive and negative direction though.
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.
And yet the Art of War tells you to attack when you are ready and your opponent is not. There are some problems that really need to "stew". Others require immediate action even when you aren't ready. It's very hard to tell the difference, but it's wrong to assert that all problems require immediate action (although I agree that if you're going to err, do so on the side of action, in general)
I am discussing a way to be disciplined, which is with decent flexion deliberately built into your self-image.
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.
This is exactly what I do. "Discipline" in the traditional sense isn't an option. I have a severe impairment with executive function and can't do specific things at specific times.
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. :)
I like the idea that "If you can master these topics, imagine what other topics you could master if you put your mind to it" — again, for the empowerment.
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).
There are infinitely many hard things. It is hard to learn Japanese. We don't require that every high school student attain basic proficiency in it though.
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.
How does a kid know that they definitely will/will not be going into a STEM field in the future? Is it better to have your school-going years slightly marred by calculus and not particularly need it after, or want to go into STEM later in life but not have the grounding necessary?
> 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
I like the idea, but I’m gonna say that (a) calculus is more than a good challenge and (b) math is actually easy.
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.
>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.
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.
> Being humble is one thing, but not realizing one's gift is another.
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.
I've found that some disciplines need calculus, others don't. I've found deeply understanding logic and binary/hex MUCH more important than calculus in my career.
Physics and gravity are behind any kind of predictable motion. But you don't need to understand those things at all to be a successful surfer. Even though surfing is entirely about physics and calculable predictions, performing the act doesn't require any detailed knowledge of either topic.
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.
Putting on a play is a very human interactive activity. Until math lets you interact with many humans in a rich experience, it won't be put on a pedestal like drama.
I saw an article where they asked a bunch of scientists and engineers if they actually used calculus in their professions. None of them did. They used Excel and the R programming language.
Calculus is needed to understand, like, basic sophomore and junior year engineering stuff. You can maybe pass the tests without understanding where the models came from if your professors are just really lazy and only give you canned problems, but that isn’t a goal we should strive for.
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.
I wonder if there’s a way to teach Calculus in a more practical way. I use the principles of Calculus constantly in many aspects of my job, but I don’t think I’ve ever done say, integration by parts professionally. Understanding deeply what an integral and a derivative are is a very useful skill though.
As an undergrad I tutored folks in a sort of “calculus for non-STEM students” class that focused more on practical stuff and applications… and TBH I think trying to shield them from the complexity didn’t really do them any favors. I often found that they had some practical algorithm memorized step-by-step and some graphical or conceptual intuition, but when the steps and intuition betrayed them we’d end up spending a while backtracking to find where they lost track of the concrete rules, and then we could work our way forward to catch up to their intuition. Practical results are good guideposts but don’t replace full understanding I think.
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?
Isn't the actual calculations of e.g. integrals the least useful part of calculus for non-stem students? Instead the basic concepts are where the easily accessible value lies.
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.
The valuable part of calculus to me was understanding the concepts of limits, differentiation, integration, and a tiny bit of differential equations.
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.
You don't use calc much when you use R, unless you're working on a problem that involves calculus, which isn't many problems, usually?
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.
im not though. I don't think you need to know measure theory and understand how to formalize probability in terms of sigma algebras to do professional stats.
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
for actual understanding, yes it is. The most basic important results, the law of large numbers, and the central limit theorem both require calculus to understand.
if you make a class without calculus, it is essentially just a bag of tricks and surface level understanding
It depends on what the definition of 'using calculus' is, as well. I use the concepts of integration and differentiation all the time in my work: it's completely integral (hah) to a good chunk of what I do (as well as complex numbers, fourier transforms, and a bunch of other 'advanced' maths). What I don't do is grind through working out the analytic solutions to odd integrals or differentials. Firstly because it's rarely useful, and secondly because I have a machine to do that for me in the cases that it is. I think it's unfortunately common to have the attitude that the latter is the majority of what calculus actually is in practice, because it makes up a lot of calculus education, but it's not the case.
As an engineer calculus is pretty fundamental to my job. It underpins all sorts of stuff - Heat transfer, fluid flow, stress and strain rates, beam deflection, fracture mechanics etc.
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.
As a chemical engineering student, it's a whole lot of formulas cobbled together to spit out calculations that would otherwise be done on hand (especially empirical methods that compute estimates where a clean answer is not possible like $\int{\frac{\sin{x}}{x}}$).
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!).
It will be amusing if we find out that all the things come in discrete quanta, even space-time, which I hear hasn't been ruled out. Calculus and real numbers might not be sitting as pretty.
There are lots of corner cases where the difference matters, like when integrals with finite bounds can go to infinity where a finite sum cannot. Boundary cases and surprise infinities seem to be a common problem problem the theoretical physicists have battled in their theories for the last century.
The issue is that one tree—calculus is pretty important and fundamental to lots of fields of study—is covered in tinsel and lights, and also for some unfortunate reason some folks have gotten it into their heads that they’ll never need it and might as well light it on fire.
Exactly this. The article isn't really at all about calculus, but rather the benefit of challenging yourself more generally. Doing challenging things that push you out of your comfort zone better prepare you to do the things you actually want to do later in life.
People (and employers) who can truly appreciate calculus wouldn't be impressed with a taken calculus course (thousands of students take calculus every year) or even a math BSc, and those who can't, how would they assess how hard your calculus course really was? There is no competition in a typical calculus course. Better ways of signaling the ability of doing hard things, especially for kids, in my opinion, are math/programming/etc contests. That's where you have a real competition and can show what you are capable of relatively to other people who participate and want to win. Also, it's important to note that passing a calculus course includes not only watching lectures or reading a textbook but also extensive problem solving. Typically "hard" college math courses are about memorization of abstract concepts, and taking any of them doesn't really prove you can do hard things (I'm not talking about PhD where you need to make novel contributions, thats crazy hard if you want your research to be competitive). Thats my experience and perspective but I'm living in my bubble.
I feel like you have missed the entire point. The point of the article was not to "signal" that you can do hard things. It was to build up the inner confidence. Competition is great, but largely orthogonal to the concept of having kids learn that they can do hard things.
I lost interest in school around the 10th or 11th grade. I never took any math classes beyond what was required to graduate way back in '96 in Florida. I also didn't go to college.
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?
You probably understand logs intuitively. Don’t worry about notation, here’s the idea: sometimes we count digits, not values.
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.
Khan Academy is great, but there are also an incredible number of great youtube videos. I would watch this 3blue1brown video to see if you like his style : https://www.youtube.com/watch?v=WUvTyaaNkzM
> go back to the basics and work my way up; I've lost a lot of it. Where do I start?
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).
A lot of math is cumulative, i.e. built on top of the prior concepts/tools. There are some things that are effectively the start of their own branches, but a lot of them then go back into a tangle of general mathematics that's all deeply interrelated, and also a subject onto itself when you get into ways to convert problems into totally different representations to use other mathematical tools on them.
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().
Just a meta comment: Don't think of calculus as hard in the same way learning a language or learning to paint is hard. Calculus is more of a gotcha moment. You fumble in the dark for a few focused hours (or minutes) and then you Get It. From then on it's relatively easy.
If I had to sort those - painting, learning a language, calculus. So, yeah, it seems hard to me. But I've never truly given it a lot of attention. That there is an "aha" moment for some, is promising.
> You know they will never use it in adulthood, outside of certain career choices.
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...
> But I recently realized there is a very good reason to take Calculus. It’s to prove you can do hard things.
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.
From that perspective, going through any kind of struggle (and building willpower/discipline) would be a good enough proxy for learning calculus.
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.
I remember when the AP Calculus test questions were released only a month or so after I had taken and aced the exam. I had no idea whatsoever how to solve the problems I had solved easily in the very recent past.
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 agree in principle that your personal story is made of the rough things you've overcome, and it's refreshing to hear it stated in a positive way (calculus) as opposed to the usual negative way (abuse, alcoholism, etc). It misses something, though: Many people will slack endlessly on doing the hard thing until it appears to us to be a challenge we've come up with for ourselves. Then and only then does doing it the hardest way possible seem not just worthy of our time, but essential to our personal growth.
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.]
It took me a while to understand what you were talking about at the end there. I think the author is referring to a grammatical period ('sorry' vs 'sorry.'), not the menstrual kind, lol.
It's a bit sad that calculus remains the stereotypical example of difficulty in most curriculums. Throughout childhood, I remember it seeming like some sort of complex, inscrutable, untouchable phantom hanging in the distance at the far end of the high school math course progression.
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.
Motivating topics for students is so important and underrated. It's also super hard I think. For one thing, everyone responds to different motivation. Some people are motivated to learn basic algebra more by the possibility of extending it into applied math so they can get a good job as an engineer. Others don't have any interest in that, but rather would be motivated by the beauty of pure math. And students themselves have no idea what they like.
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.
It’s the foundation of a lot more than just physics.
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.
Understanding calculus helps you understand much more deeply than you could have grasped form just learning algebra what kinds of numerical relationship problems it is possible to actually figure out. Ideally you also need some infinite series, some linear algebra, some combinatorics, and an appreciation of complex numbers as well, so the absence of deep coverage of those from a typical high school math curriculum in favor of putting calculus on a pedestal is more annoying.
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.
When I studied Calculus in high school, it was taught via mathematical proofs and concepts. I didn't really "get it" and struggled to keep a C average...until that one day I was working on a problem in the library about water draining from a pool it hit me that "it's about rate of change!". That was it...that concept changed Calculus from some weird math thing to something I could understand and get my head around.
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.
Don't do hard things that are unrelated to your actual goal; they're in unlimited supply, and you could be doing hard things that are relevant instead.
This essay wouldn't have impressed me in middle school; I don't know why it's on our front page.
> ... teenager asks why they need to learn calculus
> 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.
To me what's more important than Calculus being hard, and I think that's especially true for maths more broadly is that it's beautiful and one of the fundamental ways how we can make sense of the world. Everyone benefits from doing some maths.
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.
Not to distract everyone from complaining about calculus, but this reminded me of something I heard from a person with a Ph.D. in astrophysics from Caltech. They were not working in astrophysics, but they said the degree was still quite valuable to them. Whenever they had trouble learning something, rather than feel stupid, they reminded themselves: “I have a Ph.D. in astrophysics from Caltech, so I am definitely not stupid. This is just hard.”
There's some nice seeds of ideas here, but it's all a bit lost in
(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.
Yeah, but lots of parents lie far more often than would be useful for a child. But more importantly, absolutely honest is not the same as absolutely open. You can be absolutely truthful in the things you say without causing harm. If you think the truth would cause a child harm, then direct them away from a topic or tell them they aren't ready, etc.
I took calculus in High School just so I could make a comment here saying I did so. It almost cost me my diploma as I didn't want to do the homework. I paid attention in class though and learned just enough to pass the exams.
Some things which are hard for someone might be easy for someone else. Usually academic things like calculus are seen as hard, but there are many hard things in day to day life, like selling anything to unwilling buyer, handling a rude customer with calm and much more. They too are hard and calculus guys can not do them well.
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.
I think there is an similar argument for Greek and Latin in grammar schools. And the same for ancient Chinese in Chinese schools. Partly for cultural immersion and partly for brain gymnastics.
Math, and proof based Math such as Number Theory and Analysis is definitely in the same league if not for a career in academics.
>But I recently realized there is a very good reason to take Calculus. It’s to prove you can do hard things
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?
I raised the same questions as Nat, but I think he hasn't really discovered why we were asking that. The reason why I was looking for a good reason to study calculus is because I was lazy, and if I'm lazy enough then I can change the definition of what "hard" means. For instance, I could convince myself that if my could teach calculus, and him being just another dude, then surely I could learn calculus.
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.
Once you get obsessed with something, you'll see it everywhere. For you, that's 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...)
Calculus is important, because (among other things) the proper definition of limits teaches you how to manipulate logic quantifiers.
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.
Feels like many other comments here are trivial objections for the sake of sounding contrarian. Or even, humble bragging about how easy math was to the commenter. I found the article to be a great, new (to me), and widely applicable perspective on the matter.
It's almost like no one finished reading the post (emphasis added).
> 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.
> 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).
I am definitely not bragging — I’m saying the opposite, that all that pressure and elitism around math and science are unnecessary and counterproductive. I also liked the article just fine. The only controversial point I made is that science > art :)
I always felt like linking education to direct use of the knowledge is missing the point. It is irrelevant whether or not you'll need the knowlege you learned in school or at uni. What matters is your learnability, and that's the muscle you train with education. Learning smth just to use it to get paid for it is the lowest form of learning; the whole point is to acquire ability to learn whatever quickly so that you can become proficient in whatever is needed in the future, which in turn gives you an edge in the talent pool. In that sense the central point of the article is 100% valid - it's all about track record
You can choose which subjects to study in university, or even whether to spend time to get a degree. So it's optional. And these days, university departments that don't actually do something gets its funding slashed, so in that sense market forces are working well.
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".
Learning difficult concepts is only one aspect of developing perseverance - which is probably the greater ability to master. Others are finishing what you start, developing habits to complete undesirable tasks (like regular exercise or doing the dishes) and overcoming fear.
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.
As Hilbert put it, “Every boy in the streets of our mathematical Gottingen understands more about four-dimensional geometry than Einstein. Yet, despite that, Einstein did
the work and not the mathematicians."
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?
It’s really interesting how we shy away from some “hard” things, even though there is literally nothing at stake that could prevent us from doing them. Nothing life-threatening, no (or minimal) risk of physical or psychological damage, no social exclusion, no irreversable decisions about the future to make. Why can’t we just try/do it – what are we so afraid of?
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.
> It’s really interesting how we shy away from some “hard” things, even though there is literally nothing at stake that could prevent us from doing them. Nothing life-threatening, no (or minimal) risk of physical or psychological damage, no social exclusion, no irreversable decisions about the future to make. Why can’t we just try/do it – what are we so afraid of?
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.
Same reason you should be able to do pullups or deadlift twice your body weight. To paraphrase Socrates: no one should live their life without learning what they are capable of.
There will come a time when you have enough money, enough of everything, and deeper questions will arise. What is the meaning of existence? Why is there anything instead of nothing?
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 feel like math is taught from too high of a level initially. I did a second major in Logic along with CS at university, I was pretty crap at maths in HS and never really 'got it', I could manage but never had any intuition. After doing a bunch of logic papers and learning about axioms and peano arithmetic everything made a lot more sense from a foundational perspective and it really changes the way you look at numbers and the world around you
There is no end to things one cab learn, to what one can (attempt) to master. Hmm... Perhaps I am addicted to learning to avoid thinking about existential issues.
One of the dumbest things I’ve ever heard a math teacher tell me/the class is “you’ll probably never use this outside of this classroom”! And I’ve heard this story before, so it seems slightly common?
I entered college without a rock solid foundation in mathematics and it made things much more difficult.
"If you do weightlifting then sleep deprivation from a newborn is easy"
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.
I don't know anything about Calculus. My time is school was mostly spending a year in class studying to take a single test so that the school got more government money. But I will say that when a bunch of people online carry on about how hard something is, it makes me want to try it. Usually it turns out that what ever they were saying was hard or impossible is difficult and time consuming but not impossible. The hardest part is getting over how other people have built something up as impossible.
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.
There’s a difference of having to do and wanting to do hard things.
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.
I thought "doing hard things" would be something like a running Marathon, studying or working while having toddlers, learning to be a pro coder as an adult, taking care of your close ones while they are sick.. But it was about doing basic math.
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.
I am not good in calculus and that's something I wish I did pick up when it came to it.
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.
Funny, First semester of calculus I got a C i think, second semester, I failed. I thought about going back to community college and taking a calc class just to see if I can actually pass it. No reason other then that
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
So you proved to yourself that you can do hard things, what then? You don't need to continue proving the same, doesn't then that stop being a fuel for motivation? That's what happened to me at least.
It's not the point, but a lot of people are going on and on about the lack of need for calculus as a programmer. If you even touch how any machine learning works it's all basic calculus and intermediate linear algebra under the hood.
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 basic calculus and intermediate linear algebra under the hood
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?
Yes I am trained in electronics but I don't think understanding that level of detail ever helped me find bugs or improve the performance and reliability as that level is fairly static so you have to play the cards you are dealt.
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...
I don't want to be overly rude, but this is nonsense. The reason to learn calculus is that it's incredibly useful in several domains and never learning it prevents you from become a skilled practitioner in those domains which in turn reduces your future earning potential.
That is a good comic. But I am not sure "reduces your future earning potential" is really right. If you are going for future earning potential, then there are other things. Probably along the lines of - learn to code - find a way to migrate to the US - live in SF/NYC - learn algorithms and data structures - learn leet code - emotional control/resilience for putting up with those kinda jobs ... etc
I reckon there are plenty of PhDs earning less than $100k around the world, who know calculus and matrix algebra like their ABCs.
Everything can be incredibly useful in several domains, but we don't teach everything. Instead, people learn what they need when they start working in that domain.
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.
It's better to avoid waste, waste of time, waste of calories. Most of the things that most people want you to do are wasteful, pointless, better to not be done. A very valuable skill is to be able to discern bullshit, and decline 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.
I think the better way is to come out with a lot of interesting real world scenarios that require calculus or other math. Math doesn't get created out of a vacuum. It was created for a reason. I think if we can avoid diving into abstractions too quickly and focus on specific real world problems and lead the kids step by step to the solutions , that would be more interesting.
You should do calculus because it translates to actual problem-solving. Not everything is nice and chunky in discrete - calculus allows you to model and calculate continuous kinds of problems. Even if its not the area under the curve or some infinite series thing, it's a really good intro to modeling things in terms of adding up many small steps.
One of the things about doing hard things out in the world is that people often recognize you for it. With parenting, it’s the hardest thing you’ll ever do at times and most people in your life really don’t care. It’s a thankless job.
You take maths because it shapes your brain, it forces new neuron connections to be formed in the right side of your brain. Then you also have literature and arts for the left side.
The brain is like a muscle, if you don't train it, it wont grow AND you will basically be STUPID. That is it.
For some small percentage of high schoolers who can learn calculus, that knowledge will be worth ~100k a year for 40 years- and worth far more than that for their bosses. We can’t tell in advance who is in that percent, so we hedge by teaching it to all of them.
Of course, one thing a solid mathematical education will also give you is an appreciation that while ‘having passed AP Calculus’ implies ‘can do hard things’, that does not mean ‘having failed to take or complete AP Calculus’ implies ‘can’t do hard things’.
I suspect it's the justification for many PhDs. If you don't go into academia then chances are the content isn't that relevant, but the experience of having to problem solve in uncharted territory is a great confidence booster and skill to have.
> but the experience of having to problem solve in uncharted territory is a great [...] skill to have.
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).
I am not convinced „doing hard things“ should be a top priority for kids. For my own kids -or actually everyone- i hope they find a passion big enough to make a living. This takes time and lots of experimenting and cannot be forced into existance.
Grinding through „hard things“ can leave you with a great deal of confusion. It is so much easier to do „hard things“ if you love doing them.
The problem with math is: it is very hard to learn later on in life. Maybe even as hard as drawing. Schools should really focus on reading, drawing, writing and math- everything else should come secondary.
I recommend reading So Good They Can’t Ignore You: Why Skills Trump Passion in the Quest for Work You Love from Cal Newport for a different perspective on this.
Thanks. I am sure there are some valid points to be found in this book but my own experience is enough for me at this point. I have worked in jobs where i was doing great and received positiv feedback- even earned more than today. Still hated it and moved on- best decision i could have made.
This is similarly true for finishing a degree. Luckily, that occurred to me before finishing a degree, because that did feel futile. But finishing a degree proves that you can finish stuff. That's valuable proof for yourself and when you need to give an impression.
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.
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.