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.
> 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.
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.
> 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".
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".
> 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.
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.
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.
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)
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'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.
> 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.
> 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.
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.
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.
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.
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.
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.
> 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.
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.
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 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?
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.
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.
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.
constantcrying|2 years ago
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.
BiteCode_dev|2 years ago
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.
alpaca128|2 years ago
If you cannot justify to children why you force them to sit in school for hours every day you shouldn't be surprised by the consequences. Though weirdly schools tend to be very surprised in my experience. And yes, this question is absolutely asked in other subjects too, but less often as they're not as abstract and usually related to things people encounter every day.
> Justifying mathematics is easy
And thus left as exercise to the reader?
nicbou|2 years ago
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.
maccard|2 years ago
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".
paulcole|2 years ago
Are you kidding? Every tech-bro on the planet makes their disdain for literature, philosophy, etc. extremely well known.
roel_v|2 years ago
Huh? People around me asked that all the time in high school & beyond.
golergka|2 years ago
jalapenos|2 years ago
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.
yodsanklai|2 years ago
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.
rnkn|2 years ago
ed-209|2 years ago
bruce511|2 years ago
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".
szundi|2 years ago
lqet|2 years ago
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.
learningerrday|2 years ago
unknown|2 years ago
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hardware2win|2 years ago
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.
keerthiko|2 years ago
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.
lucideer|2 years ago
> 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.
Etherlord87|2 years ago
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.
raincole|2 years ago
> Since math modeling is everywhere in "modeling" industries like engineerings, financial-ish jobs and othet
In other words, certain career choices, as the OP says.
maccard|2 years ago
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?
jokethrowaway|2 years ago
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)
TheAceOfHearts|2 years ago
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.
smeej|2 years ago
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.
foofoo4u|2 years ago
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!"
bigger_cheese|2 years ago
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.
raincole|2 years ago
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.
kashunstva|2 years ago
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.
arsenide|2 years ago
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.
ethanbond|2 years ago
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.
Medh_Suk|2 years ago
But it will cost you everything if you don't."
Discipline is a muscle. Go Build it. Key is to understand different activities require different muscles.
Be mindful of picking your activities, but dont keep on waiting.
yawnxyz|2 years ago
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).
paddw|2 years ago
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.
abenga|2 years ago
savingsPossible|2 years ago
I fear that you might be right about this
wwarner|2 years ago
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.
anon-3988|2 years ago
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.
m463|2 years ago
hooande|2 years ago
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.
polishdude20|2 years ago
jimmydddd|2 years ago
TexanFeller|2 years ago
andruc|2 years ago
bee_rider|2 years ago
bobwow396|2 years ago
quickthrower2|2 years ago
vehementi|2 years ago
MaxBorsch228|2 years ago
XenophileJKO|2 years ago
WesleyJohnson|2 years ago
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?
kalid|2 years ago
When we say someone has a 6 figure salary, we are counting how many 0’s (10s) to takes to get there.
For memory, we say something has 32 bits and can have 2^32 possible values. It’s more graspable to talk about the “address size” vs the “number of possible values”, especially for things that grow fast (like storage).
I’d suggest starting with your intuitions and slowly translating them to math.
(Without being a shill, I wrote about real world logs here, it may help: https://betterexplained.com/articles/using-logs-in-the-real-...)
yCombLinks|2 years ago
ivansavz|2 years ago
You might want to check out my book "No Bullshit Guide to Math & Physics," which starts with a high school math review, and goes up to calculus. It's specifically written for adult learners (self contained + lots of practice exercises).
You can see a PDF preview here https://minireference.com/static/excerpts/noBSmathphys_v5_pr...
The concept map from the book is independently useful to check out: https://minireference.com/static/conceptmaps/math_and_physic... And you should also check out this SymPy tutorial https://minireference.com/static/tutorials/sympy_tutorial.pd... which can help you build a bridge between coding skills and math operations.
Natsu|2 years ago
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().
rramadass|2 years ago
One very good place to start is Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry by George F. Simmons (less than 150 pages!).
gniv|2 years ago
tjmc|2 years ago
chasing|2 years ago
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...
aleph_minus_one|2 years ago
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.
ssivark|2 years ago
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.
smeej|2 years ago
It's as you say. I didn't absorb it at all. I stuffed it in short-term memory and passed an exam. But that's all I learned. I learned how to stuff things in short-term memory.
I didn't learn calculus.
noduerme|2 years ago
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.]
chrfrasco|2 years ago
justsomeadvice0|2 years ago
batman-farts|2 years ago
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.
sanderjd|2 years ago
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.
fsmv|2 years ago
The formulas to do integrals aren't important but the concept of integrals is.
Honestly I think we should try to focus on differential equations instead but maybe it's necessary to do calculus first.
abtinf|2 years ago
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.
fatnoah|2 years ago
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.
esafak|2 years ago
This essay wouldn't have impressed me in middle school; I don't know why it's on our front page.
hgl|2 years ago
> 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.
Barrin92|2 years ago
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.
wrs|2 years ago
lucideer|2 years ago
(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.
stkai|2 years ago
hnthrowaway0315|2 years ago
ericsaf|2 years ago
zerop|2 years ago
spot5010|2 years ago
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.
markus_zhang|2 years ago
Math, and proof based Math such as Number Theory and Analysis is definitely in the same league if not for a career in academics.
Notatheist|2 years ago
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?
perlgeek|2 years ago
This doesn't explain why Calculus over any other number of hard things.
PartiallyTyped|2 years ago
rollinDyno|2 years ago
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.
abtinf|2 years ago
kickaha|2 years ago
Both changed everything in an epistemologically qualitative manner. Life before calculus/microwave, and life after calculus/microwave.
How did we warm up leftovers before the microwave?
What did the world look like before I learned calculus?
hnfong|2 years ago
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...)
js8|2 years ago
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.
codeisawesome|2 years ago
BWStearns|2 years ago
> 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.
wwarner|2 years ago
igorzij|2 years ago
hnfong|2 years ago
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".
tjmc|2 years ago
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.
upxx|2 years ago
8note|2 years ago
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?
phforms|2 years ago
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.
throwawybb1|2 years ago
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.
User23|2 years ago
bryanmgreen|2 years ago
codersfocus|2 years ago
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.
smfjaw|2 years ago
jononor|2 years ago
TheCaptain4815|2 years ago
I entered college without a rock solid foundation in mathematics and it made things much more difficult.
newaccount74|2 years ago
Right. Tell that to my partner, who has been nursing our newborn every two hours for the last 5 months. There is no way to wake up 2-4 times every night for months and not be tired.
Weight lifting would be zero help with that.
unknown|2 years ago
[deleted]
costanzaDynasty|2 years ago
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.
thenerdhead|2 years ago
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.
riialist|2 years ago
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.
bigpeopleareold|2 years ago
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.
tayo42|2 years ago
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
pcwelder|2 years ago
nurbl|2 years ago
jononor|2 years ago
mianos|2 years ago
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.
hnfong|2 years ago
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?
blueyes|2 years ago
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...
fortenforge|2 years ago
Basically: https://www.smbc-comics.com/comic/why-i-couldn39t-be-a-math-...
quickthrower2|2 years ago
I reckon there are plenty of PhDs earning less than $100k around the world, who know calculus and matrix algebra like their ABCs.
selcuka|2 years ago
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.
carapace|2 years ago
(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.)
tjbiddle|2 years ago
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.
derelicta|2 years ago
maxwin|2 years ago
satokema|2 years ago
xivzgrev|2 years ago
steve_adams_86|2 years ago
sourcecodeplz|2 years ago
The brain is like a muscle, if you don't train it, it wont grow AND you will basically be STUPID. That is it.
QuadmasterXLII|2 years ago
jameshart|2 years ago
kickaha|2 years ago
jebarker|2 years ago
ethanbond|2 years ago
yayitswei|2 years ago
HellDunkel|2 years ago
gjadi|2 years ago
mediumsmart|2 years ago
AnnikaL|2 years ago
LucasOe|2 years ago
beeforpork|2 years ago
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.
barrenko|2 years ago
analog31|2 years ago
a-dub|2 years ago
dangoodmanUT|2 years ago
uwagar|2 years ago