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.
Disagree. I say: Bad math teachers really filter out people in a way nothing else does.
"Here is a list of rules which can be used in specific circumstances. I am not going to tell you why you would need them and the exam will feature several corner cases. Also, I will not provide any examples or detailed explanations because what would other people think of me, then?"
This is over the top, but boils down my experience starting from highschool. I was in love with math when I was in primary and middle school. I really did.
But Highschool and university killed every last bit of enjoyment I used to have.
Okay, I did really like the first 2 semesters of math. I found an amazing online class which covered the same topics we did. Finally I understood and doing the exercises with such ease felt like cheating.
I have known many people, to whom mathematical reasoning comes as easily and naturally as flight does to a swallow, be defeated by literature. Every technical university in the world is brimming with bright young people who could effortlessly describe a Jacobian but would rather gnaw off their own hand than produce a response to a story or sonnet.
> And philosophy, mostly because of jargon and reference. Otherwise, if you break it down, it's not that complicated.
Quite the claim.
However, I hated math. HATED it. Did so poorly in it through high school I ended up in college in basic algebra. It wasn't until college I found out I loved math and it was only because I got lucky and had a string of amazing professors.
I suspect this is true for other subjects as well. I also hated exercise until I discovered what I liked doing. PE teachers ruined it for me. Runs were terrible because there was no cadence. Push ups were terrible because no one checked form. The entire exercise of pre-college education is a box checking effort to get you ready for the factory. Perhaps we simply need to stop having the unmotivated teach the motivated. This goes true for research professors forced to teach as well.
Learning Math is just like learning a language, the issue is the script/alphabet/symbols used to communicate, its dictionary, its grammar are all complicated its hard for common people to get a grip on it.
Almost all people can follow logical sequence of arguments provided they understand the language in which they are described.
This language so far is the greatest barrier to bringing Math to public. Probably a new teaching method is needed.
Yes, math is hard and definitely not for everyone, which I think is perfectly fine. But arguing that something has no use to you, because the first time in your life you actually have to engage in complex logical thinking is just lazy. The value of mathematics is independent of how hard it is.
If you cannot justify to children why you force them to sit in school for hours every day you shouldn't be surprised by the consequences. Though weirdly schools tend to be very surprised in my experience. And yes, this question is absolutely asked in other subjects too, but less often as they're not as abstract and usually related to things people encounter every day.
> Justifying mathematics is easy
And thus left as exercise to the reader?
Nobody told me why I should learn about ancient history or 20th century literature.
>usually related to things people encounter every day.
Basically nothing I did in school had anything at all to do with things I encountered every day. I played video games and hung out with friends. I didn't study European art history in my time away from school.
>And thus left as exercise to the reader?
Yes. I assumed that most people on this website understand the inherent value in abstract thiking and broadly applicable problem solving techniques.
Zing! I feel you'd want to know some of your readers get it :)
Math is a language that is required to describe how the world works. It's the official language of certain spheres of society that seriously impact your life. If you don't speak it, other people will make decisions for you that you can't understand.
It's also really useful if you deal with finance, engineering, computers, statistics etc. If you don't speak math, these doors are closed to you.
This would be a lot more evident if they had us solve real world problems with math.
Hard disagree here. When I was in school, there were constant discussions among the teenagers about "why do I need to learn a dead language (irish)", "why do I need to learn french/german, they can all speak english", "why do I need to study geography, I'm never going to need to be able to categorise volcanic vs fold mountains".
Are you kidding? Every tech-bro on the planet makes their disdain for literature, philosophy, etc. extremely well known.
Huh? People around me asked that all the time in high school & beyond.
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.
I'm convinced that justification doesn't make students more motivated anyway.
That being said, justification is needed, not for the students, but for people designing syllabus. They require arbitrage because there are so many hours in a day.
But i had better things to do. I started programming in grade 7, from a book, with a Apple 2 (circa 1982). There were no forums, no Internet just me and a thin booket that came with the computer, plus later, the odd magazine.
It was never "hard", but it was fun. It taught me how to build things, how to approach problems, how to throw something away and make things better. How to imagine.
Ultimately I would study comp Sci- people would teach me the right way to program, and 40 years later it's still my career. As I predicted at the time (somewhat obnoxiously to my then teachers, sorry Mrs Hodge), I had better things to do than raise my Geography score from a C to an A.
School is important, but finding something you enjoy, where "work" feels like "fun" is the truely "gifted child".
But there are lots of professions that can start as teenage interest. A friend of mine was obsessed with birds, he went into nature conservation, started in a bird reserve, and has had a long career climbing the ladder in that space. And also tour-guides bird watchers.
Sure, not everything works out, but if you can turn your fun into your job, then you're one off the lucky ones.
I really wish the world economy would be structured like this, with smaller entities taking more risk and responsibilities and less enormous corporations filled with people hating their lives.
That was me 100%. I passed elementary school with flying colors without ever doing any homework or actively participating in class (I had better things do to: reading comics and building LEGO contraptions) and simply continued this approach in highschool, with much, much less success. But neither did I care nor did I have any time for studying or doing homework - I taught myself programming in Batch and BASIC in the 90ies from the MS-DOS manpages, and later, when we had access to the internet, Delfi, Visual Basic and C++. I had finished a small tool with a few 100 users when I was 13, and I had to take care of them and fix bugs. There simply wasn't time left for any school work. University was just a natural extension of this childhood interest into adulthood. It didn't feel like school at all, and there I quickly discovered that math (well, computer science math at least) is actually pretty easy, and all you need is practice.
If somebody says stuff like this, then they do not understand math, imo.
Im a little bit sad whenever somebody argues for math by using "no phone available at the moment" argument.
Math is insanely powerful world modeling tool.
Starting from calculating right amount of fence for your garden, to estimation of 500km route arrival time while taking traffic statistics into the account, to data science, ML, whatever more complex.
Since math modeling is everywhere in "modeling" industries like engineerings, financial-ish jobs and other
Then you basically not only get better tools to operate (model) in real world, day2day life, but also it opens you doors to highly paid careers.
But the goal is not to have fancy jobs, but being able to do real world modeling.
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.
Mathematics is about examining things and understanding their essence - what statements can we make about all such things? How can we prove that? How can we use that to figure out other things?
But nobody ever tells kids that - they think they’re just in ‘advanced counting’ lessons.
I wonder whether a split in curriculum could help - similar to English language/literature. Mathematics needs a ‘numeracy’ program that starts in kindergarten and covers the mechanical ‘how numbers work’ stuff… then a separate mathematics program that starts in middle school and teaches reasoning and proof. Start with geometry.
The more abstract it gets, the more it warrants an introduction with "here's how it's used in real life".
I can recall the point when it started becoming mechanical - it was when I started doing Derivatives and Integrations. I was 'just solving puzzles' until I hit a chapter, tucked away way back in the syllabus, almost at the end of the 2 year arc - a chapter about Applied Differential Calculus. I still remember the feeling of "Oh... I get it now" euphoria, with a tinge of sadness - why wasn't this covered early on?
Same thing would have happened in Engineering as well, but at the time my teacher was good. They started by explaining the applications of Fourier Transform before we actually got to learn the mechanics of it.
I have the same thing with learning languages: I despise focusing on the rules of conjugation and word order and rote memorizing rules, exceptions to them and exceptions to exceptions. I much prefer to bootstrap with some vocab and then acquire by immersion, even if that means I get the conjugations wrong in the beginning. But I also find myself in the minority by far, to the extent that pretty much all language teachers I ever had (except my high school French teacher - she was awesome letting me translate MC Solaar lyrics for grades) didn't even really seem to believe that there are other ways of learning other than studying rules.
> outside of certain career choices.
As someone who did a lot of calculus in university, and definitely hit a eureka moment while integrating over vector fields that helped me conceptualise some general day-to-day stuff better in my head, would most people generally benefit from that same conceptual "grokking": of course. Would it be worth the time & effort investment for them if they're not using it in their career: no.
Abstract thinking is really useful during arguments, even about politics, religion, etc.
Yes, calculating the right amount of fence is useful, but not only (as someone pointed out) you don't need a lot of math for that, people just take take a ruler on the plan, or count steps in field, going across the entire length, then estimate, and it works.
> estimation of 500km route arrival time while taking traffic statistics into the account
Who does this? How would you convince a friend to learn math in order to do this? What people do is they remember how long a 200 km route took and so they estimate the 2.5× longer path will take 2.5× more time.
We need examples of real world math applications, because such examples are scarce across the Internet.
Who does it? Navigation app in your phone. They modeled this problem using math. While you could probably model it somehow without math, but why would you want to do it other way, when you have reliable, mature and flexible tool
I didnt write that you shouldnt leverage tech.
Do it, yet be aware how it works and why. You can leverage math in other custom scenarios
> Since math modeling is everywhere in "modeling" industries like engineerings, financial-ish jobs and othet
In other words, certain career choices, as the OP says.
(Also, while it might not be the tools needed for the average homeowner, there are plenty of optimisation problems similar to "how much fence do I need" which are most easily solved by solving the Euler-Lagrange equations)
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 forgot all my calculus after high school, had to relearn it in uni and then I promptly forgot again.
Exactly as the article says, it was more about proving we had the capacity (proxy iq test?) to learn it.
You don't need calculus in real life and I think the focus on calculus is ridiculous when we could explore other more practical areas, like category theory (which only my lucky friends who did advanced math got to play with)
I love the wikipedia intro: Calculus is the mathematical study of continous change, the same way geometry is the study of shape and algebra... that's it perfectly. And the most basic application is in everyone's life and also one of the basic physics thing: The relationship between location, speed and acceleration. I find this very essential, vs category theory at least..
I don't even know what "calculus" is really.
I have had plenty of math classes both in high school and later at university but I don't recall any significant distinction that would leave me with some concrete idea of "algebra" vs. "calculus" vs. "whatever" years later.
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 didn't find out until much later (and it still feels like much too late) that I just had an aptitude for learning the subjects traditionally taught in schools in the ways schools normally teach them. I got lucky. Everybody else was just as capable at learning some things in some ways as I was at learning school things in school ways. They just didn't have the luck that their aptitude lined up with what was measured and lauded in childhood like I did.
That meant they all got to learn how to work hard to learn things outside their areas of aptitude in school. I didn't. I didn't realize there were any such things that might matter someday.
I think learning how to learn things that are hard FOR YOU is quite possibly the most important skill in life. The sooner you master it, the better.
Framing some things as "hard" when everything is hard for some people and easy for other people undercuts the more important lesson.
I’d certainly agree it’s extremely valuable. There are some failure modes from taking it too far (like anything).
People tend to be happier when they’re very good at things. You also contribute more to the world. If you’re always doing things that are hard for you, you won’t do as much or as well, really by definition. It’s okay to do things that are easy for you. There are a lot of upsides!
Even if you choose an easy path, there will always be hard parts. So if you want to get anywhere, it’s essential to have practice navigating that. Just consider if that’s where you want to be all the time. I’ve done it myself and seen it in others, where you think you’re always challenging yourself, but you’ve actually just put yourself into a life that’s a bad fit.
I think it's important to learn early enough that things are usually hard when you get far enough into them, and that it's OK, it's not a brick wall, it just takes some effort to keep advancing.
- ADHD symptoms being mild,
- Having slept well (ties in with former point), and
- Having no imminent major worries at the moment (family/health/financial).
Any of these can make the difference between casually trying to understand quantum mechanics, and crawling under the table because I just can’t make this one rectangle on the screen do what I want.
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!"
Back when I was a high school student math (not just calculus - but the entire subject of mathematics) had this reputation as being a "hard" subject as a result scores of my fellow students just decided math is to difficult I'm not going to engage with this.
I suspect this is related to a fear of failure or kids being afraid of "looking dumb" in front of their friends - There was a definite "if I don't try then it doesn't matter if I can't do it." attitude, so they just switched off in those particular classes.
A lot of these attitudes carry forward into adulthood. I'm almost 40 and amongst my generation programming has a similar reputation. People I grew up with think if you can read or write code you are some kind of mystical wizard with powers beyond the understanding of mere mortals.
I see it today at my day job - I work as an engineer (the non software kind). I've seen my coworkers completely baulk at computer code I hear all the same things I heard back in high school. "This is too hard, I can't learn this stuff, I'm not going to bother attempting to understand it".
Fluid Dynamics was a hard subject (in my opinion), Solid Mechanics was challenging a dozen lines of python code is not on the same level.
To me it seems like a lot of the fault was with the curriculum: basically full steam ahead regardless of the class’ understanding. That’s especially bad in math when each chapter uses what the previous taught.
But the point about adult salaried professionals complaining that they supposedly can’t figure something out is disappointingly relatable. I generally believe that most people are "smart" and just don’t tend to bother using their brain as a muscle and that seems to make it doubly irritating to hear such complaints.
My daily work is 80% C# and 20% Python (to make internal Blender tools for our artists). And I'm really bad at Python. I don't know any of itertools. I don't know zip() besides its name. I don't even use lambda.
The result? My bad code can be easily understood by some of more tech-savvy artists.
for x,y in zip(['a', 'b', 'c'], ['1', '2', '3']):
print(x + y)
>>'a1'
>>'b2'
>>'c3'
Zip was simpler than I thought when I first saw it.
The easiness or difficulty of a domain or discipline is always in relation to some individual context; and that context includes variables that the learner controls. To the impatient, disinterested or undisciplined, I imagine calculus, learning the kanji, or playing the oboe all seem hard. But to the extent I can marshal patience, curiosity and discipline, the difficult domain becomes just a series of small steps integrated over time. I’m a musician and when a student complains about how hard a piece is, I ask if they can play the first note, then the second. If so, then it’s not hard. Because the process to acquire the whole thing is right there. Yes there are interpretive elements and techniques to be acquired along the way. But nothing is hard unless you are in a great hurry or you don’t really want to do the thing.
> But nothing is hard unless you are in a great hurry or you don’t really want to do the thing.
I got told this many, many times in my life, and it was incredibly frustrating when it was something I really wanted to do. I discovered after 34 years that I have ADHD, which makes a lot of stuff that can eventually become easy/easier with patience and perseverance to in practice be extremely hard.
I'm bringing this up because a lifetime of guilt and shame for not being able to accomplish something when it was deemed easy, that it "just requires some discipline", said by someone else pushed me away from a lot of things I'm interested in but wasn't able to keep motivated to do them after shame set in. It can spiral if you feel inadequate, and if you live with this you feel inadequate and "catching up" a lot of times.
Specifically, one of those things was music. I tried learning instruments when I was younger but the motivation was not in learning the instrument itself, it was music as a whole. I wanted to understand how it worked and how I could create it, not plow through guitar strumming exercises for months and months, then fingering techniques, then be able to play a few songs, and maybe in some years actually start to create something. To me what worked, in my natural branching way of thinking/learning, was to start producing electronic music some 4 years ago. Just some stupidly cacophonic basic loops in the beginning, which pushed my interest to learn the basics of music theory, learning the basics cleared to me a map I could guide myself through skills I was missing: rhythms, harmony, active listening, etc. After I started understanding what skills I needed to achieve what I wanted then it pushed my motivation to learn an instrument, the piano, and then learning the mechanical skills of the instrument made sense.
I bring this up because since I was diagnosed I had multiple conversations with people that suffered through the same as myself: being called undisciplined, inpatient, disinterested when they couldn't muster the motivation to plow through a structured path when it got boring to them. And that is not under my control, ADHD is much more about lacking motivation control than being hyperactive or actually having an "attention deficit", I get obsessed by things I'm interested in (music is an example), it's just that most of the resources to educate oneself on a discipline/domain is not tailored for people who needs to branch out, find pockets of skills that are interesting and motivating to learn, and putting the puzzle back together after acquiring some skills in a haphazard way than the usual structured learning path.
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.
I’ve used this same idea to dig myself out of ruts. When things are fucked up I’ll start paying attention to small things and deliberately “defer” progress on a few bigger things that are harder to do and more costly to fail. Each small win helps build momentum into the next-biggest challenge.
I’ve found this super useful for avoiding “habit destruction” during major life events/travel/moving.
But it will cost you everything if you don't."
Discipline is a muscle. Go Build it. Key is to understand different activities require different muscles.
Be mindful of picking your activities, but dont keep on waiting.
E.g. I know many people who go through bouts of intense fitness or diet fixations, take a lot of well-deserved pride in their discipline, and then hit a major event that temporarily precludes the fixation. They really struggle to get back on the train.
One major factor, IMO, is that they’re daunted by the intensity of what they achieved before. Obviously in fitness there’s a physical component to this, but there’s a significant mental component as well — especially outside of fitness.
Basically all I’m saying is you can (and should) gradually and deliberately dial up your sense of self-efficacy when it inevitably crosses some local minimum due to events outside of your control. You ought to build a self-image that’s robust to occasional and sometimes significant failures.
It's not about a thing being hard. Walking is hard to a paraplegic. It's about overcoming a thing and feeling good about it (instead of external rewards, like a piece of candy or good grades).
The real problem with school is that it replaces empowerment with gamification externalized rewards. You're not learning calc for the sake of understanding the world, you're doing it for a line item on a checklist. That doesn't come with empowerment.
With the mere framing of "you can do [hard thing] to prove you can do hard things" is a bad framing because it could be anything — from doing calc, to bungee jumping, to drinking a gallon of milk (please don't). This framing doesn't actually lead to empowerment (and then self-improvement).
The reason we learn calculus in high school is because it is foundational for many advanced STEM fields, and we will yield better results during university for the small percentage of students who go into those fields by forcing everyone to learn it in high school. Or, moreso, that's a viable justification for learning it today. Had history taken a different shape maybe we would learn something else, or maybe not. But the point is that calculus is not an arbitrary hard thing we learn for arbitrary reasons.
I fear that you might be right about this
To understand how things actually work, you need math, especially calculus. Deep learning? Calculus. Statistics? Calculus. Finance? Calculus. Physics? Calculus. Mech E, robotics, earth science, econ? Calculus.
Second, calculus, like all math, is easy. Like that’s the point, it’s the science of simple things. That math is competitive and presented as a cryptic challenge is beside the point — it is designed to make it possible for anyone to reason for themselves and solve problems. The sense of impatience and criticism around math is totally unwarranted and isn’t good for anyone.
I get kind of bummed when I see schools spending so much creativity and enthusiasm on art and theater. There really is no reason why science should be thought of as judgmental, difficult and painful, while putting on a play is creative, inviting and fun.
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.
Very true, but this doesn't change the fact that math is actually simple, but it is generally taught so badly that most students can't "get" it.
I did a bit of private tutoring back when I was in college (and I'm still doing it for my own children), and every person learns in a different way. It is not always easy to find the right way to convey an idea, but once you find it you can see in the student's eyes how it just clicked.
Totally anecdotal, but I once helped someone who "didn't get how percentages work" get a really high (with respect to her previous attempts) GMAT score in maths.
"If you do not believe that mathematics is simple, it is only because you do not realize how complicated life is." -von Neumann
So . . . logic and number bases? Do tell!
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.
And engineers should usually be using battle-tested models rather than coming up with their own derivations, so to some extent using the calculus day-to-day shouldn’t be necessary in many cases. But it is necessary in order to understand where the models came from. This is what separates engineers from tech-priests.
I may not be solving differential equations by hand but I'm using knowledge about calculus everytime I reason about our industrial process.
The excel part is probably referring to solvers - where you plug in boundary conditions and spits out a solution. Edit - and excel or R (or Matlab) is what you use in lieu of needing to solve this stuff by hand.
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?
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-...)
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.
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().
One very good place to start is Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry by George F. Simmons (less than 150 pages!).
Just going to register my dislike of this particular trope. Mostly because "certain career choices" is doing a huge amount of work. "Certain career choices" can give you access to a very high income. They can give you access to a deeper understanding of some pretty interesting stuff and put you in a position to accomplish all sorts of amazing things.
People use calculus in the sciences, of course, but also in business, the arts, music, politics, and beyond.
We learn calculus when we're young as a part of expanding ourselves, casting the widest possible net to find that intersection of what we're good and and what we're excited by. And to fill our quiver with as many arrows as possible for when we hit the adult world. And, of course, to build our self-esteem by showing ourselves we can accomplish hard things.
But I would highly recommend against taking the attitude that "you'll probably never use it." That's counter-productive. And the people who "don't use things" are boring...
While I do like mathematics (thus I clearly have no negative feelings about learning calculus), this argument is dubious: the longer I live, the more I realize that the capability to learn complicated scientific stuff (including mathematics) hardly does transfer to other areas of life.
Just consider this: if such skills transferred so well, one would expect that those people who make a steep career in companies have learned lots of insanely hard stuff, often hard mathematical and scientific stuff. The reality is that the people who have such a steep career are rather great "policitians" (in the negative sense), sycophants and self-promoters. On the other hand, learning hard scientific/mathematical stuff nurtures the personality trait of "no bullshitting" and developing less tolerance for claims that clearly cannot be abided by.
Thus: I really like learning hard things, but this personality trait is in my opinion often a career killer.
It’s a little sad that the best value that the OP can impute for learning calculus is masochism — I cannot imagine saying that for anything in the school curriculum that I actually learned/understood. I wonder (in good faith) whether the OP actually even absorbed calculus at all… (i.e. can they solve calculus problems even today, for example?) — if not, they’re not the person who should be making authoritative comments on the usefulness of calculus.
Calculus (Taylor approximations, perturbation modeling, error propagation, significant figures in measurement precision, gradient descent, etc — just off the top of my head) is so deeply embedded in my thinking, that it strongly shapes how I think and amplifies my effectiveness!
I disagree with the OP’s claim so fundamentally — they might as well claim that schools/education focus on literacy for the same reasons.
It's as you say. I didn't absorb it at all. I stuffed it in short-term memory and passed an exam. But that's all I learned. I learned how to stuff things in short-term memory.
I didn't learn calculus.
I'd argue that as long as someone reaches that attitude toward something that they choose, they have lived a good life. And that something doesn't need to be a high school math class. The hard thing could be trying to become a chef when you're 50, or deciding to write your next app in Assembly knowing none at all, or surviving a month in the woods, or going to a foreign country with the intention of learning the language. It has to be hard to make it worthwhile, but it has to be your own to make it valuable as an accomplishment to you, as opposed to something imposed on you which you merely endured. I think this is why a lot of people come out of incredibly hard ordeals in the military with much less personal sense of self-worth than they were sold they would get going in.
[edit: removed a critique. I had misinterpreted the words "errant period" to allude to something other than punctuation. My mistake.]
>> We’ll cry into TikTok over an errant period at the end of a text message.
My brain read that as, "At the end of a text message, we'll cry into TikTok over an errant period." lol
If somebody had told me that calculus is how you transition between dimensions, or that techniques of integration would enable me to generate 3D shapes from 2D lines, I think I would have been much more motivated to progress rapidly in math, and much less discouraged when I hit the "hard parts." Those are the answers I tend to give today when somebody asks me, "why take calculus?" Demystifying it doesn't even have to be a wholly practical explanation, like deriving acceleration from velocity.
Segregating out the "hard stuff" doesn't even necessarily lead to great learning outcomes, either. At my high school, and it seems many others, the honors kids were put on the track leading to calculus while everyone else ended up in a dedicated statistics class. The honors kids were expected to pick up statistics through supplementary assignments in their laboratory science classes, and this same approach carried over into lower-division undergrad. As an adult, I feel like that approach has only given me cause to go back and seek out a firmer grounding in statistics.
But yeah, I can think of so many examples of things that I would have been way more into in school if I understood how they mapped onto the adult world. Statistics is probably at the top of that list though.
The formulas to do integrals aren't important but the concept of integrals is.
Honestly I think we should try to focus on differential equations instead but maybe it's necessary to do calculus first.
It’s absolutely crucial to economics and business, and it is a travesty that it isn’t a required part of lower division curriculum. You cannot grasp micro/macro/applied/business economics without understanding relationships between changing variables.
But the idea that if you know how the rate of change of a thing changes over time, that that gives you enough information to understand it completely? That’s pretty important and deep.
It also underscored the poor teaching methods at my school. I was somewhat vindicated by being the only person in my class to get a 5 on the AP test. I also ended up in a major where I did almost nothing but calculus for undergrad and grad work.
This essay wouldn't have impressed me in middle school; I don't know why it's on our front page.
> But if we avoid hard things
I don't see how you can justify the former by arguing the latter. These two are orthogonal. If I were that teenager, I think what I really would want to ask is that why it has to be calculus instead of some other things that is also hard but with obvious real world application like writing a small 3D game engine.
And my answer to that question is you probably shouldn't if your were in an ideal education system. You would be taught what interesting interactions you could have with the physical world, and be induced to discover calculus or some other math tools that helps you understand how the interactions really work and demonstrates you really need such tools. You're more likely to grasp them when you're driven by curiosity.
I studied maths in uni and while I've not used it much, even as a programmer, I still enjoy doing it. My dad never had much schooling but now that he's retired he actually picked up a few of my books and slowly worked through high school to now undergraduate courses. He's having a lot of fun with it.
(a) brevity - it's a deep topic so the post length can't do it much justice &
(b) naïvety - their extremely oversimplified explainer for the “C students hire A students” trope is that the C students are actually secret A students in other areas. This assumes a utopian world where academic-style assessment of study/effort maps to career progression. C students do well for a range of reasons including-but-not-limited-to: charisma, nepotism, unconscious-bias-hiring on the basis of race/gender/accent/height/hair/appearance, normative mental health characteristics. It's not because of their secret extracurricular study: that's in direct contradiction of the message behind the trope.
Newton had discovered the law of gravitation, but didn’t publish it for a long time because he couldn’t justify how the gravitational effect of a sphere could be the same as if all its mass was concentrated at a point in the center. He had to invent calculus to prove it. Isn’t that amazing?
I know the author is using calculus as an example of a hard thing, but if someone is genuinely following their curiosity, then things do not seem hard. Personally I wouldn’t want to work on hard things just to prove to myself that I can do hard things. I’m happy being lazy.
Math, and proof based Math such as Number Theory and Analysis is definitely in the same league if not for a career in academics.
Doing hard things proves you can do hard things. Where does calculus enter into it?
>The more hard things you push yourself to do, the more competent you will see yourself to be
"The whole problem with the world is that fools and fanatics are always so certain of themselves, and wiser people so full of doubts."
>Most C students are not doing other hard things instead of school. They’re just goofing off, so they end up working for the A student.
This is ridiculous. Author goes on to imply these ("most C students") are on social media and using drugs. I'd put any money on them being much more likely to have been raised by the type of parent who believes their kid needs a reason to want to learn calculus.
Oh, and what is "hard" by the way? Is it doing something you're not good at until you're good at it? Does it need to be valuable from my perspective or that of my parent? Should it take a certain amount of time to become "good" at it? When am I good at it? Can it be a hobby I enjoy even while it's difficult, or must it be a chore? What if I'm forced to overcome some hurdle in my life for example growing up with a foolish parent with warped worldviews?
This doesn't explain why Calculus over any other number of hard things.
I would also be cautious about setting yourself up for a hard life. The takeaway resonates well with the HN crowd, myself included, because we like challenging ourselves. There's a lot of people out there who are simply looking to satisfice their lives, and you'll need another way to motivate them to learn calculus.
Both changed everything in an epistemologically qualitative manner. Life before calculus/microwave, and life after calculus/microwave.
How did we warm up leftovers before the microwave?
What did the world look like before I learned calculus?
And no I'm not kidding, a couple years ago when I was studying distributed systems, I saw the CAP theorem everywhere. Why isn't distributed systems part of the high school curriculum? It's used in basically ALL computing devices (before cloud computing, distributed system theories were applied in multicore systems...)
In any case, IMHO best advice to young people - try to learn hardest things you can. Later, there will be less time, less energy and more distractions.
> I don’t particularly care what grades my kids get once they start school. But I do care that they consistently prove to themselves they can do hard things. If Calculus is how they want to do it, fine, but there are many, many more options.
Separately, as someone just recently learning calculus as an adult now I'm digging the obliviousness of the "math is just inherently easy" folks. I get where y'all are coming from but there's a deep lack of empathy there. Go teach some people something (anything) and learn that other people's brains don't necessarily contain your knowledge.
Concerning the empathy point: I do believe that to become good in mathematics, you seriously have to believe that everything in mathematics is actually easy, and you either don't work hard enough (so you fail understanding it; solution: work harder) or you are too stupid to understand the material (which means that you should better work even harder to understand the material to compensate for this weakness).
It's a different story for education up to high school. School is pretty much mandatory these days, and basically everyone has to go through it. It's a worthy question to ask why learn X instead of Y, if X has little real application? Why glorify those who can do maths well, as opposed to those who can best remember the names of Pokemon? Or those who can play LoL the best?
Who defines which areas of knowledge counts as "education", and which areas are frivolous trivia?
In the case of maths, the mere fact that it has tonnes of applications in many technical subjects is sufficient justification for me, but if you reject that line of reasoning, I'm not sure you can convince me (or anyone with critical thinking) that it's more worthy to pursue knowledge in maths than knowledge in "Pokemon studies".
The latter is something you see creep in as you get older and it's something that's always attracted me to hobbies that require you to keep your nerve like skiing, scuba diving and flying planes. There's nothing like having to land a plane solo with a bit of a crosswind to give you confidence in yourself.
Some things are vaguely impossible, like finding a solution to fluid mechanics, or finding a theory that combines quantum mechanics and general relativity.
Others are proven to be impossible under some conditions like the halting problem.
Others still require access to some amount of resources, or some amount of leverage, like buying a $100M yacht.
Lots of these hard things are quite unlikely for me to be able to do.
I think most people refer to the resource one? And that some people have more access to resources than they think, and the scales of resources needed are smaller than assumed?
Many people (especially those in a lucky environment) can just try to draw realistically (which is actually not that hard, given a good teacher/instruction), study a “hard” topic in philosophy, get good at playing a “hard” song on an instrument or even learn how to prove mathematical statements or how to model and analyze complex systems. We can just try, there is nothing at stake except our time and energy.
When something is really interesting to me, I try not to stop myself from pursuing it just because it is considered “hard”. I have many mental issues (executive dysfunction, bad memory, etc.) but I am also stubborn and curious enough to try anything, again and again if I have to, which I never regretted. It doesn’t hurt and it can be so much fun with a humble and relaxed attitude and especially if I don’t try to compare myself to others, which might be one of the main things that is holding most of us back.
Heh, I don't know about that.
I got into motorcycles because of how dangerous they are. When I was just learning dirt riding I specifically picked out an old barely used logging road to do so. It's 2 hours from the nearest town, about 40 miles long winding road with a sheer drop on either side.
Getting there is tiring in of itself. Every time I started it, I was already fatigued from the ride out and I just go straight in without a break. But there's very little room of error. There's no cellular service, traffic is almost non existent. I don't tell my family when I got out there or where I am exactly. If I overcook a corner, loose focus for more then a second, fail to spot where the road's been eroded out, make just one little fatigued induced mistake... that's it. There's no getting out. There's no help coming. They won't even have my body to bury.
Finding the strength to crank the throttle wide open and hold it open right until the last millisecond before disaster is hard as hell when every instinct is screaming to slow down.
But in many ways, it's easier then trying to make friends or ask a someone out on a date. If and when I do make that fatal mistake, I don't have live with the knowledge that I screwed up a relationship or have to live with the regret what may or may not have not have been. Or worse; disappointing those who's opinion has weight.
Dying is easy. Living with the consequences is hard.
You will see that math plays a very fundamental role in our reality, and once you start seeing it in such a manner it may begin to interest you.
I entered college without a rock solid foundation in mathematics and it made things much more difficult.
Right. Tell that to my partner, who has been nursing our newborn every two hours for the last 5 months. There is no way to wake up 2-4 times every night for months and not be tired.
Weight lifting would be zero help with that.
But perhaps my FU-mode is stronger than other peoples. Someone on here said I'd never work for a FAANG company, so I guess that's something I'll have to do just because. Not impossible.
Most people pass a class because they have to for their degree.
Most people raise a child while being sleep deprived because they frankly have to.
When people want to do something, they don’t need to prove to themselves that they can do hard things because difficulty hardly matters to one with their mind set on something. For example anyone who decides to run a marathon one day.
Instead for the things you have to do, one could reframe the “have to” with a “get to”. Gratitude is empowering. Not everyone gets to go to college. Not everyone gets the opportunity of being a parent. Etc.
Proving to yourself or anyone else that "you can do hard things" since you did more or less math in school/collage/university will leave you trainwreck at the first real hard thing that bumps your way.
And why some people don't do the math? I guess, because they are told its boring and/or hard, but they should do it anyway. And people don't like to be told what to do.
The harder thing was keeping the pressure up to pass the tests and in total, pass a course, while dealing with the severe discomforting exhaustion of coming in early in the morning for classes, having to understand rather complex abstract notions. I still cannot think when tired or even inebriated. But going through that was really to get that good pencil pushing job reserved for college graduates, no? :D
Instead, in the end, I like what I do and if the spirit moves me or need it for something, I would sharpen those calculus skills.
I suspect I could have passed it with better teaching. I hated that I had to memorize things, which felt tedious, not hard in a good way. If you dont memorize cos,sin,tan stuff you can't take tests fast enough. The class was just how good can you memorize things. I also hated the proofs, pages of proof. No idea why, and the teacher didn't speak english well or communicate well in general
Sure most people can get a long way without understanding anything under the hood but I think I am better developer knowing assembly and architecture. The same for ML, you can get by most of the time. You can also make a whole career not doing hard things. That's not for everyone.
It's all bits and bytes under the hood, which in turn is just electric signals under the hood, and ultimately (hopefully) quantum mechanics under the hood.
Are you an expert in all of these? Why arbitrarily stop at the calculus layer?
The same with the underlying physics. I have never had any use for that and I am not sure why they taught it hat first up. Interesting all the same
Doing random hard things in order to get into college or get a white-collar job is really no different than pointless test preparation that shows you can follow instructions, and it has produced strange hierarchies such as Qing-era mandarins.
Much better to find something that intrinsically motivates you to do hard things that feel less hard to you. It'll take you further. Or you can prove to the world that you are capable of a yearlong mindless grind...
Basically: https://www.smbc-comics.com/comic/why-i-couldn39t-be-a-math-...
I reckon there are plenty of PhDs earning less than $100k around the world, who know calculus and matrix algebra like their ABCs.
The point of the article is that calculus is not taught because it might be incredibly useful for a small percentage of students, but because it measures their ability to pick up a hard subject and ace it.
(As an aside, the idea that calculus is hard is a pedagogical failure. Calculus is easy to learn if it's taught well. Most of our education systems actually make learning much more difficult than it is.)
My current explanation for this is more along the lines of being well-rounded. By being exposed to a number of different skills and learnings throughout your education, your more likely to be able to connect the dots in other areas - and just hold better conversations with people.
Don't need to ace all your classes for that, but put in the effort to try and structurally understand whatever it is you're coming across.
The brain is like a muscle, if you don't train it, it wont grow AND you will basically be STUPID. That is it.
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.