I really despise all the "why you should care about math" takes.
The question is never asked about any other school subject and only mathematics has to justify itself that way. I had to learn about categories of plants and animals, interpret 20th century literature, learn about events from a thousand years ago, I did presentations on the demographics of European countries, how certain chemicals react and much, much more. I never used any of that knowledge for anything, certainly not in my career or in university.
But somehow mathematics is the one field which needs to justify its own existence? Mathematics needs to bend itself over and "be relevant" so that people will actually learn about it? Why? Why not ask the same of any other subject.
Justifying mathematics is easy, especially such a universally applicable subject as calculus. But I see no reason why it should have to justify itself in any way.
There is no difference between learning a name in history or in biology, or some workflow.
But math reasoning really filters out people in a way nothing else does.
All other topics just require memory and basic reasoning.
Even physics mostly is hard because of math. And philosophy, mostly because of jargon and references. Otherwise, if you break it down, it's not that complicated.
But you can break down a math problem as much as you want, some of them are beyond what you can do comfortably. And a lot of students reach this limit early in their life.
I see this when I play board games with people: there is a threshold of rules and calculation power above which I lose 90% of the players. They just can't enjoy it, because it requires too much effort to play.
It's similar for me and sport. I've been doing exercise all my life, but my brother will always do more, and harder, because there is a barrier after which it's just too painful for me.
Math and athleticism can be trained, but there is a hard ceiling. And even before you reach that ceiling, closing the gap gets more and more expensive for some people, so much the ROI is difficult to justify.
> math reasoning really filter out people in a way nothing else does.
Disagree. I say: Bad math teachers really filter out people in a way nothing else does.
"Here is a list of rules which can be used in specific circumstances. I am not going to tell you why you would need them and the exam will feature several corner cases. Also, I will not provide any examples or detailed explanations because what would other people think of me, then?"
This is over the top, but boils down my experience starting from highschool. I was in love with math when I was in primary and middle school. I really did.
But Highschool and university killed every last bit of enjoyment I used to have.
Okay, I did really like the first 2 semesters of math. I found an amazing online class which covered the same topics we did. Finally I understood and doing the exercises with such ease felt like cheating.
> math reasoning really filter out people in a way nothing else does.
I have known many people, to whom mathematical reasoning comes as easily and naturally as flight does to a swallow, be defeated by literature. Every technical university in the world is brimming with bright young people who could effortlessly describe a Jacobian but would rather gnaw off their own hand than produce a response to a story or sonnet.
> And philosophy, mostly because of jargon and reference. Otherwise, if you break it down, it's not that complicated.
I can second that. I was an pretty good programmer as a kid. I'd go home and write whatever I could imagine on the old 286 box my parents had. I spent time in "gifted" programs at school, etc.
However, I hated math. HATED it. Did so poorly in it through high school I ended up in college in basic algebra. It wasn't until college I found out I loved math and it was only because I got lucky and had a string of amazing professors.
I suspect this is true for other subjects as well. I also hated exercise until I discovered what I liked doing. PE teachers ruined it for me. Runs were terrible because there was no cadence. Push ups were terrible because no one checked form. The entire exercise of pre-college education is a box checking effort to get you ready for the factory. Perhaps we simply need to stop having the unmotivated teach the motivated. This goes true for research professors forced to teach as well.
Math as a field of study might be hard but in my experience math pre grad-school is just memorization as much as any other subject. I studied for my exams by simply doing countless exercises and memorizing every problem pattern to the point where I'm just plugging numbers into a series of formulas, even if I don't fully know why I'm doing so.
>>But math reasoning really filters out people in a way nothing else does.
Learning Math is just like learning a language, the issue is the script/alphabet/symbols used to communicate, its dictionary, its grammar are all complicated its hard for common people to get a grip on it.
Almost all people can follow logical sequence of arguments provided they understand the language in which they are described.
This language so far is the greatest barrier to bringing Math to public. Probably a new teaching method is needed.
What I specifically dislike is the question "what am I ever going to use this for" or versions of it. Which I only ever hear about mathematics, but which can be asked about every other subject as well.
Yes, math is hard and definitely not for everyone, which I think is perfectly fine. But arguing that something has no use to you, because the first time in your life you actually have to engage in complex logical thinking is just lazy. The value of mathematics is independent of how hard it is.
There's a major chicken and egg problem here—is math hard and therefore it has to justify itself, or is it hard because people don't think it's relevant and so they care less to try?
> I see no reason why it should have to justify itself in any way
If you cannot justify to children why you force them to sit in school for hours every day you shouldn't be surprised by the consequences. Though weirdly schools tend to be very surprised in my experience. And yes, this question is absolutely asked in other subjects too, but less often as they're not as abstract and usually related to things people encounter every day.
>If you cannot justify to children why you force them to sit in school for hours every day you shouldn't be surprised by the consequences.
Nobody told me why I should learn about ancient history or 20th century literature.
>usually related to things people encounter every day.
Basically nothing I did in school had anything at all to do with things I encountered every day. I played video games and hung out with friends. I didn't study European art history in my time away from school.
>And thus left as exercise to the reader?
Yes. I assumed that most people on this website understand the inherent value in abstract thiking and broadly applicable problem solving techniques.
This drove me nuts when I was a student. No one gave me a solid reason, even though it was right there.
Math is a language that is required to describe how the world works. It's the official language of certain spheres of society that seriously impact your life. If you don't speak it, other people will make decisions for you that you can't understand.
It's also really useful if you deal with finance, engineering, computers, statistics etc. If you don't speak math, these doors are closed to you.
This would be a lot more evident if they had us solve real world problems with math.
Same. For K-12, at least, I believe the reason why no one could give a solid reason is because the teachers who are teaching the subject have literally no professional industrial experience with math. They have nothing to share with no real-world insights. I presume this should come to no surprise. According to GTP, the average salary range of a mathematician in the US is, "$80,000 to $100,000, while more experienced mathematicians can earn up to $150,000 or more." With salaries like these, no one working professionally as a mathematician would choose to downgrade to become a teacher, with teacher pay, lower respect and deal with uninterested, rude, problematic students.
> The question is never asked about any other school subject and only mathematics has to justify itself that way
Hard disagree here. When I was in school, there were constant discussions among the teenagers about "why do I need to learn a dead language (irish)", "why do I need to learn french/german, they can all speak english", "why do I need to study geography, I'm never going to need to be able to categorise volcanic vs fold mountains".
Teachers of calculus should have to justify why people should learn it, given 99% of them will never use it, and could be taught something else challenging that they will use instead.
But the same does also apply, and get called out just as often, regarding the humanities, especially given 99% of people can fulfil their humanities interests for free and at their leisure and on their phone with Wikipedia.
> somehow mathematics is the one field which needs to justify its own existence
I'm convinced that justification doesn't make students more motivated anyway.
That being said, justification is needed, not for the students, but for people designing syllabus. They require arbitrage because there are so many hours in a day.
This resonates a lot with me. At school I coasted, ignored all home work, never studied, and was a consistent C. I was continually told I could fo better by putting in some work (an objective viewpoint I agreed with.)
But i had better things to do. I started programming in grade 7, from a book, with a Apple 2 (circa 1982). There were no forums, no Internet just me and a thin booket that came with the computer, plus later, the odd magazine.
It was never "hard", but it was fun. It taught me how to build things, how to approach problems, how to throw something away and make things better. How to imagine.
Ultimately I would study comp Sci- people would teach me the right way to program, and 40 years later it's still my career. As I predicted at the time (somewhat obnoxiously to my then teachers, sorry Mrs Hodge), I had better things to do than raise my Geography score from a C to an A.
School is important, but finding something you enjoy, where "work" feels like "fun" is the truely "gifted child".
Yeah, it didn't hurt that my interest co-incided with history.
But there are lots of professions that can start as teenage interest. A friend of mine was obsessed with birds, he went into nature conservation, started in a bird reserve, and has had a long career climbing the ladder in that space. And also tour-guides bird watchers.
Sure, not everything works out, but if you can turn your fun into your job, then you're one off the lucky ones.
If you escape the employee mentality you can turn almost anything into a small business which provides value to the world.
I really wish the world economy would be structured like this, with smaller entities taking more risk and responsibilities and less enormous corporations filled with people hating their lives.
> At school I coasted, ignored all home work, never studied, and was a consistent C. I was continually told I could fo better by putting in some work (an objective viewpoint I agreed with.)
That was me 100%. I passed elementary school with flying colors without ever doing any homework or actively participating in class (I had better things do to: reading comics and building LEGO contraptions) and simply continued this approach in highschool, with much, much less success. But neither did I care nor did I have any time for studying or doing homework - I taught myself programming in Batch and BASIC in the 90ies from the MS-DOS manpages, and later, when we had access to the internet, Delfi, Visual Basic and C++. I had finished a small tool with a few 100 users when I was 13, and I had to take care of them and fix bugs. There simply wasn't time left for any school work. University was just a natural extension of this childhood interest into adulthood. It didn't feel like school at all, and there I quickly discovered that math (well, computer science math at least) is actually pretty easy, and all you need is practice.
This reminds me of a powerful post I read a while ago with the charter to "Half-ass [your projects] with everything you've got!". Pre-commit to what level of quality you're looking to achieve and then strive to do just what is necessary to get there: https://mindingourway.com/half-assing-it-with-everything-you...
>You know they will never use it in adulthood, outside of certain career choices.
If somebody says stuff like this, then they do not understand math, imo.
Im a little bit sad whenever somebody argues for math by using "no phone available at the moment" argument.
Math is insanely powerful world modeling tool.
Starting from calculating right amount of fence for your garden, to estimation of 500km route arrival time while taking traffic statistics into the account, to data science, ML, whatever more complex.
Since math modeling is everywhere in "modeling" industries like engineerings, financial-ish jobs and other
Then you basically not only get better tools to operate (model) in real world, day2day life, but also it opens you doors to highly paid careers.
But the goal is not to have fancy jobs, but being able to do real world modeling.
IME a big part of this misdirection stems from school focusing on the mechanics of math, rather than the intuition of math.
In the long run, the mechanics of math (how to do long division, or differentiate an expression, or expand a geometric series) are important only insofar as to help us model and predict and analyze the world, intuitively. However, students who do not intuitively see the power of mathematical intuition as a tool for understanding and modeling the world better, think that they are taught the mechanics of math just for the sake of "no phone/calculator/computer available" circumstances.
One of the problems I find with mathematics education is that we seamlessly morph the introductory numeracy lessons of ‘counting’ and ‘arithmetic’ into the subject of ‘mathematics’, without ever stopping and telling kids ‘okay, now you can count, we’re going to start doing something different called ‘mathematics’.’
Mathematics is about examining things and understanding their essence - what statements can we make about all such things? How can we prove that? How can we use that to figure out other things?
But nobody ever tells kids that - they think they’re just in ‘advanced counting’ lessons.
I wonder whether a split in curriculum could help - similar to English language/literature. Mathematics needs a ‘numeracy’ program that starts in kindergarten and covers the mechanical ‘how numbers work’ stuff… then a separate mathematics program that starts in middle school and teaches reasoning and proof. Start with geometry.
I can only speak for myself here, but my journey with Maths did start with concrete, practical relationship between concepts and usage. I still remember fascinated by Geometry and Trigonometry, because how it can be used to create graphics and video games is obvious.
The more abstract it gets, the more it warrants an introduction with "here's how it's used in real life".
I can recall the point when it started becoming mechanical - it was when I started doing Derivatives and Integrations. I was 'just solving puzzles' until I hit a chapter, tucked away way back in the syllabus, almost at the end of the 2 year arc - a chapter about Applied Differential Calculus. I still remember the feeling of "Oh... I get it now" euphoria, with a tinge of sadness - why wasn't this covered early on?
Same thing would have happened in Engineering as well, but at the time my teacher was good. They started by explaining the applications of Fourier Transform before we actually got to learn the mechanics of it.
I've never taught mathematics but I think it's much harder to teach the intuition, and I also think that more people are capable of learning the mechanics than the intuition. So from the point of view of someone designing a methodology and curriculum - you have to design for scale.
I have the same thing with learning languages: I despise focusing on the rules of conjugation and word order and rote memorizing rules, exceptions to them and exceptions to exceptions. I much prefer to bootstrap with some vocab and then acquire by immersion, even if that means I get the conjugations wrong in the beginning. But I also find myself in the minority by far, to the extent that pretty much all language teachers I ever had (except my high school French teacher - she was awesome letting me translate MC Solaar lyrics for grades) didn't even really seem to believe that there are other ways of learning other than studying rules.
You've given one day-to-day example (garden fence) which requires only extremely elementary math, and then listed a bunch of stuff related to specialised career paths... which was the OP's original qualifier:
> outside of certain career choices.
As someone who did a lot of calculus in university, and definitely hit a eureka moment while integrating over vector fields that helped me conceptualise some general day-to-day stuff better in my head, would most people generally benefit from that same conceptual "grokking": of course. Would it be worth the time & effort investment for them if they're not using it in their career: no.
There is no license required for using math. Yea, if you dont want, then you will not use math. But, if you dont have some mental block, then you may find ways to apply math to your problems.
Abstract thinking is really useful during arguments, even about politics, religion, etc.
I'm using math(s) a lot, trigonometry, matrices, algebra… But most of my friend, family, I couldn't convince to learn math just based on the argumentation they can use it in practice.
Yes, calculating the right amount of fence is useful, but not only (as someone pointed out) you don't need a lot of math for that, people just take take a ruler on the plan, or count steps in field, going across the entire length, then estimate, and it works.
> estimation of 500km route arrival time while taking traffic statistics into the account
Who does this? How would you convince a friend to learn math in order to do this? What people do is they remember how long a 200 km route took and so they estimate the 2.5× longer path will take 2.5× more time.
We need examples of real world math applications, because such examples are scarce across the Internet.
>Who does this? How would you convince a friend to learn math in order to do this? What people do is they remember how long a 200 km route took and so they estimate the 2.5× longer path will take 2.5× more time.
Who does it? Navigation app in your phone. They modeled this problem using math. While you could probably model it somehow without math, but why would you want to do it other way, when you have reliable, mature and flexible tool
I didnt write that you shouldnt leverage tech.
Do it, yet be aware how it works and why. You can leverage math in other custom scenarios
This makes the same mistake being called out in the comment you're replying to. The point isn't about the mechanics of solving a differential equation, it's about gathering the intuition about a way of approaching problems.
(Also, while it might not be the tools needed for the average homeowner, there are plenty of optimisation problems similar to "how much fence do I need" which are most easily solved by solving the Euler-Lagrange equations)
They're not everyday things for a lot of people though.
I was in a shop this weekend where the price per 250g of coffee was displayed, but the woman in front of me only had a 175g container. Neither her, the person serving her, or the other person working in the café knew how much to charge her. It's 175/250 * £PRICE_PER_250.
In supermarkets, prices of items are displayed beside each other - a sharing bar of chocolate is £x/100g, but the multipack is £y/item, and each item is z grams. Which is better value?
Cooking - I have a recipe in a book that serves 2 people, but I'm cooking for 3. How much X do I need?
I don't get why calculus is always brought as an example.
It wasn't particularly hard, the entire class had to learn it in high school. We all had it (in a slightly harder shape) in every university course (no matter how detached from what we actually needed)
I forgot all my calculus after high school, had to relearn it in uni and then I promptly forgot again.
Exactly as the article says, it was more about proving we had the capacity (proxy iq test?) to learn it.
You don't need calculus in real life and I think the focus on calculus is ridiculous when we could explore other more practical areas, like category theory (which only my lucky friends who did advanced math got to play with)
But what is the general application of category theory, outside computer science.. and even there the average programmer who hasn't some type theory experience will stare at you with huge eyes when you mention it..
I love the wikipedia intro: Calculus is the mathematical study of continous change, the same way geometry is the study of shape and algebra... that's it perfectly. And the most basic application is in everyone's life and also one of the basic physics thing: The relationship between location, speed and acceleration. I find this very essential, vs category theory at least..
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.
I hate the framing of problems or domains as hard, since it kept me from pursuing them further for many years. And the years later when I tried my hand at those problems, I found that it wasn't nearly as hard as it was being made out to be.
Historically many problems have also been hard until people figure them out, and then they stop being considered hard problems. In recent years this has been mostly true of AI-related topics.
A lot of people have achieved mastery over really hard problems and synthesized their learnings over countless hours, making the information much more easily accessible for future generations.
If you keep hearing someone talk about how some field is hard you should take that as an opportunity to challenge yourself rather than shy away from it. One field that has recently interested me is organic chemistry, which I'm interested in learning mostly because of how many people I've heard talking about how it's so challenging. May I find a worthy opponent.
Edit: This is relevant to HN when talking about C and C++. People talk about these languages as if they're some magical beasts, but in reality you can get really far with them by treating it as a serious endeavor. People will talk about how they don't have full mastery over the language, but you don't need anything close to that in order to be effective. If you know how to program in other languages you can pick up C++ just as easily and start being effective very quickly. No mastery required. It's not that hard.
I hate the framing of problems and domains as hard because I found it so confusing when they turned out to be easy for me. It gave me a false sense of competence and superiority. Because I found calculus (and everything else in school people said was hard) easy, I thought I was just exceptionally capable and other people were incompetent and/or stupid.
I didn't find out until much later (and it still feels like much too late) that I just had an aptitude for learning the subjects traditionally taught in schools in the ways schools normally teach them. I got lucky. Everybody else was just as capable at learning some things in some ways as I was at learning school things in school ways. They just didn't have the luck that their aptitude lined up with what was measured and lauded in childhood like I did.
That meant they all got to learn how to work hard to learn things outside their areas of aptitude in school. I didn't. I didn't realize there were any such things that might matter someday.
I think learning how to learn things that are hard FOR YOU is quite possibly the most important skill in life. The sooner you master it, the better.
Framing some things as "hard" when everything is hard for some people and easy for other people undercuts the more important lesson.
> I think learning how to learn things that are hard FOR YOU is quite possibly the most important skill in life.
I’d certainly agree it’s extremely valuable. There are some failure modes from taking it too far (like anything).
People tend to be happier when they’re very good at things. You also contribute more to the world. If you’re always doing things that are hard for you, you won’t do as much or as well, really by definition. It’s okay to do things that are easy for you. There are a lot of upsides!
Even if you choose an easy path, there will always be hard parts. So if you want to get anywhere, it’s essential to have practice navigating that. Just consider if that’s where you want to be all the time. I’ve done it myself and seen it in others, where you think you’re always challenging yourself, but you’ve actually just put yourself into a life that’s a bad fit.
Ah, the trap of the gifted student! When everything other students find somehow hard is easy for you, you get a delusion that nothing is hard, and that putting in some effort is not necessary. A rude awakening may come at high school, or at college. Those who kept toiling just keep toiling, and overtake you, because.you're not used to pushing through.
I think it's important to learn early enough that things are usually hard when you get far enough into them, and that it's OK, it's not a brick wall, it just takes some effort to keep advancing.
For me, being able to do hard things is not something I learned as a kid. It’s a sum of:
- ADHD symptoms being mild,
- Having slept well (ties in with former point), and
- Having no imminent major worries at the moment (family/health/financial).
Any of these can make the difference between casually trying to understand quantum mechanics, and crawling under the table because I just can’t make this one rectangle on the screen do what I want.
I've seen the enthusiasm for any subject wither away when a child or teen is told a particular subject is hard, whether that be math, a programming language, or learning a musical instrument. I was this way. In their totality, yes, the subject is hard. But what they aren't taught about any difficult subject matter is that they are achievable by breaking them up into a series of small, easier to understand concepts. Their practical utility grows as the number of these small steps are achieved. And as they are achieved, mini demonstrations of their use should be performed so the student understands the importance and gets exited to continue.
Example 1: "I learned five notes in shape 1 of the minor pentatonic scale. That took a bit of practice, but now I'm able to play a bunch of cool licks. Neat! If I continue this path, who knows what other cool licks I can pull off!".
Example 2: "I learned how to import libraries. My lesson had me register a twillio account. I imported the twillio library into my python script. And I copied some code that'll instruct the library to send me a text message. I don't quite understand these python concepts, but wow, this is really cool; I just got a text message from my computer program. The fact that libraries can give me abilities like these is neat. I can already imagine how I can build some basic automation to leverage them. Who knows what else I can accomplish if I discover more libraries and understand python better to actually build something automated!"
Rightly on Wrongly some areas of study have the reputation/stigma of being difficult attached to them.
Back when I was a high school student math (not just calculus - but the entire subject of mathematics) had this reputation as being a "hard" subject as a result scores of my fellow students just decided math is to difficult I'm not going to engage with this.
I suspect this is related to a fear of failure or kids being afraid of "looking dumb" in front of their friends - There was a definite "if I don't try then it doesn't matter if I can't do it." attitude, so they just switched off in those particular classes.
A lot of these attitudes carry forward into adulthood. I'm almost 40 and amongst my generation programming has a similar reputation. People I grew up with think if you can read or write code you are some kind of mystical wizard with powers beyond the understanding of mere mortals.
I see it today at my day job - I work as an engineer (the non software kind). I've seen my coworkers completely baulk at computer code I hear all the same things I heard back in high school. "This is too hard, I can't learn this stuff, I'm not going to bother attempting to understand it".
Fluid Dynamics was a hard subject (in my opinion), Solid Mechanics was challenging a dozen lines of python code is not on the same level.
> math (not just calculus - but the entire subject of mathematics) had this reputation as being a "hard" subject
To me it seems like a lot of the fault was with the curriculum: basically full steam ahead regardless of the class’ understanding. That’s especially bad in math when each chapter uses what the previous taught.
But the point about adult salaried professionals complaining that they supposedly can’t figure something out is disappointingly relatable. I generally believe that most people are "smart" and just don’t tend to bother using their brain as a muscle and that seems to make it doubly irritating to hear such complaints.
> People will talk about how they don't have full mastery over the language, but you don't need anything close to that in order to be effective.
My daily work is 80% C# and 20% Python (to make internal Blender tools for our artists). And I'm really bad at Python. I don't know any of itertools. I don't know zip() besides its name. I don't even use lambda.
The result? My bad code can be easily understood by some of more tech-savvy artists.
for x,y in zip(['a', 'b', 'c'], ['1', '2', '3']):
print(x + y)
>>'a1'
>>'b2'
>>'c3'
Usually you just use it to group two items you're iterating through that are the same length. You CAN do items of different lengths but then when one gets used up the rest of the other get tossed IIRC. Can use it in list comprehensions as well of course.
Zip was simpler than I thought when I first saw it.
> I hate the framing of problems or domains as hard
The easiness or difficulty of a domain or discipline is always in relation to some individual context; and that context includes variables that the learner controls. To the impatient, disinterested or undisciplined, I imagine calculus, learning the kanji, or playing the oboe all seem hard. But to the extent I can marshal patience, curiosity and discipline, the difficult domain becomes just a series of small steps integrated over time. I’m a musician and when a student complains about how hard a piece is, I ask if they can play the first note, then the second. If so, then it’s not hard. Because the process to acquire the whole thing is right there. Yes there are interpretive elements and techniques to be acquired along the way. But nothing is hard unless you are in a great hurry or you don’t really want to do the thing.
> The easiness or difficulty of a domain or discipline is always in relation to some individual context; and that context includes variables that the learner controls
> But nothing is hard unless you are in a great hurry or you don’t really want to do the thing.
I got told this many, many times in my life, and it was incredibly frustrating when it was something I really wanted to do. I discovered after 34 years that I have ADHD, which makes a lot of stuff that can eventually become easy/easier with patience and perseverance to in practice be extremely hard.
I'm bringing this up because a lifetime of guilt and shame for not being able to accomplish something when it was deemed easy, that it "just requires some discipline", said by someone else pushed me away from a lot of things I'm interested in but wasn't able to keep motivated to do them after shame set in. It can spiral if you feel inadequate, and if you live with this you feel inadequate and "catching up" a lot of times.
Specifically, one of those things was music. I tried learning instruments when I was younger but the motivation was not in learning the instrument itself, it was music as a whole. I wanted to understand how it worked and how I could create it, not plow through guitar strumming exercises for months and months, then fingering techniques, then be able to play a few songs, and maybe in some years actually start to create something. To me what worked, in my natural branching way of thinking/learning, was to start producing electronic music some 4 years ago. Just some stupidly cacophonic basic loops in the beginning, which pushed my interest to learn the basics of music theory, learning the basics cleared to me a map I could guide myself through skills I was missing: rhythms, harmony, active listening, etc. After I started understanding what skills I needed to achieve what I wanted then it pushed my motivation to learn an instrument, the piano, and then learning the mechanical skills of the instrument made sense.
I bring this up because since I was diagnosed I had multiple conversations with people that suffered through the same as myself: being called undisciplined, inpatient, disinterested when they couldn't muster the motivation to plow through a structured path when it got boring to them. And that is not under my control, ADHD is much more about lacking motivation control than being hyperactive or actually having an "attention deficit", I get obsessed by things I'm interested in (music is an example), it's just that most of the resources to educate oneself on a discipline/domain is not tailored for people who needs to branch out, find pockets of skills that are interesting and motivating to learn, and putting the puzzle back together after acquiring some skills in a haphazard way than the usual structured learning path.
It is useful to use the words hard and easy. As you mention, changing perspective around these concepts is the crux.
Hard problems or domains are unknowns. Working towards solving hard problems involves thinking through unknowns, which may or may not lead to understanding. An aversion to hard problems is an aversion to the unknown.
Self-image really is important, especially on the dimension of self-efficacy. There are compounding effects in both the positive and negative direction though.
I’ve used this same idea to dig myself out of ruts. When things are fucked up I’ll start paying attention to small things and deliberately “defer” progress on a few bigger things that are harder to do and more costly to fail. Each small win helps build momentum into the next-biggest challenge.
I’ve found this super useful for avoiding “habit destruction” during major life events/travel/moving.
And yet the Art of War tells you to attack when you are ready and your opponent is not. There are some problems that really need to "stew". Others require immediate action even when you aren't ready. It's very hard to tell the difference, but it's wrong to assert that all problems require immediate action (although I agree that if you're going to err, do so on the side of action, in general)
I am discussing a way to be disciplined, which is with decent flexion deliberately built into your self-image.
E.g. I know many people who go through bouts of intense fitness or diet fixations, take a lot of well-deserved pride in their discipline, and then hit a major event that temporarily precludes the fixation. They really struggle to get back on the train.
One major factor, IMO, is that they’re daunted by the intensity of what they achieved before. Obviously in fitness there’s a physical component to this, but there’s a significant mental component as well — especially outside of fitness.
Basically all I’m saying is you can (and should) gradually and deliberately dial up your sense of self-efficacy when it inevitably crosses some local minimum due to events outside of your control. You ought to build a self-image that’s robust to occasional and sometimes significant failures.
I like the idea that "If you can master these topics, imagine what other topics you could master if you put your mind to it" — again, for the empowerment.
It's not about a thing being hard. Walking is hard to a paraplegic. It's about overcoming a thing and feeling good about it (instead of external rewards, like a piece of candy or good grades).
The real problem with school is that it replaces empowerment with gamification externalized rewards. You're not learning calc for the sake of understanding the world, you're doing it for a line item on a checklist. That doesn't come with empowerment.
With the mere framing of "you can do [hard thing] to prove you can do hard things" is a bad framing because it could be anything — from doing calc, to bungee jumping, to drinking a gallon of milk (please don't). This framing doesn't actually lead to empowerment (and then self-improvement).
There are infinitely many hard things. It is hard to learn Japanese. We don't require that every high school student attain basic proficiency in it though.
The reason we learn calculus in high school is because it is foundational for many advanced STEM fields, and we will yield better results during university for the small percentage of students who go into those fields by forcing everyone to learn it in high school. Or, moreso, that's a viable justification for learning it today. Had history taken a different shape maybe we would learn something else, or maybe not. But the point is that calculus is not an arbitrary hard thing we learn for arbitrary reasons.
How does a kid know that they definitely will/will not be going into a STEM field in the future? Is it better to have your school-going years slightly marred by calculus and not particularly need it after, or want to go into STEM later in life but not have the grounding necessary?
> and we will yield better results during university for the small percentage of students who go into those fields by forcing everyone to learn it in high school
I like the idea, but I’m gonna say that (a) calculus is more than a good challenge and (b) math is actually easy.
To understand how things actually work, you need math, especially calculus. Deep learning? Calculus. Statistics? Calculus. Finance? Calculus. Physics? Calculus. Mech E, robotics, earth science, econ? Calculus.
Second, calculus, like all math, is easy. Like that’s the point, it’s the science of simple things. That math is competitive and presented as a cryptic challenge is beside the point — it is designed to make it possible for anyone to reason for themselves and solve problems. The sense of impatience and criticism around math is totally unwarranted and isn’t good for anyone.
I get kind of bummed when I see schools spending so much creativity and enthusiasm on art and theater. There really is no reason why science should be thought of as judgmental, difficult and painful, while putting on a play is creative, inviting and fun.
>Second, calculus, like all math, is easy. Like that’s the point, it’s the science of simple things. That math is competitive and presented as a cryptic challenge is beside the point — it is designed to make it possible for anyone to reason for themselves and solve problems. The sense of impatience and criticism around math is totally unwarranted and isn’t good for anyone.
lol, typical HN answer.
Mathematics is not "easy", it is a niche; some people are good at, some are average, some are bad.
Expecting every person to be able to do math is folly. People will fail. People already fail at memorizing math concepts and literally just applying said concept by plugging the numbers around.
Some people just don't "get" it, I know I don't "get" probability, but is pretty good at other branches of math like group theory and calculus.
To some people, deriving derivatives is basically black magic, but to me it's pretty intuitive.
Thus, if the world economy relies on people being good at probability then I am screwed.
Fortunately for me, the world economy somehow relies on people being good at writing texts on a computer to tell it what to do (programming).
I personally think programming is piss easy (its the actual problem being solved that is hard, programming is just knowing how to knock a hammer)
However, there are people out there that simply can't program, either because they are not interested or not capable.
Perhaps they are good at something else that is not entirely marketable?
Is it wrong to be that way?
Being humble is one thing, but not realizing one's gift is another.
> Being humble is one thing, but not realizing one's gift is another.
Very true, but this doesn't change the fact that math is actually simple, but it is generally taught so badly that most students can't "get" it.
I did a bit of private tutoring back when I was in college (and I'm still doing it for my own children), and every person learns in a different way. It is not always easy to find the right way to convey an idea, but once you find it you can see in the student's eyes how it just clicked.
Totally anecdotal, but I once helped someone who "didn't get how percentages work" get a really high (with respect to her previous attempts) GMAT score in maths.
I've found that some disciplines need calculus, others don't. I've found deeply understanding logic and binary/hex MUCH more important than calculus in my career.
Physics and gravity are behind any kind of predictable motion. But you don't need to understand those things at all to be a successful surfer. Even though surfing is entirely about physics and calculable predictions, performing the act doesn't require any detailed knowledge of either topic.
It's the same with calculus and almost everything you mentioned. People can create algorithms, make statistics calculations and financial predictions, build robots, etc. All without any knowledge of calculus of any kind.
The skills of all of those things are based on calculus like surfing is based on physics. Related, but not in the sense of practical application. Knowledge of the math that underlies the math that underlies the thing is neither required nor sufficient for actually doing the thing.
Putting on a play is a very human interactive activity. Until math lets you interact with many humans in a rich experience, it won't be put on a pedestal like drama.
I saw an article where they asked a bunch of scientists and engineers if they actually used calculus in their professions. None of them did. They used Excel and the R programming language.
Calculus is needed to understand, like, basic sophomore and junior year engineering stuff. You can maybe pass the tests without understanding where the models came from if your professors are just really lazy and only give you canned problems, but that isn’t a goal we should strive for.
And engineers should usually be using battle-tested models rather than coming up with their own derivations, so to some extent using the calculus day-to-day shouldn’t be necessary in many cases. But it is necessary in order to understand where the models came from. This is what separates engineers from tech-priests.
It depends on what the definition of 'using calculus' is, as well. I use the concepts of integration and differentiation all the time in my work: it's completely integral (hah) to a good chunk of what I do (as well as complex numbers, fourier transforms, and a bunch of other 'advanced' maths). What I don't do is grind through working out the analytic solutions to odd integrals or differentials. Firstly because it's rarely useful, and secondly because I have a machine to do that for me in the cases that it is. I think it's unfortunately common to have the attitude that the latter is the majority of what calculus actually is in practice, because it makes up a lot of calculus education, but it's not the case.
As an engineer calculus is pretty fundamental to my job. It underpins all sorts of stuff - Heat transfer, fluid flow, stress and strain rates, beam deflection, fracture mechanics etc.
I may not be solving differential equations by hand but I'm using knowledge about calculus everytime I reason about our industrial process.
The excel part is probably referring to solvers - where you plug in boundary conditions and spits out a solution. Edit - and excel or R (or Matlab) is what you use in lieu of needing to solve this stuff by hand.
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.
Readit News
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.
[1]https://en.wikipedia.org/wiki/Commercial_butterfly_breeding [2]https://money.cnn.com/2014/08/14/smallbusiness/butterfly-far...
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.
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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.
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> 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.
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.