Once you get obsessed with something, you'll see it everywhere. For you, that's calculus.
And no I'm not kidding, a couple years ago when I was studying distributed systems, I saw the CAP theorem everywhere. Why isn't distributed systems part of the high school curriculum? It's used in basically ALL computing devices (before cloud computing, distributed system theories were applied in multicore systems...)
Yeah, but lots of parents lie far more often than would be useful for a child. But more importantly, absolutely honest is not the same as absolutely open. You can be absolutely truthful in the things you say without causing harm. If you think the truth would cause a child harm, then direct them away from a topic or tell them they aren't ready, etc.
I took calculus in High School just so I could make a comment here saying I did so. It almost cost me my diploma as I didn't want to do the homework. I paid attention in class though and learned just enough to pass the exams.
It’s the foundation of a lot more than just physics.
It’s absolutely crucial to economics and business, and it is a travesty that it isn’t a required part of lower division curriculum. You cannot grasp micro/macro/applied/business economics without understanding relationships between changing variables.
Understanding calculus helps you understand much more deeply than you could have grasped form just learning algebra what kinds of numerical relationship problems it is possible to actually figure out. Ideally you also need some infinite series, some linear algebra, some combinatorics, and an appreciation of complex numbers as well, so the absence of deep coverage of those from a typical high school math curriculum in favor of putting calculus on a pedestal is more annoying.
But the idea that if you know how the rate of change of a thing changes over time, that that gives you enough information to understand it completely? That’s pretty important and deep.
When I studied Calculus in high school, it was taught via mathematical proofs and concepts. I didn't really "get it" and struggled to keep a C average...until that one day I was working on a problem in the library about water draining from a pool it hit me that "it's about rate of change!". That was it...that concept changed Calculus from some weird math thing to something I could understand and get my head around.
It also underscored the poor teaching methods at my school. I was somewhat vindicated by being the only person in my class to get a 5 on the AP test. I also ended up in a major where I did almost nothing but calculus for undergrad and grad work.
I suspect it's the justification for many PhDs. If you don't go into academia then chances are the content isn't that relevant, but the experience of having to problem solve in uncharted territory is a great confidence booster and skill to have.
> but the experience of having to problem solve in uncharted territory is a great [...] skill to have.
While I do love to solve such problems, in many business areas there are hardly any hard problems to solve as part of your job: either because they don't exist, or because hardly any boss would be willing to let you work focusedly for years to potentially solve one the hard problems that do exist (which is what a PhD in mathematics or physics is about).
The issue is that one tree—calculus is pretty important and fundamental to lots of fields of study—is covered in tinsel and lights, and also for some unfortunate reason some folks have gotten it into their heads that they’ll never need it and might as well light it on fire.
Exactly this. The article isn't really at all about calculus, but rather the benefit of challenging yourself more generally. Doing challenging things that push you out of your comfort zone better prepare you to do the things you actually want to do later in life.
Self-image really is important, especially on the dimension of self-efficacy. There are compounding effects in both the positive and negative direction though.
I’ve used this same idea to dig myself out of ruts. When things are fucked up I’ll start paying attention to small things and deliberately “defer” progress on a few bigger things that are harder to do and more costly to fail. Each small win helps build momentum into the next-biggest challenge.
I’ve found this super useful for avoiding “habit destruction” during major life events/travel/moving.
I 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.
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)
This is exactly what I do. "Discipline" in the traditional sense isn't an option. I have a severe impairment with executive function and can't do specific things at specific times.
So I gave up on that. I go with the flow of my brain. When I can, I build and shore up systems that will survive chaos and reduce my cognitive load. When I can't do plan A, hopefully I can manage plan B, C, or D.
This seems to work pretty well. I'm more reliable long-term when I allow myself to be unreliable day-to-day. :)
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.
> 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
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?
I think there is an similar argument for Greek and Latin in grammar schools. And the same for ancient Chinese in Chinese schools. Partly for cultural immersion and partly for brain gymnastics.
Math, and proof based Math such as Number Theory and Analysis is definitely in the same league if not for a career in academics.
I don't want to be overly rude, but this is nonsense. The reason to learn calculus is that it's incredibly useful in several domains and never learning it prevents you from become a skilled practitioner in those domains which in turn reduces your future earning potential.
That is a good comic. But I am not sure "reduces your future earning potential" is really right. If you are going for future earning potential, then there are other things. Probably along the lines of - learn to code - find a way to migrate to the US - live in SF/NYC - learn algorithms and data structures - learn leet code - emotional control/resilience for putting up with those kinda jobs ... etc
I reckon there are plenty of PhDs earning less than $100k around the world, who know calculus and matrix algebra like their ABCs.
Everything can be incredibly useful in several domains, but we don't teach everything. Instead, people learn what they need when they start working in that domain.
The point of the article is that calculus is not taught because it might be incredibly useful for a small percentage of students, but because it measures their ability to pick up a hard subject and ace it.
One of the dumbest things I’ve ever heard a math teacher tell me/the class is “you’ll probably never use this outside of this classroom”! And I’ve heard this story before, so it seems slightly common?
I entered college without a rock solid foundation in mathematics and it made things much more difficult.
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.
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.
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.
I hate the framing of problems and domains as hard because I found it so confusing when they turned out to be easy for me. It gave me a false sense of competence and superiority. Because I found calculus (and everything else in school people said was hard) easy, I thought I was just exceptionally capable and other people were incompetent and/or stupid.
I didn't find out until much later (and it still feels like much too late) that I just had an aptitude for learning the subjects traditionally taught in schools in the ways schools normally teach them. I got lucky. Everybody else was just as capable at learning some things in some ways as I was at learning school things in school ways. They just didn't have the luck that their aptitude lined up with what was measured and lauded in childhood like I did.
That meant they all got to learn how to work hard to learn things outside their areas of aptitude in school. I didn't. I didn't realize there were any such things that might matter someday.
I think learning how to learn things that are hard FOR YOU is quite possibly the most important skill in life. The sooner you master it, the better.
Framing some things as "hard" when everything is hard for some people and easy for other people undercuts the more important lesson.
> I think learning how to learn things that are hard FOR YOU is quite possibly the most important skill in life.
I’d certainly agree it’s extremely valuable. There are some failure modes from taking it too far (like anything).
People tend to be happier when they’re very good at things. You also contribute more to the world. If you’re always doing things that are hard for you, you won’t do as much or as well, really by definition. It’s okay to do things that are easy for you. There are a lot of upsides!
Even if you choose an easy path, there will always be hard parts. So if you want to get anywhere, it’s essential to have practice navigating that. Just consider if that’s where you want to be all the time. I’ve done it myself and seen it in others, where you think you’re always challenging yourself, but you’ve actually just put yourself into a life that’s a bad fit.
I agree that it wouldn't be a good idea to spend your life focused on things that are hard for you, but since, as you said, there will be hard parts no matter what you do, learning how to learn the things that are hard is the critical skill.
You don't really have to learn how to learn things that are easy for you. That's what gets them classified as "easy." But you'll never accomplish anything unless you learn how to work through those inevitable hard parts, and the sooner you can learn that, the better.
Ah, the trap of the gifted student! When everything other students find somehow hard is easy for you, you get a delusion that nothing is hard, and that putting in some effort is not necessary. A rude awakening may come at high school, or at college. Those who kept toiling just keep toiling, and overtake you, because.you're not used to pushing through.
I think it's important to learn early enough that things are usually hard when you get far enough into them, and that it's OK, it's not a brick wall, it just takes some effort to keep advancing.
This extends into life as well. I play golf (for fun [2]), which cunningly has a scoring system that tells me I'm objectively rubbish, and getting (mostly) worse.
I've found this professionally useful in combating the seductive idea that "because I'm -really- good at one thing, I'm good at everything. "
I've seen the opposite in customers sometimes. Doctors who are very good doctors, have strong feelings about UI (that are objectively just wrong[1].) But because they operate in a culture which treats their word as law, they find it hard to accept that others may have skills in other areas they lack.
If you are an expert in something I recommend including something else in your life to keep you humble. That humility allows you to be a better spouse, parent, and human being. (And ironically a better expert who's able to recognise and adopt an idea or solution that'd better than yours even in your area of expertise.)
We choose to do these these things, not because they are easy, but because they are hard.
[1] think yellow comic sans italic text on blue background levels of wrong.
[2] golf is fun precisely because it is hard. Things that are easy are not fun. The pleasure of 1 perfect shot out of 100 tries is the dopamine that keeps us coming back.
I've found that one of the best ways to dismantle an overblown ego is to play StarCraft (Brood War or SC2, the result will be the same) or really any other 1v1 game. The direct feeling of being beaten, potentially overrun completely, time and time again, especially after putting in hard work is very humbling.
Fighting games are very good for this except they've always invited a certain mental block in people where they'll declare certain things "cheap" and end up offloading a lot of the humbling experience and not take it to heart. This type of scrub mentality exists everywhere but in fighting games it seems to come extra easy to people, perhaps because they have too many potential things they can blame (tier of their character, match-up supposedly being bad, etc.).
Golf might be better because its me against me. No excuses about the other guys ability. But sure, as long as you find your fun, and your humility, it doesn't really matter where it is.
I console myself by believing that Tiger Woods is pretty bad at CSS :)
> I've found that one of the best ways to dismantle an overblown ego is to play StarCraft (Brood War or SC2, the result will be the same) or really any other 1v1 game.
Sadly no. Playing SC2 online weaned me off online competitive games. Because of all the kids calling me a stupid noob ... when I won. The ones beating me were polite.
I was always told that at the next level of school, I would finally meet a challenge. I got through a master's degree without its happening, and unfortunately that just drove the initial point deeper, that I was somehow fundamentally different from other people rather than just very luckily in an environment exceptionally well-suited to my aptitude (even though the rest of the world is not).
For me, being able to do hard things is not something I learned as a kid. It’s a sum of:
- ADHD symptoms being mild,
- Having slept well (ties in with former point), and
- Having no imminent major worries at the moment (family/health/financial).
Any of these can make the difference between casually trying to understand quantum mechanics, and crawling under the table because I just can’t make this one rectangle on the screen do what I want.
> I 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.
> 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'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!"
One of the things about doing hard things out in the world is that people often recognize you for it. With parenting, it’s the hardest thing you’ll ever do at times and most people in your life really don’t care. It’s a thankless job.
> ... teenager asks why they need to learn calculus
> But if we avoid hard things
I don't see how you can justify the former by arguing the latter. These two are orthogonal. If I were that teenager, I think what I really would want to ask is that why it has to be calculus instead of some other things that is also hard but with obvious real world application like writing a small 3D game engine.
And my answer to that question is you probably shouldn't if your were in an ideal education system. You would be taught what interesting interactions you could have with the physical world, and be induced to discover calculus or some other math tools that helps you understand how the interactions really work and demonstrates you really need such tools. You're more likely to grasp them when you're driven by curiosity.
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).
To me what's more important than Calculus being hard, and I think that's especially true for maths more broadly is that it's beautiful and one of the fundamental ways how we can make sense of the world. Everyone benefits from doing some maths.
I studied maths in uni and while I've not used it much, even as a programmer, I still enjoy doing it. My dad never had much schooling but now that he's retired he actually picked up a few of my books and slowly worked through high school to now undergraduate courses. He's having a lot of fun with it.
Don't do hard things that are unrelated to your actual goal; they're in unlimited supply, and you could be doing hard things that are relevant instead.
This essay wouldn't have impressed me in middle school; I don't know why it's on our front page.
Not to distract everyone from complaining about calculus, but this reminded me of something I heard from a person with a Ph.D. in astrophysics from Caltech. They were not working in astrophysics, but they said the degree was still quite valuable to them. Whenever they had trouble learning something, rather than feel stupid, they reminded themselves: “I have a Ph.D. in astrophysics from Caltech, so I am definitely not stupid. This is just hard.”
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.
It will be amusing if we find out that all the things come in discrete quanta, even space-time, which I hear hasn't been ruled out. Calculus and real numbers might not be sitting as pretty.
There are lots of corner cases where the difference matters, like when integrals with finite bounds can go to infinity where a finite sum cannot. Boundary cases and surprise infinities seem to be a common problem problem the theoretical physicists have battled in their theories for the last century.
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.
As a chemical engineering student, it's a whole lot of formulas cobbled together to spit out calculations that would otherwise be done on hand (especially empirical methods that compute estimates where a clean answer is not possible like $\int{\frac{\sin{x}}{x}}$).
The formulas usually come from what you study in school (e.g. the Redlich-Kwong equation in Physical Chemistry to estimate the properties of real/non-ideal gases).
What's neat is that before computers were popular in the early 20th century, these calculations (mostly empirical equations' iterations) had to be done by hand by engineers. So yeah... you can imagine how tedious it was (especially when you have tiny errors due to humans compounding) yet they were still able to build complex things (like factories!).
You don't use calc much when you use R, unless you're working on a problem that involves calculus, which isn't many problems, usually?
R is usually used to do data stuffs in my experience. Like, "take in this data from this CSV, and do these manipulations" which may involve math but not often calculus.
im not though. I don't think you need to know measure theory and understand how to formalize probability in terms of sigma algebras to do professional stats.
But I would be very skeptical of a professional data scientist that doesn't understand things like derivative, integral, limit on an intuitive level. I don't know how you would understand distributions without that knowledge
for actual understanding, yes it is. The most basic important results, the law of large numbers, and the central limit theorem both require calculus to understand.
if you make a class without calculus, it is essentially just a bag of tricks and surface level understanding
Calculus is needed to understand, like, basic sophomore and junior year engineering stuff. You can maybe pass the tests without understanding where the models came from if your professors are just really lazy and only give you canned problems, but that isn’t a goal we should strive for.
And engineers should usually be using battle-tested models rather than coming up with their own derivations, so to some extent using the calculus day-to-day shouldn’t be necessary in many cases. But it is necessary in order to understand where the models came from. This is what separates engineers from tech-priests.
I wonder if there’s a way to teach Calculus in a more practical way. I use the principles of Calculus constantly in many aspects of my job, but I don’t think I’ve ever done say, integration by parts professionally. Understanding deeply what an integral and a derivative are is a very useful skill though.
As an undergrad I tutored folks in a sort of “calculus for non-STEM students” class that focused more on practical stuff and applications… and TBH I think trying to shield them from the complexity didn’t really do them any favors. I often found that they had some practical algorithm memorized step-by-step and some graphical or conceptual intuition, but when the steps and intuition betrayed them we’d end up spending a while backtracking to find where they lost track of the concrete rules, and then we could work our way forward to catch up to their intuition. Practical results are good guideposts but don’t replace full understanding I think.
That’s just my perspective, though, I have a pretty simplistic, algebraic way of looking at math. I could never be a mathematician or see true beauty in math, it is just a bunch of little rules to me.
IMO calculus class is fundamentally just relatively abstract compared to the stuff before it. But once you’ve finished it, your engineering and science classes should be full of practical, reinforcing applications, right?
Isn't the actual calculations of e.g. integrals the least useful part of calculus for non-stem students? Instead the basic concepts are where the easily accessible value lies.
Teaching tricks without teaching the concepts not only fails in the way you describe (it prevents building ideas further). It also fails to teach anything useful at all. Because barely anyone outside of STEM wil ever have to solve any kind of integral or derivative in their life.
The valuable part of calculus to me was understanding the concepts of limits, differentiation, integration, and a tiny bit of differential equations.
Learning how to actually solve integrals or differential equations was useless other than it teaching me more about calculus and how it is useful. For calculations in practice I will turn to wolfram alpha, but it has some value to understand what wolfram alpha is doing under the hood.
I think a calculus course whose tests defocusses calculations might be very valuable for practically minded people. Knowing to think about derivatives is much more useful in practice than actually being able to calculate a derivative.
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 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.
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'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.
>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.
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.
From that perspective, going through any kind of struggle (and building willpower/discipline) would be a good enough proxy for learning calculus.
It’s a little sad that the best value that the OP can impute for learning calculus is masochism — I cannot imagine saying that for anything in the school curriculum that I actually learned/understood. I wonder (in good faith) whether the OP actually even absorbed calculus at all… (i.e. can they solve calculus problems even today, for example?) — if not, they’re not the person who should be making authoritative comments on the usefulness of calculus.
Calculus (Taylor approximations, perturbation modeling, error propagation, significant figures in measurement precision, gradient descent, etc — just off the top of my head) is so deeply embedded in my thinking, that it strongly shapes how I think and amplifies my effectiveness!
I disagree with the OP’s claim so fundamentally — they might as well claim that schools/education focus on literacy for the same reasons.
I remember when the AP Calculus test questions were released only a month or so after I had taken and aced the exam. I had no idea whatsoever how to solve the problems I had solved easily in the very recent past.
It's as you say. I didn't absorb it at all. I stuffed it in short-term memory and passed an exam. But that's all I learned. I learned how to stuff things in short-term memory.
I raised the same questions as Nat, but I think he hasn't really discovered why we were asking that. The reason why I was looking for a good reason to study calculus is because I was lazy, and if I'm lazy enough then I can change the definition of what "hard" means. For instance, I could convince myself that if my could teach calculus, and him being just another dude, then surely I could learn calculus.
I would also be cautious about setting yourself up for a hard life. The takeaway resonates well with the HN crowd, myself included, because we like challenging ourselves. There's a lot of people out there who are simply looking to satisfice their lives, and you'll need another way to motivate them to learn calculus.
There will come a time when you have enough money, enough of everything, and deeper questions will arise. What is the meaning of existence? Why is there anything instead of nothing?
You will see that math plays a very fundamental role in our reality, and once you start seeing it in such a manner it may begin to interest you.
I feel like math is taught from too high of a level initially. I did a second major in Logic along with CS at university, I was pretty crap at maths in HS and never really 'got it', I could manage but never had any intuition. After doing a bunch of logic papers and learning about axioms and peano arithmetic everything made a lot more sense from a foundational perspective and it really changes the way you look at numbers and the world around you
There is no end to things one cab learn, to what one can (attempt) to master. Hmm... Perhaps I am addicted to learning to avoid thinking about existential issues.
I lost interest in school around the 10th or 11th grade. I never took any math classes beyond what was required to graduate way back in '96 in Florida. I also didn't go to college.
I've been a professional web developer since 2005 and a development manager (who still codes) since 2017. I don't understand the first thing about Calculus or even logarithms. I'm sure if I did, I'd probably be a better developer. I've had employees try to explain to me fairly basic log notation and my eyes just glaze over. It's never impacted my abilities, nor the respect and admiration I get from them as a well-experienced and knowledgable developer, but I can't help but feel ignorant.
I need to go back to the basics and work my way up; I've lost a lot of it. Where do I start? Kahn Academy?
A lot of math is cumulative, i.e. built on top of the prior concepts/tools. There are some things that are effectively the start of their own branches, but a lot of them then go back into a tangle of general mathematics that's all deeply interrelated, and also a subject onto itself when you get into ways to convert problems into totally different representations to use other mathematical tools on them.
In your case, follow something like Khan academy through the normal grade school programs to pick up where you left off and work backwards on picking up any concepts you're weak on then pursue whatever threads interest you. Wolfram can also help you look up specific things or find necessary formulas if, e.g. you just need the formula for a cone or to know how to integrate sin().
You probably understand logs intuitively. Don’t worry about notation, here’s the idea: sometimes we count digits, not values.
When we say someone has a 6 figure salary, we are counting how many 0’s (10s) to takes to get there.
For memory, we say something has 32 bits and can have 2^32 possible values. It’s more graspable to talk about the “address size” vs the “number of possible values”, especially for things that grow fast (like storage).
I’d suggest starting with your intuitions and slowly translating them to math.
Khan Academy is great, but there are also an incredible number of great youtube videos. I would watch this 3blue1brown video to see if you like his style : https://www.youtube.com/watch?v=WUvTyaaNkzM
> go back to the basics and work my way up; I've lost a lot of it. Where do I start?
You might want to check out my book "No Bullshit Guide to Math & Physics," which starts with a high school math review, and goes up to calculus. It's specifically written for adult learners (self contained + lots of practice exercises).
Just a meta comment: Don't think of calculus as hard in the same way learning a language or learning to paint is hard. Calculus is more of a gotcha moment. You fumble in the dark for a few focused hours (or minutes) and then you Get It. From then on it's relatively easy.
If I had to sort those - painting, learning a language, calculus. So, yeah, it seems hard to me. But I've never truly given it a lot of attention. That there is an "aha" moment for some, is promising.
I think the better way is to come out with a lot of interesting real world scenarios that require calculus or other math. Math doesn't get created out of a vacuum. It was created for a reason. I think if we can avoid diving into abstractions too quickly and focus on specific real world problems and lead the kids step by step to the solutions , that would be more interesting.
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[ 3.6 ms ] story [ 262 ms ] threadBoth 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...)
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.
While I do love to solve such problems, in many business areas there are hardly any hard problems to solve as part of your job: either because they don't exist, or because hardly any boss would be willing to let you work focusedly for years to potentially solve one the hard problems that do exist (which is what a PhD in mathematics or physics is about).
Although I'd argue that just having the ability to press on in the face of challenges is a closely related and widely useful skill.
I’ve used this same idea to dig myself out of ruts. When things are fucked up I’ll start paying attention to small things and deliberately “defer” progress on a few bigger things that are harder to do and more costly to fail. Each small win helps build momentum into the next-biggest challenge.
I’ve found this super useful for avoiding “habit destruction” during major life events/travel/moving.
But it will cost you everything if you don't."
Discipline is a muscle. Go Build it. Key is to understand different activities require different muscles.
Be mindful of picking your activities, but dont keep on waiting.
E.g. I know many people who go through bouts of intense fitness or diet fixations, take a lot of well-deserved pride in their discipline, and then hit a major event that temporarily precludes the fixation. They really struggle to get back on the train.
One major factor, IMO, is that they’re daunted by the intensity of what they achieved before. Obviously in fitness there’s a physical component to this, but there’s a significant mental component as well — especially outside of fitness.
Basically all I’m saying is you can (and should) gradually and deliberately dial up your sense of self-efficacy when it inevitably crosses some local minimum due to events outside of your control. You ought to build a self-image that’s robust to occasional and sometimes significant failures.
Goal Mindset = "reach difficult state of fitness"
Process Mindset = "Time to do my gym routine, just with lighter weights bc I'm temporarily weakened"
So I gave up on that. I go with the flow of my brain. When I can, I build and shore up systems that will survive chaos and reduce my cognitive load. When I can't do plan A, hopefully I can manage plan B, C, or D.
This seems to work pretty well. I'm more reliable long-term when I allow myself to be unreliable day-to-day. :)
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
Math, and proof based Math such as Number Theory and Analysis is definitely in the same league if not for a career in academics.
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.
I entered college without a rock solid foundation in mathematics and it made things much more difficult.
Historically many problems have also been hard until people figure them out, and then they stop being considered hard problems. In recent years this has been mostly true of AI-related topics.
A lot of people have achieved mastery over really hard problems and synthesized their learnings over countless hours, making the information much more easily accessible for future generations.
If you keep hearing someone talk about how some field is hard you should take that as an opportunity to challenge yourself rather than shy away from it. One field that has recently interested me is organic chemistry, which I'm interested in learning mostly because of how many people I've heard talking about how it's so challenging. May I find a worthy opponent.
Edit: This is relevant to HN when talking about C and C++. People talk about these languages as if they're some magical beasts, but in reality you can get really far with them by treating it as a serious endeavor. People will talk about how they don't have full mastery over the language, but you don't need anything close to that in order to be effective. If you know how to program in other languages you can pick up C++ just as easily and start being effective very quickly. No mastery required. It's not that hard.
Hard problems or domains are unknowns. Working towards solving hard problems involves thinking through unknowns, which may or may not lead to understanding. An aversion to hard problems is an aversion to the unknown.
Back when I was a high school student math (not just calculus - but the entire subject of mathematics) had this reputation as being a "hard" subject as a result scores of my fellow students just decided math is to difficult I'm not going to engage with this.
I suspect this is related to a fear of failure or kids being afraid of "looking dumb" in front of their friends - There was a definite "if I don't try then it doesn't matter if I can't do it." attitude, so they just switched off in those particular classes.
A lot of these attitudes carry forward into adulthood. I'm almost 40 and amongst my generation programming has a similar reputation. People I grew up with think if you can read or write code you are some kind of mystical wizard with powers beyond the understanding of mere mortals.
I see it today at my day job - I work as an engineer (the non software kind). I've seen my coworkers completely baulk at computer code I hear all the same things I heard back in high school. "This is too hard, I can't learn this stuff, I'm not going to bother attempting to understand it".
Fluid Dynamics was a hard subject (in my opinion), Solid Mechanics was challenging a dozen lines of python code is not on the same level.
To me it seems like a lot of the fault was with the curriculum: basically full steam ahead regardless of the class’ understanding. That’s especially bad in math when each chapter uses what the previous taught.
But the point about adult salaried professionals complaining that they supposedly can’t figure something out is disappointingly relatable. I generally believe that most people are "smart" and just don’t tend to bother using their brain as a muscle and that seems to make it doubly irritating to hear such complaints.
I didn't find out until much later (and it still feels like much too late) that I just had an aptitude for learning the subjects traditionally taught in schools in the ways schools normally teach them. I got lucky. Everybody else was just as capable at learning some things in some ways as I was at learning school things in school ways. They just didn't have the luck that their aptitude lined up with what was measured and lauded in childhood like I did.
That meant they all got to learn how to work hard to learn things outside their areas of aptitude in school. I didn't. I didn't realize there were any such things that might matter someday.
I think learning how to learn things that are hard FOR YOU is quite possibly the most important skill in life. The sooner you master it, the better.
Framing some things as "hard" when everything is hard for some people and easy for other people undercuts the more important lesson.
I’d certainly agree it’s extremely valuable. There are some failure modes from taking it too far (like anything).
People tend to be happier when they’re very good at things. You also contribute more to the world. If you’re always doing things that are hard for you, you won’t do as much or as well, really by definition. It’s okay to do things that are easy for you. There are a lot of upsides!
Even if you choose an easy path, there will always be hard parts. So if you want to get anywhere, it’s essential to have practice navigating that. Just consider if that’s where you want to be all the time. I’ve done it myself and seen it in others, where you think you’re always challenging yourself, but you’ve actually just put yourself into a life that’s a bad fit.
Find a niche, do one thing and do it well. You do you.
Or if you want to be a generalist, that's very valid too, but again, you do you.
The constant glorification of growth and the massive industry built around it doesn't help.
You don't really have to learn how to learn things that are easy for you. That's what gets them classified as "easy." But you'll never accomplish anything unless you learn how to work through those inevitable hard parts, and the sooner you can learn that, the better.
I think it's important to learn early enough that things are usually hard when you get far enough into them, and that it's OK, it's not a brick wall, it just takes some effort to keep advancing.
I've found this professionally useful in combating the seductive idea that "because I'm -really- good at one thing, I'm good at everything. "
I've seen the opposite in customers sometimes. Doctors who are very good doctors, have strong feelings about UI (that are objectively just wrong[1].) But because they operate in a culture which treats their word as law, they find it hard to accept that others may have skills in other areas they lack.
If you are an expert in something I recommend including something else in your life to keep you humble. That humility allows you to be a better spouse, parent, and human being. (And ironically a better expert who's able to recognise and adopt an idea or solution that'd better than yours even in your area of expertise.)
We choose to do these these things, not because they are easy, but because they are hard.
[1] think yellow comic sans italic text on blue background levels of wrong.
[2] golf is fun precisely because it is hard. Things that are easy are not fun. The pleasure of 1 perfect shot out of 100 tries is the dopamine that keeps us coming back.
Fighting games are very good for this except they've always invited a certain mental block in people where they'll declare certain things "cheap" and end up offloading a lot of the humbling experience and not take it to heart. This type of scrub mentality exists everywhere but in fighting games it seems to come extra easy to people, perhaps because they have too many potential things they can blame (tier of their character, match-up supposedly being bad, etc.).
I console myself by believing that Tiger Woods is pretty bad at CSS :)
Sadly no. Playing SC2 online weaned me off online competitive games. Because of all the kids calling me a stupid noob ... when I won. The ones beating me were polite.
- ADHD symptoms being mild,
- Having slept well (ties in with former point), and
- Having no imminent major worries at the moment (family/health/financial).
Any of these can make the difference between casually trying to understand quantum mechanics, and crawling under the table because I just can’t make this one rectangle on the screen do what I want.
The easiness or difficulty of a domain or discipline is always in relation to some individual context; and that context includes variables that the learner controls. To the impatient, disinterested or undisciplined, I imagine calculus, learning the kanji, or playing the oboe all seem hard. But to the extent I can marshal patience, curiosity and discipline, the difficult domain becomes just a series of small steps integrated over time. I’m a musician and when a student complains about how hard a piece is, I ask if they can play the first note, then the second. If so, then it’s not hard. Because the process to acquire the whole thing is right there. Yes there are interpretive elements and techniques to be acquired along the way. But nothing is hard unless you are in a great hurry or you don’t really want to do the thing.
> But nothing is hard unless you are in a great hurry or you don’t really want to do the thing.
I got told this many, many times in my life, and it was incredibly frustrating when it was something I really wanted to do. I discovered after 34 years that I have ADHD, which makes a lot of stuff that can eventually become easy/easier with patience and perseverance to in practice be extremely hard.
I'm bringing this up because a lifetime of guilt and shame for not being able to accomplish something when it was deemed easy, that it "just requires some discipline", said by someone else pushed me away from a lot of things I'm interested in but wasn't able to keep motivated to do them after shame set in. It can spiral if you feel inadequate, and if you live with this you feel inadequate and "catching up" a lot of times.
Specifically, one of those things was music. I tried learning instruments when I was younger but the motivation was not in learning the instrument itself, it was music as a whole. I wanted to understand how it worked and how I could create it, not plow through guitar strumming exercises for months and months, then fingering techniques, then be able to play a few songs, and maybe in some years actually start to create something. To me what worked, in my natural branching way of thinking/learning, was to start producing electronic music some 4 years ago. Just some stupidly cacophonic basic loops in the beginning, which pushed my interest to learn the basics of music theory, learning the basics cleared to me a map I could guide myself through skills I was missing: rhythms, harmony, active listening, etc. After I started understanding what skills I needed to achieve what I wanted then it pushed my motivation to learn an instrument, the piano, and then learning the mechanical skills of the instrument made sense.
I bring this up because since I was diagnosed I had multiple conversations with people that suffered through the same as myself: being called undisciplined, inpatient, disinterested when they couldn't muster the motivation to plow through a structured path when it got boring to them. And that is not under my control, ADHD is much more about lacking motivation control than being hyperactive or actually having an "attention deficit", I get obsessed by things I'm interested in (music is an example), it's just that most of the resources to educate oneself on a discipline/domain is not tailored for people who needs to branch out, find pockets of skills that are interesting and motivating to learn, and putting the puzzle back together after acquiring some skills in a haphazard way than the usual structured learning path.
My daily work is 80% C# and 20% Python (to make internal Blender tools for our artists). And I'm really bad at Python. I don't know any of itertools. I don't know zip() besides its name. I don't even use lambda.
The result? My bad code can be easily understood by some of more tech-savvy artists.
Zip was simpler than I thought when I first saw it.
Example 1: "I learned five notes in shape 1 of the minor pentatonic scale. That took a bit of practice, but now I'm able to play a bunch of cool licks. Neat! If I continue this path, who knows what other cool licks I can pull off!".
Example 2: "I learned how to import libraries. My lesson had me register a twillio account. I imported the twillio library into my python script. And I copied some code that'll instruct the library to send me a text message. I don't quite understand these python concepts, but wow, this is really cool; I just got a text message from my computer program. The fact that libraries can give me abilities like these is neat. I can already imagine how I can build some basic automation to leverage them. Who knows what else I can accomplish if I discover more libraries and understand python better to actually build something automated!"
> 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.
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).
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.
This essay wouldn't have impressed me in middle school; I don't know why it's on our front page.
To understand how things actually work, you need math, especially calculus. Deep learning? Calculus. Statistics? Calculus. Finance? Calculus. Physics? Calculus. Mech E, robotics, earth science, econ? Calculus.
Second, calculus, like all math, is easy. Like that’s the point, it’s the science of simple things. That math is competitive and presented as a cryptic challenge is beside the point — it is designed to make it possible for anyone to reason for themselves and solve problems. The sense of impatience and criticism around math is totally unwarranted and isn’t good for anyone.
I get kind of bummed when I see schools spending so much creativity and enthusiasm on art and theater. There really is no reason why science should be thought of as judgmental, difficult and painful, while putting on a play is creative, inviting and fun.
The formulas usually come from what you study in school (e.g. the Redlich-Kwong equation in Physical Chemistry to estimate the properties of real/non-ideal gases).
What's neat is that before computers were popular in the early 20th century, these calculations (mostly empirical equations' iterations) had to be done by hand by engineers. So yeah... you can imagine how tedious it was (especially when you have tiny errors due to humans compounding) yet they were still able to build complex things (like factories!).
R is usually used to do data stuffs in my experience. Like, "take in this data from this CSV, and do these manipulations" which may involve math but not often calculus.
calculus is the most basic of mathematical machinery, that it is essentially a requirement to do anything else.
But I would be very skeptical of a professional data scientist that doesn't understand things like derivative, integral, limit on an intuitive level. I don't know how you would understand distributions without that knowledge
if you make a class without calculus, it is essentially just a bag of tricks and surface level understanding
And engineers should usually be using battle-tested models rather than coming up with their own derivations, so to some extent using the calculus day-to-day shouldn’t be necessary in many cases. But it is necessary in order to understand where the models came from. This is what separates engineers from tech-priests.
That’s just my perspective, though, I have a pretty simplistic, algebraic way of looking at math. I could never be a mathematician or see true beauty in math, it is just a bunch of little rules to me.
IMO calculus class is fundamentally just relatively abstract compared to the stuff before it. But once you’ve finished it, your engineering and science classes should be full of practical, reinforcing applications, right?
Teaching tricks without teaching the concepts not only fails in the way you describe (it prevents building ideas further). It also fails to teach anything useful at all. Because barely anyone outside of STEM wil ever have to solve any kind of integral or derivative in their life.
Learning how to actually solve integrals or differential equations was useless other than it teaching me more about calculus and how it is useful. For calculations in practice I will turn to wolfram alpha, but it has some value to understand what wolfram alpha is doing under the hood.
I think a calculus course whose tests defocusses calculations might be very valuable for practically minded people. Knowing to think about derivatives is much more useful in practice than actually being able to calculate a derivative.
I may not be solving differential equations by hand but I'm using knowledge about calculus everytime I reason about our industrial process.
The excel part is probably referring to solvers - where you plug in boundary conditions and spits out a solution. Edit - and excel or R (or Matlab) is what you use in lieu of needing to solve this stuff by hand.
So . . . logic and number bases? Do tell!
lol, typical HN answer. Mathematics is not "easy", it is a niche; some people are good at, some are average, some are bad. Expecting every person to be able to do math is folly. People will fail. People already fail at memorizing math concepts and literally just applying said concept by plugging the numbers around. Some people just don't "get" it, I know I don't "get" probability, but is pretty good at other branches of math like group theory and calculus. To some people, deriving derivatives is basically black magic, but to me it's pretty intuitive.
Thus, if the world economy relies on people being good at probability then I am screwed. Fortunately for me, the world economy somehow relies on people being good at writing texts on a computer to tell it what to do (programming). I personally think programming is piss easy (its the actual problem being solved that is hard, programming is just knowing how to knock a hammer) However, there are people out there that simply can't program, either because they are not interested or not capable. Perhaps they are good at something else that is not entirely marketable? Is it wrong to be that way?
Being humble is one thing, but not realizing one's gift is another.
"If you do not believe that mathematics is simple, it is only because you do not realize how complicated life is." -von Neumann
Very true, but this doesn't change the fact that math is actually simple, but it is generally taught so badly that most students can't "get" it.
I did a bit of private tutoring back when I was in college (and I'm still doing it for my own children), and every person learns in a different way. It is not always easy to find the right way to convey an idea, but once you find it you can see in the student's eyes how it just clicked.
Totally anecdotal, but I once helped someone who "didn't get how percentages work" get a really high (with respect to her previous attempts) GMAT score in maths.
It's the same with calculus and almost everything you mentioned. People can create algorithms, make statistics calculations and financial predictions, build robots, etc. All without any knowledge of calculus of any kind.
The skills of all of those things are based on calculus like surfing is based on physics. Related, but not in the sense of practical application. Knowledge of the math that underlies the math that underlies the thing is neither required nor sufficient for actually doing the thing.
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 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.
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've been a professional web developer since 2005 and a development manager (who still codes) since 2017. I don't understand the first thing about Calculus or even logarithms. I'm sure if I did, I'd probably be a better developer. I've had employees try to explain to me fairly basic log notation and my eyes just glaze over. It's never impacted my abilities, nor the respect and admiration I get from them as a well-experienced and knowledgable developer, but I can't help but feel ignorant.
I need to go back to the basics and work my way up; I've lost a lot of it. Where do I start? Kahn Academy?
In your case, follow something like Khan academy through the normal grade school programs to pick up where you left off and work backwards on picking up any concepts you're weak on then pursue whatever threads interest you. Wolfram can also help you look up specific things or find necessary formulas if, e.g. you just need the formula for a cone or to know how to integrate sin().
When we say someone has a 6 figure salary, we are counting how many 0’s (10s) to takes to get there.
For memory, we say something has 32 bits and can have 2^32 possible values. It’s more graspable to talk about the “address size” vs the “number of possible values”, especially for things that grow fast (like storage).
I’d suggest starting with your intuitions and slowly translating them to math.
(Without being a shill, I wrote about real world logs here, it may help: https://betterexplained.com/articles/using-logs-in-the-real-...)
One very good place to start is Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry by George F. Simmons (less than 150 pages!).
You might want to check out my book "No Bullshit Guide to Math & Physics," which starts with a high school math review, and goes up to calculus. It's specifically written for adult learners (self contained + lots of practice exercises).
You can see a PDF preview here https://minireference.com/static/excerpts/noBSmathphys_v5_pr...
The concept map from the book is independently useful to check out: https://minireference.com/static/conceptmaps/math_and_physic... And you should also check out this SymPy tutorial https://minireference.com/static/tutorials/sympy_tutorial.pd... which can help you build a bridge between coding skills and math operations.