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Comment by WheelsAtLarge | Hacker News Reader

“Bingo!!” said the teacher. “It’s the same thing with calculus. You’re not here because you’re going to use calculus in your everyday life. You’re here because calculus is weightlifting for your brain.”

Total BS...

There are better ways to exercise your brain that will be many more times better than Calculus. This is HW so one that comes to mind is programming. But there are so many more. Here are a few, understanding and fixing a car, understanding music and playing music, art appreciation, literature and understanding the human condition and on and on. Recent research has shown that doing daily exercise is a great way to keep a healthy body and brain, rather than sitting on your butt learning a useless subject.

Yes, there are professions where Calculus is needed and there are people that truly enjoy math. Cool, take all the math you need and want to learn. You should take it.

Anyone that tells you that Calculus is a good way to exercise your brain is just trying to justify their job. Don't for a minute believe that it's the best way to use a limited resource like your time.

Source: Me, it took me 3 semesters of Calculus to figure out that it was useless to me and 90+% of the people that take any of it.

P.S. note: Many people disagree. Good, you should not take it as absolute truth. I guess my real point is that you should question whether you really need to take Calculus. Don't just take it blindly because you are told you should. You have other options.

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Mathematics is learned in a spiral process. It takes probably 3 classes or years before you become competent in it. You are introduced to calculus in high school, but only at the very end. Really, most of the work of calculus is becoming competent at algebra and pre-calculus, which teach you incredibly important concepts like exponential growth, etc (which fundamentally depend on calculus for their development and motivation, but not needed for basic calculations).

Don't think algebra or exponential growth matter? I think these concepts are critical to basic citizenship. Real understanding of exponential growth helps you understand viscerally why, for instance, you should nip a viral outbreak in the bud, and reducing the spreadrate even slightly (R0) can make a huge impact later on, even if spread isn't totally stopped. This is all just gibberish if not learned in high school.

Algebra is used in programming and is basically an introduction to many different programming concepts. Symbolic manipulation of variables, etc, needs to be understood at a basic level to competently program anything, or even use Excel spreadsheets effectively (which almost everyone who ever works a desk job--which is most people--will eventually come in contact with), which is a type of programming.

If you learned 3 semesters of calculus, then you must have learned this in college. If your job is programming-related, then it's pretty relevant for you to understand concepts in calculus like limits, rates of change, total area under a curve, plus having a confident grasp of algebra (which is much of the actual work of calculus).

Blue collar jobs like machinists, homebuilder/carpentry, plumbing, electrician, etc have tons of need for other areas of math that utilize concepts in algebra, pre-calculus, and geometry. As things become more automated, mechatronics and g-code programming are starting to become more relevant in a lot of trades that were previously highly manual. Tuning a PID loop is a fairly normal task for some of these. And you definitely benefit from pre-calc and calculus for things like this, being literally what the I and D stand for.

> I think these concepts are critical to basic citizenship.

In all my life this was never really something I considered important, but the whole pandemic thing gave an entirely new definition to ‘basic education’.

‘I learned all this stuff in high school, why do I have to explain these basic concepts?!’ Was a very common thought.

In all these cases learning the math is but one path to the understanding. Not a requirement.

Learning the math and the mathematical notation is probably the most powerful and straightforward way to fully understand the concept.

Like many, many people who understand exponential growth, I never took a calculus class.

That's certainly possible, even without self-learning. The topic is explained in precalc, typically. But it's important enough that I think going further is helpful.

The many, many people, including those in government, who clearly didn’t, prove that you’re an outlier.

So, I don't mean to force you into something you don't want, but do you think people who don't understand exponential growth are missing something relative to you? That they could benefit from the knowledge you have?

That's what I know having understood calculus to those who have bits and pieces of the concepts (exponential growth included) but don't have a big picture. If you have the time and ability to learn more, why limit yourself? Why allow yourself to be put at a disadvantage? And worse (not saying you are, it's hard to gauge from your comment) why would you be in favor of stiffing other people from being better?

I'm not getting the impression commenters are trying to limit others. Rather, there is a strange obsession with math most individuals will never need and doesn't satisfy them, in a world where one can learn so much else.

Even the exponential growth through calculus example is obsessively nitpicky: just draw a few graphs of y=x, y=2x, y=x*x and y=2^X. Most people will grasp the idea, and it's 30 minutes at most.

> most people will grasp the idea, and it's 30 minutes at most

I think we experienced very different pandemics. I could pick 100 people off the street and I guarantee you 9/10s of them aren't able to do a logarithmic change of base. If you can't do that can you say you grasp exponential growth?

> If you have the time and ability to learn more, why limit yourself? Why allow yourself to be put at a disadvantage?

So what is that advantage for everyday people? I see some people are making it a "citizenship requirement" but except for exponent, which is not a part of calculus anyway (OP original point), there seems to be little advantage to it.

You cannot really understand the nature of exponentiation if you do not understand calculus! Whoever is saying is likely ignorant or they have a shallow understanding, and when I say shallow, I really mean it, as in their knowledge is not sufficient. You cannot know a how dependence based on a power over a linear relationship is "stronger" without talking about rate of change, which is literally calculus. Perhaps you can have an intuitive notion of change, and that's fine, but the point of education often is to either refine intuition or correct it.

And that then leads me to my point above: if your knowledge is shallow, to the point that it limits you, then why clamor to limit yourself or further to limit others?

Did mathematicians before Newton not understand the concept of exponential growth?

Interestingly, the modern notation for exponents (including variable exponents and non-integer exponents) was developed by Euler after Newton's calculus. This modern, simple notation certainly makes it easier to explain the concept to the masses... And apply it in a spreadsheet or something.

Compound interest was understood earlier, too, of course. Thousands of years ago, in fact. But not with as clear and simple notation. It was often made illegal.

To reason with agility in a domain, one must have succinct language (verbal and symbolic) to express and manipulate ideas in that domain. Such domain-specific language facilitates both communication and understanding.

Symbolic mathematical notation was the breakthrough that most greatly increased the rate of mathematical breakthroughs thereafter.

Well said!

I think that how much calculus you get in highschool depends on which country you live in, I got 3 years (though the first one was minimal), I also got more in highschool physics classes.

But we also have national exams for entrance into Uni and no "general ed" requirement because we're expected to have met that minimum requirement in highschool

I feel like calculus in high school can be so easily motivated but isn't. For what a high schooler is concerned, you can use it to model the depreciation of mobile devices, cost per day of upgrading devices or how much you save each day by waiting to buy (assuming used market), estimated lifetime revenue for each piece of social media content, the same parametrized on subscribers at time of upload, estimated lifetime additional subscribers per piece of content, etc.

This assumes most people care about making optimal financial decisions. They don’t.

You know what they care less about? The complete abstract and being forced to learn a complex subject without a single morsel of motivation.

You're right! You know how you make them care? Demonstrating over and over again how important it is. At school. A place where we are supposed to equip people with the knowledge of how to function effectively in society. To give them the tools to live the best life they can.

School is a place where we indoctrinate all sorts of ideas into students, maybe we could spend a little more time highlighting financial decisions since it is so core to quality of life?

My discrete math professor always made the problems involve food. Because college students are always interested in food, he said.

Finance has calc all over the place as well. Marginal costs for instance.

>This is HW so one that comes to mind is programming.

> > Real understanding of exponential growth helps you understand viscerally why, for instance, you should nip a viral outbreak in the bud, and reducing the spreadrate even slightly (R0) can make a huge impact later on, even if spread isn't totally stopped. This is all just gibberish if not learned in high school.

I think programming/algorithm analysis and things like discrete simulations will give you a more durable notion of basic exponential growth for things like virus outbreaks than high-school calculus which is going to focus on things the derivative of the exponential function being similar to the function and stuff about Euler's number.

I hope you do realize "discrete simulations" is an application of calculus (analysis really). The "continuous" version of calculus is a special case. Sure though, this is a problem with the way calculus is taught (too much focus on symbolic differentiation and integration, although that becomes valuable somewhat if you become a physicist or engineer, primarily).

Yeah, discrete simulations that show things like exponential growth are basically just numerical calculus.

> Don't think algebra or exponential growth matter? I think these concepts are critical to basic citizenship. Real understanding of exponential growth helps you understand viscerally why, for instance, you should nip a viral outbreak in the bud, and reducing the spreadrate even slightly (R0) can make a huge impact later on, even if spread isn't totally stopped. This is all just gibberish if not learned in high school.

Indeed. I remember talking to a doctor who worked in the ER when the first wave of COVID (brutal in my country) was brewing. She said that it wouldn't be a big deal, they had like 50% of beds vacant (or something like that) so they would be able to handle it just fine. I said that by looking at the data, I thought they would run out of beds next week. Her expression was dismissive, like "this guy doesn't work in healthcare, hasn't set foot in an ER, what does he know?"

The next week, ERs were overloaded, of course. It was in plain sight from the straight line in log-scale graphs. But for most people (including most doctors) the interpretation was (and still is) "wow, this virus is rough, it comes in sudden waves out of nowhere!". Just because they don't understand exponential growth.

Do you really think doctors don't understand, or never learned, exponential growth? I knew a bioengineering student at Cal that was pretty smart and barely made it into med school. They had to go to a DO school instead of MD. So unless the bar for doctors used to be much lower than it is today, every doctor 'knows' what exponential growth is.

Whether they apply it to the real world is another thing.

I know people that don't think raising the minimum wage basically just causes inflation. They're just wondering why apartments in undesirable areas became 3x more expensive when minimum wage went from $5 to $15.

In my country most definitely don't. They take a little calculus in high school, but practically zero at university, unless they take an epidemiology elective or something like that. I know this well because my wife is also a doctor.

It probably varies a lot by country, like many things.

You certainly don't need calculus to understand exponential growth; you'd get a good sense of it from just looking at a chart.

The ability to apply learned theory to real life in cross-domain ways is not common and not so easily taught by rote. Teaching calculus to everyone won't solve this.

> I know people that don't think raising the minimum wage basically just causes inflation. They're just wondering why apartments in undesirable areas became 3x more expensive when minimum wage went from $5 to $15.

That's a pretty poor example though as there are many factors that contribute to property prices and there isn't just a mechanical link between wages and house prices.

Low interest rates probably have a much greater effect, given that most people aren’t buying houses with cash.

> Whether they apply it to the real world is another thing.

'Knowing' it but not applying it (and in what is literally a textbook circumstance, no less) is itself a failure of understanding.

K12 math classes are not about understanding. It’s more like typing. The symbol manipulation rules you need to apply are few and straightforward. Just memorize where the keys are. Applying them is also simple. Just press. After that, the difference between an A and a C is all about hitting the keys in succession faster and with fewer mistakes. It’s a Zen thing; overthinking it will never get you there. Concepts like exponential growth and rates of change may be presented to you in lecture, but letting them in your head while doing problems is a classic blunder. Don’t think just do.

I am willing to bet that most educated people who walk around with gross misapprehensions of rates of change and exponential growth phenomena, have in fact drilled the computations just as well as anyone else.

You don’t need calculus to teach exponential growth. 99% of the public doesn’t start plotting graphs whenever there is news of a viral outbreak.

It depends.

David Epstein's Range is a good way to look at outcome based learning.

There are 'Kind Learning Environments'. Things like chess, golf, concert piano, etc. The goal is easy to define, you can rank yourself against others, and the feedback on effort is quick. In such scenarios, so argues Epstein, the 10,000 hour grind is a best way to achieve success.

There are 'Unkind Leaning Environments'. Things like tennis, jazz, business, etc. The goal is difficult to define, you cannot easily rank yourself against others, and the feedback on effort is slow or nonexistent. In these environments, Epstein says that a 'browsing' approach is best. One where you learn as much as you can about as many disparate things as possible and to still deep degrees all the same. You want as many pegs to hang a hat on as you can get, curiosity is not wasted time.

I would say that, in terms of education for the masses, learning Calculus is a great way to develop the 'browser' side of things. General/public education is inherently to be made for the 'unkind learning environment'. Specializing and 10,000 hour grind-fests obviously aren't suitable.

Calc is especially useful as it gives the ideas of derivatives, rates, limits, and integrals for your mental toolset. These are powerfully broad ideas ripe for application. Additionally, as it is traditionally taught, it helps expand the mind to true higher math and lets pupils see how deep that logic/math rabbit hole can go. Lastly, the inescapable history behind it's development is another great dive and gives another avenue for the 'browser' mentality.

I can scarce think of a better subject outside of religious texts that provides such great tutelage for the 'unkind' learning environment that is life.

Tennis has got to be closer to a kind enviornment, a well defined set of strokes, immediate outcomes, a ranking system, and full information.

The benefit of calculus isn’t about the ability to write things down on paper in a fancy code and playing a game to solve it. The benefit is that it paints your waking reality in a color you weren’t aware you could observe previously: curves, and expectation, and prediction.

Based on your take here, I’m gonna guess that you’re in but haven’t yet graduated college.

I’m not sure how you’re going to suggest “learning programming” rather than learning calculus, as calculus is a foundational element of all modern languages. For loops are a further generalization of Leibniz notation, in a rough but very real fashion.

You can only understand a car so deeply without brushing up against physics, the study of which is classically explained by (you guessed it). Sure, you can argue that “you don’t need physics to understand a car well enough to fix it.” Okay, congrats on mastering the adult version of putting the right shaped block into the right shaped hole.

Understanding music really doesn’t require a ton of calculus, unless you want to go into building instruments and music software. If you want to do sound design, you’re also fucked, because understanding Fourier transforms is an important aspect of being a good design engineer.

To me, it sounds like you failed calculus twice and now are trying to prevent people from sharing in your grief. That’s less admirable than you think—-it’s not that calculus is fundamentally hard without reward or merit, it’s likely that somebody failed to indicate to you the importance of calculus.

You are giving plenty of ways in which calculus can be useful. But the GP wasn't arguing against claims that calculus is useful. The GP was arguing against a claim that calculus was useless, but you should study it anyway as "weightlifting for your brain". Your arguments about how calculus is useful would be better directed at the teacher the GP was arguing against.

> But the GP wasn't arguing against claims that calculus is useful.

So no clue who GP is. Grandparent is my best guess. And with respect to you, that would be the guy I was responding to, but that couldn’t possibly be correct because he said

> rather than sitting on your butt learning a useless subject

From context, inferred to be calculus, esp because

> took me 3 semesters of Calculus to figure out that it was useless to me and 90+% of the people that take any of it

Soooo… who is GP

"GP" is Hacker News lingo for "the grandparent post to this one", which in the post I made responding to you before, means the post you were responding to.

> I’m not sure how you’re going to suggest “learning programming” rather than learning calculus, as calculus is a foundational element of all modern languages. For loops are a further generalization of Leibniz notation, in a rough but very real fashion.

Eh, I'm pretty much opposed to GP's assessment that calculus is useless; in fact, it is probably one of the biggest intellectual achievements of the past couple centuries and modern society would be unimaginable without it.

But I don't really see the connection with programming. Programming/CS is mostly discrete maths and little calculus (with some exceptions, like complexity theory, because it's just easier to talk about functions R -> R than Z -> Z, and numerical analysis, which is about how nice theorems break down when you have to work with messy approximations instead of the real values). Calculus is about the real numbers and we can't even encode the majority of real numbers on computers.

> I really don’t see the connection with programming

It may just be the stupid way my brain is wired. When I think about calculus I can’t help but also consider programming, and vice versa. Okay, Calculus is not a precursor to learning to code. But the DNA of calculus is definitely there.

First, how building blocks of programming and calculus are similar.

1) We can probably throw out the control flow concepts, although they vaguely map to the notion of intervals on evaluated integrals.

2) I said for loops are a generalization of Leibniz notation. I also said it’s a rough relation. You said you don’t see the parallel because you claim pure maths calculus deals with reals and computers are discrete. Yep. Real numbers are discrete at the infinitesimal limit (grab your torch and pitchforks). I hope this is enough explanation on that front. Loops roughly = integrals. It’s purely theoretical. I get that you can’t actually represent an infinite precision real using bytes.

Next, algorithms.

1) I think there’s another very loose but valuable perspective in which calculus and efficient algorithm implementation at least shop at the same grocery store, if not fool around on the down low. I can imagine every possible implementation for solving the knapsack problem as being distributed in a higher dimensional space. There are a ton of bad ones out there with dogshit runtimes. But somewhere near the middle is one that goes zoooooom. That’s an optimization problem—that’s calculus.

2) And then within solving a problem itself. The *good* solutions make use of derivative-like notions. Properties about the problem which you use to solve it efficiently are effective pseudo-derivatives. The way that you can use a derivative plus a point to approximate some next point forward, you can use problem topology plus current state to improve state a bit further until you converge upon an “answer”.

The list goes on.

Calculus is in programming and programming is in calculus. They are cross-pollenated dialects of the mother tongue of the universe.

> Real numbers are discrete at the infinitesimal limit (grab your torch and pitchforks).

I don't even know what that means. If you embed the real numbers in the hyperreals, where you do in fact have "infinitely close" numbers, those numbers aren't discrete either, as the hyperreal numbers are dense too.

> Loops roughly = integrals

I think this is the wrong way around. Loops loosely correspond to mathematical sums (finite or possibly countably infinite). We got integrals once people started asking themselves "what if you could have something like sums, but over intervals that are arbitrarily/infinitely small?". In that sense, integrals are an extension or generalisation of sums/loops - in the sense that sums are basically integrals with a discrete measure - but you don't need to more complicated, general concept to understand the simpler one.

Of course, different areas of maths (and CS is basically a branch of maths) are inter-linked. That's the beauty of maths. Some people will see calculus everywhere, others will immediately see connections to category theory, logic, topology, etc. Maybe in 50 years we will discover a new branch of maths and suddenly see that it can be found everywhere, but that doesn't mean that we needed to know that branch to do all the things we were doing up until that point.

But I don't think calculus is as fundamental to CS/programming as it is to, say, physics or much of the rest of the natural sciences. You do mention some examples (optimisation problems), but by and large I don't need to understand limits and Cauchy sequences to program a loop or even to prove statements about push-down automata or graphs or my favourite type system.

(You mention topology. Topology is an abstraction over analysis which can be used in discrete settings too, it doesn't fundamentally require the real numbers.)

> (You mention topology. Topology is an abstraction over analysis which can be used in discrete settings too, it doesn't fundamentally require the real numbers.)

Ugh I wanted to respond but this little quip puts the rest of your post into perspective about what an obnoxious person you must be.

As programmer (20+ years of experience) with a master degree in engineering, I think I can compare programming with calculus. In general, calculus is continuous, while most of programming is discrete. I dare to say that you can be a very good programmer without knowing derivatives and integrals, which are only at the very basis of calculus. Most formulas used while programming, have already been derived by a physicist or mathematic. Most (99 per cent) of programming paradigms can be fully understood and applied without calculus. I learned about exponential functions, logarithms and for loops way before knowing calculus.

Strongly disagree. To extend the original metaphor, calculus is an exercise, not a whole workout. Sure, if you only do squats, you may not end up looking as good in a tank top as the guy who does arms all day. On the other hand, you're never going to reach peak physical performance if you skip leg day.

Good luck trying to understand any modern ML paper without a solid understanding of calculus, for example.

Ok, given the number of people that take Calculus, how many will ever read an ML paper? For that matter, how many even know or care what ML is?

This only adds to my point.

Economics, Statistics, ML, Engineering all use calculus. You might be able to get by in some areas without it, but it's not as if it's useless. It's also really easy and straightforward. You would create calculus if it didn't exist.

I can’t help but feel like there’s far more people who do not need more than really simple calculus than people who do. Should we really disadvantage people by frustrating them with subjects almost useless for them just for the sake of the few who will need it? There could be other things to learn with as much or more interest for them. Calculus can be a good thing but we should stop to make mandatory for a lot of persons and refuse it to reconsider its use.

Calculus is really simple. The part some people have trouble with is that they're forced to remember all the other math they've learned to that point. Others struggle with it because the concepts are new.

There are successful studies teaching 12 year olds calculus, using software that helps them with the computational side (they don't have to remember all their trig rules or how to factor but they need to know when to apply the core ideas). These arguments that people shouldn't learn calculus (or should because they need some rigor) are all wrong. A small number of people should learn to do by hand, all the non-calculus computation they'll run into in calculus. Most people should probably just be taught the main concepts and how to apply them using readily available software that can help them with the calculation part.

From the general population? Not that many.

From this site? I'd wager a significant percentage.

Here's maybe a more generally relevant example: Have you heard about this thing called inflation?

I bet even in the HN population the number is still less than 10% for work. Including hobbyists, maybe 15%.

"understand any modern ML paper"

I believe 7 people read an academic paper on average. (1 of those is a mother).

There are going to be people in the world that are capable of reading things like an ML paper and there are going to be people who are not. Calculus (or other advanced mathematics) is part of what you'll need to understand those papers.

If having the ability to read such papers and understand such concepts isn't something you want for yourself then you definitely shouldn't take advanced mathematics courses. However, many people see the ability to understand those types of things to be a useful skill in giving them opportunities in the future.

I have read many ML papers, and even understood a few of them. My understanding of calculus is beyond weak: I know what a derivative (and a partial derivative!) is, and what integration is, but God help me if I have to do one.

I've got a pretty firm grasp on linear algebra though. I don't think calculus plays into modern ML that much.

The basic building blocks of ML are simle MAC operations, weights and offsets are applied to inputs to produce outputs. These building blocks are combined to form a neural network. To understand that, or to train or apply a model, you do not need calculus.

>I guess my real point is that you should question whether you really need to take Calculus.

There are very few classes in school that any student "really needs" yet for some reason Calculus, or math in general is the one that takes the brunt of this argument. Why?

Whens the last time you needed to know that Hydrogen has 1 proton and 1 electron for instance?

Here's an HW case about why knowing about Hydrogen will help in life.

Question: Why does Space X use kerosene and liquid oxygen rather than liquid hydrogen and liquid oxygen a better power to weight fuel.

Simply because of the size of the molecule. The Hydrogen molecule is such a small molecule that it's difficult an expensive to use vs kerosene. It's very easy for it to leak. As we have seen in the Artemis 1 rocket that uses hydrogen.

There's my use case. I've used it at least once.

How is having that knowledge any different to if you knew the mathematics involved in the building and launching of the rocket?

Knowing that didn't help you at all. If you didn't know about Hydrogen that rocket still would have launched. It's purely for your own interest.

Unless you're a SpaceX engineer of course but then the other 99.9% of the people on the planet don't "need" to know about the size of Hydrogen and we are back to the same argument.

Ah yes, I too design rockets on a daily basis.

Just above, you wrote:

>Ok, given the number of people that take Calculus, how many will ever read an ML paper? For that matter, how many even know or care what ML is?

..and now you, apparently without joking, assert that the physics info is different because it's necessary in rocket science, the one thing that is colloquially used to describe knowledge that normal people will never have to worry about?

> Here's an HW case about why knowing about Hydrogen will help in life.

*proceeds to give an example only applicable to rocket scientists ...

I think you accidentally disproved your own point.

This may be an EU-centric view, but I think the "problem" with math is that, basically, it's a requirement for "success" in life.

Selection at elite universities is mainly math-based. Sure, you're expected to have great grades in other subjects, but basically, if you suck at math, you're stuck with "suboptimal" paths.

Yes, I know many people have made it without a college degree, or by following some other path. But most of "the rich" have been through elite universities, which require good grades in math. So, it can be perceived as a kind of gatekeeping.

No one cares about chemistry.

So, since neither chemistry nor calculus are seen as "useful in day to day life", but math is used as a selection criterion, people talk about math.

The flipside is that all the well renumerated jobs are very mathematically intensive.

Finanance, technology, they all are.

The vast majority of people working in those industries don't do any more calculations or maths that the average office worker. Yes deep down in the heart of their computer systems there's a ton of analysis going on, but only a very few people have to work on the calculations. Even them most financial transactions are just adding and subtracting.

I think the point of math is to learn other more advanced math.

Which is so useful to the few that will have jobs that need it, that they want to push it as hard as possible just to give them every possible advantage, because it's hard.

They want as many people in advanced STEM as possible, because that's basically like being a billionaire in terms of the level of wealth and comfort, and things like chemistry might solve some really big problems.

Also, a really large number of people still think math is something you actually use daily. These are people that still balance checkbooks and make budgets with paper and do woodworking with fractions instead of CAD apps.

Math really is useful to anyone who isn't comfortable letting a computer do half their thinking.

I might never even own a checkbook in my life, and I've never even used basic algebra IRL. But I can see why someone who never got comfortable with a "There's an app for that" mindset would think long division was a life skill.

I highly doubt I have the talent needed to ever learn a useful amount of math (My idea of useful is enough to get an EE or CS degree), so I don't make it a super high priority to get better at it.

We should be teaching subjects which will (A) ultimately be useful to as many people as possible and (B) exercise the brain as much as possible.

I'm inclined to think statistics and programming would fulfill these requirements better than calculus.

----------

> Here are a few, understanding and fixing a car, understanding music and playing music, art appreciation, literature and understanding the human condition and on and on.

High schoolers already spend a lot of time analyzing literature. I do think they should spend more time with other forms of art as well—why teach only literature criticism, when literature is just one of many art forms?

However, this work exercises your brain in a distinctly different way than mathematics, and I do think students should learn both.

Have you people actually taken a basic statistics course that doesn't require calculus? They are all about memorizing formulas that you plug numbers in to calculate different statistical measures, they don't teach you to understand anything at all.

We teach students calculus at that age since teaching them statistics is basically a dead end, we teach statistics to those poor students who will need to calculate statistical measures without understanding them but we should not force every kid to suffer through that boring thing. Calculus is way more interesting, since kids can easily understand it and you can derive all results on your own, statistics is just plug and chug, much worse than calculus ever could be since students aren't ready for it yet.

> Have you people actually taken a basic statistics course that doesn't require calculus? They are all about memorizing formulas that you plug numbers in to calculate different statistical measures, they don't teach you to understand anything at all.

Disclaimer I should have included: I personally took statistics in high school instead of calculus.

I can't say what my experience would have been like if I'd known calculus, but loved learning statistics. I don't remember exactly what we did, but I recall it being quite conceptual. Certainly not just a ton of formulas.

There is a parallel in college physics. Typically, most colleges have two tracks: "Calc-based" and "non-calc-based" physics. Everybody finds the non-calc-based course to be an utterly bewildering exercise in memorizing formulas. Even the calc based students are baffled by it. The calc based course is widely regarded as easier and more intuitive -- if you also take calc.

Stats is the same way. I was a math major, and my college had two stats tracks: "Math stats" and "stats for scientists." The first track was 2 semesters, and we had to prove everything. Of course we used calc. The second track was 1 semester, and was an utterly bewildering exercise in memorizing formulas.

I took "math stats," but was then asked to run the discussion section for "stats for scientists." There were things that were utterly intuitive to me, but that I couldn't satisfactorily explain to the students, such as the need for different formulas and methods for discrete and continuous distributions.

Freshman economics. The professor bent over backwards to make sense of the formulas related to things like the supply and demand curves, because he couldn't use derivatives. Also, it was 1982, and yes, the professor showed us the Laffer Curve.

I had a similar experience, I took stats without calculus just because that was what was available/feasible in college, and eventually I took a more rigorous calculus based probability and stats course in my early 30s. Despite having to relearn my calculus, the difference in conceptual clarity was obvious from even the first lecture because the answer to “why does this work?” has a direct interpretation through calculus that you usually can’t get without calculus.

Feels a lot like hiking: as you get more fit you also get access to more fun.

Thinking more about this, my answer to "why learn calculus" is that so many ideas and tools of modern life are explained, and even relate to one another, through calculus. Without calculus, many simple things become unapproachable, or have to be approached by rote learning of formulas and rules.

When in grad school, a friend of mine taught a freshman course on modern history, and in the first week he introduced a general outline of calculus. The students were surprised, but he explained that calculus is a thread woven through modern history thanks to the importance of science and technology.

> since kids can easily understand it

Maybe with a good teacher. I took calculus courses in high school and college and came out clueless. I could manage the rote work to complete the course, but the big picture was left blank.

I revisited it later in life as an adult and gained a fuzzy picture, but it is still not well defined in my mind. According to many comments here some mathematical concepts I am well versed in and use regularly, if not daily, are calculus and that surprises me as they don't seem like anything that was presented in said classes.

I dare say that calculus has a marketing problem.

The first AI generated feature length film I’d want to see would be, “It’s a Wonderful Life, Calculus,” where we get to see what human history would’ve been like if we’d never stumbled across Calculus.

So the basic idea at the heart of Calculus is that when you break a problem up into tiny enough pieces, in the right way, the pieces become simpler to analyze and approximations become much more accurate. Like, if I break a complex surface into triangles, if the triangles are small enough, many of the physical properties of interest can be computed using the little triangles.

The two main manifestations of this principle are differentiation and integration. With differentiation, the simplification which comes from breaking a function into tiny pieces is that the pieces behave asymptotically like linear functions.

For integration, the simplification becomes that the tiny fragments eventually tend to have approximately uniform density, so the mass of the whole body is the sum of the masses of all the tiny bits, each of which is just the volume of the fragment times its density.

Differentiation and Integration are in some sense inverses, or opposite sides, of single greater idea. In one dimension, the derivative of the (indefinite) integral of a function equals the original function, and the integral of the derivative of a function is that function plus a constant. Almost inverses of each other, but not quite, and this nuance is where some care is required in developing your understanding of the subject.

That about matches my fuzzy understanding, but the big picture is the struggle. Like as submitted in a comment here about the venerable for loop being rooted in calculus, it makes sense when it's explicitly pointed out, but I haven't established a framework to draw those connections intuitively.

To generalize further, given a solution where the use of calculus is explained then that isolated example is comprehensible, but given a problem there is nothing that sparks a "I know, I can solve this with calculus!" I don't feel the same way about other mathematical disciplines that I have studied, even those of which I have spent far less time studying.

Perhaps it's simply harder to understand than other areas of math, but then I think that goes against the idea that is easy for kids to understand.

Calculus was quite difficult for human civilization to get a logically sound handle on; basically, it took the better part of two centuries from Newton’s original formulation of Calculus in the 17th Century to the work of Cauchy, Gauss, Weierstrass, Dedekind, Riemann, etc working throughout the 19th Century to develop rigorous foundations for Analysis (the modern name for the subject). That’s where all the epsilon-delta business comes in. But this machinery is totally overwhelming for the vast majority of children and teenagers without signicant context, motivation, and guidance.

The analogy to a ‘for loop’ in programming is pretty direct to a mathematical summation. For S := 0, i := 1 .. N { S := S + a[i] } differs only in notation from the standard sigma notation for a finite series.

You don’t need to look far for applications of Calculus. Any simulation of a physical system, such as the Solar System, navigating the DART space vehicle into an asteroid; modeling climate, nuclear explosions, fluid dynamics, structural stability, propagation of sound through matter, modeling and manipulating the properties of semiconductors through controlled diffusion of dopants, chemical engineering, thermodynamics, electrodynamics, optics, quantum mechanics, hypersonic missile flight, even Computer Graphics has the Rendering Equation at its heart …

Mastery of Calculus (and the ability to apply it to model dynamics) is pretty much the dividing line between Classical Ancient civilizations, and the modern world.

Here's how I see the relationship:

Calculus helps define the underlying rules for the higher-level (simpler by appearance) math we use daily. "I know, I can solve this with calculus!" is unlikely to ever come up, but the vague idea that there's something there you can dig into when you need to can be helpful in rare edge cases, where other people might be lost.

An example using programming languages: If all you've ever been exposed to was python, and no CS, you may never have considered why using "insert" on a list may be slow. Python presents it as a single function call, so you probably think of it as a single operation and don't go any further. That's the equivalent of the higher-level (simpler by appearance) math. But if you've been exposed to something lower-level, like C where you may well have implemented "insert" yourself on an array, or general CS concepts where you had to use big-O notation, you'll probably have in the back of your mind "yeah, that's not a single operation, it's doing more stuff in the background". Usually not something you need to think about, until you hit that edge case where it's suddenly running really slowly.

Remember very early on in education when you had to memorize various equations like area of a circle? Those equations can be generated from basic calculus. One I could never remember was area of a sphere, until one day when I was bored at my part-time job, found a pencil and scrap of paper, and decided to see if I could use what I'd just learned in class to derive it. And it worked, and I've never forgotten that equation since, because instead of it just being a series of numbers and letters to memorize, each part now has meaning.

That makes sense, but suggests that calculus is perhaps the most difficult concept to wrap one's head around, which flies in the face of the idea that is easy to teach to children. It is not clear where the breakdown occurs here.

Calculus is not easy to teach to children. We fail to teach it adequately to most college students in their first two years of study. Even a few historically noteworthy mathematicians failed their first contact with the subject.

Elementary linear algebra is far easier to understand and motivate. We can deal with finite, concrete examples without having to delve into the subtle complexities of limits, continuity, and infinity.

But it was said that "since kids can easily understand it" and it is not clear how that resolves.

That statement was clearly false. With an amazing teacher, an extremely bright student, focus, patience, time, etc., sure we can have the next Galois… but in the vast majority of cases, we should avoid setting kids up for failure by expecting them to easily grasp things which took humanities greatest minds centuries to grasp. Newton “invented” Calculus in the 17th Century, but these ideas had been percolating since Archimedes and even before going back two millennia.

Calculus in itself yes. But the statistics / probability or optimisation stuff you can execute are nice ( eg : gradient descent )

Or even linear algebra. I think it made me better at grasping highly formal stuff.

Yes, a statistics course is so much more useful. It's not emphasized in school but it will truly help through out your life if you understand it.

> Yes, a statistics course is so much more useful. It's not emphasized in school but it will truly help through out your life if you understand it.

I think this may be based on an impression of what math coursework used to be. A statistics course is a very common, if not required, part of any modern mathematics major.

yes exactly, and usually it comes after calculus or in parallel.

It provide foundation to work with the data, filling the gaps, or do a first pass on the distribution without fucking it up, and then do some stats on it.

Probabilities under normal curves or any shape probability distribution function are measured as areas under the curve. It helps to have an integral calculus intuition for comparing p(0.1<x<0.5) to p(0.5<x<0.6). It helps to have a multi variate vector distance interpretation of length for error and variance magnitude.

> But the statistics / probability or optimisation stuff you can execute are nice ( eg : gradient descent )

You learn gradient descent in calculus, it is based on derivatives...

I learned it as part of "operational research", that was some algo-y math course. No idea if that translate. But yeah, definitely closer to calculus than stats or proba.

There are a lot of people in these comments saying, "this thing is more important than calculus" and it turns out it's a concept that is fully fleshed out in analysis which is just calculus essentially. I feel like the problem is calculus as taught focuses too much on algrebraic manipulation which is only useful essentially if you become theoretical physicist and little else while the "why" behind calculus leads you to a lot of more useful results that are along the lines of approximation and optimization, which is closer to what a modern understand of analysis is.

I think the argument was that a calculus based approach to understanding those subjects dramatically improves and enriches the study of those subjects.

" There are better ways to exercise your brain that will be many more times better than Calculus. "

the ways that you list are great brain exercises but there's no good reason or research that suggests they are 'many more times better than Calculus'.

and continuing to call it a 'useless subject' is also totally unjustified. if i just that art appreciation is a useless subject because 99% of people are not going to get a job writing movie reviews for a major publication or curating exhibits for museums is that sufficient justification to say 'art appreciation is useless. source: me'?

Nah you'll find that there actually is research done into which activities stimulate the brain the most. Some of the only things that stimulate all four hemispheres of the brain at the same time include sex or sight-reading and singing at the same time.

I'm not convinced that 'stimulating all four hemispheres of the brain' necessarily has anything whatever to do with improving learning ability and developing mental skills or critical thinking.

There are better ways to exercise your brain that will be many more times better than Calculus.

Maybe. And there are probably better ways to exercise your brain than memorizing cities, mountains, seas and rivers in countries you're never going to visit.

But that doesn't make any of that useless. All that knowledge, calculus included, makes you know how the worls is structured, wether you're going to pull the levers yourself or not.

You can't decide you're not interested in something if you have no idea how that something looks, even superficially.

>There are better ways to exercise your brain that will be many more times better than Calculus. This is HW so one that comes to mind is programming.

And become a programmer who doesn't know calculus?

Some programmers will use Calculus but most, by far, will not.

And none of the programmers who don't learn calculus will use it.

I am surprised by the places where calculus comes up. I certainly didn't thing e.g. a class titled "Discrete Math" would need it, but it did. And Discrete Math is to CS what Calculus is to Physics.

They will all could have used calculus at many points, and even more so algebra, combinatorics, statistics, and geometry (in game dev, UI work in canvas an many other places).

But many wont, because they don't know it, so they'll trust some random formulas handed over by others (perhaps in Stack Overflow) for some things, or constrain their work and output to what they know.

Same way somebody who doesn't know about X technique (not even that something of the sort exists), wont know that there could be a great solution to the problem he works on based on that. So he'll use a subpar solution (in performance, memory wise, or even correctness), working around his limitation - or be beholden to this or that library that offers it as a black box.

Take a simple example: scoring systems.

It's funny how many websites use crappy scoring implemented by a developer that 'doesn't have a use for math', and e.g. naively averages scores, and ranks a movie with two 10/10 reviews above a movie with a thousands of 9.9/10 reviews...

Knuth is the poster child for the school of thought that just doing a _lot_ of math problems, especially calculus, is great mental weightlifting and we should make our young students do a lot of math homework. See this interview: https://ia600803.us.archive.org/29/items/conversation-with-k...

    I spent hours and hours studying the mathematics book we 
    used -- Calculus and Analytic geometry by Thomas.
    We were assigned only the even-numbered problems, but I 
    did every single one together with the extras in the back 
    of the book because I felt so scared. I thought I should 
    do all of them. I found at first that it was very slow 
    going, and I worked late at night to do it. I think the 
    only reason I did this was because I was worried about 
    passing. But then I found out that after a few months I 
    could do all of the problems in the same amount of time 
    that it took the other kids to do just the odd-numbered 
    ones. I had learned enough about problem solving by that 
    time that I could gain speed, so it turned out to be very 
    lucky that I crashed into it real hard at the beginning.

It's important to get down a lot of hours solving problems for the express purpose of improving your mental faculties, improving working memory. So for the finite amount of time you have, what problems do you choose to work on? I think maths and calculus is a great bet. Though I'd rather suggest a young person do Spivak's Calculus as opposed to Thomas' Calculus as Knuth used. You'll notice that most of geniuses we know were spending long periods of time solving problems. I'm kind of scared that we will have less people like Knuth in our people, people who can just go and deepthink and solve _difficult_ problems. If we keep up with this meme of 'no homework' we are in danger.

The takeaway here is that calculus was good _for Donald Knuth_.

But not every kid wants to go into math or engineering or STEM. And even the ones who do, is calculus the best use of that time?

While it's true that crafts, arts, cleaning, farming, and other jobs exist, it's also worth considering that science, technology, engineering, and mathemathics are world-revolutionizing disciplines.

Not everyone needs to change the world, but there's good argument for throwing all kids at STEM to see which ones stick, because any one of them could end up saving billions of lives.

Ah yes, arts and farming, famously things that aren’t world revolutionizing

Please offer the missing insight into how further developments in arts and farming will transform the world comparably to science, technology, engineering, and mathematics.

I mean, I can’t predict the future for sure, but I can point to examples in the past and throw out some hypotheticals.

It’s hard to imagine a more impactful invention in today’s world than modern genetic engineering and pesticides (both in good and bad ways). It’s what allows countries like China and India to feed a billion people, it’s why we have 1 farmer feeding hundreds of people, freeing other people to work on things like tech and science, it’s also causing mass ecocide and harmful biological effects. Sure, the personal computer has changed the world, but almost 40% of the world still has no access to computers of any kind, while share of people who don’t rely on genetically engineered food or pesticides is likely under 1% at this point.

Now, you might say “genetic engineering is a scientific invention”, but I don’t think you can meaningfully separate science and agriculture in that way. Scientists who invent genetic engineering techniques rely on farmers to POC and industrialize their inventions in the same way deep learning academics rely on SWEs to industrialize their discoveries.

Some important ideas in agriculture that have the potential to spur revolutions:

Permaculture - could allow people and communities to decrease their reliance on big corps, reduce reliance on oil for transportation in an era where oil is becoming scarcer, make space colonization feasible, eliminate food deserts.

Urban farming - in a revolutionary scenario, this is less about “urban” vs “rural” and more about removing land ownership as a requirement for industrial farming.

Food as leverage for political/economic movements - see Dutch farmer protests for a small example

Similarly, revolutions in art can’t meaningfully be separated from science. Art and industry/science have historically worked in kind of a call and response fashion where science attempts to concretely define human understanding, and artists theorize about things that can’t yet be explained and solved by science.

As an example artists perfect photorealism -> camera is invented -> modern art is created, which emphasizes aspects of the human experience that can’t be captured by photorealism -> discoveries about how the human brain works (e.g. modern psychiatry, neurology, artificial intelligence) -> postmodernism, a focus on individualism and societal structures -> ???

> P.S. note: Many people disagree. Good, you should not take it as absolute truth. I guess my real point is that you should question whether you really need to take Calculus. Don't just take it blindly because you are told you should. You have other options.

I agree, this should be questioned! Calculus is probably not the right choice for everyone, and I think most competent math departments are looking at how to make their curriculum more appropriate and relevant for today's world—certainly mine is. However, I think it is too easy to confuse "I don't see the relevance of this" with "this isn't relevant for me"; most students are not really in a position properly to evaluate whether a course of study is useful for them. Students shouldn't take whatever the university, or their professor, or anyone tells them as gospel—but neither should they think that their individual judgement will necessarily guide them towards the path that will best prepare them for whatever future it is that interests them, or the future that they don't yet know interests them. Perhaps most importantly, it is easy for a student to tell when they aren't using knowledge they have, but it is not so easy for a student to tell when they need knowledge they don't have!

I'll also dare to venture the suggestion that a lot of people could benefit from a much less utilitarian approach to education. College for everyone is a wonderful opportunity, but I wish it were that, an opportunity, not a requirement—and, given that it is a de facto requirement, I understand students' frustration with it. But, while a student is here, I wish they would make the most of the experience, and treat as a chance to learn things just because learning is an enriching human experience, without subjecting every bit of knowledge to pure humanitarian experience. I don't program, but I am glad for the knowledge about programming I have; I don't use history directly in any professional historical sense, but I am glad that I know the history I do. I am a professional mathematician and use my knowledge as such—but I'd be glad I learned it even if it weren't my career.

Crucial and inexplicable error, now outside the edit window: "pure humanitarian experience" was supposed to be "pure utilitarian computation".

> PS Note Many people disagree. Good, you should not take it as absolute truth.

I don’t think anybody is concerned that what you’ve claimed could be misconstrued anywhere north of mostly false. The nonsense you’re boldly peddling here is absolutely false. Seriously, leave flat earth community if you want to be taken even remotely seriously in the working world somebody bud.

> There are better ways to exercise your brain that will be many more times better than Calculus... fixing a car, understanding music and playing music, art appreciation, literature and understanding the human condition... daily exercise

These are all things we used to teach in high school. Did we stop? It's been a few decades for me.

For the last couple decades, at least, you can take all of those classes as 4 year programs at my local, public high school. In fact, you can take most of them at the same time. As it turns out, students have more than 1 class on their schedules every year!

We also have mechanical engineering, culinary, law enforcement & forensic science, marketing & business, computer science, game design, education, ROTC, printshop, photography, we even have a class that teaches students how to do their taxes! (it is literally 2 lessons) and this is in Texas where people assume the worst about our schools.

Many people that comment here about education don't really know what happens in a school outside of their memories of going to school as a teenager. When I became a teacher and looked back at the school I went to, I was surprised at all the opportunities that were available that I either wasn't interested in, or didn't notice existed. Most things that people think "schools should teach" are actually being taught at schools.

Mid-2000s:

> fixing a car

As far as I remember, working with machinery was restricted to the academy students at my highschool, which was something like 5% of the students. It only accepts freshmen, so if you didn't get in then you can't get in later.

> understanding music and playing music, art appreciation

We did have band, but I don't recall anything for the rest.

> literature

We had English classes where we did some lightweight books - I remember Animal Farm, The Great Gatsby, and The Scarlett Letter. Thicker books like Nineteen Eighty-Four and Brave New World were in the academy version of the English classes, but not ours.

> understanding the human condition

We did have one or two psychology classes.

> daily exercise

Gym was required for a year or two, after that almost all of us opted out. It was pretty terrible, and I wouldn't be surprised if it put most of us off exercise. I know I cared less afterwards (used to have fun biking and rollerblading, gym class took the fun out of it).

IDK about high school, but grades under that have cut back heavily on everything that's not math and reading, because those are the focus of tests that have major implications for funding and how good a school is judged to be. Science, history, social studies—even these "core" subjects have had cutbacks. Art, music, other electives, and even recess, have also lost time and resources in many districts.

This was my experience where I grew up:

Calculus: Yes, but not everyone took it. Those whose took pre calc the prior year would take either Calc or Stats their senior year, or pre calc if they haven’t taken it.

Fixing a car: Yes, if you went to the charter school and took that elective.

Understanding music: Don’t recall any specific classes on this, may have been possible with dual enrollment.

Playing music: After elementary school, only if you joined band.

Literature and understanding the human condition: I guess most language arts classes ended up like this, but I don’t feel like many people cared outside of getting good grades in the class.

Daily exercise: For high schoolers, one PE class was required for graduation and most people took it freshman year. People didn’t usually take any other athletic electives unless you were an athlete or really wanted to for some reason.

I think you are generally right. I used to teach physics at the university (and do research).

The actual, practical use of anything you learn at school is close to zero. There are a few life-sustaining topics that should be addressed (of my head: what is electrical power and current (to be safe and understand your bills), percentages, some biology (reproduction, and generally - human body), geography (rather detailed for your own country, tahns more and more general), etc.)

The problem is that many of these topics require at least a basic introduction that is not directly useful, but necessary to understand the rest.

You also need to account for the maturity of the student, and would like to start early.

All this means that by 15 you should have everything you need to be ready for everyday life.

I guess 16 is an average age until which education is compulsory by law. Unfortunately the education provided by then is not they useful, per your (again - reasonable) criteria.

The main obstacle I see is that except for a few brilliant students, you have a lot who start late. They are medium to say the least until 16 and then suddenly take off. A 14 yo doe snot have an idea about what to do later (and despite this we force them to make choices at 14 in France).

So overall al lot of things could be improved, but that would mean fundamentally altering the way school works (everyone is equal and gets the same education)

Indeed. I'm pretty sure that the research shows practicing music is a much, much better workout for your brain than doing calculus. Not only that, but practice in music is supposed to directly correlate with increases in mathematical ability.

The existence of a calculus class doesn't stop a student from taking music courses or being in the school band. In fact, at the schools I went to/taught at most students that took Calculus were also in the band.

The really interesting question this brings up is, is there some causal relationship between students that study music and their likelihood of taking calculus? (e.g. Assuming we call all kids that take calculus "smart kids", are smart kids more likely to also study music, or does studying music make kids into smart kids) Or, is there just some selection bias where kids from a certain background are more likely to do both?

Yeesh. The concept of marginal change (from Calculus) is valuable in all sorts of situations. It's a basic concept for business and social policy.

I don't know. I never learned calculus but I'm aware of the concepts and what they mean. I've recognized integration problems in my life... Seems like a very useful skill to me. For example, in computer programming there's game development: the physics simulation is essentially integration.

Programmer here. I wish I knew more calculus. It's really tough to learn outside of school and much older.

Why? There's so many resources now. I actually tried to relearn calculus recently. I ran into the same problem I had the first time in college, it felt like learning for learning sake and didn't have a strong enough motivation for it

Yep. Even just linear algebra. I’m trying to get into graphics programming and it’s very difficult without a solid understanding of that stuff.

“Just” linear alg, LOL. I made it through calculus 2/3 in high school but linear algebra gives me fits, especially since I only ever learned it in a data science context. Linear algebra is freakin’ tough!

It's easy, just work through a book and do practice problems.

The difficult part isn't the math, it's just finding time to do it.

I would actually start with a first pass at elementary linear algebra, before taking a serious crack at even single variable Calculus. Then let the key ideas marinate in your brain; develop some facility and intuition / insight.

Multivariable Calculus hardly makes any sense without a basic amount of linear algebra. It’s a bit of a historical accident that (Western) mathematics developed Calculus extensively prior to linear algebra taking shape as a significant subject. Don’t try to tackle Multivariable Calculus with a decent grip on matrices. Maybe do some computer graphics and learn how to build 2D & 3D shapes and animate them using matrices.

You need this book: Practical Linear Algebra: A Geometry Toolbox by Gerald Farin and Dianne Hansford. You can get older editions for pennies. The 1st edition was actually called The Geometry Toolbox for Graphics and Modeling.

You should also read Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares by Stephen Boyd and Lieven Vandenberghe: https://web.stanford.edu/~boyd/vmls/

I should also add the famous 3Blue1Brown's videos on the "Essence of Linear Algebra" - https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

Basically it's two things:

1. answering the question "how do we get the next x" for a function x=y.

If x=y, then if you increase x, you also increase y, so dx/dy=1. Figuring out dx/dy is fun when dealing with things like 3x^2+5x+7=y.

But take the game of Pong, for example. A simple Pong game has a ball and the direction of the ball can be controlled by two variables - horizonal velocity (H) and vertical velocity (V).

Each frame, you take the ball's X and Y and add H and V to it, to move the ball. When the ball collides with something, just multiply by -1 (to flip the sign) to reverse the ball.

If you divide V by something like 0.0001 each frame, you will implement gravity.

Want the ball to have a gravity or other "pull" that results in it hitting a point in a specific number of frames? Well ... someone who knew what they were doing would know what to do. That's all I got.

2. If you have a few points for x=y, you should be able to figure out dx/dy somehow. I think that's called integrals.

>If you divide V by something like 0.0001 each frame, you will implement gravity.

That'd be a wild ride.

Minor correction:

> If you divide V by something like 0.0001 each frame, you will implement gravity.

You will implement air resistance. Air resistance is proportional to the velocity, gravity is a constant offset.

V(n+1) = V(n) - drag * V(n) - gravity

Oh sure, that's simple enough. How about gravity over a sphere in 3 dimensions? How about a custom physics and collision engine? How about some portals? How about some non linear spaces?

Games get incredibly complicated very quickly.

Sorry, but that's not any excuses for you not to start learning! Looks like your problem is procrastination which we all struggle with. Old age is also not as bad we're lead to believe for learning. It's all about getting into the habit of studying. That can come after an idea that turns into a goal and is kept in motion by brute force will. Alternatively the peer pressure from school or a course can keep you going and meeting goals effortlessly.

There are now many easily accessible online resources like 3blue1brown's "essence of" series [1], Khan academy [2], or Brilliant.org's courses [3].

[1] https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...

[2] https://www.khanacademy.org/math/calculus-1

[3] https://brilliant.org/calculus/

"tough to learn outside of school and much older."

Be very careful with this way of thinking. People around me have used it to justify not taking the time to learn something as far as I can remember. In short, giving up before they try. This attitude will hinder any possible growth. I guarantee it.

What's tough is following thru, not the subject you are trying to learn. If you can't learn on your own take a class at a local college. It will force you to show up and try. But thinking that you can't learn because you are older is not true.

Obviously I've not looked at all of them but my main problem (especially with language books) is that there are so few good materials to learn from in self-study. Everything is "supporting the class room", or has some ridiculous split into (bad) contents and (worse) exercises that more often than not are the classic "draw three circles.. draw the rest of the owl".

And yes, people usually go on to recommend some vague resource that helped them understand just enough if they were starting from a decent problem already. (IMHO a good example here is vector math. If you do some stuff with 3d and games the problems are clear and most people will start from there and then learn enough to understand it. But just learning it in isolation is where people complain).

You have the first prerequisite, the desire to learn! Try to find resources which nurture your interest, until it grows into the will to learn. I find the history of mathematics and how it came to be, its role in the rise of civilization, to be quite inspiring.

Between YouTube, and maybe the right blogs or online classes, it seems like a motivated person could learn almost anything online. There’s also a social aspect, we learn by talking about things with others, sometimes as pupils, other times as peers, and even teachers. Try to teach concepts you’ve learned to other people.

Finally, you can’t learn math just by reading or listening, any more than you can learn to play soccer just by watching matches. Math is a problem solving art, you have to resist the urge to peek at the answer until you’ve exerted serious effort and put in the time.

It is not that tough if you approach it in the right manner. Here is my "guaranteed understanding" 5-step recipe for Calculus :-)

a) Learn co-ordinate systems Geometry.

b) Learn Functions and Graphs.

c) Learn Derivatives.

d) Learn Integrals.

e) Learn Differential Equations.

Focus on understanding single-variable only and ignore everything else in the beginning.

You will find the following resources most helpful.

1) First refresh the basics from George Simmons' Precalculus Mathematics in a Nutshell : Geometry, Algebra, Trigonometry (it is less than 150 pages!) - https://archive.org/details/precalculusmathe00geor (pdfs are available elsewhere on the web) In particular; read the Trigonometry section thoroughly.

2) Higher Math for Beginners (Mostly Physicists and Engineers) by Zeldovich and Yaglom - https://archive.org/details/HigherMathForBeginners/mode/2up - Read everything from the beginning including Preface/Notes/etc. (which setup the Motivation) through chapter 1 (Functions and Graphs), chapter 2 (What is a Derivative) and chapter 3 (What is a Integral).

That's it! You now "know" Calculus; the rest is mere elaboration on this basic edifice. You should be able to do this in a week by just studying an hour or two everyday.

3) For a more Pictorial/Graphical help in understanding the above, get the book: Who Is Fourier?: A Mathematical Adventure by Transnational College of Lex. This is a manga-style book which gives a very intuitive understanding of Calculus and more.

Do not wait but get started today; You can thank me later :-)

I'd like to add to this list a series of lectures on YouTube by Professor Leonard: https://www.youtube.com/watch?v=fYyARMqiaag&list=PLF797E9615...

Great!

I should also add the famous 3Blue1Brown's videos on the "Essence of Calculus" - https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...

One use of class + 1 of what is needed is that it demonstrates mastery of the previous material when it integrates such. For example, calculus shows mastery and a decent understanding of algebra and trigonometry among other topics of the level taught previously.

One thing that's eye-opening is how useful set theory could be to general reasoning. The number of basic category errors I see, even sometime on HN comments, but mostly in public discourse and political rhetoric and thinking, is astounding.

“Training the brain to not give up”, also called “grit”, is a more precise way to say it.

Programming, music, art ++ isn’t as good as calculus on this.

Learning to play a music instrument or learning to paint, well, is the definition on not giving up. So, no...

Never heard it put that way! Still, it takes grit to graps any deep formal topic, and there are those in all the disciplines you mentioned.

Harmony for music, Composition for Art and programming for systems are some examples

Grit is useless without focus. Calculus, and math in general, provides focus: it cuts through the BS.

We aren't living in an era of people suffering because they did too much calculus they don't need.

We are, on the other hand, living in an era where people are lied to, fooled, prayed upon, and duped everyday, because they can't understand math.

We also live in an era where people could do amazing things, even as amateur hobbyists with some math and science knowledge, but are drowned in BS doom-scrolling, binge-watching of crap, and the like...

I was just reading Sir Walter Scott's Waverley where he suggest that this is the main reason for a classical education as well:

> Edward would throw himself with spirit upon any classical author of which his preceptor proposed the perusal, make himself master of the style so far as to understand the story, and, if that pleased or interested him, he finished the volume. But it was in vain to attempt fixing his attention on critical distinctions of philology, upon the difference of idiom, the beauty of felicitous expression, or the artificial combinations of syntax. “I can read and understand a Latin author,” said young Edward, with the self-confidence and rash reasoning of fifteen, “and Scaliger or Bentley could not do much more.” Alas! while he was thus permitted to read only for the gratification of his amusement, he foresaw not that he was losing for ever the opportunity of acquiring habits of firm and assiduous application, of gaining the art of controlling, directing, and concentrating the powers of his mind for earnest investigation—an art far more essential than even that intimate acquaintance with classical learning which is the primary object of study.

But I think the above poster has a decent point, in that there are many places where you can train your brain to focus that will also provide you with skills that are more useful for most people. For instance, I'd say statistics, combinatorics, and probability will be more useful to most people.

Of course, if you're in a field where you need to learn calculus, then you need to learn calculus, just like if you're in a field that needs differential equations or topology, you need to learn that. But that's a small segment of the population. I've studied calculus for fun (and differential equations as well), and enjoy it, but have never once found myself running into a use for it in the real world.

I'm sorry, but "art appreciation is way better brain exercise than calculus" is just an extremely silly assertion.

Not because the opposite is what's true, but because it's bizarre to compare them in the first place.

> Total BS...

Suppose for a minute that you had to do mental weight lifting, but you could choose which discipline to do it in. If you chose math as the discipline, then calculus would be perfectly fine as the mental weight lifting in high school.

Are there other disciplines that you could choose? Certainly. Does you school give you such choices? If not, then that might be where the BS is.

IDK about you, but there's a ton of stuff that I know only because I find it fun to know and not at all because it's useful to me in any other way. This must be true for many man many people.

While I can't comment on the example(s) you provided, I agree with much time is "wasted" on unnecessary stuff.

Not even basic logic is taught! Nor any daily life useful skills like cooking...

It depends a lot more on how calculus is taught than it depends on calculus itself.

If your calculus education is: learn these formulas by rote memory and apply them to this set of problems. Yes, calculus, and in fact any and all the mathematics, will be mostly useless for you. You don't need the formulas.

But, if your calculus education is: OK, there is this limit. No, I will not tell you the proof. You will find a proof of it by yourself, and you will show it to the class.

Or: this is the volume. Imagine a way to calculate the volume! How would you do it?

Or thousands more ways to stress your creativity to the max. Writing proofs is the most intense workout of creativity I have experienced. And it feels like writing code, in a way. The difference is in a mathematical proof you are also the compiler, the syntax checker, the interpreter, and the person who writes the program.

If that's the way you learn calculus, not as a formula applier, but as a solution finder, then yes, calculus will make you a much better software developer. I say this based on personal experience, I consider my proper mathematical education, having to think about proofs in linear algebra, to have a very positive impact in my capacity to write software later.

And I had been a software developer for ten years before the linear algebra and calculus courses. So it's not related to acquiring experience as developer later.

1 - World needs world class mathematicians to continue to evolve science for the betterment of humanity.

2 - Those world class mathematicians need to get started sooner in life to learn most of whats out there so they can work on extending the reach of collective human knowledge on Math. So, these mathematicians better get to calculus before they graduate high school.

3 - I wish there were two courses like "math for bright kids" and "math for not-so-bright kids". Since you feel you didnt need to learn calculus and, assuming you felt the same way when you were in school, you would be taking "math for not-so-bright kids" .. AND THAT would not have been acceptable to your parents back then.

4 - I wish teacher had magical powers to know who would go on to be a world class mathematicians but i dont think we are there yet but I feel if there are enough mathematicians in the world, we may get there some day. Case in point - Einstein did bad in school and showed his brilliance relatively late in life.

In any case - lets stop the bickering because you had to study little bit more than what you are comfortable with, for the sake of overall evolution. Theres more good than bad here.

It is a metaphor, and as such, has it limits. Of course there are better ways to exercise your brain than calculus. That is just a way. Is the best? I don't know honestly. I personally found algebra much more interesting as a brain opener.

Basic ideas of calculus come up everyday in life: interest rates, credits, statistics. If you want to understand some physics, it is very useful. Can you understand physics without derivatives? Yes, but not as good as with them.

Like everything is a compromise. One possibility would be that each person has a couple of teachers that go with him/her in a way or learning specifically tailored. The other extreme is to teach absolutely the same thing from 1st grade until graduate to everybody... the compromise is to separate it in primary/secondary/tertiary education and let choose paths. Everybody will have to learn things that will never be used "in life". We have to get over it.

I agree that pedagogy should be flexible. Not be static/constant, but, rather, a relationship that reacts to inputs from students; changing dynamically to suit the student's needs. It'd be a great thing if people could build up a strong intuition about how their problems react to change over time, in order to maximize the value they get over time.. if only they had been taught a non-useless model for predicting how things change.. ;)

More seriously: in an abstract sense, "when are we going to use this?" is a totally valid question - but most solutions are probably about changing the "this" in question, rather than getting the students to practice the act of recognizing in real life what they're being fed at school. Maybe recognizing applicability itself is a higher priority skill than whichever skill is being taught.

The mechanisms of calculus probably don't matter, but calculus is the study of change. In particular, how things change over time in non-linear ways. Calculus is the abstraction of multiplication in the same way that multiplication is the abstraction of addition.

THIS is the real reason why calculus is important. Knowing how to study, describe, and understand changes that are not constant.

If that isn't important in real life, then neither is counting or multiplication. Unfortunately, calculus is so deeply embedded in the vocabulary and forms of mathematics that even people with a "full education in calculus" often can't describe that. So they resort to stupid metaphors. They only unconsciously understand its value and fail to describe it.

Our education system wasn't able to get rid of its scholastic roots. Even scientific establishment itself. The patterns that were developed at the time when early scientific research and teaching were evolving in medieval monasteries still keep appearing in research and education.

Something that managed to evolve in a medieval monastery, during an era of warring kingdoms, must've been significant, and we dismiss our roots at our own peril.

Of course, we should actively seek to simplify, clarify, and forget the unnecessary. But Chesterton's fence, etc.

> Something that managed to evolve in a medieval monastery, during an era of warring kingdoms, must've been significant, and we dismiss our roots at our own peril.

Nope, in such 'warring kingdoms' environments science and education were not able to prosper. The middle to latter parts of the middle ages in Europe were rather stable times to allow the development of both of those.

Even with that, its still a medieval institution. It must progress and adapt to the 21st century.

And Algebra ? Cause there are few subjects that help you quite so much to think in the abstract and when it becomes automatic, you see yourself toying with concept from any level without needing to define everything below to anchor them in reality.

My non-maths wife is so much more attached to these anchors it s hard for her sometimes to accept to discuss ideas from a high level by accepting for now not to explain how their parts came to be.

Calculus I imagine is useful to calculate an interest rate at the bank to avoid maybe stupid surprises ? But I wonder if advanced calculus is necessary indeed.

If we don't teach calculus to kids in highschool then every (non software) engineering degree will have to be a year longer.

> There are better ways to exercise your brain

They didn't assert it was the best way, but that it was _A_ way.

Seems like the grapes were very sour.

People who claim its weightlifting for the brain probably forget that the stress you are putting yourself through cramming for the hellish calc final is probably a net harm

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