For many subjects, most kids will end up never using them. But, we have no way to predict which subjects will be useful for which kids. Without the ability to do that, our priority is maximize each child's opportunity. We never want a kid to be in the situation where they would have been interested in a subject and a career path but never ended up discovering that and using it because we didn't expose it to them.

So we teach some of every subject to every kid. That way no matter which path they end up following, they are as prepared for it as we can make them.

(Also, yes, I agree that math is good general training for cognitive rigor. Also, numeric literacy is vital for all adults since we live in an ecomonic world and participate in a democracy where statistics are necessary to understand policies.)

I've used a decent amount of calculus in my life, but that was the first time I had been actually happy to have learned it.

Most teachers and professors just parrot their subject material year after year after year without EVER giving a reason what any of that is used for or where should we apply it. It's just learning for learning's sake.

I suppose it's no surprise that when people are finally given the option to learn in a practical way at the odd subject that allows for some project work most students can't seem to think of a damn thing they want to do. It's like a systematic suppression of creativity to make education more like a factory production line.

During my senior year I was able to take a course through BOCES on audio production. That course related some of the trigonometry I was struggling with to a subject I was deeply interested in.

I don't expect Math teachers to start teaching audio production, but it would have been nice if the teacher had seen me struggling and at least attempted to approach the subject from a different angle ¯\_(ツ)_/¯

I think if you put together an entire class of completely different projects that all somehow end up teaching trigonometry it would also help show everyone all the possible applications for it when discussing afterwards. I never would've guessed trig is used in audio for example.

I wonder how much of the value of the course is just in the repeated observation that the rate of change (and so on) is useful to measure

Basic math and physics education helps build intuition for it, but without people are really bad.

Erm, in some abstract ways yes - but actually people are very good at extrapolating current physical events. "It is getting hot fast? Oh not, it might even get hotter, lets look for shade."

Or throwing a ball. You would need calculus to correctly calculate the flight path of the ball, yet we can do so, without and very fast.

Where our intuition fails often, is understanding the reason why things happen. For this physics and math should be taught from very early on.

Of course we have no intuition for how planes behave.

And with Ski and co. I would argue it is somewhat intuitive, it is just a new tool that needs learning. But I do not remember learning ski or snowboard felt unintuitive. It was just hard coordinating it, but this is not unlearning to me.

I struggled for years with 'keep your weight on the downhill ski'. When I realised that it sort of meant 'lean downhill' turning on steeps suddenly became a lot easier. This was counter intuitive in that when I turned on a bike, I was invariably leaning in to the turn, not out.

It was actually learning to skate on skis that helped me make the transition to better turning.

And let’s not forget, it’s helpful to be able to augment smart devices that already exist to do things like this rather than throwing them out and buying a newer one that can do it on its own.

Also, lots of math is optional (depending on your school and career.) You may not use calc or trig regularly, but most people use some algebra and geometry.

You should read the Aims of Education speech given by Abbott; you might really enjoy it.

https://college.uchicago.edu/student-life/aims-education-add...

"Electives" is an important word there. By high school, I think you're ready to explore the things you already know you might be interested in. Much more so than what high schools typically have on offer.

I was bored out of my mind for my first two years of high school. I went to a HS at a community college for the second two, and it made a world of difference. We had English and History classes taught by HS teachers, but for all our other credits we had the whole college's course list to choose from.

Being able to choose makes learning so much more engaging.

We tend to waste a lot of time teaching subjects which they're unlikely to use, and fail to teach them about the ones that they would really benefit from. A basic understanding of criminal and civil law, along with accounting and statistics would be extremely useful to almost everyone as individuals and as citizens. Music, history, and calculus are useful to some people, but not nearly as many.

There is a lot of cool stuff you miss out on in the basic stats course because of having to dumb it down to avoid the calculus. Some I remember off hand:

- proof of the central limit theorem, which gives the shocking result that if you sum several uniform distributions you get rapidly more precise approximations of the normal distribution, which looks similar to exp(-x^2) if I recall. This central result is the foundation of all statistical sampling. This is why in real life if you see something follow a normal distribution you can guess it is probably caused by a moderate to large number of somewhat independent factors, and vice versa. This is genuinely useful, but if you don't know it you won't miss it - poisson distribution which relates the mean time between events to the probability of failures. Obviously very applicable to a lot of real life tbings

The same is the case in basic physics. No, V does not = IR, nor does F = ma. That's the simplification they tell us so they can explain a simplified version to us. In fact, the correct equations have derivatives in them and thus are differential equations.

Look, nobody needs calculus but nobody needs to read either. After all you could hire someone to read everything out loud to you. All knowledge is like this.

If there is any issue, there is an issue in how it is taught: I feel like there is too much focus on symbolic manipulation. The algebra essentially prepares you to take a physics course, and that's it really. The underlying concepts however do lead you to things like optimization and approximation (which is fundamentally what calculus is anyway) and that needs to be communicated to students somehow.

The thing I remember vaguely is when I was taught calculus first, we "took limits" by hand, including derivatives, numerically, and then we did the formulae and spent the rest of the time doing nonsense like difficult trigonometric integrals and integration by parts. The thing is as you go onto proofy classes including real analysis and such, you go back to the original concept and learn that that was the important bit and actually useful piece after all, as most of life's data cannot be well modeled by analytic solutions you can write down.

I think this is what I contend the problem is. Unfortunately, I don't have much contact with people who actually teach students high school calculus, but almost every mathematician and physicist I know (apart from the theorists may be) agrees with me, that at the end of the day, there is a lot of value to the concepts underlying calculus because they are general and help both naive models of data in your head and eventually statistical and numerical (read computational) models that vastly more people use, while the trigonometric substitutions are much less useful, and are really only useful if you're going to go on to being a theoretical physicist (or at least get a degree in physics where you'll need to do derivations).

I spent my entire university degree convinced that I was going to go into the video game industry. It took only a few months to realize that it's not what I wanted for a career, and I've spent the next 20 years loving my industry but doing anything but gaming.

I was an arrogant teenager that thought I knew what I was doing. I disrespected the arts, music, history, and focused exclusively on stuff like Math and Calculus.

Now I don't feel like a well-rounded adult, and I wish I spent more time when I was younger on music and humanities.

When I arrived for my CompSci they basically said: "We'll leave what you already learned in math behind around christmas (so after half of the first semester), no matter what kind of math you learned before".

I can't 100% grade both judgments, but I did not take the advanced math thing, but if they hadn't said these (apparently) completely wrong thing, I 100% would've taken French at school. (Which is another problem I'll not go further into, some fixed tracks of what path you need to choose in which grade)

Easy remedy. Learn an instrument. Find someone local to take weekly lessons, and practice several hours a week. If you never did this before I think you will be overjoyed by how well you play with consistent practice.

I couldn't imagine not introducing my kids to History, Music, the Classics and so on. I value them far higher than my experience with Computing, Finance, Law, what have you. What a pointless life to only have interest into things that are productive.

I had many different teachers with different approaches. Ones it was all about memorizing events and dates. Of course that is trash. But others is was about understanding why it went the way it went, why not other way. What were the key events that triggered another events, under which circumstances... alone the critical thinking that went into that, is every minute worth it.

History is a beautiful subject and a great hobby, but almost completely useless. On the other hand, every student needs to understand finance and the law, both on the individual level, and in order to be a thoughtful voter.

But if you for example analyze the rise and fall of cities, empires and civilizations, there are some things you can learn (e.g. overuse of resources). Also the the economical crisis, bubbles and inflation teaches a lot of things.

Studying totalitarian regimes in the past, can help to detect the first signs of alarm.

It is not an exact science, yes. But knowing the past helps to understand the present, and helps to not repeat mistakes. I really think it does.

The outcome? I still enjoy music despite this torture of lessons, but I never properly learned to play an instrument, and was mostly dissuaded instead of encouraged.

Spreadsheets and algorithms on the other hand I find highly entertaining. I love many board games for this same reason: it’s an opportunity to build novel algorithms in strange domains to achieve a specific purpose.

And most can see that boardgames are more similar to “productive things” you find disdainful than music.

We all knew that's not the truth. The teacher knew, but that is what the text book said so we learned it.

My point is, while history is important it shouldn't be a marked lesson where you just have to remind right and wrong facts.

In Germany I found the focus on WW2 a tad much (but that was '93 to '02). Sure, it's important, but I think they could've gone at least one year out of nine where it wouldn't come up, and instead ignoring a huge amount of epochs outright, or with a laser focus on central europe.

I see it like when I learned about programming, I was frustrated to learn about language theory, complexity, graphs, etc. I wanted to learn langages, frameworks, specifics for being ready to work right at the end of my degree but it would have made me more fragile and less versatile to future changes. Although law and economy are less likely to change as fast as the latest cool tech stack so this example is not the best.

The argument doesn't seem to apply very well to calculus though, does it?

"When am I ever going to use calculus in my life??"

You? Probably never. But we're teaching everyone on the off chance that one of you goes on to do something useful with it. Enabling that one person to find a way to make rockets more efficient or something is well worth the tradeoff of wasting the rest of the class's time, from a societal point of view.

If you want to force everyone to learn Calculus for “the good of society”, then don’t force the onerous debt of student loans on private individuals.

I mean, if you're an engineer and you don't know the relationship between position, velocity and acceleration - you're going to have a bad time.

(I am one)

I have no idea. So I will gladly defer to those who do understand math, and be glad someone does, or my career wouldn't exist.

These lessons help bring you up to speed with foundational concepts and ways of thinking that took humanity a very long time to discover and develop. Learning these things while you are young will, at a minimum, help you keep up with others and avoid being scammed, or at best, help you quickly reach the current limit of our understanding and possibly expand our capabilities.

You can also think of it like stretching and exercising your brain. You may not need to actually do that work, but it's still good for you and helps make other work easier.

However at least here in Norway, I think we spend too much of the time focusing on useless details. For example, non-trivial part of our Norwegian classes was filled with language history, like the art periods and when various authors lived and so on.

I get that it's nice to know a bit about this, to be able to place them in roughly the right period, but giving a 14 year old a "wrong answer" on a test because the kid doesn't know the exact year some author was born, or failing to list all the authors in some romantic-period clique, is frankly stupid.

Meanwhile, far to little time was devoted to practical writing. Like, say, an email. We spent just a few hours writing reports and similar non-prose, compared to several semesters full of language history, learning about the romanticism and realism periods etc.

I see so many of my colleagues and customers who couldn't write a coherent email if their life depended on it, and can't help but wonder if some of that history time at school had been better spent on practical matters. If a kid wanted to really study language history, they can very well learn this later.

We did have a bit of it, but pretty insignificant compared to the rest. When I got to high school, nobody in my class could write a half-way decent report for example. Just the basics of what a report even was and what it was supposed to contain. I got the equivalent of a D and the teacher said I had done "by far the best in class", the rest got F's and NR. None of us in class had come from the same junior high schools, so wasn't that.

Most of my colleagues seem to have no issue telling a story, but many seem to have problems forming a coherent argument, or asking a non-confusing question, in writing. Again, don't think it would have hurt to have more non-prose experience in the basic education.

Also, it's hard to know when people will need background information necessary to understand what someone is saying. I'm often blown away by what others do not know, only to turn around and find myself completely at a loss about something else.

When we see those “guess how many jellybeans” contests, I let them guess and then show them how to work out the formula for volume of the container.

I once made them do an entire ROI analysis of the Monopoly board to figure out which spaces were the best and how many houses were worth building. They’re really good at Monopoly now. :)

It helped some.

We don't know who is going to be an electrical engineering student, and of those folks even many of them might manage to get through the degree without needing calc (you can memorize lots of answers and then get a career plugging in discrete components I guess), but we do know somebody is going to have to design the antennas.

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.

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.

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.

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.

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.

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

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.

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.

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

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?

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?

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.

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?

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.

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?

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.

> > 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.

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

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.

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

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.

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.

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.

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.

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.)

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.

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

This only adds to my point.

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.

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.

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

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

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?

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.

Finanance, technology, they all are.

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.

>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?

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

I think you accidentally disproved your own point.

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.

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

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> 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.

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.

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.

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

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.

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.

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.

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.

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 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.

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.

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.

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

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.

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.

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

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'?

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.

And become a programmer who doesn't know calculus?

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.

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...

    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.

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?

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.

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 -> ???

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.

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.

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

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.

> 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).

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.

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)

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.

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

Games get incredibly complicated very quickly.

That'd be a wild ride.

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

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.

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).

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/

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 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/

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

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.

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

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...

> 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.

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

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

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.

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

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.

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.

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.

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.

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.

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

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.

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.

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

Want a fancy job? This is one of the hoops to jump through. Same as leetcode further down the line, you won't do it at work but you will do it to get work. But that's also a pretty tragic take on it.

Practice for other things, sure, that is also a way to see it. You won't bench press the other team but you will make yourself stronger. But for what? A sport you'll never play? What are you preparing for?

Here's another one. Math, especially pure math, is a thing that is totally separate from observation. It just sort of exists without being anywhere, and yet there's all this depth to it. You can get a puzzle that cannot be solved by any anything other than thoughts, and you can keep building on these puzzles that don't exist. Go nowhere and explore.

Lastly I note that it's mostly math class that gets asked this "what's the point" question. But you may as well as this about everything else you do in school, and you will mostly find that you'll have spent years to learn French for 4 weeks of actual use in France, dissected frogs for no reason, and learned how to play the recorder. All things that I'm sure you can find positives for despite the superficial benefits being quite small.