“When are we going to use this in our everyday life?” (2017)

"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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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

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

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

And become a programmer who doesn't know calculus?

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

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.

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.

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.

----------

> 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.

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 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.

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.

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.

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.

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.

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.

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

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

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?

    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.

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.

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.

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'?

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)

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.

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.

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

I think it tends to come up as a way of resisting something hard and unpleasant, and math tends to be the subject that most often feels hard and unpleasant to a plurality of young people. Of course most of us, if we had been freed from HS math as teenagers and left to our own devices, would not have gone off to do something really useful. We would have instead spent that time on something far more useless, like browsing HN. :-)

I still cannot see a value in studying classical literature. At least not one that does not have 1000 better tradeoffs for other subjects.

There are also aspects of studying that can 'nerdify' the brain and make you weaker at interpersonal skills. There are very few CEOs, influencers, actors, and musicians that are good at math. In fact, I think the artistic/athletic pathways in life can be damaged by beginning to condition someone for office work.

Attempting algebra is how you solify your knowledge of arithmetic, attempting calculus is how you learn algebra and finally master arithmetic.

And we should constantly question that...

> Same as leetcode further down the line

Leetcode is free and has proven sufficiently enough to get us a 6-figure job.

> All things that I'm sure you can find positives for despite the superficial benefits being quite small.

Except that the cost of going to school is expensive. Even if schools are free for you, it is paid by tax money. We should always aspire to teach useful subjects with decent ROIS in schools.

I don’t have to use any calculus to get a weather report, etc., because other people do that for me and give me their results - it’s part of their job.

Calculus is indispensable and is used in our everyday life - but most of us won’t use it ourselves, or need to know the specifics, or really even know the broader parts of it.

If you don’t know how other people made the things you use, then 1) you’re pigeonholed into being totally dependent on them, and 2) you’re likely to get all sorts of weird beliefs about how the stuff you depend on works (like crystal healing/homeopathy/etc in the bio realm).

It's also clearly not the reason we are educating children in calculus. We can know this because we don't teach children to do weather calculations, we don't test them on statistical analysis, and so on.

The real reason public schools teach calculus is that they started doing it at some point for some reason and then never quit because they are bureaucracies resistant to change. All the people involved have a kind of status quo bias preventing them from saying "yeah, I guess that was useless, let's teach something else."

If I'm wrong, we could imagine a test. Take a comprehensive calculus exam from senior year of highschool or freshman year of college. What grade do you think the average adult would get on this test? How about top ten percentile adults for intelligence, wealth, or whatever? If, as I do, you think the average score would be F, can you explain why it's important to teach the general population of kids something that the general population of adults demonstrably do not know?

This is like claiming David Beckham uses advanced physics to kick his free kick.

Calculus is important to the world, sure. But it's not important to regular people to spend time and money learning it. In some cases, these people take out student loan to learn calculus which doesn't help them pay back the loan.

For many kids, that's not true of all subjects. Some find certain courses very difficult.

For example, (for me), the "important things" about world war 2 is, who, why and how... what was before, what made people make decisions they did, how did it start, what happened during, and why and how it ended... the exact date when some named general attacked some small city somewhere is pretty irrelevant (atleast not a thing you should keep memorized), but a lot of history classes focus on exactly that... on which date which unit/general took over which town where did they break through, etc... I'd prefer half less memorization data and a googling class for kids to find the dates needed, and more focus on the whys and hows, because history repeats itself, while dates and names don't.

You hit the Nail right on the Head !

Given that our current society is so thoroughly intertwined with Science and Technology is why this basic knowledge is called "Education" and is said to "prepare oneself for the Modern World". Once we get into the workforce (or not) we can keep/remember/use/build-upon what is needed and leave the rest in the attic only to be brought out if and when needed.

I agree and I'll go even further:

I think that all these attitudes towards education ("it's too hard", "nobody will need this", "we should make it more fun", ...) are very typically Western and don't seem to be shared in certain Asian societies (e.g. Korea). I'm not saying we should go towards the other extreme (which has a ton of downsides too), but somehow, in certain other cultures actually applying yourself in school seems to have a higher value than it does for us. I'm not exactly sure why this is, but it seems some parts of our society have become too complacent, and I think this is ultimately dangerous for society.

except probability and stats does require calculus. maybe not at the high school level but if you are doing it in college it's almost certainly going to have some needing of calculus.

And it's also training your brain (but that can be done by other things like puzzles or games).

Only for a reasonable price. The price differs between individuals for many meaty and mindful reasons.

I was just recently giving your exact argument, that if in high school you learned only exactly what was needed to perform your job as an adult, you would essentially be a cog in a machine that requires the world to stay completely static for your entire life in order for you to not get screwed when your skills inevitably become obsolete.

The point of calculus (which imo is just the common path of achieving mathematical maturity) is not that you will use math in your day to day, but that you will be a more well-rounded and dynamic person mentally.

That being said, uninspired high school mathematics focused on memorization is not helpful for anyone.

For non-US residents, they may not realize that a university education for an American is basically indentured servitude. The American student, as of today, must decide upfront what his career path will be. Because he will be taking on a gigantic loan to train for that career. If that career does not provide financially, the American student is burdened with debt that he cannot discharge even in bankruptcy.

Yes, Americans want to prepare for their career. That is why they look at calculus and other required subjects as an unnecessary tax on their time in college and monies.

For what it's worth, I have had extensive classes in calculus and statistics at the graduate level. For most people, those classes would be useless. They would be better off mastering Algebra and Finance 101 (how loans work and net present value of money). For non-engineering types, I would argue they are better off mastering Statistics as they will be reading and reviewing scientific experiments their entire career. I find the argument you need to master calculus to master statistics disingenuous. Statistics instructors are perfectly capable of teaching all the concepts needed to understand statistics within the class including any that may have derived from calculus. If extra instruction is needed, they can offer a pre-statistics class. This argument, as it seems to me, is that calculus instructors are threatened by statistics instructors.

I have a feeling the people who were going to become math majors would do so anyway, under such a system, and the rest of the kids would learn and retain more math than they in fact do with how we teach it today—"here's 6 weeks on how you solve quadratic equations, without a hint of a reason for doing this, feeling motivated yet?"

Then in grade 11 he did physics and calculus and suddenly it all made sense! He was super excited.

Years later he says "I guess this is just more learning the alphabet" but it sounds to me like he's trying to convince himself. :-/

Everything from Newton's laws, the quantum mechanics of a single electron, bulk materials (Ohm's law), semiconductors devices, communications theory (esp. Shannon's limit, Nyquist etc), Norton and Thévenin models, logic gates, ALUs, frequency domain operations, state machines, coding theory, all of it.

It was a lecture where we basically figured out the required ADC clock jitter upper limit to get a certain number of bits at a certain sample rate[1]. At some point something fundamental like conservation of energy was invoked and I had a holy-shit moment when it all made sense.

However, I do question how much of the grinding away at the maths is necessary and how much is tradition that may have made sense in slide rule and table days. Perhaps a more holistic and intuitive method with an emphasis on "if you need to do this in detail, remember this is where you go". Personally, I can barely remember any domain equations at all, other than Ohm's law![2]

[1] It popped out as something like femto or attosecond and the lecturer said something like "and consider this when buying expensive audio files" (this was back when they were hard to get).

[2] as the same lecturer as above told us on the first day in campus: "honestly, all you need is Ohm's law, everything else we're going to teach you is just that in a dress, you just need to know how to get back to it".

Statistics is way more than formulas. It's easy enough to learn and, if you understand it, so applicable to one's life that it should be one of the principle courses in school.

Here are some basic ideas where it will help.

- helps you understand why the most dangerous part of any flight is the time you spend in a car not the plane.

- why it's useless to try to break even at a gambling establishment

- why the best way to win at the lottery is to never buy a ticket

- why birth control is good at its function but it's not absolutely secure

I can go on for hours.

Statistics needs to be part of any educational reform. It's so useful

The great thing about kids is that they'll often accept shitty arguments as long as they seem legit at first glance. I mean, so will plenty of adults, but kids especially.

Which is handy since, as an adult, shitty arguments are most of what I've got.

* computer skills * finance math * formal logic/critical thinking/rational philosophy/bias identification * electrical/plumbing/auto/construction repair skills * cooking * principles of fitness

High school math's only helpful for entertainment if you like recreational math puzzles or maybe Factorio or something.

You'll notice it takes far less convincing to get kids to understand the value of addition and arithmetic and maybe even very basic algebra. This is because they can immediately use it for play and entertainment. You're locked out of a ton of board games, even, if you can't do simple arithmetic with small numbers. "How much more money do I need to buy that video game I want?" is a question they're motivated to answer.

When it's common for people to encounter and eagerly choose to engage with entertainment the enjoyment of which is greatly enhanced by knowing how to find a second derivative, I expect math will stop being particularly prone to this kind of scrutiny.

As the years went by, each new topic that I’ve learnt seemed like some kind of revelation: the fundamental theorem of calculus, Fourier- and Laplace transforms, Cauchy-Riemann equations, the central limiting theorem, Markov chains, quaternions, Galois theory, and the list goes on. I felt like I was living in Plato’s cave before, being oblivious to this infinitely complex and fascinating world.

I still love making all kinds of art, but it is mathematics and software engineering where I feel truly at home. (the pay is also nice)

Anyway, my point is that you shouldn’t assume someone with artistic talents wouldn’t find math enjoyable, or that they wouldn’t be talented in it if they gave it an honest try. It can “click” at any point in life, not just high school - but if it “clicks” it’s going to be an awesome journey.

Lo and behold, I went to university to study electrical engineering. I needed it for everything, from differentiator circuits, to digital signal processing, to current flows. Limits, derivatives, and integrals are the backbone of EE. My foundational knowledge of Calculus was non-existent. We glossed over it on my course because the assumption is that most everyone knew it because it's part of the College curriculum. I had to go through hell and back at university, doing my coursework while simultaneously catching up on all of the mathematics I had failed to grasp in my college days.

I would have been much better served by solidifying my Algebra, Geometry, and Trigonometry. Better yet, do a pre-Calc (not really a thing in the UK afaik) course containing 101 real life examples of where this might be useful.

things that seem overtly complicated are not necessarily, things that appear simple and small could be wildly complicated, and the process of telling the difference through practice, skill, and experience;

how to think in ways that are beyond sensory;

how to build knowledge on subjects that can provide context (if you know how to perform limit functions you can more accurately estimate what a limit function would look like and then apply that concept or estimate) to the real world in abstract but useful ways.

General intelligence exercises is just a term to mean “teaching people how to think in a more abstract way”

1. Delinearization of progression. You don’t “advance grades” until you demonstrate mastery, and it’s not a competition.

2. Allowing aptitude and interest to tailor schedules. I will never need my ability to rattle off the capital of each state, and that was definitely a wasted week or two. Somebody got PAID for that, and it still blows my mind.

3. Better mental health budgeting.

These comments show a thinking, inventive mind wishing to be useful and improve things, but such a basic lack of understanding of physics. I can only think that these people weren't paying attention at High School, were (poorly) home schooled, or have some sort of incapacity to understand/believe the established laws of science.

Yet they think that they can invent simple solutions which have somehow eluded the experts in the field. Perhaps it is some form of Dunning-Kruger effect.