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Comment by munificent | Hacker News Reader

I have had to answer this question to my kids (one of whom abhors math). The explanation I gave is this:

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.)

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I think a lot of the time it is just too abstract to grasp. I think the first time in my life where I was really happy to have learned calculus for my own intrinsic benefit was a few weeks ago, when I set up Home Assistant in an effort to automatically minimize heat in my apartment. It wasn't enough to tell the shade to come down at a certain temperature, because the apartment would already be too hot. So instead I could take the derivative of the temperature of my apartment, allowing me to get out ahead of the worst part of the blast of sun. After all, if the temperature is increasing very quickly, we should act to stop it.

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.

If you hadn't learned calculus or what a derivative is, do you suppose you would have eventually figured out to measure the change in temperature and respond to that?

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

Humans has terrible intuition for these things, it was just 300 years ago humanity figured these things out but once we did we did all these things afterwards in just 300 years. Learning this one thing is the key to so many things.

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

"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.

We are (on average/intrinsically) terrible at predicting or even guessing how certain things will behave. A lot of times when you learn something like say skiing or diving, a major part of training is to untrain your brain from guessing incorrectly how things will unfold/what is the appropriate action. Try recovering a plane from a stall -- you might be able to guess why something happens, but you have to fight your brain to aim the nose of the plane down. (Again on average.)

We humans optimized for everything related to our body movement.

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.

> But I do not remember learning ski or snowboard felt unintuitive.

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.

It seems like this is a solution that should have been baked into the smart device. For example, the Nest thermostats preempts your arrival home and commences toward the desired temperature.

The problem is, the automations you might want and the combination of devices you might want them to act on is large enough that manufacturers can’t possibly foresee them all. When you want to do something ever so slightly outside the stock functionality, it’s helpful to have a little knowledge.

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.

It's called a PID controller.

You invented a PID controller! https://en.m.wikipedia.org/wiki/PID_controller

Technically just the D component :)

I bet that P is still in there too!

This is my favourite part of PID controllers, you can just arbitrarily not implement bits of them and still come out with something that approximates what you need. (Or endlessly oscillates around what you need ;) )

The real way to motivate someone to learn a thing is to give them a project or something they actually want to achieve instead of trying to absorb some drivel without a reason why. That's where self learning shines. You give a great example there. A notable one of mine would be learning vector math and quaternions through trying to make games years ago, but the list is endless and not limited to math or physics.

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.

I struggled with trigonometry in high school, to the point that I had to repeat the class twice. Each time I took the class it was the exact same lesson, and I struggled.

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)_/¯

Yeah I mean I don't really see how we could practically make this approach work in a standard classroom, but the idea that CGP Grey presents where each student would have a sort of AI-tailored personal curriculum (or "digital Aristotle" as he calls it) would potentially allow for it, since each student then gets their own interests turned into projects they can work towards (and still learning the same concepts) while the group teacher is mainly there as an observer and helper.

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 only ever really got into calculus when I decided I wanted to know how AI worked—I'm a strong believer that academic learning needs to be motivated or it simply won't benefit most students.

I don't really agree with this. It seems to be based on the assumption that the entire purpose of school is to prepare you for a job. Obviously that's important, but education also simply enriches your life. Some of the electives I took in high school and college have had a great impact on the way view things, or the way I live my life, despite having nothing to do with my career.

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.

Problem with this is that it’s not very comforting to someone who feels extremely frustrated (not enriched at all) by the experience they’re going through. That’s true even if you know with certainty they’ll feel enriched by it later on.

> Obviously that's important, but education also simply enriches your life.

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...

Imagine if I'd said "life path" instead of "career path" and the rest of my comment still holds true. We all have a finite time on Earth and we're going to spend it doing something. Most of us seem to want to spend that time doing something meaningful and interesting.

> Some of the electives I took in high school and college have had a great impact on the way view things

"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.

But that's part of the problem - not every school system lets you really choose a lot, and more importantly often does not let you choose to not take certain things, often with kinda arbitrary categories. (Speaking of the German school system here, the amount of bullshit I had to learn is astonishing, and I definitely didn't want to get rid of math or history)

>"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."

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 never liked how people say that statistics is useful but calculus is not. I do not believe that you can actually understand statistics without understanding at least some calculus. So much of statistics is about areas under curves!

The problem with this is that the first classes in calculus are usually focused on continuous functions, which don't really exist in statistical datasets. The math has a lot in common, but most people don't really see or use that to their advantage, as evidenced by the literature on "transfer of learning".

Have you actually studied calculus based probability/stayistics though? Your comment seems characteristic of my own former thinking from when I had only taken an algebra based intro stats course (AP statistics) and hadn't yet learn it the calculus based way a few years later.

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

Also, pretty much every fancy formula you learn in Stats 100/ AP stats that looks weird but is very useful, can be derived and proved using calculus. Without calculus you just have to take it on faith, and may not have as intuitive an understanding of why it's true and what the significance of the terms is.

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.

I hate to pull rank, but I've seen this enough to realize almost everyone saying "statistics but not calculus" are simply ignorant to what calculus (or analysis) entails. It is true that the classes as taught focus on continuous (smooth actually) functions, but data needs not to be continuous for you to use calculus, otherwise the field would be useless in real life applications and wouldn't even be a part of high school education like number theory isn't.

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.

I actually do a lot of discrete math (numerical analysis) and statistical analysis for work, and completely agree that there is too much emphasis on symbolic manipulation in school-math. That said, I’m trying to address the reality of what exists in high schools, and the current reality is that a little extra discrete statistics, and a lot less continuous calculus seems like a good trade-off to me.

Alright, sorry to assume less of you. It is a sentiment across the thread but you do know that it's analysis and that calculus is just analysis. Really, calculus of continuous smooth functions is a special subset of calculus.

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).

The problem is kids just don't know.

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.

Your experience is what I think of whenever someone discounts a liberal arts education. It seems like the perfect second degree!

> 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.

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 think the kids are also misled to a great degree. They basically told us if we'd want to go to uni and go for a STEM degree we should absolutely, totally do the focused math courses etc.

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)

Good luck convincing anyone here that they might be acting a little arrogantly.

I would say that these subjects are more likely to turn into vocations than the teaching of how law and economy works.

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.

We are subjected to the law everyday and we all need to know about money to support our addictions to food and shelter.

Most of us are subjected to computers everyday too.

> Music, history, and calculus are useful to some people, but not nearly as many.

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.

You're welcome to introduce your kids to all kinds of things, but more people end up in prison than with record contracts.

Aspects of music can benefit people who do not have record contracts.

I would rather analyze deep finance than listen to music. I just don’t enjoy music, at all.

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.

Introducing kids to hobbies they might enjoy is good. But the fact that we teach them upper-class hobbies, and only upper-class hobbies, in school, suggests we're not doing it solely for their benefit.

I find teaching history essential. It helps understand the present, why things are as they are, and avoid repeating mistakes.

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 absolutely fascinating (one of my favorite subjects), but not very useful. History has too high of causal density to allow drawing any clear lessons. We’ve studied WWI and WWII extensively, and nobody’s really sure what ‘caused’ either. Was it the shooting of an archduke that caused both? What about the arms buildups? The alliances? Just flukes? It’s not clear, and it never will be.

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.

That part of history, like the two WW, of course. The sample size is too small. It is not like: "I will look what happened in the past, and apply that today." That is most likely impossible, just because of context. But trying to understand what happened, really helps as a gymnastic to understand world dynamics. For example, good history teaching should have helped Europe to not be so dependent of Russian gas... I think the people making the decision there were not paying attention in history classes, or had bad teachers/professors.

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.

History in Austria is 2 years about how Hitler was the bad guy and Austria had zero fault in WW2, was essentially forced into everything.

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.

It’s always been my impression that school history is more about indoctrination than any broadly applicable lesson. I took as much as I could in high school, and loved my teachers, but I’m not sure I learned anything I can apply outside the classroom.

Would be interesting to know when or if that's still the current case.

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.

How do you define "introduce"? When I went to school I'm pretty sure I had to take the music subject for around 10-11 years. First we had to learn the flute (no singing), then our third class teacher was shocked that no one could hit a not when singing. A little later it was a mix of good and bad, but mostly getting grades for singing, without ever properly being taught anything (not talking about the first 2 grades here, also ever onward, in a class of 30) and then some mix of tests on musical history.

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.

I feel that music and calculus are very different to history. I believe that history should be a fundamental course taught all the way through, we can't understand where we're going if we don't understand where we came from.

If everyone had at least a brief understanding of game theory, maybe people would cooperate better (not just at work, but in society at large)?

That’s an interesting idea, but I’m not sure people who have leaned game theory in an academic setting actually apply it to their lives. This would be evidence of ‘transfer of learning’, which is alarmingly uncommon. If the students did manage to benefit from learning game theory, I’d support it being added to the curriculum.

It seems like what you're arguing for is to identify the most generalized, broadly applicable subjects possible. And that makes sense. Learning to read and write is probably the most obvious example, because it's about as broadly applicable a skill as one can imagine.

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

I agree with this. A less tactful way of explaining 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.

https://www.smbc-comics.com/comic/why-i-couldn39t-be-a-math-...

Funny as that comic is, it's very unclear at a young age, and even when they're a bit older it's far from obvious. Even at first degree stage, some of the apparently best qualified teenagers who turned up for their first classes this week are going to flunk out anyway, and some of the kids who struggled and seemed like they'd be lucky to get their degree will be potential Fields Medal winners in 10-15 years. Their prior record, even now they're adults, is at best somewhat predictive and nowhere close to definitive.

Something like that did happen in one of my classes and the kids who didnt want to learn it said "why dont you just teach [ smart kid ] then? If anybody is gonna design rockets itll be him.

Do you want to tell a parent that their kid has already decided not to design rockets?

Colleges attract big fish from small ponds, but most small ponds have small fish. Realistically, only big fish will have the attitude and aptitude to become something as advanced as rocket designer.

Yes, but do you want to convince a delusional parent of that? I sure don't. At best it would be a thankless, difficult, and messy task.

The problem with this way is that calculus is needed to get through, like, a basic engineering degree, I assume economics if you are doing it with any rigor. I suspect these aren't like careers for the top 1% braniac kids, they are normal B+ student fields (I mean I know everyone gets straight A's in highschool now, but you know what I mean).

Consider the number of people that go through a typical Calculus class and the debt people get into go to college. Are you sure that ROÍ makes sense?

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.

Who will grow up to routinely do calculus mentally or on pencil and paper? I guess some people will be calculus instructors. Are there any other examples?

Anyone who does a STEM degree?

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.

He said, in the room full of software "engineers".

(I am one)

Neural networks build on calculus.

Few engineers work with neural networks, even fewer build them

Nobody, but can they do whatever math they actually will need without first learning that?

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.

Perhaps also/instead:

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.

What we definitely should teach kids that isnt taught is discounted cash flow analysis as almost everyone has a loan at some point in life and few know how to calculate them

Also, understanding basics of statistics.

I definitely learned a present value calculation in high school at some point, it's not an actual DCF but does teach that fundamental principal about the time value of money.

So why can't they use calculators for that?

I absolutely agree.

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.

Do your colleagues and customers still grasp the finer points of language history? Even if school spent more time on how to write basic emails, I'd suspect it still wouldn't sink in for many children because many of them wouldn't care to apply it. It should only take a couple hours for an educated adult to learn how to write a coherent paragraph. If your coworkers still can't be bothered, then I don't see how forced education would help anyway.

Fair enough, but language history certainly didn't make them any better at it. And memorizing details can be a lot harder and thus more demotivating. It surely can't get much worse if they actually tried teaching kids to write better non-prose.

I'm not familiar with the Norwegian education system but didn't you have classes that required writing essays or at least short answers? If kids go through over a decade of schooling and still cannot write a coherent message well into adulthood, something is fundamentally flawed.

Sure, but I think non-prose is different enough that it's worth spending more time on it.

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.

I like this answer better. It also fits with my experience, mostly in the absence of training I could have received in my college days but did not because I was studying something else, but which would be very applicable to what I do now. It's hard to know where you'll end up.

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.

I try to get my kids interested in math by showing them how it can help win games.

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.

I think this is a much more honest answer.

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.

It’s a good answer. Another is that the goal of education is not always for practical application. This is one of the most important discoveries of mankind ever, and it would be a pity not to be exposed to it.

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