But also, specifically for calculus, thinking of things of the areas or slopes of other things, and how incremental changes affect them, is a very simple and powerful lens for lots of things.
Of course, lots of teachers just hammer the fiddly memorisables until the wonder is dead because they're easy to test and/or they don't have an intuitive feel of the underlying meaning themselves.
And, for calculus, the fiddlies have never been so needless to know as everything non trivial is a computer job and no one is limiting things carefully to closed forms. Few people need the chain rule specifically, they'd be much better served with knowing that there's a thing called the chain rule and what that means, rather than the exact painful calculations and lists of forms.
Surely there are ways to exercise one's brain that also happen to be useful in everyday life.
It's hard to pick something that everyone would find useful and engaging, so I understand why schools just pick an arbitrary subject and stick with it.
It would be nice if they were honest about it. If they were, they might say something like "We could train your brains with something fun like chess practice, or something useful like programming classes and statistics. But we already have calculus teachers around because some kids will become engineers or whatever, and we don't want to hire a thousand teachers for a thousand niche subjects so we'll use the teachers we already have".
> We could train your brains with something fun like chess practice, or something useful like programming classes and statistics.
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.
Even better. If you teach them basic statistics first, you can teach calculus later and they won't have to wonder why. Just tell them they need calculus to make statistics easier.
That's what my physics teacher did. Whenever he had to explain something basic about Newtonian mechanics he would say "this would be much easier to explain if you knew calculus already".
I never understood people asking those questions. High school stuff is so basic that it's less about learning a particular subject and rather more about getting to know some common language that can be used to discover the world around. I hated some of the subjects I wasn't interested in back then, like biology or history, but I'm still glad that something has remained in my head because now I can have at least some basic clue in conversations surrounding those subjects and have some reasonable starting point in case I actually decide to pursue some understanding of a given topic. I believe that's the whole point of high school education after all.
And not even talking about the fact that if you don't know <SUBJECT_NAME_HERE/>, you're simply not going to notice all the places where applying it could be useful.
The problem with a lot of high school subjects is, that you have to memorize a bunch of dates and years, random names of random plants and animals, that you then immediately forget after you pass the exam.
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.
There are certainly many ways in which education could be improved to be more effective, and the way math is often being taught isn't an exception there. Many people rely on memorization for learning math as well, which is as counterproductive as it gets.
My teachers were moving away from date memorization back in the 90s. These things were mostly approached as a lecture that talked about exactly what you wrote about WW2. Is your experience outdated or did I just get lucky? I went to American public school if it matters.
Former yugoslavia, then slovenia... I had to know every goddamn date and every goddamn village on the exams. And ok.. WWII was the start of the socialist yugoslavia... but I had to know the same for napoleon and the french revolution, and he barely passed here. Franco revolution, the same.. and soviet one too. Also a bunch of caesars too.
Geography was the same... ok, countries and capitals.. sure.. but a bunch of mountains and rivers and streams, where exactly the source is, and where and into which river it flows into... not just the major ones, even the crappy minor ones. Also stuff like, what is the greatest export of nigeria and other countries that are far enough, that I didnt need to know.
Of course I forgot all of that data probably days after the exam, and never cared for 99% of it, and googled the last percent when needed.
To be fair, rote memorization is one of the most improtant and transferable cognitive skills you can develop.
Also, even if I agree that history classes often go overboard, having some notion of the years and even dates that some things happened is important to having a general understanding of history. If you know the who, what, why of WW II but have only a vague idea of when it started and when it ended, or when some of the major events within took place, you'll have a very hard time correlating with other events. It matters for example that WW II happened only 20 years after WW I, not 5 years after, not a century after. You won't get a decent picture of the sequence of events if you don't know some rough dates at least - especially for events happening in different parts of the world, with more indirect linking.
> To be fair, rote memorization is one of the most improtant and transferable cognitive skills you can develop.
To say so is missing the whole point parent comment is trying to make. Memorization is an important skill, that is one thing but saying memorizing random stuff to build that skill is entirely a different claim. I bet there are better ways so learn and hone memory skills than memorize history place/time/dates and kill a student's interesting in learning.
I followed up that statement by explaining that anyway some level of place/time/dates learning is in fact important for history education (though I will re-iterate that entirely too much emphasis is put on that aspect of history, especially in earlier grades).
Still, I don't think that the claim that asking you to memorize (pseudo-)random things improves your skill at memorizing things is a strong claim, I think it's fairly obvious. It's not necessarily the best way, but if it's paired with fairly important education, I don't think it's that bad either.
It's also important to note that, whatever career you chose later in life, there will be lots of random factoids that you'll need to rote memorize to be effective at it - be it names, years and places in history, JavaScript frameworks in programming, diseases in medicine, or even hair styles and product names in hair styling.
FWIW, history teaching seems to have moved away from just looking at dates - at least where I am.
I graduated high school <10 years ago and most of our history classes (including WW1 and 2) were spent on what, why and how. A significant amount of time was spent looking at the leadup and aftermath of both WW1 and WW2 as well as the ideas of the time. We pretty much didn't look at troop movements, generals, battles, etc. apart from mentioning the really significant ones. Same goes for pretty much every other unit of history (mediaeval Europe, colonialism in Asia and Africa, etc.).
Maybe this is a reflection of differences in teaching styles in different parts of the world?
I graduated HS >20 years ago and did not have to memorize a single date in HS history. We did need to know the general ordering of events though. For example, we had to know that the Munich Agreement was before Pearl Harbor, and that the Korean War was after WWII, but it's hard to know anything about these events without knowing that.
OTOH I know people my age who went to different schools that had to memorize things like the exact date that Lincoln was assassinated, so there's definitely disparate pedagogy.
The most use I ever got from my high school English literature class was at a bar in college. An older, much more sophisticated English major was talking to me about her favourite line from Macbeth and I was able to finish her sentence. It felt amazing. You never know when it might come in handy.
>High school stuff is so basic that it's less about learning a particular subject and rather more about getting to know some common language that can be used to discover the world around.
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 never understood people asking those questions.
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.
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.
> Lastly I note that it's mostly math class that gets asked this "what's the point" question.
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. :-)
Also, we would be gullible to whatever new trend is invented by the people who do master those topics. I have interns upset because I don’t want to pay them in bitcoins or give them shares in the company, while we’re quietly churning 1m$ ARR with just two engineers and myself (and others are doing orders of magnitude better). The same interns getting tired after 3 lines of documentation and suggesting that every documentation page should be a video, generated by those american SAAS for a hefty price. They are basically illiterate trying to cover their lack of skills.
The divide between those who use and those who get used is getting wider. And I don’t appreciate belonging to the first group, knowing how little my wisdom is.
I think math feels hard and unpleasant to most students because the way it is taught is often extremely outdated.
In primary school for example, we learn maths by memorising times tables and solving thousands of basic arithmetic problems. This was important in a time before calculators as being able to compute functions is a skill that students might need.
Today though, arithmetic should be taught, not because it might be useful, but because from arithmetic we can discover interesting properties about numbers themselves. I think maths would have been more interesting if you showed students how properties of pure numbers have this nice association with any set of real world objects that can be ordered.
> In primary school for example, we learn maths by memorising times tables and solving thousands of basic arithmetic problems. This was important in a time before calculators...
I used to think like you on this point, until I taught students who were brought up using calculators instead of memorizing multiplication tables, etc. It turns out that many of them could not figure out how to use calculators when needed--they didn't know what to enter because they were rarely required to do any mental math. It's really important for elementary school students to count out loud (including by 2's, 3's, etc.), and count backwards, and memorize multiplication tables, etc., so they are comfortable and confident doing basic arithmetic. Calculators are for people who already understand how to do arithmetic.
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.
You're welcome to explain why you disagree with the OP and what true value can, in your view, be derived from studying classical literature.
I likely agree with you, but if you're just going to make a vaguely disparaging statement in the negative without elaborating or contributing to the discussion then you really might as well not comment at all.
I love how "brevity is the soul of wit" has become some wise aphorism, but of course in context of Polonius' rambling, it is a genuinely hilarious joke.
Reminds me of a sci-fi short story where the military leaders against an alien (?) invasion keep demanding "harder and sharper" human tools for the war. Finally they need a poet and find they don't have any any more.[1]
Though I think that the way classical literature is taught is probably enough to sicken all but the most die-hard readers. Endless dissection of things on a word-by-word basis. Shakespeare (say) wasn't a godlike superhuman imbuing every single word with dozens of layers of meaning. Sometimes it's just a fart joke.
Exactly the same as maths teachers drilling integration rules to death and having everyone conclude, not unreasonably, "this is pointless bullshit". Or history teachers listing dates and names.
You'll better understand contemporary media and culture by being familiar with the foundations they're built upon. Much of modern media are either nods or homages to, or direct knockoffs of, classics. Creators weave allusions to other works in their own work all of the time, and you won't pick up on or appreciate them without familiarity with what they're alluding to.
> But you may as well as this about everything else you do in school
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.
It's hard not to wonder then whether it would be beneficial to keep solving differential equations on a weekly basis to keep our brains in shape even long after we've finished our formal education...
It absolutely would and I really wish I had done that. Starting again after many years is incredibly hard, I feel like I would need to start much lower than differential equations to get back in shape :)
I had the experience of coming upon a differential equation, in the course of some research, which I could not solve explicitly. Mathematica choked on it and my boss and his office neighbor (both math PhD's) were unable to solve it explicitly. When I was about to give up set off to do it numerically my boss's neighbor suggested they call another fellow they both new. Two days later he delivered two neatly written sheets of paper with an explicit solution which featured a really novel (to all of us) substitution which facilitated the solution.
Now in the grand scheme of things the differential equation we were looking at might not be 'interesting' in the sense of being representative of a class of problems in a rich branch of math, but it was sure interesting to us, as it modelled the behavior of the system we were studying. We all were pretty sure there wasn't a closed form solution (but certainly weren't going to spend time proving that) and were pleasantly surprised. The solver did not get a co-author credit in the eventual paper, but he did get a shout out in a footnote.
Sure, but people manipulate them by hand plenty. The task is making them more amenable to solution by either A) converting it into one of the few closed form solutions, or B) converting it into a form more amenable to numerical solution.
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.)
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
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.
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.
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.
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.
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).
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?
>"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).
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.
> 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)
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.
> 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.
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.
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.
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.
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.
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.
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)
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.
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?
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.
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.
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 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’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.
Football players lift weights because it is known to be one of the more effective ways to build muscle strength. Do we have evidence to support the claim that learning calculus is particularly effective at improving general cognitive ability?
Good question. Maybe rigorous mathematical expositions should be replaced with visual metaphors or explanations that will get the idea across without children going through tiresome process of manipulating symbols and calculations.
"The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols."
Most people can’t do more than the simplest derivations in their head, the symbols are just a notational placeholder, and also used to communicate with others.
The abstraction is the what you have to have a knack for, not the symbols themselves.
> “It’s the same thing with calculus. You’re not here because you’re going to use calculus in your everyday life. You’re here because calculus is weightlifting for your brain.”
There are many non calculus things that are weightlifting for your brain, including many math fields that high schoolers don't even know about. Calculus is taught to teenagers for historical reasons, do not overthink it.
I've only maybe used differentiation/integration a few times in my professional career (use it more on personal projects actually). That said, having a solid intuition about first/second order derivatives, rates of change, is incredibly valuable when thinking about the world. I probably use this intuition quite a bit in day to day life without even realizing it. I do wish more probability & statistics was taught earlier on though.
I agree on the intuition. But once the intuition and the fundamentals are there, should teenagers spend months crunching calculus heuristics? It's still the way it's taught in Europe and it's incredibly inefficient.
Ironically, if von Neumann was alive today he would probably encourage kids to use some number crunching software rather than "getting used to it". In that sense civilization may actually have regressed since the von Neumann / Feynman days. Ditching pen and paper for sophisticated computing tools gave us nuclear power and moon landings within 20 years.
Calculus (and the rest of math) is taught because development of human civilization depends on some people knowing it and developing it further. And if you don't start early it's hard to catch up not to mention developing it further.
And it's also training your brain (but that can be done by other things like puzzles or games).
I'd say this used to be true, but calculus has become a historical artifact even at some engineering fields. We have become very good at building abstractions. Matrix / linear algebra on the other hand is something we unconsciously do all the time for high level tasks such as rearranging UIs.
A great many things that humans depend on every day require some understanding of calculus. If we stop teaching teenagers the vast amount of knowledge that humans have accumulated over centuries then progress will stop.
> There are literally math concepts taught in high school and middle school that are only used in extremely specialized fields or that are even so outdated they aren’t used anymore!
So a more appropriate analogy would be doing the wrong exercises for the type for the type of sport being played. It’s still exercise, so probably increases the chances of winning somewhat?
Yeah I like this better. If 99% of the people are not going to use 99% of the things taught in that class, certainly there are subjects that are equally beneficial on a problem-solving basis that are also useful.
It would be better to change the required mathematics curriculum in high school and college to focus more on statistics and less on calculus. Sure it's useful to understand the basic principles underlying calculus. But even in engineering work, only a small fraction of engineers actually use calculus. Statistics is just as good for strengthening the mind, and is more broadly applicable to many real world fields.
I've long held a notion that doing exactly the opposite of what lots of math PhDs think we should do in primary and secondary school would be the right path—take math education much farther from "real math". Focus almost completely on math as a tool for solving real problems.
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?"
You are going to love Methods of Mathematics Applied to Calculus, Probability, and Statistics by Richard Hamming (one of the greatest applied mathematicians).
I learned a lot in calculus and physics classes in high school and college that I have never used over my 35 year career. But learning those principles was tangentially beneficial in many ways. It taught me how to solve problems and think through several steps to come up with an answer. When I hear or read stories about outer space, power generation, or communication signals; I have a framework that I can build upon to understand the issue.
I have kids now in high school and when I help them with some math problems some of it comes back to me, but many of the formulas I memorized so many years ago are long gone from my memory. But that is ok.
I'm 60. I learned things in high school football - about physical conditioning and discipline, and being able to push myself - that have been valuable over the last 40 years.
In high school P.E. I learned how if you're sufficiently athletic teachers will let you bully weaker kids. On the other hand I use calculus everyday and it is fundamental to my understanding of the world.
A lot of the stuff you learn in school is basically just a peek under the hood of how something works. So in the best case you leave with kind of a shallow sample of quite a few really deep subjects.
This shallow knowledge is fairly useless by itself, for sure, beyond the very practical basics, but it gives you a bit of a hook into a variety of core disciplines that you can later - maybe much later - use to connect to other things you do go deep on, even and perhaps especially in completely unrelated fields.
I think really this is the value of an education done right, almost making you aware of what you don't know and giving you just enough context on it that it's not a completely unknown unknown, or unapproachable or unknowable 'magic'.
So by itself any one thing you learn might be pretty useless, all together as a big picture it starts to get a lot more useful. But to get to that big picture you just have to grind through the hard, small, useless seeming stuff piece by piece!
I still remember the lecture when it all lined up, like the Omega molecule in that Star Trek episode.
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".
I used to hate math up until about 8th grade when I had the realization that math problems are just puzzles and when looked at in that way can be fun and interesting. Eventually this lead to the realization that so many other things can be viewed in the same way, and that fostering this ability to change how I view things was pretty crucial to leading a happy life.
School is terrible at helping foster such an attitude though, perhaps because it is incredibly difficult to do so at scale (even at classroom scale), but also because most teachers don't have this ability within themselves.
My kid hated maths in school. I told him that unfortunately he was just "learning the alphabet" and it would just take a long time. This didn't console him.
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. :-/
I’ve never understand why people bitch about having to study math, but are seemingly fine studying history, literature, etc., which are just as useless in everyday life.
My take is that studying history and literature aids in understanding human behavior and connecting with different people, valuable in many situations, but not sufficient by itself as having hard skills/opportunity/leverage/etc are just as important.
This would make sense if they teached recent history, which was at least not part of our curriculum and not to mention even if it was, it would be even more "regulated"
But no, at least in where I lived (Turkey) the history classes are ancient history, then some Turkish civilizations in Anatolia, Ottomans and early Turkey history. Then we have like a 60-70 years of empty space. Is this different for other countries?
Various forms of entertainment are typically much improved by significant history and higher-level literacy training. People like entertainment.
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.
But even this answer is bullshit. The real answer is these are just hoops you need to jump through to get a good job. My daughter will most likely grow up to become a great artist, she has talent for it and she loves it. I can't see her ever doing algebra and beyond in her career or interests. Why do we continue to torture kids with this one size fits all? It's terrible
I was always good at art. Until the age of 14 I wanted to be an artist. I paid no attention to math - I spent most of those classes practicing graffiti lettering in my notebook. It was around this time that we got Internet in our household, and I wanted to create a custom website for my artworks, because I found deviantART lame. So I started looking into how websites are made, and ended up cobbling together a basic PHP page on a free hosting provider. I was fascinated by web programming, so I decided that I would go on to get a software engineering degree, but I still considered graphic design and illustration my main forte. The first class on the first day of university was Introduction to Linear Algebra, which started with matrices, determinants and Gauss-Jordan elimination. I vividly remember it was that first 2-hour lecture that made me realize math was actually awesome! It sounds stupid, but it was at that lecture that I realized for the first time that vectors are just lists of numbers. Like, what the hell? It all made sense, and it was beautiful!
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.
I'm sad to see this because we literally do use calculus every day of our lives. We just don't often recognize it. The weather report is made using calculus. The calculation of the minimum payment on your credit card bill is made with calculus. Calculus is used in computer animation and video games. It's part of statistical analyses that affect government and financial institution decisions. It's used in manufacturing.
It's impossible to live a day in the modern world without calculus.
It's a huge missed opportunity to liken it to working out.
You can use all of these things without you personally knowing calculus. The point of the question is that it's posed by the people who aren't going to go on to create weather reports, credit card payment systems, video games, etc.
Maybe we can try, "you have to learn calculus so you can land a job that lets you pay for things & services that handle calculus for you, so you never have to think about it again".
Except the kids taking high school calculus likely ARE going to do those things one day. Maybe not all of them, but some.
Heck, I don’t use calculus directly in my daily life. But I’m glad I took it because I recognize where it is used, and how, and that helps me understand my world better then without.
I would argue that saving money and personal financial planning uses calculus concepts, and that they are enhanced by formally knowing calculus. It makes questions like "how much money will i have after x years given my mortgage, income, and assets?" approachable. It isn't feasible for most people to hire a human financial planner, and i wouldn't want to use automated tools without understanding enough to be able to perform sanity checks.
> The point of the question is that it's posed by the people who aren't going to go on to create weather reports, credit card payment systems, video games, etc.
I don't think so. If you're in high school and you ask this question, you surely do mean something like "what activity will I possibly doing in my future career that would require calculus" and in that case the answer that you may be a financial analyst, a meteorologist, an electrical engineer, etc. is right on. It's exactly what kids want to know.
But now there's this myth that "you won't ever use calculus in real life" which is totally wrong.
But then why not just take these things in college, when you major in electrical engineering and are taking all the other highly specific classes for your field of interest? It makes no sense to make someone bound for e.g. a career in the arts to suffer through calculus. You could replace that time sink with something more productive and generally useful, like learning to program. Now you can make a website for your art portfolio without having to pay a webdev.
If the goal is to teach reasoning then I think most 101/102 level calculus fails at that. For most students (including mine when I took it), their experience is just getting through generalized homework problems or an exam than actually applying that calculus to test a scientific hypothesis. Reasoning is taught better in those sciences, such as physics, biology, chemistry, or statistics, where you are explicitly developing and testing a null hypothesis. Maybe replacing calculus with statistics in high school curricula would be a lot more useful, if the goal is to teach reasoning and critical thinking.
I think math in general fails at that. After all it's just one tool in the big ol' cognitive shed.
(This is why I think the recent brouhaha about California changing some requirements completely misses the point... but meh. Education is like healthcare, completely broken and fucked in all the ways it could be.)
You don't use it in those cases, you get what you need from someone else using calculus. In the same way you don't use cooking when you go at a restaurant.
The “when are we going to use this” question is about when “we” ourselves will directly use it - not when we will use something that uses it.
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.
You probably don't need to know how to compute a derivative, but there are tons of related concepts that are helpful for reasoning about systems in the world. You can always Google the chain rule, but having a general sense of the trend is often all you need.
For example, you don't have to remember how to derive it, but knowing that y'' = y is a positive feedback loop (exponential growth) but y'' = -y is a negative feedback loop (oscillating) is really useful in all sorts of common sense scenarios.
Learning is about concepts more than facts or algorithms.
>knowing that y'' = y is a positive feedback loop (exponential growth) but y'' = -y is a negative feedback loop (oscillating) is really useful in all sorts of common sense scenarios.
I'm not sure what sorts of situations you keep finding yourself in, but I think they're pretty atypical.
Oversimplifying leads to confidently wrong predictions based on superficial understanding. Basic intuition about differential equations doesn't meaningfully help you with the math of economic models, nor is math alone enough to understand what happens in a complex system made of people.
You may be better off not knowing anything and knowing that you don't.
Edit: Not to say it's good not to know things in general. Just that there's some minimum you need to know for it to practically help you, and sometimes it's a lot.
> The “when are we going to use this” question is about when “we” ourselves will directly use it - not when we will use something that uses it.
You don't have to use it directly for it to be useful.
Having some knowledge/experience with it means you can assume a level of trust in the result of a system that uses it, even if you don't touch it directly.
If you don't it's either blind trust (which requires quite a leap of faith) or, more probably, distrust.
By and large, there's very little of what we're taught (whether it's math, or logic, or science at large, or literature...) that we use directly in our everyday life. Nonetheless it helps build an internal compass that helps us eyeball/gut feel what we can trust or not trust.
The growing distrust in recent key events (climate change, covid...) is largely due to that compass being broken, and to me that's in good part due to a failing of education systems at large.
But for these things they are often really quite uncontroversial. Are you calculating the weather by hand to confirm NOAAs numbers? Definitely not. In the end you have to put trust in things you don't understand, because you can't learn the exact underpinnings of each and everything you face in life within the span of one human lifetime.
I agree with this 100%. The insight into Calculus that we get in high school is pretty fleeting, but you do at least get to see the ingredients that go into things like weather reports. Otherwise it just becomes a magic black box. Maybe it doesn't work for a lot of people, but it just has to stick for enough people that we can continue to tell magic apart from science at the society level.
The issue is that calculus in itself with symbolic algebra is next to useless for average person. However intuitive concepts, like area under a curve, are not.
I "solve for x" all the time, though, admittedly, outside of work, it rarely gets more complicated than a simple expression with a fraction or two.
However, what is aggressively useful is dimensional analysis. When I'm doing a calculation and need to quickly check that the formulation is right, checking the units works every time.
You’re getting a lot of answers about how you don’t need calc to use things other people have made with calc. This turns the answer into “so that you can avoid weird mysticism about how the world works.”
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).
This is like saying we use quantum physics every day of our lives because physics. It's true, I guess, but you don't have to know anything about quantum physics and the vast majority of people don't need to know anything about calculus.
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?
> I'm sad to see this because we literally do use calculus every day of our lives. We just don't often recognize it. The weather report is made using calculus.
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.
> This is like claiming David Beckham uses advanced physics to kick his free kick
David Beckham is in a highly-specialized field (professional soccer player) and this is about things everyday people use, so I guess I don't follow the analogy.
The problem with this is that people don't really retain information like that. College is 15 years in the past for me and I'd bet that if you handed me every exam I took in college I'd flunk everyone of them. And probably quite badly too. I'd wager most people are the same. So how can it be so important if we all remember so little.
You also can't live in that world without knowing meteorology, computer graphics, animation, and, well, all of manufacturing. Yet, we aren't suggesting to teach all of those things to every people now are we?
I would argue its an even bigger missed opportunity not replacing calculus with programming classes focused on using a cli and writing scripts to do work with the computer. Like it or not people get by fine in life with abstractions of more complicated things, but I think having knowledge of programming is akin to learning to read in terms of the potential it can unlock that can be relevant to every career there is. If programming became widely mandated into the curriculum, we would probably see a lot more interesting technologies and applications of existing technologies emerge in the coming decades in places you wouldn't even expect, than if we pressed forward with forcing calc down everyone's throat in high school and making them hate math for life.
That's like saying you use general relativity because you own a GPS. Understanding general relativity is only useful for the people making the device, not the people using it. You don't need to be a mechanical engineer to drive a car, a biologist to have children or a mathematician to use a credit card.
IDK if that's the reason I'd given for calculus. I might not literally solve integrals, but the base knowledge of what an integral is, what a derivative is, yes, I absolutely use those. I'm also a SWE/SRE, so … there's that. But how often I see graphs from products whose entire job is metrics that are just labelled wrong, e.g., w/ the base unit instead of the rate, or what actually use the base unit instead of the rate, making for a difficult UX¹. If the devs of those products understood … calculus (let alone stats!) maybe the products would be less garbage? As it is, I still need to know that as a user.
But yeah, I've not taking a literal integral in a while. Usually I'm doing some sort of very crude integration.
Similar w/ the CS degree and everybody in this field going "it isn't needed" and then going "why isn't the database answering this query quickly, when there is an index on those fields?²" and follow that with a discussion of how B-trees work (or rather, don't)…
And should I ever need to solve an integral, I will recognize that problem when I see it, and know what Wikipedia articles I need to page back into my brain.
¹what I mean here is, e.g., like what Azure Metrics does. E.g., there's a graph I use that measures throughput, but the unit is just "Bytes". But each point is "number of bytes transmitted during the window of time represented by that point" so it's really "bytes / 5 minutes" or something. But of course, then, you zoom, and now it is "bytes / 10 mintues" … but the axis doesn't tell you that. This has the effect that as you zoom in or out … the numbers change! Which makes no sense (obviously the effect of zooming a graph does not go back in time and alter the readings) … but only if you were properly measuring bytes/sec. (But as it is, there's a constant / divisor caught up in there.)
(And that ignores harder problems with zooming metrics, like aliasing or resolution, or other metrics problems like percentiles on aggregates or efficient computation of calculated values and where to put windows, etc. … but pfft I'm in the stone age over here.)
²and it's almost always a 2D range query or a range + exact value and the exact value is the second column in the index…
476 comments
[ 2.6 ms ] story [ 310 ms ] threadBut also, specifically for calculus, thinking of things of the areas or slopes of other things, and how incremental changes affect them, is a very simple and powerful lens for lots of things.
Of course, lots of teachers just hammer the fiddly memorisables until the wonder is dead because they're easy to test and/or they don't have an intuitive feel of the underlying meaning themselves.
And, for calculus, the fiddlies have never been so needless to know as everything non trivial is a computer job and no one is limiting things carefully to closed forms. Few people need the chain rule specifically, they'd be much better served with knowing that there's a thing called the chain rule and what that means, rather than the exact painful calculations and lists of forms.
It's hard to pick something that everyone would find useful and engaging, so I understand why schools just pick an arbitrary subject and stick with it.
It would be nice if they were honest about it. If they were, they might say something like "We could train your brains with something fun like chess practice, or something useful like programming classes and statistics. But we already have calculus teachers around because some kids will become engineers or whatever, and we don't want to hire a thousand teachers for a thousand niche subjects so we'll use the teachers we already have".
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.
Which is what the majority at 18 intends to do.
That's what my physics teacher did. Whenever he had to explain something basic about Newtonian mechanics he would say "this would be much easier to explain if you knew calculus already".
And not even talking about the fact that if you don't know <SUBJECT_NAME_HERE/>, you're simply not going to notice all the places where applying it could be useful.
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.
Geography was the same... ok, countries and capitals.. sure.. but a bunch of mountains and rivers and streams, where exactly the source is, and where and into which river it flows into... not just the major ones, even the crappy minor ones. Also stuff like, what is the greatest export of nigeria and other countries that are far enough, that I didnt need to know.
Of course I forgot all of that data probably days after the exam, and never cared for 99% of it, and googled the last percent when needed.
Also, even if I agree that history classes often go overboard, having some notion of the years and even dates that some things happened is important to having a general understanding of history. If you know the who, what, why of WW II but have only a vague idea of when it started and when it ended, or when some of the major events within took place, you'll have a very hard time correlating with other events. It matters for example that WW II happened only 20 years after WW I, not 5 years after, not a century after. You won't get a decent picture of the sequence of events if you don't know some rough dates at least - especially for events happening in different parts of the world, with more indirect linking.
To say so is missing the whole point parent comment is trying to make. Memorization is an important skill, that is one thing but saying memorizing random stuff to build that skill is entirely a different claim. I bet there are better ways so learn and hone memory skills than memorize history place/time/dates and kill a student's interesting in learning.
Still, I don't think that the claim that asking you to memorize (pseudo-)random things improves your skill at memorizing things is a strong claim, I think it's fairly obvious. It's not necessarily the best way, but if it's paired with fairly important education, I don't think it's that bad either.
It's also important to note that, whatever career you chose later in life, there will be lots of random factoids that you'll need to rote memorize to be effective at it - be it names, years and places in history, JavaScript frameworks in programming, diseases in medicine, or even hair styles and product names in hair styling.
I graduated high school <10 years ago and most of our history classes (including WW1 and 2) were spent on what, why and how. A significant amount of time was spent looking at the leadup and aftermath of both WW1 and WW2 as well as the ideas of the time. We pretty much didn't look at troop movements, generals, battles, etc. apart from mentioning the really significant ones. Same goes for pretty much every other unit of history (mediaeval Europe, colonialism in Asia and Africa, etc.).
Maybe this is a reflection of differences in teaching styles in different parts of the world?
OTOH I know people my age who went to different schools that had to memorize things like the exact date that Lincoln was assassinated, so there's definitely disparate pedagogy.
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.
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.
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. :-)
The divide between those who use and those who get used is getting wider. And I don’t appreciate belonging to the first group, knowing how little my wisdom is.
In primary school for example, we learn maths by memorising times tables and solving thousands of basic arithmetic problems. This was important in a time before calculators as being able to compute functions is a skill that students might need.
Today though, arithmetic should be taught, not because it might be useful, but because from arithmetic we can discover interesting properties about numbers themselves. I think maths would have been more interesting if you showed students how properties of pure numbers have this nice association with any set of real world objects that can be ordered.
I used to think like you on this point, until I taught students who were brought up using calculators instead of memorizing multiplication tables, etc. It turns out that many of them could not figure out how to use calculators when needed--they didn't know what to enter because they were rarely required to do any mental math. It's really important for elementary school students to count out loud (including by 2's, 3's, etc.), and count backwards, and memorize multiplication tables, etc., so they are comfortable and confident doing basic arithmetic. Calculators are for people who already understand how to do arithmetic.
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.
And that is the real tragedy of modern education.
I likely agree with you, but if you're just going to make a vaguely disparaging statement in the negative without elaborating or contributing to the discussion then you really might as well not comment at all.
My liege and madam, to expostulate
What majesty should be, what duty is,
Why day is day, night, night, and time is time —
Were nothing but to waste night, day, and time.
Therefore, since brevity is the soul of wit,
And tediousness the limbs and outward flourishes,
I will be brief. Your noble son is mad.
Though I think that the way classical literature is taught is probably enough to sicken all but the most die-hard readers. Endless dissection of things on a word-by-word basis. Shakespeare (say) wasn't a godlike superhuman imbuing every single word with dozens of layers of meaning. Sometimes it's just a fart joke.
Exactly the same as maths teachers drilling integration rules to death and having everyone conclude, not unreasonably, "this is pointless bullshit". Or history teachers listing dates and names.
[1]: edit: not aliens, and it's by Alfred Bester: https://archive.org/details/New_Worlds_029v10_1954-11/page/n...
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.
Now in the grand scheme of things the differential equation we were looking at might not be 'interesting' in the sense of being representative of a class of problems in a rich branch of math, but it was sure interesting to us, as it modelled the behavior of the system we were studying. We all were pretty sure there wasn't a closed form solution (but certainly weren't going to spend time proving that) and were pleasantly surprised. The solver did not get a co-author credit in the eventual paper, but he did get a shout out in a footnote.
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.)
"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.
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.
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.
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)
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.
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.
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.
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.
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.
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'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.
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.
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.
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.
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.
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.
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.
http://worrydream.com/KillMath/
The abstraction is the what you have to have a knack for, not the symbols themselves.
This goes far beyond math, it is applicable to most important areas of life.
There are many non calculus things that are weightlifting for your brain, including many math fields that high schoolers don't even know about. Calculus is taught to teenagers for historical reasons, do not overthink it.
—John von Neumann
I’m sure there’s room for improvement, but intuition and understanding are usually the result of repetition.
And it's also training your brain (but that can be done by other things like puzzles or games).
> There are literally math concepts taught in high school and middle school that are only used in extremely specialized fields or that are even so outdated they aren’t used anymore!
So a more appropriate analogy would be doing the wrong exercises for the type for the type of sport being played. It’s still exercise, so probably increases the chances of winning somewhat?
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?"
I have kids now in high school and when I help them with some math problems some of it comes back to me, but many of the formulas I memorized so many years ago are long gone from my memory. But that is ok.
This shallow knowledge is fairly useless by itself, for sure, beyond the very practical basics, but it gives you a bit of a hook into a variety of core disciplines that you can later - maybe much later - use to connect to other things you do go deep on, even and perhaps especially in completely unrelated fields.
I think really this is the value of an education done right, almost making you aware of what you don't know and giving you just enough context on it that it's not a completely unknown unknown, or unapproachable or unknowable 'magic'.
So by itself any one thing you learn might be pretty useless, all together as a big picture it starts to get a lot more useful. But to get to that big picture you just have to grind through the hard, small, useless seeming stuff piece by piece!
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".
School is terrible at helping foster such an attitude though, perhaps because it is incredibly difficult to do so at scale (even at classroom scale), but also because most teachers don't have this ability within themselves.
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. :-/
But no, at least in where I lived (Turkey) the history classes are ancient history, then some Turkish civilizations in Anatolia, Ottomans and early Turkey history. Then we have like a 60-70 years of empty space. Is this different for other countries?
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.
It's impossible to live a day in the modern world without calculus.
It's a huge missed opportunity to liken it to working out.
... except most of those are cheap. So. Hm.
Heck, I don’t use calculus directly in my daily life. But I’m glad I took it because I recognize where it is used, and how, and that helps me understand my world better then without.
I don't think so. If you're in high school and you ask this question, you surely do mean something like "what activity will I possibly doing in my future career that would require calculus" and in that case the answer that you may be a financial analyst, a meteorologist, an electrical engineer, etc. is right on. It's exactly what kids want to know.
But now there's this myth that "you won't ever use calculus in real life" which is totally wrong.
to reason.
it would be great to teach people critical thinking. at every age, at every year.
applied epistemology, rationality, etc. of course no need for those fancy words.
... and during those lessons at one point they could learn about the usefulness of models, and the usefulness of math, money, programming, etc.
but otherwise there's no point in ramming math/programming/finance directly into the heads of kids.
(This is why I think the recent brouhaha about California changing some requirements completely misses the point... but meh. Education is like healthcare, completely broken and fucked in all the ways it could be.)
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.
For example, you don't have to remember how to derive it, but knowing that y'' = y is a positive feedback loop (exponential growth) but y'' = -y is a negative feedback loop (oscillating) is really useful in all sorts of common sense scenarios.
Learning is about concepts more than facts or algorithms.
I'm not sure what sorts of situations you keep finding yourself in, but I think they're pretty atypical.
if you want to have a chance of understand the economic news it is a good idea to have familiarity with them.
You may be better off not knowing anything and knowing that you don't.
Edit: Not to say it's good not to know things in general. Just that there's some minimum you need to know for it to practically help you, and sometimes it's a lot.
Which I guess just brings us back to the top of the question. My bad.
You don't have to use it directly for it to be useful.
Having some knowledge/experience with it means you can assume a level of trust in the result of a system that uses it, even if you don't touch it directly.
If you don't it's either blind trust (which requires quite a leap of faith) or, more probably, distrust.
By and large, there's very little of what we're taught (whether it's math, or logic, or science at large, or literature...) that we use directly in our everyday life. Nonetheless it helps build an internal compass that helps us eyeball/gut feel what we can trust or not trust.
The growing distrust in recent key events (climate change, covid...) is largely due to that compass being broken, and to me that's in good part due to a failing of education systems at large.
However, what is aggressively useful is dimensional analysis. When I'm doing a calculation and need to quickly check that the formulation is right, checking the units works every time.
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.
David Beckham is in a highly-specialized field (professional soccer player) and this is about things everyday people use, so I guess I don't follow the analogy.
But yeah, I've not taking a literal integral in a while. Usually I'm doing some sort of very crude integration.
Similar w/ the CS degree and everybody in this field going "it isn't needed" and then going "why isn't the database answering this query quickly, when there is an index on those fields?²" and follow that with a discussion of how B-trees work (or rather, don't)…
And should I ever need to solve an integral, I will recognize that problem when I see it, and know what Wikipedia articles I need to page back into my brain.
¹what I mean here is, e.g., like what Azure Metrics does. E.g., there's a graph I use that measures throughput, but the unit is just "Bytes". But each point is "number of bytes transmitted during the window of time represented by that point" so it's really "bytes / 5 minutes" or something. But of course, then, you zoom, and now it is "bytes / 10 mintues" … but the axis doesn't tell you that. This has the effect that as you zoom in or out … the numbers change! Which makes no sense (obviously the effect of zooming a graph does not go back in time and alter the readings) … but only if you were properly measuring bytes/sec. (But as it is, there's a constant / divisor caught up in there.)
(And that ignores harder problems with zooming metrics, like aliasing or resolution, or other metrics problems like percentiles on aggregates or efficient computation of calculated values and where to put windows, etc. … but pfft I'm in the stone age over here.)
²and it's almost always a 2D range query or a range + exact value and the exact value is the second column in the index…