“When are we going to use this in our everyday life?” (2017)
476 comments·October 4, 2022
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.
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)_/¯
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.
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.
It's called a PID controller.
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.
You invented a PID controller! https://en.m.wikipedia.org/wiki/PID_controller
Technically just the D component :)
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.
> Obviously that's important, but education also simply enriches your life.
You should read the Aims of Education speech given by Abbott; you might really enjoy it.
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)
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.
>"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".
The problem is kids just don't know.
I spent my entire university degree convinced that I was going to go into the video game industry. It took only a few months to realize that it's not what I wanted for a career, and I've spent the next 20 years loving my industry but doing anything but gaming.
I was an arrogant teenager that thought I knew what I was doing. I disrespected the arts, music, history, and focused exclusively on stuff like Math and Calculus.
Now I don't feel like a well-rounded adult, and I wish I spent more time when I was younger on music and humanities.
Your experience is what I think of whenever someone discounts a liberal arts education. It seems like the perfect second degree!
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)
> 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.
Good luck convincing anyone here that they might be acting a little arrogantly.
> 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 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.
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.
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 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.
You're welcome to introduce your kids to all kinds of things, but more people end up in prison than with record contracts.
History in Austria is 2 years about how Hitler was the bad guy and Austria had zero fault in WW2, was essentially forced into everything.
We all knew that's not the truth. The teacher knew, but that is what the text book said so we learned it.
My point is, while history is important it shouldn't be a marked lesson where you just have to remind right and wrong facts.
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.
I would say that these subjects are more likely to turn into vocations than the teaching of how law and economy works.
I see it like when I learned about programming, I was frustrated to learn about language theory, complexity, graphs, etc. I wanted to learn langages, frameworks, specifics for being ready to work right at the end of my degree but it would have made me more fragile and less versatile to future changes. Although law and economy are less likely to change as fast as the latest cool tech stack so this example is not the best.
We are subjected to the law everyday and we all need to know about money to support our addictions to food and shelter.
If everyone had at least a brief understanding of game theory, maybe people would cooperate better (not just at work, but in society at large)?
That’s an interesting idea, but I’m not sure people who have leaned game theory in an academic setting actually apply it to their lives. This would be evidence of ‘transfer of learning’, which is alarmingly uncommon. If the students did manage to benefit from learning game theory, I’d support it being added to the curriculum.
It seems like what you're arguing for is to identify the most generalized, broadly applicable subjects possible. And that makes sense. Learning to read and write is probably the most obvious example, because it's about as broadly applicable a skill as one can imagine.
The argument doesn't seem to apply very well to calculus though, does it?
I agree with this. A less tactful way of explaining it:
"When am I ever going to use calculus in my life??"
You? Probably never. But we're teaching everyone on the off chance that one of you goes on to do something useful with it. Enabling that one person to find a way to make rockets more efficient or something is well worth the tradeoff of wasting the rest of the class's time, from a societal point of view.
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.
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).
Do you want to tell a parent that their kid has already decided not to design rockets?
Consider the number of people that go through a typical Calculus class and the debt people get into go to college. Are you sure that ROÍ makes sense?
If you want to force everyone to learn Calculus for “the good of society”, then don’t force the onerous debt of student loans on private individuals.
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.
Who will grow up to routinely do calculus mentally or on pencil and paper? I guess some people will be calculus instructors. Are there any other examples?
Anyone who does a STEM degree?
I mean, if you're an engineer and you don't know the relationship between position, velocity and acceleration - you're going to have a bad time.
Nobody, but can they do whatever math they actually will need without first learning that?
I have no idea. So I will gladly defer to those who do understand math, and be glad someone does, or my career wouldn't exist.
These lessons help bring you up to speed with foundational concepts and ways of thinking that took humanity a very long time to discover and develop. Learning these things while you are young will, at a minimum, help you keep up with others and avoid being scammed, or at best, help you quickly reach the current limit of our understanding and possibly expand our capabilities.
You can also think of it like stretching and exercising your brain. You may not need to actually do that work, but it's still good for you and helps make other work easier.
What we definitely should teach kids that isnt taught is discounted cash flow analysis as almost everyone has a loan at some point in life and few know how to calculate them
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.
Also, understanding basics of statistics.
So why can't they use calculators for that?
I absolutely agree.
However at least here in Norway, I think we spend too much of the time focusing on useless details. For example, non-trivial part of our Norwegian classes was filled with language history, like the art periods and when various authors lived and so on.
I get that it's nice to know a bit about this, to be able to place them in roughly the right period, but giving a 14 year old a "wrong answer" on a test because the kid doesn't know the exact year some author was born, or failing to list all the authors in some romantic-period clique, is frankly stupid.
Meanwhile, far to little time was devoted to practical writing. Like, say, an email. We spent just a few hours writing reports and similar non-prose, compared to several semesters full of language history, learning about the romanticism and realism periods etc.
I see so many of my colleagues and customers who couldn't write a coherent email if their life depended on it, and can't help but wonder if some of that history time at school had been better spent on practical matters. If a kid wanted to really study language history, they can very well learn this later.
Do your colleagues and customers still grasp the finer points of language history? Even if school spent more time on how to write basic emails, I'd suspect it still wouldn't sink in for many children because many of them wouldn't care to apply it. It should only take a couple hours for an educated adult to learn how to write a coherent paragraph. If your coworkers still can't be bothered, then I don't see how forced education would help anyway.
Fair enough, but language history certainly didn't make them any better at it. And memorizing details can be a lot harder and thus more demotivating. It surely can't get much worse if they actually tried teaching kids to write better non-prose.
Mathematics is learned in a spiral process. It takes probably 3 classes or years before you become competent in it. You are introduced to calculus in high school, but only at the very end. Really, most of the work of calculus is becoming competent at algebra and pre-calculus, which teach you incredibly important concepts like exponential growth, etc (which fundamentally depend on calculus for their development and motivation, but not needed for basic calculations).
Don't think algebra or exponential growth matter? I think these concepts are critical to basic citizenship. Real understanding of exponential growth helps you understand viscerally why, for instance, you should nip a viral outbreak in the bud, and reducing the spreadrate even slightly (R0) can make a huge impact later on, even if spread isn't totally stopped. This is all just gibberish if not learned in high school.
Algebra is used in programming and is basically an introduction to many different programming concepts. Symbolic manipulation of variables, etc, needs to be understood at a basic level to competently program anything, or even use Excel spreadsheets effectively (which almost everyone who ever works a desk job--which is most people--will eventually come in contact with), which is a type of programming.
If you learned 3 semesters of calculus, then you must have learned this in college. If your job is programming-related, then it's pretty relevant for you to understand concepts in calculus like limits, rates of change, total area under a curve, plus having a confident grasp of algebra (which is much of the actual work of calculus).
Blue collar jobs like machinists, homebuilder/carpentry, plumbing, electrician, etc have tons of need for other areas of math that utilize concepts in algebra, pre-calculus, and geometry. As things become more automated, mechatronics and g-code programming are starting to become more relevant in a lot of trades that were previously highly manual. Tuning a PID loop is a fairly normal task for some of these. And you definitely benefit from pre-calc and calculus for things like this, being literally what the I and D stand for.
> I think these concepts are critical to basic citizenship.
In all my life this was never really something I considered important, but the whole pandemic thing gave an entirely new definition to ‘basic education’.
‘I learned all this stuff in high school, why do I have to explain these basic concepts?!’ Was a very common thought.
> Don't think algebra or exponential growth matter? I think these concepts are critical to basic citizenship. Real understanding of exponential growth helps you understand viscerally why, for instance, you should nip a viral outbreak in the bud, and reducing the spreadrate even slightly (R0) can make a huge impact later on, even if spread isn't totally stopped. This is all just gibberish if not learned in high school.
Indeed. I remember talking to a doctor who worked in the ER when the first wave of COVID (brutal in my country) was brewing. She said that it wouldn't be a big deal, they had like 50% of beds vacant (or something like that) so they would be able to handle it just fine. I said that by looking at the data, I thought they would run out of beds next week. Her expression was dismissive, like "this guy doesn't work in healthcare, hasn't set foot in an ER, what does he know?"
The next week, ERs were overloaded, of course. It was in plain sight from the straight line in log-scale graphs. But for most people (including most doctors) the interpretation was (and still is) "wow, this virus is rough, it comes in sudden waves out of nowhere!". Just because they don't understand exponential growth.
Do you really think doctors don't understand, or never learned, exponential growth? I knew a bioengineering student at Cal that was pretty smart and barely made it into med school. They had to go to a DO school instead of MD. So unless the bar for doctors used to be much lower than it is today, every doctor 'knows' what exponential growth is.
Whether they apply it to the real world is another thing.
I know people that don't think raising the minimum wage basically just causes inflation. They're just wondering why apartments in undesirable areas became 3x more expensive when minimum wage went from $5 to $15.
I feel like calculus in high school can be so easily motivated but isn't. For what a high schooler is concerned, you can use it to model the depreciation of mobile devices, cost per day of upgrading devices or how much you save each day by waiting to buy (assuming used market), estimated lifetime revenue for each piece of social media content, the same parametrized on subscribers at time of upload, estimated lifetime additional subscribers per piece of content, etc.
My discrete math professor always made the problems involve food. Because college students are always interested in food, he said.
This assumes most people care about making optimal financial decisions. They don’t.
Like many, many people who understand exponential growth, I never took a calculus class.
The many, many people, including those in government, who clearly didn’t, prove that you’re an outlier.
That's certainly possible, even without self-learning. The topic is explained in precalc, typically. But it's important enough that I think going further is helpful.
So, I don't mean to force you into something you don't want, but do you think people who don't understand exponential growth are missing something relative to you? That they could benefit from the knowledge you have?
That's what I know having understood calculus to those who have bits and pieces of the concepts (exponential growth included) but don't have a big picture. If you have the time and ability to learn more, why limit yourself? Why allow yourself to be put at a disadvantage? And worse (not saying you are, it's hard to gauge from your comment) why would you be in favor of stiffing other people from being better?
K12 math classes are not about understanding. It’s more like typing. The symbol manipulation rules you need to apply are few and straightforward. Just memorize where the keys are. Applying them is also simple. Just press. After that, the difference between an A and a C is all about hitting the keys in succession faster and with fewer mistakes. It’s a Zen thing; overthinking it will never get you there. Concepts like exponential growth and rates of change may be presented to you in lecture, but letting them in your head while doing problems is a classic blunder. Don’t think just do.
I am willing to bet that most educated people who walk around with gross misapprehensions of rates of change and exponential growth phenomena, have in fact drilled the computations just as well as anyone else.
Finance has calc all over the place as well. Marginal costs for instance.
>This is HW so one that comes to mind is programming.
> > Real understanding of exponential growth helps you understand viscerally why, for instance, you should nip a viral outbreak in the bud, and reducing the spreadrate even slightly (R0) can make a huge impact later on, even if spread isn't totally stopped. This is all just gibberish if not learned in high school.
I think programming/algorithm analysis and things like discrete simulations will give you a more durable notion of basic exponential growth for things like virus outbreaks than high-school calculus which is going to focus on things the derivative of the exponential function being similar to the function and stuff about Euler's number.
I hope you do realize "discrete simulations" is an application of calculus (analysis really). The "continuous" version of calculus is a special case. Sure though, this is a problem with the way calculus is taught (too much focus on symbolic differentiation and integration, although that becomes valuable somewhat if you become a physicist or engineer, primarily).
I think that how much calculus you get in highschool depends on which country you live in, I got 3 years (though the first one was minimal), I also got more in highschool physics classes.
But we also have national exams for entrance into Uni and no "general ed" requirement because we're expected to have met that minimum requirement in highschool
David Epstein's Range is a good way to look at outcome based learning.
There are 'Kind Learning Environments'. Things like chess, golf, concert piano, etc. The goal is easy to define, you can rank yourself against others, and the feedback on effort is quick. In such scenarios, so argues Epstein, the 10,000 hour grind is a best way to achieve success.
There are 'Unkind Leaning Environments'. Things like tennis, jazz, business, etc. The goal is difficult to define, you cannot easily rank yourself against others, and the feedback on effort is slow or nonexistent. In these environments, Epstein says that a 'browsing' approach is best. One where you learn as much as you can about as many disparate things as possible and to still deep degrees all the same. You want as many pegs to hang a hat on as you can get, curiosity is not wasted time.
I would say that, in terms of education for the masses, learning Calculus is a great way to develop the 'browser' side of things. General/public education is inherently to be made for the 'unkind learning environment'. Specializing and 10,000 hour grind-fests obviously aren't suitable.
Calc is especially useful as it gives the ideas of derivatives, rates, limits, and integrals for your mental toolset. These are powerfully broad ideas ripe for application. Additionally, as it is traditionally taught, it helps expand the mind to true higher math and lets pupils see how deep that logic/math rabbit hole can go. Lastly, the inescapable history behind it's development is another great dive and gives another avenue for the 'browser' mentality.
I can scarce think of a better subject outside of religious texts that provides such great tutelage for the 'unkind' learning environment that is life.
Tennis has got to be closer to a kind enviornment, a well defined set of strokes, immediate outcomes, a ranking system, and full information.
The benefit of calculus isn’t about the ability to write things down on paper in a fancy code and playing a game to solve it. The benefit is that it paints your waking reality in a color you weren’t aware you could observe previously: curves, and expectation, and prediction.
Based on your take here, I’m gonna guess that you’re in but haven’t yet graduated college.
I’m not sure how you’re going to suggest “learning programming” rather than learning calculus, as calculus is a foundational element of all modern languages. For loops are a further generalization of Leibniz notation, in a rough but very real fashion.
You can only understand a car so deeply without brushing up against physics, the study of which is classically explained by (you guessed it). Sure, you can argue that “you don’t need physics to understand a car well enough to fix it.” Okay, congrats on mastering the adult version of putting the right shaped block into the right shaped hole.
Understanding music really doesn’t require a ton of calculus, unless you want to go into building instruments and music software. If you want to do sound design, you’re also fucked, because understanding Fourier transforms is an important aspect of being a good design engineer.
To me, it sounds like you failed calculus twice and now are trying to prevent people from sharing in your grief. That’s less admirable than you think—-it’s not that calculus is fundamentally hard without reward or merit, it’s likely that somebody failed to indicate to you the importance of calculus.
You are giving plenty of ways in which calculus can be useful. But the GP wasn't arguing against claims that calculus is useful. The GP was arguing against a claim that calculus was useless, but you should study it anyway as "weightlifting for your brain". Your arguments about how calculus is useful would be better directed at the teacher the GP was arguing against.
> But the GP wasn't arguing against claims that calculus is useful.
So no clue who GP is. Grandparent is my best guess. And with respect to you, that would be the guy I was responding to, but that couldn’t possibly be correct because he said
> rather than sitting on your butt learning a useless subject
From context, inferred to be calculus, esp because
> took me 3 semesters of Calculus to figure out that it was useless to me and 90+% of the people that take any of it
Soooo… who is GP
> I’m not sure how you’re going to suggest “learning programming” rather than learning calculus, as calculus is a foundational element of all modern languages. For loops are a further generalization of Leibniz notation, in a rough but very real fashion.
Eh, I'm pretty much opposed to GP's assessment that calculus is useless; in fact, it is probably one of the biggest intellectual achievements of the past couple centuries and modern society would be unimaginable without it.
But I don't really see the connection with programming. Programming/CS is mostly discrete maths and little calculus (with some exceptions, like complexity theory, because it's just easier to talk about functions R -> R than Z -> Z, and numerical analysis, which is about how nice theorems break down when you have to work with messy approximations instead of the real values). Calculus is about the real numbers and we can't even encode the majority of real numbers on computers.
> I really don’t see the connection with programming
It may just be the stupid way my brain is wired. When I think about calculus I can’t help but also consider programming, and vice versa. Okay, Calculus is not a precursor to learning to code. But the DNA of calculus is definitely there.
First, how building blocks of programming and calculus are similar.
1) We can probably throw out the control flow concepts, although they vaguely map to the notion of intervals on evaluated integrals.
2) I said for loops are a generalization of Leibniz notation. I also said it’s a rough relation. You said you don’t see the parallel because you claim pure maths calculus deals with reals and computers are discrete. Yep. Real numbers are discrete at the infinitesimal limit (grab your torch and pitchforks). I hope this is enough explanation on that front. Loops roughly = integrals. It’s purely theoretical. I get that you can’t actually represent an infinite precision real using bytes.
1) I think there’s another very loose but valuable perspective in which calculus and efficient algorithm implementation at least shop at the same grocery store, if not fool around on the down low. I can imagine every possible implementation for solving the knapsack problem as being distributed in a higher dimensional space. There are a ton of bad ones out there with dogshit runtimes. But somewhere near the middle is one that goes zoooooom. That’s an optimization problem—that’s calculus.
2) And then within solving a problem itself. The *good* solutions make use of derivative-like notions. Properties about the problem which you use to solve it efficiently are effective pseudo-derivatives. The way that you can use a derivative plus a point to approximate some next point forward, you can use problem topology plus current state to improve state a bit further until you converge upon an “answer”.
The list goes on.
Calculus is in programming and programming is in calculus. They are cross-pollenated dialects of the mother tongue of the universe.
Strongly disagree. To extend the original metaphor, calculus is an exercise, not a whole workout. Sure, if you only do squats, you may not end up looking as good in a tank top as the guy who does arms all day. On the other hand, you're never going to reach peak physical performance if you skip leg day.
Good luck trying to understand any modern ML paper without a solid understanding of calculus, for example.
Ok, given the number of people that take Calculus, how many will ever read an ML paper? For that matter, how many even know or care what ML is?
This only adds to my point.
Economics, Statistics, ML, Engineering all use calculus. You might be able to get by in some areas without it, but it's not as if it's useless. It's also really easy and straightforward. You would create calculus if it didn't exist.
There are going to be people in the world that are capable of reading things like an ML paper and there are going to be people who are not. Calculus (or other advanced mathematics) is part of what you'll need to understand those papers.
If having the ability to read such papers and understand such concepts isn't something you want for yourself then you definitely shouldn't take advanced mathematics courses. However, many people see the ability to understand those types of things to be a useful skill in giving them opportunities in the future.
From the general population? Not that many.
From this site? I'd wager a significant percentage.
Here's maybe a more generally relevant example: Have you heard about this thing called inflation?
"understand any modern ML paper"
I believe 7 people read an academic paper on average. (1 of those is a mother).
I have read many ML papers, and even understood a few of them. My understanding of calculus is beyond weak: I know what a derivative (and a partial derivative!) is, and what integration is, but God help me if I have to do one.
I've got a pretty firm grasp on linear algebra though. I don't think calculus plays into modern ML that much.
The basic building blocks of ML are simle MAC operations, weights and offsets are applied to inputs to produce outputs. These building blocks are combined to form a neural network. To understand that, or to train or apply a model, you do not need calculus.
>I guess my real point is that you should question whether you really need to take Calculus.
There are very few classes in school that any student "really needs" yet for some reason Calculus, or math in general is the one that takes the brunt of this argument. Why?
Whens the last time you needed to know that Hydrogen has 1 proton and 1 electron for instance?
This may be an EU-centric view, but I think the "problem" with math is that, basically, it's a requirement for "success" in life.
Selection at elite universities is mainly math-based. Sure, you're expected to have great grades in other subjects, but basically, if you suck at math, you're stuck with "suboptimal" paths.
Yes, I know many people have made it without a college degree, or by following some other path. But most of "the rich" have been through elite universities, which require good grades in math. So, it can be perceived as a kind of gatekeeping.
No one cares about chemistry.
So, since neither chemistry nor calculus are seen as "useful in day to day life", but math is used as a selection criterion, people talk about math.
The flipside is that all the well renumerated jobs are very mathematically intensive.
Finanance, technology, they all are.
Here's an HW case about why knowing about Hydrogen will help in life.
Question: Why does Space X use kerosene and liquid oxygen rather than liquid hydrogen and liquid oxygen a better power to weight fuel.
Simply because of the size of the molecule. The Hydrogen molecule is such a small molecule that it's difficult an expensive to use vs kerosene. It's very easy for it to leak. As we have seen in the Artemis 1 rocket that uses hydrogen.
There's my use case. I've used it at least once.
How is having that knowledge any different to if you knew the mathematics involved in the building and launching of the rocket?
Knowing that didn't help you at all. If you didn't know about Hydrogen that rocket still would have launched. It's purely for your own interest.
Unless you're a SpaceX engineer of course but then the other 99.9% of the people on the planet don't "need" to know about the size of Hydrogen and we are back to the same argument.
Just above, you wrote:
>Ok, given the number of people that take Calculus, how many will ever read an ML paper? For that matter, how many even know or care what ML is?
..and now you, apparently without joking, assert that the physics info is different because it's necessary in rocket science, the one thing that is colloquially used to describe knowledge that normal people will never have to worry about?
Ah yes, I too design rockets on a daily basis.
> Here's an HW case about why knowing about Hydrogen will help in life.
*proceeds to give an example only applicable to rocket scientists ...
I think you accidentally disproved your own point.
I think the point of math is to learn other more advanced math.
Which is so useful to the few that will have jobs that need it, that they want to push it as hard as possible just to give them every possible advantage, because it's hard.
They want as many people in advanced STEM as possible, because that's basically like being a billionaire in terms of the level of wealth and comfort, and things like chemistry might solve some really big problems.
Also, a really large number of people still think math is something you actually use daily. These are people that still balance checkbooks and make budgets with paper and do woodworking with fractions instead of CAD apps.
Math really is useful to anyone who isn't comfortable letting a computer do half their thinking.
I might never even own a checkbook in my life, and I've never even used basic algebra IRL. But I can see why someone who never got comfortable with a "There's an app for that" mindset would think long division was a life skill.
I highly doubt I have the talent needed to ever learn a useful amount of math (My idea of useful is enough to get an EE or CS degree), so I don't make it a super high priority to get better at it.
We should be teaching subjects which will (A) ultimately be useful to as many people as possible and (B) exercise the brain as much as possible.
I'm inclined to think statistics and programming would fulfill these requirements better than calculus.
> Here are a few, understanding and fixing a car, understanding music and playing music, art appreciation, literature and understanding the human condition and on and on.
High schoolers already spend a lot of time analyzing literature. I do think they should spend more time with other forms of art as well—why teach only literature criticism, when literature is just one of many art forms?
However, this work exercises your brain in a distinctly different way than mathematics, and I do think students should learn both.
Have you people actually taken a basic statistics course that doesn't require calculus? They are all about memorizing formulas that you plug numbers in to calculate different statistical measures, they don't teach you to understand anything at all.
We teach students calculus at that age since teaching them statistics is basically a dead end, we teach statistics to those poor students who will need to calculate statistical measures without understanding them but we should not force every kid to suffer through that boring thing. Calculus is way more interesting, since kids can easily understand it and you can derive all results on your own, statistics is just plug and chug, much worse than calculus ever could be since students aren't ready for it yet.
There is a parallel in college physics. Typically, most colleges have two tracks: "Calc-based" and "non-calc-based" physics. Everybody finds the non-calc-based course to be an utterly bewildering exercise in memorizing formulas. Even the calc based students are baffled by it. The calc based course is widely regarded as easier and more intuitive -- if you also take calc.
Stats is the same way. I was a math major, and my college had two stats tracks: "Math stats" and "stats for scientists." The first track was 2 semesters, and we had to prove everything. Of course we used calc. The second track was 1 semester, and was an utterly bewildering exercise in memorizing formulas.
I took "math stats," but was then asked to run the discussion section for "stats for scientists." There were things that were utterly intuitive to me, but that I couldn't satisfactorily explain to the students, such as the need for different formulas and methods for discrete and continuous distributions.
Freshman economics. The professor bent over backwards to make sense of the formulas related to things like the supply and demand curves, because he couldn't use derivatives. Also, it was 1982, and yes, the professor showed us the Laffer Curve.
> Have you people actually taken a basic statistics course that doesn't require calculus? They are all about memorizing formulas that you plug numbers in to calculate different statistical measures, they don't teach you to understand anything at all.
Disclaimer I should have included: I personally took statistics in high school instead of calculus.
I can't say what my experience would have been like if I'd known calculus, but loved learning statistics. I don't remember exactly what we did, but I recall it being quite conceptual. Certainly not just a ton of formulas.
> since kids can easily understand it
Maybe with a good teacher. I took calculus courses in high school and college and came out clueless. I could manage the rote work to complete the course, but the big picture was left blank.
I revisited it later in life as an adult and gained a fuzzy picture, but it is still not well defined in my mind. According to many comments here some mathematical concepts I am well versed in and use regularly, if not daily, are calculus and that surprises me as they don't seem like anything that was presented in said classes.
I dare say that calculus has a marketing problem.
Calculus in itself yes. But the statistics / probability or optimisation stuff you can execute are nice ( eg : gradient descent )
Or even linear algebra. I think it made me better at grasping highly formal stuff.
Yes, a statistics course is so much more useful. It's not emphasized in school but it will truly help through out your life if you understand it.
Probabilities under normal curves or any shape probability distribution function are measured as areas under the curve. It helps to have an integral calculus intuition for comparing p(0.1<x<0.5) to p(0.5<x<0.6). It helps to have a multi variate vector distance interpretation of length for error and variance magnitude.
> Yes, a statistics course is so much more useful. It's not emphasized in school but it will truly help through out your life if you understand it.
I think this may be based on an impression of what math coursework used to be. A statistics course is a very common, if not required, part of any modern mathematics major.
> But the statistics / probability or optimisation stuff you can execute are nice ( eg : gradient descent )
You learn gradient descent in calculus, it is based on derivatives...
There are a lot of people in these comments saying, "this thing is more important than calculus" and it turns out it's a concept that is fully fleshed out in analysis which is just calculus essentially. I feel like the problem is calculus as taught focuses too much on algrebraic manipulation which is only useful essentially if you become theoretical physicist and little else while the "why" behind calculus leads you to a lot of more useful results that are along the lines of approximation and optimization, which is closer to what a modern understand of analysis is.
I learned it as part of "operational research", that was some algo-y math course. No idea if that translate. But yeah, definitely closer to calculus than stats or proba.
I think the argument was that a calculus based approach to understanding those subjects dramatically improves and enriches the study of those subjects.
> 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.
Be careful, it sounds like you're describing Common Core Math and several states made that illegal
I would say that you don't really master the most advanced topic you learn.
Attempting algebra is how you solify your knowledge of arithmetic, attempting calculus is how you learn algebra and finally master arithmetic.
> 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.
There are also some counterpoints to it.
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.
> I still cannot see a value in studying classical literature.
And that is the real tragedy of modern education.
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.
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.
: edit: not aliens, and it's by Alfred Bester: https://archive.org/details/New_Worlds_029v10_1954-11/page/n...
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.
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.
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.
> 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.
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.
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?
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.
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.
> 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.
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.
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 most of those are cheap. So. Hm.
> 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.
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.
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 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.
Yeah, the idea sounds like a black-or-white fallacy. The choice isn't, "calculus or nothing". There may be things we could teach that would be equally important that people would be more likely to use.
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.
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.
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.
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.
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.
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.
> 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.
How can one be a citizen if they don’t understand stats, and how to cheat them? The citizenship should only be automatic if you pass that class.
Which is what the majority at 18 intends to do.
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".
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.
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.
“Young man, in mathematics you don't understand things. You just get used to them.”
—John von Neumann
I’m sure there’s room for improvement, but intuition and understanding are usually the result of repetition.
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.
I think the illiterate part is a bit harsh but I generally agree with the rest of your sentiment.
I was just recently giving your exact argument, that if in high school you learned only exactly what was needed to perform your job as an adult, you would essentially be a cog in a machine that requires the world to stay completely static for your entire life in order for you to not get screwed when your skills inevitably become obsolete.
The point of calculus (which imo is just the common path of achieving mathematical maturity) is not that you will use math in your day to day, but that you will be a more well-rounded and dynamic person mentally.
That being said, uninspired high school mathematics focused on memorization is not helpful for anyone.
>Don't you want to understand a fundamental part of how the universe in which you exist works?
Only for a reasonable price. The price differs between individuals for many meaty and mindful reasons.