I mean over continuous domains. In the real world, you have finite (or, at least, discrete) domains. The rate of change is just a subtraction.

In fact, because there are finitely atoms in the universe, everything is fundamentally finite. Thinking about infinite sets is useful, as an abstract… Continuous domains are… Well… it is not such a useful abstraction for most people.

It turns baby-level mathematics into fancy abstractions that most people will never grasp. The fundamental theorem of calculus, for example, is entirely trivial in the real world… if you compute a prefix sum, and then you compute successive differences, you get back the original data. That’s entirely intuitive for everyone… But we make it super complicated by using continuous domains… even though nobody has actual data over continuous domains… that’s not even possible in principle.

Ok, it’s clearer now, and I can see where you’re coming from.

Replace ‘nearly everyone’ with nearly every CS student, and I can agree that maybe a large portion of CS students don’t need to take calculus because they intend to be professional programmers.

But it seems unlikely that most people who go to college take calculus classes: don’t most people take non-STEM majors? The STEM disciplines that do require it involve either reasoning that requires understanding continuous dynamic quantities (economics, any kind of engineering, the hard sciences, etc.) or continuous probability spaces (anything involving nontrivial statistics, like biology, medicine, psychology). And of course math majors need to know everything about math, if not everything period 🙂

If anything I think the problem with calculus would be in the way it’s taught and motivated—e.g., you need to know how to manipulate this specific integral for the test—, rather than the fact that it is taught at all.

But it seems unlikely that most people who go to college take calculus classes: don’t most people take non-STEM majors?

Where I live, calculus is a requirement for anyone going to college in management, health, science, engineering, economics…

Granted, the humanities don’t have to take calculus, but they do not make up anything close to a majority.

I’d like to meet an engineer who can testify that knowing how to differentiate x tan(x) is a sought-after skill in industry. Of course it is not! It is about as useful as knowing latin.

Samsays:

Daniel,

As an engineer, I would say that calculus is a necessary skill (and also underrated) in the industry. My work depends on modelling systems based on their underlying physics, for which calculus is indispensable. As for your example, I would say differentiating x, and tan(x) is useful, as is knowing the rule for differentiation of a product f(x)*g(x). Now, for an engineer who knows these 3 rules (as they should), differentiating x*tan(x) can be broken down and becomes trivial. Saying that the knowledge x*tan(x) is not useful, and therefore calculus should not be taught is a bit like saying the word ‘farraginous’ is not useful and hence English should not be taught. (I’m exaggerating, but you get the point)

Nonetheless, all this is nitpicking details, and I agree with the basic premise of your post that the most important skill is the ability to think critically. Picking different skills to teach while dismissing this crucial ability is missing the forest for the trees. This ability to question things despite how obvious they may seem is something that is missing from our education system.

Alex Gittenssays:“hardly anyone ever needs calculus over continuous functions”

What does calculus over continuous functions mean— surely you don’t mean the calculus of variations?

Daniel Lemiresays:@Alex

I mean over continuous domains. In the real world, you have finite (or, at least, discrete) domains. The rate of change is just a subtraction.

In fact, because there are finitely atoms in the universe, everything is fundamentally finite. Thinking about infinite sets is useful, as an abstract… Continuous domains are… Well… it is not such a useful abstraction for most people.

It turns baby-level mathematics into fancy abstractions that most people will never grasp. The fundamental theorem of calculus, for example, is entirely trivial in the real world… if you compute a prefix sum, and then you compute successive differences, you get back the original data. That’s entirely intuitive for everyone… But we make it super complicated by using continuous domains… even though nobody has actual data over continuous domains… that’s not even possible in principle.

Am I clearer?

Alex Gittenssays:Ok, it’s clearer now, and I can see where you’re coming from.

Replace ‘nearly everyone’ with nearly every CS student, and I can agree that maybe a large portion of CS students don’t need to take calculus because they intend to be professional programmers.

But it seems unlikely that most people who go to college take calculus classes: don’t most people take non-STEM majors? The STEM disciplines that do require it involve either reasoning that requires understanding continuous dynamic quantities (economics, any kind of engineering, the hard sciences, etc.) or continuous probability spaces (anything involving nontrivial statistics, like biology, medicine, psychology). And of course math majors need to know everything about math, if not everything period 🙂

If anything I think the problem with calculus would be in the way it’s taught and motivated—e.g., you need to know how to manipulate this specific integral for the test—, rather than the fact that it is taught at all.

Daniel Lemiresays:@Alex

But it seems unlikely that most people who go to college take calculus classes: don’t most people take non-STEM majors?Where I live, calculus is a requirement for anyone going to college in management, health, science, engineering, economics…

Granted, the humanities don’t have to take calculus, but they do not make up anything close to a majority.

I’d like to meet an engineer who can testify that knowing how to differentiate x tan(x) is a sought-after skill in industry. Of course it is not! It is about as useful as knowing latin.

Samsays:Daniel,

As an engineer, I would say that calculus is a necessary skill (and also underrated) in the industry. My work depends on modelling systems based on their underlying physics, for which calculus is indispensable. As for your example, I would say differentiating x, and tan(x) is useful, as is knowing the rule for differentiation of a product f(x)*g(x). Now, for an engineer who knows these 3 rules (as they should), differentiating x*tan(x) can be broken down and becomes trivial. Saying that the knowledge x*tan(x) is not useful, and therefore calculus should not be taught is a bit like saying the word ‘farraginous’ is not useful and hence English should not be taught. (I’m exaggerating, but you get the point)

Nonetheless, all this is nitpicking details, and I agree with the basic premise of your post that the most important skill is the ability to think critically. Picking different skills to teach while dismissing this crucial ability is missing the forest for the trees. This ability to question things despite how obvious they may seem is something that is missing from our education system.