Anyone who has a critical mind and who attended college long enough, knows not to trust textbooks. They are full of mistakes. Because textbooks tend to copy each other, you cannot even trust a fact that appears in multiple textbooks. Mistakes reproduce.

Some mistakes are conceptual. For example, hardly anyone ever needs calculus over continuous domains, but this still get taught as if nearly everyone needed it. But many mistakes are also factual.

This is made worse by the fact that human being a very sensitive to arguments from authority: if someone important said or wrote something, it must be true, surely? And influential people are not immune to this fallacy, so they sometimes fall like dominoes, quickly embracing the wrong ideas… leaving all of us stuck with a mistake.

From at least 1921 up to 1955, it was widely believed that human DNA was made of 24 pairs of chromosomes. This came about because a distinguished zoologist called Theophilus Shickel Painter estimated that this must be the case.

Then, in 1955, a relatively undistinguished research fellow (Joe Hin Tjio), who did not even have a PhD at the time, recognized the mistake for what it was and published a short paper in a relatively modest journal that dared to contradict the established wisdom.

This was enough, in this case, to shift our view, but… think about the fact that there were countless researchers at the time that were better equipped, better funded and more established. None of them dared to question the textbooks. It took someone like Tjio.

There are countless such examples, but we are quick to dismiss them as they do not fit well in our view of the world. Yet I think we should always keep in mind Burwell’s famous quote:

“Half of what we are going to teach you is wrong, and half of it is right. Our problem is that we don’t know which half is which.” (Charles Sidney Burwell)

Daniel Lemire, "The “consensus” is sometimes wrong," in *Daniel Lemire's blog*, September 23, 2015.

“hardly anyone ever needs calculus over continuous functions”

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

@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?

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.

@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.

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.