Ask HN: Why is math often taught without explanation? | Hacker News Reader

Ask HN: Why is math often taught without explanation?

5 pointsmettamage6 comments

I'm brushing up my high school math skills and am following a course from a Dutch university (derivatives and integration). I have some knowledge gaps in my basic math skills (especially integration), and I am plugging them all.

It often explains things like as follows.

This is the product rule: f(x)=g(x)h(x)⟹f′(x)=g′(x)h(x)+g(x)h′(x)

And then an example follows. And that's it!

Why is it on the students to go to websites such as [1] to get an intuitive sense of how it works? That geometric explanation saves so much time. It's much easier and by understanding it, I don't need to commit anything to memory (at least not to rote memory by drilling it).

[1] https://betterexplained.com/articles/derivatives-product-power-chain/

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In my experience, it's because of the order in which math is taught. I didn't _really_ learn about proofs until college, and even then, not until after my first basic calculus course.

Then, if you wanted an explanation for your example, it'd be a proof. The proof is not hard ( see [0] ), but how can one understand a proof if they've never been taught about proofs?

This is what I've seen over and over again. If we want folks to understand what they're looking at, the meat is waiting in the fridge for us. It seems to me that instructors don't have the time or inclination to teach these things at all, which is unfortunate because of how foundational it is to really understanding what you're looking at when it comes to math.

If you're interested in learning how proofs work, and you've missed out on a course like I did, I'd recommend "Mathematical Proofs", by Albert D. Polimeni, Gary Chartrand, and Ping Zhang. You should be able to find a paperback for some bucks.

Usually mathematics texts are thick and difficult to follow for me, I'm not the brightest, but this one was easy to follow and I worked through the brunt of it in a month of concentrated effort by myself. It really set me up for understanding how proofs are structured and why, and how to reason about them on my own.

[0] https://tutorial.math.lamar.edu/classes/calci/DerivativeProo...

Because otherwise it'd take too long. East Asian cultures figured this out a long time ago. Learn by rote, then have it gel over young adulthood.

Differentiation is a mechanical process. You follow the steps and you get the right answer. Performing differentiation is mostly an exercise in following rules. Understanding the proofs of the rules does not help in knowing how to apply the rules and for the vast majority of calculus students they need to learn how to differentiate rather than how to prove the rules of differentiation.

Also, the geometric idea behind the proofs is nice but does not constitute what we consider rigorous. Such geometric intuition quickly fails in more complicated situations. If one has to learn the proofs they are better off learning the analytic proofs and how to operate with and reason about limits.

On the first encounter with the subject almost no one is able to appreciate the intuition behind it. Sometimes people go back to freshen up their skills and are able appreciate and desire a deeper understanding. They wonder about the approach taken to teach the subject and ask why we do things the way we do them. They don’t see that they are no longer at all like the average student who is 18 or so learning it for the first time.

I actually appreciate the way math is structured at the university level in the states. I think introductory math (calculus) should focus on use rather than addressing the "why", as long as it's clear that there is a "why" that can be learned later. In your example, I would hope there would be more than just a formula (i.e. some hint at justification).

There are several reasons for this. 1) Misplaced rigor, especially at the wrong level, can cause students to lose interest, 2) Addressing "why" concerns does not necessarily bring any intuition on use, 3) people can develop intuition at a higher level (I'm sure you have experience using programming functions which you don't necessarily understand), 4) the economics of education (there is only so much time you can be in the classroom).

This is the case for nearly every subject in school. Teaching the why of everything would require a lifetime of study, so that is often only made clear once a student narrows their focus of study (typically in college).

Lossy compression. Educators must squeeze knowledge humanity collectively developed over ~6000 years of written history into ~15-30 years of formal education. So lots of details end skipped over. Mortality sucks.

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