From what I’ve seen, the beauty side tends to come from mathematics teachers and professors who have developed a passion for the subject and genuinely want to share that passion with their students. The practicality side, on the other hand, seems to be driven by industry, administrators, lawmakers, parents, and others involved in the curriculum-setting process.

Ultimately, both sides struggle mightily for students’ attention and a typical result is a collective sigh from the class. If were to make any progress I think we need to recognize that student motivation comes from within. Some students are genuinely attracted to the beauty of mathematics while others are interested in different subjects (science and engineering) for which mathematics is only a prerequisite. Perhaps one approach for dealing with this disparity might be to offer students a choice of projects with differing mathematical styles (proof vs calculation).

As I'm sure you know, group theory arose as a novel yet natural way of investigating the conditions under which polynomials can have closed-form solutions, and when Galois began to sniff this out it allowed him to get to grips with the impossibility of a closed-form solution to the general quintic polynomial: a problem which had beguiled mathematicians for centuries, solved by a theory which crystallised a more structural way of approaching abstract mathematics. Even though Galois theory is a relatively niche topic within the broad context of academia, for me it draws out the true character of group theory in a way that matrix groups don't, and neither do contrived examples formed from the natural numbers under various quotients. And it clearly solves a problem that required a new way of thinking.

There's certainly a balance to be struck here. If you fixate on historical origins then you aren't engaging with the reasons why the topic is still relevant today, so you risk missing the point. It also slows down the pace at which you can absorb the tools needed for applications. But if you only engage with the modern approach you risk building a disconnected, lifeless archipelago of knowledge, unable to see the beautiful links that unify so much of mathematics.

Years later I surprised myself spending time on new maths problems thanks to YouTube maths contents(3brown1blue, Michael Penn, mindyourdecisions,etc...) Which make me reboot my knowledge. The challenge here is how is it inseminated during your scholarship.

From the what I remember, complex number theory were introduced to me as a rule that need to be acknowledged ( with no incentive to look further).

Welsh lab vid below made me reconsider my own shortcomings.

https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...

That being said, it is difficult to find a motivating example for much of elementary abstract algebra -- which is doubly hard, because for many students abstract algebra is a new level of abstraction, math without numbers and it takes something to make that leap.

To know this gives us some perspective on our own biases/limitations and not take things as gospel.

This is sometimes prescribed in such form as: "Fraction X of your reading should be of authors writing at least 100 years ago."

Similar: In defense of the reading of old books by C.S. Lewis ...

https://www.youtube.com/watch?v=jFHQDIE7CSw

"All contemporary writers [of any period] share, to some extent, the contemporary outlook."

But "practice" and "drills" of complex math as done in typical USA schools don't really help many students truly learn the advanced topics. That's why you end up with students who get an 'A' on the test because they memorized the mindless rules to multiply polynomials or calculate the determinant of a matrix but still have no understanding of the underlying math. It was just some weird rules in Algebra to memorize and pass the test. "Mindless plug-n-chug" as people call it.

Repetition of drills do seem to work well for elementary students and basic math like arithmetic: counting, addition, multiplication tables.

But drills is a tool to teach understanding has major shortcomings for later advanced math like Algebra, Calculus, Differential Equations, etc. -- even for the students that got an 'A' on their math tests.

That's why you often get comments on Youtube math explanation videos with variation of, "I studied Fourier Transform in school and somehow pass my exam but this video finally explains what it is! Thank You!!!"

Clearly, there's a huge gap that practice exercises do not address.

As a concrete example, if someone asks “how do we know the Earth is round” or “how do we know the planets orbit the sun”, one could go through ways to figure this out with math and minimal technology. But one could also point out that we have spaceships and probes and have actual pictures! You can see the Earth from a satellite or the ISS or the moon, and you can see the planets from space!

I agree that a process of rediscovery can be a useful teaching technique, but that doesn’t mean that rediscovering something the way it was first known to have been discovered is the right approach.

It would be trivial to skip the historical story and just "yo here's a FET transistor, a solar cell, and a vacuum tube, and the math just works, so get used to it" but nobody is ever taught that way.

Would be interesting for someone to attempt to write a non-historical physics textbook.

On the other hand, in IT, this style of learning is considered normal. "This is how the kool kids do things today. The past does not matter and probably didn't exist anyway. The end."

I recall some prominent physicist saying they've never read Einstein's original papers on relativity; and they don't have any real desire to, since we understand it much better now, and have better examples, notations, etc.

* “here’s what the computer came up with” doesn’t really build as much intuition as doing the equations out by hand

* there’s enough material in “how to plug it into a computer and make it run quickly” for a whole degree

And yet nand2tetris is one of the most common suggestions for people who actually want to figure out how computers work. Because you actually are posed with problems that real world engineers used to struggle with, and are allowed to play with the problem before finding out the answer. It builds a much deeper understanding than "it works like this because it is"

The history is useful there, I think. First off, it is motivating — that’s a pretty big question, really shows what math can accomplish without much hardware. Second, it provides some sort of useful constraints if you are trying to work out the computation yourself. In a textbook the problems are always a bit artificial, some info will be given, some hidden. “Anything an Ancient Greek would have” is a bit more real-world, you can see where the constraints come from.

He's also famous for the sieve of Erastosthenes for finding prime numbers and figuring out the earth's axial tilt.

But in philosophy, students do read everyone from Plato to Rawls in the original. There are (lots of) supplementary texts, of course, but they're companions.

I think that shows an interesting divide that the way we actually think about philosophy is not just a catalogue of arguments and positions one could take, but that the famous works are things worth reading unto themselves. There are a few math papers like that ("God Created the Integers" tries to anthologize them) but they are the exception rather than the rule.

But maybe there's another explanation?

1. Being able to tease apart language from ideas can't be 'taught' you have to do it

2. People disagree much more in philosophy than math

While calculus is actually easy to develop and motivate, the way we "discover" it today is nowhere close to how it was originally discovered.

In my opinion, calculus is most easily motivated via physical examples. Here is a curve, it represents some information, and what are some things we'd like to know about this curve and thus the information it represents? What are some things we can do with that information? Such motivation can be more abstract and not rely on physical examples, but I think the physical examples help tie it to the so-called real world and somewhat mirrors, in spirit, how and why calculus was originally developed.

This has to be the world's worst analogy ever used.