[earlgray]

Learning names for the sake of learning names is never helpful. However the history of mathematics can help to contextualise the material and its motivation. For example, group theory has myriad modern applications. It can be (and usually is) presented in the standard mathematically-polished form, beginning with the axioms and perhaps using matrix groups or simple geometric symmetries as examples. For many learners this is a perfectly sensible way of approaching it and they wouldn't be interested to know more, yet I found it rather clinical and dry.

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.

[flashfaffe2]

I totally second this. For someone who has a tortuous relationship with mathematics, I always lovedthe beauty of som concepts but to a certain point, got lost mainly due to knowledge base not really understood. Getting to know some historical facts would have help to be more critical to myself ( rather than just copy and paste formula).

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...

[JackFr]

While I agree with your sentiment, studying abstract algebra at the undergraduate level we never got more than a broad sketch of a proof of the general insoluability of the quintic. It more seemed an opportunity for the professor to tell the life story of Galois -- mathematical genius and political firebrand killed in a duel over a woman at age 20.

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.