Why doesn't he just teach them highschool calculus...

Why doesn't he just teach them highschool calculus? He could literally boost them out of the dark ages and get famous on his own.

He is busy dying.

He's a dropout. I don't think he's all that great at math.

He's dumb.

He needs to know how to prove them first

Depends. If they don't already have an organized academic system of mathematicians, he might be able to get away with practical proofs.

A bigger problem would be proving the utility of calculus - this doesn't seem like the sort of world that's going to have a nice list of Unsolved Problems.

I mean, there's gotta be a research department in the kingdom somewhere. Surely, he could show them his cellphone or something or teach them algebra at the very least. All he has to do is ask Emilia.

Chasing after half-elf clunge is higher on his priority list.

The witch literally railroads him into doing what she wants to further her goals so there's no way he can even try.

Here's a better idea:
Since things clearly aren't working out for him here, why doesn't he try to figure out how to get back home? And while he's at it he could even figure out why the hell he has Return by Death in the first place? Even if he doesn't do that there's still almost a million better things he could be doing instead of chasing some half-elf pussy that's probably not worth the trouble.

There is a reason the protagonists of these kind of shows are always loser NEETs who have no connection to the real world

No she doesn't. She rail roads the check points. He can go off and be retarded all he likes. It's just that when he dies he comes back and has to repeat shit until he sorts out what she wanted. That and she can allure him to certain things.
Honestly he has quite a lot more wriggle room than you're making out.

Not necessarily. That's not the sort of thing you see with any regularity until at least the Renaissance, when you have wealthy nobles/patricians/whatever providing patronage to, among others, inventors. Then you have to largely wait until the Enlightenment for true research.

I'm not saying it COULDN'T exist, just that it isn't guaranteed. I'd bet against it if the historical analogue holds at least roughly true.

Any idiot could explain the basic concept behind a gun, he should just trade the secrets of modern warfare (which would result in the worlds strongest military by a grossly gratuitous margin) in return for priscillas hand in marriage

>Marry Priscilla
You going to die before you tap that ass. Better to get Crusch pussy.

But the basic concept has been around forever; it was a series of incremental improvements that made them practical.

Is he going to know about corning gunpowder, or the composition of gunmetal alloy, or how to build a reliable flintlock ignition mechanism?

>a million better things he could be doing

He could learn with each stroke of the sword, for example.

That reason isn't to self insert.

That too, but I suspect it has to do with the fact that writing a smart protagonist is far harder than a stupid protagonist.

Writing a hero who is proactive and looks at his surroundings and tries to manipulate them to suit his purpose is hard, because it requires at least as much thought from the author as it would the character. You have to think through the situations you've created.

No, I mean that the reason is that a loser NEET wouldn't try to look for a way home because being stuck in a fantasy world is an improvement for him.

I'm waiting for him to suicide then do a thousand lives worth of training with the old man, killing himself after it everytime, then Kirito his way through all the cultists.

The show then turns as boring as SAO after the first half of season one.

For what purpose?

In a world with magical bullshit does mathematics necessarily work the same way?

Yes. Maths is discovered anyway. He could basically introduce advanced advanced (relative to period) mathematical conceptualisation and let them model it onto real world phenomena as required.