Mathematical map - diagram illustrating what you need to know to learn

[ThereIam]

I was curious if anyone is aware of anything resembling a mathematical map, or learning tree. By this I mean a diagram that illustrates the sequence in which one must learn particular topics. I ask because I find myself encountering topics (today it was finite element method) and clicking the Wikipedia article, and finding that that article uses explanations dependent understanding other topics. Thus I work backwards through a series of articles until I find familiar subjects, and then work forward again. It would be nice to start with the stuff I know, and then move forward.

Maybe this is a ridiculous idea.

 

[Mandelbroth]

I was curious if anyone is aware of anything resembling a mathematical map, or learning tree. By this I mean a diagram that illustrates the sequence in which one must learn particular topics. I ask because I find myself encountering topics (today it was finite element method) and clicking the Wikipedia article, and finding that that article uses explanations dependent understanding other topics. Thus I work backwards through a series of articles until I find familiar subjects, and then work forward again. It would be nice to start with the stuff I know, and then move forward.

Maybe this is a ridiculous idea.

Imagine the most complicated subject ever created. Intricate, powerful, unbending. A language that can uniquely describe anything in the universe down to every detail. A very specific language that gives order to chaos and clarity to randomness.

Now add on to that a bunch of stuff that can't even apply to the real world. Things that seem to be almost nonsensical to our world, like the idea that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Now add even more. Stuff that we don't even comprehend ourselves yet. Unsolved problems, unproved theorems. Things that don't quite fit in the margins of books.

Poetically, that's mathematics.

If you can find a map like what you're looking for, there won't really be anywhere to start. It probably isn't small enough to singularly fit on a page. The best way is to just learn backwards. See it as a game. Have fun with it.

At least, that's my opinion.

 

[symbolipoint]

I was curious if anyone is aware of anything resembling a mathematical map, or learning tree. By this I mean a diagram that illustrates the sequence in which one must learn particular topics. I ask because I find myself encountering topics (today it was finite element method) and clicking the Wikipedia article, and finding that that article uses explanations dependent understanding other topics. Thus I work backwards through a series of articles until I find familiar subjects, and then work forward again. It would be nice to start with the stuff I know, and then move forward.

Maybe this is a ridiculous idea.

Not a ridiculous idea; but just not an absolute idea.

Consider some mathematical education far, far less advanced than the Finite Element Methods that you mentioned. If you could gather 100 mathematically educated and experienced people and ask each of them to create a mathematical learning roadmap, you could expect at least 20 different such roadmaps... and maybe some of these people will sooner or later make changes to their roadmap designs. No single roadmap would fit every student the same. The order in which topics can be studied is not absolutely fixed, in that some topics can/will be learned out of sequence of the roadmap.

Consider: Sure, learn addition and subtractions of whole numbers before studying fractions; but learn long division before polynomial division? Maybe, maybe not. A student might very well understand polynomial division BEFORE he really understands how to perform regular long division. Learn fractions before high-school Algebra? Maybe, maybe not. For some students, high-school algebra will give the missing understanding for what to do with computations of rational expressions and therefore regular rational numbers such as typical elementery school fractions.

One can create a roadmap of what to learn in what order, but this one roadmap will fit everyone.