Frustration when trying to learn a new topic deeply

Question

When trying to learn a new topic in mathematics I consistently find myself extremely frustrated because I find that far too often, definitions are given without providing context for why they are chosen as such.

For example, I don't think I have ever come across a text which gives significant preamble to roots of the Normal distribution. It simply appears as:

Definition: A random variable $X$ has a Gaussian distribution with mean $\mu$ and variance $\sigma ^2$ if it's p.d.f is given as:

$$f(x) = \frac{1}{\sqrt{2 \pi \sigma ^ 2}} e^{ - \frac{(x-\mu)^2}{2\sigma^2}} $$

I've never seen this distribution derived, for example, from information-theoretic methods or by the CLT, or by analogy to a diffusion process so that the concept has some significance.

Currently, I'm an undergraduate student and I find that I waste an inordinate amount of time searching for the origins to many concepts to guide my learning. I find myself extremely disgusted by a course's requirement to memorize any formula and it makes it extremely difficult to learn in these kinds of courses.

Am I wrong to approach learning math with such an outlook? Is it possible, in general, to find the kind of resource that I'm looking for? Or, is it recommended that I just painfully memorize my way through undergraduate courses? If so, when does it get better?

Answer 1

This is related to "A Mathematician's Lament," by Paul Lockhart, although that's more skewed towards the education system as a whole. I suppose demand informs supply, insomuch as unintuitive fact dumps dominate the textbook industry.

In general, I've found that math textbooks conform to one of three structures: the pedagogical, the reference manual, and the vestigial (you know the one, chock full of useless TI-89 commands and unhelpful figures). The reason it's generally difficult to find the first type of these is because the other two are more geared to classrooms, and a class requiring some textbook is the primary impetus for a textbook purchase. If your textbook were pedagogical, what use is there for a teacher?

To answer your question, seek out self-study texts. In the comments, @Moo mentions seeking out older books, and particularly recommends Parzen and Feller. I've had a good experience with Dover paperbacks; they are inexpensive reprints of classic material, the contents are usually well suited for the mathematical autodidact. Sometimes, lecturers provide their class notes, and I've found those are a treasure trove for understanding the purpose of definitions.

Another option is to start with the problems. Difficult problems often have created entire branches of mathematics. Apropos to that, look for math history books, which will often detail the intuition (although not necessarily the details) behind equations.

If all else fails, you can always ask for the intuition here on the Mathematics StackExchange.