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?

  • $\begingroup$ Your complaints are definitely valid, but the thing is, most of the time this pure derivation takes to much time, which is of the essence when it comes to teaching. web.sonoma.edu/users/w/wilsonst/papers/Normal/default.html - random first google result for normal pdf derivation. God bless the 21st centuary means. Everything is available on the internet $\endgroup$ – Makina Dec 20 '18 at 18:16
  • $\begingroup$ Side question: Is H(XY) referring to the entropy of the product of independent random variables, or the joint entropy? Do you have a reference? $\endgroup$ – Manano Dec 20 '18 at 19:07
  • $\begingroup$ I feel your pain! So much... $\endgroup$ – littleO Dec 21 '18 at 1:51
  • $\begingroup$ @Manano I believe OP meant the joint entropy $H(X,Y)$, which is equal to $H(X) + H(Y)$ if and only if $X$ and $Y$ are independent. $\endgroup$ – angryavian Dec 21 '18 at 2:52

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.

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    $\begingroup$ I would add another thread to the post. I have seen that older books would contain more detail on specific things, but as time progresses, they remove those details to cram in a bunch of perceived improvements and newness. There are many wonderful older probability books (Feller, Parzen, for example) that are very readable. So, search out the older books in the college library and then use the newer text for the class. $\endgroup$ – Moo Dec 20 '18 at 20:49

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