First, let me apologize if this question has already been answered. I'm new here (first post!) and while I've been searching for similar questions, I haven't found any that match my problem. I also understand this is a general question so I wont mind if this gets closed for being as such, though hopefully some answers come through before that happens.
I'm looking for some direction that can help me "fill in the gaps" of my mathematics knowledge. The problem is, I'm finding it hard to describe what exactly these gaps are. For the most part, it's a general feeling that I'm not seeing things the way others who are more astute at math do. Or that there is some general foundation in quantitative thinking that I'm lacking. The best way I can describe my struggles is through a couple examples.
Let's take the area of a triangle. Up to a certain point in my life, I just memorized the formula $\frac{1}{2}bh$ without understanding why it is that way. It wasn't until I learned you're simply taking the area of a square and cutting it in half that it clicked. (I know, this is a gross simplification)
Another example comes from a stats class I'm taking now. The text gives us the standard deviation for a sample:
$$s = \sqrt{{\sum_{k=1}^{n}(x_{k}-\bar{x})^2}\over{n-1}}$$
I can see why we are summing all of the samples but why are we subtracting $\bar{x}$ from it, squaring it, dividing it by ${n-1}$, and taking the square root of all of it?
These illustrate a fundamental problem I have: there is a disconnect between seeing these symbols on a page and understanding what they are describing in real life. Sure, I can memorize how to compute the answers, but in the long run I want to truly understand what I'm doing. Not just because I find it interesting but because I need that understanding to be able to read more advanced topics. So my questions for ya'll are:
1) Are there recommendations on resources that can help with this?
2) Do you have any insight from your own experiences?
Maybe not the specific question that you mentioned, I would highly recommend the videos of 3 Blue 1 Brown and Mathologer to get these intuitions that you are talking about.
Also, there are books that often give you these ideas to understand what symbols represent. For example, if you read Herstein's Group Theory, it gives you so much geometry behind things. I'm sure, similar books are available for statistics as well (although I'm not a stat student).
3 Blue 1 Brown - https://youtube.com/c/3blue1brown
Mathologer - https://youtube.com/c/Mathologer
Also, there are some things which doesn't have any specific geometry- they are just very useful algebraic quantities. For example, I asked our Linear Algebra professor about the geometry of the trace of a matrix. He said, it's easy to force some geometry using divergence or curl or something similar. But, the best thing is to just look at it as the sum of the diagonal entries.