Novantix said:
However, I feel like when I am spending hours pondering these trivial questions that I am somehow wasting my time and also feeling guilty for doing so, or that time could be better spent just simply reading and doing more maths instead of sitting and thinking or taking a walk to jog my thoughts more.
This depends on so much unknowns, that I find it hard to answer. E.g. some people are good at learning many rules and algorithms to solve problems, others are better in seeing the crucial points fast. It is certainly better for understanding to go to the heart of a problem. Things you finally figured out by yourself are much harder to forget than quickly learned formulas. Time management on the other hand is equally important as you don't want to spend too much time on a minor issue. It's like always in life: the truth lies somewhere in the middle.
Novantix said:
In the same fashion, how might you suggest I try confirming these queries to check whether or I am understanding the "why"/"how"? Should I simply try to prove it to myself mathematically, or should I ask on these forums or do both?
Again, this depends on case, time and number. Before you spent too much time on something, come on over and ask here. However, what you've written by yourself is easier to remember than what you've read or seen on videos. There is another advantage of posting questions here, which is far too often neglected. People often use the internet to find quick answers for a certain question and don't bother the way to the answer. They only want to overcome the hurdle "homework exercise 4.c". But if you're forced to explain something to others and have to think about how to do it and what exactly your difficulty is, then you'll often find the answer on the way in doing so. Teaching (or here explaining) is a good method to learn something. E.g. if you want to know why ##2x=0## implies ##x=0## then you'll probably won't take the effort to type it in on PF. On the other hand, it would force you to think about why ##a\cdot b = 0## implies ##a=0## or ##b=0## and that this is not always the case (cp. my example with the light switch above). So areas in which this holds are already more special than others. This in return means: If you want to solve ##2x=0## with ordinary numbers, then first try to figure it out by yourself. If you want to know, what it is, that this property holds or what it's called, then come over and ask. Also if you get stuck somewhere, it's better to ask than to endlessly try. You will be told, if you're questions are that simple, that you should answer them by yourself. And again, being forced to explain a situation often already reveals a way to the answer.
By performing these "thinking sessions", are you of the opinion that It will be beneficial for me having questioned and answered these facts vs a student in school who takes said statement as fact and does not take the time nor liberty to question these facts?
Let me give you another example. Problem: Solve ##2x^2-10x+12=0##.
I have learned the formula ##x_{1,2}= - \dfrac{p}{2} \pm \sqrt{\dfrac{p^2}{4}-q}## where the equation has to be brought into the form ##x^2+px+q=0## first. And as I'm better in learning what to do than learning formulas, I remember it by ##x_{1,2}= - \dfrac{p}{2} \pm \sqrt{\left( -\dfrac{p}{2}\right)^2 -q}\, : \,##take ##-p/2##, then ##\pm \sqrt{}## and put the first squared minus ##q## under the root.
I think more common, given ##Ax^2+Bx+C=0\,,## is ##x_{1,2} = - \dfrac{B}{2A} \pm \sqrt{\dfrac{B^2-4AC}{4A^2}}##
But in any case, the principle is the same:
##Ax^2+Bx+C=Ax^2 + 2 \cdot \sqrt{A} \cdot \dfrac{B}{2\sqrt{A}} \cdot x + \left(\dfrac{B}{2\sqrt{A}}\right)^2 - \left(\dfrac{B}{2\sqrt{A}}\right)^2 +C = \left( \sqrt{A}x + \dfrac{B}{2\sqrt{A}} \right)^2 - \left(\dfrac{B}{2\sqrt{A}}\right)^2 +C = 0## and then solve for ##x##.
Once you've understood this, you can still use (the faster) formula, but can always rollback to this principle as in case you're not sure, which I just did to derive the second formula that I haven't learnt.
This should illustrate the difference between learning and understanding. Wikipedia is better to look up the formula, we are better to understand why this formula holds.
Here's another read which might help you:
https://www.physicsforums.com/insights/10-math-tips-save-time-avoid-mistakes/
