I am currently a Junior in High School. I am in an Intermediate Algebra class, but my teacher does not always explain things in a way I can understand. I like to learn Math intuitively, but my teacher seems to just throw us formulas and rules, and expects us to use them without understanding why they work.
For example, when learning exponential properties, we learned $a^na^m=a^{m+n}$. I asked why this is, and my teacher gave me the old "It is what it is" explanation. I played around a bit with these variables later, and I intuitively reasoned it out. I figured since $a^n$ is just $a \times a \times a \dotsc$ ($n$ times), and $a^m$ is just $a \times a \times a \dotsc$ ($m$ times), then by the commutative axiom I could just combine them into one big $a \times a \times a \times a \times a \dotsc$ ($m+n$ times).
Now that I am getting into more advanced topics, I am worried I will not be able to reason some properties or concepts out myself. I realize that most kids do not really need or want to know why these axioms and theorems hold true, but I am a bit more inquisitive than others, and I have a desire to pursue a degree in Mathematics or possibly Physics.
When you learn new information, is there a certain way you make it intuitive for yourself? Is there any website or textbook than can give you a proof or a visual display of why certain properties work? (other than things that are completely obvious/self evident such as $a=a$ and $a\times0=0$) Are there certain techniques you can recommend me for understanding my newfound knowledge? Is there study techniques I should practice? Should I really care about Mathematical intuition and philosophy — am I being too curious?
I really have a drive to learn, but I feel like if I cant grasp why certain properties hold true in Arithmetic and Algebra, I will not be able to intuitively understand concepts that I am taught later on when I take Calculus or Statistics.