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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.

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closed as off-topic by user296602, Will Jagy, user147263, Chris Godsil, Harish Chandra Rajpoot Mar 16 '16 at 3:48

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Seeking personal advice. Questions about choosing a course, academic program, career path, etc. are off-topic. Such questions should be directed to those employed by the institution in question, or other qualified individuals who know your specific circumstances." – Community, Will Jagy
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Also as a side note, I do not know Latex or Mathjax (I am planning to learn soon). Please excuse my formatting, and if anyone wants to kindly edit my equations and format them properly, I would be grateful. $\endgroup$ – user323239 Mar 16 '16 at 2:22
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    $\begingroup$ It sounds to me like you've been getting some awful teachers. Especially with high school topics, its important to understand why things are the way they are. If you're having trouble with some topics, you can always feel free to post them. $\endgroup$ – Kaynex Mar 16 '16 at 2:26
  • $\begingroup$ I'd say, intuition comes from yourself. No one can teach you that. Also for understanding you need to be somewhat smart. However, you can be assisted in understanding and that is sometimes necessary - may you be the smartest in school. So, if your teacher is not capable of this, I think there are only two possibilities to deal with that: either you find somebody else to talk to (parents, another teacher, class mates, this site, etc) or you read books. What about the book you use in school? But there are surely many other books on the market. Unfortunately, I cannot recommend you any. $\endgroup$ – Friedrich Philipp Mar 16 '16 at 2:27
  • $\begingroup$ I have edited your question to introduce $\LaTeX$ formatting. Please do learn it, it will greatly help you in the future, considering your desire to go in this field, besides its syntax is plain intuitive, especially for simple typesetting like for your formulae. You can view the results of the edit to see how simple it's done. Additionally, you may consider following this tutorial to familiarize yourself with more advanced techniques. $\endgroup$ – dbanet Mar 16 '16 at 2:30
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    $\begingroup$ People often don't explain the intuition, for some reason. But the intuition is out there, you just have to search for it (or find it yourself). You're not being too curious. $\endgroup$ – littleO Mar 16 '16 at 2:38