They taught me just to calculate,
and not to think or ask.
So now I just regurgitate,
and bullsh*t just to pass.
They taught me how to integrate,
but not the reasons why.
As well, I learned to derivate,
but not how to derive.
They packed my head with mindless tricks,
and told me I was wise.
But when I tried real mathematics,
I found, they taught me lies.
I am a university math student with the math skills of a four year old. I was taught math incorrectly for the past 16 years. I was never taught proofs or given conceptual understanding of math concepts. I was taught to memorize formulas and algorithms instead of understanding concepts.
To further elaborate how I was taught math, let's take derivatives as an example. I don't have a clue as to what a derivative actually is (both geometrically and conceptually), but I can solve differentiation problems by approaching them algorithmically, meaning I simply follow a series of steps that I memorized to solve the problem. I'm sure you have heard of the trick called "power rule". I know that for power rule type problems, you bring the power down in front of the variable and you subtract the power by one. I don't know where these steps come from and I have no idea about the meaning of the steps, but it didn't matter because I could solve the problem without thinking. This is what math has been to me for my entire life. Just a bunch of tricks and algorithms.
You may be wondering, "how are you a university math student if you don't even understand basic concepts?" I have gotten A's in all my math classes by doing enough problems. By that I mean I did every problem at the end of each section in the textbook. That way, I came across nearly every possible problem I could possibly see on an exam.
Obviously, the way I was taught was wrong, so now I am looking to relearn all of math up to Calculus 1. I am self learning so I do not have anybody to talk to (besides SE). I want to be able to learn a concept and apply it to new situations, rather than do hundreds of problems and memorize the algorithm for each case. However, I have never studied math, only memorized it. This is what I need help with: I want to learn how to learn math from a textbook.
The reason I'm asking this question is because I got a book called Basic Mathematics by Serge Lang and I am struggling to understand how I can take the text from the book and apply it to new problems. For example, the book asked me to prove that -(ab) was equal to (-a)b. I have never done a proof before in my life, but I attempted the problem anyways. I looked at the text preceding the problem, it talks about the distributive property and multiplicative identity property, so I tried to apply what the text said to the problem. However, no matter how much I read or thought, I had no idea how to approach the problem. I gave up and looked up the proof on SE. The proof was much more complicated than I thought it was going to be. I immediately knew that it didn't matter how much time or thought I put in, I was never able to solve the problem to begin with. It wasn't clear how the distributive property played a role in the proof just by reading the text. However, after I looked up the answer, I got a feel for using the distributive, identity and inverse properties to prove things. After I looked up that problem, I was able to solve the other problems in the section.
However, I'm frustrated that I needed to look up an example before I understood how to approach these types of proof problems. In fact, I don't feel like I learned anything. I feel more that I memorized a pattern. Apparently what I did was wrong: you're not supposed to look up answers ever according to some users on SE. However, I don't see how I could have possibly solved that problem and similar problems without seeing an example first. So is it wrong that I didn't get the problem by myself? Did I miss out in the learning process?
Another concern I have is that I'm going to miss something while studying from Basic Mathematics. For example, the multiplicative inverse property is never explicitly stated in the book. The first time I heard of that property was from the proof on SE I linked earlier. I'm concerned that I'll miss things that will be important later on. How can I make sure I don't miss anything? Should I supplement this book with other resources such as Khan Academy? But what if Khan Academy is also missing some things? It was only by chance that I happened to come across the multiplicative identity, what if this book is missing other pieces? What can I do?