What is the correct way to self learn from a textbook?

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.

-/u/ejk314

Background:
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.

Questions:
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.

Experience:
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?

Solution 1:

My advice: Do not get too distracted by your own shortcomings. Study mathematics with passion, curiosity, and enthusiasm (whether it's mathematics or any other discipline, these are three characteristics that will take you a very long way in life). I will say, however, that the pursuit of mathematics does require a large degree of passion.

If you strive to "master" mathematics, then I would suggest teaching yourself from the classical works of those who laid the foundations of the various branches of your interest. For instance, if you are interested in "Number Theory", then study the works of Gauss (see, "Disquisitiones Arithmeticae"). If you told me you are interested in "Set Theory" then I'd tell you to study the original works of Georg Cantor that laid the foundations of set theory. You can easily search the internet to find various PDF's of original papers. However, a good book is "God Created the Integers" by Stephen Hawking. If you wish (and my advice), pick a random "mathematician" from this book and read his works. From there, supplement it with modern literature and practice until you fully understand it. You will be well versed if you proceed in this manner.

Despite the branch(es) you may be interested in (and I strong recommend against the idea of "specializing"), your primary focus should be to understand the classical works while supplementing this objective with modern texts. This has the benefit of not only allowing you to recognize the intuition that served as the basis of these theories but you will also come to realize first-hand that contemporary proofs are largely simplified. By that I mean, do not become discouraged by the degree to which a proof appears to be "trivial" in a modern text; the originator's version was much longer and more complicated, I assure you.

Now, regardless of what branch of mathematics you happen to be studying at a given time, I've always found that it is supremely beneficial to find a geometric interpretation for things that are purely algebraic and similarly, when working with geometric notions, I try to find the algebraic interpretation of those notions. This is not always an easy thing to do, but IF/ WHEN you do it, the solution of many complicated problems will become more intuitive.

Generally speaking, you want to find a practical application for every mathematical concept you come across. Newton's genius was his ability to not only develop the calculus but to translate it to various properties of the universe. My point though, is that calculus seems to be more "obvious" once you find this application. Most of mathematics works this way.

None of what I'm saying here is "absolute". This is mere opinion. However, I hope it helps you in your future endeavors. The most important aspect though, is your own degree of passion and curiosity. None of the above is relevant without that.

Solution 2:

Let me give you my own advice, Movers. In college, I was a creative writing major and for the first year or so had little interest in mathematics. I took calculus in high school, but I didn't have any passion for the subject. That changed for me when my freshman literature teacher taught us (or, more aptly, struggled to teach us) basic propositional logic. It was interesting to me, but I didn't think much of it at the time.

Later that year, actually, we read David Foster Wallace's essay "Consider the Lobster" and, interested in Wallace's writing style, I researched his background and noticed that he specialized in modal logic during his college years at Amherst. It was then that I started to truly find a passion for mathematics.

It might seem unrelated, but what I'm trying to communicate is that, at least for me, mathematical logic is at the heart of all of mathematics. I started appreciating calculus and real analysis more when I analyzed the rigorous definition of the limit. Having experience analyzing and manipulating logical formulae is invaluable when faced with difficult theorems. For example, sometimes it helps to consider the contrapositive of an implication because it is more intuitive (of course, what is and isn't intuitive is a personal preference). You should certainly take a course in logic, but I am not suggesting that you specialize in logic, especially if it doesn't interest you at all. But, this skillset is something that you should not be without, and I trust that it will serve you well in the future.

Another piece of advice: never get embarrassed if you do not immediately understand something that seems simple. I read lecture notes frequently and try working through proofs. Almost always, I encounter a "proof" stating that it's trivial. Sometimes, I just sit there and stare blankly. Then, after a while, I feel embarrassed, convince myself that I see it, and move on. I try to remind myself that the only embarrassing thing is convincing yourself that you know something to be obvious and then get it completely wrong in front of someone. When you study on your own, nobody will know if you struggled with something. So, I say that you should always continue to revisit the basics. When I reach a problem that I don't understand, and I realize it's because I don't fully understand some basic concept presupposed in the notes, I revisit that concept.

I can't give you as much help as others probably can, but I hope my experiences inspire some hope and confidence.