Reading Maths Books
Finding Texts for Study
You start first with a topic you want to learn about. Then you research texts to study from. Broadly speaking, they are:
- Easy-reading high school books. Good if you are very short on time.
- Undergrad textbooks, such as Springer undergraduate books. They are a good intro to a subject, or if studying an advanced book then you will want one or two of these as supplementary material for understanding difficult concepts.
- Graduate level books usually are the best but require a lot of effort put in. Concepts and questions will need to be looked up and cross referenced with other materials. Examples include the yellow Springer books.
Usually you will follow one main text on a topic, but with a few other supplementary books as backup. Often you get stuck on a concept in the main text, and the supplement books will assist you to make sense by looking at things from a different explanation. Re-phrasing the same idea using different words can make a big difference in dicephering some theorem or object.
There are many high quality online courses following important texts. They explain the main core forums, focusing your attention on the key ideas and explaining things in an intuitive non-formal manner.
- Elliptic Curves by Alvaro Lorenzo. This course uses the Springer book on Elliptic Curves by Silverman.
- Harpreet Bedi
- Zvi Rosen
Getting Excited, Taking a High Level View
Take a look at the contents. Familiarize yourself with the structure of the book. Make note of topics that you will learn and master. Get excited about the truths that you will unlock. You will come back here every periodically to remember why you are studying and where you are going.
Make a lesson plan. Often the first chapter of a new topic is important, but if you're already familiar then maybe you can jump to advanced material.
Be aware if you struggle too much at the advanced level, and make no progress at all then it's a signal to swallow your pride, be humble and go down to a lower level before moving up again. We take shots, but sometimes we have to take a few steps back. The tortoise beats the hare.
However you must struggle. Don't be a weakling. Fight to rise up. Give it your focus, dedication and attention. Get into the zone, or rausch. You evolve because it is hard.
Reading the Chapter
Now you've chosen your chapter. Do a light first-pass read through it. Focus not on the details but the main theorems and structure of what you're learning. Try to understand from a conceptual level the main ideas and how they will fit together.
It's normal for the end of the chapter to feel increasingly cryptic and unintelligible.
Now return to the beginning of the chapter and begin seriously reading it. Make sure to follow the logic of ideas and understand what new objects are. You might get stuck on a difficult idea or long proof. Feel free to skip over these and return back to them after. Many of the concepts will be new, and you will be awkward in your dealing with them. Do not worry as the more familiar you become with this subject, your understanding will become solid.
As you work through the chapter towards the end, you are learning where all the theorems, definitions and proofs are. You will likely return back to these as you try to solve questions.
While you're reading through, you will likely pass back over theorems you tried to understand earlier but skipped over. If they still don't make sense, then it's fine to again put them to the side and return back to them again after.
In this way we are reading a chapter in several passes, going back through past material as we go forwards or try to solve questions. We also might sideline material in the beginning and decide to look more into them later.
Eventually our familiarity with the chapter is strong, and everything (more or less) makes sense.
When you are stuck, feel free to ask others in the team, or post questions on math stackexchange if nobody knows.
You will need to research things, searching the web and studying the supplement books.
I tend to slightly prefer books with solutions to questions for self study.
You should always do questions. As many as possible. For core subjects, always attempt to do all or most of the questions, unless there are far too many.
When you are shorter on time or studying a subject on the side, you may choose to pick out a sample of questions with a mix of important looking topics and others which grab your attention or pique your curiosity.
After reading the chapter, be sure to do a quick review and write down any theorems and proofs that caught your attention. You may wish to write them on flash cards or on a special notebook so later you can come back to them.