How to study Mathematics

Here are some pointers that underlie my approach to teaching and learning Mathematics. In my experience, understanding these aspects of Mathematics education will significantly ease your journey. 

Mathematics is a skill!

Mathematics is a skill - it's something you learn to do, like swimming or riding a bicycle.

You learn Mathematics just like how you would learn any other skill, i.e., by observation, imitation, and practice.  You cannot merely read it off a book as you would literature or history. Mathematics teachers sometimes couch this wisdom in the saying, "Mathematics is not a spectator sport" and "you learn Mathematics with a pencil and paper at hand".

So, learn Mathematics actively. Redo the derivations, fight with the theorems, work through the problems, perhaps even plot graphs or write programs to crunch the numbers. Moral of the story: you must do Mathematics actively, not just read it passively.

Learn to learn!

You must deliberately work on your learning skills (i.e., study skills). This is not explicitly taught in our educational systems, but imbibing a large body of knowledge and skills efficiently and effectively is a critical skill that we can get better at.

Keep the broader perspective in mind

When you are learning a topic, always ask yourself why you are learning that topic. Why is the topic personally meaningful to you? What is that topic really about? Why should you care?

Your teachers might be able to shed some light on this topic. I certainly try to. But ultimately, this is a question that you have to find an answer for yourself. A lot of this depends on the kind of person you are and the priorities you have for your learning goals. Someone who sees an intrinsic intellectual merit in learning Mathematics might have different reasons for learning it than someone who is of a more pragmatic bent of mind.

In particular, you should have a really firm idea of why you are spending time learning Mathematics at all. Studies have shown that people who have a strong reason to study the subjects they are trying to study are better at learning. Thus think about why Mathematics might be important to you in whatever plans you may have for your life.

Do not mistake unfamiliarity for inherent difficulty

The Mathematical topics you learn at the A-levels have a distinctly practical and applied bent to them. You won't be learning to prove theorems in your studies, which are the bread and butter of working Mathematicians. Instead, you will be learning to use mature Mathematical tools (such as calculus or probability and statistics) to solve real-world problems. This is very much an Engineer's or Physicist's perspective than a Mathematician's. 

To become adept at any tool or instrument requires you to spend some time with it. For instance, to learn how to drive a car requires you to spend time driving a car. The same is the case with pre-university Mathematics, which are after-all, intellectual tools. 

When you first start using a tool, it might present you with formidable difficulties, but this only arises because you are unfamiliar with the tool, not because the tools are inherently difficult. Do not mistake your unfamiliarity with the tool for its inherent difficulty. Give Mathematical concepts time to take root in your mind after you have been introduced to them. Stay with the concepts for a while, familiarizing yourself with them. Over time, they will become as natural to you as integers or alphabets themselves (which were also new to you once upon a time!). 

Learn dirty!

When you systematically read a specific source material, you are learning "clean". When you read articles or watch videos from here and there in an unsystematic, seemingly haphazard way, you are learning "dirty".

Do not underestimate dirty learning! We can't always sustain a high degree of focus on our learning tasks. There are times when we can be fully focused and attentive, but there are also times when our attention is more diffused and relaxed.

One way to make use of the latter times is to learn dirty. For instance, you could read about the non-technical or high-level views of the subjects you are learning. These readings could be the human or historical aspects of a specific topic, such as how the idea of the vector cross product was discovered by Hamilton after a long and arduous search, or how Mathematicians resisted the idea of imaginary numbers for a long time. Perhaps you could watch a high-level overview of the role that vectors play in modern Physics and Engineering, or how they are generalized.

Generally, try to attack your learning project from several angles and in several ways. All these efforts will enrich your learning experience.

Understand the derivations

There is an unfortunate tendency among many students to side-step derivations and get to the results right away.

However, this is a mistake! You don't have to memorize derivations of results and you most likely will not be asked to derive the results that you use, but derivations can shed a great deal of light on the nature of the topics that you are learning. It can also take away some of the befuddlement and randomness from the topics you are learning, and make them more meaningful. Finally, derivations themselves shed light on how to solve several types of problems using the concepts involved.

So do not take derivations lightly! At least go through them once, and ensure that you have some idea where the results you use come from.