It is very important to approach self-study in this day and age very carefully as the internet is an endless sea of information and that information can cause such issues as analysis paralysis and pull people into new delusional phenomena such as the “vortex of false understanding.” Therefore, I will outline three important techniques for the beginner to approach self-studying of math and to avoid possible pitfalls that come with the amazing technology that is the internet.
Choose the Right Book
Books give you the tried and true structure of learning that is absolutely crucial to keeping most people (including myself) on task. The right book is the difference between passionate motivation and utter despair. As humans, we like to intuitively solve puzzles and get great pleasure in solving them with minimal help. A book will give you the needed information to approach problems and also exercises to flex the brain muscles you just acquired from working through the text. Furthermore, the right book will approach the material in a way that you are comfortable with in an accessible manner to you. This is absolutely crucial in the beginning as different people learn dependent on their own experience.
Luckily, this is where math shines, as there are a multitude of books specifically made for various people with various backgrounds. Do you want to learn Linear Algebra and are a proof heavy, person of the analytic type? There’s a book out there for you? Do you come from a computational background and love algorithms? There’s a Linear Algebra book for you. Are you a Physics buff and love experimentation and real world practicality? I promise, there is a Linear Algebra book out there for you too. And you can replace Linear Algebra with Calculus, Algebra, Trigonometry, etc. The amount of books out there for a wide range of personalities is astounding.
This brings me to an important topic; the choosing of books. After you’ve chosen the type of math you’d like to learn, you need to go through a selection of books to find out which one does indeed click with you. Quora and the Stack sites are great places to get insight into a list of books you may find enjoyable. You can perform searches there like, “best algebra book from a proofs background” or “good linear algebra for a person with no calculus” and if the results aren’t good enough, you can post your own question and expect a great response time from a very relevant user (this is why I avoid Yahoo! Answers).
After you’ve created a little list, spend the next few days pushing through each book’s first chapter. You are sure to find the right book for you, and the extra time invested in choosing that book will be well worth it when you are able to absorb it better than the others. You may even like two of the books and feel they are a good supplement for each other. This may allow you to skip the next portion, and avoid what I call the “vortex of false understanding.”
Supplement Your Learning with Videos
YouTube is an absolutely wonderful resource when it comes to gaining intuition into non-intuitive instances in math. Sometimes it’s difficult for words in a text to convey something very simple visually and so a great many accounts have been created to help learners of math and other subjects on YouTube. Adding to that the amount of free online lectures from prestigious colleges around the world, there is really no way you can get away from gaining intuition of math concepts. But there is a dark side to YouTube, and a reason I put it second in importance to getting yourself the right book. You can very easily get sucked into the “vortex of false understanding.” This vortex, in its simplest form, begins when you watch a video and feel you understand the material because it has been explained so well. The intuition the person has given you was so compelling and the examples so intuitive that it feels as though the subject is that simple and so you move on to the next video; and deeper in the vortex.
This is a very real situation, and I remember distinctly when I was first caught in it. I was learning Trigonometry through the Khan Academy and Sal was so good at teaching the subject, I avoided trying any problems on my own because I felt that I completely understood the material. Luckily the Khan Academy website allows you to attempt generated problems on an example portion of the site, and I was taken aback by my lack of understanding. Personal anecdotes aside, it is very important to get a book to help you structure the lessons and practice the examples at the section’s end.
Relate It to What You Enjoy
Math was mostly created to help us get a better understanding of the world around us, and so the majority of it focuses on real-world problems and practicality. This means that you can relate it to most anything you enjoy. If you are a history buff, there is probably no better history to study than the history of math and all of the very interesting actors that have played a role in it’s transformation and growth, from Pythagoras to al-Khwārizmī to Newton. The history will also take you all around the world if you are a culturephile or linguaphile, such as myself. In the sciences, the connections are obvious, and therefore lend themselves toward motivation and passion in the math subject you are learning.
It is important to connect these synapses and relate math to your everyday life as much as possible. This will make the learning of math more enjoyable and easier to absorb. This is probably the best skill to get good at as it takes a bit of critical thinking and analysis. Like anything, it becomes easier, and will improve many facets of your life.
I hope this has helped anyone who comes across it, and I hope to update and expand it as I learn more about self-study and grow myself.
The Unavoidable Art of Chronology 