How to read a mathematics textbook

Reading a mathematics textbook is perhaps one of the most difficult types of reading there is. It can be slow, frustrating, and discouraging. BUT if done correctly, it can also be an incredibly rewarding experience and open up new horizons of study.

Here are some tools that you can use to get the most out of your maths book

1. Patience

If you have to leave this article and remember just one thing it is to be patient. Maths books are hard, and they’re hard to read for everyone. You have to go into the reading without expecting to breeze through the pages, but rather to spend a lot of time on each page, reading and re-reading sections.

Set the bar as low as you can. Tell yourself, I am going to spend 30–60 mins on this page alone. If you manage to get through a few pages in that time, you’ve gone above and beyond! The difficulty which comes with the extremely slow reading style can be partially confronted simply with a change of mindset, attitudes and expectations of the reading.

2.Skipping

When you learning mathematics, it is completely fine and even beneficial to skip sections which are (1) marked as less important (2) more difficult than other sections in the text (3) less interesting or relevant to you.

Very very few people read a textbook cover-to-cover as you would read a novel. Remember that the textbook is a resource which should be used in a way which benefits you the most.

If you’re really keen on learning a subject as deeply as possible with the least possible omissions, you may still find it more efficient to read the book with some skipping when necessary and then going back over to fill in the gaps

3.Complementary resources

This step is crucial. One of the best things that you can do for yourself is complementing your reading of the textbook with as big a variety of additional resources as you can. This can include YouTube videos, additional books, or simply reading random articles on the internet.

There are always many different ways to say the same thing, and if you find yourself spending too much time on one idea, it could be that you just need to hear the same thing expressed in a different way.

The other good thing about this step is that you can easily complete in small windows of time throughout your day. Are you waiting in line for your coffee? Instead of mindlessly scrolling through your social media or your emails again, why not read the Wikipedia article on partial derivatives or Lagrange’s theorem? Your not trying to understand it in depth, your just increasing your exposure to these ideas so that when you come back to them later they don’t feel as strange as before.

4.The Million Dollar Question: Exercises

Exercises are a huge point of debate when it comes to the correct way that you should read mathematics textbooks. There are some people who claim that doing every single exercise in the book is necessary and gives you the strongest understanding of the topic. Others see them as merely optional add-ons for the eager reader.

The role of exercises varies greatly depending on the author, and often their importance will be commented upon in the preface of the book. However, as a general rule, what I like to do is to treat the examples in the book as the beginning of the exercise section. Before reading the solutions, have a go by yourself. If you are successful then compare your solution to the author’s and if you are struggling then don’t worry, remember you’re not even meant to be doing the exercise! You’ll gain a deeper understanding at the end of the chapter even before you begin the exercises. In the exercise section, I would generally recommend not to try and do every single problem because it will be extremely difficult to work your way through the book. Rather, pick a few exercises and dedicate a set amount of time to them.

Many of the questions in mathematics textbooks seem completely incomprehensible and you’ll have this feeling of not having any idea on how to even start approaching the question.

Don’t let that stop you!

Rather, set yourself a 15–30 minute timer and dedicate that time just for that question. It doesn’t matter what you’re doing, as long as you’re spending time thinking about the problem. Write down equations or theorems you think may be useful, try assuming the result is true and working backwards, draw diagrams, simplify the problem, it honestly doesn’t really matter. You’ll be incredibly surprised how much progress you can make just be randomly trying out things with no expectation or clear path in mind. And if it doesn’t work out, then that’s ok as well. At the end of your allocated time, try to find a solution and see if you understand the explanation. Then, think about what you could have done to lead you to discovering the solution by yourself.

5. Involve other people

This may be difficult for some people, but learning mathematics becomes a lot more interesting, fun and engaging if you can involve other people in the process. Learning mathematics is by nature a communal experience, at the very least its interaction between you and the author. If you have people around you that may be interested in learning as well, then try to get them on-board! Otherwise, there are a lot of community forums online where you can talk to people struggling with the same ideas, or ask specific questions that you may have about the content.

In moments of frustration and despair, remember that you are not alone. Remind yourself that this feeling shared by everyone who learns mathematics, and that it doesn’t mean that you’re not good enough or that you’re not able to learn what’s in front of you. Take a break when you need to, and return to the problems with a clear mind and a fresh attitude.