Math textbooks, though daunting, are invaluable study companions when it comes to studying math. Whether you wish to have a taste of some advanced math, or you are cramming for an upcoming test, you should literally hit the books!
Here are some tips on how to make the best use of your math textbooks:
1. Set your goals
Depending on your objective, you should focus on different aspects of your textbook. Choose what you want to accomplish and at what pace.
For example: say you are previewing a textbook that you will be using in class next semester. You can read one or two sub-chapters and complete two or three exercises, especially those focused on new definitions and concepts, in each sub-chapter every day. Change the number of chapters, number of exercises, or the weekly frequency of reading as suitable for you.
Here's a different scenario: say you are practicing for an upcoming quiz. Use summaries and chapter-wide exercises to figure out what you need to read or practice more. Read as needed while minding your time constraints. Choose a reasonable number of exercises that would aid your weaknesses. Re-assess now and then to have an updated sense of what you need to do next.
You might need a bit of trial and error to find the right pace for you. If your textbook has instructions on how to use the book, those can help you make an informed decision in this process. They often include the recommended order to follow the chapters depending on your background and goals. Some authors tell you more about the exercises, like the labeling of difficulties or where to find solutions.
2. Read the words, and write in words
It should not be mistaken that equations are the main actors in the ‘story’ of math books. Good textbooks tell the story in plain words, and the equations only aid the flow. You should carefully read the plain words at least once so that you follow the author’s motivation behind taking each step and later be able to thoughtfully reproduce it.
When reading, go over each line carefully. Of course, not everything will make perfect sense and that is fine. However, try not to skip a sentence without really processing the meaning of it. At least try to remind yourself of new definitions and concepts appearing in each line.
This will require a lot of focus when you are less used to it. Reading it out loud can help. When in a quiet space, copy the words on a different piece of paper or follow the words with an eraser. As you repeat reading like this, the full meaning of each sentence will be clear to you faster and faster, to the speed where quickly reading with your eyes will be enough to understand everything.
Take this even further and WRITE in plain words. Taking notes in plain words will help you reinforce mental connections between concepts and improve articulating logical reasoning. Talking about math in plain words is an important goal of learning math. Many exams with free-response questions assess this kind of proficiency, too.
I recommend that your notes focus on your learnings rather than a mere summary of the textbook. Making a good, useful summary requires some advanced skills, so many textbooks provide chapter summaries for your benefit. Here are some helpful things to keep in your notes.
While reading:
- What was confusing and how was that resolved?
- Striking insights that you found most useful
While working on exercises:
- Common types of mistakes you made and how to prevent them
- Problem-solving strategies that you found complicated and how these were clarified
3. Examples, figures, diagrams, and exercises
Yes, math textbooks contain a very dense set of information, and some parts are even more condensed than the rest.
Examples, figures, and diagrams visualize core concepts and demonstrate important insights. Textbooks will often instruct how these are constructed. It is always a good idea to follow these on a separate sheet of paper, blackboard, iPad, etc. Doing this the first time can easily feel like copy-and-pasting, so make sure you carefully think through each step.
Once you are familiar with the concepts and logic used behind these examples and visuals, you are ready to work on the exercises. Exercises and their solutions will train you, usually through repetition, how to apply the intuition learned from the examples and visuals to problems. There is not much more than pure grind to working on problem sets.
However, there is one crucial point: If you get a problem wrong, always take note of it. Go over the solutions again, and practice applying the solution strategy. Textbooks with extensive problem sets will usually have more problems that require a similar approach. If you can’t find one, do not hesitate to repeat the same problem. Many math students, even at college and PhD level, go over the same exercise a few times to make sense of it.
Bonus: Study with friends!
It will feel less lonely and speed up the learning process. First of all, you can read the text together. Take turns reading out loud, and save your voices. As questions arise, ask them, and you can attempt to answer these questions and convince each other.
Here’s a hack: Divide yourselves into groups and assign different exercises to each group. Then, discuss among each group to gather important takeaways. For example:
- Which concepts and strategies did you use in the assigned exercises?
- Which questions had any disagreeing answers? What were the roots of wrong answers?
- Which questions were difficult? Was there any particular trick to make them easier?
Share these with other groups, and also hear what other groups learned from other exercises. You can learn like you worked on all of them while putting in way less work.
What will you be reading next? Go ahead and try these tips now!
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