Active learning vs Passive learning in Math
The issue is not particularly specific to mathematics:
- passive learning is being taught, writing down what you have been taught, and learning what you have written down
- active learning is answering questions which require you to apply what you have been taught
So in preparation for an exam, a passive learner will spend most time going through notes while an active learner will spend most time practising with previous editions of the exam. Most people do both.
I want to emphasise something that was briefly touched in other answers, but which I think is the core of active learning.
Taking notes and reading on your own is, in my opinion, still quite passive. In fact, mindlessly taking notes during lectures can actually be a disengaging process. The only way to really learn actively is by actually struggling with hard problems.
Now, this is something many people end up passing on. I have many colleagues who just go to the lectures, do a couple of mandatory assignments, nod at the teacher, but when it gets hard they just give up (and often come ask questions here...).
Now, learning by struggling with hard exercises is awfully time consuming. You can easily spend 10+ hours more per chapter in a book like Munkres or Rudin, but at least you'll be able to actually use what you've learned actively in the future.
For me the best reference for training this kind of learning are books like Putnam and Beyond, which pose very challenging questions related to most major undergraduate topics. Unless you're extremely gifted, working through the first chapter in 'basic techniques' like induction and proof by contradiction will leave you feeling sorry for your own stupidity. But when the solution does come it is usually very rewarding.
I think the other answers here are great, but I thought I would provide one more perspective: "the best way to learn is to teach someone else". When you are learning passively by reading or listening to lectures, you are guided by the hand and it is easy to convince yourself that you understood everything. A good way to test this, is to try to explain what you learned to someone else or apply your knowledge. Here are three ways I recommend:
- Explain what you are learning to an interested friend, one that has enough background to ask you questions. Of course, sometimes your friends don't have the time, so
- Explain what you are learning by writing a blog. Even if nobody reads your blog, it is better that private notes because even the though someone might read makes many write more carefully and prod their own understanding, it works magics for me. It also helps you to learn how to communicate to the mathematical community, which is an essential skill if you go on to graduate school and one that is seldom explicitly taught. Finally,
- Participate on Q&A sites like Math.SE. Don't simply write your homework problems here, that is useless, but as you read, if something doesn't make sense, try to restate your confusion in your own words. Sometimes formulating a clear question can be incredibly enlightening to your own understanding. I have solved many of my problems by simply trying to write down a precise SE question and thus solving my own confusion without even asking. On top of this, as you become more comfortable, you can try your hand at answering questions. On a site like this, you effectively have a whole community of tutors that will give you positive reinforcement when you do well (+1 votes) and correct you when you are wrong. Of course, as you do this, respect the community norms and try to learn something from every answer you give.
Quoting my own Professor here regarding this topic. He does research into the most effective way to learn mathematics, and also effective teaching strategies:
Your lecture notes will be your most valuable resource. You will refer to them when you do homework, or prepare for a test or an exam. So:
during a lecture, take notes
later, read the notes; make sure that you have correct statements of all definitions, theorems, and other important facts; make sure that all formulas and algorithms are correct, and illustrated by examples
fill in the gaps in your notes, fix mistakes; supplement with additional examples, if needed
add your comments, interpret definitions in your own words; restate theorems in your own words and pick exercises that illustrate their use
write down your questions, and attempts at answering them; discuss your questions with your colleague, lecturer or teaching assistant, write down the answers
It is a waste of time to try again and yet again to understand the same concept. so, when you are sure that you understand a particular definition, theorem, algorithm, etc., write it down correctly, in a way that you will be able to understand later. This way, studying for an exam consists of re-calling and not re-learning; re-calling takes less time, and is easier than re-learning.
Keep your notes for future reference: you might need to recall a formula, an algorithm or a definition in another mathematics course.
Attend lectures regularly! Concentrate and follow lectures, take notes, and fix them later (how?). Ask questions, respond to questions from the instructor, discuss material with a colleague.
for each section that is covered in class:
study solved examples from the textbook ... don't just read through a solution! Hide the solution, work on your own; if you get stuck, first understand why you got stuck, i.e., what's the problem. Only then look at the solution in the book to see how that problem was resolved. If you just read through a solved example, most probably you will miss the point. of the exercise/example; when you encounter similar situation, you will not know what to do.
work on suggested practice questions; do as many as you need in order to feel that you know the material; mark the exercises that you are not certain about, or do not understand, discuss these with your colleagues and/or with your TA
work on homework problems to see what homework you are supposed to do; solutions to homework will be posted regularly on the course web page.
Weekly tutorials Be prepared for your tutorial: study sections that were covered in class; ask questions, discuss theory or assignments with your colleagues, ask your instructor to explain things you are unclear about. http://ms.mcmaster.ca/lovric/lovric.html