Taking notes in a math class is one of the most basic ways to record the information given in a lecture for use later when trying to piece together the subject matter of a course in a comprehensive way. When working on homework problems, or studying for exams, a good set of notes allows for the information from the lecture to be reviewed and placed in a context where one can reflect on difficult concepts over time. The professor may cover material not directly in the text, provide alternative techniques or insight to understanding material in the text, or offer particularly good exmaples not found in the written material.
But how to take notes effectively is difficult to teach and even more difficult to figure out on ones own.
Some tips I find useful (since I am male, I will refer to the professor as a "he"):
- Try to record what the professor is saying and not just what he may write on the board.
- It is difficult to listen and write at the same time. Be terse in your note taking, so that you do not spend too much listening time simply writing.
- Don't work so hard at being neat, but do work hard on being clear.
- Leave space in your notes to fill in obvious gaps after the lecture. This goes back to the point that you cannot listem and write at the same time. Why waste time writing something obvious when you can listen instead and fill in the gap later.
- Provide space to fill in the details of examples or proofs "left to the audience". When a professor says "this is a good exercise", he almost always means it. You should leave space in your notes at that point to do the exercise later.
- Directly after a lecture (meaning sometime soon afterwards), go through your notes and fill in any gaps, thinking about what the professor said, what you remember of questions from other students, and other thoughts you have and still have fresh in your mind. Do this in another color pen or pencil (these extra markings may become critical to understanding a concept clearly).
- Add your own insight ot your notes. Even if it seems obvious at the time, if you make a connection not mentioned by the professor, write it down.
- As you go through your notes, you will find places which don't make sense, even after some thought. In yet another color, mark these places with a big question mark. Talk to other students, the TA, or the professor and get these question marks resolved as soon as you are able. Write in the resolution into your notes.
- Don't worry about rewriting notes, or trying to prepare them for publishing. The purpose of the notes is not to impress. The purpose is to use them to gain understanding.
- Compare your notes with a friend or an acquaintance in the class, and add to your notes things the other student recorded but you did not.
Lectures are a critical part of the course experience. The purpose of a professor is to, in some sense, organize the material into a logical story line, allowing you to make much progress by taking steps that follow each other naturally, and partially digest the material, making it edible brain food for you. Since it is a live interaction, he can gauge your (as a class) understanding of the material and alter his presentation to fit the needs of the class. or at least he should....
All of this winds up sitting in your notes. And winds up being the backbone of the course information structure.
I hope this helps.