What is the act of taking notes? The answers all mentioned why we should/should not take notes, or how we should take notes, but did not clearly explain what the act of taking notes meant. I believe the subject has this experience. If you are already familiar with the idea of a certain step, you don't even want to take notes to help you remember, because you would rather feel the process of ruminating in your mind and take it out at any time. Taking notes will only make you feel distracted or even annoyed, which actually reduces the efficiency of learning.
Therefore, it is necessary to consider the scheme of "one theorem and one example" in similar answers. If you have a good feeling about the form and reasoning of the theorem, then finding examples may be limited to ideas; In the same way, whether it is necessary to write down the "part" of the proof and the key to the idea actually depends on your feelings about the proof. Many times, people will be willing to ruminate all the information of these structures. The process of reading or listening to proof has become the process of "looking up the dictionary", looking for the correct details to prove your ideas. In this case, understanding is basically in your heart. At this time, the best way to assist understanding is probably not to take notes, but to use draft paper to assist thinking, write down relevant symbols, and then calculate and demonstrate the definition of concepts. If you need to write something that is easier to understand, write it down carefully and then do the reasoning and calculation you want. Therefore, taking notes is not a question of whether to do it or not, but a question of how you understand its relationship with you. When you are thinking, think more about what the writing means and what it will be, so that you can gradually adjust and form your own ideas about taking notes.
When talking about the motivation of taking notes, the topic owner said that "I feel that many things have not been learned thoroughly", indicating that the purpose of taking notes is not to remember, but to sort out my thoughts and understanding in the process of taking notes. To understand this problem, first think about how to get stuck in a certain step. If you don't understand a sentence, first start with the word in the sentence, then explain it yourself, list the definition of the word, what kind of nature it represents, and write down the steps to get closer to the goal.
In fact, the effect you want to achieve by taking notes is the same as this process. You can find out for yourself what a picture that conforms to the theorem and definition is and how you describe it, and then your intuitive grasp of this picture includes the concepts, theorems and so on you want to obtain. In a concrete practice, I provided my own answer on Zhihu: Why is there no formula for finding the roots of addition, subtraction, multiplication and division from the quintic equation? -Cai Yixin's answer. Another method is to find a truth (usually expressed in natural language) or method that you are most familiar with (it can be natural language or some operations and reasoning that you are used to), and then make an analogy with this simple truth to deduce the theorem or concept you want to think about. For example, discuss the most basic in mathematical analysis.
Language, one of the best metaphors I can think of is the "quasi-dead chess" in chess, that is, the opponent is not going to die, but will die no matter which step he takes, which perfectly corresponds to "any radius is
Will the community be numbered? Break? The definition of "coming in". Let's take linear space as an example. At first glance, the definition of this thing is only related to operation, and we don't know what it is for. But if we come into contact with many function classes, and so many members in a class can find a coordinate frame as a base like a plane, then we are more eager to contact the linear space. Taking notes at this time is to find these suitable pictures or natural reasons. Writing them down can help you find a route, pull the concept back to your cognitive range, and the concept or theorem can be grounded.
There is also a situation where I feel that I have not learned it thoroughly, that is, I feel that the reasoning of theorems can only be passively followed by the ideas in books, as if those ideas can only be mastered by "memorizing" or "practicing", and I have not found the freedom to come and go freely in computational mathematics. Taking notes at this time can also play a similar auxiliary role. Don't try to write things down clearly, leave yourself more blank to try and organize more ideas. If you have a good habit of looking for familiar forms or truths, you can constantly interpret your intuitive impression of theorems, think about how to express them smoothly, and even design lemmas yourself, just as my mathematical analysis teacher told us, to achieve the degree of "theorem creation". Similar to Lagrange's mean value theorem, after such a long period of preparation (Rolle's theorem, mean value theorem, etc.), you can completely summarize and reorganize yourself. ) Before entering this theorem. So many steps may make you spread out all the relevant details on a piece of paper, and this kind of note-taking is certainly more interesting. Relatively speaking, the former's notes focus on how to explain concepts and theorems, while the latter's notes basically help you get through various joints.
This topic mentioned Euler and China. First of all, it must be explained that works and notes are actually two different things, and works need to be further sorted out than notes. Then the notes at this level should actually be further expanded on the basis of understanding. Specifically, you should record a lot of calculations, attempts, proofs and ideas that you have done independently. Euler found so many wonderful formulas, most of which were the result of changing formulas in his usual manuscripts; Gauss must also have a lot of notes and manuscripts, and the law of quadratic reciprocity was shocked by him n times; Needless to say, Ramanujin's famous notebook is still the research object of many papers. Similar notes are your own independent exploration, which may come from the refinement of usual exercises, or the related calculations you made when you saw other materials, or you may play by yourself and find some interesting forms or ideas. But you need to make some personal explanations, such as what you are optimistic about this theorem, which step or lemma in the proof is more exciting than the theorem itself, and you want to practice it and make various variants, which can be used in other topics in the future. Such notes are very interesting, and it is strange that you can't understand them when you play them.