[Math] What are good “math habits” that have improved your mathematical practice

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I currently feel like I am not doing maths the best way I could; that is, I'm not making the most out of my time when I'm working on maths problems.

The main thing I feel is that I'm not organizing my mind and my derivations as clear as I could, because I don't have the best "math habits". I feel like if I could develop better math habits, I could significantly improve both my time efficiency and the quality of my thinking.

To show what I mean, I'll compare it with the skill of writing: I used to write in a very unstructured way: I simply started writing with some vague idea of what I wanted to write. Then after having written a paragraph, I would generally be somewhat confused. After 2 paragraphs I'd be more confused. Eventually I didn't have a clear idea of what to write because my mind was so cluttered, as if all my neural pathways were firing un-synchronously, creating a senseless mess. I have now solved this by developing better habits: I started making bullet point lists of my papers that contained the central argument, before I wrote the actual paragraphs. I then wrote one paragraph at a time, focusing only on what that particular one had to convey. Also, I developed a more structured way of structuring paragraphs: rather than just "writing it", I thought about the first sentence separately, and then its relation to the second, and so on… After developing these better habits, I felt like my brain had a much more "lean" and "uncluttered" process it was following, as if my neural pathways fired synchronously, in harmony.

I feel like right now with maths, I am in a similar stage that I used to be with writing. I understand math concepts, and I know how to do many of the methods, and I'm progressing. But whenever I'm working on a math problem, I feel like I'm getting confused, not just because the problem is new and difficult, but because my mind is cluttering and confusing itself, as if I don't have a "process" that is optimized for figuring out new math.

One way this shows, though I don't know if its a cause or a symptom, is that my derivations look like a plate of spaghetti. Yet if I try to write things more structuredly, I'm held back even more, because it puts me into a very "fearful" and paralyzed state of mind (fearful to write something wrong).

So I'm looking for habits that I can develop that will, just like I did with my writing process, turn my "cluttered" mind, into a "harmonic" one. That doesn't mean math will suddenly be easy, but at least the difficulty will be due to the complexity of the math, rather than due to me working against myself.

So I'm interested if any of you have experienced this same thing, and
whether there have been specific habits or other things that have
helped you overcome this.

To give an example of something that recently has actually helped me somewhat: Whenever I now derive an intermediate result, I write big boxes around it, with a big dense filled circle in the corner, in order to signify that it is an important result. This somewhat declutters my mind, because I no longer have to wade through all the intermediate steps, looking for the important stuff.

ps. I hope this question is not too general or subjective. I know that subjective questions are not the purpose of math.stackexchange, but I thought: there certainly are some objective principles behind what kind of habits work and don't work. And I wouldn't be surprised if I'm not the only one who could benefit.

Thank you for all the great answers! Many of these are actually things I will immediately apply.

Here is a suggestion: There is a certain topic that the answers haven't addressed, so maybe someone can address this with another answer:

How, in a very practical sense, do you write down your derivations, and how do they help make you more effective?

  • For example, do you have two separate pieces of paper for intermediate results and for details?

  • Are there any specific ways of organizing your derivations on paper, or in notebooks, that help clear your mind?

  • Do you write everything linearly, from top to bottom of your notebook, or do you go back and forth on your scrap paper, only writing it linearly when you've found the result?

  • Do you scratch formulae completely if you've made a mistake, and start over, or do you just correct the formulae?

  • Do you write derivations quickly on a scratchbook, until you've found the final answer, or do you write them neatly from start to finish?

Best Answer

I think this is a great question and you've already made an important step in addressing the problem - realizing that you are not satisfied with your math working process and searching for ways to improve it. Here are some ideas and suggestions which I found helpful:

  1. Understand well the basic objects of the game. This means that you should be able to give many interesting examples and non-examples of the objects you work on. Make a (mental or physical) list of such examples. What are the most important examples of vector spaces? Of subspaces? Can you give an example of something which is not a subspace? What kind of constructions generate subspaces? What kind of integrable functions are there? What do you know about them? And so on.
  2. Make sure you understand everything about the statement of the problem first before trying to approach it. If you don't, go back and review what you have learned. There is no point in trying to solve an exercise about nilpotent linear operators if you can't give an example of a nilpotent operator and an example of a non-nilpotent operator. This will only cause you to halt and feel depressed.

  3. Play with simplified models. This is something I really learned in graduate school and I wish I would have been told explicitly much earlier. If you are facing a problem that you have no idea how to approach and you feel paralyzed, try to work on a simplified (even trivial) model. For example, let's say you need to prove some statement about a linear map $T$ on some vector space $V$ and you have no idea what to do. Can you solve the problem if you assume in addition that $V$ is one-dimensional? Even better, if $V$ is zero-dimensional? Can you do it if $T$ is diagonalizable? If you are asked to prove something about a continuous function, can you do it if the function is a particularly simple one? Say a constant one? Or a linear one? Or a polynomial? Or maybe you can do it if you assume in addition it is differentiable?

    Applying this idea has two advantages. First, more often than not you'll actually manage to solve the simplified problem (and if not, try to simplify even more!). This will increase your self-confidence and help you feel better so that you won't give up early on the harder problem. In addition, the solution of the simplified problem will often give you some hints on how to tackle the general one. You might be able to perform an induction argument, or identify which properties you needed to use and then realize those properties actually apply in a more general context, etc.

  4. When working on a problem, try to drop an assumption and see what goes wrong. Often this will help you to identify the crucial property which you need to actually solve the exercise and then you can review the theorems and results you learned to see if it actually holds.
  5. Try to have some mental image associated to any important object and concept you meet. This way, when you'll work on a problem which involves various objects and concepts, you'll already feel familiar with them and won't halt and feel paralyzed. Review the images as you make progress and make adjustments as necessary. For example, for the notion of a direct sum decomposition you can hold in your head the image of $\mathbb{R}^3$ decomposed as the "sum" of the $xy$-plane and the $z$-axis. This is, of course, a particular example of a direct sum decomposition but it helps you to feel much more at ease with the concept.
  6. Build a mental (or physical) map of relations between various results and concepts. For example, let's say you want to determine whether a series converges or not. A useful thing to realize is that it is easier to determine whether a series with positive terms converges than an arbitrary series because there are more tests available for this case. Another useful thing to know is that if the series converges absolutely, it also converges; so in some cases even if the series doesn't have positive terms you can reduce it to the easier case. Knowing all those relations and results before you start the problem will help you to decide on a good strategy to attack the problem. Not knowing them in advance will often cause to to go astray.
  7. Don't be afraid of writing something wrong. Be hesitant of writing something that you don't really understand. It's not that bad if you write something like "All operators are diagonalizable, hence $X$" because once you understand that not all operators are diagonalizable, you'll immediately see the error. But if you write a convoluted argument two pages long which uses somewhere the fact that your operator is diagonalizable, it will be much more difficult to discover and learn from the error.
  8. Develop decent computational skills. Math is hard enough without being bogged down in computation errors and wrong applications of techniques. For example, when learning how to solve a general linear system of equations, sit down and solve $7$ different systems. If you got a wrong result in $5$ of the $7$ cases, something is fishy. Identify clearly the origin of the mistake in each case (is it an arithmetic error? did you apply the algorithm incorrectly?). Then repeat with $7$ other systems until you get at least $6$ correct.
  9. Try to work on math problems with other people. By that I don't mean asking other people for solutions to exercises you couldn't solve. Try to find someone which is more or less your level and has good communication and interpersonal skills and work together with them all the way through a few problems. Be active, propose some ideas, listen to the other person's ideas and work together. This way, you'll get exposed to techniques that work for other people, their mental maps and ideas about the concepts involved and you'll be able to adapt and implement what you learn as part of your own skill set if you find it helpful.
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