What are good "math habits" that have improved your mathematical practice?

Question

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.

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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.

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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?

Answer 1

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.

Answer 2

EDIT: I misunderstood the OP at first, and the first half of my answer gives advice on how to approach proving an unknown problem. I then tie this into the organizational question the OP is really asking below the line.

Examples, examples, examples! For me, pretty much all of mathematics is driven visually and by example.

Every time you see a theorem, first seriously commit yourself to finding a counter example. Find almost-counterexamples that show why every assumption in the problem is necessary. Then for each of those almost-counterexamples find an example that is extremely similar, except satisfies the assumption the counterexample was missing. Now you're ready to prove the theorem or read its proof, and in all likelihood you're already close to the proof.

There's a great anecdote about this by Keith Kendig about Hassler Whitney

One day in his office, I happened to mention Bezout's theorem which basically says that two curves of degree $m$ and $n$ respectively intersect in $nm$ points. He says he never heard of it and seems galvanized by it. He jumps up and heads to the blackboard, saying "Let's see if I can disprove that" Disprove it?! "Wait a minute!" I say, "that theorem is nearly two centuries old! You can't disprove anything... really..." As he begins to working on some counterexamples at the blackboard I see my well-meant words are simply static.

His first tries were easy to demolish, but he was a fast learner, and ideas soon surfaced about the complex line at infinity and how to count multiple points of intersection. After a while it got harder for me to justify the theorem, and when he asked "What about two concentric circles?" I had no answer. He argued his way through and eventually found all four points. Finally he was satisfied, and the piece of chalk was given a rest. He backed away from the blackboard and said "Well, well - that is quite a theorem, isn't it?"

I think I mostly kept my cool during all of this, but after I left his office I realized I was pretty shaken. I remember thinking to myself. "Golly, Kendig, you just saw how one of the giants does it!" He'd taken the theorem to the mat, wrestled with it, and the theorem won. I'd known about that result for at least two years, but in 15 or 20 minutes he'd gained a deeper appreciation of it than I'd ever had. In retrospect, it represented a turning point for me: I began to think examples, examples, examples. Whitney worked by finding an example that contained the essential crux of a problem, and then worked relentlessly on it until he crack it.

Doing mathematics by example has taught me how to feel the shape of a theorem, to naturally divide mathematical objects into collections based on how the theorem divides them into "examples" and "non-examples" and those lines that the theorem draw shows you how to prove the theorem. This will also massively help you recreate the proof in the future.

Thinking about the theorem from the "wrong direction" will teach you to think in unusual ways help you falsify conjectures and assumptions easier. People have a strong bias towards looking for confirmation of facts, but struggle to remember to look for disconfirmation. This can make it hard to understand theorems, because $\mathbb{Z}[\sqrt{-5}]$ tells you a hell of a lot more about the nature of prime factorization than $\mathbb{Z}$ does. There's a famous quote about the importance of this kind of thinking about the philosopher and logician Wittgenstein (source):

Tell me," Wittgenstein asked a friend, "why do people always say, it was natural for man to assume that the sun went round the earth rather than that the earth was rotating?"

His friend replied, "Well, obviously because it just looks as though the Sun is going round the Earth."

Wittgenstein replied, "Well, what would it have looked like if it had looked as though the Earth was rotating?

Here's a MathOverflow link about counterexamples to get to know and love.

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Now, to tie this into the actual questions in the OP:

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

That depends on the flow of the proof. I'm a big chalk-boarder, and often sketch out my proofs and my examples on chalkboards before committing them to paper (or TeX, more commonly). If proving an intermediate result seriously disrupts the flow of the proof (which tends to mean "requires more than a paragraph") then in my proof sketch I'll just write "by Padding Lemma" or whatever and then prove the "Padding Lemma" on a different pane of the chalkboard. This is largley because I want to be able to look at my examples and proof sketch without getting bogged down in the combinatorics of this lemma I happen to be using. In the actual write up of the proof, learning how to properly organize your "digressions" is a very important part of learning to do academic writing, but is heavily contextual. Like organizing stories or essays, it's more of an art than a skill.

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

For me, the process of designing a proof (and in fact, thinking in general) is like a conversation. I create an interlocutor in my mind and argue with them. I walk them through example and non-example, explaining why in each case the theorem holds or fails. I find that doing so helps me focus on the similarity between the examples and find the underlying logical thread that contains the "real reason" that the theorem is true. I might draw diagrams or do a few calculations on paper if I can fit them in my head, but I generally don't start really writing the proof until I know what's going to happen. At that point, I chart out a few examples that have been particularly enlightening and that can guide my thinking on a chalkboard. This is usually a few pictures and equations, the equations written with the pictures, and each example separated in space. Then I'll sit down somewhere that I can see the whole chalkboard and begin to write.

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?

As I mentioned, I sort my thoughts by example, with each example being an explication of why the theorem is true for that example. I tend to separate examples horizontally, and organize their explication either vertically (especially for formula-heavy problems) or circularly (especially for graphic-heavy problems). I don't worry too much about the arrangement of my chalkboard though, and just place things "where they obviously fit."

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

I prefer working on a chalkboard and erasing mistakes and fixing them. In particular, I avoid slashing through or using other markings to indicate that a term is incorrect, because my equations are often adjacent to arrows and other markings that indicate how they fit together, and I use crossing out to indicate cancellation.

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?

My handwriting is not that neat, and I ususally TeX the final. If it's legible to me (and any collaborators), that's all that's necessary for scratch work.

Answer 3

Everyone is confused when learning a new subject, so don't get discouraged if you don't pick things up immediately. My advice would be: instead of letting the confusion scare you away, harness your confusion for all it is worth.

There are many different kinds of confusion. Sometimes it may be just a sort of general haziness around a subject - this might just indicate you need to re-read the textbook because you don't remember the details.

But there is another kind of confusion, one that is potentially useful. Often you will have some sort of cognitive dissonance. That is, you're learning something new - but it does not jive with your current understanding. Something does not feel right. One difference between an excellent student and a mediocre one is that the excellent student will simply refuse to let this feeling go until it is resolved. It's very easy to accept what you've just learned, even if it doesn't make sense to you. But if you're feeling this kind of cognitive dissonance, it means something is wrong in your understanding, either of the previous material or of what you're currently learning.

The trick is to be able to precisely pinpoint what the issue is. At first what you might feel is only an emotion, a vague uncertainty. But this uncertainty may be meaningful. If you double down on this, and keep thinking about it, eventually some more concrete questions will bubble to the surface. It's okay to not have the answer immediately - the important thing is to sharpen the confusion into a very precise question. This will often involve taking away all of the extraneous details surrounding your question and focusing only on precisely where the issue is. If you're having an issue with a very general theorem or idea, try coming up with a specific, concrete example that demonstrates the problem.

Once you've funneled the haziness into a very specific question, you'll often find that the answer is not as difficult as you thought. If you can't come up with it after some thought, you should ask your a classmate, instructor or a TA - or ask here on Stack Exchange. Generally speaking, I think teachers appreciate getting very well-formed questions that show the student has put a lot of thought into it.

It is possible to take this too far, getting bogged down in minor issues instead of moving on with the material. Sometimes it's necessary to table a question so you can get on with your studying. In that case, my advice would be to write down your confusion and remember to come back to it later. Sometimes you just need to take a walk, or sleep on it. Channeling confusion into a meaningful question is a difficult skill that takes time and practice.

Edit:

Here's an illustration of the process. Many students don't actually get from Step 3 to Step 4. (Cartoon due to Fan Wei, found on the page for Richard Stanley's Enumerative Combinatorics.)

Answer 4

I had a conversation with a friend of mine who at one point felt similarly about writing. While I could never hope to do his view 100% justice, I can attempt to explain how he conquered his situation with writing, and how it helped him to grow mathematically. There are a few caveats. For example, without knowing how you and he compare in mathematical ability (you now compared to him then), I can't say for sure if this advice will be useful to you. You may also just find that what worked for him does not work for you. I regard my friend as one of the most disciplined people I know, and so he has quite a bit of time in his day to spend on mathematics, and in thought, where I personally would feel tired for the day.

With that said, my friend's position was more or less the following: the art of learning is a lost art. At one point in our history, we learned first by memorizing. Now memorizing has quite a bit of stigma associated with it, as if it's not real learning. This is kind of true, if the only thing you do is memorize. There are many students in schools these days who have memorized a great many theorems who understand virtually nothing, let me say that this is not what I advocate. What is true though, is that having a large class of theorems and proofs that you have memorized will be the foundation of your mathematical ability.

When my buddy wanted to learn how to write, he had a similar problem. His writing was disorganized, and reflected a certain amount of aimlessness. He found that without a clear purpose or a target audience for who to write to, his writing would suffer greatly. It was also his experience that although he knew how to compose sentences, he did not know how to be a good writer, or even how to really improve.

The solution he came up with was to begin by copying. He would learn to write first by imitation, the way one learns an instrument. Before one composes a masterpiece, they learn other masters' works inside and out, and learn from their styles. Find a passage of a text that spoke to him, and just copy it word for word. Read it out loud several different ways, find the voice with which the passage was written. Then, when the sentences start to stick to your brain, try to summarize the piece. Try to re-express the same thoughts in your own words.

At first, when you do this, the things you copy and your summaries will be pretty shoddy. They will have little additional insight (although you might find in your summaries things you don't actually understand about the passage, and this is very important both in writing and in math), and they will sound rather dry. But if you do this with many authors and many passages, you will develop what I can best describe as taste in writing. You'll learn certain kinds of things that just sound right to your ear, and this will be your writing voice.

This is also what it is like to learn math. You start by copying and reproducing proofs over and over until you have them memorized. Many people like to do some of the things outlined in levap's answer, like dropping hypotheses. When you reproduce the proofs, try to fill in the proof with some prose about what you are doing and why, like you are lecturing. This is similarly your mathematical voice, and learning to write mathematically is not all that different than writing prose, even if the styles may be different.

Personally, I find it incredibly helpful to write about mathematics to help me process the ideas. I find that graduate school does not afford me the time I would like to do all the writing I would do otherwise. Of course, one cannot write a book about every subject one takes a class in, but when I venture off in my own direction, I find it helpful to write notes in my mathematical voice, filling things in as best I can.

Anyway, I feel in some sense that I have not 100% addressed your question. I guess what I'm really shooting for here is that the process of memorize -> explicate is a fundamental process for learning math. It will equip you with tools to say what's on your mind, which seems to be something you are struggling with, and it will also help you improve your mathematical writing too. So I will say that as much as time permits, you should try to reproduce proofs, at first word for word, but eventually in your own voice, and maybe even you will be able to give alternative proofs using your newly refined intuition.

Answer 5

I've noticed since I started to get really into math (just last year when I took my algebra and analysis sequences), my skills with logic have grown as well. So, I think the practice of taking facts at face value and viewing the problem as simply as possible has certainly helped a lot. In general, if you're able to explain the problem to someone who has a small mathematical background, then you sincerely understand the problem yourself.

I also enjoy drawing 'inspirational pictures' for problems! I think this goes along with trying to understand the problem in its most simplest form.

Answer 6

This is a really excellent question, that is likely to get a lot of varied answers. Here's what I've taken to doing that's helped me along the way to getting my bachelor's as well as in my own research endeavours.

A bit of advice I got from one my professors in my first advanced mathematics classes I had (took a 300 level w/ him, then a 2-term 400-level series on linear and multilinear algebra with the same professor): Know the definitions. If there is anything that is worth practically rote memorizing from your mathematics text, it is the definitions for things, because if you don't know the definition for the thing, how can you expect to solve a problem involving that thing? (This is especially important for closed book tests, which are common in mathematics curriculum!)

One thing I took increasingly seriously the more I studied math: Don't be satisfied with a 'half-assed' understanding of a result or argument, keep interrogating the argument until you fully understand it. (The same applies to each and every part of a definition, I vividly recall putting in non-trivial effort to understand why the definition of quotient topology contained an iff instead of just an implication in part of it. I eventually figured out why, and later on at the end of term, when we were reviewing for our final from material someone else had prepared, I spotted an error that the rest of the class and even our professor missed because of understanding that very subtlety to the definition of quotient topology; details matter!)

Ask 'what if' as much as possible: This is really helpful to understanding definitions, as well as the reasons for assumptions in theorem statements. It gets at a general principle that is already a recurring theme in what I've said: 'why is always the most important question' If you can't answer the question why, you do not yet understand it (it's easy to know something without understanding it).

To address the addendum added by the OP: What I found to be useful when working on homework near the end of my undergrad, was to work first on a board, putting together a 'skeleton' of what would eventually become my complete and rigorous proof of what it was I was to show. After getting a skeleton in place, leaving space between lines because I would probably need it soon, I would go back over my skeleton and flesh out every single bit of the argument, filling in all the details, and would refine it in this manner until I was completely satisfied that my proof used only results I was sure were true (usually if they were nontrivial I would explicit reference the theorem, or show it in detail) and that the logic of my arguments made no leaps, no implicit or accidental assumptions, and was entirely correct and valid. Once I had a complete proof that was entirely satisfactory, I would then copy it onto the paper I would eventually had in. Working on the board first meant no erasing and re-writing on my paper that I would hand in, and made it easy to write out a clean and complete proof to hand in.

Mathematics is complex, and at times very difficult, and the details matter, hence it is important to get used to being scrupulously attentive when working on math, and being gratuitously thorough as well. (It's a good habit to check your work as you go, and again once you think you may be finished, it's amazing how easy it is to miss a simple arithmetic error in the middle of an involved proof, and you can be almost assured that such an error will end up mattering later on in the argument).

Answer 7

I don't think this was mentioned in any of the other answers:

Be patient

When I encounter a nontrivial problem, I work on it hard for a few hours. If I don't solve it right away, I put it away for a day, and come back to it later. This lets me see it with fresh eyes which always helps me, and sometimes, I end up "solving" it when I am not even working on it - I suppose my subconscious does not put it away.

If it takes more than a few days, repeat this process. Don't stress yourself out about it taking a while, math is not a race.

Answer 8

well for a long time I faced the similar problem to yours but now it's mostly solved.

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

when I want to solve a question mostly I first write my idea and the probable solution to the question on a scratch piece of paper and after becoming sure about the plan I start to write it on my main paper, textbook etc. and in this step I do the calculations because many math problem are not just about putting number in a formula.

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

well there's a method which had helped me a lot specially in many hard problems after I get disappointed from solving the question. I have a small white-board and start to solve the problem there and also try to explain myself the solution out loud it's a bit funny but it helps.and also while working on paper when I reach some important result which will help me in the last solution I draw a circle and put a number by it to avoid forgetting that(even my teachers do this).

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

I usually go back and start again because sometimes changing a number can change many factors which you may not recognize first and it may lead to wrong answers.

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?

sometimes I don't even reach the answer on my scratch paper and I just try to identify the idea behind the question so my struggling on the scratch paper can only be read my myself which sometimes put me in trouble so I recommend you a combination of not focusing on being neat on your scratch paper and just trying to find the idea and also not being so much messy that puts you in trouble. and one last thing: don't be afraid of making mistakes mistakes should teach us where we are facing a problem and after recognising them they should become a stair to help us become more professional in solving problems.

Answer 9

I haven't done the following things while practising math ever in my life (in all these 15 years of my life), but do consider these advices seriously (and please don't consider this answer hypocritical 😅)

1. Keep in mind what you have and what you want to get. If you have a clear sense of your path, it will ease your road ahead.
2. Questions and problems pertaining to the topics have to be practised regularly ( I used to do that, but not very much nowadays). As the people say, 'Practise makes you perfect'. You will fix concepts in your mind, if you practise.
3. Apply those concepts you learn in some topic you want to explore (in my case, I am using the bit of calculus I learnt from an entrance coaching centre in my search for a general way to find zeroes of a polynomial of degree $n$ [ which is, in fact, something that can't be got that easily]). When you keep using a tool, you get accustomed to it. The effect is the same as it would be in practice, but it provides extra happiness if you are able to stumble upon some unfound stuff (or something that will benefit you when you require it).
4. Talk to your friends and teachers about the things you end up while doing those discoveries. Why , you can talk those here itself if you prefer ! ( I mean, MSE ; I talked of teachers just because I used to talk to the teachers who taught me there at the coaching centre about whatever extra I tried to learn). Discussions help you correct yourself where you go wrong and confirm your thoughts where you stumble upon something new (this is nothing much to qualify as a math practice, but of course this is something that will benefit you).