How long to do math each day?

Question

I have seen some posts math.SE (mkko's answer) indicating that it is the norm for (undergrad?) math majors to study 70-80 hours per week. I'm a little bit shocked by that. For some background on me, I'm not very advanced (only finished calculus 1-3 and taking my first DE class). However, personally if I am trying to solve a tough problem or prove a theorem, I can't work on it for more than an hour at a time without killing my ability to think creatively, which is the most important skill we mathematicians should cultivate, right? After about one or two hours, I can't engage the theorem in deep thought, so trying proving it becomes more of just an unproductive guessing game. If I just come back the next day, I feel like I have digested the problem much better and gained more perspective. When there is new material, I do spend multiple hours just trying to learn the material and become familiar with all the definitions and intricacies. However, I feel that it's most important to focus on problem-solving and proofs.

So my question is, what do these 80 hours a week consist of for typical math students? Is most of that time spent on trying to just learn the material? Is it actually spent on cultivating problem-solving skills but I have just have a really low tolerance for focusing? Is spending no more than 1-2 hours per day on the same problem optimal, but just a luxury that students can't afford once they reach a certain level?

Answer 1

I doubt anyone needs to put in 70-80 hours a week to get a math BA/BS, some people just really really like math and can't help but study it all the time.

If you find yourself burning out after an hour maybe try a different strategy. I put in probably about 50 hours a week and do micro breaks. Basically periods of deep concentrated study/thought, broken up by 10-15 minute breaks whenever my brain needs a quick rest, usually I'll check reddit or Math.SE or watch a little bit of some show.

Sometimes if I'm trying to understand something especially abstract or conceptually deep I'll go for a walk around the block and just ponder. These walks can be very enjoyable.

At your level you really just need to do lots and lots of exercises, however once you've done a couple thousand proofs, and gotten through the standard graduate level material, then it's more like exploration, and proofs and rigor become less important.

Everyone's different, and the study strategies which work for me may not work for you, but this is what works for me.

Answer 2

My personal experience is that breaks are really important. As you mentioned it's really difficult to stay focused for a certain amount of time, especially when someone is engaging harder problems which require a high amount of focus. 70 to 80 hours seems unrealistic if you only count the time you are indeed studying. If it's the time spent at the university it's still very unlikely this is right, or at least I don't see how one is able to study efficiently for than 10 hours a day while doing math.

Answer 3

Since this is kind of a subjective thing, I'll pitch my own experience.

I'm a senior undergraduate taking two graduate level classes. Typically, I work for about 3-4 hours a day books open, laptop open, pen and paper all over the place, basically working on either my thesis, or homework problems for my graduate classes. I do this basically every day, sometimes a little more, sometimes a little less. I have a little bit of ADHD, so I take short, but frequent breaks to do something else, stretch my legs, get a drink, or just scroll Facebook for a minute or two. Off the bat, that's somewhere between 20 and 30 hours each week. I think this is an underestimate for what typical people in two graduate classes go through - last semester I worked much harder than this semester. Maybe as much as 6 hours a day.

But this is not really what I would call my 'studying.' I think that, for me, studying is a much broader thing. I spend all my time walking to and from other lectures thinking informally about problems, homework or not. These comprise another hour a day, again sometimes a little more if I have to walk to my tutoring job that day. So here are maybe 5 or 10 more hours.

But I also do a good number of problems completely separate from any class I'm in. I have a very large reading list for mathematics, that I hope to someday have made real progress on. I'm at that stage in my mathematical development where everything I come across is really cool, and no matter how cursory I need a result, I really want to investigate it and internalize its contents. Right now, I'm working through Sets for Mathematics by Lawvere, and Basic Algebraic Geometry I, by Shafarevich, each with a separate buddy.

These serve purposes for me. I've been seeing a lot of category flavored arguments lately, and so I've been trying to sink my teeth into that subject. I also am really interested in geometry, but have never really looked at anything algebraic outside a first course in algebra. I probably spend about two hours a day reading or working on problems of these types. Sometimes I just skim these books really quickly and make notes on my PC about things that catch my eye. Other times these problems look like my homework problems - I take them very seriously.

I figure these are 15 to 20 hours more each week then.

And of course, during exam season, I spend less time on these side problems (but still not $0$, I find it helpful to take my mind off my courses just like everyone else), and spend a lot more studying. I can't estimate this time well - for my topology class last semester, I probably was studying all day every day for at least a week and a half before the exam, punctuated with my usual breaks.

So... typically it looks like I work on something mathematical either on paper or in my head maybe 7 hours a day, on a typical non-stressful day. That's about 50 hours a week.

Now, if you count the time that I'm in lecture or meeting with my adviser, we can add another 10 hours, so we're up to 60.

So I think 70 is probably a lot. I consider myself a pretty driven student - I really want to be successful at mathematics, even though I'm probably not very good at it compared to my peers. I guess some people probably could do it though.

One last comment, regarding optimal studying. I think that it is not about how much time to spend thinking on a problem. Some days I think all day about the problem, some days not at all. There's at least one or two problems that float in my head for a long time, because I don't have the right notions yet to digest them. But I still think about them and try to connect up what I've been doing lately with them... see if I can have any insight, and this has been fruitful in the past. More importantly than spending any amount of time with material is avoiding burnout and frustration. If you are happy with the state of your work, and you feel you can, raise the bar. If not, it is better to stay happy with mathematics and yourself then to struggle and lose interest. That's why I like to tinker with a bunch of different things at once - I can always put down what I'm doing, and look at something else fascinating too!

Answer 4

Speaking as an undergrad from the UK, I am assuming it is all fairly similar, throughout the week I try and get to the library at about $9$ and leave about $7$, then SWITCH OFF(or try). Then on the weekend I will do a little bit,maybe just some thinking or tidying up some notes.

This method,so far, has worked for me (Over 90% average... so for.).

If you do too much you WILL get burned out, you will then need to shut off for a while which is a nightmare.

So I think, if you count the "thinking about problems" then 70 hours is not that far away.

Thoth nailed it though.

Answer 5

Too long for a comment. +1 for recognizing the value of alternating hard thinking with letting your subconscious do the work. This from Poincaré via Hadamard and wikipedia (https://en.wikipedia.org/wiki/Henri_Poincar%C3%A9#Philosophy)

Poincaré's famous lectures before the Société de Psychologie in Paris (published as Science and Hypothesis, The Value of Science, and Science and Method) were cited by Jacques Hadamard as the source for the idea that creativity and invention consist of two mental stages, first random combinations of possible solutions to a problem, followed by a critical evaluation.[64]

Although he most often spoke of a deterministic universe, Poincaré said that the subconscious generation of new possibilities involves chance.

It is certain that the combinations which present themselves to the mind in a kind of sudden illumination after a somewhat prolonged period of unconscious work are generally useful and fruitful combinations... all the combinations are formed as a result of the automatic action of the subliminal ego, but those only which are interesting find their way into the field of consciousness... A few only are harmonious, and consequently at once useful and beautiful, and they will be capable of affecting the geometrician's special sensibility I have been speaking of; which, once aroused, will direct our attention upon them, and will thus give them the opportunity of becoming conscious... In the subliminal ego, on the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of discipline and to disorder born of chance.

Answer 6

In the book Outliers, author Malcolm Gladwell says that it takes roughly ten thousand hours of practice to achieve mastery in a field.

If you want to (Bachelor + Graduate + PhD) this makes roughly 5/6 hours per day. That's it. I find this estimate quite reasonable. If you want to accomplish the same result before graduation You have to switch to the 10 hours mode.