My motivation to ask this question actually comes from here, receiving negative reactions for 12 hours a day or even 7 hours spending time on Mathematics.

A person usually spends 4.5 years to accomplish a BSc plus a MSc coursework. Books that I found to study in pages (BSc+MSc) approx. are:

  • Calculus (760)
  • Algebra (450+940)
  • Topology (510+540)
  • Analytical Geo (210)
  • Real Analysis (610+520)
  • ODE (440)
  • Complex Analysis (480+350)
  • Differential Geo (510+420)
  • Logic (260)
  • Number Theory (480)
  • PDE (320)
  • Total: 7800 pages.

Supposing spending 30 min per page of studying and 2 hours per page of exercises, so it takes (4/5)×7800×0.5+(1/5)×7800×2 = 6240 hours = 5.5 hours a day in 4.5 years (excl. Sat+Sun) for a student.

The problem is:

1- I think that a Mathematics student has to learn more than I have listed, for example other subjects in Mathematics.

2- a Mathematics student has to spend on other than studying texts, like attending classes, other courses (e.g. computer) as well.

How a typical university student in Mathematics spends a few hours in average on studying and knows a lot Mathematics and is able to start research, which I am not capable of? It's a paradox, or maybe I am not smart as other?

Thank you.

PS I am not officially a uni student yet, but I had to spend 12 h/d on average to study the mentioned books which I am pretty sure are less than what a uni student covers in their studies.

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    Also keep in mind that maths is easy for some and hard for others (I don't know why). The same topic/problem may take one person 1 hour and another 10... - in the end both will get the degree but one found it significantly harder than the other. Spontaneously I'd say the more you like maths and the easier it is to you (including abstract concepts) the easier a maths degree (and research) will be to you - but I guess that's obvious. Only personal experience will give you a definite answer. (Side note: I have a UK maths BSc) – DetlevCM Jul 17 '15 at 11:39
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    Those page counts seem very high, compared with my experience. Are you including large exercise and solution sets in those numbers? In my experience, mathematics books are often quite low on the page count, compared to other disciplines, but very dense: it can take months to digest even a 100-page book. – Federico Poloni Jul 17 '15 at 12:12
  • @Federico Poloni: None of them has solutions for exercises (not examples) in the same book and I didn't count any solution book at all. Maths-books are usually 300-500p in ave. but it can also be 940p (Foote's book of Abstract Algebra) or more, e.g. Spivak's Differential Geometry >2000p. – MKR Jul 17 '15 at 12:23
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    How much of those books is included in the program of an average class? Typically there is more than what is actually covered, so that an instructor can choose which chapters to leave out to personalize the course. – Federico Poloni Jul 17 '15 at 12:46
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    I wish it still only took me 2 hours to do a whole page of exercises! – coldnumber Apr 17 '16 at 2:52

I am afraid that your computation is not even wrong. You might as well ask how many hours a day you need to practice dribbling in order to be a professional basketball player.

Mere knowledge of mathematics is just one component of being a professor, even a professor of pure mathematics. To become a successful mathematics professor (or any type of professor, really), you need a lot more than just to have studied many textbooks: you need scientific creativity, you need vision, you need to be able to communicate and write well, you need to be able to teach, you need self-discipline, you need some broader understanding of how mathematics fits into the larger world, and you need many more things besides.

Moreover, one of those many things that you need is skill at extracting knowledge from texts. The deeper your understanding becomes, the more that you will be able to see the structure of the content that you are reading, to analyze which parts are critical and which are accessory, and the less that the computation you present makes sense.

In short: yes, it takes a lot of effort on mathematics to become a professor, but the process and challenges are very different than you think.

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  • 32
    Let me put this more plainly. Most mathematicians don't know all that stuff nearly as well as you might think. Most of it we know only vaguely in outline and know how to look it up and learn it (or just figure it out for ourselves) if we find we need it (and we also know how to tell when we need it). – Alexander Woo Jul 17 '15 at 4:47
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    I didn't realize the truth of this until later, but one professor told us that the benefit of coming to class when we could just have read and worked through his script was to hear/see him thinking out loud to deduce all that stuff. You can learn a lot from that. Most importantly, you'll learn that it isn't important to know everything. You need to know the basic tools of your field, but you can deduce or look up the rest when you need it. – Sumyrda - Reinstate Monica Jul 17 '15 at 8:29
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    There's a lot of textbooks which are very bulky (I bought plenty when I was in MKR's position). In extreme cases, you might be able to master a 1000 page tome by studying 40 pages intently, and ignoring the rest. The successful student is the one who can work out quickly which 40. – Julia Hayward Jul 17 '15 at 9:57
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    @MKR: It's possible because of the fundamental flaw in your reasoning, that has been pointed out and that you're continuing to overlook. An undergraduate course in abstract algebra does not cover every single detail in that book. It probably doesn't cover half of it. In my mathematics degree, all the material that a student actually needed to know could be hand-written by that student during lectures and homework. We did not do the equivalent of copying out an entire textbook per course. Degrees just don't work how you think they do. – Steve Jessop Jul 18 '15 at 12:20
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    ... on the other hand, an able student who has taken the undergraduate course might able to deduce for themselves much of the detail that the book goes into and that there was not time for in the course, because they now understand the topic. So if the need arises they can do it despite never having seen the particular case before (or even the particular technique). Pretty much the whole point of mathematics is that you can know a relatively small amount and figure out the rest as you go along. – Steve Jessop Jul 18 '15 at 12:25

One should not think in terms of a necessary (or even worse, sufficient) number of hours of work needed. What you should set yourself are goals: learn the courses of your curriculum, practice them to make them become natural to you, ask yourself questions about them (seek examples of definitions, counter-examples of theorems with an hypothesis removed, etc.), and very importantly get enough sleep (and possibly sport) for your body and brain to work efficiently. Studying for hours can be less efficient at getting good at maths than sleeping more and be more acute during the fewer hours of work.

Working hard is an important part of success, true. Working well is of even greater importance, so be prepared to change your way of working for a more efficient one.

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I would say your calculation 5.5 hours a day in 4.5 years is close to real based on my own experience, the link @jakebeal provided and others' experience.

I think one of the differences between your case (not officially a uni student yet) and others is that they have professors and TAs to help them. They attend Math classes in school. They learn what's important to read and what important exercises to do in the class. In other words, they learn Math more efficiently and more effectively than you do.

This is why attending formal trainings in universities is always encouraged. Not only you have professors and TAs to help you, but also you have peers (classmates) to discuss the problems with. So, you learn faster and better. You eventually graduate from school (BS or MS or PhD). Hopefully you'll become a professional Mathematician someday.

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I my experience most students do most of their learning from lectures, courses and problem sheets rather than reading textbooks.

I fact I will go so far as to say most students read few/no textbooks cover to cover. While, I am probably not the best example of a hard working student during my undergrad (in Physics) I can think of only two subjects where I got a textbook to look at: Quantum field theory and general relativity. Even then I had a pretty good idea of the subjects so only looked at the bits I was uncertain of.

Other people definitely used more textbooks than me but I don't think they really read them cover to cover. This strikes me as a very slow and difficult way to learn.

Having said that your estimate of 5.5 hours/day to learn a subject is probably not too far wrong. My undergrad (Physics in UK so YMMV) had approximately 25 contact hours per week, including labs etc. so ~5hours per day plus another couple of hours per day for problem sheets. I think the course load for Maths (and any other STEM subject) was similar.

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  • 3
    Indeed, my maths BSc did not rely on any textbooks at all - and even if you need to look something up, Wolfram Mathworld is a quicker resource... Although this may also be a cultural thing because just 20 years earlier in Germany, Professors would prescribe books as essential reading and the library would have a sufficient amount for an entire course - so you could go to the lecture or learn from the book (and double check with friends/colleagues if you have all information). – DetlevCM Jul 17 '15 at 11:43
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    I also agree, I got undergraduate degrees in both mechanical engineering and mathematics, and I am truly reading my first textbooks right now, three years post graduation, studying for my qualifying exams for my PhD. Textbooks are good references and sources of practice problems, but for learning new material, there are better ways. – Godric Seer Jul 17 '15 at 15:30
  • @GodricSeer "there are better ways" - I think it depends on the person and may even change over time for each individual. But agree with you many students today are more used to interactive materials. The problem may be relying only on one single type of resource - only textbooks, or only Mathworld, etc. – ALAN WARD Jul 17 '15 at 20:00

It is different for each person. I know people who spends less than 2 hours a day on math and gets higher grades and I know people more than 5 hours to do the math and get a passing grades.

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