Advice on Mathematics

Here are some thoughts that I have gathered over the years. Some of what I will say may be controversial to people, but I hope that my perspective may be of some value. This is mainly geared towards undergraduates and graduates, but may be helpful for others. A lot of what I say may be applicable outside mathematics.
  1. The natural state of things is struggle:
    Mathematics is one of the hardest things you can do. While everything is easier when you have a proper foundation, it will never be easy. For me almost every day is a challenge. Learning new things and problem solving is painful, uncomfortable, even though I've been doing maths for many years. However I know that it is hard, and so this fact does not dishearten me: know that the pain that you feel is a reflection of the intrinsic difficulty of the subject, and not a reflection of your aptitude.

  2. The major determinant of success is hard work and accumulating deep understanding:
    I'm of the controversial position that anyone can become very good at maths (a position shared by the great Terry Tao :P). However it takes an unreasonable amount of work to achieve mastery. Every successful student works extremely hard, and anyone who tells you they get good grades without studying is a liar.

    However hard work isn't enough: It's easy to work very hard and achieve little. You need to emphasize rigor, and understanding every aspect of what you do. Never be vague, flippant, lazy, hand-waving, etc. Leave no stone unturned. I recommend my students study for 6+ hours per week, per subject, and to study over the summer break if they're ambitious.

    In the short term, it is quicker to have a shallow understanding of what you're doing: It may take up to 5 times longer understand the material properly. You may be able to muddle through your homework assignments, but progressing without a rigorous foundation will come back to haunt you, and prevent you from progressing any further. A steady accumulation of deep understanding saves time in the long run, and is the only viable path to long-term success in mathematics. There are no shortcuts.

  3. Don't directly attack the problem:
    What? What are you supposed to do then? Over a long time, I have experienced far more success if instead of attacking a problem, I sought to simply understand all of the background theory of what I was doing. I would explore every avenue, and then eventually, through doing this I would incidentally solve the problem with ease. Directly attacking a problem is often fruitless and frustrating in the extreme. When you have sufficient understanding of the background, the problem will become easy, and the solution will become obvious to you.

  4. Learn, re-learn, reformulate, draw:
    I will often spend an entire weekend simply trying to understand a single page of a book! Here is a process that I go through – I'll read the proof of something. I'll reprove it myself, following the author's proof as I go. I may reprove it in different ways, from different perspectives, etc. I'll draw pictures of the flow of the proof – What leads to what? I'll ask questions – What is the fundamental idea of the proof? What are techniques used, and how did I miss them? How would I recognize the way forward if I saw something similar in the future?

  5. Articulate your thoughts on paper:
    You should be constantly writing: If you aren't writing, you aren't learning. I use tremendous amount of paper as I try to develop ideas and try to understand things. I am constantly drawing diagrams, arrows, comments, scribbles, etc. Even as an advanced PhD student, I find it far more effective to learn and research on paper. Often using a computer is unavoidable, but for the last 10 years, I have always carried a pad of paper with me wherever I go. I've also observed that I remember things that I write down far better.

    Apart from computers being inherently distracting, I think that any barrier or layer you put between you and the core cognition of what you are doing has a detrimental effect. And paper allows you to more flexibly notate your thoughts, in a way that more closely follows the structure of how a human brain actually thinks: nonlinearly, and in graphs or networks.

    Writing helps focus your thoughts, and augments your short-term memory. Being able to see your thoughts on paper means that your short-term memory is free to be put to better use. But thinking without paper is useful for more free-form thought. Both methods compliment each other, and should alternate.

  6. Never compare yourself to others:
    This is an easy trap to fall into. There is an inherent bias in observation: You only observe peoples' successes and never their failures (and people are not super eager to inform you of failures either!). You could call this the Facebook effect. When someone intimidates you, you should always remember that the buildup to where they are may have taken decades, and that you will eventually reach their level over time. Topics you have now mastered seemed so difficult when you were introduced to them long ago: But now they are easy. The same will become true of what you are struggling with now, and what you will struggle with next.

  7. Don't work too long:
    You can only really concentrate on something difficult for about 75 minutes. I have asked many people including professors, and universally, people seem to get work done in small chunks of about an hour. It is possible to work much longer, but observe yourself and you'll realize that you are distracted, and have ceased to make any real progress. Take a walk, take a nap, it's more useful than continuing.

    It's a trope – but it's also true! – You're going to think of your best ideas when you're sleeping under a tree, brushing your teeth, walking around the block. Staying religiously glued to your desk, apart from being miserable, will inevitably lead to narrow thinking, falling into a rut, going around in circles, frustration, despair. Free your mind, get some sunlight, and go for a cycle!

    Work that is difficult, like learning new things, research, problem solving etc. I refer to as “core cognition”. Over about a decade, I have never consistently been able to get more than about 3.5 hours of productive core cognition per day, and many other people have similar experiences. It appears to be a finite resource that needs to be replenished over a long time. It's possible to do simple things like write slides seemingly forever: However this is not the case for core cognition.

  8. Get adequate sleep:
    Your brain is a sensitive instrument: It needs to be well rested, and nourished to function properly. It has been said that Americans don't remember what it feels like to be completely alert. People who are sleep deprived, ironically, rate their concentration and aptitude higher, but perform much worse in tests. When you get adequate sleep, everything just seems easier! Things that seemed daunting are no longer so hard. If in doubt, I always take a nap and work later: I have never regretted doing so because I've always returned to what I was doing with renewed vigor.

    However you have to get in the habit of sleeping more: You have to force yourself to stop, have the discipline to say “no more tonight”. When you see the positive benefits of doing this, it becomes easier.