# How to improve mathematical skills(University level)?

I am doing Ph.D in Mathematics, I feel I lack few of the skills, if I can improve those skills I think I can do better as a Math scholar. I need some suggestion on these following(below I am talking about problems or theorems in University level text books or some competitive exams),

1. When I try to solve a problem(or learning theorems) I just go by my intuition and creativity, sometimes that works, many times confused and got stuck and restart again. Is there any formal way of approaching the problems?
2. Similarly, on making arguments I without knowing I use my intuition as a correct statement(many times it is a false) here and there in the flow of arguments. How to practice to go neatly?
3. This is also same point as the (2) but while doing with pen and paper(problems in differential equation, I mean kind of algorithmic way problem solving), I do silly mistakes and skipping steps(doing a intermediate step in mind). How to overcome this?

I can only help with (3).

A. This behavior is not unusual and not just with intuitionists. It's good to be able to do things in your head, but you need to "know your head" and when you will have issues.

B. In general, when doing pen and paper you should try to write down all the steps prone to an error. Of course there's a balancing point. But clearly, you are doing too many computations in your head as per the result. SHOW the algebra. If you do it often, you will at least get fast at writing it.

C. Work lots of drill problems and do those that have answers available (to check yourself). You need volume to build automaticity. (Find and read the Feynman quote about this "bing bing bing". Calculus, and diffyQs is just integral calc on steroids, is an opportunity to do a lot of algebra...to solidify that skill.)

D. WhenEVER you make an error (of any sort, missing a constant, sign, anything), this means you have to do the whole problem over again. Yes, even if you instantly get why you did it wrong.* When reworking the problem, flip a little switch inside yourself and work the problem "as if from scratch". Do the whole thing, not copying your old solution or just the part wrong. But the whole thing as one piece from start to finish and performed correctly. Do it as if forcing yourself to do a musical piece over again after a memory error, or a gymnastics routine over again after a single error. (If you are punished, even mildly, for errors in algebra, by having to do more work, you will learn to stop making them. Very similar to physical conditioning feedback in sports coaching. I know you think of math in men as some spark of God to Adam in the painting...and in some ways it is...but in other ways we are dogs trained by Skinner and Pavlov.)

P.s. While I don't have the complete answer for 1 and 2, I think building some stronger algebraic muscles (like long series and the like) during 3, will rub off and help you on 1 and 2.

Don't worry, your "spark" of intuition ability won't go away because you get a better mechanical ability. In fact, you may find yourself able to address some situations more easily because of having worked so many problems that you see patterns. "Tricks" and complications can repeat themselves.

Now I wouldn't also say that training mechanically will build the genius puzzle solver (you need to solve lots of puzzles to build that muscle). But it won't hurt, will help what it's meant to help and will even mildly help your intuition.

So don't be lazy. Don't be proud. Buckle down. An easy one is to just do "all the homework problems", not just the assigned one. Don't even worry too much about time efficiency. It will be hard at first. But as you build the calculational muscles, you will find yourself getting faster and stronger. And even the time efficiency will become decent (not a bad thing to have when tested).

*I advise to work the entire "set" of homework (say the section, if covering one section a night) and then check all the results. And rework the missed parts. It gives you a teensy bit of time in between the attempts so that you get more value from them. Also, it forces you to address the entire set of homework without the crutch of reinforcement at each problem. But still, you get rapid reinforcement by seeing it that evening, vice days later. I think you will also find yourself starting to play the internal game of trying for 100% perfection on the section homework. Obviously when you start doing that, or close to it, you have corrected the issue in (3).