The nice thing about math is that once you reach a certain level, self-assessment becomes extremely reliable. From that point on you can learn math on your own

The nice thing about math is that once you reach a certain level, self-assessment becomes extremely reliable. From that point on you can learn math on your own without worry! The key is to always have some concrete procedure available to check the results of what you studied. If you are solving a system of equations, the concrete procedure is to check if your answer actually solves those equations. If you are taking a derivative, the concrete procedure is to numerically compute the expression (f(x+h) - f(x))/h for a really small value of h, and check if it is approximately the same as your computed value of f'(x). If you are integrating a function, the concrete procedure is to take the derivative of your answer and see if you got the original function back, or to numerically compute a Riemann sum. If you are writing a proof... that is harder to verify on your own, but there are still ways to do it. You could think of a particular example and follow the steps of the proof in that particular situation and see if they work as advertised. You could try to rephrase the proof as a method for doing computations, and check that the method of doing computations gives correct results. You could attempt to transcribe the proof into a formal language, or even all the way to an interactive theorem prover. (Of course, someone who can successfully use an interactive theorem prover probably doesn't need advice on verifying proofs - they aren't very usable pieces of software yet.) When learning a new area of math, your first question should be: how do I concretely check whether basic claims people are making are true or not? For this you need to know a few examples that illustrate different behaviors that can occur, and you need to understand these examples like the back of your hand. You also need at least one reliable method of concretely computing with whatever objects are being studied. The moment you don't feel like the subject is grounded in something concrete anymore, stop, try to find some examples, look for another book, maybe just put the topic to the side and study something else.