I have been thinking, is it possible, or a good thing to do when learning from a textbook without ever writing anything down. But have to verbally give out the solution from beginning to the end with explaination of how the solution works and just records the solution in audio files. Is this a good way to do math? Because writing down is so slow I think this could be an alternative. Should I try this? I aware that this could be harder to find mistakes in solutions after it is done though. But when doing the recording, I think when you get stuck you will easily points out where you are not good enough.
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4$\begingroup$ Personally, I can talk myself into all sorts of nonsense. The conclusions of which are usually correct. However, the chain of reasoning is often full of nonsense which I only confront as I write things on paper and wrestle with details which are hard to appreciate without writing. That said, I have noticed many of the best mathematicians are able to communicate much without so much as an equation. To me, the ability to do such effectively is the mark of a truly exceptional mathematician. So, I guess your practice would probably be a device to impress the likes of me. $\endgroup$– James S. CookSep 14, 2018 at 0:22
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Mathematics has a highly optimized writing style. It is like shorthand, except that there are widely agreed standards for how to read and write it. This means that hand-writing numbers, formulas, and equations can be faster than saying them.
Written mathematics is much easier to check than verbal mathematics. Most mathematical transformations have symmetries and near-symmetries. With a few exceptions, it is easier to notice mistakes when the transformations are written out, rather than when they are spoken aloud.
The exceptions are things like negative signs, and the difference between exponentiation and multiplication.
If you want to convince someone of a mathematical argument, it helps if they (think they) can check it. Since it is much easier to check a mathematical argument that is written in standard notation, most people are suspicious of anyone who is not willing write out "the steps" of any mathematical argument that requires more than a couple of simple arithmetic steps.
It is possible to perform mathematics verbally, or "in one's head". This is useful for practicing math problems, and for becoming faster at doing math. But it has the same risk as any other form of practice -- if you practice the wrong thing, you will reinforce bad habits. It is also useful for impressing people, if you can come up with the right answer "in your head" faster than they can compute it using tools. But there are better ways to win friends and influence people.
Because it hard to keep track of many details "in one's head", performing mathematics verbally is best done with relatively simple problems. For example, first-order approximations and second-order approximations.
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1$\begingroup$ Mental mathematics need not be verbal, e.g. I can do blackboard computation on a virtual mental blackboard image much faster than I can write/speak such. If you train yourself to do such then it is just as accurate as writing and much faster. This is best done at early ages when the mind is still somewhat plastic so you can repurpose visual capabilities (as chess masters do for chessboard patterns, e.g. see de Groot's book: Thought and choice in chess. $\endgroup$ Sep 14, 2018 at 15:54