Is memorization a good skill to learn to master mathematics?
As you say, mathematics is a language. In any language, memorizing vocabulary very quickly becomes useless, if it is not paired with active use. Certainly, memorizing an entire Spanish dictionary before attempting to formulate one's first Spanish sentence is not the right way to learn Spanish. You should learn a few basic words, and practice using them in all sorts of combinations until you are comfortable with their properties; perhaps you must take a few things on faith from a fluent speaker. Then learn a few more words, plus maybe a grammar rule, and practice them intensively as well, using them in sentences together with the words you learned earlier, looking at correct and incorrect instances of that grammar rule. Each time, you expand more and more, bringing in more intricate aspects of the language, learning more vocabulary as a natural part of expanding your general facility with the language.
The most important part is that you practice using the language, expressing your own thoughts with it. Memorizing a dictionary will never teach you the language, nor will memorizing books or articles that use it. Reading with understanding is much better. Reading with understanding, together with using the language in conversation and writing as often as you can, is ideal. It is the best way to practice the vocabulary and grammar you've learned from a teacher or book and checking whether you really understand it. After all, consciously trying to memorize anything will only go so far; if you want to reach the point where you can use the language fluently, you have to practice it, so that each time you learn something new, it becomes ingrained, second-nature. This unconscious memorization is ultimately much more important.
In mathematics, we do not use our mouths, but our minds. Think of exercises from a book as conversation prompts. You should answer in the language of mathematics, as best you can, using the vocabulary and grammar you've learned so far. Note that knowing lots of vocabulary and grammar do not, in and of themselves, let you participate in even the most rudimentary conversations - you have to have something to say first, and you have to practice expressing it in the language. Those are not things one can practice passively. So, without a doubt, learning mathematics requires an immense amount of active thinking.
You might find this post helpful: How Do You Go About Learning Mathematics?
Here are some relevant quotes:
"What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises." - G. C. Lichtenberg
"The only way to learn mathematics is to do mathematics." - Paul Halmos
"Keep in mind that there are millions of theorems but only thousands of proofs, hundreds of proof blocks, and dozens of ideas. Unfortunately, no one has figured out how to transfer the ideas directly yet, so you have to extract them from complicated arguments by yourself." - Fedja Nazarov
"Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?" - Paul Halmos
There are roughly two kinds of memorization: with sufficient understanding and without. Without enough understanding, our memory tends to fade more quickly than we would like it to. So I would say: try to memorize with understanding as much as we can.
From chat by t.b. about "memorizing" theorems and their proofs:
Always ask: what are the hypotheses used for? How does this assumption enter the argument? What is the crucial point of the proof, what do I need to remember in order to re-prove that result? What is just standard technique, what is new to me?
Well, one could certainly ask for an outline of the proof, I believe. There are a few main ideas that one can try to isolate. I mean these six pages aren't six pages of pure calculation, they certainly are divided in some natural steps. Try to partition the proof in such a way that it looks natural. This takes a few hours to do, but if for some reason you know that this theorem is considered important for your exam, you probably need to really grasp these ideas.
If one proof doesn't help me a lot, I need a different angle of looking at things. If I find a place where things are presented the way I like it, I can then go back and see what the other author emphasizes and thus get a more complete picture.
When we are driven by the deadlines of homework and exams and don't have time to digest the ideas, we usually try to memorize them in a hurry. Courses are sometimes arranged in ways that they contradict their original purposes. That is sad...
I think it depends on what your priorities are and what you're trying to achieve. If it is indeed something resembling mastery of maths then my answer would be a resounding no to memorising in the conventional sense being the way to achieve this. If however your requirements are just recollection in the short term then this kind of memorisation is a tried and tested reliable method and perhaps the safest means.
I'd say there are different levels of understanding spanning from just being able to read the words, to following the logic and then to the highest levels where you reconcile with what the mathematician who originally came up with the proof might have been thinking and begin to consider how this could influence your future thoughts on these subjects. If you're under pressure to memorise proofs for a lot of theorems then a way which I think is effective is to break the proof down algorithmically and memorise the algorithm for the proof. Such an algorithm may include steps like "define a such and such set", "sum elements and apply some property". In this way when the time comes to recall it, you can at least do it in a way that still gives you the feel of "doing maths" to some capacity as opposed to just rewriting word for word. It will also help you to develop mathematical technique and you may be able to get through things quicker as there is less to memorise this way.