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