This thread makes me really happy, since I'm a 3rd year math phd student myself
Just getting ready to take the generals in the summer, then only the dissertation remains
About memorization. A lot of people are echoing the sentiment, "rote memorization is bad". I believe this is a rare example of mathematical dogma. What started as a loose rule of thumb-- try to understand things, not just memorize them-- is often taken way overboard until memorization is demonized.
People need to put more faith in their subconscious minds. When you're enthusiastic about a subject, memorizing it will AUTOMATICALLY lead to understanding it.
Compare it to languages. If you try to learn, say, Japanese by carefully understanding every detail of the grammar, you'll never learn the language. Even if you already had all the vocabulary, consciously thinking about grammar is just too freakin slow when tongues are flapping and words are flying. The only way to learn the language is to get the grammar in your
subconscious mind through tons and tons of exposure and practice-- in other words, rote memorization.
I wrote an article about rote memorization in math:
Rote Memorization In Mathematics