Back in the *good* old days, before reform (and the associated strife) had muddied the waters, memorizing certain things was the *only* way that a practitioner of mathematics could practice certain things. (Please--no jokes about how long one has to practice before one is permitted to perform.) Unless, I suppose, said practitioner carried around multiple resources in the quaint, but bulky, form of books. Even with books as resources, the practitioner usually needed to have acquired a certain level of facility with the techniques most of us memorized.
Chanting, and similar strategies, are really quite effective. I used to teach the quotient rule for derivatives by telling students to write it out on a 3-by-5, carry that card around with them, and pull it out and read it *out loud* every time they remembered that they had the card with them. (Well, I told them that they were allowed to mumble it under their breaths if they were in a situation, like, say, church, where it wasn't socially acceptable to say it out loud.) If they do this, they find that they aren't really reading the card after a few days; it's become automatic. From then on, when one wants the derivative of a quotient, one need only ask one's Mouth what do to. The Mouth will reply with "The derivative of a quotient is..." perfectly, and all that one need do is listen carefully to the instructions it gives.
Lest Ralph complain, I hasten to add that I do not consider such memorization "doing" mathematics. Lest Mike complain, I hasten to add that I don't know whether or not I am describing present necessity.