From Saskatchewan, here's a page of "mental math strategies" - reasoning to help make elementary mathematics seem sensible to the student.
It is important that most students have mastery of basic facts. It is equally important that they make sense of number combinations as they are learning these facts. Here are some strategies to help with this understanding.
There are rules for add 0, 1, 2; the commutative property, aka "turn-around facts"; adding 10, 9, 8; doubling numbers, and "near doubles" (e.g 4+5). With all of those techniques firmly in hand, the addition grid that I had to memorize has only a handful of holes in it that you can't figure out.
What I remember being awesome for learning math facts was competitive dice games - Yahtzee in particular.