Seeing a lot of criticisms back and forth, how these tricks aren't really learning it, how anyone can and should be able to memorize their times tables if they just apply themselves, etc. While it may be human nature to universalize our own experiences, it's true that some people really do have extraordinary difficulty with tasks that are simple for most, such as memorizing math facts, and for those, having any hook or peg to hang the thought on, to attach it to something else, whether a mnemonic, or this finger method, can be helpful. So what? If a person who could never remember what 6X9 was through more traditional methods, was able to do it with this one, that hurts no one, and helps the person it helps.

As for it not being an authentic representation of what 6X9 is in terms of arrays, well...that depends. If you closely examine the 9 method on fingers, you will see it does indeed represent something real going on with multiplication of 9s. Even if you draw arrays, you see the same pattern that 9 is always one less than 10, so that however many 9s you have, you have that many more ones, fewer. 9X3 is 3 instances of something being 1 less than 10, so it's only 1 less than 10 the first 10, then the next 10, it's 1 less again, for a total of two less (18...2 less than 20!) and then for each iteration, take again another "one less than 10" cumulatively (27...3 less than 30). I see the same pattern evident in the finger method for the 9s, at least.

In mathematics, there can be many ways to solve the same problem, and still be correct in the end, and all are valid. In fact, the more comfortable you are with finding more than one way to solve a problem, the stronger your overall understanding is.