Visualization, semantic encoding, memory castles, etc. are great for memorizing sequential primarily disconnected information. They are effective for that purpose. Ideally math isn't disconnected. Math works best integrated. Numerical patterns work extra math skills and their usefulness reoccur outside of just say solving timed times tables. Discovering that adding the digits of a number divisible by nine gives a number that is also divisible by nine has more uses than the image of a basketball star surfing on a pizza box.
One of the old MOEMs problems that my son was looking at was:
What is the remainder of 111,111 divided by 27.
If the go-to skill is memorized multiplication tables, the path to the answer may be slow grinding. With some of the patterns a kid may discover in exploring multiplication, this can be much quicker. And these skills continually to build and diversify with more multiplication problems.
With either path the end result is automaticity. On one of the paths there are also tons of other patterns, methods, and insights earned.
