I really struggle with arithmetic and times tables. Don't get me wrong, I learned everything I was taught, and did okay on school tests, but I retain almost nothing now. Why is that? The problem actually stems from poor teaching, at least as much as it does from my declining memory.
You see, when learning my multiplication tables, I usually had to learn it piece by piece. So first we'd do the 1 times table then the 2 times table, and so on. Just like we have pictured in this image below. Did you have the same experience?
Unfortunately, the above image is not a 'table' in any reasonable sense of the word. What the times table actually is, is what you see below. It is also known as the 'table of Pythagoras', which should give you some indication that this is no recent invention, but it's actually at the very route and beginning of the history of mathematical education. Teachers, like Pythagoras, actually used this table thousands of years ago.
As you can see, the above table does not contain any wasted space or information. Yet it is perfectly intelligible. Furthermore, there are no ready made answers here, rather we have to think for ourselves a little to do the multiplication and spot the patterns.
It's been noted that students would each have had a copy of this table at hand during their school days, which they could consult. However, they eventually found they had memorized the information, as if by accident.
Because the table is arranged so neatly, there are many patterns that a discerning student naturally notices, since the human mind is wired to find similarities and patterns. As you can see above, the data is symmetrical - the numbers are the same on both sides of the imaginary diagonal line. This simple truth is just not visible on the 'wrong' table we showed further up.
When we spend any amount of time focusing and meditating on these numbers, we start to notice plenty of little patterns all by ourselves. The beauty of this is that the brain is far more impressed by patterns it notices itself than those which it is merely taught.
Another way of saying this is that we learn better through experience than through instruction. Instruction is important, but stronger memories are made by trying things out ourselves.
It would be very interesting to find out whether the schools in your local area use these real Pythagorean multiplication tables, because I know my school didn't. In my opinion, the kind of learning that I was subjected to was not very interesting at all, but I can really picture being excited at learning with the real times table!