What are the different methods of learning arithmetic called?

[Gretl]

And what are schools using these days, ie what is current thinking?

I ask because ds has been slogging away with pigging numberlines for ages and seems to count on his fingers - so if the sum is 7+9 he will first count out seven on his fingers and then he gets stuck, obviously.

It slightly concerns me that this isn't being rectified. (He's 6 btw)

SO yesterday I taught him that if you think of 7+10 it's really easy, then think 10-1=9 so take away 1 from 17 and you get the answer. Result. He came home and voluntarily did three pages of a maths book we bought him, with ease.

Then I panicked slightly as I'm sure I've messed up some sort of system they're using.....

That method is certainly one that they will be using at some point (10+7 then take away one), or similarly, to change 9+7 into 10+6 by adding one from the seven on to the nine first, then that leaves six more to add on. (Easier to explain with balance scales or counters or whatever). But they do a variety of different methods at different times.

The counting on method is also used, but children need to get to the point where they can realise that they can start at seven, and just use their fingers to count on nine more, which is not immediately obvious to them. Also, it takes a bit of multi-tasking for a child to be able to count "8, 9, 10, 11, ", whilst also trying to make sure he gets to nine fingers. Sometimes I have a child show what nine fingers looks like first, so they have a sort of visual impression, then do the counting start at 8 or whatever and know when to stop when their fingers look like they did before. It can take a lot of practice to get the idea, and so a lot of work with counters, number lines, and so on is usually done first, so that the child might eventually discover some of those methods naturally. If a child is only at the stage of needing to count to seven first, before he can add nine, then I would be working on methods that let him discover that he doesn't need to do that every time. (Like adding two on to various numbers, as that's usually quite easy to keep track of the fact that you've said/used fingers for two more numbers, whilst at the same time saying other numbers aloud). If the child is good at counting, starting with larger numbers can help, because they aren't tempted to count them on their fingers first (like 13 plus 2, easy to show two fingers as you say 14, 15).

Rather than teaching new methods at this stage, it might be better just to play around with lots of concrete materials, and let him discover new ones himself.

Thanks.
I'd have thought they'd be teaching him that you'd already have the 7 in your head and then go on to count out the 9, but they don't seem to have done so, and tbh he's been getting discouraged.
Maybe it's more a question of pace. He's the sort of kid who doesn't do slog (!) so if he doesn't get something near the beginning, he kind of stops trying. (NO idea how to remedy that, but we'd blinking well better try before he gets any older....)
Interesting that they do all sorts of methods - that makes me feel a bit better.