I'd learned The Monkey and the Mirror, and then started playing it from memory, first with the music as an aide memoir, and then looking down at the keys. This was a revelation - the way ahead is to memorize these simple pieces, and just carry on with that as I get onto more complicated ones.
I played around the octaves with it, putting the hands opposite end of the keyboard, then one around middle C, and the other elsewhere. And then I did the baby version of the first bar of We Three Kings, (that link's to a slightly more sophisticated version), and realised I could do all of it's five notes in one hand, and that was great fun, getting the velocity right, and trying to keep the fingers flat on the keys.
There's another app we've got, PianoMeter, which looks interesting - purely for music-reading purposes: probably that's a bit down the road.