I'm a Y4 TA, and I say rote learning too.
I would find out which tables he's confident on, and then focus on the next one along (we teach in the order 10s, 5s, 2s, 3s, 4s, 8s, 6s, 7s, 9s, 11s, 12s, so if he was, for example, confident with his 10s, 5s and 2s, the next one to work on would be 3s, and nothing else until he's confident with them).
I "do" times tables with my class every morning for the first 20 minutes whilst the teacher does mental maths stuff, I see them individually to practice times tables - I write the numbers 1-12 on my whiteboard and then ask them in a random order - 2x6, 7x6, 11x6 ect, rubbing off the numbers when they've been able to quickly tell me that fact.
If there's one they're struggling on I remind them how they can figure it out from facts they know, like 9x6 is one lot of 6 less than 10x6, so 9x6=60-6, or reminding them that 6x7 is the same as 7x6, if they've just moved on from their 6s and are used to hearing it that way round. If there's a particular fact they're just not getting, I have stickers which say "ask me" on them, which I write the question on and they put on their jumpers, so all day everyone they see keeps asking them "what's 9x9?". I also have been known to ask them the same fact about ten times in a row to get it into their heads, which they find funny - "what's 9x9?" "81." "What's 9x9?" "81." "Hey, Jake?" "Yeah?" "What's 9x9?" "81."
They have to be able to answer my questions twice, and then they do a 46 fact challenge which is all the facts from the times table all four ways round - ie 7x6=, 6x7=, 42/6=, 42/7=. If they get over 40 in this (allowing for some silly mistakes/rushing/writing 12/4=48 because they misread it as 12x4) they move onto the next table.
When they move onto the next table they get a little bookmark with the times table on it, and a little note to parents on the back asking them to help practice them out of order, and also the related divison facts. The first time I see a child in the morning for a new times table, we talk about which facts they already know for this table, like in the 7s, they already know 10x, 5x, 2x, 3x, 4x, 8x and 6x7, from knowing those tables, so they now only have to learn 7x7, 9x7, 11x7 (which is easy) and 12x7.
It does seem to be working. This year group came up from KS1 with notoriously low data, and 5 in my class already know all their times tables, 2 more are on their 12s, and over half the class are on their 7s or above, which is where we reckon they should be at this point in the year to know all their times tables by the end of the year, with tables up to 4x supposed to have been taught in Y3, and learning a table every half term.
To help with knowing the divison facts, I keep on telling them that 56/8 is just asking you how many times 8 is 56.
Specific facts - as has already been mentioned 7x8=56 can be rearranged so that 56/7=8. 8x8=64 = I ate and I ate and I was sick on the floor - they don't forget that one in a hurry.
9s are great - I try to avoid teaching them the fingers trick as they tend to fall back on that instead of remembering them, but do show them that the tens digit of an answer in the 9x table will be one less that the number in the question, so 6x9 will be 50 something, 9x9 will be 80 something. Then you can figure out the ones, since the two digits in the answer always add up to nine (except 11x9, which adds up to 18, which adds up to nine). So this is a quick way to figure out one you don't know/have forgotten - 3x9 = twenty... (what adds to 2 to make 9?) ...seven.
(Sorry, this turned out longer than I had planned!)