The question you opened
Four cups of oats, half a cup each
A recipe uses half a cup of oats for one bowl. You have 4 cups of oats.
How many bowls can you make?
8 bowls
Every cup fills two bowls, because two halves make one cup. Four cups therefore fill eight bowls.
You have 4 cups of oats and each bowl takes half a cup. You can make 8 bowls. The quick way to see it is to forget the word divide and picture the cups. Pour out one cup. It fills two bowls, because two halves make a whole. Now do that four times. Four cups, two bowls each, eight bowls in total. The answer that catches people is 2, and it comes from halving the four instead of counting halves inside it.
Halving and dividing by a half sound almost the same when spoken quickly, and they do opposite things. Halving 4 gives 2. Dividing 4 by a half gives 8. Say the question in words and the confusion disappears. How many halves are there in 4. Answer: eight of them. Dividing always asks how many of the small thing fit inside the big thing, and when the small thing is less than 1, more of them fit, so the answer gets bigger.
That last sentence is the rule worth keeping. Divide by a number smaller than 1 and the result grows. Divide 6 by a third and you get 18, because there are three thirds in every one. Divide 10 by a quarter and you get 40. It shows up in the kitchen constantly. How many quarter cup scoops in 3 cups: twelve. How many 200 gram portions in 1 kilogram: five.
How many half hour lessons fit in 3 hours: six. Each one is a division by something smaller than the whole, and each one gives a bigger number than the one you started with. If the wording ever traps you, swap in a picture. Cups on a table, scoops going in, bowls filling up. Counting the bowls gives the same answer as the arithmetic and never feels backwards.
Technique: Dividing by a half doubles the count