The elementary school example of division, say 12:4 is saying that you have to share out 12 cookies to 4 kids. However, another (only slightly less intuitive) way would be to ask how many times 4 "fits" into 12. I am not a native English speaker but I guess that's the intuition behind the (mostly American ?) expression and notation of dividing "4 into 12".
My question is the following. What would be an intuitive way to explain why these two intuitions of division are in fact equivalent?
You hand out 12 cookies to four kids. The cookies are not chocolate chip cookies, as they first appear to be -- they are oatmeal raisin cookies. The kids throw the cookies back into the jar. How many cookies does each kid throw back into the jar?