When you teach children division, there's an issue of what $a \div b$ means. Here are two answers:
1) Take $a$ things and arrange them in $b$ groups of equal size. $a \div b$ is the number of things in each group.
2) $a \div b$ is the number of times that $b$ goes into $a$.
Answer 1) corresponds to the fact that $b \times (a \div b) = a$. (In other words, $b$ of $a \div b$ equals $a$.) Answer 2) corresponds to the fact that $(a \div b) \times b = a$ (In other words, $a \div b$ of $b$ equals $a$.) It's only because multiplication is commutative -- which I believe is not obvious, until you draw the right picture -- that 1) and 2) give you the same number.
Do you recommend explaining to children that division has these two interpretations, and that they always give you the same answer?