# Teaching the meaning of division to children

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?

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There is at least one more interpretation, that division is the inverse operation of multiplication, or that division is the same thing as multiplying by the inverse. Of course that last one works best after the student has learned fractions. I like to keep bringing up as many interpretations of any concept until it seems like the student has found the one that makes the most sense to them. We want them to "own" it. If you're working with a group, you have to balance flexibility versus confusion and you could try to go on group feedback. –  Todd Wilcox Nov 3 '12 at 17:19
Also, shouldn't the commutative nature of multiplication be actually obvious to any student who knows their times tables? I would want a student to know their times tables before doing division. More important would be to emphasis that division, like subtraction, is NOT commutative. Which brings up an alternate interpretation of division - adding:multiplying::subtracting:dividing. –  Todd Wilcox Nov 3 '12 at 17:22
Thanks for your reply. Why do you say that knowing the times tables makes it obvious that multiplication is commutative? Yes they can observe that multiplication appears to be commutative, and they can conjecture that multiplication is commutative. But I don't think it's obvious that, say, $17$ groups of $19$ things is the same amount as $19$ groups of $17$ things. At least, it's not obvious until you draw the right picture. –  littleO Nov 3 '12 at 17:28
(The picture I'm referring to is just a $17$ by $19$ array of dots. Column by column you have $19$ groups of $17$ dots. Alternatively, row by row you have $17$ groups of $19$ dots. I don't think drawing this picture is that obvious before you've seen it. But we are all so used to the commutativity of multiplication, we forget that it's surprising.) –  littleO Nov 3 '12 at 17:29
From the point of view of an eight-year-old, knowing that every entry in the times table for 1-9 (or 1-12) has an answer equal to another entry with the order reversed (except for the squares), would seem to be "proof" that multiplication is always commutative. For students taught that multiplication is an extension of addition, that impression would be reinforced both intuitively and mathematically. I wouldn't expect a child of that age to require a rigorous "adult" proof of that to assume it's always true. Just my take. –  Todd Wilcox Nov 3 '12 at 17:58