## Notes on Notation – Division

In the course of normal school, students encounter various notations for division, and boy, are there many.

We’ve got 15 / 5, and then there is also 15 ÷ 5 as well as $\frac{15}{5}$.  You may even have come across 15 : 5, though in the United States this doesn’t seem common and is often regarded as a notation for odds instead.

And then, when students are taught long division, they are shown a form in which the 5 comes first, 5 | 15 (usually with a horizontal line over the 15 as well).  This one is sometimes pronounced as “5 into 15”.  Students naturally think of this as yet another notation for division.

Will the real division please stand up?

I have not done a deep historical analysis of the notations, something I think would be appropriate here.  Short of that, I can fancy that $\frac{15}{5}$ is the older notation, and that ‘/’ and ‘÷’ found widespread use as horizontal alternatives, that is, versions that allow the entire division to be written on a single line.  If you look at 15 / 5 slightly slanted, you can imagine the 15 being on top and the 5 being on the bottom, and the two numbers being separated by a somewhat horizontal line..  Similarly, in 15 ÷ 5, you can imagine the horizontal division line being shown with two place holders for the top and the bottom, indicated with a dot each.  On the left, you see the value that goes in place of the placeholder on top, and on the right is the value that goes in place of the placeholder on the bottom.

All these multiple notations, do they do any harm in math class and beyond?  Most teachers observe that kids are confused between “15 divided by 5” and “5 into 15” and confuse their respective notations as well.  Typically, most kids attempt to keep them straight by relying on a rule such as “the big number goes on top”, or “the little number goes outside of the box”.  These rules, later in school, break down when kids encounter fractions or decimal divisions.

From what I’ve seen, kids are amazingly good at dealing with multiple notations for the same thing.  After all, these are kids who learned to talk when they were two or three years old, without any textbooks to guide them.  I have yet to encounter a student who confuses the horizontal line in a divison for a subtraction.  That’s pretty impressive, I’d say, when you think about it.

I contend that, instead, kids are so strong at keeping notational conventions straight that they often run into the opposite trap.  When kids are introduced to fractions, like $\frac{5}{15}$, they tend to treat it as an entirely different beast from division.  Fractions are in a different chapter of the book, the pictures in the book are suddenly about slices of pizzas, and we now have a big number on the bottom.  That the division “3 divided by 5”, written as 3/ 5,  is directly related to the fraction “three fifths” written as $\frac{3}{5}$ sometimes takes years for kids to get clear.  Isn’t that amazing?