Schoolwide News

I want to talk with you today about long division, which is a confusing topic for adults and children alike. Part of this challenge comes from the fact that division can mean two different things, and part of it comes from the fact that while different cultures seem to have very similar-looking standard algorithms for addition, subtraction and multiplication, the standard algorithm for division varies significantly depending on where you learned it.  

If I start with 12 ÷ 4 = 3, it could mean two different things:

1. If I divide 12 into 4 groups, how many in each group? (Partitive division)
2. If I divide 12 into groups of 4, how many groups do I have? (Quotitive division)

When we see a bare calculation like 12 ÷ 4, with no story attached, most of us naturally picture one of these two meanings, but not necessarily the same one as our friends, family members or classmates. That can lead to real misunderstandings when we talk about splitting quantities into equal groups versus groups of equal size.

The second reason division is challenging for adults to talk about with their kids is the different standard algorithms that we've learned.  Each of these three standard algorithms for division was shown to me by students I have taught in Switzerland, Kazakhstan and Brazil.  

I've spent time working out how the steps of each one line up with the others, but it isn't always easy to see. I once had a parent tell me that I was teaching their child to divide “upside-down”, so they would never learn to do it properly.  Although there are many different ways to set up a standard algorithm, they are all doing the same thing: breaking the number into hundreds, tens and ones, sharing out each place value in turn, and trading any leftovers for smaller pieces.  Here is a breakdown of what a standard algorithm is doing when solving this problem:

447 ÷ 3

First, we split 447 into its place values: 400 + 40 + 7.  We don't do this in the algorithm, but when we first see how many times “three goes into 4”, we are actually saying, “when we split 400 into 3 groups, how many in each group?"

In the model below, the green squares each represent 100.  The blue bars each represent 10, and each small orange square is a unit, or 1.  We are walking through dividing each colored block into 3 groups, and regrouping the leftovers into the smaller place value. 

Splitting up the hundreds first, we see that we have 100 in each of the three groups, and 147 left over.  Since we can't split the last 100 into 3 groups of 100, we split it up into 10 groups of 10.  Added to our four groups of 10, this gives us 14 groups of 10 to split up into our 3 groups:

We can then divide up the 10's into our three groups, and add these to our 3 groups of 100:

This shows us that we now have 140 in each group with 27 left over.  Since the two groups of 10 (blue bars) that are left don't divide evenly into three groups, we split them into 20 units, then divide these 27 units evenly into the three groups, giving us the outcome of 149 in each of our three groups:

So we can see that when we divide 447 into three groups, we get 149 in each group. 

Now look back at the three algorithms. The third one shows these steps almost exactly: subtracting 300 is giving each group 100, subtracting 120 is giving each group 40, and subtracting 27 is giving each group the last 9. The second does the very same thing in shorthand: the 3 it subtracts is really 300, the 14 is 14 tens, and the 27 is 27 ones. And the tiny 1 and 2 in the first algorithm are the leftover hundred and the leftover two tens being traded down to the next place, just like the carried 1 in addition, which, as I wrote last month, isn't really a 1 at all.

Sharing the different ways to divide with each other celebrates the diversity we have at our school, honors the thinking of all of our students, and allows us to meet each student where they're at. There are so many ways to wrap our heads around mathematics, and sense-making is deeply personal. Allowing our students to use the method that makes the most sense to them can build genuine mathematical confidence and deepen their conceptual understanding. When students see that their intuitive strategies are valued alongside traditional algorithms, math transforms from a rigid set of rules into a flexible, empowering tool for problem-solving.