Long Division

The other day I saw this TikTok by the inimitable Howie Hua:

@howie_hua

Long division CAN be understandable! #math #maths #mathtok #mathematics #mathtrick #teachersoftiktok #mathteacher

♬ original sound – Howie Hua

This is a common topic: What is the long division algorithm about anyway? I’ve likely even seen Hua talking about it in the past, but this time, something clicked in my mind.

I don’t personally recall struggling with long division. I suppose there was a time in early grades where it was a struggle, but for as long as I can remember, it’s made perfect sense to me. Even polynomial division follows so naturally for me that I’ve struggled with how to properly teach it to my students in a way that makes sense to them (I’ve mostly settled on synthetic division, which is even more of a mystical “use it because it works, don’t ask questions” algorithm).

I think I know a big part of why long division makes sense to me, as an older person; it has to do with how it’s taught now, versus how it was taught back in my day. I’ll get back to that later.

Before we get there, though, what is division in the first place?

Quotative and Partitive

There are two basic ways to see division, just as there are two basic components to multiplication. With multiplication, we tend to gloss over that distinction because it’s irrelevant to the actual operation of multiplication, but with division, it’s more important if we want to understand what’s really going on.

Consider \(3 \times 5 = 15\). We can see this as “three groups of five” or “five groups of three”. Either way, the final count doesn’t matter: If we have three groups of five books, or five groups of three books, the total number of books is the same. There are certainly real world contexts where the distinction matters, but on one key level mathematics is about stepping away from real world contexts and functioning in a world of abstracts.

There are purists who would insist that \(3 \times 5\) must be \(3 + 3 + 3 + 3 + 3\) or must be \(5 + 5 + 5\), but operationally speaking, it’s not that important.

Now consider \(15 \div 3\). There are two basic ways to see this: We have fifteen items we want to distribute into three equal piles, or we have fifteen items and we want to know how many piles of three we can create. The first concept is called “partitive” and the second “quotative”.

To divide partitively, we create three buckets and then drop one item in each bucket cyclically until we’re out of items. If, at the beginning of a cycle, we have fewer items to distribute than we do buckets, we stop and declare this the remainder.

For instance, with \(15 \div 3\), we create partitions like this, where the leftmost stack is what we’re starting with: \[\begin{array}{|c|c|c|c|}\hline 15 & 0 & 0 & 0 \\ \hline 12 & 1 & 1 & 1 \\ \hline 9 & 2 & 2 & 2 \\ \hline 6 & 3 & 3 & 3 \\ \hline 3 & 4 & 4 & 4 \\ \hline 0 & 5 & 5 & 5 \\ \hline \end{array} \]

Quotative division relies on the notion of division as repeated subtraction. In this case, we repeatedly remove the target number of items until we don’t have enough in the original amount to fill a bucket: \[\begin{array}{|c|c|c|c|c|c|}\hline 15 & & & & & \\ \hline 12 & 3 & & & & \\ \hline 9 & 3 & 3 & & & \\ \hline 6 & 3 & 3 & 3 & & \\ \hline 3 & 3 & 3 & 3 & 3 & \\ \hline 0 & 3 & 3 & 3 & 3 & 3 \\ \hline \end{array} \]

Conceptually, it feels like we tend to focus (albeit not exclusively) on partitive division these days, but both concepts are important for a full understanding of division. And, in my opinion, the long division algorithm is easier to understand from a quotative perspective.

Long Division: Then and Now

Here’s an example of how I’m used to seeing long division performed. The question is to evaluate \(2,654 \div 7\):

This is a screenshot of page 118 of Holt School Mathematics Book 5, from 1974. In isolation, this may look confusing. What’s going on? Where does 55 come from? What about 64? And why does this even work?

But on the previous page, there’s a comparison diagram:

This illustrates that the current standard version is a shortcut: First we find out how many 30s are in 225 (7), then we find out how many 3s are in the remainder of 15.

Here’s a more explicit example, from Holt School Mathematics Book 5, from 1980, pages 104-105:

This shows three levels of explicitness in the detail. The form with the arrows and the missing zeroes is presented basically as a shortcut.

Compare this to 1995’s Addison-Wesley Mathematics, Grade 5 (p. 186):

and 2015’s enVision Math Common Core, Grade 5 (2015):

Admittedly, I haven’t scoured those later curricula to see if they offer the scaffolded versions in earlier grades, but given what seems to be a significant uptick in people not understanding long division (including teachers), I wonder how much this plays a role. (All screenshots were taken from the Internet Archive.)

How It Works

Even what Holt calls the long form of long division is itself a shortcut for quotative division. Consider \(2653 \div 7\). We could just subtract 7 repeatedly from 2653 and count how many times we’ve done so, but that’s a lot of work. We know that 2100 is a multiple of 7, because 21 is a multiple of 7, so why not just start there?

So \(2653 – 2100 = 553\), and \(2100 = 7 \times 300\). Now we’re left with a more manageable 553, but even then, we know that 490 is a multiple of 7, because 49 is a multiple of 7, so we can use \(553 – 490 = 63\) and \(490 = 7 \times 70\). Finally, we have \(63 = 7 \times 9\).

That is, \(2653 = 7 \times (300 + 70 + 9)\), so \(2653 \div 7 = 379\).

This is idea behind long division: Repeatedly remove the largest powers of ten you can, then move to the next lower power, and so on, until you have a remainder that can’t be distributed at the level of precision you want.

But if we don’t teach the part with the zeroes first, the algorithm looks mysterious.

What About Multiplication?

While there is some confusion about long multiplication, it’s not nearly as deep. But, unlike with division, we still teach writing the zeroes. Here’s 1974’s Holt Book 5 again (p. 90):

1995’s Addison-Wesley, Grade 5 (p. 124):

and 2015’s enVision, Grade 5 (p. 71):

With multiplication, we call this method “partial products”; with division, it’s the same basic method, but we call it long division. In fact, when I Googled “partial quotient”, the first hit I got was this video, which explicitly distinguishes between “partial quotients” and “the standard method”:

And that standard method? It involves going “digit by digit” in a method that, if I didn’t already personally understand long division, would confuse the heck out of me:

Meanwhile, the same instructor writes place-holder zeroes with multiplication (as is the common convention):

Conclusion

It seems to me: When we wrote the zeroes with the partial quotients in early grades scaffolding, there was less confusion about the algorithm. We don’t need those zeroes as we mature in our understanding of the process, but if students are having far more trouble with long division than with “partial product” multiplication, I wonder how much of that is because of the shift in pedagogical strategies regarding division.

Also, since division is the inverse of multiplication, I’m not sure why we wouldn’t teach them as such: Why would we drop the place-holder zeroes with division, but keep them with multiplication? I feel like the ideal is to teach both concepts with those zeroes, and then teach students with deeper understanding that they can leave them off if desired.