The Beauty of Algorithms
“Why do we have to know this?" This question is the bane of every math teacher's existence. It gets asked in math class more frequently than in any other subject area (Latin teachers, please form a queue if you wish to lodge a complaint against that claim). But many math teachers don't know how to the answer it.
I probably asked it as a child more frequently than I've been asked it in my career as a math teacher. I was a curious child, but I was also a bit of a pain. My elementary school career took place in the 1980s and I graduated high school in 1994, and my teachers all had the same built in excuse: "You're not always going to be able to use a calculator, you know? You need to know how to do this stuff by hand!"
I am unsure if my teachers thought that the resources used in calculator construction were rare, or perhaps that there would be some sort of apocalyptic event wherein all technology would be rendered useless and there would be high demand for brave men who could extract the square root of a number by hand. But the fact is calculators are more common today than anyone could have imagined. If you own a smart phone, you have access to a free graphing calculator, higher quality and easier to use than the $100 department-store version. If you don't own a phone, you can probably find a free scientific calculator with a company logo on it that was used as a promotional giveaway. A basic four-function calculator can be found at any of those stores where everything costs a dollar. Essentially, the opposite of what I was told as a child is true: You are, in every conceivable instance, able to use a calculator these days. And calculators are increasingly able to handle more and more advanced mathematical concepts, including long division, a concept everyone is familiar with.
The Long Division Example
Most of us remember learning "long division" sometime in elementary school. It is one of students' first exposures to what I call an "ugly algorithm." Long division has a lot of steps to remember and the format for writing those steps is really unlike anything else that students learn before that — or even after! The only algorithm with a similar format is the algorithm for extracting a number's square root. You will note that this particular skill is no longer taught in most schools and has been abandoned as a mathematical standard altogether. These days, long division really is a "stand-alone" algorithm. Because there are so many opportunities to make small computational errors, it's hard to learn, hard to remember, and hard to achieve success with. Long division truly is an "ugly algorithm" — on the surface.
Underneath the surface is a different story. Because this is not a "pro" vs. "con" article. I am very "pro" teaching long division in late elementary and on into early middle school. I believe the motivation for teaching this skill is critical, however. It would be better to skip this standard with kids altogether than to teach it for the wrong reasons. It can do more long-term harm than good.
The algorithm on its own is at best a utilitarian model of obtaining answers to division problems. But the beauty that is revealed within the steps — what it reveals to us about our number system, how we utilize place value, how our understanding of numbers developed over time, and more — is worth the time it takes to learn the algorithm.
So with this is mind, here is one bad reason - and a few good ones - to teach things like long division.
One Bad Motivation For Teaching Long Division
Because kids need to know how to do long division by hand to survive as an adult.
This simply isn't true, and what was a mildly far-fetched scenario back in the 1980s is now ludicrous to the point of ridicule. From now until the end of civilization, anyone with the means to receive a formal education will have access to a device that can perform division. And this device will do so far more efficiently than any pen and paper method.
That's it. That's the only reason not to teach long division. Unfortunately, that is probably the main reason that it is still taught. Its only possible rival is because of its inclusion on the Common Core State Standards (CCSS.MATH.CONTENT.6.NS.B.2). I would bet, though, that most teachers who claim CCSS as their motivation for teaching long division feel that the reason it is included is for the same reason mentioned above — because kids just need to know how to do long division. And truly? They don't. At all.
Instead of debating the merits of the long division algorithm, I'd rather spend my time imagining how even the stalest of algorithms could be used to instill awe and wonder? What if they could be tools to seek beauty and truth in this world?
Good Motivations For Teaching Long Division
The algorithm provides learning opportunities
Quick, without looking it up: Why does the algorithm for long division work? Let's say I asked you to divide by hand right now. How many of the following questions about this example of long division could you answer easily?
- The first "2" you write at the top. Why? 2 of what?
- Why do you then multiply 2 by 27 to get 54? What is that multiplication accomplishing?
- Why would you subtract 54 from 74? Why don't I add or do something else there (or every time)?
- How can 20 just "become" 206 when I "drop down" that 6? What does "drop down" even mean? Why don't I "drop down" more than just that 6?
- Why can I just keep adding zeroes to this number after the decimal point? When should I stop adding zeroes? Why do I sometimes not have to add any zeroes?
- What's with that 8 at the bottom? 8 whats?
You can probably answer some of these questions, and some you probably can’t. I would argue that not only should all teachers know the answers to all of those questions, but students should too. The "trick" of long division is valuable in that it opens up so many other learning opportunities. Opportunities for learning place value. Opportunities for learning about factors and multiples. Opportunities for learning about different types of numbers (fractions, decimals, their relationships, etc). Opportunities for learning about Rational numbers (how do you know you will eventually find a pattern or a stopping point in that decimal?)
Simply knowing the algorithm for long division isn't enough. I want students to understand why it works, why the steps they take are useful, what category (or place value) each number belongs in, and what information they are communicating every time they make a pencil stroke.
The algorithm is our cultural heritage
It's pretty much a given that if you were schooled in North America, you learned the algorithm included above. Have you ever wondered why that is the case? The division question is simply "How many 27's fit into 7460?", but we would all attack it the same way. Why were we all taught the same way to approach that question?
There's a certain elegance to the fact that this is the agreed-upon algorithm for answering difficult division problems. Few students would describe it as "user friendly." Can you imagine what was rejected if this is the algorithm that was accepted? What did the different iterations of this development look like? Did people argue about it?
These would be great questions to discuss with students. They could research other methods for division. They could debate pros and cons of each method. This research would give them a connection to the history of mathematics and the story of how humans came to understand and work with numbers.
Truth, Beauty and Goodness
Students who pursue an understanding of the algorithm and its place in our shared cultural heritage will have no choice but to see something that transcends even mathematics. They'll see the beauty of our place value structure and how it can be utilized effectively. They'll see that there are so many ways to represent numbers, but that they all point to the same underlying system that's at work in our brilliantly created world. Maybe they'll fall a little bit in love with numbers and the breathtakingly exquisite system we've developed to represent our understanding of them. And ideally they'll start to realize that they are free to create their own systems to help understand the world around them. Because this is the real work of mathematics: Finding new ways of describing the beautiful creation that we experience every day.
Once students have grasped the beauty and goodness of the long division algorithm, once they set out on the real work of mathematics, then let them use a calculator. Imagine what they can accomplish with all that extra time.
by Cheryl Swope
by Angelina Stanford
by David Kern
by David Kern
by David Kern