So Long, Long Division
With
the possible exception of fractions, no elementary math topic or skill
stresses students out more than long division does. And it's so
unnecessary--not just the stress, but the skill. Think about it. How
often do you use long division? For that matter, how often do
mathematicians use it?When I observe classes where students are working on long division, there are always a few kids who have the routine down and get one correct answer after another. No stress for these students because they're feeling successful. But ask them what they're doing, and they're likely to refer to a mnemonic like "dad, mom, sister, brother" (divide, multiply, subtract, bring down). Ask them how they knew where to place the two numbers, and they'll tell you "the bigger one always goes inside the house." These students can do division, but do they understand division? And how's their number sense?
Then there are those students who not only don't get division but also can't do it. Some of these students avoid long division problems by disrupting class, going to the bathroom, or putting their heads down. But other students try and try and try. And they fail and fail and fail. On the one hand, it's great that they're persistent, since we want students to engage in productive struggle. We also want them to make sense of problems and persevere in solving them, per Practice Standard #1 of the Common Core State Standards (see my last post, Engaging Math Students in Productive Struggle).
Long division, however, isn't a problem-solving exercise for students. It's a procedural exercise. And if a procedure doesn't make sense for them or they lack the skill to be successful with it, they're not engaged in productive struggle. They're engaged in unproductive suffering.
I'm reminded of a student who was grimacing as he stared at a problem for a few minutes, pencil in hand but nothing on his paper. I approached him and noticed the problem he was stressing over was 700 ÷ 20. After a few words of comfort, I said what I often say to math students: "please put your pencil (or pen) down, and look at me." I then asked him, "What's 100 divided by 20?"
"Five," he said without hesitation, and then picked up his pencil and wrote seven 5s in a column on his paper. He then counted by fives, and wrote 35 as his answer. And he was grinning rather than grimacing.
So, who understands division better: the arithmetically challenged student in this example or a student who gets the right answers with the aid of a mnemonic? Of course kids need to master division in order to be successful in math. But instruction that focuses on procedures helps some students do math, while helping few students know math.
A better approach is to advise students to put their pens and pencils down (calculators too), and estimate quotients before trying to compute them. Estimation before computation is an important first step in the problem solving process for a few reasons: it compels students to read and think about what's being asked in a problem; it helps students develop number sense; and it gives students ballpark answers to compare their precise answers with, prompting them to reconcile any significant differences. What's more, estimation is a far more practical skill--for math and life--than paper and pencil computation.
After students have estimated, ask them to keep their pencils down and identify the most efficient strategy for a given problem. Encourage them to use mental math or play around with the divisor or dividend to make a problem more manageable, as in the example above where I asked the student, "What's 100 divided by 20?" Pushing students to think before resorting to a procedural approach (or reaching for a calculator) helps deepen their conceptual understanding. It also creates opportunities for them to learn other math concepts/skills. (I just worked with another arithmetically challenged student who discovered the distributive property after breaking a dividend up into the sum of two numbers.)
What do you think? Am I missing something here? Do you have a good reason for students to learn long division that I'm overlooking? If not, please join me in saying "so long" to long division.
