Do you remember that episode of Friends where Ross, Chandler and Rachel try to move a couch? They reach a point on the landing of the stairs where the couch doesn’t fit, and Ross keeps yelling for Chandler and Rachel to “pivot.” The more Ross yells the same word over and over, the more we laugh, mostly because we know he’s not improving the situation. That is exactly what we are doing when we focus our division instruction on just the long division steps.
Knowing Long Division Isn’t the Same as Understanding Division
It’s not uncommon for teachers to double-down on mnemonics when teaching the long division algorithm. Whether it’s “Daddy, Mother, Sister, Brother” or “Does McDonald’s Sell Burgers?” the internet is full of similar phrases to aid memorization. Each letter stands for one of the long division steps: divide, multiply, subtract and bring down.
Knowing the names of the steps can be helpful; I’ve even used them myself with my fourth graders. The problem is that we can easily become Ross yelling, “Pivot!” over and over. We get so focused on the procedure that we lose sight of the important part: the concept of division itself.
Save the Mnemonics for Facts
Mnemonics support the rote memorization of facts. They are great for remembering the names of planets or the order of the colors of the rainbow.
However, the development of a concept like division can’t be reduced to steps. Sure, students can get the right answer to 103 ÷ 4 using long division, but do they understand that they are breaking up 103 into 25 groups of 4 with 3 left over?
Word problems often highlight this gap in understanding. Do students know how to reason out how many whole dollars they can make out of 103 quarters? Do they grasp the meaning of the remaining 3 quarters? Give your students a similar word problem. If they aren’t sure what they are being asked to do or if they go straight for the long division algorithm but can’t explain their answer, they have missed a huge concept.
What Long Division Steps Don’t Teach
The problem is mostly with the long division steps themselves. The first step to long division is to “divide,” which is unhelpful as far as steps go, and things only get wackier from there.
Take 103 ÷ 4. We tell students to divide 10 by 4, but the one and the zero in 103 don’t represent 10. They represent one hundred and zero tens. The long division steps are efficient but they don’t require students to make sense of the numbers they are dividing.
The “multiply,” “subtract” and “bring down” steps are similarly confusing. And the placement of the numbers that make up the quotient at the top aren’t much help either. I find students saying things like, “You just put a zero here” without much explanation as to why.
Teaching the long division steps is like programming a computer. The computer does what you tell it to do, and can even get the right answer, but it can’t explain why its answer makes sense.
Change the Way Students Learn to Divide
Instead of focusing on “Daddy, Mother, Sister, Brother,” the alternative is to have students work with smaller numbers and reason out the quotient using word problems and manipulatives.
Take 103 ÷ 4 as an example. When presented as just digits and a division symbol, students aren’t given any context to bring their own reasoning into the situation. They aren’t being engaged enough to make sense of division for themselves.
Instead, present 103 ÷ 4 as a real-world situation, such as the number of candies each of four siblings would get if they shared a bag of 103 Skittles. Or ask how many dollars you would have if you had 103 quarters. They are the same division problem, but the different situations force students to get down to the business of actually dividing.
Without a series of steps to follow, students will come up with their own strategies. For example, a student might make four circles, each representing a sibling, and “distribute” 103 Skittles by making a mark for each one. As their teacher, you might suggest that giving out 10 at a time would be more efficient or have students share their methods with one another. This way, they develop an understanding of division without steps.
How do you move beyond the long division steps in your classroom? Share how you teach division without the mnemonics.