Wednesday, November 21, 2018

Conceptual Approaches to Teaching Division

When observing a fifth grade teacher the other day, I noticed that while the students were engaged in many positive ways, they relied upon DMSB -- Daughter, Mother, Sister, Brother -- or some other mnemonic to remember the procedure for long division. They were also ignoring the value of the unit they were dividing. This is very common, but there are better ways, as Singapore Math shows us! 

Why avoid mnemonics like these to help divide? 

          ●  They lead to place value errors, because all digits are treated the same. 

          ●  It’s no easier to remember these than an actual procedure. 

          ●  Students can remember them in the wrong order. 

          ●  They reinforce the idea that math is “magic,” 
following rules that no mortal can understand. 

How can division be taught conceptually?

There are multiple approaches that can lead to the understanding of division. In early grades, students take a number of objects and split it into equal groups. A counter such as the unit cube in Base Ten blocks can be used, and the Matholia tool can help with this.

Work with arrays and place value disks can reinforce the idea of division as repeated subtraction as well. So can using bar models, as illustrated in the article Bar Modelling For Two Types of Division.

Some math approaches undercut conceptual strategies by always representing the problem horizontally or in division’s case, using a division bracket. Starting by representing the problem horizontally allows more flexibility in thinking and does not guide the student into immediately jumping into an algorithm. The problem can always be rewritten in algorithm form if desired.
Once the students reach a solid understanding of what is happening in division, the division can be taught in multiple ways. What is commonly considered  the  division algorithm is just one of several, and not even the most efficient. Two algorithms that can be used to scaffold the understanding of the common long division algorithm are:

Partial Quotient Division -- layout 1:
In this style, students use whatever math facts they know to divide the numbers. They then continue to subtract and perhaps repeat the same fact multiple times in order to reach their goal. Then the partial quotients are added to find the total quotient.

Partial Quotient Division -- layout 2:
This style resembles traditional long division more closely. Eventually they will be able to combine steps to find the total quotient.

With practice, students are able to estimate closer to the product they are trying to reach, and they can eventually practice long division as traditionally taught, but while keeping the value of the unit in mind and not needing mnemonics. Teachers can assess understanding by asking the student to explain his or her work using place value language.

The traditional algorithm is usually considered most efficient. However, for students with a command of multiplication, short division is even more efficient. An example is shown here. Again, teachers can assess understanding by requesting an explanation of the thinking involved.
Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at

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