A Florida researcher says elementary teachers need to reconsider the way they’re teaching basic division.
Last week at the NCTM annual conference, Zachary Champagne led a session based on work he’s done at the Florida Center for Research in Science, Technology, Engineering and Mathematics at Florida State University. In the presentation, he illustrated the difference between “partitive” and “measurement” division situations, and argued that teachers give far too many partitive word problems in the early grades.
A partitive word problem asks students, “How many are in each group?” explained Champagne. For example:
There were 6 friends who wanted to share 54 cookies equally. How many cookies will each friend get?
Partitive situations are “easier to model, easier to think about,” said Champagne, and tend to be used more often in classrooms.
In measurement problems, on the other hand, the number of groups is unknown. The cookie example, in that case, might say:
A bag can hold 6 cookies. If Troy had 54 cookies, how many bags can he fill?
These types of problems are “less common in textbooks,” the presenter said. But in the long run, he argued, they “allow [students] to see the relationship between multiplication and division more clearly.”
In teaching multiplication, Champagne wrote in a follow-up email, teachers tend to explain 3 x 4 as “three groups of four” to help students visualize the problem.
However, when they get to division, teachers tend to talk about “sharing” and pose partitive word problems. It’s much more difficult to read a division equation as a partitive situation. “Basically for a naked computation problem such as 12 ÷ 4, we would have to think, ‘How many would each person get if 12 were shared equally among four people?’ And that connection to multiplication becomes less clear.”
Instead, he said, teachers should describe 12 ÷ 4 as “How many groups of four are in 12?” That would constitute a measurement problem. It would also demonstrate the inverse relationship between multiplication and division.
Champagne, who was selected as Duval County, Fla.'s teacher of the year in 2010, said he himself was guilty of using partitive problems during direct instruction. “It’s no wonder I was confusing my kids because I’d just taught them about sharing, and now I’m asking a different kind of question.”
Champagne’s team at FSU has been working on developing a formative assessment system for mathematics, which you can find more about here. The system is free for teachers, and the website also has professional development modules and common-core-aligned lessons and tasks.
A version of this news article first appeared in the Curriculum Matters blog.