The trouble with teaching division……
WOULD YOU ACCEPT THESE RESPONSES FROM A STUDENT WHO IS ASKED TO EXPLAIN MULTIPLICATION ?
I do not show these examples to have you laugh at, or be amused by, students’ lack of knowledge or communication. I find these examples to be a sad testament to our lack of understanding. These were offered by excited and well intentioned grade 5 students.
They are not funny. They scare me.
Teachers so often ask me to help them with division and my answer is always if students do not understand multiplication and are not able to recall some small multiplication facts you should not be teaching division. It makes no sense.
LET’S BE CLEAR….
Division is the inverse of multiplication. That means without multiplication, there is no division or without division, there is no multiplication. They are one and the same.
While you could learn to divide first, then learn to multiply, teachers never have this option because from the moment they start to talk about numbers, well meaning adults are encouraging your children to memorize silly little facts like 2 x 5 = 10 and 10 x 10 = 100. This is not understanding multiplication.
The Alberta curriculum is quite clear in Grade 3. The outcome begins with the statement:
Demonstrate an understanding of multiplication as equal groups and arrays.
“And” in math means they go together so teach them together.
Area models built with square tiles or square grid paper make visible the commutative connection between “basic facts” like 3 *4. By simply viewing the array from a different perspective you can see there are 3 groups of 4 or 4 groups of 3.
This understanding is a key to division.
The Alberta curriculum is also quite clear in GRADE 3 that students are to
- represent and explain division using equal sharing and equal grouping
There are multiplication equations to describe each array “and” for each of those 2 multiplication equations, there are 2 ways to express and describe a division.
If the array above is 12 cookies, you could share out in sets of 3. How many groups or sets of 3 can I make?
When I label an array as 3 by 4, the three is labelling the sets of 3 that are running horizontally. There are 4 of them. Therefore 12 ÷ 3 came to be interpreted by many teachers as “how many threes are in 12?” You can see there are 4.
that 3 at the top of the array also signifies there are 3 equal groups in 12. See the 3 columns or groups?
If I change my story a little, I have 12 cookies and 3 bags. If I want the bags to be equal I can put 4 in each. Do you see where the 3 bags are in the array? Do you see how I know each bag will have 4 cookies?
The Alberta curriculum make clear in the outcome in Grade 3 that division is to be taught and interpreted in two ways.
What if the array we started with represented 12 cookies and I wanted to give out 4 to each friend? How many friends can I feed?
Looks like I can feed 4 friends with 12 cookies because 4 x 3 = 12. Same array, different interpretation.
What if I had 12 cookies and wanted to pack them into 4 boxes. How many would go in each box?
We cannot and must not teach division as just set of facts to memorize or a list of rules to follow. Teaching for understanding means understand FIRST…. not solve and memorize equations and facts FIRST.
The evidence is clearly established that once you think you “know how” to do something you are much less likely to care how or why it works. When we focus on memorizing meaningless facts and multi step algorithms like “long division” before we develop with students an understanding of how multiplication and division are related both visually and spatially students do not come to understand. That is what makes teaching division so difficult. They have no way to make sense of what they are doing or why.
They “get” answers that have no meaning.
My BERCS cards are set up to be puzzles. Puzzling attracts the brain. Puzzling takes the pressure off “know the answer” quick as you can or fill in the blanks by moving the focus to hmmmm I wonder what is going on here? Puzzling engages students in thinking, talking, comparing, connecting all the components of LEARNING.
Is this solved with multiplication? with division? or with understanding both?