January 21 - Rethinking Division in Elementary School

We have been whizzing through our division unit.  The students are excited to use new strategies that they have invented on their own, and have been learning from their classmates.  They particularly liked the fact that division is actually the inverse of multiplication.

Here is the thought process that one of my students shared with the class during our math congress (almost verbatim):

Me: Okay, today we are going to solve the problem 342/6.  Anybody have any ideas where to start?

Student; So, 342/6 is the same as 6 times something equals 342.  Well, I know that 6x50=300, that's easy. And I also know that 7 times 6 equals 42, so if I add the 50 and the 7, the answer should be 57 with no remainder.

Even if the question does have a remainder, it is still a simple process to figure it out mentally.  Of course, once you start getting into bigger and bigger units, 2 and 3 digit divisors, then this gets more complicated.  Then again, so does the traditional algorithm.

Speaking of the traditional algorithm, we looked at it and tried to break it down.  I have no problem with students using this as one of their strategies to solve division problems.  The more methods the better.  What I am concerned with, and this is something that we have been doing a lot of this unit, is how you think about the question while you do it and the language you use to describe your thought process. Here is what I mean:

Lets look at the problem of 387/7.

If we use the traditional way of doing long division, this is how we would say it.

7 can't go into 3, so we put a 0 up top and move onto the next digit.  7 can go into 38, 5 times, so we put the 5 up top and subtract write 35 below the 38, and then do 38 - 35 and we get 3.  Next, we bring down the 7 and we get 37.  7 can go into 37, 5 times, so we put the 5 up top and write 35 under the 37, then subtract, which will give us a remainder of 2.  So the answer is 55 with a remainder of 2.

This works.  However, I do not believe that the student is understanding the numbers as they are using them.  Rather, they are just following a rote set of instructions designed to get them through the task at hand.  I introduced the following idea to help my students to talk their way through math problems.  I have a policy in my math classes that you are encouraged to talk out loud to yourself as you do problems.  Math talk is the one element of math that I stress the most in elementary grades.  Math class should be noisy, not filled with quite kids sitting at desks and working on problems alone in their own personal math bubble.  Anyway, rant over.  Back to my point.

Introduce this way of thinking about numbers; preferably at the beginning of year when you cover place value.  I use the metaphor of a shipping company.

• Hundreds are boxes; inside each box is one hundred units

• Tens are cartons; inside each carton is ten units

• Ones are unpacked units; loose, individual

Lets start with 387.  These numbers represent a package of something that is being shipped to a destination (candy, cans of juice, baseball cards, whatever).  There are 3 boxes and each box has 100 cans in it (hundreds).  We also have cartons, and each carton has 10 in it, so therefore we have 80 units in our 8 cartons.  There are also 7 individual items in the shipment that are not wrapped.  If you want to get into thousands you can introduce the term palette, and for ten thousands, we could use shipping container (and so on, continue the metaphor).  I would recommend letting children come up with their own metaphor for numbers, something that makes sense to them.  When students think of the number in this way, they are conceptualizing the number, and not just plugging it into a rote formula to be memorized.  Lets go back to our original problem and see how the answer sounds this time in Math Talk.

We need to share this shipment among seven people.  There are 3 boxes, so nobody gets a full box because 7 people cannot share 3 whole things, so lets open up those boxes and share the cartons.  Once we open up the boxes, we have 38 cartons, because each box has 10 cartons.  If we have 38 cartons, each person will get 5, and we have 3 cartons left over.  Now, we open up the cartons and get 30 individual items, plus the 7 other items makes 37.  Each person will get 5 each, with 2 left over.  Therefore each person gets 5 cartons and 5 individual items, which equals 55 total.  There are 2 items left that nobody gets.

The written record will look exactly the same as the example above, but the thinking that went into it is completely different.  In the second example, the kids are actually thinking about the number and what it means.  They are applying the concept of sharing to the concept of division, and they are creating a metaphor to understand numbers as real objects.  They are thinking conceptually, and not just memorizing a set of rules and facts.

I know the obvious dilemma here is that this is much harder to assess. True.  But who said teaching was easy?

1. I like this system. My main reason why is because it gives students better intuition about what to do with the numbers. The only switch I would do is to actually have the students start with the hundreds position because this is what you would actually do if faced with the problem you describe. No one would start by dividing the small part of the loot up first.

Next, I'd recommend getting some actual shipping cartons, filling them with the right materials, and well uh, using them to solve division problems. It might get messy but it will be totally worth it.

2. DwyerTeacher21.1.11

Yeah, we drew pictures of shipping containers, boxes and cartons. I would love to get the real thing, but.....

David Wees said:
The only switch I would do is to actually have the students start with the hundreds position because this is what you would actually do if faced with the problem you describe.

I think we did start with the hundreds position, didn't we? Maybe you could clarify what you meant.

3. Yeah, you did. I confused the two 7s. My bad.

4. Blair Miller21.1.11

I like this approach as well. Another way to think of it that might fit better with your students description is that you are splitting the shipment up into custom carton sizes that only take 7 items. Therefore, from the boxes holding the 300 you can make 42 cartons of 7 with 6 items left over, from the 86 (80 in cartons + 6 left over from the boxes) you can make you can make 12 cartons of 7 with 2 items left over, and from the 9 loose items (7 loose + 2 left over from the cartons) you can make 1 carton of 7 with 2 items still remaining. So you've made a total of 55 cartons of 7 with 2 left over. I realise this doesn't fit as neatly with long division, but is a useful conception of division as students start thinking about dividing fractions.

Here are some online virtual manipulatives that might be helpful in representing up to 1000s, though similar physical manipulatives are available and would be handy to have as David mention above.
http://olc.spsd.sk.ca/de/math1-3/baseten-1.html