Wednesday, January 6, 2016

A Concrete-Diagrammatic-Symbolic Development of Division

It is no secret that students find division the most difficult of the four operations to understand. I don't just mean the procedure of the long division algorithm, I mean the concept of what division is as an operation; how it relates to the other operations and how it is used in different contexts.
Here are some thoughts on how we can get our students to truly understand division, based on the Concrete-Diagrammatic-Symbolic continuum. Whilst one of the end-products of this will be that students can use an algorithm to divide two numbers, I truly believe that just teaching an algorithm will not get our students to understand what division is. It must be pointed out too, that a huge part of understanding division necessarily involves seeing how it links to multiplication. Graham Fletcher has created a great video on this. And we should also remind ourselves that knowing your multiplicative facts is more than knowing the answer to 7×8. It is also knowing the answers to such questions as 24÷a=8 and a÷8=7 or even 48÷a=b.

Concrete Division
This can begin in primary grades. Kids might well have real-world experiences of sharing situations (e.g. sharing candy, dealing out cards, cutting a pizza into equal slices). They might not have real-world experiences of grouping situations though so it is a good idea to give questions like 'If an egg box hold 6 eggs, and you have 24 eggs to pack, how many egg boxes are needed?' or 'There are 24 kids in our class and they need to put in groups of three. How many groups will there be?' It is vital that students experience this tactile sense of what division (either sharing or grouping) is. As they become more familiar with this, we can give them problems that will involve remainders so that they can consider what effect this has on their answer (e.g. if an egg box holds 6 eggs, how many are needed to hold 32 eggs?)

Diagrammatic Division
With enough experience of this, they can then represent the division action using diagrams. By this, I don't mean that they need to draw pictures of the actual objects that are in the question but rather use this method which I call Spoke Division:
Suppose you have to do 517÷4.
Firstly, write 517 with four spokes radiating outwards:

Now think of a friendly number that you could put into each spoke. In this case, 100 seems to be a good choice. After taking four hundreds out we are now left with 117:

Now think of a friendly number that you could put into each spoke. Some students might say 10, some might say 20.  Both of these work but will take a little longer. I myself will use 25:

Now we have 17 left, so I can put four more into each spoke leaving a remainder of 1.

Since each spoke has 129, we can say 517÷4=129 R1 or, if you prefer, 129.25.

Initially, I'd be careful about what numbers to use; friendly at first, then building complexity. What I like about this method is that it connects with the students' concrete representation of division and, as such, still feels like division. There will come a point when the spokes method can be developed into something more powerful.

Symbolic Division
Although I learned a version of the standard algorithm growing up, it is not the one I would initially show my students. Instead, I would use the following method, often called partial quotients. 

See how it connects nicely with diagrammatic division. Also notice how it allows the student to use friendly numbers to get the answer. It is not so easy to use friendly numbers in the standard algorithm.

With regards the more abstract standard algorithm, it should be pointed out that different countries have different versions of what this looks like (see this entry in Wikipedia). I myself learned something which I believe is called short division and spent a long time focusing on single digit divisors. When I first saw long division, it seemed (to me at least) to involve an unnecessary amount of writing.

Here in Ontario, not too many teachers seem to have seen 'short' division. It does require the user to mentally compute the remainder at each step (e.g. there are 6 sixes in 40 with a remainder of 4). Will this be tricky for students? I don't believe so, especially if we gradually build up the complexity of such questions.

Double digit division is problematic (unless the double digits are 'friendly': the advantage of working with single digit operators that standard algorithms usually have vanish when trying to do something like 7054÷82. Mentally, I'm trying to do 705÷82. It's doable for sure but potentially time consuming and open to error. Yet using partial products, a student can use friendly numbers:

Some students are fine at using the standard algorithm for double digits. For those who aren't, get them to try partial products; my experience is that this is a game changer for these students.

This concrete-diagrammatic-symbolic development of division takes a long time, years even. It is not to be rushed unnecessarily; I am not convinced that the best way of teaching students about the operation of division is to jump straight to the algorithm of division. 
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Recently, I gave a Math Night for parents at one of the schools in our Board during which I shared our Board's vision of what good Math education looks like. Afterwards, a parent approached me seeking some guidance as how he could help his daughter with long division. He said, "They have to do something like 'Dragons Must Suck Blood' and, to be honest, I don't understand what any of this means. And neither does she." As we chatted more, I gathered that the Dragons Must Suck Blood was an mnemonic to help students remember certain steps of the algorithm (Divide, Multiply, Subtract, Bring Down): clearly it wasn't working. At this point, rather than explain the algorithm in a different way, I showed him how I would develop the concept of division and how this needs to be in place before we try to make sense of the algorithm. It was neat seeing his eyes light up when I showed him the spokes method and the partial products methods and hear him say, "I actually understand those ways!"

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