I have, on several occasions, had a student walk into my grade 9 class on the first day and state “I’m going to fail this class!”. When asked why, I get responses such as “I suck at math”, “I’ve never been good at math” “I can’t do fractions”….etc.

“I can’t do long division.”

This one surprises me most. Really? Long division? You think that is what is going to prevent you from passing math? Long division is certainly not high on my list of key ideas in mathematics!

Last week Jessica and I had the opportunity to work with a great group of elementary teachers and to observe a grade 5/6 class. It was a great learning experience for us as most of our teaching experiences have been in secondary math with some time in grades 7/8. The students had just finished working with multiplication and exploring the array model for multiplication. We decided to see how they might tackle a “division” problem and chose one from Van de Walle.

” Aidan has a bag with 783 jelly beans. Aidan and his four friends want to share them equally. How many jelly beans will Aidan and each of his friends get?”

Having spent very little time in junior classes I wasn’t sure what to expect in terms of strategies. The teacher mentioned that most of the grade 5’s would NOT have seen the standard algorithm for long division but the grade 6’s would have explored it last year. We also wanted to see if students could make a reasonable estimate before they began as the teachers wanted students to work on their ability to judge the reasonableness of their answers.

We saw students approach the problem using a wide variety of strategies. Some tackled it by using a strategic guess and check approach using either repeated additon or multiplication. Others approached it by “distributing” the jelly beans to the five students (i.e. let’s give everyone 100, now how many more do we have….now lets give everyone 10, etc) until they ran out.

I have been thinking a lot about this lesson since that day – specifically about the benefits of student generated algorithms vs. standard algorithms.

There were a few students that used the standard algorithm for long division successfully. What was interesting is that these students had a hard time thinking about the problem in any other way or explaining how the algorithm worked. They were very successful in executing it (right down to extending it to decimal values) but there seemed to be little understanding as to why it worked. In fact, one of the groups that took this approach had an estimate that was very low (5 – 20 jellybeans).

There were a few groups that tried the traditional algorithm, likely because they remembered it vaguely from last year and thought they should use it. I found looking at their misunderstanding of the algorithm interesting.

What was nice was that these students recognized that their answers using this approach didn’t make sense and decided to try a different strategy that actually made sense to them. They are in a classroom where alternate approaches are encouraged and celebrated and the ability to “make sense” of what they are doing is what is most important.

One pair of students that I watched took an approach that I find fascinating and although it looks a bit chaotic on paper, it was quite strategic.

It is somewhat difficult to follow on paper but essentially these students broke up 783 into more “friendly” numbers and then divided each of those. The teacher helped the other students follow what these students were doing by writing it this way.

When asked how they knew that 3/5 was .6, they explained that they first tried 0.5, but when they counted up by .5’s they ended up at 2.5. They then realized that since they were 0.5 away from their “goal of 3” that they just needed to add .1 to .5 and it would work.

I have had many conversations, mainly with colleagues who have children in elementary school, about why they aren’t teaching math “the way we were taught”. “Why aren’t they teaching them how to do long division properly – they have this new way that students make up themselves – don’t they have to eventually learn it the right way…”.

So I wonder, which of these groups of students that we watched have a better understanding of the problem they were working on and number sense in general? The students that did it the “right way” – using long division – or this group that broke 783 down into more friendly numbers?

Some people would cry “but the standard algorithm is more efficient!” I’m not sure the students that developed their own method took any longer….

Tricia BCertainly something to think about – unreasonable answers that are sometimes determined from algorithms can be terrifying to me as a teacher. Great examples from the students! Thanks.

David WeesI think that once kids have a good understanding of what division actually is, you can help them move toward more efficient algorithms, but like you said, I’m not hung up on ensuring that they learn the standard algorithm for division. If they need it later (for polynomial division in university) they can learn it then.

Nicole.ParisI love the different methods that students bring to seventh grade for dividing…the one problem I’ve seen them encounter is how to extend some of those methods for working with decimals.

JanetWhile I fully agree that students need a hands-on understanding of long division, I also think that we must teach the long-division algorithm to go along with it. I have seen too many students in algebra, pre-calculus and calculus classes who cannot carry out the needed polynomial division mentioned above. The algorithm itself is just a written expression of this hands-on understanding of grouping! So, teach it! Don’t leave these students stranded in their later courses!

themathletesPost authorI agree Janet that the algorithm is a written expression of the grouping but I’m not sure the connection is always made for students or that students are given the opportunity to make the connection. As a result it becomes just memorizing steps and what I call “doing things to numbers” without any sense of why or what. My hope is that students would have both – an understanding of division and a procedure(that makes sense to them) to help them work through it efficiently.

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