Should we scrap long division?
The primary school curriculum asks us to make mathematics a real life experience. In the main, this is nice and easy as the concepts taught in primary schools are basic enough to transfer into real life. 2 + 2 = 4 can be shown easily as a real life situation: Take 2 objects and another 2 objects, put them together and count them – voila. We can do this for all of the other simple operations.
By third class, children approach multiplication and even then things are ok. We can see multiplication with concrete objects. Even when we move to the abstract, children can understand the idea behind numbers. For example 24 x 5 can easily be seen as 20 x 5 and 4 x 5 and with a bunch of Dienne’s blocks or pictures, the concept of long multiplication can be seen in its entirety. When we do long multiplication and we pop that zero down before we start multiplying the tens, we can explain why this is and show how we’re really multiplying by a group of tens rather than the single digit number.
Short division is a similarly easy concept to show. Throw out a bunch of items and share them equally amongst the crowds. Again no problem. Then we meet long division.
Back in the “olden days”, children had to learn concepts like “carry the one” or “borrow and payback” to learn addition and subtraction respectively. Thankfully, this has changed and we can use place value to rename numbers to make the experience real and concrete. Children no longer have to learn off a formula. In fact, almost every concept in the Irish primary maths curriculum has a concrete explanation – look at calculating area, adding fractions, etc. The biggest exception is long division.
In order to do long division, you need to learn off a formula. It goes along the lines of Divide, Multiply, Subtract, Bring Down then start again until you can divide no more then you have your answer.
Is there a way to show this formula in action in a pictorial or concrete way? I haven’t found anything except for a bunch of web sites that make the process more colourful or an assortment of acronyms to help one learn the order in which to complete each step, (see above). However, I have never seen anyone try and draw this process on paper. For me, this goes against the spirit of the primary curriculum.
Far closer to real life would be the repeated subtraction method where one can repeatedly subtract the divisor from the number. It is easy to express them in a concrete way and certainly in a pictorial way. When I was studying this, I came across the concept of “super repeated subtraction.” I’ll use the following problem to show you how it works: 2694 ÷ 36.
It would take a very long time to take 36 away from 2694 repeatedly so let’s take a look at this concept which should be easy to picture in your head.
Let’s take bunches of 36. 2 bunches of 36 is 72 and so on. I’ll put this in a list.
- 1 bunch: 36
- 2 : 72
- 10: 360
- 20: 720
- 2,694 -> take 720 (20 bunches)
- 1,974 -> take 720 (20 bunches)
- 1,254 -> take 720 (20 bunches)
- 534 -> take 360 (10 bunches)
- 174 -> take 72 (2 bunches)
- 102 -> take 72 (2 bunches)
We can’t take any more bunches away because we have shared them all out and the 30 remaining are left out (the remainder). If we add the bunches that we shared out, we come back with 74 with the remainder of 30.
At primary level, it is unlikely that long division gets more difficult than this and this is a “real life” way of solving it and it takes roughly the same time as doing it the other way. While I accept that the other method can be a shorthand way of doing it, at primary level, I argue it is better to keep it real! Some will argue that long division has the advantage of getting pupils to practice a number of operations. I would argue that this is useless if they don’t explain in concrete terms the problem that we’re solving.
So should we scrap long division in its current form at primary level? I can’t see why not.