There is a point at which, when studying maths, you get to the stage where you can be introduced to something of real beauty. For my students it was usually about halfway through their Further Maths course. Euler’s formula has been referred to as “our jewel”:
If you want to understand this equation more then you can look here or here. If you have a peek then you will see some of the complexity behind the equation. Much of the proof was beyond the students at that stage. The reason to show them was by that stage they were often flagging and needed something to help them through to the end. Euler’s formula was that thing. The idea that Maths could be that beautiful helped them dig in and carry on.
For mathematicians this equation is the Mona Lisa, La Boheme and the Pieta all rolled into one. This is what Mathematics is, an art form.
There are, however, bits of mathematics which are perhaps a little more artisan that art. Say hello to arithmetic. In the same way that Leonardo needed to mix his paints to make the right colour, mathematicians need to know how to add and subtract, how to multiply and divide. They need to know what the purpose and effect of these operators are. Their understanding of any higher level of mathematics depends on it.
Adding and subtracting have algorithms that require knowledge of number bonds to 20. But they are fairly simple algorithms which are easily mastered and frequently required for higher level algebraic mathematics. Multiplication has a slightly harder algorithm which requires times table knowledge (either by rote recall or counting up). Division of numbers of any size requires a more complex algorithm, which is time consuming to carry out.
What is the purpose of long division? No secret here. Its to divide one number by another and get the result. And therein lies the rub. It has no higher purpose. The very best that could be said for it is that it is a good diagnostic of times table knowledge.
There are times when we need to divide one number by another. For small numbers it is useful to be able to this by mental methods. For larger numbers there are ways to divide without a calculator. Long division, of course. Other pencil and paper methods include chunking (which in my view helps children understand the numbers they are working with much better). If you want to get arcane then you could use a slide rule. Again, you will learn much more about numbers this way. However, for larger numbers any non-calcualtor method will usually take longer than it would to locate and use a calculator of some kind.
All that said, why is long division important? Well, it tells us a bit about how the draft national curriculum has been constructed. Long division is what politicians call a “dog whistle” issue. Most people would not be able to read the draft maths NC and decide if it was good bad or indifferent. But they do know they had to do long division themselves at school, so if it is in there then it must be right.
So, thats why long division is important. It shows that at least some elements of the NC are not there because they mathematically necessary. They are there because they are politically necessary. That this part is suspect undermines the integrity of the whole.
Long division was a tool created to perform a function. We don’t need it any more, we have better tools. I have never in a 30 year working life ever used long division for a purpose. Learning how to use it isn’t even an interesting intellectual exercise. Long division is a dead end. Insistence on its inclusion in a mathematics national curriculum has nothing to do with improving the mathematical capability of children, and it will certainly not get any of them closer to appreciating the beauty of Euler’s formula.
It has been suggested that one mathematical rationale for long division is that it supports algebraic long division.
Whether a (possible) small gain at the algebraic end for a minority of students is worth a (known) large downside at the arithmetic end for the majority of students goes to the heart of the debate. Is the purpose of teaching long division to benefit all students or to weedle out those who won’t be studying maths in several years time. I would very strongly argue that chunking provides a solution to the problem of dividing large numbers which can be more readily used by a larger number of people than long division and provides a suitable basis for developing the skills necessary to divide algebraic terms later.