Recently I spent a day with some close friends and their family and partners. The family is quite competitive, but even I didn’t expect to be sitting at one point with four people in their late twenties asking them long multiplying questions – their choice not mine! The four covered a wide range of jobs – a sales manager, a teaching assistant, a doctor and a builder.
The first question I offered was 37 × 27 (because it comes to 999!) and only the TA was able to answer it. It was no surprise that she knew how to do it, because she had just had to take a numeracy test as part of her application for teacher training. None of the others could remember how to long multiply! This, of course, didn’t mean they have no use for numbers – they just had access to other means of working out what they needed.
That all just made me wonder where we’re going with the Government drive towards “traditional methods” ? One of those four mentioned above – knowing I had used alternative methods - actually asked me about the grid method and seemed to prefer it!
I am a retired maths teacher and I volunteer two mornings a week in my local primary school. I have to say that I am starting to lose interest. Don’t get me wrong – I don’t blame any staff at the school. It’s just that the focus is moving away from mathematics and more towards arithmetic. I seem to be asked more and more to go over boring arithmetic methods which have been introduced into the KS2 SATs tests – and which appear to be the ONLY methods the students are allowed to use!
You also shouldn’t get the idea that my lack of enjoyment means I am bad at times tables! Part of my interest in times tables comes from patterns that you can find from them – it was actually my interest in algebra that rekindled a knowledge of times tables. Noting that 6×6 is one more than 7×5 – and that is true for all similar examples like 28×28 being one more than 29×27 – gave me a greater interest in arithmetic as well as furthering my understanding of the use of algebra.
The use of the example 28×28 is deliberate. When I told a small group I was teaching that I knew all my tables up to 37, one student immediately asked me what 28×28 was – and was surprised when I said 784 straightaway ! I had chosen up to the 37 times table because of a certain unique quality about that number – 37×3 is 111, 37×6 is 222 up to 37×27 being 999.
Come to think of it, an easy way to look at 28×28 would be to think of it from the fact that 28²-2²=(28+2)(28-2)=30×26. 30×26=780, so 28×28=780+2²=784. That will perhaps give you some idea of how a lot of my arithmetic stems from algebraic rules! This does all perhaps suggest that a good time to be learning tables would be during the introduction of more complex algebra in secondary school, so that teachers can use knowledge of algebra to improve students’ arithmetic.
Look how many answers are accessible through simple algebra. As soon as you know any square, for example, it will lead to many other calculations. 10×10 leads immediately to 11×9 and then 12×8 – the latter because (10+2)(10-2)=10²-2²=100-4=96 – and even, following on from that, 13×7=100-3²=91.
You can reverse the process. 10×4=40=(7+3)(7-3)=7²-3². If 40=7×7-9, then 7×7=40+9=49. Similarly, 10×6=60=(8+2)(8-2)=8²-2². If 60=8×8-4, then 8×8=60+4=64.
An alternative would be to highlight patterns when learning tables in primary school, with the aim of leading on to algebra rather than just rote learning. Noting things like …
might be a useful introduction into the world of patterns – and hence the world of algebra – and which would probably improve thinking skills far more than the rote learning of tables would.
Come to think of it, exercises on patterns can lead to many other, more advanced, pieces of mathematics. I was concerned to see one of those reality TV programmes being advertised by a schoolgirl saying “when will we ever need Pythagoras?” To me, it was always the learning process in addition to the outcome that was important. Hence, in the 80s, the maths book I remember is “Thinking things through” by Leone Burton. In the case of Pythagoras’ Theorem, it might be useful to look at patterns of right-angled triangles with whole number sides. Look at this pattern...
3, 4, 5
5, 12, 13
7, 24, 25
9, 40, 41
and note that the difference between the square of two consecutive numbers is (n+1)²-n², which is just the sum of the two numbers n + (n+1). This is always odd, so squaring ANY odd number above 1 can create the lengths of the sides of a right-angled triangle.
For even numbers, it gets a bit more complicated.
4, 3, 5
6, 8, 10
8, 15, 17
10, 24, 26
If you halve any even number and square it, then the numbers on either side of that answer give you the two longer sides of the triangle. This is because (n+2)²-n²=4(n+1). If you square any even number, the answer will divide by 4, hence this will give you a result for any even number.