# Is learning the times tables by rote really helpful?

Recently I was watching a TV advert in which the cost of a product was £323.99. This prompted a thought from me which, I am sure, is not the thought that most others would have had when watching the same advert.

My reaction was “32399 is 32400 – 1 or 180² - 1². Therefore 323.99 is 18.1 × 17.9.”

My strength with numbers hasn’t come from rote learning of times tables but more from identifying algebraic rules which have arithmetic consequences. This, of course, is in direct contrast to the likes of Nicky Morgan, who don’t see the link between arithmetic and algebra. I received a letter in reply from her from which I make a direct quote:

"It is, ultimately, for teachers to decide the methods they use to introduce concepts such as algebra. Mathematics is, however, a discipline with a language and formal methods of its own, and it is very important for pupils to master these methods if they are to succeed in later mathematics studies."

Unlike most maths teachers, Nicky Morgan would appear to suggest that somehow algebra is not part of real mathematics and therefore arithmetic has no connection to it. Let me explain how wrong that is and how much algebra can be used, for example, to learn tables.

If a student knows, for example, that 5 × 5 is 25, then my example above tells us that 6 × 4 will be one less, so 6 × 4 = 24. Similarly, if you know that 10 × 10 is 100, then it is a short step to knowing that 11 × 9 = 99.

This process can be reversed. If, for example, you know your 5 times table then you know that 7 × 5 is 35. This means that 6 × 6 will be one **more**, so 6 × 6 is 36.

The point about algebra, of course, is that once you know an algebraic rule you know that it ALWAYS works. I have great fun with primary students finding out the above rule – and then being able to work out that, if 20 × 20 is 400 then 21 × 19 will be 399 … and then that 31 × 29 will be 900 – 1 or 899 ! Goodness, even 51 × 49 is immediately calculated !!

One way to use this knowledge before learning algebra is to offer examples such as the following one at various ages in primary schools and see how the students react. Note that:

2 × 2 = 4 3 × 3 = 9 4 × 4 = 16 5 × 5 = 25

and that 3 × 1= 3 4 × 2 = 8 5 × 3 = 15 6 × 4 = 24

and see if they spot any pattern emerging.

Then as they become acquainted with tables of higher numbers, continue the pattern:

6 × 6 = 36 7 × 7 = 49 8 × 8 = 64 9 × 9 = 81

7 × 5 = 35 8 × 6 = 48 9 × 7 = 63 10 × 8 = 80

8 × 4 = 32 9 × 5 = 45 10 × 6 = 60 11 × 7 = 77

9 × 3 = 27 10 × 4 = 40 11 × 5 = 55 12 × 6 = 72

etc., which could then prompt a discussion as to why the difference between the answers when compared to the top answer form the pattern 1, 4, 9, 16 etc.

You can then point out to students that we know this pattern ALWAYS works because later they will actually learn to prove it.

Note also that the pattern also highlights reverse patterns. For example, if a student knows that 10 × 6 = 60 - having gone 2 up and 2 down from 8 × 8 – then they might be able to make sense of the fact that 8 × 8 is simply 4 more than 10 × 6.

Part of the reason for my concern over the issue of times tables is the memory of discussions with sixth form maths groups about tables. One student commenting that she had not enjoyed maths until she started learning algebra – and adding that arithmetic and times tables in particular had been really boring – made me raise the same issue with other groups over the years – and that “tables boring” factor did arise many times.

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