A)If you had to multiply 13 × 5 in your head, which of these two methods would you use?
- I can separate 13 into 10 and 3, so 10 × 5 = 50, 3 × 5 = 15, so total is 65
- 3 × 5 is 15, carry 1. 1 × 5 is 5, plus the carried one is 6, so 65
B) If you had to multiply (n + 3) by 5, which of these two methods would you use?
- Like a) 1), 5 × n = 5n and 3 × 5 is 15, so the answer is 5n + 15
- Like a) 2), 5 × 3 = 15, carry 1. 5 × n = 5n plus the one you carried so 6n + 5.
Clearly, one of my answers in b) is wrong. I have used the method of “carrying”, which clearly does not work in the transition from arithmetic to algebra.
So …. next question? Which of the methods above, 1) or 2) has been effectively banned by the Government?
Method 1) works like this
You can see a clear similarity between its use in arithmetic and its potential use in algebra. However, method 1) is the one banned by the government. Well, effectively banned. What they have said is that in KS2 SATs in Year 6, if you use this method – grid multiplying – you will only get marks if your answer is correct. You’ll get no method marks for a wrong answer – but you do get method marks for a wrong answer using “traditional” multiplying methods such as 2).
In responding to my e-mails about this, I was initially informed that grids led nowhere while long multiplying led to further mathematics ….. which, as you can see, it clearly doesn’t! When I made this clear, in my next e-mail the only reason for sticking to “formal methods” was their use by “high-performing” countries. No longer was a use in transition from arithmetic to algebra of any value!!
OK – rather than continue to express my views, let me just go over what the Government itself – and Government bodies – have had to say about grid multiplying.
In 2011, Ofsted brought out a report called "good practice in primary schools - evidence from 20 successful schools"
Here is a direct quote from the report:
"One of the advantages of learning the grid method is that it can be used later for work on multiplying decimals, and for secondary mathematics topics including multiplication of algebraic expressions such as (2x + 3)(x – 6) and numerical expressions involving square roots, for example (√3 – 1)(2√3 + 1). It is particularly valuable in emphasising the four products, thereby tackling the common error where only the first and last terms in each bracket are multiplied. Moreover, while able pupils can often move to more succinct methods of expanding and simplifying such products, this method of recording is accessible to middle-attaining secondary pupils. It also provides insight into the reverse process, factorisation, which pupils generally find more difficult."
I couldn’t put it better myself ! Over the years of volunteering in my primary school, I have come across many examples of bright students wanting to calculate 13 × 13 or 23 × 23 quickly in their head. Inevitably, the answers they come up with are 109 and 409. This highlights the comment in that report about “emphasising the four products”, as I’ve shown in the grid below.
Another interesting Government offering came just as Michael Gove gave way to Nicky Morgan. The DfE released a video called “the school revolution”. The DfE decided to start the video with a Maths Free School Head showing us a simple example of grid multiplying – 35 × 35 – then finish the video with the same person using grid multiplying to show that there is a rule for squaring ALL numbers ending in a 5.
I should at this point tell you that Nicky Morgan has a different view of this. In response to my comment to her, she said that …
“The teacher in my department’s video The School Revolution is using the grid method to illustrate the nature of multiplication as a concept, he is not teaching it as an effective method of calculation.”
That told me!!
In the same response, she had said “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.”
Part of my problem with her comment is that “later mathematics studies” include the algebra that her own body Ofsted says can benefit from grid multiplying!
I just feel that primary students are suffering just because the Government wants to appeal to traditionalists and that the school system is suffering as a result of such a limited outlook.