Core arithmetical functions

Did you spot this little gem in Michael Gove's White Paper, the Importance of Teaching, published this week? In paragraph 4.19:

“As part of the review of the National Curriculum, we will also make sure that children are expected to master the core arithmetical functions by the time they leave primary school.”

Core arithmetical functions? Now there's an interesting concept! I wonder what he has in mind? I don't actually – to be honest, I think I know what he has in mind. I have heard rumours from three reliable sources that there are plans afoot to require teachers to teach children in primary schools formal, written, standard algorithms. Less of all this informal, mental stuff and ad hoc strategies based on number lines and understanding. And out with the dreaded 'grid method' for multiplication, so scorned by that expert on primary mathematics teaching, Carol Vordeman. We shall get back to the 'core functions' of arithmetic: column addition and subtraction for 7-year-olds, and the condensed form of long multiplication and long division for 11-year-olds. It's all based on thorough research, of course – namely, this is what we were taught when we were at prep school and it never did us any harm.

The phrase 'core arithmetical functions' is a new one on me. It's actually an appealing idea and I'll offer a few suggestions for things that ought to qualify under this heading in a subsequent post. I have a feeling that long division is not going to be near the top of my list, or – since I really cannot remember when I last needed to do a long division calculation – even on it.

Solution to last week's problem

What will be the first date from now which when written as XX/XX/XXXX (day/month/year) contains eight different digits?

That was the challenging little problem thrown at the students on University Challenge last week. It's not straightforward! Obviously (mathematicians use this word when we can't bothered to explain our thinking) we have to start by getting the earliest possible year, then the earliest month in that year and finally the earliest day in that month.

My immediate thought was that the earliest possible year, using four different digits, would be 2013. But 2013 will not work because the month must use either 0 or 1 for its leading digit, so we can't use both of these in the year. I then tried 2034. But the problem with that is that the leading digit of the day can be only 0, 1, 2 or 3, so I now realised that we need at least two of the digits 0, 1, 2 and 3 for the first digits in the day and month. So, I then went for the first possible year that uses only two of the digits, 0, 1, 2 and 3: 2045.

I was confident this would work! But, it doesn't. The only month that does not use 0 or 2 is November which repeats the digit 1! So, we can't have both 0 and 2 in the year.

OK, so surely the solution must be 2145? No, that doesn't work either! The month would have to use the 0. So, having now used 0, 1 and 2, the day would have to be 'thirty-something'. But the only possibilities are 30 or 31, which are not available because we have used the 0 and the 1.

Welcome to the year 2345! This works. We can use the earliest month in that year, June (06) and the earliest day in that month (17). So, the solution is 17 June, 2345. But how on earth were those University-Challenged students supposed to work this out on the spot?

Challenging maths problem

This was asked by Jeremy Paxman to the puzzled students on University Challenge last Monday! Unsurprisingly in the time available none of them could do it. Not could I, to be honest. But then I don't suppose Jeremy Paxman could either. Here's the question:

What will be the first date from now which when written in the form XX/XX/XXXX (day/month/year) contains eight different digits?

I'll give the solution and explanation next week. It's not actually that difficult, you just need more than 30 seconds to think about it.

Mathematical creativity lecture

Well, I am off to the North-West of England tomorrow to give a lecture at Edge Hill University. It takes most of a day to get there from Norwich by train, going via London and Liverpool. I've been asked to talk on 'Creativity in mathematics' to a group of primary school teachers on the Edge Hill Mathematics Specialist Teacher Programme.

This topic is one of my personal enthusiasms and one that I always enjoy sharing with primary school teachers. My involvement in classroom-based research in this area goes all the way back to my 1984 PhD, which I did part-time at London University. Over the years I have been privileged to explore the topic further with various groups of more able children in primary schools, particularly at Cringleford Primary, a local school on the edge of Norwich close to the University. So, I have plenty of material from interacting with children to share on this subject – and I hope this will entertain and stimulate the troops in Ormskirk.

The following weekend I shall be doing a repeat performance in Birmingham, with another cohort of mathematics specialist teachers.

I see that Ian Sugarman is doing a session after me at Edge Hill on place value and calculation. He's always good value – likewise, Ian Thomson, who is doing the same session at the Birmingham event.

Less fuss, fewer complaints

We don't often go to Morrisons supermarket, but we called in today. The sign at the checkout said, '15 items or less'. Did my hackles rise and did I vow never to desert Waitrose again? Well, actually, no. I'm getting a bit bored with all the fuss about the distinction between less and fewer. As a writer, I have to get it right, of course. If I don't my copy editor would change it anyway!

And it's not actually that difficult. Here are three simple rules.

1. Less goes with a comparison of two measurements or two quantities – quantities like weight, volume, population. Fewer goes with cardinal numbers – numbers describing a set of things. So I had less cereal for breakfast this morning, but there were fewer cereals to choose from than usual.

2. Use less if the sentence used to make the comparison uses a singular verb and fewer if it uses a plural. So, there is less traffic, but there are fewer cars.

3. Use less if you are talking about how much, and fewer if you are talking about how many. So, how much money do you have? Less than last week. How many coins do you have? Fewer than yesterday.

But does it matter? I enjoy correcting my friends when they get this 'wrong' (annoying them, actually), but I would be reluctant to correct a child. By the time they grow up fewer will probably be archaic anyway.

It's really odd that in most contexts the opposite to less and the opposite to fewer are the same word: more! So, we would not have a problem with saying 'more traffic' and 'more cars'. So, why do we insist on 'less traffic' and 'fewer cars'? If I can ask for more cheese and more biscuits today, why do I have to ask for less cheese but fewer biscuits tomorrow?

It's really just an accident that we have two words available for making comparisons that focus on the smaller number. Other European languages manage with just the one word. In French, for example, the word for less is moins and the word for fewer is moins.

So, I'm starting a campaign to make less fuss about this – and less complaints.