676 is the new 243

Where is the mathematics in the new mathematics curriculum for primary schools in England?

Some of us in mathematics education will remember the significance of the number 243: not just because it is 3 raised to the power 5, although that helped us to remember it! Paragraph 243 in the hugely influential Cockcroft Report (1982) was very significant in defining what constituted a good and balanced mathematical experience for learners. This one paragraph had a great impact on mathematics teaching and learning and was for many years a positive influence on the mathematics curriculum. Do an internet search on 'Cockcroft 243' and you will see what I mean.

However, 243 is a long way back in history now, so instead I would like you to welcome and celebrate with me 676! I will explain.

The new primary mathematics curriculum for England, being introduced later this year, has an excellent statement of the purposes for learning mathematics. It also has a statement of aims that includes aspects that I would recognise as being real mathematics: in particular, the development of the distinctive ways of reasoning in mathematics and solving problems using mathematics (Aims 2 and 3). But these are then followed by pages of specific learning targets in the programmes of study that just do not seem to match the laudable aims and purposes. My fear has been (and still is) that teachers will focus just on the details in these programmes of study and real mathematical experience for our children will be neglected in favour of rote learning of routines, such as long division and calculations with fractions.

In particular, since there is no longer anything comparable to the 'using and applying' strand in the programmes of study and attainment targets for the current curriculum, there seemed a real possibility that the national assessments at the end of Key Stage 2 would focus just on the specific detail in the programmes of study.

The good news is: they won't!  Of the three papers, one (27% of the marks) will focus on context-free calculations, but the other two will contain a genuine focus on reasoning and problem-solving. And to make sure this happens we have paragraph 6.7.6 in the Key Stage 2 Mathematics Test Framework for National Curriculum Tests, published recently by the Standards and Testing Agency. This paragraph defines in detail what is expected of children at the end of Key Stage 2 in terms of mathematical reasoning and problem solving. This is where the real mathematics is in the mathematics curriculum! Maybe not in the programmes of study, but clearly there in the assessment criteria. And the hope is that if all this is going to be assessed it might just be taught!

Section 6.7.6 Solving Problems and Reasoning Mathematically

Children working at the expected standard are able to:

·            develop their own strategies to solve problems by applying their mathematics to a variety of routine and non-routine problems, in a range of contexts (including money and measures, geometry and statistics) using the content described above

·            begin to reason mathematically making simple generalisations, using mathematical language and searching for solutions by trying out ideas of their own

·            use and interpret mathematical symbols and diagrams, and present information and results in a clear and organised way; for example:

·            derive strategies to solve problems with a two or three computational steps using addition, subtraction, multiplication and division and a combination of these

·            solve problems involving numbers with up to two decimal places

·            select appropriate strategies when calculating depending on the numbers involved

·            use rounding and estimation to check their answers and determine, in the context of the problem, appropriate levels of accuracy

·            identify simple patterns and relationships, and make simple generalisations

·            draw their own conclusions and explain their reasoning in simple contexts using mathematical language

·            make simple connections between mathematical ideas

·            solve problems involving data 

So, hail 676, say I! Three cheers for 676! 676 is the new 243! Let's make sure that our primary school teachers are aware of 676. It justifies – and indeed requires – a proper focus in their mathematics teaching on genuine mathematical experiences for our children.

And, to cap it all, 676 is an interesting number, being the product of the squares of two primes (2 and 13)!

Four 4s problem

With the summer holidays approaching, here's an engaging diversion for a long train journey or a long wait in an airport departure lounge. I remember first coming across this problem when I was about [( 4 × 4) - (4 ÷ 4)] years old. I see from an internet search that it is still a much-discussed problem; so readers of this blog will have to resist the temptation to look up solutions elsewhere!

The challenge is to see how far you can get in constructing whole numbers, starting with 1, each using up to four digits 4, but no other digits.

You are allowed to use standard mathematical symbols: +, –, ×, ÷, √, for example. So √4 is a way of writing 2. And the [( 4 × 4) – (4 ÷ 4)] written above is a way of writing 15. 

You can use two of the 4s to make 44. You can use decimals without a leading zero (for example, .4) and powers, as long as you don't use any other digits. So, for example, you could also make 15 by writing (√4)⁴ – (4 ÷ 4).

Very useful is the mathematical symbol for 'factorial', which is an exclamation mark. So, for example, 4! = 4 × 3 × 2 × 1 = 24, and 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. 

So, yet another way of making 15 might use 6! (720) and divide this by 48. To get the '6' we can use (4 + √4) and to get the 48 we can use (4! × √4). This gives the following very complicated way of constructing the number 15: (4 + √4)! ÷ (4! × √4). 

So, here's a start ...

1 = 4 ÷ 4

2 = √4

3 = 4 – (4 ÷ 4)

4 = 4

5 = 4 + (4 ÷ 4)

6 = 4! ÷ 4

So, over to you ... you should find it fairly plain sailing up to 50. Now, remember, no cheating. Googling 'four 4s' is not allowed!

Luke's challenge

Grandson Luke set Grandad (me) this intriguing little mathematical challenge.

Find a 5-digit number, which when multiplied by 4 gives a 5-digit result with the digits of the original number in the reverse order.

It took Grandad about 10 minutes to crack this (without a calculator to hand).

I then started thinking whether there are other problems like this ... a 5-digit number when multiplied by something other than 4 giving a 5-digit result with the digits reversed? A 4-digit number version? A 3-digit number version? Driving home in the car yesterday I convinced myself that there is not a 2-digit version.

Something to investigate further when I have a few minutes free to do something pointless ...

Mathematical problem 1: 29 rounds

Now and again I shall offer some interesting mathematical problems that my blog readers might like to solve – and perhaps use in the classroom, if they are teaching.

Here's a little mathematical problem that could be used with bright Year 6 pupils (10 to 11 years).

I was watching a snooker tournament final. This consisted potentially of 29 'frames' (rounds). But when the score was 15 to player A and 9 to player B then A was declared the winner, even though only 24 frames had been played. Why?

That's easy, of course. But I wondered how it would work if there was some kind of game of 29 rounds between 3 players (A, B and C), in which only one player can win each round. When would the match be over and one of the players declared the winner? Let's use a, b and c for the numbers of rounds won so far by A, B and C.

If a = 12, b = 9 and c = 6, then A has won and the match is over. Why?

But if a = 11, b = 9 and c = 6, then no player is certain yet to win and the match must continue. Why?

Can you find a general rule for when we can say that A has won? (Assume that A has the highest score and C the lowest).

You know what's coming next ... what about a rule for 4 players playing 29 rounds? 5 players?

Can you find a generalization for any number of players playing a game consisting of n rounds?