News of 4th edition of Mathematics Explained

You may have seen from the advance publicity (www.uk.sagepub.com/authorDetails.nav?contribId=100852) that the fourth edition of my book Mathematics Explained for Primary Teachers is due out later this year.

Everything is going according to schedule! The new primary national curriculum, statutory from September 2011, (www.dcsf.gov.uk/newprimarycurriculum) was published at the end of November 2009. I then had two months to get the new edition finalised, to ensure that it matched the new curriculum and reflected its emphases appropriately. This involved writing some new material, including a couple of new chapters. I'll tell you about these in a later blog.

So, I sent off the new edition to my publisher, Sage Publications, on 31 January 2010, as promised. Remarkably, there were no queries from the copy editor! So the book is now with the 'typesetters' – they're still called that, curiously. I expect to see the page proofs on 6 April and I will then have three weeks to check them over. Experience suggests that the main problems at this stage are likely to be getting the figures right and on the appropriate pages.

All being well, it will then be all systems go. I hope to see the new book some time in July!

Mathematical Problem 2: Page numbers probability

Here's a variant of a well-known counter-intuitive question about probability.

You choose eight different books randomly from a library shelf. If you turn to the same numbered page in each book (for example, page 5), how likely is it that on at least two of these pages the text will start with the same letter? (Exclude dictionaries!)

Choose one of the following:

• highly likely (probability about 0.9)
• fairly likely (probability about 0.65)
• about evens (probability about 0.5)
• fairly unlikely (probability about 0.35)
• highly unlikely (probability about 0.1)

This is a good problem to use with primary school children because it is easy for them to gather data to enable them to estimate the probability using relative frequency. Each of eight children has one of the books. They all turn to page 5 and call out the first letter on the page. They just record 'Yes' or 'No', depending on whether or not there is a letter that turns up twice. They then do the same for page 6, page 7, and so on. They can quickly gather data for a sample of, say, 50 pages and calculate the relative probability of 'Yes'.

Try it! The result may surprise you.

Mathematical Problem 1: Solution

Here's a solution to Mathematical Problem 1, posted on 13 March 2010.

With 2 players, A wins a match of 29 rounds when A has won 15 rounds. Even if A loses all the remaining rounds B will only have won 14.

The principle is that A would still win even if all the remaining games were to be won by B (A's closest rival).

With 3 players, A wins when a is greater than b added to the number of rounds still to be played (assuming B has the second highest number of wins so far). The number of rounds still to be played is 29 – (a + b + c). So, algebraically:

a > b + [29 – (a + b + c)] which simplifies to 2a > 29 – c.

With 4 players, A wins when

a > b + [29 – (a + b + c + d)] which simplifies to 2a > 29 – (c + d).

The general solution , with any number of players and n rounds, is that A wins when

a > b + [n – (a + b + c + d + e + ...)] which simplifies to 2a > n – (c + d + e + ...)

The surprise in this result is that b does not appear in the simplified rule! Provided B has won more rounds than C, D, E and so on, the actual number that B has won does not make any difference to when A is declared the winner.

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?

The Housekeeper and the Professor: book recommendation

Yesterday morning I was chairing the West Midlands regional meeting for the TDA's Student Associates Scheme at Staffordshire University. I left at about 1.20 pm, and, thanks to a broken down train somewhere around Derby, got home to Norwich at about 8 pm!

However, this gave me the ideal opportunity to finish the novel I was reading: a charming, subtle book that Jenny, one of my two daughters, astutely chose for me at Christmas. So I recommend to you: Yoko Agawa (2009, translated by Stephen Snyder) The Housekeeper and the Professor. London: Harvill Secker.

It's the story of a housekeeper and her son who look after a Japanese professor of mathematics, who has lost all but 80 minutes of short-term memory in an accident. But deeply embedded in his memory is his love of number theory, a world into which the housekeeper and her son are gradually drawn. It's truly enchanting.

To get the most out of it you may need to brush up on multiples, factors, primes and number patterns: Chapters 11 and 12 of the current edition of Mathematics Explained for Primary Teachers!

Rising Star: Peter Haylock

Did you spot another Haylock in the Times Educational Supplement Magazine this week (5 March 2010)? Peter Haylock, deputy head responsible for assessment and intervention at Fulham Cross Girls School, London, was featured as a 'Rising Star'.

Peter is my nephew. Just thought I'd mention it and maybe get a bit of reflected glory from the rising star in the family!

Minimum or maximum temperature?

The woman running the food hygiene course that my wife, Christina, attended last week said that food in the freezer should be kept at a minimum temperature of minus eighteen degrees. Christina pointed out that this should be a maximum of minus eighteen degrees. But the instructor just could not get this!

Why is this so difficult to understand?

Error and ambiguity in reporting percentage increases

The following appeared in a report in The Times on 1 March 2010: 'The number of candidates awarded a grade C or above (in GCSE science) was predicted to rise by 2.4 per cent …'.

There are examples here of a common error and a common ambiguity in the use of percentages.

First, the error. The reporter (Greg Hurst, Education Editor) clearly cannot be referring to an increase in the number of candidates awarded a Grade C or above. This could simply be a result of changes in the population from one year to the next. He must be referring to an increase in the proportion of candidates.

But even with this understanding it is not clear what the 2.4 per cent relates to. Is it 2.4% of the proportion who were successful or 2.4% of the entire population? Here is the ambiguity.

As an example, assume that in the previous year 50% of candidates were awarded grade C or above. Does the 2.4% increase mean 2.4% of the 50%? In which case, the proportion becomes 51.2%. Or does it mean the 50% has increased to 52.4%?

I suspect that it means the latter. To make it clear the reporter should write: 'The proportion of candidates awarded a grade C or above (in GCSE science) was predicted to rise by 2.4 percentage points …'

This ambiguity can always arise when we start talking about percentage changes in data given in percentages. We need to be clear about this distinction between an increase of so much per cent, and an increase of so many percentage points.