Thursday, 5 November 2015

Back to back?

The BBC news announced recently that New Zealand had won two rugby world cups 'back to back'. Well done, New Zealand. But not so well done, BBC.

How has it happened that 'back to back' has come to mean 'in succession'? Given that this question is about the language we use to describe spatial or temporal relationships, I think that, as a mathematician, I am entitled to a little light-hearted rant.

I hear people talking about watching successive episodes of a TV programme they have recorded and saying that they watched them 'back to back'. That's just daft. The back of the first episode (i.e. the end) is not followed by the back of the second episode, but by the front of it (i.e. the beginning). So, if anything, they are watching the two programmes 'back to front'.

The pastor in our church announces that we are going to sing two songs 'back to back'. It would clearly make more sense to say that we'll sing them 'back to front'! But 'in succession' or 'one straight after the other' would perhaps be less ambiguous.

When I was a lad, growing up on a council estate in London, 'back to back' described houses, where the back of one house faced the back of another, often with very little space between them for our 'back yards'. Those were the days when the phrase 'back to back' actually meant something; when we got the geometry right.

This phrase 'back to back' seems to be associated with the image of books or DVDs stacked on a shelf, cover to cover. But when they are stacked like this the back of one book is touching the front of the next book.  This is related to a well-known little mathematical puzzle ...

There are ten paperbacks, each width one centimetre, standing neatly and upright on a bookshelf, with no gaps between them. Working left to right, a bookworm starts at the beginning of the first book and chews horizontally through to the end of the tenth book. How far does the bookworm travel?

No, it's not 10 cm!

I enjoyed Tim Vine's one liner about this irritating phrase: 'my wife and I watched two episodes of Downton Abbey back to back; fortunately I was the one facing the television.'

By the way, the bookworm travels about only 8 cm.

Wednesday, 7 October 2015

Less Than Zero

The Times has been running a feature called 'Grammar for Grown-Ups', written by John Sutherland, Emeritus Professor at UCL. As we would expect it generates the usual battles between the traditional pedantics who believe passionately in the validity of randomly imposed and sometimes archaic rules (such as the one about split infinitives) and the more liberal modern linguists who are happy for 'rules' to be determined by usage, rather than the other way round.

I think I know most of the so-called rules and, to avoid confrontation with my copy editor, I try to stick to them in formal writing – even though I know in my heart of hearts that many of them are unjustified and unnecessarily rigid. And, if I do ever pick people up about dangling participles or hypercorrection of 'you and me', then it is done (usually) tongue in cheek. You can now challenge me about starting a sentence with the word 'And'!

But I must take issue with John Sutherland about his claim that if you follow the rule about the distinction between 'fewer' and 'less' then the title of Bert Easton Ellis's book, Less Than Zero, should be Fewer Than Zero.

His argument is that zero is a 'counting number' and therefore 'fewer' is correct. He's wrong.

Zero can be a counting number describing an empty set, but it can also be an ordinal number describing a position; it is an integer separating negative integers from positive integers on  various measurement scales; and it is a real number representing a unique point on the number line. Only when comparing an empty set with another set would the word 'fewer' be associated with zero. [A has zero marbles, B has 7 marbles, A has fewer than B.]

But if we are referring to a number as an abstract entity, not as an adjective attached to a set of things, then it is a singular noun. So we can form sentences about, say, 7, that begin '7 is ...'. Examples would be: 7 is greater than 5; 7 is a prime number; 7 is a factor of 21; and so on. So, we would correctly say, 7 is less than 9. Likewise, zero is less than 7. Or, indeed, 'negative three is less than zero'.

We always talk, correctly, about negative numbers being numbers less than zero. They are not cardinal numbers that describe sets of things, so it is incorrect to use the word 'fewer' when making statements about negative numbers. 'Negative three is fewer than zero' sounds bizarre: it seems to imply that there is a set of 'negative three things' that is being compared with an empty set. In none of the contexts in which negative numbers describe actual things (temperatures, bank balances, heights above sea level, and so on) does it make any sense to use 'fewer'. Would anyone say 'my bank balance is fewer than zero'?

Finally, we should note that 'fewer' can only be applied to whole numbers (because they can describe sets).  It cannot be used with non-whole numbers. Is there a meaningful sentence that begins '0.2 is fewer than ...'. So, a number 'less than zero' could be the number –0.2. Once again, this number is never going to be described as 'fewer than zero'.

Saturday, 26 September 2015

Measuring scales

One of the important principles to understand about measurement is that there are different kinds of measuring scales. And what you can do with a particular set of measurements will depend on what kind of measurement scale is involved.

For example, you can compare two lengths (say, a = 6 metres and b = 2 metres) by their ratio (a is 3 times longer than b). So, 'length' is what is called a ratio scale. One of the features of a ratio scale is that 0 represents 'nothing'. So, '0 metres' is no length at all.

Temperature measured in degrees celsius is not therefore a ratio scale, because '0 degrees' does not represent 'no temperature at all'. So, you could not compare, say, a temperature of 12 degrees with one of 4 degrees, by saying that the the first one is three times hotter! To compare these you could only use the difference in temperature: the first is 8 degrees hotter than the second. This is an example of an interval scale.

A more primitive type of measurement, often met in everyday life, is an ordinal scale. Essentially all this does is to put various entities into order, according to some criteria.

An interesting example of the misuse of an ordinal scale is provided by the Amazon customer reviews for a book. To give a book 3 stars means simply that it is judged better than 2 but not as good as 3, by whatever criteria the reviewer is using.

The thing to note is that you cannot do arithmetic with ordinal scales. You cannot say that a 4-star rating is twice as good as a 2-star rating; or even that the difference between a 2-star rating and a 3-star rating is in some sense 'the same size' as the difference between a 3-star rating and a 4-star rating! So, you definitely cannot add up all the customer ratings and divide by the number of them to calculate an average rating (i.e. the arithmetic mean). That could only be done if the interval between various ratings represented the same amount of something! If there was something that could be measured that we could call, say, a 'unit of worth', so that 1 star meant 1 unit of worth, and 2 stars meant 2 units of worth, and so on, and 0 stars meant 'absolutely worthless', then we would have a ratio scale and we could merrily calculate averages and so on. But customer ratings are not like that!

I mention this because I noticed that my book, Mathematics Explained for Primary Teachers, is given an average rating of what looks like four and a half stars. This is the result of doing some arithmetic with 21 five-star reviews, 3 four-star, 1 three-star and 1 one-star. [This 1-star reviewer wrote 'not fore parents' (sic).] If you were to treat these as measurements on a ratio scale, the arithmetic mean would be 4.65, which appears in an icon as 4.5 stars.

But, as explained above, you cannot do arithmetic with an ordinal scale, so the use of the arithmetic mean is unacceptable to find an average. What you could use for an average ranking is either the median or the mode, since these require nothing more than putting items in order or putting them into a set number of categories.

If you use the median as an average you line up all the reviews from the lowest to the highest and choose the one in the middle to represent the whole set. In this case the median value is 5 stars!

If you use the mode then you just select the most commonly occurring grade: which again is 5 stars.

So, clearly, if the maths is done appropriately, using either of the legitimate averages for ordinal measurements (median or mode), then my book should get an average ranking of 5 stars! I rest my case.

Tuesday, 25 August 2015

Shortage of Mathematics Teachers

Recent press reports have highlighted concern about insufficient mathematics graduates being recruited into the teaching profession. The Times headline invited us to be horrified that many maths classes in secondary schools in this country are being taught by PE teachers!

My immediate reaction was: so, what's new? My second reaction was that this seemed to suggest that PE teachers must be quite incapable of teaching anything other than forward rolls and cartwheels. I am a mathematics graduate, but I am also a teacher. In my time I have on occasions taught music, RE, Games (cricket/hockey) and science at secondary school level, and just about everything at primary level. So, in principle I can't be horrified at the thought of PE teachers filling in some gaps in the mathematics lessons timetable!

The key question, of course, is, why are there not more mathematics graduates going into teaching? Of course, with a good maths degree there are more lucrative professions than teaching available to a new graduate. Early on in my career I was very nearly recruited by a major computing company; but at the last stage I declined their offer of a very well-paid job. My reasons were simply that I loved teaching, that I looked forward to going to work every day, that helping youngsters learn with understanding and enjoyment gave me great satisfaction, and that I really felt that I was making a difference.

Teaching was a respected profession and I felt privileged to be part of it. In my view this is what has changed. Our press and our politicians are constantly running down the profession and undermining the morale of teachers. Government ministers interfere in educational practice at a level of detail that is intolerable. For example who decides what proportion of marks in the end-of-KS2 mathematics tests should be awarded for the context-free written calculation papers? Not teachers! Not mathematics educators. Not any professional body with expertise in the field. No, a government minister decides and imposes their decision on the testing agency. And how much attention was given to the view of the profession in the consultation over the National Curriculum? (That's a rhetorical question!)

If we want to recruit more able graduates in mathematics – and other subject areas where the financial rewards elsewhere are much more attractive than that of a career in education – then a major priority is to restore the professionalism of teachers; to make teaching a profession which is respected; to trust teachers and to recognise their professional judgement. Teachers need professional conditions of employment, with the expectation for properly-funded professional development opportunities throughout their career. Educational policy related to the practice of teaching and children's learning must be informed first and foremost by the profession itself. Teaching should be a profession that bright young people will be proud to be part of.

Now, back to the real-life mathematics problem I set in my previous posting. I was charged £10.99 for the lawn feed, which with my 10% membership discount (£1.10) came to £9.89. But the lawn feed should have been sold at 80% of £10.99, namely £8.79, which with my membership discount would have cost me £7.91. So, the refund should have been £9.89 – £7.91, which is £1.98. I was cheated. The garden centre should have given me a refund of £1.98, but they gave me a refund of only £1.10. They owe me 88p. Next time I go I might consider stealing a daffodil bulb.

Tuesday, 18 August 2015

Discount problem

Seeing a 20% reduction in the usual price of £10.99 for a pack of Scotts Lawn Builder in my local garden centre I was overtaken by a moment of untypical enthusiasm to do something about the state of the fading green patch in the middle of our small plot. With an additional 10% discount for having a membership card, I was well on track for a bargain here!

Unfortunately the 20% reduction had not been entered into the system and my 10% membership reduction was applied to the full price of £10.99. Because I had purchased some other items as well, I did not immediately spot this and paid what they asked.

When I looked at my receipt, I saw the error and drew it to the attention of the garden centre staff. They checked and agreed that an error had occurred.

While the queue behind me grew longer and longer the cashier struggled to work out my refund. The problem proved to be beyond the smartness of their computerised system and the mathematical ability of the cashier. In the end they gave me a refund of £1.10, which I accepted meekly, knowing that this could not possibly be right!

What refund should I have received? More than that? Less than that?

Ironically, several weeks later the pack remains unopened in my garden shed and the lawn looks as bad as ever.

The solution will appear on this blog in about a week or so.

I can't promise when the lawn will be treated.

Thursday, 16 July 2015

Australian mathematics curriculum

My apologies to anyone out there who might read this blog from time to time. I have sadly neglected you for the last three months or so, such is the busy nature of my life as a 'retired' academic! I am now back from a wonderful holiday in Provence and able to give some time to writing the occasional post.

At my publisher's request I have spent quite a bit of time mapping the Australian primary mathematics curriculum to the content of the 5th edition of Mathematics Explained for Primary Teachers, to help them with marketing the book in that country. I must say I found the curriculum considerably more intelligent in its construction and content than the shoddy and embarrassing curriculum that we now have in place in England. It has quite clearly been written by people who actually know something about mathematics and mathematical pedagogy.

I drew on this Australian material when talking about algebra at a conference for primary school teachers in Cambridgeshire a few weeks ago. We noted that the English curriculum does not recognise algebra until Year 6! By contrast the Australian curriculum has an algebra strand all the way through from the start of primary education.

For example, all these are given as examples of algebraic reasoning developing in earlier primary years:

·      Sort and classify familiar objects and explain the basis for these classifications
·      Copy, continue and create patterns with objects and drawings
·      Describe patterns with numbers and identify missing elements
·      Solve problems by using number sentences for addition or subtraction
·      Describe, continue, and create number patterns resulting from performing addition or subtraction
·      Explore and describe number patterns resulting from performing multiplication
·      Solve word problems by using number sentences involving multiplication or division
·      Use equivalent number sentences involving addition and subtraction to find unknown quantities
·      Use equivalent number sentences involving multiplication and division to find unknown quantities
·      Continue and create sequences involving whole numbers, fractions and decimals
·      Describe the rule used to create the sequence

My contention has always been that the most fundamental component of mathematics education is not doing harder and harder calculations (the message in the English curriculum) but algebraic thinking and reasoning. This is where real mathematics emerges, powerful, widely applicable and creative. So, a proper mathematics curriculum must recognise and seek to develop algebraic reasoning right through the primary years; and not see it just as an add-on in Year 6. So, well done, Australia!

But I still hope you lose the Ashes.

Wednesday, 22 April 2015

Solution for Cheryl's birthday

It is 16 July! Below is the explanation of this very pleasing problem.

Cheryl has told A and B that her birthday is one of these ten dates:

May 15, 16, 19

June 17, 18
July 14, 16
August 14, 15, 17

She has told A the month and B the number of the day, and they both know this.

A says, 'I do not know C's birthday.

This tells us nothing, because whatever month he had been told A would not be able yet to deduce the birthday.

But B would know C's birthday if he had been told that the day was, say, 19: because 19 occurs in the list of options only in May. Likewise B would know the birthday if had been told that the day was 18: because this occurs only in June.

So, then A says:

'... but I know for sure that B does not know either.'

A would not be able to say this, if the month he had been told was May or June, because for each of these two months there is the possibility that B actually knows the birthday already.

From this B (and we) can deduce that the month of C's birthday is neither May nor June.

B replies: 

'At first I did not know C's birthday, but I do now.'

This means that the number of the day (which B knows) must occur in either July or August but NOT in both these months. For example, if B knew the day was 17 then he would know it had to be August 17. Likewise, B would know the birthday if the day was 16 (July) or 15 (July).

A has deduced all this as well! He knows that the day is 15, 16 or 17. But A also knows the month. And he then replies:

'Now I know as well!

Now A could only know the birthday if the month (which he knows) has only one of these three options in the list: 15, 16 or 17. That month is July.

This is how we (and B presumably now) know that Cheryl's birthday must be 16 July.