Friday, 4 March 2016

Roman numerals

Here are three questions about Roman numerals that genuinely puzzle me! 
First, why, in the twenty-first century, do we persist in using Roman numerals in particular contexts, such as the hours on clock or watch faces, and dates on buildings or at the end of a movie? 
Second, in such a technological age, can anyone really justify the inclusion of Roman numerals in the statutory English primary mathematics curriculum and in the associated national assessment of mathematics? 
Third, there’s something odd I’ve noticed recently about Roman numerals on clock and watch faces. It is important for teachers to be aware of this, because there are limited contexts for assessment items in Key Stage 1 and 2 national mathematics tests in this country, so clock and watch faces with Roman numerals turn up often. 
In an early form of Roman numeration, the numbers we call ‘four’ and ‘nine’ would be represented by IIII and VIIII. A later development was to represent these more concisely as IV and IX, the convention being that when a letter representing a smaller value is written in front of another letter, then the value is to be subtracted, not added. So, XC would represent 90 (100 subtract 10). So, here’s what we have noticed: in most cases where Roman numerals are used on a clock or watch face, the four is written using the early system (IIII) and the nine is written using the later system (IX). Check this out and see if we are right. The question that puzzles me is, simply, why?

Thursday, 25 February 2016

The Haylock Bookmark

When I am in the middle of reading a book I need a bookmark that not only helps me to find the page I am on, but also exactly where on the page I should start to continue my reading! I offer readers my design for an effective bookmark for this purpose and a simple example of mathematics applied to another area of the curriculum!

First measure the height of the page of the book (h mm) and the height of the smaller of the top and bottom margins for the text on the page (x mm). Cut a rectangular piece of firm card so that it has a length that is equal to h/2 + x. The width can be whatever you wish, say, about a third of the width of the book.

For example, for a standard paperback novel I find that h = 198 and x = 19. So we need the bookmark to be 99 + 19 = 118 mm long.

Then on one side of the card draw a double-headed arrow x mm from the top (i.e. 19 mm for our standard paperback). The diagram shows a bookmark made for this standard paperback.
That's it! Now, when you have finished your reading session place the bookmark in the fold of the book on the opposite page to the one you are on, with the arrow showing and pointing to your place on the page. This might involve rotating the bookmark, of course, depending whether you are on the top half or the bottom half of the page. Then just close the book. When you open it next time just make sure you have the side of the card with the arrow visible, and remember that the arrow tells you where on the opposite page you should start reading again!

With this system any starting point on either the even-numbered page (verso) or the odd-numbered page (recto) can be marked, without the bookmark sticking out of the closed book (which I don't like).

The diagrams below show the bookmark placed inside the book to mark (a) a point half-way down the left-hand page; and (b) a point near the bottom of the right-hand page.



Well, it works for me. And it is a good application of simple measurement and spatial reasoning!

Tuesday, 9 February 2016

Mastery and understanding mathematics

In the context of the challenge to raise standards in mathematics in schools in England the word ‘mastery’ has recently become prominent in the vocabulary of the English mathematics curriculum (NCTEM, 2014, It is reassuring to note that the way in which the word ‘mastery’ is being used is entirely consistent with the approach to children’s learning of mathematics that I have promoted in my own writing.
Mastery is seen as children developing fluency in mathematics alongside a deep understanding of mathematical ideas and processes. So, for example, teaching approaches for mastery should ‘foster deep conceptual and procedural knowledge’ and ‘exercises are structured with great care to build deep conceptual knowledge alongside developing procedural fluency’ (op.cit.). This is a key principle in teaching mathematics to young children: that mastery of the subject is not achieved simply by repeated drill in various procedures. Instead, the focus is on the development of understanding of mathematical structures and on making connections.
Making connections in mathematics – a recurring theme in all my books – ensures that ‘what is learnt is sustained over time, and cuts down the time required to assimilate and master later concepts and techniques’ (op.cit.). Nearly all mathematical concepts and principles occur and can be applied in a wide range of contexts and situations. Because of this, the deeper understanding central to mastery in mathematics is facilitated by a wide variation in the experiences that embody mathematical ideas.
For example, mastery of the 5-times multiplication table by Year 2 children is not just a matter of memorizing a chant that begins ‘one five is five, two fives are ten …’ – although that is part of it. It would also involve, for example:
·       connecting each result in the table with a collection of 5p coins and the total value;
·       articulating the pattern of 5s and 0s in the units position in the odd and even multiples of 5;
·       explaining how to get from 4 fives to 8 fives by doubling;
·       explaining how to get from 6 fives to 7 fives by adding 5;
·       counting in steps of five along a counting stick;
·       knowing that, say, ‘3 fives are fifteen’ is what you use for the cost of 3 books at £5 each;
·       constructing patterns with linked cubes that show 1 set of five, 2 sets of five, and so on;
·       filling in the missing number in number sentences like ‘6 × = 30’.
To teach for this kind of mastery teachers themselves need a deep structural understanding of mathematics, an awareness of the range and variety of situations in which a mathematical concept or principle can be experienced, and confidence in exploring the connections that are always there to be made in understanding mathematics. Any teachers looking for this? I can recommend one or two books.

Thursday, 31 December 2015

Three halves make a whole?

Our Prime Minister, David Cameron, came in for some criticism this year for his use of the word 'half', when, in answer to a question about what were the Houses of Parliament like, he replied, 'half museum, half chapel, half school'. 'You can't have three halves!' was the outcry.

Obviously, what he said is not intended to be an exact statement of proportions, but even so this looks as sloppy as the often-heard 'I'll have the bigger half'.

However, on reflection, Cameron is OK in what he says here. You can have three halves of a set, for example, if the halves overlap. For example, a teacher could say, half of the children in my class are Asian, half of them are male, and half of them walk to school.

So, if, we were to assume that there are 100 quintessential qualities or features that characterise the Houses of Parliament, it is quite possible that 50 of these are shared with those of a museum, 50 of them are shared with those of a chapel, and 50 of them are shared with those of a school. This is because the quintessential qualities of museums, chapels and schools are not unique to each of these kinds of places.

So, it seems alright to say that Parliament feels like half museum, half chapel and half school!

There! It's not often you get me supporting our Prime Minister; but fair's fair.

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.