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.

New editions published!

New editions of Mathematics Explained for Primary Teachers (5th edition) and the accompanying StudentWorkbook (2nd edition) have now been published and are already on the shelves of the bookshops. Sage Publishing has done a great job with the new editions. The turnover time these days is remarkable. It was less than a month between signing off the final proofs of the main text and printed copies of the book arriving for me in the post!

This is, of course, all related to the rapid development of publishing technology. This has also enabled us to include with the 5th edition of the textbook a one-off electronic download of the book for mobile study, obtained by a unique code provided inside. Plus loads of electronic features, accessed either by clicking on icons in the e-book version or by going to the accompanying website. These include a group of multi-choice self-assessment questions for each chapter, videos of yours truly introducing each section, and about 30 audio-visual clips sprinkled around the book providing explanations and examples of various mathematical procedures. There are electronic links to the relevant sections of the new English National Curriculum for Key Stages 1 and 2; links to Sage journal articles; and links to further practice questions.  A new feature in the book is that for each of chapters 6–29 my colleague and friend, Ralph Manning, has contributed some wonderfully creative ideas for inclusion in lesson plans, related to the content of the chapter (for Years 1–2, 3–4 and 5–6).

For someone who has never possessed an ipad or an ipod or an iphone or even an ipatch, and who thought that an 'APP' was an Accredited Purchasing Practitioner and that getting information from a tablet was what Moses did when he was given the ten commandments, this has all been quite an experience!

Since the fourth edition of this book a new National Curriculum for primary schools in England has been produced and this will be taught in schools from September 2014. This new edition has therefore been expanded and revised to ensure that the content is in line with the mathematics programmes of study for Key Stages 1 and 2 (children aged 5 to 11 years). In doing this I have ensured that the primary mathematics curricula of other countries in the United Kingdom are also covered comprehensively.

If you are familiar with a previous edition of this book I hope you will be pleased to see that I have continued in my commitment to focus on what has always been the key message of Mathematics Explained: the need for priority to be given in initial teacher training and professional development to primary school teachers developing secure and comprehensive subject knowledge in mathematics, characterized by understanding and awareness of the implications for teaching and learning.

Probability chapter rescued

I am delighted to day that at the last minute my publisher has agreed that we should reinstate the chapter on Probability in the 5th edition of Mathematics Explained for Primary Teachers (see my post on this site on 23 December). We were alerted to the fact that the enlightened Welsh and Scottish primary curriculums still include probability - and the book serves the other UK countries as well as England. And, of course, there are plenty of academies and free schools who do not have to follow the National Curriculum (ironic, isn't it!), so we want to make sure that teachers in these schools have the opportunity to understand this most important and interesting application of mathematics and to consider exploring it with the children they teach.

Now here's an intriguing use of numbers that turned up at home recently. I said to Mrs H, 'What's four plus one?' She replied, 'Thirteen'. She was right, of course. Can you explain?

And, finally, just to grab a little reflected glory: the amazing Jon Haylock doing astounding things around the Welsh coast with Griff Rhys-Jones on ITV this evening (8 pm) is my nephew.

How large is the largest known prime number?

In editing Mathematics Explained for Primary Teachers I had to update the bit about the largest known prime number, because clever people with even cleverer computers keep discovering larger ones. So far every new edition of the book has required an update! 

At the time of writing, to my knowledge, the largest known prime number was ‘2^57,885,161 − 1’. This means 57,885,161 twos multiplied together, minus 1: which produces a number with over 17 million digits.

How large is this? It's very difficult to conceive of a number with 17 million digits. So, I came up with the following idea. Imagine trying to write it down. The fifth edition of my book will contain over 400 pages. Just to print all 17 million-plus digits in the largest known prime number would require ten books of that size!

Bald criticism

In a posting on http://conservativehome.blogs.com/localgovernment/2012/02/teach-tables-and-long-division.html John Bald complains that in my best-selling book Mathematics Explained for Primary Teachers I do not describe or subscribe to long division, and that I do not explain how to teach multiplication tables.

His criticism is based on a quotation from an old edition of Mathematics Explained for Primary Teachers. In the current (4th) edition, published 2010, I do actually outline the steps involved in long division – although I continue to encourage the use of other methods that can be taught with understanding rather than learnt by rote. The fact that John Bald has managed to drill a dyslexic 12-year-old into reproducing the long division algorithm does not undermine my position. When I said that I had been unsuccessful in teaching the method, I was using the word 'teach' in a sense that does not just mean 'instruct' and that is in relation to classes of children in schools, not individual drill-and-practice tuition.

The first obvious question is whether Mr Bald's pupil will still be able to carry out this procedure in a year's time, without spending more valuable learning time continuing to rehearse it with further practice examples at frequent intervals.

The second question is whether the experience will have helped this young man to learn how to learn mathematics in a meaningful way? It is more likely that it will have reinforced a rote-learning mind set in the learner.

The third question is whether it was worth all the effort! I would be pleased about the teacher's success here if he had given the young man something that would be really useful for him. But in fact I feel sorry that this young man has had to spend so much of his time mastering something that is of such little value. Perhaps he can now move on to learning how to extract the square root of a number? Then on to how to calculate the cost in pounds, shillings and pence of quantities measured in hundredweights and stones?And then how to hunt sabre-tooth tigers?

What is it about long division that gets some people, particularly non-mathematicians like Bald, so heated? Do they not want children to have every opportunity to learn with understanding? Do they really think that learning to reproduce this one particular algorithm is the pinnacle of achievement in primary school mathematics? Do they really want children to spend so much of their time in primary schools mastering a technique that is not required in any of the questions in the end-of-Key-Stage 2 National Tests (SATs) for mathematics? Is there not enough mathematical material much more interesting and helpful and meaningful to teach anyway? There are 183 pages in Mathematics Explained for Primary Teachers on understanding number and calculations!

John Bald does concede that I advocate children being taught to memorise the multiplication tables, although again I stress that they do this with an emphasis on understanding the relationships involve – and I give examples of how this can be done. But Mathematics Explained for Primary Teachers is not a book that sets out principally to tell people how to teach. Although it contains numerous teaching and learning points, the main focus is on helping primary teachers themselves to understand the mathematical concepts and principles that underpin what they teach. The huge sales of the book and the continued positive feedback from teacher-trainees suggest that they find this approach really helps them to feel more confident in their teaching of a subject about which many of them had previously felt insecure. They tell me that to their surprise they discover that mathematics is a subject that can be understood, and that it is is not just about memorising meaningless rules and recipes for doing various kinds of questions.

Carol V's Report

Well it was a shrewd move to get Carol Vordeman to be the public face of the group reviewing mathematics in schools. Her celebratory status has guaranteed plenty of media coverage – and who could complain about what is proposed by her group?

Who would not be in favour of:

• a greater emphasis in schools on the applications of mathematics to the real world in which youngsters are growing up;
• more opportunities for students not particularly turned on by pure mathematics to engage with genuine and realistic problem-solving in real world contexts;
• more young people studying mathematics in some form or other beyond the age of 16;
• primary school teachers having a greater depth in mathematical subject knowledge and greater confidence in this area of the curriculum?

The last of these points has, of course, been the focus of my personal research, teaching and writing. Working with primary school trainees – many of whom have done no mathematics beyond scraping through a Grade C at GCSE – has revealed to me that many of them have to address worrying gaps in their subject knowledge, misunderstandings and lack of confidence in mathematics.

Requiring a grade B in GCSE mathematics as a minimum for entering teacher training would clearly help – and would enable teacher trainers to focus their efforts more on developing the 'pedagogical subject knowledge' of trainees. This is more than just knowing how many faces in a dodecahedron or being able to add fractions. It is the knowledge of mathematics needed to be an effective teacher, which would include being able to:

• anticipate and analyse pupils' errors and misconceptions;
• interpret learners' incomplete thinking;
• predict their responses to mathematical tasks;
• evaluate alternate ideas suggested by pupils and novel responses;
• give clear and well constructed mathematical explanations;
• evaluate and choose appropriately from different mathematical representations;
• sequence mathematical material appropriately for learning.

Link:

http://www.guardian.co.uk/education/2011/aug/08/maths-taskforce-gcse-split

Mathematics Explained companion website

The new companion website for Mathematics Explained for Primary Teachers, 4th edition, is now up and running. There's an interview with the author, which you may find mildly interesting, information about the book, a table of contents, a compendium of all the key Teaching and Learning Points chapter by chapter, and some useful weblinks. A really useful inclusion is a complete glossary of all the key terms that appear in the book at the ends of the chapters, combined into one alphabetical list. There are also masses of freebies to support teachers and trainee teachers who use the book. Sage Publications have been very generous in offering so much free additional material for readers.

You'll find links for each chapter of the book with the mathematics National Curriculum programmes of study and level descriptions. When the National Curriculum does finally get revised I will update this. I've done the same with the new Scottish Curriculum for Excellence (more on this interesting approach to the curriculum in a later blog) – and I intend to do the same for the Welsh curriculum fairly soon.

You'll also find five free chapters from Haylock and Thangata (Key Concepts in Teaching Primary Mathematics) that you can download, plus masses of additional further practice material, taken from Numeracy for Teaching to help trainees with the Numeracy Test.

There's even a video of a lecture on Creativity in Mathematics that I gave recently to primary PGCE students at UEA. I think this just about works in this format, although occasionally the integration of my powerpoint slides into the video is just a little bewildering. My favourite moment in the lecture is when I manage to get nearly every student in the room to give the answer 'three' to the question, 'how many lines do you need to draw to cut a rectangle into four equal pieces?'