Saturday, 12 April 2014

Context-free calculations

So we now know that the national tests for children at the end of Key Stage 2 from 2016 onwards will include one paper (of the three) that will consist of context-free calculations.

That seems a strange idea to me. Mathematics – as far as I understand it as a humble mathematician –
does not generate calculations without a context. Calculations only ever occur in a context. Normally this would be a practical context in which mathematics is being applied, and where the numbers are likely to be attached to sets of items or units of measurement of some kind. Even in pure mathematics, on the rare occasions you might want to do a calculation it would be to investigate the relationships between two numbers that have some property – like finding the ratio of successive terms of the Fibonacci  sequence (1, 1, 2, 3, 5, 8, 13, 21, ...) – so there is a context which provides the need to do the calculation.

Yet our children will be given a test paper consisting entirely of context-free calculations, to be done by prescribed formal written methods, the argument being that these calculations are 'the fundamental processes' of mathematics.


Wednesday, 19 February 2014

Mathematics national curriculum oddities

The statutory requirements for the new primary mathematics curriculum (to be taught from September 2014) have some odd aspects for which it is difficult to find a rational justification.

Here are two examples.

1) Under algebra, children (the curriculum prefers to refer to them as 'pupils') in Year 6 are required to learn how to

·    find pairs of numbers that satisfy an equation involving two unknowns

but nowhere are they required to find solutions to equations with one unknown!

2) In learning about geometric transformations, children will meet the concept of 'rotation' in Year 2, but then the word does not appear again in the primary curriculum. So, after Year 2 there's nothing at all about recognising and describing a transformation of a shape in terms of rotation and there's nothing anywhere in the curriculum about rotational symmetry.

Monday, 20 January 2014

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.

Tuesday, 24 December 2013

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 ‘257,885,161 − 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!

Monday, 23 December 2013

Fifth edition of Maths Explained

I am aware that I have rather neglected this blog for the last month or so. The reason has been that I have been in the last stages of revising Mathematics Explained for Primary Teachers, to be published in its fifth (!) edition in 2014. I am pleased to say that I managed to get it all sent off to Sage Publications around the  middle of December, only a couple of weeks late.
My major task was to accommodate Michael Gove's new primary mathematics curriculum – without compromising my principles regarding the teaching and learning of mathematics. I am pleased with what I have managed to do in this respect. Yes, I have had to put in a lot of extra material on fractions and standard algorithms for multiplication and division. I feel an obligation to the future teachers who will have to teach this stuff to primary children, to ensure that they have the chance to understand what they are teaching. And, yes, I have regretfully had to ditch a whole chapter on probability, now removed from the primary experience. It must have been too much fun. But the core material of the book, and its characteristic approach and style are retained.
The book will appear in both electronic and paper formats, and will have have loads of clever electronic features in it. I will not say more about these at present, other than to say that I am having fun preparing these!
And the Student Workbook that accompanies the main text book has to be revised as well, so I am currently working on that. I may take a day off for Christmas.

Tuesday, 12 November 2013

The heart of mathematics:

In an interview about the new primary mathematics curriculum, Debbie Morgan, Director of Primary at the National Centre for Excellence in Teaching Mathematics, has stressed that mathematical reasoning and problem-solving are to be at the heart of children's experience of mathematics.

She argues for the prominence of the three aims given at the start of the mathematics section of the document in determining what children learn and how they learn it. These three aims are, in summary, (1) fluency based on conceptual understanding; (2) reasoning mathematically; and (3) problem-solving.

All this would be great, if it were not for the fact that the detail in the subsequent pages and pages of statutory requirements focusses almost entirely on the first of these and contain very little to indicate how precisely the other two are to be developed.

Two factors that will be crucial in this are the end-of-key-stage 2 national tests (the so-called SATs) and Ofsted's approach to inspection.

Over the years those responsible for the national tests have at least identified aspects of 'using and applying mathematics' as defined in the current curriculum that can be assessed in the context of written tests. This has been possible because, for all its deficiencies, the current curriculum does actually  contain specific statutory requirements for children's learning in this respect. I fear that the likelihood is that there will be political pressure on test developers for the 2016 tests onwards – when Gove's new curriculum will start to be assessed – to emphasise disproportionately the assessment of written, formal arithmetic skills and to assess only the detailed statements in the programmes of study. Is there any hope that aims 2 and 3 will be not be overlooked entirely in the national tests? If they are then teachers will overlook them in their classrooms as well. And Debbie Morgan's laudable aspirations will prove to be a fantasy.

And is there any hope at all that Ofsted will give the highest endorsements to those schools who seek to embrace all three of the aims in the experience they provide for children? Will they be checking that children are engaged in genuine mathematical reasoning, following a line of enquiry, conjecturing relationships and generalisations, developing arguments, applying their mathematics to non-routine problems? Or will they just be checking that Year 4 children can multiply a 3-digit number by a single-digit number using the formal layout, that Year 5 children can multiply a mixed number by a whole number and that Year 6 children can divide a fraction by a whole number?

Tuesday, 15 October 2013

1729 and Futurama

Simon Singh has written a very enjoyable piece on the BBC website today about the appearance of the famous 'number on the taxicab' (which is 1729) in the programme Futurama. For anyone who enjoys a bit of number theory, it's well worth reading. Here's the link: