Wednesday, 23 July 2014

Learning curves

If I hear another person say that they have been on a steep learning curve I shall explode! Here is another example (see previous post) of a phrase from a field of knowledge being taken up by those who do not understand it, and then misused – and, of course, over-used! So people tell me they've started a new job and they've been on a learning curve, when all they mean is they have been learning.

'Learning curve' is a term from the field of the psychology of learning. It refers to a graphical representation of the progress of an individual's learning over time. So, if you are learning some new skill your progress in learning can be represented by a graph, with one axis the cumulative learning achieved (going from 0% to 100%) and the other axis the time devoted to learning.

So 'being on a learning curve' really does not mean anything other than 'learning'! So, why not just say 'I have been learning'!!? Being on a learning curve does not mean that the learning has been especially difficult, demanding, rapid or slow! Anyone who is learning anything can have their progress in learning modelled by a learning curve! Here's an example.

The learning curve simply attempts to model the progress of the learning. In this case, initially the learning curve is shallow. This is because to start with the learner makes slow progress, probably because learning is difficult. Then the rate of learning picks up, as learning gets easier. Then it gets shallow again, as towards the end of the process there is not much more to learn and a lot of time is given to consolidating what has been learnt already.

Here's another example: to start with this learner is on a 'steep (part of a) learning curve'!

Initially this learning task is found to be easy. Learning is rapid, and the learner makes quick progress. Again there is a tailing off in the rate of learning towards the end, as in the previous example.

So, the learner in the initial stages here might well say (correctly) 'I have been on a steep part of a learning curve'. But in saying this they would mean that the learning has been really easy! But people talk about 'steep learning curves' as though the steeper it is the more challenging has been the learning!

So, listen up!
1. Please don't say 'I've been on a learning curve' when you just mean 'I have been learning'!
2. Don't say 'I have been on a steep learning curve'. No learning curve in practice is 'steep' in every part.
3. You may say 'I have been on a steep part of a learning curve'. But be aware that this probably means the learning has been really easy for you!
4. If you want to imply that the learning has been difficult for you, say 'I have been on a really shallow part of a learning curve'!

Better still, just forget all about learning curves. Just talk about learning.

Tuesday, 22 July 2014


What has happened to the powerful mathematical word 'parameter'? More and more we hear people, particularly those engaged in management speak, who clearly have not a clue as to what the word really means, and using to it to mean 'perimeter'! So they talk about 'having to work with certain parameters', meaning within certain boundaries. This has become so commonplace that some dictionaries seem to have conceded this as a secondary meaning. This is all very irritating to a mathematician! Either leave our concepts alone please or use them properly! 

Parameters are not things that you work within. The word is a mathematical term for what is sometimes called, enigmatically, a variable constant. To take a simple example, the equation y = mx is the general form of the equation for a straight line passing through the origin. The x and y are the independent and dependent variables, respectively. But m represents the slope of the line. For any given line, it is a constant. But m can take any value, so it is also a variable. So the value it takes determines which straight line through the origin is being considered. This m is a therefore a parameter. 

If you want to transfer this idea to a business or social context, then you could say that the parameters are the variables whose values determine precisely the nature of the situation you are in. It would be perfectly acceptable, for example, to say that 'two of the the parameters for an architectural project are budget and completion time'. But not to say that 'the parameters of the site are clearly marked on this map'. That's talking about perimeter!

Tuesday, 15 July 2014

"Come in Number 24, your time is up!"

So, education bids farewell to Michel Gove!

In the years of my career in education I have now seen off 24 secretaries of state for education (or variations on the title of the office)! Michael Gove is number 24 on this list, ordered both chronologically and in relation to my personal enthusiasm for his policies!

Looking back, it's quite a list! There are such enduring political names as Margaret Thatcher, Shirley Williams, Keith Joseph, Kenneth Baker, Kenneth Clarke, David Blunkett, Ed Balls ... all have come for a while, dabbled in education, and moved on.

So, we know who would be at the bottom of this list of twenty-four education supremos. But who would I put at the top?

Surprisingly (to me at least), probably the secretary of state who made the most significant impact on education in this country was Kenneth Baker, who in the 1988 Education Act brought in a national Curriculum for the first time and instituted in-service training days for teachers (still called Baker days). He has since won browny-points for his outspoken criticism of Gove, accusing him of basing his policies too narrowly on his own experiences. And he is currently overseeing what looks like a first-rate initiative in education: the university technical colleges.

Sunday, 29 June 2014

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.

Wednesday, 21 May 2014

676 is the new 243

Where is the mathematics in the new mathematics curriculum for primary schools in England?

Some of us in mathematics education will remember the significance of the number 243: not just because it is 3 raised to the power 5, although that helped us to remember it! Paragraph 243 in the hugely influential Cockcroft Report (1982) was very significant in defining what constituted a good and balanced mathematical experience for learners. This one paragraph had a great impact on mathematics teaching and learning and was for many years a positive influence on the mathematics curriculum. Do an internet search on 'Cockcroft 243' and you will see what I mean.

However, 243 is a long way back in history now, so instead I would like you to welcome and celebrate with me 676! I will explain.

The new primary mathematics curriculum for England, being introduced later this year, has an excellent statement of the purposes for learning mathematics. It also has a statement of aims that includes aspects that I would recognise as being real mathematics: in particular, the development of the distinctive ways of reasoning in mathematics and solving problems using mathematics (Aims 2 and 3). But these are then followed by pages of specific learning targets in the programmes of study that just do not seem to match the laudable aims and purposes. My fear has been (and still is) that teachers will focus just on the details in these programmes of study and real mathematical experience for our children will be neglected in favour of rote learning of routines, such as long division and calculations with fractions.

In particular, since there is no longer anything comparable to the 'using and applying' strand in the programmes of study and attainment targets for the current curriculum, there seemed a real possibility that the national assessments at the end of Key Stage 2 would focus just on the specific detail in the programmes of study.

The good news is: they won't!  Of the three papers, one (27% of the marks) will focus on context-free calculations, but the other two will contain a genuine focus on reasoning and problem-solving. And to make sure this happens we have paragraph 6.7.6 in the Key Stage 2 Mathematics Test Framework for National Curriculum Tests, published recently by the Standards and Testing Agency. This paragraph defines in detail what is expected of children at the end of Key Stage 2 in terms of mathematical reasoning and problem solving. This is where the real mathematics is in the mathematics curriculum! Maybe not in the programmes of study, but clearly there in the assessment criteria. And the hope is that if all this is going to be assessed it might just be taught!

Section 6.7.6 Solving Problems and Reasoning Mathematically
Children working at the expected standard are able to:
·            develop their own strategies to solve problems by applying their mathematics to a variety of routine and non-routine problems, in a range of contexts (including money and measures, geometry and statistics) using the content described above
·            begin to reason mathematically making simple generalisations, using mathematical language and searching for solutions by trying out ideas of their own
·            use and interpret mathematical symbols and diagrams, and present information and results in a clear and organised way; for example:
·            derive strategies to solve problems with a two or three computational steps using addition, subtraction, multiplication and division and a combination of these
·            solve problems involving numbers with up to two decimal places
·            select appropriate strategies when calculating depending on the numbers involved
·            use rounding and estimation to check their answers and determine, in the context of the problem, appropriate levels of accuracy
·            identify simple patterns and relationships, and make simple generalisations
·            draw their own conclusions and explain their reasoning in simple contexts using mathematical language
·            make simple connections between mathematical ideas
·            solve problems involving data 

So, hail 676, say I! Three cheers for 676! 676 is the new 243! Let's make sure that our primary school teachers are aware of 676. It justifies – and indeed requires – a proper focus in their mathematics teaching on genuine mathematical experiences for our children.

And, to cap it all, 676 is an interesting number, being the product of the squares of two primes (2 and 13)!

Wednesday, 7 May 2014

MG enthusiasts?

"I noticed recently a logo on my next-door neighbour’s van indicating that he was an MG enthusiast. I found the idea of an MG enthusiast an intriguing concept, particularly because I would have thought that they were rather few and far between. There certainly aren't many in the teaching profession! 

So, I asked him about these MG enthusiasts and I discovered that there were more of them around than we might expect. I learnt that MG enthusiasts tended to be rather reactionary individuals, a bit right wing politically, wedded to traditional ways of doing things,  dedicated to a way of life fifty years out of date, motivated by nostalgia and memories of life when they were young, committed to the restoration of a vehicle that has been out of production in this country for years, and happy to drive and preserve something that ignores the computer-based technology of the 21st century - but also excited to know that the object of their desire is still in production … in Shanghai!"

The above was my introduction to a talk about the new primary mathematics curriculum last week to a conference at UEA Norwich. I wanted to make clear at the outset that I was not an MG enthusiast, even though I was going to go on to identify some good things in MG's new curriculum (more of that in a later post on this blog).

Unfortunately, some members of my audience missed the subtlety of what I was saying (initially) and had to have it explained to them later!

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.