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 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?

Yet another error in new Mathematics Primary Curriculum: 'irregular'

Call me pedantic if you wish, but I do think that mathematical terms used in the Mathematics National Curriculum should be used correctly. The misuse of the word 'irregular' is my latest find. This occurs in the measurement section of the Year 5 Mathematics Curriculum programme of study:

'Calculate and compare the area of rectangles (including squares) ... and estimate the area of irregular shapes.'

This statement seems to imply that a non-square rectangle is a 'regular' shape. The word 'irregular' is used here for two-dimensional shapes where you cannot calculate the area exactly using a formula, such as the outline of an island or that of a fried egg. These are irregular, but so are all rectangles that are not squares, all triangles that are not equilateral, and so on.

So, let's tidy this up. A regular two-dimensional shape is one where all the sides are equal in length and all the internal angles are equal. So, the only quadrilaterals that are regular are squares. All others are irregular, including non-square rectangles.

All these shapes, for example, are irregular polygons:

The National Curriculum needs another way of describing the other kinds of 2-D shapes that it has in mind for which Year 5 children should learn to estimate the area. I usually call them non-standard shapes or non-geometric shapes.

Error in Primary Mathematics National Curriculum

Come with me to the Year 5 Mathematics Programme of Study for England, where we read that pupils should be taught to ...

"know and use the vocabulary of prime numbers, prime factors and composite (non- prime) numbers"

This was clearly written by someone who doesn't!

'Composite number' is not synonymous with 'non-prime number'.

A composite number is a positive integer with two or more factors greater than 1.

So, composite numbers are: 4, 6, 8, 9, 10, 12, ...

The error in the National Curriculum is to overlook the fact that 1 is NOT a composite number. Of course, it is not a prime number either. So, 1 is the only integer that is neither prime nor composite.

OK, so it is only a small error. But it is still embarrassing to have a mathematical error in our National Curriculum: to add to all the pedagogical errors.

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)!

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!

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.

http://www.youtube.com/watch?v=I4aDZEZaF_A&feature=youtu.be

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?

Let's hear it for the Blob

Michael Gove has decided to pour scorn on the academic education community who oppose his imposed curriculum reforms by referring to them as 'The Blob'. It is incredibly arrogant of our Secretary of State for Education just to assume that his prejudices about education are necessarily right and thus to dismiss as worthless the views of those who have devoted their professional careers to researching and reflecting on the nature of learning and teaching, children in schools, curriculum theory, the aims of education, how to cater for the range of abilities and learnings styles, the nature of understanding, how to develop higher order learning, effective teaching methodologies, the promotion of positive attitudes and values, subject knowledge pedagogy, how to assess learning, creative and critical thinking, and so on. How come this branch of academic study is suddenly worthless and irrelevant and totally misguided?

Of course, I do not agree with all my colleagues in education about everything all the time. Sometimes, I do not even agree with myself. The study of children learning in classroom contexts is not a science and the practice of teaching is not a branch of technology. So there will always be different shades of emphasis. I recall, for example, being criticised by hard-line constructivists for calling one of my books 'Mathematics Explained for Primary Teachers' because, they argued, mathematics is not learnt by a teacher explaining but by children constructing their own meanings from their experiences. Yes, I agree with the constructivist viewpoint to some extent, some of the time, and in some respects, but it is not the whole truth. (Apart from anything else children like teachers who explain things clearly and help them to make sense of their experiences.) The debate and the sharing of different perspectives are helpful and important. Academic education conferences can be quite lively at times, with probing questions and challenges to various positions being adopted.

But for all our different enthusiasms and insights, there is a remarkable consensus in the field of education that Gove's reforms are driven by prejudice and reactionary opinions and that his new curriculum is a disaster, that it fails to put the needs of the child at the centre of learning, and that it has the wrong balance between learning facts and the development of higher order learning. In the proposed mathematics curriculum for primary schools, for example, most of the negative feedback in the consultation process has been ignored. There remains, for example, an over-emphasis on the rote-learning of formal written calculation methods, at a level which, in a technological age is obsolete and anachronistic. There remains a glaring mismatch between some of the laudable aims stated in the introductory paragraphs and the details of the actual content of the statutory curriculum. Mathematics is conceived in this curriculum as being mostly about doing calculations, the harder the better. This is arithmetic, not mathematics. For mathematicians the subject is about the beauty of pattern and generalising, making connections, equivalence and transformation, application, problem solving and creativity. Where is all that in this curriculum?

For Mr Gove to dismiss this consensus within the field of academic study of education just by labelling his opponents 'The Blob' and not to be prepared to listen to those who do not agree with him is arrogant, prejudiced, pig-headed, ill-mannered and, sadly, – for the next generation of children who have to live with his arid, ill-conceived, dull, narrow, inappropriate, constricting, reactionary and antideluvian curriculum – disastrous.

Let's hear it for the Blob! Somebody has to go on challenging the Govian prejudices and presumptions that are driving the current education policy in this country.

National Curriculum howler

Oh dear! There are two embarrassing statutory requirements about 'time' in the proposed Key Stage 1 Mathematics curriculum, which have led me to send the following message to the Department for Education:

There are two requirements in the Key Stage 1 Mathematics programmes of study for Measurement for pupils to be taught to 'draw hands on clock faces'. This is totally out of place as a statutory requirement. It might be a teaching device adopted by some teachers, but the objective is surely that children should learn to read the time from the hands on a clock face. My experience is that for many children the drawing of the hands on a clock face is much more difficult than the interpretation of it, so it is not even a good teaching method – because it is harder than the skill being taught! It would be embarrassing for these statutory requirements to remain in our national curriculum. 

While I was about it, I took the opportunity to request one other piece of editing (see my previous posting):

The notes on the Year 2 requirements for Addition and Subtraction state: 'Recording addition and subtraction in columns supports place value ...'. I am not aware of any evidence to support this assertion, although there is evidence to the contrary. Again, it would be an embarrassment for an unsupported assertion such as this to appear in our national curriculum. Please consider removing the words 'supports place value and' from this sentence. It would be acceptable to say that 'Recording addition and subtraction in columns prepares for formal written methods with larger numbers.'

Column addition and subtraction

I have just spotted an extraordinary assertion in the (non-statutory) notes and guidance in the proposals for mathematics in the new primary school curriculum. This relates to the programmes of study for Year 2 – that's children aged 6–7 years by the way. Alongside the requirement that children should add and subtract numbers using concrete objects, pictorial representations and mentally, we get this: 'Recording addition and subtraction in columns supports place value ...' They are talking here about additions and subtractions such as 27 + 4, 37 – 20, 24 + 25, 3 + 5 + 9.  So, apparently, children should do these calculations by informal methods, but then record them in column form, because this recording supports place value! No, it does not! There's no evidence at all to support this assertion. I can only assume that it reveals the unjustified preconceptions about the proper ways of doing arithmetic that are held by whichever individuals are driving this curriculum.

I refer the reader to the research of Ian Thompson (Educational Review, 52(3), 2000). Few children up to the age of 9 have a good understanding of the sophisticated concepts of place value. The kind of calculations they are doing in Year 2 require only a grasp of quantity value – which is that, say, 27 is made up of 20 and 7 – and where 27 comes in the sequence of counting numbers. Column recording encourages children to think of, say, 27, as being made up of a 2 and a 7, not a 20 and a 7, so it actually undermines their development in the early stages of understanding place value.

Probability and breast cancer

The newspaper headlines today provide yet another example to support what I posted yesterday about how the concepts of probability and risk have become central to our decision-making. The National Institute for Health and Care Excellence (NIHCE) has been reported today as saying that tamoxifen or raloxifene taken daily for five years can cut breast cancer risk in women by 40%.

Now the risk of contracting breast cancer is itself always expressed as a percentage. For example, across the entire population of women in the UK the current probability of a woman contracting breast cancer sometime in her life is given as 12%. This means that, on the evidence of current statistical data, it is estimated that on average about 12 out of 100 women chosen at random from the entire population will develop breast cancer.

Now within that population there are subsets of women who, because of genetic and other factors, have a higher or lower probabilities than this 12%. So, consider an example of a woman for whom the risk of getting breast cancer in her lifetime is calculated as 50%. What does it mean for the NIHCE to say that the risk is cut by 40% if she follows a particular medication regime? Sadly, it does not mean that the risk is reduced to 10%.

There is always a difficulty in understanding statements about probability that are based on percentages of percentages. This has been made clear by some of the comments made on today's report. In this example, to reduce a risk of 50% by 40% reduces the risk to 60% of 50%, which is 30%. So, in this example, the woman taking the prescribed medication has her risk of contracting breast cancer reduced from 50% to 30%. That's worth doing, of course, but she still lives with a higher-than-average risk.

So, I repeat my argument: that understanding probability is so central to real-life decision-making that the sooner we start getting children to understand the basic concepts of probability and risk the better. 

There's research evidence (Schlottman, 2001) that children as young as 6 years can intuitively understand the idea of risk and can simultaneously take into account both the likelihood of an outcome and the reward or penalty associated with it. So, it will be a real pity if teachers cannot build on this intuitive understanding of functional probability within the primary school mathematics curriculum through learning experiences that help children to construct a better understanding of such a hugely important topic.

Schlottman, A. (2001) 'Children's probability intuitions: understanding the expected value of complex gambles', Child Development, 72(1): 103–22.

Probability a great loss

Michael Gove's new primary curriculum proposals for mathematics have removed 'probability' from the programmes of study, presumably to make room for lots of extra hard, abstract and pointless calculations. What a shame! Probability is a really important application of mathematics that is fundamental to all kinds of discourse: political, medical, sociological, educational, and so on, and so on. And it's also great fun to introduce to children and helps them to make sense of a world where decisions are based continually on assessment of risk.

It is also an area of mathematics that people get wrong. Today's example was provided by Jack Straw, the former Home Secretary, commenting on the Stephen Lawrence enquiry on Radio 4. I think he meant to say that it is very unlikely that anything like the actions of the police 20 years ago could happen now. In fact, what he said was: 'the chances of that happening today are infinitesimally smaller than they were 20 years ago.' So, no change then!

Efficiency of a calculation method

The proposed new curriculum for primary mathematics stresses the importance of children learning what they refer to as the 'efficient' method of doing a particular kind of calculation. That's an intriguing concept. The 'efficiency' of a device is defined as the work achieved divided by the effort put in to achieve it.

On this basis, doing a division like 784 ÷ 18 on a calculator must surely be the most 'efficient' method: very little effort put in and the best chance of the output being the correct answer = high efficiency. For some reason, they seem convinced, though, that long division must be the most efficient method of doing a calculation like this: which involves a great deal of effort put in (so lower efficiency), and a significant chance of getting the wrong answer (which would be zero efficiency!)

OK, so they don't want children to do calculations like this on a calculator; I have no idea why, but they don't. But why would long division be more efficient than, for example, this approach:

10 × 18 is 180, so 20 × 18 is 360, so 40 × 18 is 720 (so far only multiplying by 10 and doubling, which is easy; not much effort expended!)

I now need to get from 720 to 784.

Adding 18 at a time, I get 720, 738, 756, 784.  I needed a further three 18s. The answer is 43.

I recognise real mathematical and creative thinking when 10–11-year-olds share different ways of doing multiplications and divisions, How sad if the new curriculum suppresses this and sends out the incorrect message that there is only one proper way of doing a multiplication or a division.

So, here's a challenge for any reader. Can you come up with 12 ways of calculating 75 × 12?

I'll give you the two most efficient to get you started:

1) You just happen to know your 75 times table because you are an avid watcher of Countdown, so you just write down the answer, 900.

2) Use a calculator.

OK, now find 10 more ways of doing it ...

The vital building block of mathematics?

Elizabeth Truss, Education Minister, spoke last week at the North of England Education Conference about the teaching and learning of mathematics and the current Government's plans. She set what she had to say in the context of some laudable assertions about the importance of people being able to think mathematically and logically, and to speak 'the language of mathematics'. She topped and tailed her main points with references to this country's great mathematical heritage, particularly the contribution of Alan Turing.

And then her own logic fell apart – in order to state yet again the government's determined agenda to ensure that all primary school children are drilled in the algorithms of long multiplication and long division. Here's what she said ...

'... the growing importance of maths shows we need to do more to make sure children speak that language too.'

Who could disagree with that? But, look out, here comes the non sequitur. Next sentence ...

'That is why we are redesigning the primary maths curriculum to focus on mastery and fluency of the vital building block of mathematics, which is arithmetic.'

What? How does that follow from the previous sentence? It's clear from what she goes on to say that what she means by 'mastery and fluency of ... arithmetic' is drill and practice in long multiplication and long division. This is total nonsense! Who in the field of mathematics or mathematics education would ever identify being able to do calculations like 49 × 32 using the traditional long multiplication algorithm and 867 ÷ 17 by the long division algorithm as 'the vital building block of mathematics'? Or even a slightly important building block?  How on earth is spending so much of their time mastering and practising long division going to help children to 'speak the language of mathematics'?

Recently on the television programme Countdown, a highly gifted young man studying mathematics at Cambridge took the easiest option in the numbers game, because, as he said, 'I may be studying maths at Cambridge, but I haven't seen a number for two years!' From where do these politicians get the idea that developing mathematics is first and foremost about doing calculations? And that the key to success in mathematics is to be able to do harder and harder calculations? (Or, as Elizabeth Truss calls them, 'sums' – how embarrassing to have an Education minister who misuses the word 'sum' when outlining policy on the teaching of mathematics!)

Does Elizabeth Truss really think that being able to do long division was even the tiniest bit significant in Alan Turing's work? And don't tell me that learning this taught him how to think algorithmically. What he had to be able to do was to devise an algorithm to solve a category of problems, not learn by rote the steps of an algorithm. Teaching children to master by drill recipes for doing calculations is not going to develop their ability to think algorithmically, to solve problems, to break problems down into their component parts, to reason mathematically. Ironically, the algorithms developed by Turing and his successors have led to the redundancy of formal written calculation methods consequent on the availability of the calculator on the mobile phone in the pocket of just about every 11-year-old in the country!

The crunch in the speech is that the Government has devised an underhand way to ensure that primary schools teach their favoured written methods for calculations. In the end-of-Key-Stage-2 national tests for mathematics, children will be awarded method marks for a question involving a calculation if they get the answer wrong only if 'their working shows they were using the most efficient method'! 'The most efficient method' is government-speak for the traditional formal written methods.

What, I wonder will count as 'the most efficient method' for, say, a question that involves finding the product of 49 and 32? Surely, the most efficient way of doing this is to find 32 fifties (easy, because it's the same as 16 hundreds) and then subtract 32: giving 1600 – 32 = 1568. In this case, would a child who tries to use long multiplication and gets it wrong be ineligible for 'method marks', because they have not used 'the most efficient method'? Logic says they should, but then logic is not the strong point of the present government's approach to the primary curriculum.