## Wednesday, 13 August 2014

### Golden Ratio and Beethoven's 5th

I had an email yesterday from a student in Spain that began:

'I am currently carrying out a Research Project about the Golden Ratio. In fact, what I am particularly interested in is its appearance in classical music and its use by some musicians in their compositions. It has been brought to my attention that you have been studying this matter. If it is not too much to ask, given your expertise, it would be of great help to me if you could share some of your thoughts about this particular topic ...'

In September 1978 (yes, 36 years ago!) I wrote a brief one-off article that was published in Mathematics Teaching (volume 84) on this subject, mainly focussing on some interesting occurrences of the 'golden section' in the structure of the first movement of Beethoven's 5th symphony. My suggestion was that this might make an interesting investigation for arts sixth-formers doing general studies courses. I have never returned to this subject and not written any more about it. But that little article – from my perspective probably the least significant piece I have ever had published – continues to get cited and references to it turn up often in internet searches. So, 36 years later once again I get someone contacting me as though I am an expert in this field! Which I am not.

However, this is what the article was about.

First some revision. A 'golden rectangle' is one with the lengths of the sides in a particular proportion: approximately 1.618 (or, precisely, half of '1 plus the square root of 5').  So, for example, imagine a rectangle with sides 1 cm and 1.618 cm, a shown. This is a golden rectangle. If you cut off a 1-cm square, as shown, the piece that you are left with has sides of 1 cm and 0.618 cm. The ratio of the lengths of these sides is 1 ÷ 0.618 which, remarkably, equals 1.618! So this smaller rectangle is also a golden rectangle!
This is the defining property of a golden rectangle: that if you cut off a square, as shown, you are left with a rectangle with sides in the same ratio as the original rectangle. From this, with a little bit of algebraic manipulation, you can work out that the ratio of the sides is 1.618 to three decimal places. Dividing the rectangle like this - or the division of any quantity into two bits that are in this ratio - is also called a 'golden section'.

For example, if I had 1618 counters and put them into 2 piles with 1000 in one pile and 618 in another, then I would have applied a golden section to the set of counters. If I were then to apply a golden section to the 1000 counters, I would get 618 in one pile and 382 in the other. And so on.

If the smaller golden rectangle in the diagram is divided into a square and an even smaller golden rectangle, the 1 cm length is divided into 0.618 cm and 0.382 cm. So a golden section can be thought of as dividing 1 unit into two parts in the ratio 0.618:0.382 (or, to two decimal places, 0.62:0.38).

This is a magical ratio that keeps turning up in all kinds of situations, in mathematical contexts, but also in art, architecture, the natural world, psychology, and, indeed, music. Bartok, for example, deliberately incorporated the golden ratio into his musical composition.

My discovery was that there are a number of occurrences of this golden ratio in the first movement of Beethoven's fifth symphony – that's the one that starts with the famous motto theme: 'da, da, da, daah'! In Beethoven's original score there are 600 bars before the final statement of the opening motto. But a statement of the opening motto also appears at bar 372. So, we have this structure for the three main statements of the motto: motto starts ... 372 bars ... motto starts... 228 bars ... motto starts.

Now what do we get if we split the 600 bars into 2 section using the approximation to the golden section 0.62:0.38? Well, amazingly, 0.62 of 600 is 372, and 0.38 of 600 is 228. So the three statements of the opening motto theme begin at points that divide the score using a golden section!

Is this just something that happened by accident, or did Beethoven do it deliberately? We shall probably never know. I have no view on this. But there are other instances in this movement. I will mention just two.

1. The 'exposition' in the first movement (the statements of the two main musical subjects) is in three sections: 24 bars of the first subject rounded off with a version of the motto; then 38 bars with an extended restatement of the first subject rounded off with a version of the motto; then a further 62 bars that begin with the statement of the second subject. 24:38 is a golden section of the first 62 bars; and 38:62 is a golden section of the last 100 bars!

2. The movement has the most extraordinary 'coda'. A coda is usually just a few bars of music to round off a movement. In this case Beethoven gives us a coda that is 129 bars long! Divide this coda up using the golden section and you get 49 bars and 80 bars. So what happens after 49 bars of the coda? This is the point where Beethoven actually introduces a completely new tune that has not appeared in the movement so far. Before Beethoven no-one would ever introduce new material in a coda! So this is a very significant point in the coda. Is Beethoven signalling his piece of radical creativity in this very long coda by linking it with the golden section? Again, we shall never know. But either way, it is interesting.

## Tuesday, 5 August 2014

### Number or amount

Here comes another little rant about language. This one is about the distinction between 'number' and 'amount'. Almost every day I hear the word 'amount' used where the word 'number' is what is required (never the other way round). I am getting increasingly irritated by the number of people who talk about 'the amount of people', for instance.

The distinction is so simple – and actually significant in mathematical terms. Some variables are discrete and measured by counting the number of items in a set. The value of such a variable is 'the number of items'. Other variables are continuous and measured by the size of the quantity, often in units of measurement such as grams or litres. The value of this kind of variable is 'the amount'.

It's easy to spot the difference. If you are using a plural noun then it is 'the number' of items. If you are using a singular noun then it is 'the amount'.

The number of bottles of wine, but the amount of wine in each bottle.
The number of cups of coffee, but the amount of coffee in each cup.

Here are some other examples.

I would talk about the large number of slices of toast I had for breakfast (slices: plural); but the huge amount of toast (toast: singular).

I would talk about the number of people in my garden on Sunday lunchtime (people: plural); but the amount of food that they ate (food: singular).

I would talk about the number of cars on the road (cars: plural); but the amount of traffic (traffic: singular).

## 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

### Parameter

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)!