John Bald/language and literacy

Sir, what's pi? An attempt at an explanation.

In a maths class in Educating Essex,  a girl asked this question, and didn't get an answer. In one sense, no-one knows. π, or pi, is the circumference of a circle divided by its diameter. It is a constant figure, but one that cannot be precisely described in terms of a number. People have calculated it to an almost infinite number of decimal places, and will no doubt continue to do so.

A history of π , from David Wilson of Rutgers University Maths department is here, including the speculation that it was first discovered by ancient Babylonians using lengths of rope. This could usefully be repeated using a hula hoop, taping string carefully round the outside, and comparing its length with a string stretched across the diameter. The outside string will probably be a little over three times as long as the string stretched across the diameter.

The next stage in the demonstration is to draw a line on a piece of squared paper, of a reasonable length. We don't need to measure it, and we can call it r.  We can then draw a square with r as its base, and we know that the area of the square is r².   Draw another square beside it, using the side of the first square as one of its sides, and the area of the two squares is 2r. Do the same below, so that we have four squares the same size, a little like a block of four postage stamps, and we have four squares with the same central point, and so 4 lots of the original area - 4r².

Set a pair of compasses, carefully, to the same length as r, and draw a circle on the mid point of the squares. The circle will touch the edge of the square once at each side, and cut off the corners. It's easy to see that the area cut off from the corner is the same each time, so multiplying r² by four is going to give an area that is too big. How much are we cutting off? We know that it's the same amount each time - not only can we see it, but the compass is set to the same length all the way round the circle (or it wouldn't be a circle). We also know that it's less then 4. 

We might repeat our experiment with the hula hoop to putting string round other large-ish discs and coins, and find that the longer string is always a little over three times the length of the shorter one. Again, this corresponds to what we can see in the circle - the area is always that much smaller than the area of the squares. But how long is a piece of string, and how accurate are we in measuring it? π provides the answer. We might not know what the ratio is exactly, but we know that it is the same for any circle. Multiplying r², the are of one of our squares, by π, or, for the figure used for GCSE foundation, 3.14, will give us the same reduction from the area of the squares to that of the circle.

This explanation has been helpful to students taking GCSE (and some parents), who remembered the formula, but did not know how to apply it, for example calculating (πr)² and not seeing how this produced an error. There may be clearer explanations available, and I'd be pleased to hear of them.

 

 

Churchill, language and education, with implications for foundation examinations and Further Education.

Three of Churchill’s comments, respectively on education, communication, and examinations, can be of great help to us in solving current problems.   The first two are his tribute to his teacher of grammar,  who showed him how to construct a sentence, and his statement that, “Broadly speaking, the short words are the best, and the old words best of all”. Children do not, for the most part, know enough Latin and Greek to understand the basis of terminology derived from them, and this terminology too often creates a mystique that obscures learning instead of promoting it.

In geometry, for example, the difficult word Isosceles is just Greek for two equal legs, and a theorem is something that we can prove on the basis of things we already know. I was taught many theorems at school without anyone ever explaining to me what such a thing actually was, and similar problems persist. Maths compresses language, frequently inventing its own vocabulary and symbols to do so. All of this needs to be unpacked and practised if most children are to learn maths – going past anything that is not understood creates a block that can last a lifetime. Churchill’s principles of communication are as important and effective in education as they were in politics.

Of examinations, Churchill said that, “the questions … were almost invariably those to which I was unable to suggest a satisfactory answer. I should have liked to be asked to say what I knew. They always tried to ask what I did not know. When I would have willingly displayed my knowledge, they sought to expose my ignorance. This sort of treatment had only one result: I did not do well in examinations.”

In the security queue at Westminster some weeks ago, I met two ladies who were principals of Further Education colleges in London. One told me that she had over 2,000 students retaking maths and English at GCSE. That seemed a lot, until the second announced that she had 4,000. A friend’s son, in charge of maths in a home counties college, has 3,500 students. Each of these figures is larger than almost any school.  The unfortunate lecturers have around eight months to get to know the students and turn round a lifetime of disappointment, with predictable results – the failure rate last year was 70 per cent, the highest I’ve ever heard of in a public examination.

The students are acutely aware of their situation and, unsurprisingly, hacked off with it. My friend commented that his son could cope with all sorts of difficulties in learning maths, but was now faced with sustained unwillingness to work. The reason is not hard to grasp. Almost all of us, setting out to achieve something important – and everyone knows that maths is important – give it our best shot, and are prepared, like Robert the Bruce’s spider, to keep trying if we don’t get it right first time. But if we’ve been failing for eleven years, and are given material that we have no hope of understanding, because we don’t know the elements on which it is based, or understand the language and symbols in which it is expressed, we are worse off than the spider. We don’t have a thread to climb.

Unsurprisingly, the two college principals felt that their sector was a poor relation, and that their lecturers were hard-working people facing an impossible task. By co-incidence, one of them was head of the college that had asked me to lead a revision day last year, so that I had seen the truth of their situation for myself. Sir Michael Wilshaw was asked at the presentation of his final annual review if the requirement for all students to resit GCSE “wasn’t working,” and he replied diplomatically that it may be worth considering “some form of test of fundamental skills.”

Sir Michael was right, and his comment brings me back to Churchill. The foundation level of these maths papers, which most resit students take, has far too many questions that insert a twist to force students to apply what they know in a situation that is not obvious. An example from a sample paper is a question asking them to prove that a triangle in a diagram is Isosceles, when it is deliberately drawn without two equal sides. Another is to ask them to calculate the cost of providing some cakes for a party, involving five separate calculations, ending with the amount of change they’ll receive.

These little steps and twists add appropriate rigour to higher level papers, but prevent foundation students from showing what they know in a straightforward way, and shove their limitations down their throats. GCSE foundation papers, and not just in maths, need to be redesigned on Churchill’s principles.

This posting first appeared on Conservative Home.

Geometry without tears: Laying the ghost of Euclid.

The tears - or, more accurately the four-hour sessions of homework, ending in complete incomprehension and occasional migraine - were mine. This was maths, and therefore important, so I had to do it and couldn't. The late Bill Shankly said that football was not a matter of life and death, but something much more important. So was maths. The cleverest pupils could do in ten minutes what I couldn't do in four hours. Across Europe, I'm quite sure that millions of people are in the same position. Hundreds of thousands of FE students in the UK are set up to fail maths yet again, with no hope of passing, and in a state of despair or passive revolt. 

Euclid appears in illustrations as an old man with downcast eyes, sparse hair and a straggly beard. His work is available, free,  here, with explanations from Professor David E Joyce of Clark University, Worcester, Massachussetts. I am extremely grateful to Professor Joyce for enabling me to lay the ghost of a man I saw as my tormentor. Professor Joyce describes Euclid's elements as "beautiful and influential". For most of those who approach it, it does not have these qualities, because, with the best of intentions, it is mistaught.

To begin at the beginning, Euclid used a plain straight edge,  compasses of some sort, and some means of writing, probably akin to a pencil. He did not own a protractor, or a marked ruler. These items are universal in secondary schools, and were my undoing. They encourage people to approach problems by measuring them, and this runs counter to Euclid's system of logic - you can't be sure that your measurement is completely accurate. You can, on the other hand, be sure of the logic that stems from Euclid's simple axioms - for Professor Joyce, Common Notions - and postulates. The only angle measurement in Euclid's work is the right angle - others are described in relation to it. The key to his logic is is that if two items are equal to a third item, they are equal to each other. Mathematicians have pointed to the limitations of Euclid's logic, as these suppositions are themselves not proved. For example, Euclid says that all right angles are equal to each other, but does not set out to prove it, perhaps because it is not possible to prove it - he would have had no way of proving that right angles drawn on two pieces of papyrus, were in fact equal to each other.

Proposition 1  shows the method to perfection, and is easy to teach. Draw a straight line, not too long for your piece of paper, but do not measure it. Set your compasses to the length of the line, and draw a circle centred on one end of it. Do the same from the other end. Both of your circles are equal to each other, because they have the same radius - the distance from the centre of a circle to its edge is always the same, or it isn't a circle. So, joining the two ends of the line, to the point at which the circles intersect, creates a triangle in which all sides have this radius, and are therefore equal. It works every time, and is a perfect introduction to Euclid's system of logic. The rest follows.

Professor Joyce has performed a most important service, in making this work freely available to those of us who have needed it - pro bono publico, of the first order.

Conclusion: Protractors and marked rulers should not be used in the teaching of Euclidian geometry. They are a worse distraction than mobile phones. A phone, on the other hand,  can provide an unmarked straight edge...

 

Mathematical Mindsets, Jo Boaler. A review.

Professor Boaler has become a standard bearer for progressive education in mathematics.  She believes in mixed ability teaching and is opposed to textbooks, “procedures” (standard methods of calculation) and homework, which she considers inequitable as some children don’t have anywhere quiet to do it. She has been quoted as saying that multiplication tables cause  “crippling anxiety”, and is opposed to testing them, though she does not say that they are “not worth knowing”. This is an important point, to which I'll return. If they are worth knowing, how should we teach them?

 Many of the teaching tasks in this book are quite brilliant, and I have started to use them. They are interesting and challenging, promoting mathematical reasoning and the flexible application of knowledge. Her criticism that textbooks interfere with this by presenting only the most simple examples – eg only hexagons that are regularly shaped – is often true. She is also correct in saying that school systems too often put sheer speed before understanding, and that this penalises some gifted mathematicians, including the Fields Medal winner Laurent Schwartz. It also penalises slower-learning children in mixed ability classes, as they do not receive the additional guidance and practice they need in order to learn effectively. Professor Boaler tries to combat this by using tasks with a low starting point and a great deal of extension - "low floor, high ceiling". This is a good idea in itself, and worth developing, whether the groups taught are of mixed ability or not.

Professor Boaler's politics are expressed in her reply to emails, Viva la revolution, and her methods are based on the Hollywood principle, "first simplify, then exaggerate". She starts in the introduction, where she describes the OBE as the “greatest honor bestowed in England, given by the queen”. It isn't, and Professor Boaler must know this. The OBE is third in rank out of four in the Order of the British Empire, and at least two other civil honours - Companion of Honour and Order of Merit - rank higher than any of these.

The same fast and loose attitude applies to the presentation of studies and data. A UK school dubbed Phoenix Park is compared with another, Amber Hill (amber – slow down and stop) on the basis of work that took place in the 1990s, though the dates don’t appear in the text. An interesting critique of this study is here. Professor Boaler’s flagship “Railside School” in California, is equally controversial. Her account of the work has been attacked for selective use of data by a group of California academics, whom Professor Boaler has in turn castigated on her university website for alleged bullying and intimidation. I hate bullying, but in a 2008 paper cited in a US Supreme Court case, she concedes her critics’ main point – Railside pupils did very well in algebra, but not so well in other aspects of maths. Professor Boaler says that this may be because of the pupils’ lower attainment in “language arts” – English, to us – but the available evidence suggests that it might equally well be the result of paying more attention to algebra than to other aspects of maths, particularly calculation.

On reflection, this is the basis of Professor Boaler’s “revolution”. She wants to replace arithmetic with algebra, and a little geometry, as the foundation of school mathematics. The trouble with this is that the mathematics of civil society. politics, law and everyday life, from the annual budget to a gym subscription to a supermarket offer, are based on arithmetic, and every citizen needs to know how to use it. So why has she not devoted her considerable ability as a teacher to ensuring that everyone has these skills?

A final point on brain research. Professor Boaler cites the well-known study of London taxi drivers, showing enlarged hippocampus in London Black Cab drivers following the knowledge of London test, a finding that was not repeated in bus drivers who had only learned simple bus routes. She also points out that neurons fire when children make mistakes, and concludes that making mistakes is a positive thing. This is misleading. Kandel's Nobel Prize winning research shows that repetition results in the growth of new connections in cells, while it is established that repeated activity, such as learning music, results in layers of myolin being deposited or formed along connections that speed things up. The fact that neurons are firing indicates no more than that the person is alive and that the brain is active. Some mistakes are, no doubt, creative. Mistakes on simple number facts, such as the 2x table, are anything but.

Multiplication tables are a good example. If they are worth knowing, they are worth teaching, and most schools do not do this well.  Some schools teach children to count in multiples instead of teaching the table, and some as a preparation for it. In either case, I have always found that this makes learning the table more difficult, and see it as an problem of co-ordination rather than maths, as nearly everyone can add two to a given number. What they can't do without practice is get used to one column advancing one step at a time, while another advances in multiples. My approach is to build the necessary co-ordination by teaching 2x very carefully, with plenty of extra practice where needed to make it automatic. I don't move to 3x until 2x is very clearly in place, as this provides a template for the others. I then find I can teach the remainder in three to six months, and that they help pupils enjoy maths.

 

Geometry - learning from worked examples, a window on the mind.

At school, geometry caused me severe distress, untold hours of wasted time and migraines. I did not know how to do it. Give me a protractor to measure an angle, or a pair compasses to bisect it, and I was fine. Give me a problem to solve that some people could do in two minutes, and I was up until eleven at night, having started at 5.30 or 6, and completely, hopelessly, despairingly confused. I did not know what constituted a proof, or how to apply whatever theorems we'd been shown. The misery caused by having to do something, and not being able to do it, has, I'm told, given me some understanding of how people feel when they can't read or write properly, but that, at the time, was a long way off. All I knew was that geometry homework gave me serious grief, and I dreaded Thursday nights.

My school's solution at the time was to introduce a new option at GCSE, statistics, which amounted to illustrated arithmetic, that I enjoyed and could do.  I learned the principles of geometric (Euclidian) proof twenty years later through Davis and Hersh's The Mathematical Experience, but that was too late to do anything about the initial problem. I couldn't do geometry, and, like many others, concluded that I was useless at maths.

Sahar Bokosmaty's thesis, open access here, shows how giving students worked examples before expecting them to solve problems on their own, helps free them from their own instinctive approach to problem solving - my protractor - and enables them to understand the principles of mathematical thinking. Part of the process involves reducing the load on working memory by building up longer term knowledge and understanding. This is important work, and has been published in article form here, though unfortunately only the abstract is available freely on the internet. I suggest that people interested read it via a copyright library, as I have thanks to the excellent service at Cambridge University Department of Education Library.

Peter Gross, Inspirational Mathematics Teacher, Enfield.

This article and video from the Enfield Independent shows the work of Peter Gross, who has just received an inspirational teacher award for maths teaching. We now need to know more about how he does it.

Knowledge frees the brain to think.

My 11 year old pupil, who began last summer not knowing the two times table, and reading at a level around that of an eight year old, has just got L5c in a maths practice test, and L4a in English. For readers outside the UK, the nationally expected standard is L4, or 4b to be sure of it. We still have a lot of work to do to build fluency and confidence in writing and spelling, but the improvement began with building up his knowledge of patterns and procedures in basic maths, and of English grammar. We did not accept approximations, but explained everything clearly to him first and then made him practise, even when this was not his idea of fun.

The result is that, faced with, say, calculating the perimeter of a shape, or its area, he knows how to do the additions and multiplications required without expending his mental energy on them, and can instead think about what the question is saying and how to answer it. He knows what verbs are, why simple grammatical terms are as they are, and how they function in writing. In other words, he can do his work with confidence, approach secondary school knowing that he is not set up to fail, and enjoy his football without thinking he should be doing more homework.

Perhaps most important, he now agrees with me that he is an intelligent boy - not necessarily the quickest, but intelligent - and no longer thinks he is stupid. 

Good progress in Year 6

Ten year old who came to me 9 months or so ago not knowing his 2x table has just been assessed at L4a by his school - his teacher was so pleased she came to visit his mother to tell her at home. This success has been based on building up his knowledge of tables and basic facts, such as breaking down 100 and 60 into halves and quarters, so that he has spare brain capacity to read the questions properly and think about them. 

This pupil's literacy skills are harder work. He had not got into a habit of reading, and so his vocabulary was largely limited to the everyday. We are working on Susan Elkin's Junior English course, and he is reading a page of the Dorling Kindersley Children's Encyclopaedia either to his mother or to me each day. We unpack and explain anything he doesn't understand, and practise anything he hesitates over. He has managed L4 on mock grammar tests, but his reading is held back by sheer lack of knowledge. However, he is sticking at it, and not forgetting what he has learned. He is also more serious in his behaviour, concentrating rather than pretending that it is funny not to know things. More will follow.

Professor Richard Cowan, UCL Institute of Education, Inaugural Lecture.

Professor Cowan's inaugural lecture takes place on 4th March 2015 at UCL Institute of Education. Professor Cowan has investigated the links between knowledge of number facts and progress maths as part of a major ESRC project that provides support for the approach of the new national curriculum for mathematics. A ppt presentation can be downloaded here. Professor Cowan's latest publication can be downloaded here (at foot of page).

First notes on algebra.

These notes are for a pupil in Y10, who was not sure how to proceed in a test of basic algebra. This is a first draft, and there will be additions and revisions.

First Notes on Algebra.

Algebra is usually a way of finding out something we don't know. This is usually written by a letter.  We are given other information that we have to manipulate to work out the value of this letter.

Algebra does not usually involve big numbers, but we have to make decisions about what we do, and to be careful - often we have to do the same thing twice.

If there's more than one thing we don't know, it becomes complicated.

So, a formula such as y = x + 3 can only give us a range of answers, depending on knowing what either x or y is. If we don't know, we can substitute a number for x and do the calculation. The question will usually tell us what to substiture, so that, if x = 1,  y is 1 + 3,   if x is 2, y is 2 +3.  We can plot the results on a graph, which for this type of formula gives us a straight line.

If we simplify an expression in brackets, we unpack it by carrying out an operation for each item in the bracket.  So, (3x⁴)²  becomes 9x^6.  - we have to square both the 3 and the x⁴

An equation gets its name from the word equal. Usually, equations contain letters and numbers, and we can manipulate the numbers to find a value for the letter. Both sides have to be kept equal, and we do this by doing the same thing to each side.

But how to decide what to do.  Look closely at the equation, and then

If there is a fraction in an equation, multiply both sides by the denominator (whatever is below the line).  This turns the fraction on one side into a whole, and multiplies the other side to make up for this change.  

If both sides contain a number that will divide by a common factor,  (eg 2x =4), divide both sides by that factor, so that x =2.

If it contains a negative number, add that number to each side. This will bring the negative number to 0, and keep balance by adding it as a positive to the other side.  X - 5  = 0  becomes x =5.

If it contains a positive number, bring that number to zero by subtracting it from each side.     X + 5  = 0  becomes x = - 5.