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# 'Maths Notation Problems'

**22nd April 2016**

After several delays, I finally got the basic numeracy workshops started at the beginning of April. We have had three sessions now and I am starting to get a feel for the way to approach them.

Firstly, it is always a pleasure to be working with adults. THe first session was devoted entirely to introducing ourselves to each other and then to discussing how people can spend maybe eleven years in school, perhaps three hours a week, and learn so little. It was fascinating to hear what they had to say and I was very struck by people's openness and sincerity. I have attempted to impose strict rules that they must speak up if anyone says anything that they don't uderstand and that no-one should ever show any impatience or disapproval. They have accepted this and we have a very supportive atmosphere. I am not sure that thsi could easily be achieved with children or teenagers.

What has surprised me is the significance of maths notation. As a maths teacher, I have assumed that if I write things on the board using conventional maths notation I will be understood. Not so. While discussing the commutative nature of addition and non-commutative nature of subtraction (not in those words), I wrote 3 + 4 = 4 + 3 = 7 . This caused much consternation because they thought that the equals sign meant "here's the answer". This, of course makes the statement that I wrote compeletly baffling. No wonder that these people struggled with maths throughout their schooling.

After we ahd cleared up what equals actually means, one of the students remarked "It's just like learning a foreign language." Well, of course, that's exactly what it is. As teachers we need to pay a bit more attention to this.

# ‘Basic Numeracy Workshops’

**21st January 2016**

I am organising a series of workshops with the support of my local Labour Party. I am looking for up to 12 people who have problems with numeracy to attend for six one-hour sessions. Although I will do some teaching based on the content of Numbers Explained, the primary objective is to examine the reasons why they have difficulty with Maths and give them the confidence to have another go at it.

I am going to make these sessions as unlike classroom lessons as I can. They will be in the round and we will drink tea and talk a lot. We will start by discussing their early experiences. The authors of 'Children's Mathematics - Making Marks, Making Meaning' have kindly given permission for me to use some of the childrens' drawings shown in their book and to quote their ideas as my contribution to this topic. We will then go on to discover just how much maths the participants already know. I am anticipating it will be quite a lot - certainly more than they think.

You can read more about this project **here** and I will report in a future blog how the workshops went.

# ‘Table Testing For 10 & 11 Year-olds’

**4th January 2016**

All new secretaries of state for education look for some way of making a difference – some enduring memorial of their reign. The latest initiative from the DfE to test knowledge of multiplication tables is probably a good thing as far as it goes, but that is the trouble. It doesn’t go very far.

We can all agree that everyone should know the multiplication bonds for single digit numbers up to 9 times 9. It is also handy to know the tables for some of the larger numbers such as the 15, 16 and 25 times tables, but it isn’t essential. Why single out the 11 and 12 times? When we had old money with twelve pence to the shilling, there was some point. Now, everything is decimalised. Could the reason be that Nicky Morgan and some of her DfE officials had to learn them like this when they were at school?

Will the new test determine how well the tables are learnt? This would imply some timing element to the test questions. It’s no good reciting the tables while counting on your fingers until you reach the required one. By the time you’ve reached the answer you will probably have forgotten why you needed it. When confronted by 7 and 8, the number 56 has to instantly pop up in your mind. Not only that, it is necessary to know the tables literally backwards, so that when you see 63 you just can’t help thinking 7 and 9. Why? Just one of several reasons: it is very difficult to work with fractions unless you can factorise numbers on sight. So, just how rigorous will the test be?

Next question: why is the knowledge of multiplication number bonds being selected for special testing in this way? It may be a necessary requirement for further progress in maths, but it is hardly a sufficient one. In any case, knowing multiplication facts is not an end in itself but a component of more useful number skills. Are the existing assessment tests not checking these?

Finally, what happens to students who fail the test? Do they get siphoned off into special classes to be drilled until they pass? Sent to summer holiday boot-camps? I think we should be told.

There seems to be an underlying assumption that testing, by itself, will magically lead to learning. I await the unveiling of the strategies and resources that will support this initiative, although I am not holding my breath.

If this is Nicky Morgan’s bid for glory, she should think again. If she really wants to make a lasting contribution to education in our schools, she should dust down a copy of the 2008 Williams Report on early years practice and read it. Then she should restart the abandoned programme of in-service training that the report recommends. This would not bear fruit overnight. She should seek cross-party support for a lengthy upheaval in maths education in the early years. Revolutionise the early years teaching of maths and the need for special tables testing later on would largely disappear. Now that would be a lasting memorial for an education secretary.

# ‘Please, Nicky Morgan, Don't Do It’

**5th November 2015**

I hear with dismay that Nicky Morgan, the successor to the lamentable Michael Gove, is continuing the tradition of meddling in the affairs of schools by wanting to reintroduce the formal testing of very young children in English schools. The details are not clear at the moment. Some reports say that reception and Year 2 pupils will be tested, others say that it will be at the start of Key Stage 1. I also wonder if all schools will have this imposed on them or only those under local authority control.

Let us consider Nicky Morgan’s qualifications to make such decisions. She attended an independent school, went on to Oxford University to study law and became a corporate lawyer. Please note the lack of any study of education or experience of teaching.

Now consider the impact of early testing on maths education. We know that children have a natural interest in mathematical ideas and that this develops in ways that differ from one child to the next. The ways that they express their mathematical ideas and the ages at which develop them are highly individual and need to be sensitively managed by skilled teachers. Sadly, this is not often what happens in our schools. The results are disastrous for many children and for our economy. A reliance on the early drilling of formal methods bewilders most children who quickly decide that maths is too difficult for them. The result is widespread innumeracy.

As a private tutor, I spend much time trying to convince people that their problems with maths are not their fault. Their brains are not wired up differently. They are victims of an approach to early maths education that is demonstrably failing.

Testing does nothing to remedy this situation. It makes it worse by pressuring schools to achieve a particular standard of achievement at an age which is frequently far too early. This encourages the teaching of algorithms instead of knowledge. Even those who achieve mathematical ability in spite of this dull, sterile approach are being robbed. Learning about maths should be an enjoyable awakening, but the word most people associate with maths is ‘boring’. Finally, the many children who fail the tests are being given a powerful and discouraging message – you don’t measure up to requirements.

Once children have got it into their heads that maths is boring and too difficult for them, the damage is done. Later on in their school careers, the heroic attempts of maths teachers to reawaken their interest and get them to engage with the subject are largely doomed to fail.

Please, Mrs Morgan, remove the threat of yet more cruel and pointless testing. Put your energy and money into providing the skilled teaching staff that early maths education demands.

# ‘Chinese Teachers Meet Sophie’

**26th June 2015**

If you haven’t seen ‘Are Our Kids Tough Enough? Chinese School’, do have look at it. You can see it on BBC iPlayer at http://tinyurl.com/qf7dbds .

I didn’t know whether to laugh or cry as these Chinese teachers locked horns with fifty typical English Year 9 students. The teachers, it seemed, were used to youngsters who actually want to study and who also show respect to adults in general, and teachers in particular. Not only that, they expected the students to listen to what they say. So ‘students’ who see no particular reason to stop chatting and pay attention came as a bit of a shock.

It was also a shock for the students as they struggled to cope with a twelve hour school day, even with an extra meal break thrown in. This is normal in China, it seems.

I have to say that simply lecturing to a group of fifty fourteen year-olds is not what I would regard as good practice. Some of the more able students liked it because it stretched them, but in this mixed ability group (apparently there is no setting by ability in Chinese schools), the majority switched off and started to play up, to the distress of the teachers. No surprises there. One of the Chinese admitted that the concept of classroom management was new to him. I can sympathise.

It wasn’t all bad. Starting the day with massed stretching and bending on the playing field went down well after some initial sniggering. They were basically nice kids who mostly responded to the evident sincerity of their new Chinese teachers. They also liked the more practical sessions on Chinese culture, even if Mandarin lessons were clearly going nowhere.

I can’t see this experiment ending well though. Right from the start, a handful of students saw their opportunity to have a bit of a laugh. After she was chucked out of the classroom, Sophie, grinning with satisfaction, explained that she liked school for the social side of things but wasn’t really interested in learning anything.

There you have it in a nutshell, the basic problem that bedevils our schools. There are too many Sophies. Mouthy, complacent, insistent on their ‘rights’ and eager to challenge their teachers, they soak up the energy of staff and continually disrupt lessons. Students who want to work are held back by the Sophies in their class. If we could sort out this problem effectively, it would revolutionise our schools.

My answer would be to show them the door until their attitude changes. Everyone should be entitled to the same number of years of free education, but this should be contingent on them actually engaging with it. The message should be, “Go and come back when you want to work.”

A consequence would be that many classes would have a small number of older students, possibly adults. That would be interesting. It would reduce the’ classroom management’ task and place a quite different pressure on teachers to provide high-quality lessons. A few adult students would not tolerate the disruption of their precious lessons by silly behaviour.

Well, you can dream, can’t you? Meanwhile I shall be fascinated to see how the Chinese teachers and their increasingly fractious students get on in next week’s instalment.

# ‘Count Us In - Yes, But Significant Changes in Early Education Are Needed’

**26th June 2015**

Another report on numeracy in Britain is published, this time by the British Academy which looks out for the Humanities and Social Science. ‘Count Use In’ concerns itself with the ability to process and understand data and draws some gloomy conclusions.

A taster: “… the UK faces a major challenge in relation to its population’s quantitative skills. In 2011, a government survey found that three quarters of 16- to 65-year-olds in employment in England had a level of numeracy which might not be sufficient 'to compare products and services for the best buy or to work out a householdbudget'. The maths skills of this country’s graduates were reported as among the worst in the developed world by the Organisation for Economic Co-operation and Development in May 2015. In recent years, our school pupils have generally ranked only in the middle of developed nations in mathematics. And, as confirmed in the *State of the Nation *evidence review, employers frequently identify deficits in the facility of people with numbers. This has implications for the future of the UK’s status as a world leader in research and higher education, for the employability of our graduates, and for the competitiveness of the UK’s economy.”

You can get the report here and it is worth a read. I cannot argue with any of it, but I think that it misses an important issue.

What is missing from the report is any appreciation of the mindset of all these people who lack numeracy. It isn’t that they are unaware of their deficiencies. The problem is that they don’t believe they can do anything about them. They believe that there is something wrong with their brains that prevents them from understanding arithmetic. In most cases this is quite untrue and their beliefs stem from bad experiences in their early education. In the worst cases there is a fear of anything to do with numbers. Unless these deeply held beliefs and fears can be overcome, nothing is going to change.

The situation is not hopeless but, firstly, any programme of adult numeracy education has got to recognise this attitude problem and address it.

Secondly, the report does make a nod in the direction of improving early education provision, but I looked in vain for recognition of the work of Worthington and Carruthers described in their inspiring book Children’s Maths Education: Making Marks, Making Meaning (see my previous blog about this). **Without significant changes in early maths education, the current depressing state of affairs will just continue**.

I would like to think that this report will signal a new dawn in numeracy education, but I am not holding my breath.

# ‘The Unfair GCSE Maths Question’

**8th June 2015**

Now that the twitterstorm has died down, perhaps a more considered view of the “unfair” question in last Thursday’s Edexcel GCSE Maths Paper is possible.

In case anyone reading this has missed the row, the question combined elementary probability theory with a bit of algebra. We are told that a bag of *n* sweets consists of *6* orange ones and the rest yellow. A girl takes a sweet at random, eats it and then takes another one at random. If the probability that she has taken two orange sweets is one third, show that *n ^{2} – n – 90 = 0*.

If you haven’t seen the solution and want to have a go yourself, break off from reading now.

It isn’t hard. The probability of choosing an orange sweet first time is *6 / n* and the probability of choosing an orange sweet the second time is *5 / (n-1)* . The two events are independent of each other, so you can multiply their probabilities to get the an expression for their combined probability: *30 / n(n-1)*. If you set this equal to *1/3* and do a little algebraic juggling, you get *n ^{2} – n – 90 = 0 .*

What puzzles me is that apparently the question didn’t go on to make the obvious final step: what is n? All you have to do is solve the equation by factorising it to *(n + 9) (n-10) = 0*. Then, either *n + 9 =0* and *n = -9* (which solution is obviously impossible and is discarded), or *n – 10 = 0* and *n = 10*.

So there were *10* sweets in the bag, *6* orange and *4* yellow.

All of the above is within the syllabus of the Higher Maths GCSE, including my extra bit. So why did some students think it impossible, unfair, etc., etc..

My theory is that the fault lies with the modular way that maths is experienced by students today. They will have been taught about probability in one set of lessons and then tested on it. At other times they will have been shown how to manipulate algebraic expressions and equations and then tested on that. They will regard probability and algebra as entirely separate topics and are consequently thrown by a question that brings them together.

The attitude of regarding maths as a series of unrelated hurdles to be overcome separately is a consequence of the SATs regime that students experience earlier. Many students concentrate on scoring well in the next SATs test and then forget what they have ‘learnt’ because it won’t be needed in the following test. Teachers also teach to the test, because that is what is required of them. It is a brave teacher who strays far from the plans laid down in the National Curriculum.

Another source of difficulty is that the equation describing this bag of sweets problem employs fraction notation and requires the student to clear the denominators by multiplying through by *3n(n-1)* . I have observed while teaching Year 11 students that this kind of algebraic fiddling about causes much angst. This is because a lot of them have little practice in working with numerical fractions and they don’t really understand what they are doing. So fractions that include variables are regarded with deep suspicion.

So, many students saw a probability question and a scary looking bit of algebra and under the very real pressure of an exam, they panicked.

It is very interesting that some of these students sought to blame the examiners rather than themselves. Do they think it the responsibility of the examiners to ask them questions that they can answer without breaking sweat? Are all students entitled to an A*?

I cannot resist one last observation. Probability was not always in the syllabus in the days of GCSE ‘O’ Level, but if it had been, and a question like this one had been included, it would certainly have finished up with “hence determine the value of n.” It might even have skipped the intermediate stage of “show that *n ^{2} – n – 90 = 0” *and just asked you to “find n”, requiring you to sort out the equation for yourself and solve it. No hand-holding.

You tell that to young people today and they don’t believe you.

# ‘Up to 28% Reduction. Big Deal!’

**31st March 2015**

President Obama just disclosed that the USA will reduce its emissions of carbon by “up to 28%” as its contribution to preventing global warming. This is a very easy promise to make, because “up to 28%” includes zero. Presumably it would also allow a negative reduction, that is to say an increase, in carbon dioxide emission.

If he had wanted to say something meaningful, he would have said “at least 28%”, thereby binding the USA to do something significant. He could then have said that the USA aspired to do better than 28%, say up to 50%.

Do people really not understand the difference between “up to” and “at least”? Here is a diagram that should make it clear:

I don’t mind too much when advertising men and their kind abuse the English language and the meaning of mathematical expressions, but I do think that politicians should try to do better and stop treating us as if we are all stupid.

# ‘15 + 6 = 17 and The Wedding is Off’

**16th March 2015**

In the Indian village of Rasoolabad, Monar Singh had arranged the marriage of his daughter, the delightfully named Lovely, to another local man, Ram Baran. Word reached his ears that Ram Baran was not as well educated as he claimed to be, so just before the wedding ceremony, Lovely asked Ram Baran “What is fifteen plus six?” The unfortunate Ram Baran answered, “Seventeen” and the wedding was off.

This is a sad little story, but the immediate reaction is to laugh and conclude that Ram Baran must be some kind of idiot. None of the reports has included details of his schooling, but he certainly was, at best, ill-educated. After all, 15 + 6 = 17 is a terrible howler – how could anyone think that? If you were to stop people on a British high street and give them the same test, you would get a shock.

## Everyone Has a Sense of Number

There is considerable evidence that we all have an innate sense of number. Small children quickly develop and display an interest in counting. If this is encouraged, it can become a full understanding of arithmetic. In particular, the standard ways of representing numbers and performing calculation can become second nature. It is easy to forget that these standard symbols and algorithms are only one way of representing these powerful ideas. There are others, equally valid, and small children show great ingenuity in inventing their own symbols and algorithms.

## Early Years Teaching Methods

The task of early years maths teachers is to manage each child’s transition from their first ideas about numbers to the standard methods of modern arithmetic. There is no strong consensus about the best way of doing this. I have just been reading about the ‘Shanghai’ approach which is extremely formal (see http://www.theguardian.com/education/2015/mar/13/chinese-teachers-bring-the-art-of-maths-to-english-schools). It is being held out as a shining example for our schools to follow. It gets results, even if I question how much these results reflect real understanding that can later be extended successfully into further maths. The striking thing about the Shanghai system is that all children, without exception, are expected to achieve basic numeracy. There is no setting or streaming and all children get exactly the same treatment at the same age. Apparently, it works.

## Are There Ram Barans in Britain?"

We can hardly boast about the superiority of British schools when so many leave with arithmetic skills pretty well on a par with Ram Baran. I doubt whether he had eleven years of free compulsory schooling. I am going to guess that he had none at all. If so, his innate sense of number was never developed. I suspect that he would never have made such an error if confronted with fifteen actual physical objects and asked to add another six of them. Asked to do this in the abstract, using unfamiliar language, he probably panicked and gave the first number that came into his head. Note that he did know that the answer was another number and that it had to be larger than fifteen. He gave the sort of answer that I would expect from a small child who is just beginning to explore numbers. Not a fool then, just disadvantaged.

We need to stop laughing and ask ourselves why many of our own citizens are just as disadvantaged after eleven years of expensive education.

# ‘The Multiplication Tables Fetish’

**2nd February 2015**

Why didn’t I think of that? Eliminate innumeracy at a stroke! All you have to do is force children to recite multiplication tables up to 12 x 12. Job done.

Actually, why stop there? If 12 x 12 is good, then 13 x 13 would be better.

## Politicians - Don't You Just Love Them?

I don’t know who is advising Nicky Morgan or if this latest pronouncement is merely the product of her own expertise in maths education, but she might take the time to consider the report on Early Years Maths by the All Party Parliamentary Group for Maths and Numeracy. I am assured by the APPG chair, Caroline Dinenage MP, that this report has been submitted to her and discussed with her. She might then decide to find out a bit more about maths education before going public. It might also make her consider how her benchmark of multiplication up to12 x 12 could be achieved. She hasn’t said a word about that.

We are, of course, in the run-up to an election and Nicky Morgan is really only a caretaker responsible for platitudes about education that go down well with the right-wing press. I wait trembling for Tristram Hunt’s pronouncements. Perhaps Labour will go for the 13 x 13 times table target? Or perhaps we could lower the age for 12 x 12 to the end of reception class?

## Why Multiplication Tables?

Why this fetish with multiplication tables? Is it because, secretly, a lot of people aren’t too sure themselves what eight sevens are? And why is it such a problem to remember?

## Is It Numbers That Cause the Trouble?

At a supermarket checkout the other day, the young woman held up an aubergine and called across to her colleague on the next checkout “What’s this?” Without hesitation, back came the bar-code – remarkable feat of memory involving numbers. I can remember a similar display of memory for numbers years ago buying motor cycle parts. On asking for some obscure part for a BSA C15, the young Hell’s Angels mama behind the counter would know the part number without looking it up. I conclude that remembering facts involving numbers is more to do with context and relevance than anything else.

## Learning Multiplication Number Bonds

There are about 28 multiplication number bonds that are non-trivial and need to be learned by heart. You can do this by getting someone to test you, by using a pack of flashcards with the questions on one side and the answers on the other, or by most children’s favourite method: endlessly drawing table squares from memory. Alternatively, you can resolutely refuse to learn them and rely on repeated addition to get the answer – a tedious and error-prone method but still a popular approach.

If you do enough arithmetic, eventually you will decide that the inconvenience of not knowing multiplication bonds outweighs the trouble of learning them. Incidentally, you do literally need to know them backwards. It is very difficult if not impossible to do fraction operations without being able to break numbers down into factors.

Moreover, if you do enough arithmetic, you will probably get to know quite a lot of multiplication tables not on Nicky Morgan’ radar, such as the 25 times table. If you don’t, it isn’t a problem: use some ‘long’ multiplication method to work it out.

## Multiplication Table Drills

This is also true: if some well-meaning adult attempts to drill a child into remembering ‘their’ multiplication tables (why is it always their?) before they understand what multiplication is, the child may achieve a temporary proficiency. They will also learn that maths is difficult and bewildering and not for them. The key to good maths education is to work with the child’s natural ability and curiosity about numbers and to promote understanding. The formal work comes later when the child understands the significance of what they are doing.

## Child-Centred Teaching

This is a lot easier to say than to do. A truly child-centred approach to maths in the early years requires a great deal of skill and sensitivity. There now exists some fascinating research on the way that small children develop their understanding of maths. It needs to be better known and put into practice. If Nicky Morgan really wanted to make a difference, she could start by ensuring that the recommendations of the 2008 Williams Report were implemented. The programme of in-service training that resulted from it has apparently withered on the vine, but that is where she needs to start if she wants to promote numeracy.

# ‘Maths and Numeracy in the Early Years’

**20th January 2015**

On the 25^{th} November last year, I attended a rather strange event at the Palace of Westminster. The All Party Parliamentary Group for Maths and Numeracy launched a report on Maths and Numeracy in the Early Years. I have been scratching my head about it ever since.

## What is an APPG?

I hadn’t been aware that there were such things as all party parliamentary groups, still less that there was one concerned with numeracy. It certainly seems a good idea that at least some MPs should worry about the appallingly low standards of numeracy in Britain. Good, I thought.

This APPG was formed last year, and one of their first acts was to commission a report on maths in the early years of schooling, because regrettably, many children start their maths education by learning that they are ‘no good at maths’. The study was kindly funded by the Institute of Chartered Accountants of England and Wales, who obviously have a strong motive to promote numeracy. There were six experts appointed to carry out the study, which they did with admirable speed, and there is an excellent summary of their findings on two sides of A4 . You can get a copy online at http://www.nationalnumeracy.org.uk/userfiles/Documents/APPG_paper_-_EYs.pdf. It is a good read.

I cannot fault any of the report’s findings, recommendations or its conclusion. It says (and I paraphrase) that the early years curriculum needs sorting out, that most early years teachers don’t know what they are doing in Maths and need training, and that parents attitudes are often unhelpful if not downright damaging. All true. It says (and here I quote) that “there are a number of practical steps that should be considered now that could help effect change in the long term”. Also true.

## The Launch

Apparently, a report like this needs to be ‘launched’. This means wine and nibbles and networking followed by speeches. The speeches by the experts were very good, there was an enthusiastic if rather incoherent contribution by the event’s guest celebrity, Johnny Ball, and there was a closing speech by Nick Boles, the Minister of State for Skills and Equalities, who was standing in for his boss Nicky Morgan, He gave us lots of encouragement but it seemed to me that he didn’t know very much about early years numeracy. I even wondered if he had read the report thoroughly.

The event was all very useful from my point of view, as I need to promote my book on numeracy for adults. I made some good contacts among interesting people who are well-informed, experienced and committed to improving numeracy. But I was left with a feeling “OK, now what?” I have a suspicion that the answer is “Not much.” There was definitely a sub-text to some of the speeches that went “We’ve told you what to do. Now go away, good people, and get on with it.” In fact, Nick Boles explicitly told us that people don’t trust politicians to make changes, so it was up to us. How extraordinary!

## It's up to us, apparently

Is this the official response of the Department for Education to the report? Are we, as charities or businesses working in this area supposed to do all the heavy lifting and reform early years maths education while the government looks on? Who is going to manage it all and what authority do we have? Who pays for it? The report includes eight recommendations. Seven of them require funding and four are long term projects. Is the ICAEW expected to cough up for this too? If we consider just one of the recommendations: “All early years practitioners, both new entrants and the existing workforce, should be trained in children’s mathematical development.” That is a substantial training requirement for at least 100 000 people if we include teaching assistants. Has the APPG fully understood the implications of the report?

Clearly, the APPG has more work to do if it is serious. It must secure the backing of the major political parties for the report, secure funding and get the DfE to manage the changes, which will be long-term and far-reaching. The co-chair of the APPG, Caroline Dinenage MP, did meet with the Minister for school reform, Nick Gibb MP and his team prior to the publication of the report to discuss the findings and attempt to influence their thinking. The report has been forwarded to the Secretary of State for Education, Nicky Morgan MP and the Opposition’s education spokesman, Tristram Hunt MP. The APPG also welcomed Nick Gibb to its AGM to discuss the report and how early years learning can be improved. It will be interesting to see if the report’s recommendations are reflected in any party manifestos in the run-up to the general election.

## So what might happen?

The question remains: will the DfE really accept the report? Some of the proposals run counter to trends in education that are well established. The demands of the National Curriculum, SATs, OFSTED and school league tables require maths teachers to conform to a fairly rigid scheme of work and a particular way of structuring lessons. My experience of this is limited to 11-16 comprehensive schools, but I would expect similar pressures to exist in the early years. What is the point in training teachers to be aware of the needs of young children and in equipping teachers with the knowledge to provide for those needs if they are not free to exercise their professional judgment?

## What would I like to see?

Reform of early years maths education has to take into account the wide range of development among pre-school children. It must be child-centred and not driven by a requirement to achieve a certain standard by a given age. The report says all the right things, but it is asking for some huge changes because providing an appropriate mathematical learning environment for every child in the early years is currently undervalued. It is a very difficult and demanding task – much harder than teaching ‘A’ level maths. It can only be achieved by highly trained professional teaching staff who have the knowledge and freedom to plan appropriate activities. Also, it has to be accepted that some children will achieve a reasonable standard of numeracy several years later than others. This makes school organisation complicated and difficult to assess. It means that you have got to trust the professionalism of the teachers and stop trying to micro-manage them.

## What do I expect?

The last thirty years have seen many reforms in our schools. As a retired teacher, I can now say that I disagree with nearly all of them. The reforms proposed by the APPGs report are a breath of fresh air, but as they run counter to much of the thinking that currently prevails within education, and there appears to be neither the authority nor the funding to implement them, I think the APPG has an uphill battle on its hands.

# ‘Gee, I AM a Swan!’ or How Maths Suddenly Made Sense

**26th September 2014.**

## The importance of the primary school

One of the main causes of poor number skills is the standard of maths teaching in primary schools. While there are pockets of excellence to be found, most primary school teachers have a weak understanding of maths themselves, particularly arithmetic. Despite this weakness, they are responsible for introducing children to the subject and so they play a crucial role in forming understanding of maths and attitudes towards it. Sadly, many children are completely turned off the subject. They learn that they are no good at maths.

## It’s not their fault

I have much sympathy for the teachers that I am criticising. They are under pressure to get results (as laid down by the National Curriculum) so that their school does not suffer in league tables. They may be able to get some help if their school has a good maths coordinator, but mostly they are out of their depth and no-one seems to care.

It is also easy to see how this situation comes about. Young people who are good at maths are more likely to go into numerate professions such as science, engineering or accountancy. If they decide to teach, they are more likely to opt for secondary than primary teaching.

The Williams Report in 2008 and recommended that all primary schools should have at least one specialist maths teacher and, as a consequence, the Maths Specialist Teacher (MaST) programme was establshed. A review of this programme was conducted in 2010. Those teachers that undertook this training were very enthusiastic, but only 1268 completed it.

## Tricks for ticks

There is a strong tendency to teach algorithms - do this, do that and there's the answer - also known as 'tricks for ticks'. There is no attempt to foster understanding because the teacher often doesn't understand the principles that lie behind the algorithm in the first place. Sometimes the algorithm itself is actually wrong.

Here is an example. I am giving private maths tuition at the moment to a thoughtful and diligent lad in Year 9 who has had some disastrous maths teaching in his primary school. He has been taught to do multiplication using a weird variation of the standard layout that doesn't work as soon as you multiply by a number with more than one digit. He has never been taught how to do division at all. Consequently, he was totally dependent on a calculator. (You may well ask why this has not been remedied in two years of secondary maths teaching and I will discuss this in a later posting to this blog.)

## Formal methods

A feature of poor primary school teaching is a tendency to teach formal written procedures far too early.

All teachers learn about Piaget’s research and that children’s brains work differently to adults. They know that abstract ideas about the world develop as a result of play and experiment and that it is vital to allow children time to do this. The concept of number arises from activity in the real world. It takes time and patience and practice for this activity to connect to the abstractions of formal arithmetic. If you try to push it too far or too fast, you will cause failure and frustration.

An awful example of this is the national obsession with learning multiplication tables as early as possible. Small children are required to learn them whether or not they understand what multiplication is and sometimes before they are secure in counting. What is this weird emphasis on ‘learning your tables’? I am not saying that people do not need to learn them at some point, but in the end there are at most 28 facts about numbers that need to be learnt by heart and there are a lot of patterns that assist the memory. Anyone can learn them provided that they haven’t been taught how frightfully difficult and important it is by attempts to ram them home at far too young an age.

## Mathematical Mark Making

Children need a time and facilities to explore number and shape through play. An important part of this is an activity that has come to be known as 'mathematical mark making'. The child makes drawings in which mathematical ideas such as numebr are represented using symbols of the child own choice. Theses drawings, seen afterwards, often look nonsensical, and to understand their significance you need to discuss them with the child, preferably while thay are drawing them. Children who go through this developmental stage arrive at a deep understanding of mathematics and readily adopt conventional abstract symbols used in maths. If they encounter standard maths symbols before they have done this, any understanding will be superficial.

If you want to know more about this, you can do no better than read 'Children's Mathematics: Making Marks, Making Meaning' by Maulfry Worthington and Elizabeth Carruthers.

## The baleful influence of the National Curriculum, SATs and school league tables

Many official publications, including the National Curriculum Early Years documentation, acknowledge the importance of play in the development of children and that it may take a varying amount of time at different ages. In practice, there is a requirement that children achieve target X by age Y and it is up to the school to make this happen. Since the results of tests determine school league table rankings, there is pressure to make this appear to happen. I believe that this is highly damaging to children's chances of understanding and enjoying maths, but there seems little that the teaching profession can do unless there is a huge change in the way our schools are organised, led by politicians and the DfE. The evidence is there. Is anyone listening?

## It’s not you, it’s the teaching

I strongly believe that very few people are really incapable of understanding basic arithmetic and that understanding, together with a little practice, leads to competence in handling numbers. The chief barrier is that people believe that their difficulties with arithmetic arise because their brains are somehow wired up differently, rather than because they have been badly taught.

## It's all about confidence.

I have had many private students, often in the run up to GCSE, who have needed to be taught the basics of arithmetic. It only takes a few sessions and what has been for them a buzzing heap of confusion collapses down into a few orderly ideas. With that new understanding comes what I call the "Gee, I am a swan!" experience. Suddenly, algebra makes sense - after all it is just arithmetic with variables. Despair is replaced with confidence.

So, what a difference it would make if primary school teachers could all be given a fresh understanding of basic numeracy and the confidence that comes with it.