DYSCALCULIA IN SCHOOLS:

What it is and what you can do

by

Tony Attwood C.Ed., B.A., M.Phil

T1628

British Library Cataloguing in Publication Data. A catalogue record for this book is available

from the British Library.

ISBN 1 86083 614 3

© 2002 Tony Attwood

The right of Tony Attwood to be identified as author of this work has been asserted by him in accordance with the

Copyright, Designs and Patents Act of 1988.

Published by Multisensory Maths at First and Best in Education Ltd, Earlstrees Court, Earlstrees Road, Corby, Northants NN17 4HH

www.firstandbest.co.uk

About the author Tony Attwood has taught for the ILEA, Brent LEA, Haringey LEA, Dorset LEA, the Algerian Government, the Open University and Dartington College of Arts. He is the author of over 50 educational books. Tony’s other dyscalculia works to date are • Tests for Dyscalculia • Methods of Teaching Maths to Pupils with Dyscalculia More details can be found at www.dyscalculia.org.uk

Reproduction licence This book is supplied as a photocopy master and as a text on disk. The disk may be run through any standard word processor. The purchasing school is permitted to reproduce any part of the book for its own use but must ensure that no part of the book is offered for distribution in any way to those not directly associated with the school. In particular no part of the book may be copied or distributed in any way to teachers or administrators who are associated with any school other than that purchasing the book. Where the book is purchased by a teachers’ centre or any other type of organisation other than a school or college, it is provided on the strict understanding that the purchasing individual or institution will not make any copies of the book and will take all reasonable steps to ensure that it is not possible for any individual or group of individuals to make copies of the book. It is specifically forbidden for any individual, group or organisation to copy all or part of this book and allow that copy to be sold or otherwise exchanged.

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What this book offers This book: • Explains the meaning of the term “dyscalculia”. • Explains the main reasons why some children have particular difficulty

with maths. • Explores the key difference in handling maths between the dyscalculic

pupil and the non-dyscalculic pupil. • Considers the link between dyscalculia and dyslexia. • Examines the way in which multi-sensory teaching and learning can help

pupils who have not grasped the basic functions of maths. • Shows why self-esteem is a fundamental part of the issue of under-

achievement in maths, and how self-esteem can be raised alongside the teaching of maths.

• Suggests ways in which parents, governors and other members of staff

should be informed about dyscalculia and the ways in which the school is seeking to overcome it.

• Provides sample policy documents on dyscalculia

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Contents 1: The concept of dyscalculia 2: Maths and dyscalculia: is dyscalculia a side-effect of dyslexia? 3: Why are some children bad at maths? 4: The best ways to teach children who appear to be dyscalculic 5: Overcoming dyscalculia: the systematic approach and the holistic approach 6: The systematic method of approach: the 3 steps to solving the dyscalculic problem 7: Policy documents on dyscalculia 8: The inchworm and the grasshopper: how dyscalculic children learn maths 9: Strategies adopted by dyscalculic children when faced with maths problems 11: Multi-sensory approaches to overcoming dyscalculia 12: Self-esteem as a method of overcoming dyscalculia 13: Check to praise – a method of motivating dyscalculic children 14: The maths and self-esteem special group: raising self-esteem and overcoming dyscalculia together 15: Self-esteem, maths and behavioural problems 16: Information for parents 17: Information for governors and members of staff Bibliography Other Resources for Teachers and Parents of Children with Dyscalculia

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1: The concept of dyscalculia Dyscalculia is the name given to the problem suffered by people whose ability to handle mathematical concepts is significantly lower than we might expect it to be, when taking into account that individual’s age and intelligence. This definition makes it clear that dyscalculia is not a term intended to be used for everyone who is poor at maths. But as we will see later, many of the methods of teaching that have been devised for helping pupils who are dyscalculic will work with all pupils who have difficulties with maths, irrespective of the cause of the problem. However just because the best ways of working with children who have difficulties with maths are almost always the same, this does not mean that we should be too ready to use the word “dyscalculia” in cases where it does not apply. It is obvious that everyone sits somewhere on a continuum of mathematical ability, which we might arbitrarily number from 0 to 100. Anyone at position 100 is a mathematical genius. A person at position 0 has no mathematical ability whatsoever. A person at point zero has no concept of number and therefore no ability to see how a number such as “2” relates to “4”. From this point it is possible to devise a spread of mathematical ability for all age groups. If (for the sake of this exposition) we accept that the level mathematical ability in each age group is a normal curve then we will have most people with an ability rating of 50 and ever smaller numbers of people as we get further and further away from position 50. Even if reality doesn’t quite match the standard pattern suggested above, we would expect to have a small number of people at each end of the curve, and we might choose to give a special name (such as dyscalculia) to the people who are at the low end. The giving of a name in this way will inevitably have the appearance of separating out this low achieving group, but the name does not of itself signify that there is anything particularly different about these people, other than the fact that they are at the lower end of the spectrum. The most likely factor that binds together those who fall towards the end of our 0 to 100 scale at any particular age is intelligence. It seems reasonable to start from the belief that at any given age, the lower the level of intelligence, the lower the level of mathematical ability. If the match between intelligence and mathematical ability were perfect there would be no need for a concept such as dyscalculia. However, what we do find is that there are some children who appear to have a mathematical ability which is much lower than we would expect given their intelligence. From this finding we can begin to see that there might be a specific learning difficulty which some children have in relation to maths. If dyscalculia represents a specific learning difficulty one might expect that some people who appear at the bottom of the ability range in mathematics might well be at or near the top of the ability range in other areas of study. From such a view we might speculate that some of those people who are at or near the bottom of the scale in mathematical ability within a certain age group suffer some sort of genetic

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dysfunction which makes it difficult or impossible for them to learn mathematics in the way that others of the same age will do. At first sight this seems very reasonable. Belief in such an approach has been heightened by the belief among researchers into dyslexia that dyslexia itself is genetic in origin. The definitions and ideas above make dyscalculia seem like a mathematical version of dyslexia. We might also be encouraged by a body of neuropsychological research which has shown that mathematical ability is particularly associated with specific areas of the brain. What’s more, we also know that there is a link between higher levels of ability in maths and a particular ability in music – and this suggests that a particular part of the brain might well be responsible for the ability to handle specific types of abstract activity, in this case mathematical and musical concepts. A failure in this area of the brain can then lead to a much lower than expected ability in maths. Thus we might expect to find that a percentage of the population is significantly worse at maths than we might otherwise expect when IQ score and age is taken into account. Dyscalculia becomes the behavioural outcome of a genetic disorder. We might note that those involved in working with children who have dyslexia speak on occasion of 10% of the population being dyslexic, and we might assume that a similar figure applies with dyscalculia. However we must also note that for every ten children who are diagnosed as dyslexic there is probably only one for whom the term dyscalculia is applied. Does this suggest that our opening hypothesis is wrong? Or could it be that there are other factors at work here? Or does dyscalculia simply affect fewer people than dyslexia? Or is it simply that we are less open to the notion that failure in maths is a sign of a specific learning difficulty? Certainly far fewer educational psychologists are administering tests for dyscalculia than there are administering tests for dyslexia. A review of the literature shows that there are only a handful of books on the subject of dyscalculia, as opposed to thousands of texts on the subject of dyslexia. Even as we put this book together the Times Educational Supplement carries a front page story about a supposed “cure” for dyslexia. Such claims, such stories simply do not exist for dyscalculia. In part this might be explained by the impact being dyscalculic has on a pupil, as opposed to the impact of being dyslexic. A dyscalculic pupil fails at maths – but this may have no impact on the child’s ability in history and geography, where the use of number is very limited. But a dyslexic pupil will fail at history and geography because written English is the very medium of those subjects. It is therefore possible that there has simply been less pressure for support for dyscalculic pupils because their disability affects the teaching and learning in a smaller number of subject areas. Issues relating to the impact of dyscalculia, and the publicity that it gains in the educational and national press, are quite possibly part of the explanation for the lack of awareness and action taken on behalf of dyscalculic pupils. But these pointers do not help us understand what dyscalculia really is. Is it a specific learning difficulty? Is it genetically based? To answer these points we now turn to the link between dyscalculia and dyslexia.

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2: Maths and dyslexia: is dyscalculia a side-effect of dyslexia? Some esteemed writers on dyslexia have suggested that the genetic disorder which leads to dyslexia also leads to failure in maths. They put forward a number of reasons as to why this is likely. Firstly it is suggested that a number of the problems associated with dyslexia (for example difficulty with short term memory, and difficulty with sequencing) are as likely to affect the ability to cope with mathematical concepts as they are to affect the ability to handle the written word. Clearly there is a logic in this. Issues such as understanding multiplication tables which are at the heart of mathematics are sequencing issues. Numbers are dealt with as sequences – it is logical to see dyscalculia as a dyslexia problem. A child who has particular difficulty with multiplication tables often expresses the view that she can’t remember where she was in the table – a clear indication that there might be a short term memory issue at work. Secondly it is noted that in the UK we have traditionally taught and tested mathematics in a way that leaves the subject surrounded by written English. Many mathematical questions are posed as written questions, requiring the student to be competent in English before he or she can handle the maths. Maths is not just a question of knowing what symbols such as + actually mean. It is also a case of being able to read and understand and remember the meaning of “sum of”, “add”, “total”, and “equals”. From arguments such as these we might begin to believe that a dyslexic child is also a dyscalculic child. Once a child has been diagnosed as dyslexic then support should be given in maths as well as written English. Reviewing the evidence Unfortunately there is some evidence to suggest that the link between dyslexia and dyscalculia is not as straightforward as we might hope. A few studies have suggested that instead of all dyslexic students being poor at maths, about 25% of dyslexic students are apparently above average at maths. This suggests that contrary to the above argument dyscalculia is a separate issue from dyslexia. In the light of this evidence we might feel more inclined to say that dyscalculia and dyslexia are separate, but that some pupils – perhaps the majority of pupils – suffer from both dyslexia and dyscalculia. Unfortunately the evidence which might translate these views which appear to be reasonable, into views supported by experiment and data, is hard to come by. Genetic science is not far enough advanced to give definitive answers and, where tests have been carried out on dyscalculic pupils, the number of people tested has often been extremely small. This latter point is particularly worth remembering as it has bedevilled testing in relation to dyslexic students as well as dyscalculic students. The number of pupils and students examined in the research which is reported is often no more than a

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dozen or so – a very small sample indeed and one from which it is very hard to draw serious implications. But such evidence as there is does seem to keep coming back to the fact that while many dyslexics do also show signs of dyscalculia there is this sizeable minority (the 25% figure is regularly quoted) which, far from being way below the expected level in mathematical ability, is actually considerably above that level. However we should also note that although some research has revealed this figure to be correct in surveys on dyslexic pupils, this is a figure which has now started to be handed down as if it is an absolute proven fact – when it is not. Again this is a problem that afflicts dyslexia as much as dyscalculia. It is commonplace for organisations that seek finance or support for dyslexics to quote “facts” such as the view that Einstein was dyslexic. This like many other such claims is based on virtually no evidence but is handed down from one speaker to another along with many other urban myths. Steeves 1983 cites a boy who in a few seconds could calculate the number of squares on a piece of squared paper, but who was profoundly dyslexic and this is often cited as “evidence” of the separation of dyslexia and dyscalculia. Along similar lines Griffiths in a paper in 1980 cited the issue of a university lecturer in physics who in his 50s could not repeat his six times table. In this case the suggestion is that dyscalculia is therefore a separate concept from difficulty in understanding scientific principles and formulae. Indeed I could add my own contribution to this type of debate. One of my daughters has been diagnosed as profoundly dyslexic by suitably qualified educational psychologists on two separate occasions. She suffered the traditional problems of having difficulty with maths, but it was perfectly evident when one spoke with her on a one-to-one basis that she had a perfect grasp of mathematical principles. What she had was some difficulty with sequencing, and a significant amount of difficulty reading the text of the maths questions she was given. Because of these difficulties she was placed in a group at secondary school that was entered into maths GCSE to take papers which could at best only give the pupils a Grade C pass. Such was her grasp of the subject that she was awarded the rare distinction of a B from the marks gained on these exams. Meanwhile it was considered that her dyslexia was so bad that in other subjects she was allowed a reader in the exams who would read her the questions! I cite the case of my own daughter not to prove that dyscalculia is separate from dyslexia, but rather to show the dangers of accepting much of the “evidence” that currently surrounds the subject. Evidence from individual cases cannot be used as proof that there is independence of dyscalculia from dyslexia for with the simple use of two observations the opposite can be argued: Firstly, we may note that a number of people who are dyslexic are drawn in their work and interests to activities which use the written word. It is not at all unknown for dyslexic children to select for A level study the very subjects that we might expect them to seek to avoid, such as English and history. It is not unknown for people who are writers by profession to show all the signs of being dyslexic. It is not at all impossible for my own interest in maths to have impressed itself on my daughter, and that she determined to sort out her maths problems with me. In other words she was dyscalculic, but through the application of good teaching and learning techniques, she overcame it.

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Which brings us to the second point: just as there are methods of teaching dyslexics which can be highly effective and which can remove almost all signs of the disability from people who have undergone the training, so there are methods of dealing with dyscalculia which can remove all the effects of dyscalculia. Put these two points together and we could, despite the evidence above, argue that dyscalculia is a side-effect of dyslexia – that is, all dyslexic pupils also suffer from dyscalculia. But we may also note that some pupils have a strong desire to work with mathematical concepts, and have become particularly successful at overcoming the dyscalculic part of dyslexia, to such a degree that they become above average in their mathematical ability. This over achievement may have been caused by good teaching, home support or an interest in a topic that relates to maths, or requires a mathematical ability to engage in it. In exploring this possibility, one interesting field of research has looked at the specific problems that supposedly dyscalculic pupils reveal when undertaking maths tests. In some cases it was found that pupils had particular difficulties with mental maths tests, as might be expected through the suggestion that dyscalculia is related to sequencing difficulties. But other pupils did better than expected in this field – almost as if the opportunity to do mental maths was a liberation (for them) from the tyranny of the written word and symbol. In other research some pupils were found to be mathematically able in terms of concept awareness and understanding. But some of these pupils suffered from particular problems with addition and subtraction. Indeed this latter scenario is one that often crops up in reports about dyslexics with mathematical problems. What is clear however is that a considerable number of writers on this topic have failed to spot that this is an issue which often reoccurs in reports on dyslexic pupils. It is not so much the fact that dyscalculic pupils cannot “do” mathematics or understand mathematical concepts as such, but that they simply find addition and subtraction confusing. Obviously any inability to handle any of the four basic functions of maths is bound to disrupt all attempts to solve mathematical problems. But such a fundamental failing can be easily misread as an inability to understand all types of mathematical issues rather than a specific problem relating to one or two of the fundamental concepts of maths. Therefore we are now left with three possibilities: Firstly we can argue that dyscalculia is a separate issue from dyslexia, but it is a specific learning difficulty that affects a significant number of dyslexics. Secondly it can be argued that dyscalculia is a side-effect of dyslexia, but that some dyslexics do not show the symptoms because they have worked to overcome the problem. Thirdly it can be argued that dyscalculia is nothing other than a failure of learning in relation to the basic functions of maths. This may be due to short-term memory problems, to sequencing difficulties, or simply due to poor teaching or poor learning. Once remedial teaching has been undertaken the child will be able to undertake work in maths which is as expected, given the child’s age and intellectual ability.

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All three arguments are sustainable given the current state of our knowledge. Fortunately, as we will see later, whichever theory one accepts, the way of overcoming dyscalculia remains the same.

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3: Why are some children bad at maths? Because some children who are dyslexic appear to be good at maths, those who argue that dyscalculia is a side-effect of dyslexia have had to develop more fully the explanation of what has happened to those children who do appear to excel in maths despite showing the obvious signs of dyslexia. This is useful, because if we can see exactly why dyslexic children who are good at maths, are good at maths, then this should be able to help us work with those who are showing all the signs of having a significant specific learning difficulty in relation to maths. Among the most obvious reasons why some people are unexpectedly poor at maths are:

a) Poor learning. In this case the child has been taught basic mathematical concepts well, but has failed to learn, perhaps because of time off from school, severe behavioural problems, personality difficulties, attitude problems or similar difficulties. In this scenario there is no underlying genetic reason why the child should not learn mathematical concepts – the failure of the learning has an explanation which in all probability has nothing to do with maths at all. Children with these difficulties have low self-esteem in relation to maths and will do almost anything to avoid having to confront mathematical problems.

b) Poor teaching. Here the child has not been taught in a manner that allows

the child to learn the mathematical concepts dealt with in the lesson. This might be through disruption to primary school education through teacher illness, a lack of suitable teachers, or the use of unsuitable supply teachers. Typically this results in a child who is later asked to tackle more complex mathematical problems (such as multiplying fractions) without having a firm grasp of the underlying principle (such as what fractions are, or even what multiplication is). Remedial teaching of the basic functions using the best methods of teaching can readily resolve such difficulties.

c) Inappropriate teaching. Either through genetic disorders or some other

reason, the child does not respond to the standard methods of teaching maths in the classroom which work for most pupils. The teaching methodology is therefore inappropriate to this child and an alternative method is required. This is certainly a situation worth looking at and we later look at the fact that pupils who appear to have dyscalculia do appear to favour a different approach to handling mathematical problems from that adopted by non-dyscalculic children. What makes this problem worse is that the approach used by many dyscalculic children is one that does not readily lend itself to ease of marking and assessment by teachers. Thus teachers who are not aware of the implication of what they are doing, might then suggest to dyscalculic pupils that they adopt the alternative (mainstream) approach. Unfortunately rather than helping dyscalculic pupils this can make the matter far worse.

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4: The best ways to teach children who appear to be dyscalculic When one considers these three possible underlying reasons for observed failure in maths the options for solutions are clear. Firstly, for those children who suffer from low self-esteem in relation to maths there needs to be a programme of teaching maths which is combined with self-esteem enhancement. Where this happens there can be huge improvements in apparent mathematical ability very quickly. Unfortunately, although many countries in the English speaking world take self-esteem development as a central part of school work, this is not the case in the UK, and self-esteem enhancement remains a fringe activity (the first area to have its funding cut in times of difficulty) as opposed to a central aspect of the work of the school. Secondly, there needs to be high quality teaching which focuses on the difficulties that the child has. Clearly for the classroom teacher the belief or knowledge that a child is failing at maths because the child has a genetic disorder adds little to one’s ability to help the child. However knowing that the child is failing because the child has difficulty sequencing, or is confused over the basic terminology of maths, or because the child simply cannot work with maths in the conventional way, but is able to adopt an alternative approach, allows the teacher to construct a remedial programme which can overcome difficulties very quickly. This leads us to the approach to dyscalculia adopted in this book and indeed in all the materials produced by Multisensory Maths. We do not seek to resolve the argument about whether dyscalculia is part of dyslexia, or whether it is something unique which also happens to affect some dyslexics. Rather, having recognised that the source of dyscalculia is not completely clear, we focus on the methods of teaching maths that have been shown to help virtually all children who have difficulties with maths. Firstly, we need to know if the child has failed to grasp certain basic topics in maths. For example, if the child is unclear about the meaning of division, then there is not much point exposing the child to a lesson on prime numbers. Such a child might learn that a prime is a number that cannot be divided by any number other than itself. But that phrase will be meaningless until the concept of division is grasped and can be used. To locate such problems the child needs to be subjected to a number of tests which will show quite clearly where the child’s difficulties lie. Since the child might have problems with written questions, but not aural questions, with written answers but not oral answers, it is necessary to conduct the tests in various ways to find out exactly where the difficulties lie. Secondly, when the areas that are problematic are located the best methods of teaching that child have to be selected, so that the child can be helped to learn about the problematic concepts. For virtually every child the best method of teaching is a multi-sensory method in which the child will say the numbers and functions, touch and move counters or other objects which the numbers represent, write down the numbers as the action takes place, and see the counters etc move to new positions in accordance with the maths. Likewise when working with such topics as fractions the child will have a clear physical representation of the area to be divided into

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fractions, and will undertake this division at the same time as writing the mathematical symbols. Thirdly the teacher has to examine the ways in which the child handles maths problems to locate the specific methodology adopted by the child. As we show later in this book two significantly different approaches have been observed among children: the grasshopper approach and the inchworm approach. It is vital that the teacher is aware of whether the child is adopting the grasshopper approach, and adjusts teaching accordingly. Fourthly the teacher must be aware of any specific difficulties that the child has. If the child is shown to have sequencing difficulties or short-term memory problems, then these can be overcome through suitable training. This training itself may also be necessary to overcome difficulties the child might have with literacy, and the satisfactory completion of such work can have a major impact on the child’s subsequent achievements at school. Finally the teacher will need to consider the self-esteem issue. Unfortunately, whilst this is the area that can have the biggest impact on the child’s ability, it is the area that is most likely to be ignored by teachers, not least because many teachers feel uncomfortable with tackling the concept. However as the work of Lawrence has shown, teachers who have high self-esteem themselves and who are warm and positive in their approach to children are normally just as effective in increasing pupil self-esteem as are trained counsellors and educational psychologists.

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5: Overcoming dyscalculia: the systematic approach and the holistic approach We have already specified that overcoming dyscalculia falls into three steps:

1. Identification of the areas of difficulty (for example, the failure of the child to understand and be able to operate the concept of division)

2. Identification of specific difficulties with methodology of maths (for

example, the child who insists on adding the tens before the units in a sum) 3. Identification of the best methods of teaching, so that the child can

overcome the difficulties rapidly. In considering each child who is exhibiting the signs of dyscalculia the teacher will begin to draw up a profile for the child. Many children will be found to show a combination of problems but will also exhibit an emphasis on one of the six key areas of failure in maths outlined below:

a) a failure of understanding certain fundamental mathematical concepts (such as addition)

b) a failure of understanding of the words used to express mathematical

concepts (that is, the child understands the notion of subtraction, but hasn’t linked this to words such as “subtraction”, “take away”, and “less”)

c) a short term memory/sequencing problem which means that the child

cannot hold basic mathematical information in his or her head while performing a mathematical calculation

d) a problem in writing numbers down

e) a problem in reading the information surrounding the mathematical problem f) a lack of self esteem in maths - children who repeatedly comment on their

own inability to be successful at maths or who blame others for their value in maths.

Combining these two sets of observations together we can move into solving the child’s specific mathematical problems (as opposed to self-esteem, sequencing and short-term memory problems) either through a systematic approach which involves an analysis of the child and its maths, or through a more holistic approach. The holistic approach will be used by the teacher who knows the children well and has a clear feel for each child’s area of difficulty. The systematic approach will be used where the child is new to the teacher, or where the teacher is puzzled by a particular difficulty that the child has experienced. In the systematic approach the teacher starts by giving the child a series of tests to find out exactly where the child’s difficulties are located. Multisensory Maths has published Tests for Dyscalculia which contains all the most used tests for this type of work. These tests are not normed against a national average for each age group,

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but are to be used by teachers who already have a clear awareness of the type of concepts they would expect the child to be able to grasp at a particular age. Secondly, the teacher following the systematic approach will look for specific oddities in the methods of working that the child shows in a particular area. This is often undertaken by watching the child attempt a task in which the child regularly gets the wrong answer. Finally the teacher adopts the appropriate method of teaching. The various methods available to special needs and maths teachers are detailed in “Methods of Teaching Maths to Pupils with Dyscalculia.” The teacher using a more holistic approach will normally identify the child as having a difficulty in one of the six key areas of failure in maths identified earlier, and will quickly move to an assessment and appropriate teaching. Below we look at how this can work in practice by considering a suitable response to each of the six key areas.

a) Children who have a failure of understanding mathematical concepts. These children simply do not grasp the concept of (for example) addition or multiplication. In a group where virtually every child understands and can answer the question, “I have five beans in my hand, and now I pick up three more, so how many do I have?” the child is completely lost. These children can usually be helped by a multi-sensory approach in which all the problems are solved tactilely, visually and orally at the same time. In more advanced work we see children who have grasped some concepts, such as addition, but cannot translate these into the addition of larger sums. It is quite possible that a child understands the notion of addition, but will fail with something as complex as 143 + 89 = ____ while the rest of the class are able to solve the sum. Here the child understands one concept (simple addition) but not a subsequent concept (the adding tens and units within one sum). Again multi-sensory work can help. Additionally in such a case the child can often be helped by having a teacher watch the child as the child calculates the sum. Where a repeated error (such as attempting to add the tens before the digits) can be spotted the problem can be resolved by working through a large number of the problems with the child until the correct procedure is secured. However it will be necessary to bring in an awareness of the inchworm and grasshopper approach. For such children, intense work in small groups with other children with similar problems, and with support from home, can resolve the difficulties very readily. These groups can also be used as the foundation for self-esteem improvement in relation to maths, which can bring about huge, and often wholly unexpected results.

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b) Children who have a failure of understanding of the words used to

express mathematical concepts These children can grasp the mathematical principles when presented in everyday language and solved in a multi-sensory environment. However they can get confused by the use of words such as “the sum of”, “add”, “plus” and so on. Usually a simple pattern of repetition in which only one of the optional words is used until the meaning of that word is utterly secure, can help. From this point on we then follow with repetition of the relevant sums, gradually introducing all the variant words in order that the child can build up the knowledge that is missing. Once again multi-sensory work can be beneficial here as in this type of routine: • The child selects four counters from the table and says “four”. • The child selects the + symbol and says “plus” or “add” • The child selects three counters from the table and says “three” • The child selects the = symbol and says “equals” or “makes” • The child moves the two groups of counters together, after the = symbol

and says “seven”. For children who are confused about the various words that can all mean “+” the work should be repeated with one option (eg “plus”) being used over and over again until that is secure. Then the activity can be repeated with “add” instead of “plus” and then with “add” and “plus” alternating, before moving on to “the sum of”. c) Children who have a short term memory/sequencing problem. These children cannot hold basic mathematical information in their heads while performing a mathematical calculation. Thus when attempting to say a times table they can get lost in the progression – having got to seven times six they can simply lose track of the fact that eight times six is next. Children with this type of problem need practice in sequencing activities of all types, and an understanding from teachers that their problem is primarily one of sequencing and not of maths. These children are very likely to have literacy problems and be described as dyslexic, for much the same reason. However although the problem is not primarily a mathematical one, mathematical sequences are ideal practice for these pupils. The child needs practice in putting numerical sequences in order and should then write them and say them simultaneously. d) Children who have a problem in writing numbers down These children simply do not link the concept of (for example) six as in six buttons on a table, or six tables in a room, with the symbols “6” and “six”. Such children are best helped through a multi-sensory approach which links the concept with the symbol with the child actively counting the six items, saying the word “six” and then writing down both “six” and “6”.

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There can also be a sequencing issue here, as with the child who writes eighty-one as “18” or who looks at “81” and says “eighteen”. Again the child needs constant practice and reward. As the child begins to grasp that both “18” and “81” are spoken with the sound “eight” at the start, they begin to differentiate between the two numbers. With buttons on the table the numbers do look very different, and children can then move from the visual display of buttons over to saying the number and then writing it down as a number and, if appropriate for the age, as words. Finally the child can be shown both numbers and can put out the right number of counters for each number, say the number and copy it down. Where confusion continues additional aides-memoire can be given which expand the child’s awareness of the number. “The first number is always the strongest” was one child’s way of learning “81” is bigger than “18” (81 having 8 strong numbers while 18 only has one.) Combined with the visualisation of those 81 counters, the child finally understood the difference. e) Children who have a problem in reading the information related to

maths.

These children may well be dyslexic, and may have combined their dyslexia with any of the problems above. But their key problem is that they simply cannot read the words they are given. This can lead to quite a confusing situation. A child may be able to answer the written question “6 x 3 = __” perfectly because the child fully understands the concept of six times three equalling 18. But the same child may get the question wrong when it is written as “there are six groups of three children in the playground, how many children are there” because the child cannot read the word “groups”. Thus the child may grasp the concept but get an answer wrong because the test being used is one that includes words that the child cannot decipher. Once it is established that the child’s difficulty is with the words and not with the maths, then a series of sessions using standard dyslexia related techniques will ensure that the child learns the specific words as required. We should never underestimate the power of this problem. We have seen children fail tests which speak of a group of children with the simple question, how many boys were there and how many girls. The child could not read “girls”. f) Children who repeatedly comment on their own inability to be

successful at maths or who blame others for their value in maths. These children have a self-esteem problem in relation to maths. They may well need support and help as outlined in sections a) to e) but these will need to be delivered within the context of self-esteem enhancing sessions. Fortunately this is not difficult, and details are given later in the book.

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6: The systematic method of approach: the 3 steps to solving the dyscalculic problem The three step approach to solving the dyscalculic problem in children involves:

1. Identification of the areas of difficulty (for example the failure of the child to understand and be able to operate the concept of division)

2. Identification of specific difficulties with methodology of maths (for

example the child who insists on adding the tens before the units in a sum) 3. Identification of the best methods of teaching, so that the child can

overcome the difficulties rapidly. The systematic approach to identification of areas of difficulty involves the child undertaking a few tests to find out where the problems lie. The teacher needs to have available a set of tests so that appropriate tests can be given. Even if the teacher has not known the child for long the number of tests to be given should not be huge, for the teacher should quickly get an idea of where the problem lies. Ideally the child should not participate in more than 3 or 4 tests in a short period of time. Here is a list of the principle concepts on which the teacher may well wish to test a child . Obviously, the exact concepts chosen will be selected according to the age and intellectual ability of the child. • Add, sum of, +, = • Adding decimals • Adding fractions • Area • Counting • Decimal values • Division • Dividing decimals • Dividing fractions • Errors in copying (short term memory) • Estimations • Grouping numbers, sequences • How many…, adding, total • How many…taking away • Improper fractions • Long division • Mental arithmetic: take away, minus, less • Mental arithmetic: add, plus, sum • Mental division: divided by, ÷ • Mental multiplication: times, multiplied by, times tables • Money • Multiplication (x, =) • Multiplying decimals • Multiplying fractions • Number, lower, higher, high, low, small, big

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• Odd numbers, even numbers, odd one out • Rearranging sequences • Reciting sequences • Recognising numbers • Reducing fractions • Shapes • Subtracting decimals • Subtracting fractions • Take away, less, minus, - • Written addition, subtraction • Written division, multiplication • Written subtraction tests (-, =) Once the teacher has identified that the child has a difficulty with a particular concept then the teacher can move on to step two: the identification of specific difficulties with methodology of maths. There are a number of specific difficulties with methodology, and these are identified below: • False learning • Fixed thinking • Mathematical terminology • Memory problems – long term • Memory problems – short term • Reading ability • Reversals • Self-esteem • Sequencing issues • Spatial awareness • Speed of working • Visual perception.

We now deal with each of these in turn. False learning One of the great problems with dyscalculia is that regular practice of mathematical concepts can actually prove harmful to dyscalculic students. If the student learns the basic concept wrongly (for example misunderstanding the methodology of the subtraction of one number from another where both numbers are over 100) then regular repetition through doing lots of examples, will only make matters worse. The child will internalise a totally wrong method of work, and then when the work comes back with the fact that every answer is wrong, the child will suffer a decline in self-esteem. Since it is often the case that the teacher will set a number of questions to be answered in class and homework, and (through the need to deal with a large number of pupils) will not look at the individual pupil’s work until later (as opposed to checking it in the lesson) the child can be left to do a series of exercises wrongly. This will drive the false learning deeper into the child’s memory, and cause significant difficulty in the unlearning stage that must follow.

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It is therefore essential that dyscalculic children have their work checked immediately upon doing the first worked example of a technique – to ensure that they are not using a false methodology. They might then be allowed to do two examples, and if these are correct, they can then go on to four examples, and so on. This approach enhances self-esteem and stops false learning. Fixed thinking If a child adopts a fixed approach to maths (as in “this is how I do it” or “this is how I was told to do it”) then the child is in danger of failing to find his or her own best approach to solving maths problems. Furthermore, if the teacher has a fixed view of the only or best way to solve mathematical problems, then the teacher will not be able to help a child explore alternative approaches. The approaches will be dismissed whether they lead to the right answer or not. Many children exhibit the fixed thinking approach out of what we may, when moving away from politically correct educational language, call “sheer bloody-mindedness”. Parents experience this when, on seeing a son or daughter struggling with a homework, offer help, suggest a variation in methodology and then find themselves deeply involved in a row with the child who claims that the teacher said “we must do it this way”. Rather than helping they find that they have made matters far worse. An awareness of the problem of fixed thinking leads to the view that part of the teaching of dyscalculic students must involve an openness of approach in which the teacher works with the child to discover the right approach for that child. Not only will this help the individual solve the problem, it will have two other interesting effects. Firstly, it can show the child that flexible thought is of much greater value in maths, and indeed in all learning, than rigid thought. Secondly, the child may well produce a methodology which the teacher will wish to pass on to another dyscalculic child at a later date. Furthermore, if the child produces a methodology which is interesting but not accurate, the teacher might be able to devise a way of amending the methodology so that it gives the right answer, and retains its attractiveness to the child. Mathematical terminology It is surprisingly common to find that children who have not benefited from multisensory learning are confused about the many words that mean the same thing in maths. It may seem fundamental that + means “add”, “sum of”, and “total of”, but it is often the case that dyscalculic children do not know this. Therefore we strongly advise that all children with dyscalculic tendencies undertake work on the vocabulary of maths, working with the terminology in one form at a time, until each has been fully mastered. In the area of terminology we would always say, “never assume an understanding”.

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Memory problems – long term This problem reveals itself when children appear to have learned something, and then find that the knowledge has vanished some days later. A dyscalculic child with long term memory problems is going to find it very hard to learn a multiplication table in such a way that it rests within the long term memory permanently, unless that child is given a lot of support in terms of multi-sensory work. In the multi-sensory approach the child’s memory is enhanced by having each element to be learned experienced in through several senses at once. To take the most common example of the seven times table, the child is given a set of counters and a grid. The grid has two columns as in the table below. In the left hand column the child puts down one counter, and says “once seven”, and then in the right hand counter puts down seven counters, and says “is seven”. Then two counters are put in the next row in the left column followed by fourteen in the right column with the chant “two sevens are fourteen”. One counter Seven counters Two counters Fourteen counters Three counters Twenty one counters Of course the process is slow – the child has to count out a lot of counters, but it starts the process working and shows the child the link between something real and the abstract world of multiplication. When the child starts to make some progress, perhaps being able to deal with the table up to 5 x 7, the process can be changed so that the child writes down the numbers while saying them. In another version the child puts down counters in the left column but does not deal with the right column. The issue of asking the child what 6 x 7 equals is not even considered until the child is secure over a period of several weeks in handling the table and the lower numbers at the start of the multiplication table. What one must remember here is that the long term memory is a memory based on meaning. When we are told a story we remember the meaning of the story, not the individual words and sentences. If the child does not make any meaning out of the maths to be remembered then it will not go into the long term memory. Memory problems – short term Short term memory problems often become apparent when the child is attempting to handle multiplication tables or mental arithmetic. The child may simply lose track of where she or he got to in the table – following 4 8’s with 6 8’s or simply stopping totally. Similarly the child may switch from one table to another, or go back to a secure moment in the table and work forwards again from there. In mental arithmetic the child may be perfectly able to do the sum, but simply loses track of elements already resolved. When questions are set by the teacher, the child may appear to be unable to answer when she or he actually knows how to do the work, simply because the question has

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been forgotten due to short term memory problems. Likewise hints and tips on how to solve a problem might well be lost because the child once again loses track of these issues because of short term memory problems. In such a case the pupil needs to be helped to learn each individual element of the problems that are giving difficulty. Using a multi-sensory technique usually helps at this point and aids the passing of the information from the short term memory into the long term memory. Techniques also need to be taught for overcoming memory problems such as focussing on one issue at a time, making notes, setting out work clearly and giving additional meaning to otherwise meaningless elements within questions. There is a section dealing with many aspects of memory in relation to maths in Methods of Teaching Maths to Children with Dyscalculia published by First and Best in Education Ltd. Reading ability Maths as taught at school uses written English. If the child has a problem with written English – as will be the case if the child is dyslexic – then that problem will transfer into an apparent problem with maths. Of course the temptation will be to try to find a route of teaching maths without using written English, but this can put a great demand on the short term memory, and if the child has problems here (as is very likely if the child also suffers from dyslexia) then one has not taken matters much further forward. It is much better that the child’s literacy problems are tackled at the same time as the mathematical issues. Reversals The fact that with a sum we work from right to left (units, then tens, then hundreds) whereas in long division we work the other way round, is something that doesn’t occur to most of us – until we find a dyscalculic child going about things the wrong way. The fact that we write “15” but actually say “fifteen” (in the first putting the tens first, in the other putting the units first) is likewise something that is happily ignored by non-dyscalculic people. Where a child does reverse numbers this may be a simple misunderstanding about the process, but if the child also writes numbers in a form of mirror writing, turns 81 into 18, or has a profound problem with the concept of the value of decimals then there may be a reversal problem occurring throughout the child’s understanding of maths. In order to solve reversal problems it is necessary to examine all the common areas where the reversal is taking place, and to deal with them one at a time, making a clear rule for each issue. You may recognise that four separate areas of difficulty are all reversals, but there is little point in telling the pupil this. Rather the

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knowledge should be used to inform yourself and your colleagues that every area of reversal must be given special attention. Therefore time must be set aside to focus on each area of reversal, rather than simply pointing out each error as it occurs. It is no good saying, “Now come on Alan, think about this, how do we write the number 13” when the error occurs. Time must be given for writing the numbers and saying them as they are written, until such time as the child is able to do this without making an error. One then moves on to the next reversal issue. Many children who suffer from dyslexia are helped with the regular use of wooden letters which they take from a pile and lay out in alphabetical order. The same approach can be used with numbers where reversal is an issue, with the child being required initially to lay out the numbers from 1 to 9, and later 0 to 19. Self-esteem Dyscalculic children, along with dyslexic children, almost always suffer from low self-esteem. The reason is fairly obvious – these children are failing in a high-profile way in an area where almost everyone else is succeeding. Try as they might, they cannot make sense of the world around them in the same way that their fellows seem to do fairly effortlessly. Such a scenario is not a recipe for feeling good. And yet the need to have a high self-esteem is beyond doubt the most important need in a child’s social education. No matter how much love, warmth and affection is given to a child by loving parents, doting grandparents, and professional colleagues, for the dyscalculic child it is all stripped away by the daily round of impossible maths problems. Children with a high self-esteem generally show fewer behavioural problems than children with low self-esteem. Children with high self-esteem focus more clearly on the task in hand in the classroom. They do not enter a maths lesson believing that they cannot achieve anything, that they are “poor at maths”. Instead they enter the classroom believing that with the right sort of help they can master anything. Through the adoption of specific self-esteem programmes (which are now commonplace in America, although still rarely used in the UK) children’s social and emotional development can be enhanced, and this will make the teaching of maths to dyscalculic children a much more rapid, much less painful process. Sequencing issues Just because a child can recognise some sequences it does not mean the child can recognise the meaning inside all sequences. For example the child might well be able to see that 5, 10, 15, 20 is a sequence but not recognise that 6, 11, 16, 21 is a sequence with the same attribute. Likewise many children with dyscalculia may well be able to deal with the sequence 6, 7, 8, 9, 10 but have problems with 10, 9, 8, 7, 6. Where sequencing is a problem it should be taught as a subject area in its own right – not as an add-on to another topic. Thus the child should spend five or ten minutes a day working on numerical sequences in a structured way. Depending on the child’s area of difficulty, one would start with a sequence that the child can cope

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with (for example 2, 4, 6, 8) and then move on to extending the sequence (14, 16, 18) and then move towards a closely linked, but insecure sequence (1, 3, 5, 7) and so on. By working through each area the child gradually puts together a knowledge of, and security in dealing with, mathematical sequences. Spatial awareness The signs of spatial awareness problems are usually not difficult to pick up. Children with this type of problem lose their place when reading, jump from one question to another, and have particular problems with issues such as geometry, area, volume etc. Children with spatial awareness problems need help in a variety of ways. Firstly, we can help by setting out problems on their paper as clearly as possible, ideally with just one question per page. Secondly, we can help them organise themselves, by using a ruler or finger when reading across a line, and when keeping track of which question is being answered. Thirdly, additional help should be given in all areas of maths that have any particular spatial element within them. Fourthly the child can be helped with specific maths problems which have to be set out on the paper. A sum involving the addition of two large numbers can be very difficult for the child with spatial awareness unless it is set out with a lot of space between the numbers, allowing the child to focus on one column at a time. Finally, and leading on from the fourth point, the child needs to be shown how to focus on one element of the spatial issue at a time. Most children who suffer from spatial problems also suffer from a feeling of being overwhelmed by the problem they are facing – they have no method of dividing the question into units. However with regular help on this topic, they can be shown that each question is nothing more than a set of little questions. 654 + 297 may look horrific to the child when presented as 654 297 But if it is presented as a set of steps of which 7 + 4 is the first, then suddenly it takes on manageable proportions. Speed of working Dyscalculic children present a range of problems in relation to speed of working. Some children slow down their work rate dramatically in an attempt to deal with mathematical problems. This sounds as if it might be a good idea, but there has been some research to suggest that slowness in handling maths questions can result in further difficulties – and many dyscalculic children who do slow down lose track of the essence of the question while apparently adopting the perfectly reasonable strategy of slowly trying to handle one element of the problem being faced. This suggests that the notion of giving dyscalculic children extra time to answer maths questions may not be as helpful as we imagine. Other children can be seen

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flying through questions at a very high speed, finishing long before the allotted time, and making no use of the extra time allocated – and usually doing very badly in the test as a result. The answer appears to be that dyscalculic children can be helped to use extra time and to work at a speed that balances their need to proceed more slowly with the problem of losing track of the question. This invariably involves working in a step by step approach with the child in some situations, while encouraging the child to make larger leaps in others, all the while watching how the question is answered, and ensuring that the child does not keep going backwards and forwards with the question, but proceeds in a logical manner. For many children, part of the solution can involve helping the child write down elements from the question in a way that is not only meaningful when they are written down, but which also remains meaningful and can be used a few minutes later. Thus a dyscalculic child might have no difficulty understanding the problem which says, “I have 192 widgets and want to sell them for £23. However my customer only wants to buy 24 widgets. How much do I charge him?” But as she or he starts working the issue gets confused by the fact that instead of writing: 192 widgets cost £23.00 24 widgets cost £????? The child instead writes: 192 = 23 and then loses track of what each number represents. But we should not dissuade the dyscalculic child who has learned that as long as he or she keeps the cost on the right hand side of the equation and the items being costed on the left, there is an unvarying direction to the numbers within the equation. If, knowing this, the child writes nothing other than: 24 x 23 192 we should not worry about the lack of working to get to that point. Visual perception Some dyscalculic children readily confuse written symbols. These children also are known to report that the letters on a page with black print on a white background “jump around”, and find that coloured glasses can be a great aid for them. For such children the way in which numbers and symbols are presented is of great importance, especially at the time when they are working on concepts which are difficult for them anyway.

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Double spacing can help, as can the use of coloured paper instead of white, and blue print instead of black. Obviously it is not possible to change the maths books and exam papers for these children, but if this area of difficulty can be removed before the child tries to tackle the mathematical concepts which the child finds difficult, then the route towards mathematical understanding can be made a lot easier.

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7: Policy documents on dyscalculia

1. The recognition of dyscalculia This school recognises that there is a specific learning difficulty known as dyscalculia. It affects children in such a way that their attainment in mathematics is significantly less than we might otherwise expect it to be, given their age and their intellectual abilities. We also recognise that it is our duty to accommodate the disability that these children suffer from, and to ensure that they are offered teaching of the appropriate quality and style such that they can overcome this disability. It is our view that almost all children who suffer from dyscalculia can be helped to overcome their problems, and can reach at least the average level in mathematics for the school for their age group and intellectual ability.

2. The school’s approach to dyscalculia The school adopts a three phase approach to dyscalculia. Firstly, where staff agree that there is a chance that a child might be dyscalculic the child is given one or more of the tests that the school has available. These will indicate areas in which the child has particular difficulties with maths, and will highlight the area(s)for remedial action. Secondly, the relevant member of staff will work with the child to discover those areas where the child has a particular problem with the methodology of solving mathematical problems. Finally, based on these two understandings the teacher will develop appropriate remedial teaching techniques to help the child overcome the dyscalculic difficulties.

3. Tests for dyscalculia The school has in its possession a series of tests which are administered on a one-to-one basis with any child who, in the view of appropriate members of staff, might be suffering from dyscalculia. The teacher responsible for selecting and administering the tests is: _ _ _ _ _ _ _ _ . It is the school’s policy that the parents of any child undertaking the tests will be informed that the tests are taking place, and that these are tests operated by the school, and not part of the government’s testing programme, nor tests administered by an educational psychologist. At the same time the parent will be given information on dyscalculia. Once the tests have been administered the teacher administering the tests will write to the parent again, explaining the results and the work that will now be undertaken with the child.

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4. Recognise that methods of learning vary

It is inherent within our approach to dyscalculia that we recognise that different children learn mathematical skills in different ways. While many children and teachers prefer to adopt a step by step or serial approach to maths, many children with dyscalculia prefer to adopt a holistic approach. We recognise that although it is helpful for dyscalculic children to be shown the benefits of the serial approach to mathematics, it is not helpful for them to be forced to abandon their own holistic approach. Rather they need to be helped to find the areas in which their own approach does not provide correct solutions to problems, and at such points to adopt elements from the serial approach. We recognise that it is difficult to work with children who adopt holistic approaches, since their work does not follow set patterns, and is not always easy to follow their methodologies. However we also recognise that this is by far the best way of helping dyscalculic children, and we therefore follow this route.

5. The relationship between special needs teachers and maths teachers Dyscalculia as a subject is one that is of deep concern to teachers of mathematics and to the special needs co-ordinator. We ensure that teachers from both areas are fully informed about the children who are considered to be dyscalculic in this school. Furthermore we ensure that all teachers involved in this area work together using the same three phase approach.

6. Keeping all staff informed

Mathematics is a subject area that can have an impact in many other subjects. Science, design technology and geography all deal with mathematical concepts within their subject areas, as do other school subjects to lesser degrees. We therefore adopt a policy of ensuring that all staff in the school know which children are considered to be dyscalculic. Furthermore all staff receive in service training to remind them that comments about the mathematical ability of any of these children should be handled with great care.

7. Liaison with parents

We have no doubt at all that where parental co-operation can be gained, the dyscalculic child can overcome his or her difficulties much more readily than where there is no parental support. We therefore adopt a policy of keeping the parents informed about our work with dyscalculic children and of ensuring that they are invited into the school to be given guidance on what the children are doing to overcome their dyscalculia.

8. Homework

Children with specific learning difficulties overcome these difficulties far more rapidly if they undertake regular homework relating to their difficulties. It is therefore our policy that we encourage dyscalculic children to undertake regular

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mathematical exercises five nights a week at home, and we encourage parents to be part of this work.

9. Recognition of the role of self-esteem We recognise that most children with dyscalculia suffer from self-esteem deficits in relation to maths, and therefore we work to help the children raise their self-esteem. This approach takes place in four ways:

a) We encourage parents to be positive about their child’s abilities, and to be aware that dyscalculia can be overcome.

b) We encourage the children to be aware that their problems with

mathematics can be overcome, and that their dyscalculia has given them insights into ways of working which other children do not have.

c) We ensure that no members of staff make any negative comments

on the mathematical failings of children with dyscalculia.

d) Self-esteem enhancement takes its place as an important part of the work with dyscalculic pupils.

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8: The Inchworm and the grasshopper: how dyscalculic children learn maths In 1984 Bath and Knox developed the idea that there were two extreme ways in which students learn maths – the grasshopper approach and the inchworm approach. This notion has proved to be very attractive to teachers of dyscalculic pupils and has been popularised in this country by Chinn and Ashcroft who have related the methodologies in several books on dyslexia and mathematics. Bath and Knox have not suggested that children follow either the grasshopper or the inchworm approach. Rather they have suggested that children tend towards one or other of these approaches, and that some (but only some) children do adopt an extreme position. It appears that there is a tendency for dyscalculic pupils to have a propensity towards the grasshopper approach. In the broadest sense the inchworm is a person who is interested in each individual part of the problem, and who attends to each specific detail in turn. If a problem is presented as a unified whole then the inchworm strives to break it down into parts, seeking out the relevant facts, and searching for known formulae that can be applied. We can see this as a step by step approach which works from the start to the end. As such it can seem highly attractive to parents and to teachers – it has a clearly defined logic about it, and if the method used is transparent the teacher will quickly be able to see where the child has gone wrong if this has happened. The weakness of the inchworm approach however is that it does restrict the child to one method only, and if the appropriate formula needed to solve the problem is not available, the whole method can simply grind to a halt with the child proclaiming that she or he does not know how to do it. Furthermore, once the solution is completed the child has no available option to allow checking of the work. If checking is to be done, the child will simply check in the same way as the problem was solved. Thus if an error appears in the logic, this will not be picked up in the checking. Therefore we can see that inchworm techniques are utterly method orientated – if the method is missing or inaccurately remembered, no progress can be made. If the child does not realise that the method is faulty then the child is unlikely to realise that the answer is wrong. Grasshoppers see things holistically. They look for estimates and ways of including or excluding possible answers. They explore problems and allow for and accept a range of different types of answer. In some cases they “play” with numbers, putting them into more manageable forms, and often undertaking mental mathematics during the course of the problem solving. They look for ways of verifying results in which a different method is used, in order to validate the method as well as the computations. Chinn, writing in Miles and Miles “Dyslexia and Mathematics” gives an example of how an inchworm and a grasshopper would approach the same problem: in this case 235 – 97.

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The inchworm would most likely want to write down the problem in the classic manner: 235 - 97 ___ The 7 would be taken from the 5, with the adjustment from the tens column. The 9 becomes 10, which forces the 3 to become 13 and so on. This method will give the right answer of course, but will depend on the child having a reasonable short term memory in order to keep track of the work as it progresses, and on the child understanding exactly the concept of moving numbers from the units columns to the tens column etc. The grasshopper approaches 235 – 97 in a very different way. Looking at the numbers, the grasshopper recognises that 97 is just 3 off 100, and that 100 tends to be a very easy number to handle. 235 – 100 gives no problem at all – the answer can be calculated in the head, it is obviously 135. But that is taking 100 off, and the question only demanded taking 97 off. We have taken away 3 too many, so we now add three back in. 135 + 3 = 138, which is nice and simple, and the correct answer. To verify this the grasshopper who has remembered a bit of the inchworm approach might remember that to take 7 away from 5 one has to make the 5 into 15. 15 take away 7 is 8, so the answer ends in 8. The quick route to the answer gives 138, which also ends in 8, which suggests the quick route is correct. From the dyscalculic student’s point of view the grasshopper approach is very attractive. Obviously it does not rely on remembering complex rules. Also it doesn’t rely nearly so much on short-term memory. But it tends to result in correct answers which magically emerge from a collection of doodled numbers. For the teacher trying to mark the child’s calculations, it can appear as if there are no meaningful calculations at all. When pressed for an explanation, grasshopper dyscalculics often find themselves in difficulty because, by its very nature, the grasshopper approach means there is no one unique approach. Faced with the same problem again the child could approach it in a totally different way. What most writers have noticed is that most people (children and teachers) adopt a position on the grasshopper / inchworm continuum, while occasionally adopting approaches from other parts of the continuum. However those people who feel most comfortable with a right hemisphere approach to maths will always tend towards the holistic, while those who are more comfortable with the left hemisphere of the brain will use the serial style of work. Such evidence as has been gathered on this subject clearly suggests dyscalculic students like to work as grasshoppers. The first and most important consequence of this view is that teachers must be highly flexible in their review of the approaches of their students. This can be difficult, because for a committed inchworm the grasshopper approach does look wrong in every regard. Conversely the full-time grasshopper simply cannot understand why the inchworm cannot see the obvious short cut available.

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Where the teacher seeks to engage the pupil in discussion over methodology the inchworm has a distinct advantage – the methods adopted are invariably clear for all to see. But it is the dyscalculic inchworm that is most likely to run into difficulty, for the inchworm will suffer significantly if any of the following factors apply: • False learning – in which case the wrong rule will be applied, or the right rule

will be applied wrongly. • Memory problems – particularly short term – which will mean the child might

lose track of the earlier steps in the chain before the last step is completed. • Reading ability – where the child has difficulty through a problem with one or

two words, and is unable to work around them to grasp the overall concept being questioned.

• Reversals – which can happen as the child works step by step. The problem is

that because the child is following rules, the fact that the final answer makes no sense in terms of the question overall may be missed, and thus the child will not be on the look out for a reversal.

• Speed of working – a problem because serial working is much slower than the

grasshopper approach. Grasshoppers who are dyscalculic have far fewer problems, providing that they can hit on a way of resolving the problem. If their holistic approach does not produce a solution then they cannot proceed, and will normally revert to inchworm tactics. However, having done this they might well then “see” a grasshopper holistic way of solving the problem emerging out of the inchworm approach, and this will get them going again. There is some research which suggests that an overwhelming majority of children are inchworms when it comes to maths, and that the majority of maths teachers tend to adopt more of an inchworm approach to teaching than a grasshopper approach. This sounds fine, except for the finding that we have noted above that most dyscalculic students appear to be grasshoppers. If this is true - and there does not appear to be any research which suggests otherwise - dyscalculic children seem to be in a double fix. On the one hand they have difficulties in handling maths. However they have (perhaps intuitively) adopted a method of dealing with maths that is more in keeping with their needs, and which given a fair wind can produce the right answers. But, unfortunately, this approach then puts them at odds with the approach adopted both by their non-dyscalculic classmates and their teachers. At this point there is a temptation for the teacher to fall into the trap of thinking that he or she has discovered the reason for the child’s poor performance at maths – the child is adopting a bizarre holistic lateral-thinking grasshopper approach to maths instead of the clear step by step approach which the rest of the class have adopted with such success. If only the child can be moved away from the holistic style of working into the serial style, all will be well. This view looks highly logical on the surface – and many teachers of mathematics have been confounded by the fact that attempts to move children who are failing at maths to the inchworm approach have actually made things worse.

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Our view is that grasshopper students do need to learn some of the key aspects of the inchworm techniques. They must learn to document what they are doing. They must learn how to handle the formulae, and break down problems into constituent parts – not least for those moments when no holistic approach springs to mind. But, and this is a most vital point, they must learn these approaches in addition to, rather than as a replacement for, their own holistic skills. Solving the inchworm / grasshopper dilemma using multiplication as an example Many children with dyscalculia benefit from being shown how to employ a combination of holistic and serial approaches to maths. When a maths topic is being taught dyscalculic children can find it most reassuring to be shown both the overall topic, and then be taken to the individual elements that makes up that topic. They like being allowed to explore their own solutions – indeed they often find the encouragement to experiment with a solution a rare treat in mathematics. But they are also fascinated by the inchworm approach as an alternative – as long as it is not forced upon them as the one and only way to solve the problem. If we consider the issue of a child who has only the slightest idea of what multiplication is about, and who cannot utilise times tables with any certainty of accuracy, it is obvious that we need to go back to the very basics of multiplication. But at the same time the child will know that he or she has failed to grasp something which others seem to be aware of and able to deal with. In this situation the child may well be very reluctant to accept a “back to the beginning” inchworm approach and may well anticipate a further failure. For this reason alone a different approach is needed. When one considers the grasshopper and inchworm scenarios presented above we can expect also that traditional teaching methods will not work. However a series of sessions with the child which covers the overall issue of multiplication with attainable units of learning, can provide a breakthrough. To give the overview first, the child could be presented with a multiplication square as in the example below (a larger scale version which can be photocopied for the child to have is printed at the end of the chapter) 0 1 2 3 4 5 6 7 8 9 10 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 10 2 0 2 4 6 8 10 12 14 16 18 20 3 0 3 6 9 12 15 18 21 24 27 30 4 0 4 8 12 16 20 24 28 32 36 40 5 0 5 10 15 20 25 30 35 40 45 50 6 0 6 12 18 24 30 36 42 48 54 60 7 0 7 14 21 28 35 42 49 56 63 70 8 0 8 16 24 32 40 48 56 64 72 80 9 0 9 18 27 36 45 54 63 72 81 90 10 0 10 20 30 40 50 60 70 80 90 100

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The child is now told that within a short while she or he will be able to take a blank table like this and fill it in. When the child can do this, the child will also know all the main multiplication tables – because everything is on that piece of paper. Although the collection of numbers might seem daunting it is nevertheless reassuring that the whole problem of multiplication is something that can be written on one grid. It is also an opportunity for the child to look for patterns and relationships within the table. Not all dyscalculic children want to follow this route, but some are fascinated by the patterns that they discover. Having introduced this concept at the start of the first remedial lesson the child can be asked to choose a line that looks the easiest one to learn. Perhaps for the first time the child looks at a series of numbers from the point of view of a set of relationships. The child will usually pick out the 0 row or column. Fine says the teacher, here is an empty grid. Now write in the 0 row or column. (A copy of the empty grid appears at the end of this section, for use in photocopying). The teacher then points out that one whole section of the grid is now known to the child. What’s more a second section can be filled in – the child completes the zero row or column that was not completed before. Securing the information comes next, for it is certainly not enough to tell the child that she or he has learned the Nought times table. At this point multi-sensory methods must come into play. The child sees a collection of counters on the teacher’s side of the desk or on one side of a divide. Questioning then proceeds along these lines: “How many counters are there here?” “Lots.” “How many on your side?” “None.” “Fine – can you see that ‘none’?” “No there is nothing to see.” “Good – but I want you to imagine it is there – put your hands around the ‘none.’ Now how much have you got?” Child begins to look incredulous and doubts teacher’s sanity, but to humour teacher says, ‘None’. “Now I am going to do the same, and give you another lot of none.” Teacher carefully cups hands and delivers a handful of air to the child, next to the child’s hands, still cupped holding the ‘none’. “So you now have two lots of none. Right?” “Yes” says the child, wondering how long it is until break. “So what is two lots of none?” Child is blank. “I have given you two lots of none. How many are you holding?” “None!” “Exactly. And if I do it again,” (teacher does it again), “what is three lots of none.” “None!” The child starts to wonder if this is a case for reporting to Childline. “So what is three lots of none?” “None.” “Imagine I go on, with four lots of none.” “None!!!!” “Five lots of none?”

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“None!!!” “Ten lots of none?” This is the testing point – because despite the procession of zero answers, the child might still jump to ten as an answer. But if the answer is indeed another vehement ‘none’, then the teacher can return to the grid, and show that indeed the grid reports exactly the same answer – none. That concludes the lesson. In the second lesson the child is given the blank grid and asked to fill in what was learned last time. If the child can’t recall the stream of vertical and horizontal zeros then revision is needed, but if all is well, the child can instead proceed to search for another easy table – this time the 1s. Having played the game of “none” the child can this time work with a physical set of counters. The child now looks at the table again, and is asked to choose another very easy column. This time the child chooses “1” noticing that the table builds up in the most obvious sequence of 1, 2, 3 and so on. For this activity the child can be given a grid of ten boxes which can be set out on the table. This can be a printed piece of paper with this design Or it can be laid out as a physical entity on the table – for example with pencils as the boundaries for each box. “I’ll give you one counter. How many have you got?” Recognising from the last lesson that the only way to proceed is to humour the teacher the child says, “One.” The child is instructed to put the counter down in the first box on the left, and then receive a second counter. This second counter can be set down in the second box on the table. “How many boxes have got a counter in?” asks the teacher. “Two,” says the child. “How many counters are there?” asks the teacher. “Two,” says the child. So the activity progresses. At the end the child is asked a couple of questions at random from the table, along the lines, “If I have five one’s how many have I got?” If the answers are correct the child now returns to the blank grid and is asked to fill in the 0 times table and the 1 times table, both as a column and a row. Now is the time to introduce the school’s preferred way of stating multiplication tables – as in “Once one is one, two one’s are two” or whatever terminology is preferred. The child says it with and without reference to the table. If the child has particular sequencing problems at this stage the child may get lost in the sequence when there is no printed reference point. So the child may get to “seven one’s are seven,” and then not remember that eight comes next.

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If this is the case, but the child can cope with the table when looking at the chart, allow the child to continue with the chart, moving a finger across in order to keep track. Meanwhile undertake sequencing activities with the child as a regular part of the lessons. The next lesson moves on to twos. From this point on the question “how many twos have you got?” becomes more meaningful since they are clearly there in front of the child. The following question, “how many does that make in total?” is now hard enough to cause the child some deeper thought. Having got to this point it is vital that the child must be linking the physical attribute of the number with the written symbol. A new grid can be used: 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 12 14 16 18 20 As the child sees the counters in the grid he or she says the numbers and writes them down in the table as above. So the child can say: “No twos are none” writing 0 in the top line and 0 below it. “One two is two” writing 1 in the top line as the “one” is said and “two” in the bottom line as the “two” is said. As the lessons proceed the child may start to find the patterns that exist within tables. For example, The three times table alternates odd and even numbers. The last number declines in groups of three: 3, 6, 9 becomes 12, 15, 18, and then 21, 24, 27. The four times table can be discovered to be every other entry in the two times table. The five times table has solutions that only end in 0 or 5, making it one of the easiest tables to learn. The six times table shows another interesting pattern: 6 – these alternate final numbers go up in twos – starting with 6 12 – while these alternate final numbers go up in twos – starting with 2 18 24 – then four 30 36 – then six 42 48 – then eight 54 60 The seven times table has its own logic. Firstly being an odd number, the answers alternate between odd and even 7 – the final number descends by three, starting with 7

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14 – then 4 21 – then 1 28 – and the circle goes around again, taking 3 off 11 and getting 8 35 42 49 – also the two numbers added together go down in pairs – 9+4 = 13 56 – 5 + 6 = 11 63 – 6 + 3 = 9 70 – 7 + 0 = 7 The 8 times table is not only double the four times table but also: 8 – the final number descends by two throughout, starting with 8 16 – then 6 24 – then 4 36 42 48 56 64 72 80 The 9 times table has every answer adding up to 9: 9 18 – 8 + 1 27 – 2 + 7 36 – 3 = 6 45 54 – Also from this point on the answers are the reverse of the earlier answers 63 – 63 is the reversal of 36 72 – 72 is the reversal of 27 81 90 Which takes us to 10, which is almost as easy as the 1 times table. The question is now quite reasonably put, is it worth looking at these odd effects of certain times tables such as 7. The answer must be that, if the dyscalculic child is interested, then it is.

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The times table grid Multiply a number in the top row by a number in the left column and the answer appears where the lines cross.

0 1 2 3 4 5 6 7 8 9 10

0 0

0 0 0 0 0 0 0 0 0 0

1 0

1 2 3 4 5 6 7 8 9 10

2 0

2 4 6 8 10 12 14 16 18 20

3 0

3 6 9 12 15 18 21 24 27 30

4 0

4 8 12 16 20 24 28 32 36 40

5 0

5 10 15 20 25 30 35 40 45 50

6 0

6 12 18 24 30 36 42 48 54 60

7 0

7 14 21 28 35 42 49 56 63 70

8 0

8 16 24 32 40 48 56 64 72 80

9 0

9 18 27 36 45 54 63 72 81 90

10 0

10 20 30 40 50 60 70 80 90 100

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Fill in the times table grid 0 1

2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

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9: Strategies adopted by dyscalculic children when faced with maths problems We have noted that dyscalculic children tend to adopt a grasshopper approach to maths rather than an inchworm approach. This should be seen as the broad approach that the child adopts. Within this approach the child will then use various strategies for handling maths problems. As an example of this we now turn to the major strategies that can be observed in children when faced with problems involving addition or subtraction. There are in fact only four major strategies for solving a maths problem that involves adding or subtracting.

1. Counting – often using the fingers. If a child is asked to add nine to six, the child says “nine” and then counts each additional number, often while raising another finger, until six fingers are raised. Unfortunately, although this approach may appeal to the dyscalculic child, and indeed may be highlighted by a teacher as a method to use “when all else fails” it can be exactly the wrong process for the dyscalculic child. It is a process than uses up large amounts of short term memory – which is exactly where the dyscalculic child may have a problem.

2. Looking for the nearest ten. When faced with any question the child who has discovered this method tries to find a ten somewhere in the question, and work around that. In this case the child might think that nine is close to ten, so he takes the nine as ten, and then thinks “what’s ten plus six – sixteen.” Then, he takes one off because he added one on to get to the ten in the first place, and has the answer “fifteen”.

To undertake this approach the child must have gathered one central fact – that in adding two numbers, as with multiplying two numbers, it doesn’t matter which one comes first. Thus one can change either side into a ten, and as long as one remembers to do the adjustment in reverse later, all will be well.

3. Looking for doubles. Still trying to add six and nine, the child opts to add six and six, and knows that to be twelve, and then adds the three which takes one of the six up to nine. This system can work well providing of course that the child recognises the double of six. Perhaps surprisingly, doubles is one of the first things that children who are making progress in maths, actually learn – especially where the concept of the two times table is linked to the notion of doubling. This reminds us that it is always worth teaching and re-teaching “doubling” as a concept linked to the two times table. It is not at all unusual for a child who has been taught this idea to need to have it reiterated over and over again.

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4. Recall from a previous event. The child is asked to write down the answer to six plus nine and writes “15”. When asked how she knew the answer the child simply says she “knew it”. This can be very frustrating for the teacher, since it gives no insight into the methodology adopted by the child. However it must be recognised that there are a significant number of children who do just “know” the answers, just as they “know” how to spell certain words.

To help dyscalculic children find the right strategy, and to move them away from inappropriate approaches such as counting up, children must be reminded regularly that in adding and multiplying the order of the numbers is irrelevant, while in subtraction and division the order is extremely relevant. As for the numbers themselves it can be very helpful for children to have the opportunity to play with physical representations of numbers, and see how they can rearrange them. The child might therefore be given eight counters and asked how many there are. The answer, the child will discover through counting, is eight. The child is then shown that the eight can be represented in all sorts of other ways, such as: • Seven and one • Six and two • Five and three • Four and four At this point the child may be tempted to continue with the theme and say “three and five”, and here again the point can be made that in addition it does not matter which order the numbers go in. Three and five is exactly the same as five and three. Having established that numbers are themselves the results of other additions it is possible to explore an addition square in which every possible combination for addition is explored. An example is given on the next page, followed by a blank square for the child to complete. This begins to show the child that there is a structure to all numbers. Any number from within the box can be taken and then found to be made up of two numbers added together. Nine is six add three, and is also two add seven, eight plus one and so on. Where problems still arise these facts from the square can be verified through the use of counters – “let’s get out eight counters and see if this really does work – is it six add two as the square says?” Having started to explore the patterns of the box, it is worth considering the way in which even and odd numbers fall into place. It is then possible to use the square to consider simple subtractions – and here the opportunity arises to show that subtractions are dependent on direction.

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Addition and subtraction square 0 1

2 3 4 5 6 7 8 9 10

0 0

1 2 3 4 5 6 7 8 9 10

1 1

2 3 4 5 6 7 8 9 10 11

2 2

3 4 5 6 7 8 9 10 11 12

3 3

4 5 6 7 8 9 10 11 12 13

4 4

5 6 7 8 9 10 11 12 13 14

5 5

6 7 8 9 10 11 12 13 14 15

6 6

7 8 9 10 11 12 13 14 15 16

7 7

8 9 10 11 12 13 14 15 16 17

8 8

9 10 11 12 13 14 15 16 17 18

9 9

10 11 12 13 14 15 16 17 18 19

10 10

11 12 13 14 15 16 17 18 19 20

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Fill in the addition and subtraction square 0 1

2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

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Looking for the nearest ten Children who find this approach helpful can use the square to find the nearest ten to any pair of numbers added together. For example eight add seven might be solved as ten add seven less two. This can be tracked through a movement across the square – something that makes the process more multi-sensory. If this is done at the same time as working with counters then the child can begin to cement the whole process into the long term memory. Using counters the child might lay out seven counters and eight counters in two groups. The child moves two counters across from the seven group to the eight group, and says, while handling the counters, “this group now has ten, and this group now has five”. Then looking at the two groups the child says, “ten plus five is fifteen”. The child now takes a copy of the square grid and draws the route taken in doing this sum. The line starts at eight, moves across to ten, comes down to seventeen, and back left to fifteen. Looking for doubles Taking any pair of numbers the child looks for a double as a route to a solution. For example, if adding six plus nine the child solves six add six (which is known to be 12) and then adds the extra three. This can be counted out using the counters once again, with the six being merged with six of the nine, the child now saying “12” and then adding in the remaining three. Finally the route can be charted once again on the square. Number bonds for 10 The child should take a blank number table and then mark up all the places where ten occurs as the result of a sum. Then moving across to the counters the process should be repeated with the counters, with the child stating the answers as he or she moves the counters around. Finally, using either the counters or the number grid the child should be able to recite all the ways in which ten can be created, with the counters being moved or the grid being completed as the child moves through the problem. Subtraction Children perceive subtraction as a harder problem than addition, but once they realise that the direction of the problem is all important, and that this is what makes subtraction different, it can become as easy as addition. The process of the counters and the grid of numbers can be used again.

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With the child knowing that all numbers can be broken down into their constituent parts, then subtraction does not present a problem. The link between the numbers six, nine, and fifteen is unbreakable. The child starts to see that no matter what else happens in the universe… • Six add nine is fifteen • Nine add six is fifteen • Fifteen take away six is nine • Fifteen take away nine is six The one thing that must be stressed here is that, unlike addition, the link only works in subtraction when the biggest number comes first. Through these processes the terminology must be kept consistent. If the child is struggling with subtraction there is no benefit in suddenly changing the name of the problem to “minus” or “take away”. These additional words should not be used until the process is secure. These strategies are effective not just because they take the child to the heart of the mathematical issue, but also because they are multi-sensory in their execution.

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10: Sequences Children learn about sequences as part of their normal every day development. As their intellectual capacity grows they come to understand that summer always follows spring, that the next number in the sequence “1, 2, 3,” is “4”, and later that there are sequences such as “-2, -1, 0, +1, +2”. Some sequences relate to events – the order in which clothes are put on or taken off. The order of events of taking a bath (in that you normally run the bath before you get in it). Children also learn about local sequences – the sequence of the walk to school, and the sequence of lessons in the school day. But for some children sequencing is not learned so readily. When asked to explain how they get to school, or the order of events in taking a bath, they have difficulty. While their fellows of a similar age can quite happily recite the months of the year without thinking, many dyscalculic children cannot get this sequence right. The only way to help children to overcome sequencing problems is to give them a greater awareness of sequencing, and to teach them what sequences are, and how to learn them. It is usually good to think initially about a sequence of events that the child understands, such as the journey to school or having a bath. Of course if the child travels to school by car you cannot expect the child to know anything about the journey initially, but you can still deal with the issue of coming out of the house (front door or back door), walking to the car (do you turn left or right), getting into the car (front or back seat), setting off (do you pick anyone else up en route), dropping off, going into school (which way to walk), and so on. You may, in passing, find that the child has difficulty distinguishing left from right. This may well have to be addressed as well. Once the journey or event has been talked through in outline, questions of the type shown above can be asked and then the child asked to talk the event through again. After the child is fairly secure in the details the child can be asked to set down a series of numbered points covering every aspect of the sequence. This list can be marked against all the details revealed before. What this sort of work teaches is that there is a logic within sequences – a set of details which need to be covered in the right order if the sequence is going to make sense. You must include the issue of turning left or right, because without it the sequence cannot continue. After a few such explorations the child is starting to get the idea of sequencing as a concept that itself makes sense. Now you can ask the child to explore a sequence which has thus far not been made apparent. You might, for example, return to the idea of the journey to school, and have the child note down the key elements of the journey. This need not be every left and right, but it can be a description of the major aspects of the route – leaving a village, travelling through a district, stopping at a certain location and so on.

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With the word “sequence” being used regularly throughout the activities, and the activities themselves having as much fun within them as possible, one can turn to real world sequences such as the seasons, months of the year, days of the week. These should be taught not just as a list but also as a set of real periods of time that have their own flavour or style. Summer follows spring not because we have arbitrarily decided that is how it is, but because the leaves on the trees can’t be out in full, and the plants can’t be in full bloom, if they haven’t gone through the growing part of the cycle first. If the child has difficulty with the four seasons then four pictures can be used showing the attributes of each season in terms of (for example) the trees. The winter picture might show snow, but it must also show trees without leaves. The spring picture shows some leaves, and in the summer picture there are leaves everywhere. The logic of this must be emphasised to the child. The days of the week have less logic, but there is a clear differentiation about the weekend, and this is a place to start. “Which days do you not come to school?” is of course answered by Saturday and Sunday. “What is the name of these two days together?” is answered by the Weekend. And so a chart is built: Saturday Weekend Sunday Weekend This is quite probably the first graphic representation that the child has ever seen that shows the weekend at the end of the week. Now the table can be expanded: 1 2 3 4 5 6 Saturday Weekend 7 Sunday Weekend So the days are numbered – there is a sequence – the child is reminded that this activity is about sequences and he or she should try to remember sequences. The concept of sequences can be revised at this point – like the journey to school, how you prepare for a bath, the seasons of the year. After this it is time to move on to naming the five days of the week. The fact that there are five of them is mentioned , and when the page is revised the question is asked – how many weekend days are there, and how many week days? Next the week days are listed – but this will not be enough, because the child needs to have a multi-sensory association with each day. At this point a final chart is evolved:

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1 Monday Weekday Back to school. 2 Tuesday Weekday Double maths!!!!! 3 Wednesday Weekday Sport afternoon 4 Thursday Weekday Special maths 5 Friday Weekday End of week. 6 Saturday Weekend 7 Sunday Weekend The themes in the right column could be anything but once set up should not change. It is vital that the entries in the final column are real to the child. They might be some association from home or from lessons. If the child is having difficulty here, or you suspect that the child might not remember the correct day of the week on which something specific happens, it might be worth writing a note home to get the days verified. It might also be possible to use days on which particular programmes are on TV. Once this sequence has been established the child should be asked to write the days of the week out at the start of each lesson so that they become established. Remember, just because the child seems to have mastered it in one lesson, it does not mean that it will be remembered next time round. Having dealt with days of the week the child can now look at months of the year, again giving a multi-sensory association to each month 1 January New Year 2 February Short month & leap day 3 March My birthday 4 April April fools day 5 May May day, may poll 6 June Longest day 7 July Summer holiday starts 8 August School holiday 9 September Back to school new class 10 October Halloween 11 November Bonfire Night 12 December Christmas Once the idea of sequences is established you can return to the sequences that are found in numbers. The nine times table is an interesting sequence for many children, as shown in section 10.

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11. Multi-sensory approaches to overcoming dyscalculia The multi-sensory method was developed by Montessori as a method of teaching all children through allowing them to have real world experiences of everything that they learn. It is because the method of learning is so powerful that it works so well with children who are dyscalculic. At the simplest level, making maths multi-sensory means encouraging the children to touch, see, hear, and say numbers as they use them. Thus the child might pick up four buttons and say “four” and write down “4”. Then the child will pick up another group of four buttons and while putting them with the first group will say “two fours are eight”. The child then writes 2 x 4 = 8. The process may be slow, and the child may object to what is seen to be a babyish procedure, but the child should nevertheless be encouraged to proceed in this way simply because the approach can be so beneficial. Particularly helpful can be cards which cover mathematical facts. The child looks at one side of the card, repeats the information, gives the answer, and then turns the card over to check that it is correct. The front of the card might well say the four times table and the reverse gives the answer. The particular advantage of this approach is that it overcomes the difficulty of the child with short term memory problems failing to remember where he or she had got to in the table. Because the cards are in order the child can read the next line. Once the child has a firm grasp of the table in question the cards can be mixed up and handed to the child as a test. The child can still see the questions, and so not worry about what comes next, but will also still need to be secure in finding the answer. Cards that are known can be put to one side while cards that cause difficulty can be used again. Cards that cause a particular difficulty (as with 8 x 7 for example) can be personalised by the child. The child takes the 56 on the reverse and evolves a picture around it. The same picture can also be drawn on the front of the card around the 8 x 7. The child now has a further link between 8 x 7 and the answer. If the child can remember the picture the child can then also remember the two issues associated with that picture – 8 x 7 on one side and 56 on the other. An approach like this will only work if the image is particularly memorable to the child and if the image is not used in any other situation. Such an approach does not work for all children, but where it does work it can be very helpful. Some teachers find it useful to give one problematic question of this type at the start and end of every lesson. Others always use the same question when handling wooden numbers at the start of the session. In short, there is no methodology that helps every child. Rather where there is an opportunity to experiment it should be taken.

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12: Self esteem as a method of overcoming dyscalculia

It is easy to confine self-esteem to the fringes of the topic of dyscalculia. But this would be to miss a major opportunity. Just how important self-esteem is within the concept of dyscalculia can be seen from some remarkable research undertaken by Lawrence in Somerset over a 20 year period. It is probably the most comprehensive research into child self-esteem in schools in the UK, and the results are quite extraordinary. While Lawrence was not specifically involved in maths, his work with children who are dyslexic can undoubtedly be applied to children who are dyscalculic. What Lawrence did was to divide a group of children who were experiencing difficulties with reading and writing into four groups. Each group then received a different teaching approach to help them overcome their specific learning difficulties. Group A pupils received no special help at all. They carried on with their normal lessons. Group B pupils received self-esteem support, as suggested by Lawrence. Group C pupils received additional literacy help in the form of multi-sensory learning support. This approach is used in a number of schools for pupils with dyslexia or symptoms relating to that and can make a significant difference to the pupils’ literacy. Group D pupils received both multi-sensory literacy support and self-esteem support. The prediction was that the literacy of Group D would improve the most, followed by groups B and C, with group A lagging behind. In fact this did not happen. Group B pupils – the pupils who received self-esteem support but not special literacy support, improved their literacy ability the most. Group D came second, group C third, and Group A last. Thus only Group A, the group that got no extra support or help with their literacy difficulties ended up in the position predicted. The question immediately posed is, why should the course in self-esteem support result in better literacy achievement than the course providing both self-esteem support and literacy training? We can understand that self-esteem training might in itself be more powerful than training in multi-sensory learning methods, but surely both together must be more effective than just self-esteem training. The answer appears to be complex. The self-esteem training happened in different classes with a different teacher from the literacy training. There is the possibility that the two forms of teaching were in some degree contradictory in the sense that while the self-esteem programme suggested to the children that they were successful and able young people, the multi-sensory learning programme told them that they had specific learning difficulties, and were failing to make the same

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progress in literacy as their colleagues. Thus the self-esteem support on its own proved more effective. We therefore propose that self-esteem training should take place as a significant part of the support and help given to pupils with dyscalculia. That is the essence of our approach here. The two programmes – maths support using multi-sensory learning, and self-esteem support – should be completely integrated and taught as one, by the same person. We make this a major part of our approach to dyscalculia because of the work done not only by Lawrence, but by many other writers in the United States and Australasia. All the evidence we have seen suggests that raising pupil self-esteem can have a significant effect on their ability to learn. Furthermore, Lawrence has shown that most teachers can implement self-esteem programmes as well as psychologists and counsellors. Definitions of self-esteem Self-esteem relates to the difference between what an individual feels he or she would like to be (known by psychologists as the ideal self), and how the individual feels he or she actually is (the self-image). There is virtually always a difference between the ideal self and the self-image. If an individual showed no difference between the two, that individual would be quite poorly adjusted to the world and would not be striving for improvement or change. Such a person would probably appear to be very apathetic. Although for most people there is a difference between their ideal self and self-image this is not particularly important on a day to day basis. It has some relevance to their every day lives, but it is not a central concern every day. Most people don’t worry about the difference between the two facets that make up self-esteem – or if they do, they quickly leave that worry to one side and get on with every day life. What in fact happens is that we build in mechanisms for coping with this disparity. We raise our self-image by taking on new activities or achieving more in areas where we already work. We associate with people who consider us to be of worth, and who through their comments boost our self-image. We lower our overall aims – recognising perhaps that we are not going to be film stars, astronauts or world famous musicians. But at the same time we also recognise that this does not diminish our validity as people. It is when the gap between ideal self and self-image becomes a source of regular worry that the individual needs support. So a pupil who feels he or she can’t live up to the expectations of a parent might well start blaming himself or herself. Unfortunately such blame not only makes that pupil feel bad, it also makes coping with the situation, and improving on it, even harder than it was before. Low self-esteem therefore is self defeating. Most children suffer low self-esteem at some stage in their childhood and teenage years – parents and teachers of teenagers will recognise all the signs of teenage angst. Most people gradually adjust to their place in the world over time, but when low self-esteem continues into adulthood there can be problems. What tends to happen is that the adult finds ways of coping with this low self-esteem but these can be rather unsatisfactory both for the individual and those around. They can involve

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large amounts of criticism of others, the blaming of failure on bad luck or unreasonable behaviour elsewhere, and a tendency towards paranoia. To summarise it would seem that all children should have demands put on them, both to encourage them to explore the world and make the most of their opportunities, and to encourage a certain imbalance between ideal self and self-image. But these demands should not be so strong as to stop the child making progress to reduce this difference between the self-image and the ideal self. Within this period of psychological development the key players are the child and the people who react to the child’s performance. It is people such as parents, peers and teachers who help define both the ideal self and the self-image, and therefore it is these people who are able to overcome low self-esteem. Clearly the peer group is in a curious position here because they are also adjusting their own self-esteem at the same time as influencing the self-esteem of their fellows, and therefore as we might expect the thrust of our work should be in influencing self-esteem via parents and teachers. We may at this point ask why this matters. Are we not making too much out of self-esteem – almost raising it to the level of a basic need? As you will gather from what has already been written, our answer is a categorical no. In fact many writers do see self-esteem as a basic need, which arises once other needs, such as shelter, food and water, are already met. Self-regard is seen as an important issue for every individual in 21st century western society. Reduce self-regard too much and the individual finds it almost impossible to function in contemporary society. Children with low self-esteem lack belief in their own ability to do well, and subsequently do not do well. In order to cope with this these children avoid interacting with these situations in which they expect to fail. As William James said over 100 years ago “with no attempt there can be no failure; with no failure there can be no humiliation.” Thus the students with low self-esteem come to prefer doing nothing, rather than trying and failing. Punishment for a failure to do the work at all is better than the humiliation of yet more failure. This is a fundamental psychological finding. It is also a profoundly significant explanation for the continuance of dyscalculia in some children, despite the best efforts of those around to help them overcome the problem. It explains why many dyscalculic children appear to have little willingness to try to overcome their problem. Children who fall into this situation tend to exhibit varying types of behaviour, of the three types outlined below:

a) Compensation and avoidance Some children will appear outward going, boastful, full of apparent self-esteem and bravado, as a way of avoiding all the implications of their failure. Others withdraw and fail to engage in any way with anyone when the topic arises. Both approaches represent a form of compensation and avoidance. Which approach the child chooses relates primarily to the personality of the child – the introvert is obviously more likely to opt for the withdrawal approach, the extrovert will go for the boastful approach.

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It is unfortunate that a natural reaction to this sort of problem can be to tell the child to settle down to some proper work, or “buck your ideas up”. Such an approach will not work at all. Indeed very little will work with such a child until the self-esteem is raised. b) Lack of Motivation Our behaviour comes out of our view of the world. We behave in the way that seems best suited to the world around us. We also behave as others expect us to do. If we expect ourselves to fail, and if others expect us to fail, then we will fail. We know we will fail, so what is the point of putting any effort into trying to overcome the failure? It is easier just to fail. As a result children who have already failed at maths lack the motivation to try again. They fail to respond to requests from teachers and parents to try again, because it is parents and teachers who have failed to help them in the past. c) Avoidance of change Self-esteem is not an easy concept to change. To the individual the ideal self and the self-image appear natural and obvious facets of reality. If questioned the child with low self-esteem might well say that it is obvious that he is no good at maths, and it is obvious that life would be much better if he were. But that’s not how it is, it isn’t his fault, and there is nothing much that can be done about it. The way they see the world is the way that it obviously is. “I am useless at maths, so I don’t like maths. People tell me to try harder, and I do, but it doesn’t work. I just can’t do it.” Such a statement is not open to every day debate. It is obvious and true, and therefore questioning of this view appears to the child to be as daft as questioning the existence of day and night. Thus to challenge such a view is to challenge the very fabric of the individual. What we do know is that for most people it is much easier to cope with what we know than what we don’t know, and therefore the present situation, however unpalatable is therefore preferable to what we don’t know. Changing means taking risks, and people with low self-esteem are notoriously poor risk takers.

Self-esteem and different aspects of life What we must also recognise is that it is quite possible to have low self-esteem in one area of life, but high self-esteem elsewhere. Maths might be our focus of low self-esteem, and so we avoid every possible interaction with maths. If asked to buy something with money we pretend we don’t have any. If asked to give the time, we pretend we don’t have our glasses on. A maths exam brings on a migraine. But none of this affects our view of the world when it comes to playing defensive midfield for the county football team. The local Football League team has already opened negotiations and there are stories about a Premiership scout being seen watching a few training sessions. In football, this same person whose self-esteem in maths is as near zero as makes no difference, is a total star. For such a person life is simple. Play football, avoid maths. Only by interacting with maths is there likely to be a problem. The fear of the wonderful new world of football being infiltrated by the terrible world of maths (“how can you be so good with the ball when you can’t say your five times table?”) reinforces the desire to avoid maths at all costs.

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Can we really improve self-esteem? The answer is yes. It can be done, and it can be done by teachers. We have known for many years that short simple courses for groups of children run by counsellors and psychologists can result in major improvements in self-esteem for years to come. What Lawrence has given us is the awareness that the same applies when these courses are run by teachers. The difference in the results gained in experiments in which some pupils were helped by teachers and some by trained counsellors were small, providing (and this is the key) two things applied. Firstly the teachers must follow the basic rules of self-esteem enhancement which those who constructed the programmes laid down. Varying the order of activities, or responding too much to events within the group and thus going off in different directions, can result in much lower self-esteem enhancements than one might expect. Also early curtailment of the course, particularly where this is seen by the pupils as abandonment of the self-esteem programme for “more important” subject work such as maths, can have a disastrous effect. Secondly, the teacher running the programme must have the following attributes himself or herself:

a) High self-esteem b) An ability to communicate readily with the children c) A warm, kind welcoming personality d) A belief in the possibility of changing a child’s self-esteem e) A belief in the benefit of changing a child’s self-esteem.

Lawrence has repeatedly highlighted the fact that teacher self-esteem and pupil self-esteem correlate – as he says, “Children of high self-esteem who are in regular contact with teachers of low self-esteem will gradually themselves develop low self-esteem, with associated low attainment levels. On a more positive note, the converse can also occur with low self-esteem children raising their self esteem through regular contact with high self-esteem teachers.” Further evidence to this effect comes from Burns (1982). We can see the enormous problem here. Children who suffer from dyscalculia have low self-esteem. They can be helped very quickly if they work with a teacher who has high self-esteem. But we don’t have any regular approach to measuring teacher self-esteem in a school. It is a matter of pure chance as to whether the children work with high or low self-esteem teachers. Research has shown that teachers are in a particularly strong position to influence the self-esteem of children. We have to be certain however that they are influencing the pupils in the right way. How do we know what someone’s self-esteem actually is? People with low self-esteem (be they teachers, students or pupils) exhibit a particular style of behaviour which can suggest a self-esteem problem. These include:

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1. A lack of confidence. This can be revealed in many ways but may exhibit itself only in certain situations. An individual may have low self-confidence in undertaking maths, but a high self-confidence when playing cricket. We may therefore look for a significant difference in confidence in the differing activities which the child carries out. 2. An appearance of apathy. As with confidence this can relate to specific situations. It can be extremely frustrating to be faced with a child or teenager who doesn’t go out to see friends because she or he “couldn’t be bothered”, but this apathy is not something that can be dealt with through admonitions of “pull yourself together”. Apathy is a major side-effect of low self-esteem, and apathy in situations where one has the opportunity to learn or experience new activities does suggest that there is a self-esteem issue that has to be addressed first. 3. Unwillingness to take risks, or try out new approaches. People with low self-esteem will often come up with all sorts of arguments as to why a new approach should not be tried, which can lead to some seriously convoluted suggestions to the effect that the new approach will cause problems and difficulties, and that the current approach is working satisfactorily anyway. Where such an argument is deployed, but the low self-esteem, low risk-taking individual is then over ruled, this low self-esteem person will often act in ways that almost ensure the new approach will fail. It can also be witnessed on occasion that the individual will continue with his or her old approach in parallel with the new approach. Quite what the benefit of this is on a practical level remains unclear, and it certainly causes extra work for the individual - a factor which should remind us that self-esteem is not an issue which can always be addressed with logic. On the psychological level the individual is saying, “my original method is best, and the fact that I have to run my system alongside this new system, thus causing myself more work, proves this to be the case.” 4. Self-disparagement or repeated boasting. Depending on the personality type of the low self-esteem individual we can see one of these two attributes. The occasional boast, as with the occasional moment of self-disparagement is not normally something we notice, but when it continues regularly despite negative comments made by others, it can be a clear indicator of low self-esteem. 5. Unexpected stress in certain situations. Personal experiences of stress and anxiety which are running out of control are often related to low self-esteem. Stress is a medical condition which can be countered, and anyone suffering unduly high levels of stress may well be suffering from low self-esteem. 6. Constant requests for help and reassurance. Being asked by an individual to offer some help can be a positive feeling for many of us – it enhances our self-esteem by showing that the individual in question values our abilities and judgement. However repeated requests for help and/or reassurance can become annoying, and if we experience this we are probably facing someone with low self-esteem. This is particularly so where some of the questions asked relate to how others see one. A person who regularly worries about his or her popularity can be revealing a lack of self-esteem. 7. Lack of integration with the peer group. This is another regular sign of an individual with low self-esteem.

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8. Blaming others and failing to accept responsibility. This is one of the most common signs of low self-esteem, and can be seen both in low self-esteem children and teachers. The low self-esteem child blames the teacher for “not liking me”, while the low self-esteem teacher blames the parents, the children, the headteacher, the lack of resources etc. Of course we all blame others on occasion – and often with justification. Sometimes it really is the fault of another person. But what we should be on the look out for is the person who repeatedly fails to accept responsibility and who continuously and immediately passes the blame elsewhere. 9. Problems with ascertaining self-esteem. One of the great problems with identifying low self-esteem is that it is not something that is open to statistical analysis. People with low self-esteem are often unable to describe their feelings very well – indeed many would say that there are people with a reasonably high self-esteem in the UK who have difficulty in describing their feelings! What’s more, we do not observe self-esteem directly, but rather we infer self-esteem levels from observable behaviour. Our purpose here is not to ascertain exactly if a child or teacher does suffer from low self-esteem but rather to suggest that this can be the case, and to note that we should be aware of this. What is needed is:

a) An awareness that having dyscalculic children taught maths (either in maths lessons or in sessions organised by the special needs department) by a teacher who has low self-esteem in relation to maths or in relation to teaching in general, can make matters far worse for the children. We may believe the extra lessons are helping the children, but in fact they are actually making matters far worse.

b) Many dyscalculic children will have low self-esteem in relation to their

perception of their ability in maths, and steps have to be taken to raise their self-esteem. Simply teaching them maths is not likely to be sufficient. It is the rise in self-esteem that triggers a willingness on the part of the child to look at maths afresh and put in the necessary energy that will allow learning to take place in a subject area where it has clearly not taken place before.

Who should teach dyscalculic children The teachers to whom dyscalculic children respond best are teachers who exhibit certain key personality traits. These overlap, but the full list is worth setting out. The desirable approaches and traits are: • An ability to listen without jumping to conclusions • Acceptance of the child, even if the behaviour needs correcting • Disapproval of bad behaviour, not of the child • Distraction coping (a personality trait whereby, when the teacher is working

with an individual child, the teacher is not sidetracked by the numerous distractions that seem to be part of classroom life today).

• Easy to talk to • Empathy – understanding the problems of the child • A focussed approach and a clear view of what they are doing and why • Friendliness towards the children

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• Genuineness – appreciated by the child as being genuinely interested, not “faking it” as part of the job of being a teacher

• Liking children • Liking teaching • Liking the school • More likely to smile than scowl • Non-judgemental • Open to ideas and thoughts • A positive attitude towards life in general and the school in particular • Respecting the child • Taking the child’s issues and problems seriously • Understanding • Warm A teacher who possesses most of these attributes is a teacher who puts across the high self-esteem model, which the pupils then begin to adopt as their own. It is interesting to note that many of the attributes recorded above will be picked up by the child as non-verbal signals. They will “read” the teacher’s view of themselves through eye contact, body language, facial expression, and gestures as much as through the way in which the teacher speaks. This is why an attribute such as genuineness cannot be faked. The teacher who says that she or he is seriously interested in the child, when the body language says otherwise, will not be believed, no matter how much the teacher says, “but I told you I liked you.” Once the teacher shows the child that she or he approves of the child, then the teaching of mathematical concepts can proceed apace. However as the maths teaching begins the teacher must be certain that he or she is giving the pupil a realistic self-concept, not an invented one. It is clearly pointless telling a student who has not grasped the basic functions of maths that everything is in order and that he or she is excellent at maths, when all the evidence points to the opposite. Rather the teacher must be saying that “you can succeed in maths, if you will work with me on this.” With such a statement at the heart of the relationship, the teacher has full control over the developing situation. Thus when the student fails to undertake the designated work at home, or tries to avoid attempting some mathematical problems by pulling faces at another member of the group and causing general disruption, the teacher will be able to reprimand the pupil’s behaviour, without losing the goodwill of the pupil. One approach that many teachers find extremely useful at this point is that of talking to students about their self-esteem – that is to say, they teach self-esteem enhancement in the maths or special needs lessons. Although this might seem so off-topic as to be non-acceptable, we must remind ourselves again of Lawrence’s findings that the pupils who had the self-esteem enhancement programme without the literacy training, made more progress in literacy than those who had the literacy training without the self-esteem work. We cannot over-estimate the power of enhancing pupil self-esteem.

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The self-fulfilling prophecy Thousands of psychological study programmes over the past fifty years have confirmed that by and large people behave as we expect them to behave. In education this can mean anticipating success or failure according to the predictions of other teachers, and one’s own past experiences. Pupils pick up on these messages and then act in ways that are expected. Much of the research has suggested that the power of the self-fulfilling prophecy can be as great as the impact on the classroom of teaching and learning. Because of the emphasis in our society on science and the acceptance of the idea that such attributes as IQ and personality are established within the individual and permanent, rather than transitory and under the influence of outside events, we often feel unhappy with the notion that the way in which we expect someone to behave actually does influence their behaviour. However it is often the case that what we expect, does happen. If we expect dyscalculic children to be almost unteachable in maths, that is how we will find them. If we believe that they are simply pupils who have missed out on some important aspect of the fundamentals, and that this failing can be corrected quickly, then this is what we will find. This view relates closely to the notion of self-esteem. What many teachers find is that simply by making people feel important you can increase productivity and success. By treating dyscalculic pupils as interesting people in their own right who bring a new and interesting perspective on mathematics, their level of mathematical learning can go up. This effect can be enhanced through target setting. As long as the target is realistic then what we can find is that once a target is set the pupils will work towards it. Higher expectations can then be transmitted to pupils, and then the expectations can be met. Of course the expectations have to be realistic, but when they are, and when the movement towards them is accompanied by use of the best methods of teaching, then progress will be rapid. When all these factors come together dyscalculia can be overcome very readily.

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13: Check to praise – a method of motivating dyscalculic children In many ways communication between the student and the teacher is the key ingredient in improving mathematical learning, as it is in doing anything to make the school more effective. There are many ways of motivating people within an organisation, but all of them include a strong element of communication. If you want pupils to develop a new and positive attitude to maths you need to talk to them about developing a new and positive attitude towards maths. However going over this idea with the pupils just once is not going to be enough. Teachers of dyscalculic children have to be involved in this sort of communication all the time. You have to come back to it, review it, consider it again, ask what people have done so far… Indeed to re-motivate pupils vis-a-vis maths there needs to be a massive amount of communication within the school at all levels. One way to achieve this is through the continuous use of the “Check to Praise” system. This is one of the most important ways of motivating dyscalculic children at the very start of the process. If you don't use it you are making life very hard for yourself. Check to Praise is probably one of the most valuable systems that can be introduced into any organisation with value in both management-teacher and teacher-pupil contacts. Some schools already use it and we have never seen any organisation where it is used regularly where it does not generate a major benefit. It does not guarantee complete success but it takes us a long way along the road. Check to Praise has the virtue of being a most simple system. It involves the teacher asking for or suggesting (politely and reasonably) something to be done. This something should be an immediately identifiable task - preferably a single item rather than a generality - and not a negative. This might be the learning of some facts established in class, coming up with ideas for a way of learning something that has caused a problem in the past, implementing a trial run through of a particular idea with a parent at home, or insisting that all homeworks are done on time. None of this is revolutionary - it is simple discussion of projects, and the making of polite everyday requests. Where Check to Praise comes into its own is with the advent of the checking. Five minutes, an hour, a day, or a week later (as appropriate to the situation) the teacher who has initiated the request asks, “Have you done / How have you been getting on with / What happened when you / Did you do / How did it go when...?” If the child has complied with the request, then thanks are given (the praise part of the equation). If the individual has not done that which was asked, the request is repeated with another (closer) deadline. The deadlines are very important in the system, since otherwise the person given the task can quite legitimately say, “Sorry I haven't had time yet.” In order to make sure that you leave no way out for the reluctant individual, it is also worth saying at the time of request, “Will you be able to get that done by…?” In other words, you get the individual to agree to the deadline. A record book is obviously necessary to see what needs to be checked. The value of the system is that:

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♦ It involves personal contact with the child, and the making of specific requests which the child understands.

♦ It is based on praise. ♦ It shows a continuing interest from the teacher to the pupil. ♦ Because every request is related to a deadline it allows no way out for the

reluctant pupil. In Check to Praise you do not accept the failure to do something in any way other than through setting a new closer deadline. The benefit once a task is completed comes from the praise. We all like to be praised. We all want to feel we are doing the right thing. After a short while Check to Praise leads us to do what is expected of us – it is a mechanism towards the self-fulfilling prophecy. The teacher expects something to be done, and it is done. One of the greatest advantages of Check to Praise is that it can be used with just one pupil whom you feel needs additional motivation. You can single out one pupil whom you feel is de-motivated, whom you ask to do something quite straightforward by a set date – preferably only a day or two away. You then go back on the set date and check, and if it is not done, set another date – perhaps now just one day away. If you then still get a refusal to comply you have a direct issue to deal with – but this should not lead to disciplinary action. What you have gained is confirmation that the self-esteem of the pupil is so low that the child will place the avoidance of work at the highest possible level. You might well need to set aside time to do the work with the child. But even if it comes down to this, you should still then praise the child for doing it. Next time round you will probably get compliance with the first or second request. When you do get compliance, you can praise, and start on the next Check to Praise activity, and then the next. Eventually you get a cycle of improvement and motivation, through this step by step approach.

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14: The maths and self-esteem special group: raising self-esteem and overcoming dyscalculia together In an ideal situation you should aim to work with a group of about six pupils who need specific help in maths. Tell them from the start that this is going to be a set of maths lessons unlike any that they have ever experienced before. As soon as possible introduce the notion that you are going to think and talk about how they feel about maths, and about school, as much as you are going to do maths with them.

1. Give positive feedback.

Every time the dyscalculic child makes a positive step forward with maths, give lots of positive feedback. Many pupils with low self-esteem get very little positive feedback, and this can be a very exciting experience for the pupil. The pupil may not be used to any sort of positive comment in maths, and so may feel slightly uncomfortable with this at first, which is why you need to continue with such a programme of positive reinforcement for some time, so that it becomes the normal process of the lesson.

2. Give opportunities for spontaneity.

For example, when the child is asked to contribute ideas towards the issue of a multi-sensory approach to learning a particular topic, the child with low self-esteem may well feel that he or she has little to offer. Earlier experiences which have shown the child that his or her spontaneous ideas are treated with derision can linger in the memory for a long time. Thus you will need to encourage the child to come up with the first idea that she or he thinks of, and not feel worried about saying it. The child needs to learn that it is perfectly satisfactory to have different ideas from others, because many people learn maths in a different way. The point of this is that not only is the inchworm vs the grasshopper concept relevant, but that if the dyscalculic child is, like some dyslexic counterparts, more readily in touch with the right hemisphere of the brain than the left, these spontaneous ideas can be very useful and apposite indeed.

3. Explain to the child that part of the programme relates to feeling good about yourself and feeling good about maths.

Don’t suggest to the child that this is a maths lesson, where the other things we talk about are incidental. Make self-esteem as central a part of the work as maths.

4. Start by working on feelings about maths.

Make it clear to the children that it is perfectly OK in this session to describe your exact feelings about maths. Sometimes this will slip over into the children expressing dismay about individual maths teachers. This too has to be accepted – because what the pupils are being asked to express are their feelings about maths. What they get in return is the awareness that

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feeling negative things about maths is perfectly acceptable and expressible, and is not going to result in anything bad happening to them.

5. Move on to the broader expression of feelings about lessons in school.

Ask the children when and where they feel happy at school. When and where they feel afraid. When and where they feel foolish, embarrassed, proud, upset, glad…. Keep on with this line of discussion, bringing in other scenarios. Not only will it help the children, it will probably also help you and your colleagues understand more about the school through the pupils’ eyes.

6. Return to points four and five for the start of several lessons early in the

course.

Whatever you do, don’t feel that you must rush through the self-esteem element in the lesson and get on with the real maths. If you think that “I have to move to the maths now, otherwise I won’t get through it all,” you are missing the point. The children will overcome their dyscalculia when they have addressed their self-esteem problems, not when you have taught them enough maths.

7. Ask the children to make positive and negative comments about school in

general.

No child is allowed to criticise another child’s comments, and no one can interrupt. Ask the pupils to think of a moment at school when they have been happy. Ask them to talk about when they have been unhappy at school.

8. Ask the pupils to recall good times at home, at school, anywhere.

9. Ask the pupils to make positive statements about good times in maths.

This can only happen once you have started to make some progress with the pupils from a maths point of view. Obviously you can only do this once you know that they have had some recent positive experiences with you, in managing to learn some basic maths which had previously been a mystery to them.

10. Ask the pupils to think about the person sitting next to them and say

something positive about this person.

Children will often feel strange about having a statement like this made, especially if they do not normally have anything positive said about themselves at home or at school. Remember you can only undertake this activity once the children are used to dealing with you and with each other. You must not get into this type of work unless everyone is likely to be able to have something positive to say.

11. Ask the pupils to talk about themselves.

The individual pupil says something clear about himself or herself, which is positive and straightforward. This might be to the effect that I am happy, or I have friends, or I find maths less frightening than I used to. This exercise

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can be repeated over and over again. As time goes by it can be related more and more to maths – as with the instruction, “say something positive about yourself in relation to maths” in which the child might say, “I know it doesn’t sound much, but I now understand multiplication – and I never did before.” Or they might say how much they enjoy these extra maths lessons.

As time goes by the morale of the group should rise. The more the group feeling improves the more individual self-esteem will improve. The success of any one individual in the group should be celebrated as both a success for the individual and a success for the whole group. This celebration of success should build on the notion that the success has come because what happens to each individual is under the control of that individual. “I have learned my nine times table and become successful at it because I put the effort in and I have been supported by the group” is the position we want to get to.

12. Ask the pupils to think about their own experiences in learning.

How can things be improved? What works, what doesn’t? This is a chance for pupils to blame everyone else but themselves, and they may well take that chance, but the teacher should respond by asking gently how the situation may be improved. Let’s accept the world is as you say, but what can you do? Even if the child has no answer, a simple change can be suggested, and a promise extracted from the child that this change will be tried this week. Gradually the low self-esteem child should become responsible for his or her own behaviour.

Problems that can arise with the maths and self-esteem group

• It is possible that some people will simply just not “get on” in the group. This is most likely either because group friendships will already have formed which exclude one or more individuals, or because the person in question is particularly introverted and the rest of the group is extrovert. If this happens it may be best to split the group in two with the most extrovert in one group and the most introverted in the other.

• Friends should be allowed to be together, providing that the friendship does

not exclude others in the group. The group must, above all, be welcoming to every member of the group. Everyone is going to be asked to expose their worries and fears, as well as share in their triumphs, and so a welcoming approach is essential. If you fear it will take too long to develop this, split the friends and form two groups.

• Pupils can respond to the challenge of this type of work by claiming it is

“stupid” and “not real maths”. This is a common way of avoiding engagement with the very deep issues of self-esteem that this group will bring forth. Let the group talk and discuss, and allow open exchanges. Just because a child says “this is stupid” it does not mean that this is exactly what the child feels.

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15: Self-esteem, maths and behavioural problems There is a close link between a child’s self esteem and a child’s behaviour. Children who receive self-esteem training invariably improve their behaviour, become less disruptive, are less likely to bully and become less extremely introverted or extrovert. However not all children who have low self-esteem have behavioural problems and we should not always expect the two to go together. Children can become disruptive or withdrawn as a way of avoiding engagement in maths, but they can also become disruptive or withdrawn for many other reasons. It is also the case that children do not automatically know how to behave in certain situations and so can behave inappropriately. This is particularly likely to be the case in relation to self-esteem activities which will be quite new to the child. Behavioural problems in the form of giggling or refusal to participate can arise in early self-esteem sessions, which is why the opening sessions should be taken slowly. What we have to try to do is disassociate behavioural problems with maths, in a scenario in which the two may well have already become very closely associated. In such circumstances the teacher needs to be a model of high self-esteem, a warm person interested in the children, wanting and able to help. Thus it can well be that children will find the self-esteem part of the programme difficult to adjust to – which will initially appear to exacerbate the very issues that we are seeking to overcome. If this is the case they must be given time to come to terms with this different approach to education. Just because children do not settle into maths and self-esteem classes straight away does not mean that they cannot attend the lessons thereafter. We must also recall that a child with dyscalculia may also be suffering from dyslexia, and so may be failing not only in maths but also in English and (because of this) many other lessons. Indeed if the child is not talented in some area of work outside of English and maths then the whole of the schooling experience may well be one of failure for the child. To make matters worse, there is research which suggests that a significant number of children who suffer from attention deficit disorder also suffer from dyslexia and therefore may also be dyscalculic. If you feel that you are faced with a child who does suffer from ADD or ADHD it is quite possible that this child may not be able to benefit from self-esteem discussion groups until the attention deficit disorder is addressed. Details of ways in which attention deficit disorder among children in school can be addressed are given in other publications, listed in the bibliography.

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16: Information for parents One of the great problems in helping dyscalculic children is that although the child’s parents may be more than willing to help overcome the child’s apparent disability, the parent or parents may approach the issue of dyscalculia with the wrong attitude. Many parents take the view that being poor at maths is understandable and even excusable. The phrase, “Don’t worry, I was never any good at maths,” seems to excuse everything. It is said with the best of intentions, but actually has a harmful effect. Such a view, or even the view that “some people are good at maths and some aren’t” gives no help at all to the pupil. Such views have the effect of countering any activity the school is engaged in with the aim of raising the child’s self-esteem. Since a self-esteem programme will undoubtedly include a significant element of “I can do it”, this parental approach is unhelpful to say the least. What the pupils need to hear is the view that “everyone can do maths – there may be a problem at the moment but we can solve it.” For this reason we have prepared a simple handout to be given to the parent or parents of any child whom you consider to be dyscalculic. The aim is to put across a very simple message to the parent:

a) Your child is having difficulty with maths b) We have looked into this and have a very good idea how to overcome this c) We are going to be doing our bit but we need your help d) Please ensure that your child believes that she or he can succeed in maths e) Please do the regular exercises that your child will be given. If you can do

these exercises for ten minutes a night, five nights a week with your child, then your child will make rapid progress in maths.

This message combines raising the child’s self-esteem in regard to maths with regular practice of the basic concepts that the child has not mastered. The letter that follows may be copied and used for parents. As it appears not only in this book but also on the computer disk supplied with the book the text can be copied on to a word processor and amended to suit your particular needs, without having to retype the whole letter.

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Sample letter to parents It has become clear recently that your child is having a greater than expected difficulty in maths lessons. For this reason I have been giving your child some tests to discover exactly where the problems with maths lie. These are not formal tests which your child has to take – they are not part of the normal testing procedure of the school. Rather they are special tests which have been produced specifically for children who have difficulty with maths. Obviously maths is a vital subject at school, and therefore now that I know a bit more about the problem, I am taking some specific steps to help your child overcome this difficulty. Let me say straight away that I am not suggesting that the problems that your child is having are, in any way, your child’s fault. I am not writing to you to suggest that your child is misbehaving or not trying to do the work. Rather, your child naturally finds maths difficult to do. There are many possible reasons for this. I am very happy to discuss these with you, if you wish, but the most important point for me to make to you today is that we have a way of helping your child which will work, no matter what the cause of your child’s difficulty. To ensure that your child does catch up in maths we need your help in two separate ways. Firstly, it is really important that you adopt a very positive attitude towards maths with your child. It is sometimes tempting to sympathise with a child who has difficulty in maths, perhaps agreeing that you also found maths difficult at school. But what I would ask is that you focus on the fact that we are now using a special programme to help your child, and that improvement will soon be seen. We all need to be saying that your child can and will succeed at maths. Secondly, your child will be bringing home some additional maths homework in terms of maths cards. We need your child to work through the pack of cards each night after school. It should only take a few minutes, but it does need to be done five nights a week – both during school term and during the school holidays. Your child will show you what is to be done, but if you are uncertain of any aspect of this work, please do not hesitate to get in touch.

Yours truly,

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17: Information for governors and members of staff Just as it is important for parents to understand what you are doing in terms of dyscalculia, so it is important for other members of the school to understand the issues with which you are dealing. There will be a number of members of the school community who have not heard of dyscalculia. Unfortunately there may also be one or two who feel that this is just another attempt to explain away the failings of certain children. Such people might argue that all the children need to do is settle down and apply themselves to the problem. A simple handout to other members of the school may not change the minds of those people who take up such an extreme position, but it can help those with more open minds understand what the issues are and how you are tackling them. Some schools, we find, welcome the distribution of information by one member of staff to the rest of the school. Others feel that all such information should pass via the headteacher. Obviously this will be a matter for you to resolve, and as with the message to parents, the information that follows can be found on the computer disk, and modified to meet your particular situation, without having to re-write the whole piece.

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Sample handout for governors and members of staff

Approaching dyscalculia Dyscalculia is a specific learning difficulty with mathematics. Children who suffer from it find maths much harder than we would expect when taking their age and their intellectual ability into account. Expert opinion is divided as to whether dyscalculia is closely linked to dyslexia or whether it is a specific learning difficulty in its own right. Those who claim that dyscalculia is separate from dyslexia point to the fact that around 25% of dyslexic children are actually above average at mathematics. Those who claim that dyscalculia is a part of dyslexia argue that dyslexic children are well known for their ability to find strategies which can overcome their difficulties, and that these dyslexic children who do well at maths are simply children who have found the right learning strategies to cope with their disability. Although the theoretical differences between the two groups are profound both sides agree on the methods that should be used to help dyscalculic children. These methods involve a three stage process:

1. Identification of the areas of difficulty (for example the failure of the child to understand and be able to operate the concept of division). This is important because not all dyscalculic children suffer from difficulties in the same areas of maths. Simple tests exist which reveal where an individual child’s problems are. These tests do need to be administered in small groups – and sometimes on a one to one basis – but they do not need to be administered by an educational psychologist.

2. Identification of specific difficulties with the methodology of maths (for

example the child who insists on adding the tens before the units in a sum). Children with dyscalculia often have as many problems with the working out of problems as they do with the basic concepts, and these need to be identified and rectified.

3. Identification of the best methods of teaching, so that the child can

overcome the difficulties rapidly. Although it has been known for many years that children with specific learning difficulties are best helped with a multi-sensory approach to teaching, many people agree that children with dyscalculia find it hard to use the methods of learning maths that most children use. Therefore it is important to work with the child and explore exactly which method of learning each child feels comfortable with.

Not surprisingly many dyscalculic children have very low self-esteem when it comes to maths, and research has shown that if this self-esteem problem can be dealt with then raising mathematical standards becomes much easier. Therefore I am undertaking a programme of work with the dyscalculic children which involves a considerable amount of self-esteem enhancement. This can mean that some sessions with these children focus almost entirely on self-esteem issues rather than on maths. I would assure you that such preparatory work is not a case of my getting side tracked! In fact not only does this work help the children with their maths, it can have a huge impact on all their work in the school.

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You may also wish to note that as part of this programme I am also writing to the parents of the pupils concerned to let them know exactly what is going on. Most children who have dyscalculia can be helped with a course of intensive support, and there is no reason why almost every child with this specific learning difficulty should not be able to reach the average level in mathematics for the school. I attach a confidential list of the children whom I have identified as having a particular problem in this area, and with whom I am now working. While this programme is continuing I should be very glad if you would do everything possible to avoid exposing them to any situation in which they might be called upon to solve a mathematical problem in front of other children. If you would like to know more about this area of work, please do let me know. I have a number of photocopiable resources available in relation to dyscalculia and I shall be most pleased to pass these on. Yours

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Bibliography Attwood, T. “Literacy Skills and Self Development” (School Improvement Reports, First and Best) Attwood, T. “Self Development & Secondary School Pupils” (First and Best) Attwood, T. “Tests for Dyscalculia” (Multisensory Maths, First and Best) Attwood, T. “Methods of Teaching Maths to Pupils with Dyscalculia” (Multisensory Maths, First and Best) Attwood, T. “Attention Deficit Disorder: Practical Activities in School” (First and Best) Attwood, T. “Attention Deficit Disorder: The Parents’ Support Book” (First and Best) Burns, R. B. “Self-concept Development and Education” Holt, Rienhart and Winston. 1982 Chinn, S. and Ashcroft, J.R. “Mathematics for Dyslexics” Whurr 1998 James, William. “Principles of Psychology” Vol 1. Holt. 1890 Lawrence, D. “Self Esteem in the Classroom” Paul Chapman Publishing 1996 Lawrence, D. “Improving Reading and Self-esteem” Educ Res. Vol 27 No 3 pp194-200 Lawrence, D. “Improving Reading through Counselling” Ward Lock 1973 Miles, T. R. and Miles, E. “Dyslexia and Mathematics” Routledge 1992 Sheppard, A. “The Breakthrough: Reclaiming Your Child from A.D.H.D.” (First and Best) Sheppard, A. “Attention Deficit Disorder: a Guide” (First and Best)

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