Articles and papers by Steve Chinn

These articles have been published in a variety of special needs publications and provide background information on dyscalculia and mathematical learning difficulties (for a selection of these see 'Articles').

MATHS TOPIC NUMBER 1

'Tips for supporting a student with dyscalculia' Steve Chinn BDA Handbook 2016

Top 10 Tips

  • Tip 1. Watch and listen.
  • Tip 2. Be empathetic (especially to factors such as short term memory).
  • Tip 3. Don't rely on rote learning.
  • Tip 4. Manage failures.
  • Tip 5. Engineer some meaningful successes.
  • Tip 6. Show patterns.
  • Tip 7. Revisit everything.
  • Tip 8. Revisit again. And again.
  • Tip 9. Explain and show why. Don't rely on, 'Do it like this.'
  • Tip 10. Identify and interpret errors. Don't just say, 'Wrong.'

There are tips and tips. For example, some people think that showing seven times eight as, 56 = 7 x 8 (5678) is a tip or 'I ate (eight) and ate (eight) until I was sick on the floor (sixty four)' so 8 x 8 = 64, is a tip. It could be argued that they are indeed tips, but they are one-off mnemonics to access single facts. They don't explain what the symbols 8 x 8 = 64 mean and why they make numerical sense.

So my big daddy, overall, super-top-tip is, 'Teach them to understand.' Let me explain why, so that there is an understanding of teaching to understand.

One of the key problems for dyscalculic learners is retention of basic facts and maths procedures in their long term memory. But do note that, like most of the learning problems in maths, this is not exclusive to dyscalculics. There will be a spectrum of abilities to memorise maths information. For example, I often ask when lecturing for teachers, 'How many of your ten year olds do not remember all their times tables facts?' This is now a big sample of teachers. The modal answer is 60% to 70%.

There are many programmes on-line and books in shops that claim to 'teach' time table facts. I think that all, or almost all, rely on rote learning, a belief that somehow a catchy tune or a cute illustration will do the trick. Problems with dyscalculia are, to paraphrase Ben Goldacre, 'a little more complicated than that.' As a parallel, it would be unrealistic to expect to teach a dyslexic perfect spelling by a series of quick fixes (though some of the snake oil persuasion have tried to sell this enticing prospect). And there is an added issue, 'purfection' is not perfect, but it is close enough for recognition. 7 x 8 = 64 is just 'wrong'. If used within a complex calculation, 'close enough' is not good enough. Maths works in a very unforgiving way.

Intervention needs methods tailored to the needs of dyscalculics and it takes time.

Key indicators of difficulty

An over-reliance on counting in ones.

Use manipulatives, such as base ten blocks to provide visual images of quantities. Use recognisable and consistent patterns based on 1, 2, 5 and 10. For example;

Children can be shown how 'chunking' can help them work beyond ones.
Difficulty in counting backwards.
Counting backwards requires working memory, so it can be much harder than many teachers and parents recognise. It sets the foundation for subtraction. Use materials and visuals again. It's a good opportunity to introduce and demonstrate vocabulary such as 'take away' and 'subtract' and one of my favourite questions, 'Is it bigger or smaller?'

A poor sense of number and an inability to estimate.

Again objects set out in recognisable patterns create the foundations for developing this skill. Try comparing objects set out in patterns with random sets. That question, 'Is it bigger or smaller?' is useful again.

Difficulty in understanding place value and its role in arithmetic.

A child can learn to count to ten, but writing ten as 10 is a very sophisticated task in terms of understanding how that communicates 'ten'. This can be demonstrated as a cognitively developmental sequence, using base ten blocks on a place value card as the starting point and then weaning the child down to using the symbols (digits) on their own.

The processes of multiplying and dividing by 10, 100, 1000 and so on can be demonstrated with a similar process.

Poor recall of basic facts, but better with 2x, 5x and 10x facts.

All facts are useful, but some are more useful than others. Luckily the more useful ones are the ones learners are most likely to learn. Children can be shown how to link the patterns (and their symbols) to demonstrate addition and subtraction facts, for example,

The principle can be extended to multiplication facts. Understanding is developed as well as strategies for accessing facts. Multiplication is linked to addition, so 7 x 6 is explained and demonstrated as repeated addition of six sixes, 6 + 6 + 6 + 6 + 6 + 6 + 6. The sixes can be chunked as (6 + 6 + 6 + 6 + 6) and (6 + 6), that is 5 x 6 and 2 x 6, making 30 + 12, making 42.

This strategy can be applied to many other examples and it teaches the child about multiplication and how it is linked to addition. It sets the foundation for later, more challenging work on multiplication.

Slow speed of working.

There are a number of contributing factors that combine to make dyscalculic children slower to do maths problems, for example, slow retrieval of basic facts and slow and uncertain recall of procedures. So they do fewer examples and gain less experience. They become more anxious and that makes the situation worse. So they need less examples, but carefully selected to provide a sufficient breadth of learning experience.

The culture of maths to provide answers quickly is counter-productive for many children. Any help in reducing this (often unrealistic) expectation will be good for the child.

Very weak skills for mental arithmetic.

There are people who think that mental arithmetic skills are an essential basic for learning maths. I would challenge that belief. Two skills you need to be good at mental arithmetic are short term memory (to remember the question) and working memory (to work out the answer). Dyscalculic children often have weak short term and working memories. A reasonable adjustment would be to show the question, not just say the question. Another reasonable adjustment would be to give more time and that means more than an extra 25%.

Task avoidance.

If a child (or adult) predicts that they will get a task wrong, then they will, quite reasonably, avoid it. This is a fear of negative evaluation. Most of us experience this about something. If you don't try you don't fail. That requires setting a classroom, and home, ethos that allows a child to fail without that failure creating another step up the withdrawal ladder. High levels of anxiety, usually specific to maths.

Anxiety can be facilitative. That rarely happens for dyscalculic students. There is sound research that has demonstrated that anxiety makes working memory less effective and thus reduces the ability to do maths. In my school we used attributional style to help tackle fear of failure, anxiety and low self-esteem

Forgets maths procedures and formulas.

If long term memory for maths is poor, then the child needs support to remember the often quite complex formulas we use in maths. It is a huge help if the child understands what he is doing and why. The steps should make sense and the task as a whole should make sense. Since maths builds, this will usually require a sound understanding of the pre-requisite work. Hence the need to revise, again and again. It is of no use what-so-ever to say to a child, 'I've told you this ten times, you should remember it.' Ten times without understanding is a waste of times. Use materials and visuals (alongside the symbols and digits) to develop understanding.

It is, as Margaret Rawson said, a matter of teaching maths as it is to the child as he is.' You need to understand both.

More details on methods for intervention can be found in the BDA book, 'Maths Learning Difficulties, Dyslexia and Dyscalculia' and Steve's video tutorials, 'Maths Explained', both available via the BDA.

MATHS TOPIC NUMBER 2

Some comments on an article from Telegraph.co.uk. 18/02/2014 'A former Schools Minister says.' Steve Chinn

A former Schools Minister says:

'Children may be falling behind their Chinese counterparts in maths because of a failure to learn times tables by heart.'

It may be a tad more complicated than that. Sometimes 'times tables' are categorised as 'basic facts' which begs a definition and/or explanation as to what makes a 'basic fact' 'basic'. A basic fact, it could be argued, is one that one really, really needs to know in order to work out other facts (thus introducing a conceptual component to learning).

A former Schools Minister says:

'This includes the times tables up to 12 by the age of nine.'

It is doubtful that a strong case could be made for including the 12 times tables as basic facts and, if you could make that argument, then why not make it for 13 for those of us who remember the 'baker's dozen'?

A former Schools Minister says:

'Believe it or not (the 'not' implies that there is an element of choice for us, dear reader), these are controversial proposals amongst many in the educational establishment.'

Before I try to analyse these comments in more depth, I should outline my overview on learning facts:

I do think children and adults have to be able to access maths 'facts'. Access can be taught and it has side benefits. Access is about teaching strategies that are mathematically developmental. Maths is an intellectual subject. It should be treated as such. In the USA 'number facts' are sometimes called 'number combinations' in recognition of children's use of strategies, using facts they do know to work out facts they don't know. This is conceptual learning and extends the pupils' abilities to access other facts such as the 15 times tables. It removes a sense of helplessness, because it gives learners a way to work out the 'facts' that they do not have in their memories.

OK. Let's take that look at this collection of (three) sayings. There are many factors involved here. For example, the times table facts are unchanged and thus parents recognise them and can even practise them with their children. This certainly will benefit many parents even if it doesn't benefit their children. And you can buy CDs with great tunes that have lyrics that sound remarkably like times table facts. Ah! Those happy school runs. 'It worked for me' is the mantra.

But what if the child fails to master this task? There is a substantial amount of evidence to suggest that many may not. Add in the pressure to retrieve facts at speed and the success rate may drop even lower.

But what if the child does master the task? Does this automatically predict future success in maths? Or does it give an illusion of learning? These 'facts' are part of a concept, that is, the concept of multiplication, which in turn is linked to the concept of addition and extends to the concept of division. Merely learning the words to the song may not develop those concepts. It's not that the proposal is controversial. It is that it is unrealistic and not helpful for many children, even those who do succeed in the task.

I researched an extremely efficacious method for rote learning back in the mid-80s. However, despite its power to deliver retention, it didn't work for everyone. That is such a downer in education that 'nothing works for everyone'.

My informal research of many hundreds of teachers is that a noticeable number of children begin to give up on maths at age 7 years. A factor in this de-motivation is likely to be a fear of failure and one of the early sources of failure is an over-reliance on rote learning.

A former Schools Minister says:

'Children are falling behind in maths because of 'strong resistance' to traditional teaching methods in the classroom.'

I suspect that teachers, as the default position, are likely to use the methods that they were taught and that their teachers, in turn, did much the same. Deviations from the 'traditional' are likely to be rare.

A former Schools Minister says:

'Pupils struggle to understand basic mathematical concepts following a decline in the use of mental arithmetic and rote learning at a young age.'

Maybe mental arithmetic is not the best entry point for defining a pedagogy. Daniel Kahneman, in his best-selling book, 'Thinking, Fast and Slow' turns to mental arithmetic as early as page 20 to illustrate how we think when faced with the problem, 17 x 24. It is not as simple as the Minister thinks. (There is a possibility that she was using her thinking system 1 here). Mental arithmetic has been part of the National Numeracy Strategy since the late 1990s, so it is hard to claim a decline in its use. It's complicated. Again.

Two of the key functions that are required for mental arithmetic are a good short term memory in order to remember the question and a good working memory in order to carry out the steps to solve the question. These memories are not given out equally to children. Those with weaker memories will need differentiation if metal arithmetic is to be of any conceptual benefit and certainly of any benefit to motivation.

A former Schools Minister says:

'With a lot of practice, children not only become fluent and confident in calculation they also develop an understanding of the concepts underlying those calculations as familiar patterns emerge.'

Patterns do not always emerge for all children. Should those patterns emerge incorrectly, then those misconceptions will stay with the child. It's a dangerous assumption. If practice worked for everyone, I would be a sub-3 hour marathon runner.

A former Schools Minister says:

'Large numbers of schools rejected the (traditional?) approach (algorithms?) in favour of 'chunking'.'

Arithmetic is about chunking. The place value system is about chunking (into tens, hundreds, thousands and so forth). Traditional long multiplication is about chunking. It's not the process of chunking, it's the chunks you choose to use. And for that you need good number sense and operation sense.

There is now a large body of evidence about what works in maths education. It is not all sourced from that mythical 'educational establishment', but from psychologists, researchers and educators from around the world, including the Far East.

PS. A former Schools Minister mentioned Shanghai:

It just might be that there is a significant cultural difference in attitudes to maths in Shanghai than in England, but it is a fascinating place to visit.

MATHS TOPIC NUMBER 3

Dyscalculia and Maths Learning Difficulties. Steve Chinn BDA Handbook 2015

This article is written in the light of two influences, the Julian Elliott one-man debate and the depth of knowledge shared by the collection of authors who contributed to my 2015 international handbook on maths learning difficulties and dyscalculia. I refer to their work throughout this article.

I have been working in the specific learning difficulties field for some 35 years and have heard many debates about topics such as labels, definitions, neurology, comorbidity and even the occasional debate about the role of education in addressing the needs of that significant proportion of the population who experience learning difficulties with specific cognitive skills.

My own interest has focused on the difficulties some students and adults experience in learning aspects of mathematics, most frequently arithmetic. The research in this field lags somewhat behind the research into dyslexia. The dyscalculia debate also lags behind … for now. This may be an advantage in that those of us who study dyscalculia and maths learning difficulties may be guided by parallel experiences as what to do and what not to do.

Whilst key researchers such as Ansari, Xhou and Reeve might still be wary about a definition of dyscalculia, particularly when considering the aetiology, there is no doubt that the UK experiences very high levels of innumeracy. Two basic concepts seem apparent to me as a teacher of children with significant difficulties in learning maths. Firstly, they are a heterogeneous bunch. Secondly, their difficulties lie on a spectrum. Where they lie on that spectrum may be greatly influenced by the curriculum they experience.

The BDA's checklist for dyscalculia and maths learning difficulties (available, free, via the BDA website) indicates the range of contributing behaviours that handicap achievement in maths. Children face some of these challenges very early in their lives. The ability to persevere with a subject that seems bewilderingly confusing has never been quantified, and is probably impossible to quantify, but it is a major issue for many children and an issue that perseveres into secondary school and adult life as Geary's longitudinal studies have found.

There are some characteristics that are frequently found in the population that struggles with learning maths. These are often difficult to remedy directly. For example, working memory, which is vital for efficient and fluent mental arithmetic skills, may remain weak in adulthood. An individual's processing speed may remain too slow for the 'do it quickly' culture of maths. Luckily for those with slow processing abilities (a relative term anyway), subjects such as history do not demand that ability. I am not sure that there is any connection, but some teachers may feel that saying things slower, and often louder, addresses this problem. That style of intervention may be somewhat simplistic and certainly not sophisticated or learner-aware.

The curriculum, obviously, has an influence on learning. It is possible to design a curriculum that handicaps some learners, though, one hopes, not intentionally. Arithmetic as a subject is pretty much a given in terms of content. There has to be some adding and subtracting, some multiplying and, as a big challenge, some dividing. Plus fractions, decimals, percentages and measurement. What the curriculum can change are factors such as pace of progression through the topics, the emphasis on mental arithmetic and requiring children to rote learn stuff that is of debateable value and often at the expense of understanding concepts.

An emphasis on mental arithmetic, without an understanding of the factors that will make this an experience of persistent failure is unlikely to generate motivation in many learners. Johnston-Wilder considers this to be 'an unwitting form of cognitive abuse'. Among the skills and knowledge the child or adult needs to develop skills in mental arithmetic tasks are: a good short term memory in order to remember the question, an accurate and swift retrieval of basic facts, recall of the procedure required to solve the question, a good working memory to manipulate a quantity of information and the ability to do all this quickly. Sadly, a problem with even one of these factors is likely to create failure.

Thus a curriculum needs to acknowledge the needs of what CAST (the Centre for Applied Special Technology, USA) calls the 'outliers'. In fact, learning about the factors that can contribute to efficient learning and minimise inefficient learning is going to help boost standards across the whole of the learning spectrum. It may also reduce the demand for intervention.

So, what do we look for? Among the indicators of difficulties are: Learners who have a difficulty in remembering information in both the short term and the long term. Those who find remembering information in sequence exacerbates that problem. Children, and adults, whose processing speed is below the average. Learners who have no concept of place value and the base ten number system. Children who have poor (mathematical) language skills. Learners who exhibit significant anxiety, sometimes to the point where they withdraw from even starting a maths task. Learners who fail to generalise and see patterns (making the load on memory even greater). Children, and adults, who can neither retrieve basic facts from memory nor use compensatory strategies.

These learners need early intervention if they are young or they need to be taken back to basics (empathetically) to re-learn concepts if they are older. They need a mix of cognitive style instruction from teachers, with overviews and reviews complementing the detailed use of formulas and procedures. For me this includes the use of appropriate (to the topic and vocabulary) visual images. It is unlikely that a mini-step-by-step teaching programme will allow them to catch up … such programmes are inherently slow… nor generate better recall as the information comes in so many steps and thus so far apart for easy recall.

The current buzz word in the new maths curriculum is fluency. This is something that is a challenge for some learners unless they are taught by methods that are appropriate to their learning profile, and this might be called 'dyscalculia' or a 'maths learning difficulty', but what it comes down to is summed up by the wise words of Margaret Rawson, one of the USA's great pioneers of dyslexia, 'Teach the subject as it is to the learner as he is.'

'The Routledge International Handbook of Dyscalculia and Mathematical Learning Difficulties' (2015) edited by Steve Chinn

MATHS TOPIC NUMBER 4

Dyscalculia. An Overview - Steve Chinn

Dyscalculia is usually perceived of as a specific learning difficulty for mathematics, or, more appropriately, arithmetic.

Currently (June 2015) a search for 'dyscalculia' on the Department for Education's website gives 0 results as compared to 44 for dyslexia, so the definition below comes from the American Psychiatric Association (2013):

Developmental Dyscalculia (DD) is a specific learning disorder that is characterised by impairments in learning basic arithmetic facts, processing numerical magnitude and performing accurate and fluent calculations. These difficulties must be quantifiably below what is expected for an individual's chronological age, and must not be caused by poor educational or daily activities or by intellectual impairments.

Because definitions and diagnoses of dyscalculia are in their infancy, it is difficult to suggest a prevalence, but research suggests it is around 5%. However, 'mathematical learning difficulties' are certainly not in their infancy and are very prevalent and often devastating in their impact on schooling, further and higher education and jobs. Prevalence in the UK is at least 25%. I would suggest that there is a spectrum of difficulties.

Developmental Dyscalculia often occurs in association with other specific learning difficulties such as dyslexia, dyspraxia or ADHD/ADD. Co-occurrence is generally assumed to be a consequence of risk factors that are shared between disorders, for example, poor working memory. However, it should not be assumed that all dyslexics have problems with mathematics, although the percentage may be very high, or that all dyscalculics have problems with reading and writing. This may well be a much lower percentage.

The roots of dyscalculia

Quite sensibly we expect children to develop at different rates. This is especially the case for young children. Thus it is difficult to target an age where a teacher might realistically suspect a child has dyscalculia. However, the problems with maths start with the earliest arithmetic, for example recognizing small quantities and attaching the correct name and symbol to quantities.

Progression to counting can create an illusion of learning. A child might learn to count to twenty, but in that (conceptually under-estimated) task, the child has met place value and an introduction to addition (in ones). The task of counting backwards is more challenging and too many children fail to automatise this task. It involves working memory, reversing a sequence and subtraction.

In these early stages (as indeed do older learners) children need appropriate materials and visual images, linked explicitly to the symbols and concepts they represent. It should be a sophisticated matching.

Typical symptoms of dyscalculia/mathematical learning difficulties

  • Difficulty when counting backwards.
  • Difficulty when counting backwards.
  • A poor sense of number and estimation.
  • Difficulty in remembering 'basic' facts, despite many hours of practice/rote learning.
  • Only strategy used to compensate for lack of recall is to count in ones.
  • Difficulty in understanding place value.
  • No sense of whether any answers that are obtained are right or nearly right.
  • Slow to perform calculations.
  • Forgets mathematical procedures, especially when complex, for example 'long' division.
  • Addition is often the default operation.
  • Avoids tasks that are perceived as likely to result in a wrong answer.
  • Weak mental arithmetic skills.
  • High levels of mathematics anxiety.

Because mathematics is very developmental, any insecurity or uncertainty in early topics will impact on later topics, hence to need to take intervention back to basics.

What help is available?

Currently there are very few teachers or tutors who are specifically trained to work in this field. There are a number of resources, many of which are listed in the dyscalculia and mathematical learning difficulties section in the BDA website

MATHS TOPIC NUMBER 5

Why Maths Education Isn't Working for All Pupils: Children giving up on maths. Steve Chinn

This article was published in the NASEN magazine 'Special' in January 2010. It remains pertinent today.

I lecture on learning difficulties in maths around the UK and around the world. Lately I have been asking the teachers at my lectures a question, 'At what age are there enough pupils for you to notice, in your school, who are giving up on maths?' The responses are so very similar wherever I am in the world, 'Seven or eight'. Some teachers say six years. This worrying information raises several questions, such as 'Why is this?' and 'How do these pupils survive another eight or nine years of compulsory maths?'

Why children give up on maths

I have asked some teachers to rank my speculations on why pupils might be giving upon maths. The items ranked as most influential were:

  • having to answer questions quickly
  • memorising facts and
  • mental arithmetic.

Although these responses were not focusing on pupils with special needs, there can be no doubt that pupils with special needs will feature heavily in the 'giving up' group. This will be in part down to the fact that the three key contributors to de-motivation in maths are three problems that impact especially heavily on pupils with special needs.

Some learning characteristics of children with special needs

Children with special needs have more problems with aspects of mathematical long-term memory, notably retrieving basic facts from memory. They are slower to retrieve and process information. Also, they often have poor short term and working memories, both of which are key pre-requisite skills for mental arithmetic and indeed mathematics in general.

These three deficits combine to make a powerful obstacle to learning maths. Unfortunately, it is hard to find comments from policy makers that acknowledge this fact. Instead they tend to say 'more mental arithmetic' and 'children will learn the times table facts at an earlier age'. And then, to make sure learning really is a problem there is within the culture of arithmetic the expectation that it has to be done quickly. Mathematics memory

Maths is often taught as an exercise in memory... 'This is what you do'. For example, for division by a fraction, 'Turn upside down and multiply'. Apart from the quaint image this generates, it is a highly attractive solution to a difficult process in maths because it works and is easy to remember.

In Bloom's taxonomy of learning, recall and retrieval is the lowest level of intellectual activity. An informal survey of the Irish equivalent to GCSE maths found that the majority of questions on exam papers were at this level. The same was true of the national Grade 12 assessment in South Africa in 2008. I suspect that this is another universal trend. By taking this 'remember and retrieve' approach with all children we make the assumption that it is effective for all children. Not surprisingly, children who do perform badly in maths do not remember facts and processes as effectively as their better performing peers. This problem is exacerbated by their lack of effective compensatory strategies. Their strategy for overcoming the deficit is counting and counting is a low level cognitive skill with many disadvantages. Better performing peers use linking strategies which are based on understanding numbers, operations and how they relate. Lower performing pupils seem far less able to employ these significantly useful strategies. We disadvantage our special pupils by not being aware of their reliance on counting and then not placing more emphasis on developing strategies that take them beyond counting. The attitude, even if not deliberate, of keeping them at the lowest cognitive level is patronising.

I am sure that success in maths motivates and that failure de-motivates. One of the issues with learning basic facts is that pupils cannot avoid knowing that they have failed. A consequence is a sense of helplessness. In the 2007 DFCS document 'Getting back on Track' a 13 year old girl is quoted, 'I don't get stuck in other subjects – only maths. When I'm doing English I can always get on with my work, If I'm not sure about a spelling, I can just have a go and still get my work done. But I can't do that in maths. If I'm stuck I can't do anything but wait for help.' Some students who have better memories can survive maths for quite a while simply by remembering. Sadly this will not last forever and then they begin to fail. So a better memory can actually hide a potential maths LD.

Slow processing

This is the second example of a clash between the learning characteristics of the student and the demands of the subject. I am unconvinced that doing maths quickly is a necessary demand. I heard that some Australian pupils are encouraged to play the 'gunfighter game'. Two pupils face each other, pretending to be gunfighters. The teacher says a maths fact, for example, 'Six times seven' and the pupils who draws his (pretend) gun and shouts out the right answer wins the gunfight. Not much differentiation there.

One consequence of having to do a challenging task quickly is increased anxiety. Increased anxiety has a negative impact on working memory and so the pupil has less ability to perform the task. Students use working memory to process information mentally. There is an interaction with memory for facts. The pupils who can retrieve facts from memory tend to be much quicker than those who use counting. If pupils can learn linking strategies such as 'doubles plus one' they will improve their number sense, but they will be slower. Thus they may still suffer despite developing an understanding of number.

Short-term and working memory.

I should first explain what these memories are by giving an example. If you ask a pupil to repeat a series of single digits, said at one second intervals, they will be using their short term memory. Most adults can manage about 7 digits. Many pupils with special needs will struggle to recall three. The working memory task involves digits again, but now the pupil has to repeat them in reverse order. This means they have to hold the sequence in their short-term memory and reverse them. It is a harder task and scores will be lower.

These two memories have a significant impact on learning. For example, many research papers equate poor working memory with poor arithmetic skills. Indeed it seems quite logical that a good working memory is a pre-requisite skill for mental arithmetic. Starting any lesson with mental arithmetic tasks, unless there is very careful and empathetic differentiation, will create anxiety and failure for many pupils with special needs. I think that every teacher should know the short-term memory and working memory capacities of their pupils. Of course, for most pupils, the memory capacity will increase as they get older. Unfortunately, this is not so for every pupil.

Understanding to learn

Students with low achievement levels in maths often have only one strategy for accessing facts... counting. Counting is an inefficient, often inaccurate and a very slow strategy particularly for working out multiplication facts such as 6 x 7. If learners can't count backwards (and many cannot) then they can't subtract. So the double whammy is: can't remember, bad at counting.

We need to reassure pupils that rapid retrieval is great, but not achievable for every fact for everyone. Pupils should be encouraged to learn accessible facts such as those involving 1, 2, 5 and 10. They can then relate these to other facts and thus help to build a sense of numbers, operations and how they inter-relate. The level of success will, obviously, depend on the pupil, but then introducing something that can lead to success is better than practising over and over again something which does not.

Insecure learners look for consistency and patterns. Consistency is very reassuring. Patterns help memory. Of course, patterns and consistency help all learners.

Outcomes

There is enough evidence around to confirm the importance of mathematics in everyday life and in employment. Much of what is needed post-school is about number sense and estimating. Memorising without understanding will not develop number sense nor will it give lasting skills. Our memories are designed to forget what we do not practise. If students and adults avoid maths their skills will become less and less. Avoiding maths when you are seven is not a good thing!

MATHS TOPIC NUMBER 6

This article was published in SEN magazine, May/June 2017 and by Learning Difficulties Australia in their Winter Bulletin, 2017

Subtract the negatives

Beliefs about the way maths should be taught discriminate against pupils with SEN, writes Steve Chinn

There is much about maths that makes it a great subject to study. It is logical. It is developmental. You can use what you do know to work out what you do not know.

But, there is much that makes it a bad subject for many learners, most especially those with specific learning difficulties (SpLD). This is not really the fault of the maths, but the fault of the beliefs that influence the way it is taught and how many of those beliefs discriminate against learners with dyslexia, dyscalculia, speech and language difficulties and developmental coordination disorder (dyspraxia).

A current example is the Government's plan to test children on their ability to retrieve times table facts from long-term memory. Behind this worrying addition to our testing regime is the belief that all children can learn these facts - a belief usually based on the notion that, 'It worked for me, so it will work for everyone'. Of course, being able to access these facts (quickly, which is another belief) is useful, but it does not pre-determine success in maths. It will be difficult for the Standards and Testing Agency to create a test format that is pragmatic in terms of how it is administered and yet accommodates children with specific learning difficulties. For example, if the time allowed for each fact to be recalled is four seconds, then 25 per cent extra time takes that to five seconds, which will not improve the situation for the pupil. The pressure during the test will build for many pupils, creating anxiety which will further diminish the ability to retrieve the facts and the test will succeed only in confirming for many pupils that maths is not for them. It is likely that one of the main consequences of this “innovation” will be the need to restore motivation in pupils.

Tabling the question

In the past decade of lecturing to teachers around the UK, I have asked the question: 'At age ten years, how many children do not know all the times table facts?' The most common answer, from a sample that now runs into thousands, is '70 per cent'. This could be used as an argument to increase the pressure on children to master this task, or it could be interpreted as a judgment on the efficacy of using rote learning for these facts.

Now that I have mentioned speed I can focus on that demand as a part of the discriminatory culture of maths. It is a pervasive demand in many topics within maths, in particular, metal arithmetic. Even without focusing on SpLD, there will be a normal distribution of speed for performing this skill and some pupils will not be able to match the arbitrary demands made of them.

So, think about the belief that mental arithmetic should be done quickly. But, also think what it asks of a child (or adult). To succeed in this area of maths pupils need a short-term memory capacity that is adequate enough to remember the question, a working memory capacity to perform the steps involved and a long-term memory for the procedure and facts needed for the task. Working memory capacity is reduced by anxiety, so any fear of failure will make success even less likely. And on top of this there is a demand for speed. Yet there is a belief that mental arithmetic is an effective 'warm-up' exercise. It is a warm up that starts the lesson with a high risk of failure unless it is managed for success.

Turned off maths

I am concerned, and I have another large-sample informal survey to back up my concern, that too many children are withdrawing from maths at a young age. I think that the factors I have mentioned in this article are major contributors and I think they combine with another factor - a fear of negative evaluation - to exacerbate that situation.

Early arithmetic is harsh in that answers are right or they are wrong. For example, 7 x 8 equals 56. 54: although close, it won't do. It is wrong. It seems like a reasonable human behaviour to avoid continuing failure, especially in judgmental situations. In my research into maths anxiety, 'Waiting to hear your score on a maths test' ranked highly for my dyslexic and for my mainstream sample. This is another example of the impact of a fear of negative evaluation. How teachers give out marks - whether orally, in writing or via stars - will have an impact on insecure learners.

Setting a risk-taking ethos for a maths class or an intervention session will enhance motivation and involvement. Dealing with the fear of negative evaluation and building self-efficacy will help maintain motivation.

Teaching by example

Another belief is that apparatus, such as Cuisenaire rods, should not be used by older children and 'older' is often interpreted as being around eight years old. Yet, when I taught physics (many years ago), not using demonstrations would have been judged as bad practice. This belief, as is the case for most beliefs, is absorbed by pupils and sets up a resistance to using apparatus, materials and visual images to aid understanding.

The lessons I learned from moving, after fifteen years of being a successful mainstream teacher, to thirty plus years of teaching, researching and writing about special needs is that maths education would become more efficacious if it paid attention to what works with the outliers. Help the outliers and you help all learners. It’s almost like inclusion!

PROBLEMS WITH MATHS

Steve Chinn, May 2016

I recently assessed a 9 year old boy who was having severe problems with maths. He was just about hanging on to some motivation in school. The time I spent with him, doing low-key diagnostic work, was very tiring for him. To his credit he did not let that tiredness create irritation with me, just polite withdrawal.

I hate to see children who are so low about maths, their ability to experience any success in the subject and the pervasive impact of a loss in self-worth and self-belief which often spreads to other subjects as well. Inevitably those are the children who end up spending diagnostic time with me.

Maybe I felt so sad about this because the boy, and his Mum, were so keen to do better, trying to generate low stress expectations, yet with such a sense of helplessness as to what to do. I knew the school had lost him, that they were setting inappropriate work and that they were making his situation worse. I also knew that this downward trend was going to continue and that finally, the move to secondary school was likely to complete the process of total withdrawal and maths depression.

A disturbing thing for me was that the key issues for this boy are the key issues for so many other pupils in primary schools. So, what can be done to address and reduce the problem?

First, let's look at four of those key issues:

  • A poor short term memory
  • A poor working memory
  • A poor long term memory for basic facts and procedures coupled with no efficient ways to compensate
  • A slow speed of working (compared to 'normal' expectations).

The reality is that there are no drugs and no magic interventions to address these issues directly. We can't inject these children with some magic medication that will, for example, improve their working memory. So we have to turn to the way maths is taught. This we can change, but there are many resilient barriers. However, I do think that the rewards will be significant. By making the instruction more cognisant of the way children learn, and fail to learn, maybe schools will stop applying a 'one belief suits all' philosophy.

It would be my experience, and the wisdom of many researchers, that we teachers can learn so much about effective teaching by studying the 'outliers'. What works for the children who find maths difficult will work for many, many more pupils, partly because when working with this population we do not assume that what we tell or explain will be absorbed instinctively.

If the key issues that create maths difficulties cannot be addressed by direct intervention, then we have to modify the way we present the work so that it becomes accessible to as many children as is possible.

I think I should explain that I write these observations from decades of teaching children with special needs, particularly dyslexia. I have written much about what I have found that works for children and why it works (for example, The Trouble with Maths, 3rd edition) and on how using maths to teach maths reduces the load on long term memory for seemingly isolated facts. My website mathsexplained.co.uk shows how I use visual images to help understanding and to act as the central core of teaching maths in a developmental way. So, through such work, I have tried to explain how teaching maths can be made more mindful of the learner. It's a tad too complicated to address in one article.

To end on a truly positive note, today as I write this, there was an announcement that the Government will make SENs part of Initial Teacher Training. This is potentially stunningly good news. As ever, it depends!