Responding to the needs of pupils with dyscalculia

It is somehow much more acceptable to be poor at maths than it is to be a struggling reader or have limited writing skills. There is less shame attached to confessing about ‘not being good with figures’ than there is in admitting ‘I can’t read very well’. This is one reason why teachers have, for a long time, accepted children’s lack of progress in maths when they would go to great lengths to improve their literacy skills. Yet a recent report from the Basic Skills Agency found that poor numeracy is more of a handicap in getting and keeping a job than poor literacy. Furthermore, a significant proportion of the prison population has problems with numeracy and the British Dyslexia Association cite the case of an inmate of Pentonville who was so embarrassed by his inability to calculate money that he found it ‘easier to nick it’ than ruin his street cred by admitting his weakness. No one had ever taught him in a way that he could learn.

Definition

There are many reasons why children struggle with mathematics, including ineffective teaching, home circumstances, poor attendance at school and other areas of difficulty that affect learning, such as language acquisition and ADHD. About 40% of children identified as dyslexic will also have significant difficulties with maths. But research now shows that a genetic anomaly may result in a specific deficit in the learning of numerical skills. This specific difficulty is termed ‘dyscalculia’ and is used to describe pupils who score well on intelligence indicators that are not mathematically based. It is thought that approximately 5% of children have some degree of dyscalculia, with equal numbers of boys and girls being affected. They have been taught in the same way as their peers and engaged in the same mathematical activities – and yet they encounter distinct difficulties in mastering the basics of mathematical thinking. As a result they often fail to acquire the essential concepts that underpin a true understanding of mathematical procedures. In its most severe form, dyscalculia can mean that a child cannot learn to tell the time, know the date, shop competently or do very simple arithmetic.

Although there is on-going debate about the precise nature of dyscalculia, the Department for Education and Skills (2001) describes dyscalculia as:

‘A condition that affects the ability to acquire arithmetical skills. Dyscalculic learners may have difficulty understanding simple number concepts, lack an intuitive grasp of numbers and have problems learning number facts and procedures. Even if they produce a correct answer or use a correct method, they may do so mechanically and without confidence.’

Identification and causes

There is a strong genetic influence on the development of mathematical skills. One family will have parents and children who are all very capable mathematicians, while in another, mathematical difficulties are very common. Shalev and Gross-Tur (2001) found that about 50% of the siblings of a pupil with dyscalculia can be expected to have it as well. Parents and siblings of a pupil with dyscalculia are ten times more likely to have dyscalculia than members of the general population.

Identification can be problematic, but research has shown that difficulties with recognising and comprehending numbers (number processing) are common to children with dyscalculia, and this is the basis of Butterworth’s Dyscalculia screener (2003), which identifies deficiencies (indicated by slow reaction times) in two processes: counting dots and comparing the value of numerals. The implication is that an abnormally slow speed of response indicates a neurological impairment in number processing. The response times for the processes have been standardised for different ages through large-scale testing (1,500 pupils aged 6-14 years) and a reasonably high correlation found between poor performance on these two tests and poor mathematics performance as measured by the nfer-Nelson mental mathematics tests (Clausen-May, Claydon et al 1999).

Researchers such as Geary (2004) and Dowker (2003) emphasise the variety of difficulties shown in cases of dyscalculia, and the panel opposite lists the signs associated with dyscalculic learners.

Early language is the starting point for mathematical thinking

Much of a young child’s understanding of mathematical concepts will be tightly bound up with their language development. They will learn about words and phrases such as ‘more’, ‘less’, ‘bigger’, ‘longer’, ‘twice’, ‘before’, ‘after’, ‘the same as’, ‘enough’. They will learn to count and name shapes, often before they start school. Pupils with inadequate language skills may have general problems or they may have particular difficulty with the language related to mathematical concepts such as position, relationships and size. Moreover, language is a very important vehicle for thinking. It is extremely difficult to deal with new ideas, understand abstract concepts, manipulate information and ideas, solve problems and remember previous learning without using appropriate language. Language is important as a way of carrying thinking forward: ‘I’ll have to work out how many apples there are and then I can divide that number by the number of people to find out how many each one can have.’

Children with dyscalculia may not understand the language they recite: all too often they have learned a script that is a meaningless incantation – Six take two you can’t do, so borrow a ten – when they have no idea what all of this actually means.

Teaching

Children can come to think that mathematics is about learning a lot of unrelated facts, processes and rules that have no overall pattern, logic or practical use. In fact, mathematics is all about patterns, connections and applications. But unless they are shown the connections, pupils may not realise that adding up is not just something that you do as part of a numeracy worksheet, it is also something that happens when you work out how many people are going to be sitting at the table and how many plates to put out.

Children with dyscalculia often find abstract calculation very difficult. If they can understand the link between real life and seemingly abstract procedures, it can make all the difference to the way in which they learn. Both visual imagery and language are important parts of internal mathematical thinking. Connecting with real objects and events helps to trigger a visual image and/or language that really does make sense.

The use of multisensory teaching methods will help children to understand and remember, providing more vivid associations that will help memory by attaching meanings to otherwise arbitrary words and symbols. ‘Perceptual gestalt’ images, such as five fingers on one hand or five dots on a die, are widely used. The Slavonic abacus, which has 100 beads in rows of 10, shown in fives by two colours, is recommended by teachers of dyslexic children, such as Grauberg (1998). Another useful resource is Numicon (Wing 2001), which provides Stern’s ‘ten frame’ images for number bonds, to visually represent part-whole relationships. A word of caution though about the use of the many ‘manipulatives’ commonly used in classrooms – counters, wooden blocks and other ‘concrete’ apparatus. Pupils may learn to use the apparatus but not be able to transfer what they do to any understanding of what it actually represents. They may be able to work out a written algorithm with their Cuisenaire rods, but never make any connection with the broader concept as it applies in everyday life.

Mathematics, perhaps more than any other subject, is hierarchical and so it is especially important that teachers ensure that children learn essential skills and concepts as they go along; for instance, if children do not associate values with numerals they may not go on to understand place value, and if they do not understand multiplication they may not remember multiplication facts. Some children might require more explicit teaching of the principles of counting, the meanings of number symbols, how to memorise facts, or how to check and monitor procedures. Effective monitoring of skills and understanding is essential because if gaps in their learning are not identified and addressed early on, difficulties may accumulate over the years, with assumptions being made about understanding or previous learning. This can result in compounding a child’s difficulties and lead to negative experiences in the classroom that create a fear of failure and anxiety.

Reasoning seems to help children who can use logic and have good verbal skills, but poor spatial skills. Learning number facts as derived facts, related to patterns and relationships with other number facts, is more likely to be effective than learning by rote. Memorising times tables, for example, with similar numbers next to each other can cause confusion. Butterworth et al (2003) recommend computer programs for learning number facts, as part of a broader programme of teaching for understanding, using a variety of approaches. Reasoning that 4 x 8 is double 4 x 4, which is double 4 x 2, sets up a train of associations that can be rebuilt if forgotten. Learning facts that are related through reasoning can therefore provide a reliable means of checking. It seems that an emphasis on reasoning may also help children to generalise what they learn. Explaining why procedures work and when to use them helps pupils to apply them correctly in different situations.

Dyscalculia tends to be associated with a negative self-image as a learner of mathematics, so anything that builds confidence and self-esteem is likely to be helpful. (The interactive, whole-class teaching recommended in the National Numeracy Strategy can disadvantage dyscalculic pupils, causing them embarrassment when they can’t answer a question or demonstrate how they worked something out.) Building secure knowledge and understanding at the required pace is important, as well as ensuring enjoyment and providing a safe environment where learners can take risks. It is likely that pupils with dyscalculia will need focused, one-to-one teaching to support what is taught in the classroom.

References and further reading

Butterworth, B (2003) Dyscalculia screener: Highlighting children with specific learning difficulties in maths. London: nferNelson.

Clausen-May, T, Claydon, H (1999) Mental mathematics 6-14 test series. Windsor: nfer-Nelson.

Department for Education and Skills, (2001) Guidance to support pupils with dyslexia and dyscalculia (DfES0521/2001). London.

Dowker, A (2003) ‘Interventions in numeracy: Individualised approaches’ in Thompson, I, Enhancing Primary Mathematics Teaching. Buckingham: Open University Press.

Dowker, A (2003) ‘Brain-based research on arithmetic: Implications for learning and teaching’ in Thompson, I, Enhancing Primary Mathematics Teaching. Buckingham: Open University Press.

Geary, DC (2004) ‘Mathematical disabilities: What we know and don’t know’, www.ldonline.org.

Gifford, S (2005) ‘Young children’s difficulties in learning mathematics: Review of research in relation to dyscalculia’. QCA/05/1545. www.qca.org.uk.

Grauberg, E (1998) Elementary Mathematics and Language Difficulties. London: Whurr Publishers.

Hannell, G (2005) Dyscalculia: Action plans for successful learning. London: David Fulton.

Kay, J and Yeo, D (2003) Dyslexia and Maths. London: David Fulton.

Shalev, RJ, and Gross-Tsur, V (1993) ‘Developmental dyscalculia and medical assessment’ in Journal of Learning Disabilities, vol 26(2).

Wing, T (2001) ‘Serendipity, and a special need’ in Mathematics Teaching, vol 174.

Yeo, D (2003) Dyslexia, Dyspraxia and Mathematics. London: Whurr Publishers.

Identifying pupils with dyscalculia

There are various warning signs that a child or adolescent may have dyscalculia:

Numbers

• pupils may read and write numbers competently but numbers over 1,000 can prove difficult
  
• there may be an impaired sense of number size, affecting the comparison of numbers etc.

Slowness

• in giving answers to mathematics questions
  
• in working, compared to others in the class.

Difficulties with mental calculation

• relies on tangible counting supports such as fingers, tally marks
  
• uses the ‘counts all’ method instead of ‘counting on’ for addition
  
• finds it difficult to estimate or give approximate answers.

Difficulties with the language of mathematics

• finds it difficult to talk about mathematical processes
  
• does not ask questions, even when he or she evidently does not understand
  
• finds it difficult to generalise learning from one situation to another
  
• makes mistakes in interpreting word problems
  
• gets mixed up with terms such as ‘equal to’, ‘larger than’.

Poor memory

• difficulty in remembering basic mathematics facts
  
• cannot remember what different symbols mean
  
• forgets previously mastered procedures very quickly
  
• recites the entire multiplication table to get an answer for 9 x 3 =
  
• completes multiplication tables by ‘adding on’
  
• forgets the question before the answer can be worked out.

Ineffective use of visual images

• may not be able to locate a number on a number line without searching up and down
  
• will not notice visual patterns such as the 0 in 10, 20, 30, 40 etc
  
• cannot relate to visual representation of fractions/decimals such as circles divided in half etc.
  
• Difficulties with sequences
  
• loses track when counting/reciting tables
  
• has difficulty in navigating back and forth, especially in twos and threes etc
  
• has difficulty remembering the steps in a multi-stage process.

Difficulties with position and spatial organisation

• is confused about the difference between 21 and 12
  
• puts numbers in the wrong place when redistributing or exchanging
  
• poor setting out of calculations and of work on a page
  
• may not understand the importance of working left-right or taking the bottom number away from the top.
  
• scatters tally marks instead of organising them systematically
  
• gets confused with division:
  Is it 3 into 6, or 6 into 3?
  
• in tens and units takes the smaller number from the larger, regardless of position
  
• finds rounding numbers difficult
  
• finds telling the time on an analogue clock difficult
• is easily overloaded by pages/worksheets full of figures
  
• copies work/shapes inaccurately
  
• measurement and understanding of space may cause problems
  
• symmetry, tessellation and geometry present a real challenge.

Reliance on imitation and rote learning instead of understanding

• can ‘do’ sums mechanically but cannot explain the process
  
• sometimes uses the wrong working method such as treating a ten as a one (or vice-versa) in exchanging or redistribution
  
• cannot decide what arithmetical operation is required
  
• cannot build on known facts, eg they may work out that 3 + 4 = 7 but not realise that, therefore, 4 + 3 = 7 as well.

[Amended from Hannell, 2005]