How do pupils with dyscalculia learn mathematics differently?

Dyscalculic learners:

 often have difficulty counting objects.
This affects basic ‘number sense’. They need clear instructions on how to count in an organised, meaningful way. Numbers need to have a meaning, magnitude and a relationship. They should first of all learn the skill of subitising before moving to counting.

 may have difficulty processing and memorising sequences.
Dyscalculic learners may be slow to learn a spoken counting sequence. Counting backwards is particularly difficult. They need additional practice in counting orally and need to continue oral counting into higher value sequences. Support can be
provided by presenting sequences such as 0.7, 0.8, __, __, as 0.7, 0.8, __, __, 1.1, 1.2. The use and recognition of pattern is important and can be used to circumvent some of the problems with memory. Dyscalculic learners need support counting through transitions, e.g. 198, 199, 200, 201 or 998, 999, 1000, 1001, and practice structuring from one count to another, e.g. from counting in tens to counting in ones.

 need extra support in counting forwards and backwards.
Use a clearly labelled number line, or counters placed in recognisable clusters, as on dominoes. Teen numbers are an example of the inconsistencies of our number system. For example, thirteen should be ten three, but it is said and written as three and ten. By contrast, the word twenty-three is in the same order as the digits, even if twenty is an irregular word (compared to two hundred). Careful teaching can minimise these difficulties as well as introduce the more regular pattern of larger numbers – sixty-six, seventy-six, etc. Dyscalculic learners may find the transfer of a learned sequence, say 90, 80, 70 ..., to a modified sequence 92, 82, 72 ..., challenging. Base ten blocks or coins may help illustrate which digit changes and which remains constant.

 often have difficulties understanding place value.
Language uses names to give values when counting (ten, hundred) while numerals use the principle of place value – the relative places held by each digit in the number (10, 100). Pupils who have not mastered the name value system may say that nine hundred and ninety-nine is bigger than one thousand. Language demands are greater in writing numbers in words. Numbers that feature zeros, such as 5006, will need careful teaching, using practical materials and focusing pupils on the ‘top value’ word: five thousand and six has four digits because the top word is ‘thousand’. A place value chart might be useful. Place value cards can also demonstrate the structure of numbers at a more symbolic/abstract level.

 find it difficult to learn number facts ‘by heart’ but can usually
work within a manageable target and can learn to use strategies. Number bonds to 10 are fundamental and the key to so many more facts that they should form the focus of quick recall. Patterns need to be taught using multi-sensory approaches. Use memory hooks to help relate new facts to learned facts. Visual imagery, e.g. showing the links between 5 + 5 and 5 + 6 with coins or counters, will also support non-Dyscalculic pupils in the class.

Facts that may be accessed through rapid mental recall are stored as verbal associations in exact sequences of words, such as ‘8 plus 5 equals 13’ or ‘7 times8 is 56’. Dyscalculic learners find it difficult to remember such verbal associations. Facts that have been successfully stored as verbal associations may be accessed very slowly. Learners should be encouraged to maximise the use of key number facts, e.g. ‘10 ×’ facts can be used to deduce ‘9 ×’ facts, as in 9 × 7 = (10 × 7) – 7. Short sequences of step counting from ‘5 ×’ can lead towards a ‘partial products’ approach in which, for example, 7 × 8 is seen as (7 × 5) + (7 × 3).

 fail to remember the variety of fact-derived strategies or mental calculation methods.
The sequence of steps in a calculation is difficult to remember for Dyscalculic pupils because of a poor working memory. Weak number concepts and a lack of flexibility hinder multi-path reasoning and learners may become confused or feel overburdened. Some see too many methods to learn and remember. It is important to concentrate on strategies that can be generalised, such as partitioning, rather than ‘one off’ methods, as these skills can then be more widely used across a range of calculations.

 have problems recording calculations on paper.
Learners who have performed well in mental mathematics may fail to cope with written methods of calculations. This is due to the increased load on the working memory of having to remember a more formal written procedure, plus difficulties in writing the calculation. Mental calculations often favour working with the most significant digit first. It may be useful for some to continue this approach with written calculations.
Working with base ten materials should support the introduction of written calculations, as these can illustrate the written method. Area, using squared paper, is a good model for multiplication.

 may have problems using calculators.
Calculators may help to overcome difficulties and help learners access more mathematics. But a calculator will only facilitate work in some stages of the question and thus not act as a total problem-solver. Also, once a Dyscalculic learner has selected the appropriate calculation, they may then have difficulty between the stages of reading it on a page and transferring it to a calculator keyboard.

 may need more clues to recognise, develop and predict patterns to help them solve problems.
Word problems are likely to be a source of difficulty. Teach the use of a ‘problem-solving frame’:
– read the problem;
– identify the key information and write it down or draw pictures;
– decide which calculation is necessary;
– use an appropriate calculation method: mental, written or calculator;
– interpret the answer in the context of the problem.

Pupils may learn how questions are constructed if they invent their own word problems. The use of materials or images to interpret word problems can increase success.

 may be unsettled by the insecurity of estimation.
Estimation requires risk-taking and insecure learners avoid risk. Visual models may help pupils see ‘closeness’.

 find the sequencing of time difficult.
Sequences of days of the week or months of the year are not easy to learn, and the introduction of simple clock time may also be a problem. The language of time is potentially confusing, with deceptively simple changes such as saying 7.10 as ten past seven (reverse order) creating problems. Using a clock face with pupils moving the hands and specifically
relating the language to the image may help. The introduction of digital representations may be supported, in the first instance, by a set of personal sequencing cards.

 may confuse left and right, hindering work on position, direction and movement.
Left and right are difficult to anchor to a fixed image. Learners need to spend time involved in physical activities using direction cards and possibly learning a simple mnemonic to help remember left and right: e.g. ‘write with my right hand and the one that is left is my left’. Clockwise and anti-clockwise may present similar problems, although they can be anchored to a visual image. ICT equipment, including the use of programmable toys, may help.

 may have problems understanding the different types of averages.
The teaching and use of the terms mode, median and mean is difficult as they all begin with the same letter. When teaching, it might be useful to use separate coloured index cards with the words and their meanings written on.
e.g. mode – most frequent
median – middle
mean – average
range – ‘biggest minus smallest’

Department for Education and Skills(2001) with certain sections modified in line with current research and practice (2009)

Dynamo Maths is a remediation programme that considers the above symptoms of Dyscalculia.