Understanding number concepts and basic mathematical skills is important for many everyday activities in modern societies. Little is understood about the numeracy abilities of people with Down syndrome. At present, it appears that numeracy is an area of relative difficulty and that progress with more complex mathematical understanding is slow. However, some teaching approaches that seek to utilise certain relative strengths to communicate number concepts seem to be useful in practice. Further research is needed to define the precise difficulties experienced by children with Down syndrome and to evaluate teaching strategies.
Buckley SJ. Teaching numeracy. Down Syndrome Research and Practice. 2007;12(1);11-14.
One definition of numeracy is 'to use mathematics effectively to meet the general
demands of life at home, in paid work, and for participation in community and civic
life' . Few would argue with the view that
a basic understanding of numeracy will enhance all aspects of the quality of life
and independence of young people with Down syndrome.
Basic numeracy could be defined as being able to count and to calculate with numbers
to 100. Most countries have decimal systems for money, measuring and weighing, therefore,
knowing numbers to 100 would allow individuals to have a working understanding of
these systems. Adding and subtracting are the most useful operations to understand.
When learning to multiply, some relationships may be more useful than others - for
example knowing how to count in tens, fives and twos would enable a young person
to be competent with the UK money system, which has coins in these values. (To stress
the point, learning to count in threes or sevens would be less useful). Counting
in fives will also be relevant for telling the time.
There has been very little published information on the numeracy abilities and achievements
of individuals with Down syndrome, but the papers that have been published indicate
that most teenagers and adults do not reach this level of basic competency.[2-4] In a survey of the skills of 46 teenagers
with Down syndrome in the UK[5-7] the teenagers
who had been educated in inclusive classrooms had better numeracy skills than those
educated in special education classrooms, but even so, very few understood place
value (tens and units) and very few could multiply or divide numbers up to twenty.
There is considerable variability in the individual rates of progress reported,
with some young people with Down syndrome enjoying maths and doing quite well but
most struggling (for a review and case examples see REF 8). One interesting fact is that
the numeracy achievements of children with Down syndrome are typically at a lower
level than their literacy achievements even when they are receiving good instruction
in inclusive classrooms. In a recent longitudinal UK study of 24 pupils (ages 6-14
years) with Down syndrome their number skills were, on average, two years behind
their literacy skills.[8-10]
At present, therefore, we know that numeracy is difficult - and more difficult than
literacy - for most children with Down syndrome but we do not know why. There have
been very few studies which have attempted to find out what the stumbling blocks
are. One recent UK longitudinal study compared the progress of children with Down
syndrome and typically developing children, matched for non-verbal mental age, as
they learned to count.[11,12] To learn to count,
children first have to learn the 'number words' or 'count sequence', that is, to
recite correctly the numbers one to twenty. This task requires good short-term memory
skills - the words must be learned and always kept in the correct order, before
the child can use them to 'count' objects. Children then learn to count small numbers
of objects and as they become proficient at this, they begin to understand that
we count to find out 'how many' we have. Once they know that the last count word
they use tells them 'how many' they have, they have achieved what is called cardinality. They are now beginning to understand that numbers
represent quantity. In the longitudinal study, the children with Down syndrome had
learned a smaller section of the 'count sequence' than the typically developing
children but they could accurately count the same set sizes and understand cardinality
as well as the comparison group. (All beginning counters know more words in the
count sequence that they can use accurately when counting; e.g. they may be able
to correctly recite numbers to 14 but only count sets up to 9 items correctly).
This UK study is encouraging because at this point the children with Down syndrome
are doing well. Why, then, does research suggest that they do not continue to learn
number skills at a mental-age appropriate rate?
One possibility is that the number concepts become more difficult - a child does
not need to know that 6 is twice as big as three to count six items correctly -
or indeed that 9 is bigger than 5. However, to progress in learning to calculate,
a child does need to know the relative sizes of number and the relationships between
them and this may not become apparent just from practice at counting.
In addition, language becomes important and the child needs to understand concepts
such as 'bigger than', 'smaller than', 'more than', 'less than', 'the same as'.
Thirdly, verbal working memory becomes important for learning longer number sequences
and multiplication tables, for example, and for doing mental arithmetic. Many research
studies report the specific speech, language and working memory delays (relative
to non-verbal mental age) that are usually associated with Down syndrome[14,15].
At present, then, the best guides that we have to adaptations that might make learning
about number easier for children with Down syndrome will take account of their known
language and memory delays. This means being aware that number concepts - even simple
ones such as 'more than' - may need to be taught. It means that methods which provide
support for working memory and allow the child to work out number problems using
tangible materials for each step may help to reduce memory demands. The usual adaptations
for children who learn more slowly such as breaking tasks into smaller steps and
giving the child the opportunity for more practice should also help.
There is some evidence from a recent group study that children with Down syndrome
learn early counting and cardinality skills more quickly when taught using computer
software than when spending the same amount of time on similar pencil and paper
tasks. This could be interpreted
as using a method which makes full use of visual supports and plays to the children's
strengths. Computer programmes are under the child's control so that they can operate
them at the speed they wish - the computer may be more patient than a person and
give the child enough time to process the information and respond. In addition,
the child is not required to speak the answer. The child uses the mouse to make
a response, which may be easier for many children with Down syndrome than using
Tactile and visual strategies
In addition, systems which can visually illustrate the relationships between numbers
and help children really understand the value of whole numbers should be beneficial.
Several papers in this issue[17-20] describe
experience of using number teaching systems which offer some or all of the adaptive
strategies that we may predict will help children with Down syndrome. However, at
present, we have only a small amount of hard data beyond case studies to provide
evidence that they do actually help.
Studies designed to evaluate the effectiveness of different approaches to teaching
number to children with Down syndrome, as well as studies which may increase our
understanding of why they find number difficult should be a priority for research
funding. Meanwhile, parents and teachers need to know what methods may help their
children. Numicon, Kumon and Stern methods and materials are three widely promoted
approaches. Therefore we invited articles from developers,
trainers and users[17,20] of these approaches to provide information
on each approach allow some comparisons of them.
Stern Structural Arithmetic
A programme for learning number concepts that is based on reasoning and insight
into mathematical relationships, rather than rote learning and counting.
A teaching programme that emphasises a step-by-step approach, building on success
and learning through practice.
A multi-sensory approach to arithmetic teaching that uses patterns that are structured
to encourage the understanding of number and number relationships.
The Kumon method of teaching numeracy provides the child with daily repetitive paper
and pencil practice of small learning steps in order to try to ensure success. This
worked for Sam and he achieved a grade in the UK national standard school assessment
in mathematics at 16 years old. Interestingly,
Sam's mother, who is also a primary school teacher, comments that Sam benefited
from the rote learning approach - he could use numbers by learning through this
method - and suggests he may have actually benefited from not manipulating concrete
materials as he learned. 'Sam was neither distracted nor confused by trying to extrapolate
abstract from concrete'. This observation warrants further research as, if this
is generally true, it would lead to significantly adapting current numeracy teaching
approaches for children with Down syndrome. Much more time could be spent on rote
learning and less on learning through experience. However, it should be noted that
Kumon is not used in schools - it is an extra programme which parents enrol their
children in outside school time and any child using the Kumon materials will also
be experiencing other approaches to maths in school classrooms.
Stern and Numicon
The Stern and Numicon methods for teaching number have some benefits in common -
both sets of materials help children to gain meaningful concepts for whole numbers
and to understand the relative sizes of numbers - and draw on similar original ideas.
The materials actually represent the relative sizes of numbers and give children
concrete materials to work with to understand counting and calculating operations.
The introduction to the Stern article emphasises
that the "materials were designed to enable children to learn from their own
discovery and thus gain insights into the number system". This is the opposite
approach to the rote learning one of Kumon - it advocates using materials to illustrate
number concepts. In describing the way in which children learn with the Stern materials,
Vikki Horner emphasises that they address a number of the areas of difficulty that
may be holding children with Down syndrome back - the materials provide visual illustration
of numbers, they reduce memory load and can be used to teach language concepts.
She gives clear illustrations of how children's understanding can be taken forward
in many different ways with the materials and the Stern programme of activities.
The Numicon materials represent whole numbers in the same pattern shapes as the
Stern materials but, in many ways, the similarities in approach and materials end
there. The Numicon designers have used the
shapes to teach children to understand number in a variety of ways that are different
from the Stern approach. Readers will be able to see the differences in the detail
presented in the two articles. The Numicon patterns can be fitted together in a
way that the Stern wooden version of the patterns cannot. The boxes and other materials
in the Stern materials provide a number of ways of teaching that are not directly
provided by the Numicon materials, and vice versa. The Numicon designers have brought
their recent experience as maths teachers and a current understanding of how children
learn maths to their development of the Numicon step-by-step activities which can
be started as early as 18 months.
It is clear that much thought has gone into both Stern and Numicon approaches and
that both offer many fun ways to help any child learn about the world of numbers.
The use of the Numicon approach for children with Down syndrome has begun to be
evaluated in comparative group studies and these indicate a modest benefit for Numicon
when compared with typical UK numeracy teaching in schools. The Stern approach has
only recently been introduced in the UK and case reports describe its value but
group studies are not yet available. Additional guidance is available on the use
of Numicon with children with Down syndrome.
The Numicon approach is being used in many other countries worldwide and has been
translated into other languages.
The fourth article on numeracy, which
describes the excellent progress of Katrina, aged ten years really stresses that
the three methods described all have their strengths and can be successfully used
together. Katrina has used them all. One method may work better for one stage or
step of learning, and another at another stage. Katrina is doing exceptionally well
at maths for her age but her parents make very clear that this level of competence
has required many hours of learning and practice to consolidate her understanding
and let her move to the next stage. There are some difficult messages here as all
the case studies suggest that the children will not reach their potential without
additional work and practice outside the school day. This highlights the need for
teachers and families to work together in planning work for children and the need
for materials and schemes which make learning at home fun and relatively easy for
both children and families.
Sue Buckley is at
Down Syndrome Education International,
Portsmouth, Hampshire, UK.
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