# Number and arithmetic skills in children with Down syndrome

#### Sophie Brigstocke, Charles Hulme and Joanna Nye

It is clear that arithmetic and number skills are areas of particular difficulty for individuals with Down syndrome. Studies of arithmetic development in typically developing children suggest that a pre-verbal “number sense” system and counting skills provide two critical foundations for the development of arithmetic. Studies of children with Down syndrome suggest that the development of both these foundational skills present difficulties for them, though these conclusions are based on relatively small samples of children. It would seem that further studies of arithmetic and number skills in children with Down syndrome, involving larger samples of children and broader ranges of measures, are badly needed.

Basic number skills, such as knowing how to count and solve simple arithmetic problems, are essential for everyday independent living. It is clear that arithmetic and number skills are areas of particular difficulty for individuals with Down syndrome. So far, we have quite limited understanding of the cognitive bases of the problems with number skills seen in children with Down syndrome. However, major advances have been made in research on the development of number skills in typically developing children and it appears that these advances offer the prospect of better understanding the problems seen in children with Down syndrome.

## Levels of attainment in arithmetic in children with Down syndrome

Studies of arithmetic attainment in individuals with Down syndrome consistently
report very low levels of attainment. Carr reported that more than half of her sample
of 41 individuals aged 21 years, could only recognise numbers and count on the Vernon's
arithmetic-mathematics test^{[1]}. Buckley and Sacks
surveyed the number skills of 90 individuals with Down syndrome aged between 11
and 17 years and found that only 18% of the sample could count beyond 20 and only
around half of the sample could solve simple addition problems^{[2]}.
This pattern contrasts with the increasingly positive achievement levels in reading
skills that children with Down syndrome are attaining (e.g. ref
3). Indeed the most consistent finding in the literature is that reading
accuracy is significantly higher than arithmetic attainment^{[1-6]}.
Age equivalents on standardised number tests are typically reported to lag age equivalent
reading scores by around two years in children with Down syndrome (e.g.
ref 1). Brigstocke et al. report measures of arithmetic ability from a group
of 49 children with Down syndrome^{[7]}. The
sample ranged in age from 5:06 to 16:02, and had an average British Picture Vocabulary
Scale (BPVS) standard score of 60 (range 39-91). Of this sample 45 children had
measurable single word reading skills on the British Ability Scales (BAS) reading
test (average standard score 67, range 55-115). However, only 27 of the sample could
score on the BAS basic number skills test, and for these children their scores were
very low (average standard score 62; range 55-111). It is clear therefore that in
this sample, like others studied, number skills are much weaker than reading skills.

In typically developing children education and age are strong predictors of arithmetic
performance^{[8]}. Language skills are generally predictive of variations
in arithmetic ability, and in line with this, children with specific language impairment
typically have low arithmetic achievement despite average IQ^{[9]}.
Nonverbal ability is also associated with arithmetic achievement in typical and
atypical development (e.g. ref 10). Working memory and
knowledge of number facts are also important predictors of arithmetic performance^{[11,12]}. Number facts are learned from associations
between the problem and the retrieved answer, which is a function of experience,
and working memory capacity. These findings are particularly relevant given the
cognitive profile typically noted in Down syndrome in which language skills^{[13,14]} and verbal short-term memory^{[15,16]}
are weak relative to nonverbal abilities.

In common with typical development, number skills in children with Down syndrome
appear to improve with age ^{[17,18,19]} but this
is not always the case and wide individual differences are noted in all studies.
The relationship between achievement levels and mental age in Down syndrome is not
clear so far. Findings are inconsistent (e.g. refs 17,20,21) but this may reflect the variety and generality
of the measures used to assess number skills. Floor effects on standardised IQ tests
can also be a problem (e.g. ref 22). Moreover, general ability
is a wide measure and the mechanisms that govern the relationship between IQ and
mathematical achievement are not clear even in typical development, which makes
interpretation difficult. Language skills are related to achievement in number skills
in children with Down syndrome (e.g. refs 5,23,24).
For example in recently collected data there is a strong correlation between verbal
ability measured by the BPVS and both single word reading standard scores (*r*
= .69) and BAS arithmetic standard scores (*r* = .63)^{[7]}.

Education clearly has a positive influence on achievement levels in arithmetic.
This is indicated by the success of small intervention studies (e.g. refs
19,25,26). Children with Down syndrome in mainstream
schools have better attainments in arithmetic than those in special schools; but
this is likely to be confounded with selection biases^{[1,17,27,28]}. Nye et
al. note that individual differences in response to an intervention using Numicon
were related to the quality and quantity of teaching^{[26]}.

The Numicon approach to teaching numbers skills and arithmetic^{[29,30]}
is based on a system of structured visual representation first developed by Catherine
Stern which makes clear the stable order of the number system, and how different
numbers are related^{[31]}. One of the key features
of the scheme is it provides children with representations of whole numbers which
are used to develop mental imagery of numbers, and makes an explicit connection
between the preverbal number system, counting and arithmetical operations. Therefore,
the scheme uses the perception of whole numbers to support mental arithmetic, rather
than using counting as the basis for arithmetic, as is often the case in UK numeracy
teaching. Whilst the scheme has been developed for all children to use it was thought
to be particularly appropriate for trialling with children with Down syndrome as
it complements their particular cognitive profile (e.g. having strengths in visual
processing), and targets many of the areas of numeracy that they have difficulty
with.

The recent Portsmouth project followed the development of number skills in 16 children
with Down syndrome (aged between 5 and 14 years) over 2.5 years whilst they were
taught using Numicon, and their performance was compared to archive data of children
with Down syndrome who had not used Numicon^{[32]}.
A small but non-significant gain was seen in numerical performance (as measured
by the BAS Basic Number Skills sub-test^{[33]})
in the children who had used Numicon compared to those who had not. Qualitative
analysis of the children's profiles, including data from observations of lessons
and non-standardised detailed number assessments indicated that Numicon is of particular
benefit to children for developing both early numerical concepts and those who are
starting to work with arithmetical operations. Regular use of the materials and
creative adaptation of the scheme to meet the needs of the individual children were
both found to be critical in effective implementation of the scheme. Whilst the
main finding from the standardised measure was non-significant statistically, it
should be noted that the was a wide range of gains seen in the children, and the
gains made may still make a considerable different to children's numbers skills
and resulting quality of life.

In summary, individuals with Down syndrome find number skills very difficult in comparison to their ability to learn to read but respond positively to tuition. However, while studies investigating the cognitive correlates of general mathematic tests are a useful starting point, they give little insight into the underlying processes involved.

## The cognitive bases of arithmetic in normal development and the origins of mathematical difficulties

In the last 20 years or so there has been a good deal of research concerned with
understanding the cognitive bases and development of human numerical abilities.
It appears from studies of animals and pre-verbal human infants that some basic
numerical skills exist in the absence of language. This pre-verbal numerical system
is probably somewhat imprecise and can only deal with small numbers of objects.
Nevertheless, it has been suggested that such a preverbal "number sense"
may form a foundation for more complex verbally elaborated number skills in humans^{[34,35]}. The possible role of such a putative
nonverbal number sense system in mature numerical processing remains controversial,
but in one view, this nonverbal system provides the semantic underpinnings for understanding
number since numbers, fundamentally, signify magnitudes.

One other important skill that also develops in the pre-school years is counting.
By the time children go to school they are generally proficient at counting, at
least for numbers up to ten, and these counting skills form a foundation for the
development of arithmetic skills. Counting is fundamentally a form of measurement,
and one that is more flexible and precise than the form revealed in studies of animals'
and infants' preverbal numerical abilities. Learning to perform basic addition,
which is the earliest arithmetical skill to be taught in school, can be seen as
a natural extension of counting. At first, children use a simple 'count all' strategy
to solve addition problems. By the age of 6, most are using a 'counting on' strategy
in which they start with the smaller number and count on from this (the *min*
strategy). Later, as they learn the number bonds, they can begin to retrieve these
automatically. Development involves a change in the mix of strategies that are used.
Importantly, the creation, in long-term memory, of an association between the problem
integers (e.g. 3+4) and the answer that is generated (7) requires practice in the
execution of basic computations. With each execution, the probability of direct
retrieval of that number fact or bond increases. This direct retrieval strategy
is rapid and highly efficient but only develops after the child has performed many
less automatic computations of the relevant sums.

## The possible cognitive bases of difficulties with arithmetic in children with Down syndrome

Such studies of arithmetic in typically developing children suggest it is important to understand the integrity (or otherwise) of the pre-verbal number system in children with Down syndrome, and the development of counting skills, as these two skills appear to provide two of the foundations for the development of arithmetic in typically developing children.

### Preverbal numerical systems in children with Down syndrome

Magnitude comparison tasks have proved a very useful paradigm for investigating number skills and cardinal number understanding in typical development. Some authors interpret the ability to discriminate between magnitudes as a behavioural indicator of the operation of a basic "number sense" (e.g. ref 34) that underlies later number skills. It has been suggested that difficulty judging between magnitudes may underlie the difficulties that typical children with dyscalculia have with mathematics (e.g. ref 35).

In a typical numerical judgement task participants are presented with two stimuli
(either digits, or arrays of dots differing in numerosity or squares differing in
size) simultaneously on a computer screen and asked to indicate which is larger
as quickly as possible. The simplicity and non-verbal requirements of the task make
it ideal for individuals with Down syndrome. Findings from studies on numerical
magnitude comparisons in typical adults and children have proved remarkably replicable.
As Moyer and Landauer observed in their seminal study, the time required to compare
the numerical magnitude of pairs of digits decreases as the numerical distance between
stimuli increases (1 vs. 9 is a much easier judgement than 1 vs. 2)^{[36]}.
This is referred to as the symbolic distance effect (SDE). When the distance is
held constant, discrimination of numbers becomes more difficult as their magnitude
increases. This is referred to as the magnitude effect. This pattern of results
mirrors that observed in comparison of physical magnitudes such as length and is
the opposite pattern to that predicted if counting strategies were used. Recent
work suggests that speed in making magnitude comparisons predicts individual differences
in addition ability in typically developing children^{[37]}.

This symbolic distance effect (SDE) has been observed across all ages from 6 years
upwards, supporting the idea that the effect is relatively independent of educational
influences and cognitive ability^{[38,39,40]}.
The hypothesis that it is independent of language and does not rely on counting
is supported by findings that children with specific language impairment who have
significant difficulties with the verbal count sequence demonstrate typical performance
in numerical comparison tasks^{[9]}. Since individuals with Down syndrome
are observed to have relatively preserved visuo-spatial abilities, and numerosity
judgments are independent of language and general cognitive ability, this suggests
that they will demonstrate normal performance on numerosity comparison tasks, provided
they are sufficiently familiar with the count sequence and digits.

Paterson, Girelli, Butterworth and Karmiloff-Smith investigated the distance effect
in infants and older individuals with Down syndrome and Williams syndrome^{[41]}. They also administered a battery of number
tasks hypothesised to rely on verbal abilities to the older groups. Eleven infants
with Williams syndrome and 18 infants with Down syndrome, matched on chronological
and mental age plus 16 mental age and 14 chronological age typically developing
controls were tested on a preferential looking paradigm. Infants were familiarised
with arrays of 2 objects. In the test phase they were presented simultaneously with
one card displaying new objects but the familiar numerosity and one card with three
objects i.e. a new numerosity. Cumulative looking times were measured. It was found
that there was a significant difference in mean looking time between the familiar
and novel numerosity in all groups except the Down syndrome group. This suggests
that the infants with Down syndrome were unable to distinguish between 2 and 3 items.

However performance in a numerosity comparison task with older individuals showed the reverse pattern. Eight older children and adults with William's syndrome, 7 with Down syndrome, 8 typically developing controls matched for mental age using the British Abilities Scales, and 8 typically developing controls matched for chronological age to the clinical groups took part in the experiment. Participants were asked to indicate the larger of two dot arrays presented simultaneously on a computer screen. Reaction times and accuracy were measured. The numerosity of the arrays varied from 2 to 9 and the numerical distance between the arrays was classified as small (a difference of 1 to 3) or large (a difference of 5 to 7). Although reaction times were slow in the Down syndrome group, individuals responded more quickly and more accurately to arrays that had a large difference between them than those that had a small distance between them. A significant effect in the same direction was noted in the control groups but this distance effect was not observed in the William's syndrome group. Analysis of errors revealed that the Williams syndrome group was the least accurate of all the groups. The results of this study support the conclusion that language skills do not support performance in magnitude comparison tasks.

Participants also took part in a detailed battery of number tasks that assessed rote counting, dot and numeral seriation, matching dots to numerals and reading numerals aloud as well as single digit addition, subtraction and multiplication. Performance in the clinical groups was below that of the control participants who performed near ceiling. The William's syndrome group displayed considerable difficulties when compared to the Down syndrome group on all the tasks except rote counting from 1 to 20 and reading single digits where performance was good in both groups. Both groups found matching numerosities to Arabic numerals difficult.

No correlations were found between performance in the dot comparison task and the
number battery task except in the performance of the Down syndrome group on the
matching dots to Arabic numerals task. This could suggest a link between the ability
to discriminate numerosities and the ability to associate Arabic digits with their
underlying quantity representation^{ [41]}.

An unpublished study conducted at York investigated the pattern of reaction times
obtained by 16 children with Down syndrome, with a mean age of 13; 2 years (SD 24.44
months) and a receptive vocabulary level of above 5 years, on three computerised
comparison tasks and a timed pencil and paper single digit addition task^{[7]}. Each computerised task comprised 54 trials
and required participants to identify the larger of two simultaneously presented
stimuli. There were three sets of stimuli: dot arrays (matched for surface area),
Arabic digits and horizontal lines. The order in which these stimulus types were
presented was counterbalanced between participants.

Performance of the individuals with Down syndrome was compared to that obtained
by typically developing children in Year 1 and Reception classes matched for receptive
vocabulary level. The children in Year 1 demonstrated typical distance and magnitude
effects in all tasks. The speed with which they made magnitude comparisons using
line and digit stimuli correlated with their performance in the addition tasks (*r*=.
49). This correlation between comparison speed for numeric and physical stimuli
and addition skills suggests numeric representations in this group are underpinned
by analogue magnitudes representations, which in turn support addition skills. The
children with Down syndrome and children in Reception also demonstrated typical
distance and magnitude effects in all tasks, although five children with Down syndrome
had to be excluded from RT analysis of the numeric comparison task because of high
error rates on the task. Although group sizes were too small to make firm interpretations
of the pattern of correlations achieved in these groups, intriguingly, speed in
making magnitude comparisons using dot arrays was the only correlate with addition
performance (reception: *r*=. 72; Down syndrome: *r*=. 69). This pattern
suggests that children with Down syndrome may have typical representations of numerosity
but raises the possibility that the ability to link digit representations to magnitudes
may be immature in children with Down syndrome (as in much younger typically developing
reception year children).

### The development of counting skills in children with Down syndrome

The development of counting has been examined in some detail in individuals with Down syndrome. Counting is an important skill that is often claimed to underpin a number of later mathematic skills (e.g. ref 42) such as children's early attempts at addition. Counting involves not only learning the number words, their sequence and how to tag number words to individual objects, but also requires understanding of the cardinality principle. This refers to the fact that the final count word refers to an exact quantity – the cardinal value or magnitude of the set. The cardinality principle means that the order in which items are tagged is irrelevant to the cardinal value of the set. Understanding of the order-irrelevance principle is used to assess whether children understand the purpose as well as the procedure of counting.

Gelman and Cohen reported the first detailed study of count production and understanding
in ten children with Down syndrome with a mean chronological age of 10:06 years
and mental ages ranging from 3:06 to 6:08years compared with younger typically developing
children broadly matched for social economic status^{[20]}.
All of the children with Down syndrome attended special school. Children were assessed
on rote counting and object counting knowledge as well as a task designed to test
knowledge of the order-irrelevant principle. Children were presented with a line
of objects and asked to count them in a non-linear order. For example, they might
be asked to label the middle object, "the one". The children with Down
syndrome performed better than controls on rote counting and object counting but
worse on the order irrelevance counting task. On this basis, the authors concluded
that the children with Down syndrome performed rote counting with no conceptual
understanding of number. However, the instructions for the order-irrelevant counting
task and the feedback involved very complex language. In contrast, Caycho, Gunn
and Siegel found no difference between 15 children with Down syndrome (mean chronological
age of 9:07 years) and 15 typically developing children (mean age of 4:06 years)
matched for receptive vocabulary level^{[23]},
on a simplified version of the Gelman and Cohen task^{[20]}. In this task
the children presented with a row of items and asked to count them in a non-linear
fashion but their finger was guided to the start item and they were told it was
"one". The language and feedback used in the task were simplified. Caycho
et al. concluded that conceptual understanding of counting is related to receptive
vocabulary levels^{[23]}.

A longitudinal study by Nye investigated performance of a group of children with
Down syndrome and typically developing children matched for non-verbal mental age
on a variety of counting tasks^{[43]}. A striking
similarity was found between the counting skills of the Down syndrome group and
the typically developing group matched for non-verbal mental age, both in terms
of object counting and understanding of cardinality. While counting skills have
not been found to be a particular problem for children with Down syndrome in previous
research, what was particularly surprising here was how these skills developed in
line with non-verbal mental ability (see ref 23). Even
more surprising was the lack of a difference between the Down syndrome and typically
developing groups in terms of their cardinal understanding; this would not have
been predicted from previous research^{[20,23]}.
The only difference between the two matched groups was in count word vocabulary
and sequence production, which were both significantly greater in the typically
developing group, though by no means lacking in the Down syndrome group. However,
any limitations that the children with Down syndrome had in production of the count
word sequence did not seem to impact on their ability to count or give sets of objects,
as evidenced by the lack of a difference between them and the typically developing
children on these tasks. These seemingly positive findings of fledgling number skills
in young children contrast strongly with the poor levels of achievement reported
in older children although success was limited in these studies to very small arrays
of objects (up to 18 items) so these positive results only extend to very basic
skills, typically achieved by pre-school children.

### Conservation

In order to progress to any form of number skills without the use of concrete props,
understanding of the relative value of number and conceptual understanding of the
number system that goes beyond the perceptual characteristics of a given array of
items is essential. Conservation tasks that manipulate the surface characteristics
of an array but keep the underlying value the same are often used to test this in
typical development. The only studies to use traditional conservation tasks in Down
syndrome were conducted by Lister and Lee and Lister, Leach and Riley^{[44,45]}.
They studied number and length understanding in 48 individuals with Down syndrome
between 5 and 26 years. The tasks involved the subject creating or agreeing the
initial equality of two stimuli. One of the stimuli was then transformed and the
individual asked to judge whether the remaining quantities were still equal. None
of the participants succeeded on all the conservation of length tasks, although
five succeeded on all of the number conservation tasks. This is the pattern of development
observed in typically developing children^{[40]}.
No information is provided on the counting ability of the participants. Given the
wide age range of the sample it is likely that this is a significant factor in performance.
Consequently, no clear conclusions can be drawn about the understanding of conservation
in Down syndrome without further research.

## Conclusions

It is clear that children with Down syndrome show severe difficulties in mastering basic number skills as assessed by tasks that include size and numerosity judgements, counting and simple arithmetic. There are suggestions that a pre-verbal "number sense" system may show atypical development in Down syndrome, but so far the group sizes studied preclude strong conclusions. It is clear that learning to count is difficult for children with Down syndrome, though there is no evidence that the development of counting follows a qualitatively different path to that seen in younger typically developing children. It appears that problems in the sphere of arithmetic show strong correlations with language skills in the Down syndrome population though such correlations may in part reflect limitations in children's ability to understand the arithmetic tasks they are required to complete. It would seem that studies of larger samples of children with Down syndrome that assess their pre-verbal number sense skills as well as counting and basic addition skills are badly needed.

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Sophie Brigstocke and Charles Hulme are at the Department of Psychology, University of York, UK

Joanna Nye is at the Department of Experimental Psychology, University of Bristol, UK

Correspondence to Sophie Brigstocke • e-mail: s.brigstocke@psych.york.ac.uk

Paper prepared from presentations and discussions at the Down Syndrome Research Directions Symposium 2007, Portsmouth, UK. The symposium was hosted by Down Syndrome Education International in association with the Anna and John J Sie Foundation, Denver. Major sponsors also included the Down Syndrome Foundation of Orange County, California and the National Down Syndrome Society of the USA. Information about the symposium can be found at http://www.dseinternational.org/research-directions/

doi: 10.3104/reviews.2070

Received: 14 January 2008; Accepted: 21 January 2008; Published online: 2 July 2008

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