Numerical ability, general ability and language in children with Down syndrome
Joanna Nye, John Clibbens and Gillian Bird
The aims of this study were to investigate the relationship between numerical and general ability and the contribution that receptive language makes to numerical ability in children with Down syndrome. Sixteen children with Down syndrome were tested on the following measures: two nonstandardised tests of numerical ability, two standardised numerical tests, two measures of receptive language and an IQ scale. Only one of the sixteen children attained a score on the IQ measure so the relationship between general and numerical ability in this population could not be assessed. All four numerical measures significantly correlated (positively) with each other, and receptive grammar (but not vocabulary) was found to significantly correlate (positively) to numerical skills. Details of the children's performance on the two main numerical measures under investigation are presented.
Nye J, Clibbens J, Bird G. Numerical ability, general ability and language in children with Down syndrome. Down Syndrome Research and Practice. 1995;3(3);92-102.
The study reported here investigated the numerical ability of children with
Down syndrome, and the relationship between numerical ability and general
ability in these children. The contribution of receptive language to numerical
skills was also examined.
Numerical Processes in the Typical Population
The study of numeracy covers a wide range of abilities making use of a variety
of experimental methods and populations (see
for review). Areas of research include investigation into the adult representations
of numerosity, formal calculation processes and the role of notation within
these two areas. The developmental processes that lead to a fully operational
numerical system have also been extensively researched. In these fields
there are many unresolved issues.
One of the main controversies in this area is how much numerical processes
are influenced by general cognition, or whether they are discrete modules
Shipley and Shepperson, 1990;
Gallistel and Gelman, 1990). A wider debate also exists about the nature
of general cognitive ability and its relationship to specific abilities
(see Anderson, 1992).
Gelman and Gallistel (1978) assert that the development of children's counting
is guided by domain-specific principles, whereas
Shipley and Shepperson
(1990) argue that the general ability to process discrete physical objects
enables them to identify what to count (preceding the acquisition of counting
principles) and also operates in some aspects of language acquisition. This
work has been enhanced by studies of patients with brain-lesions (e.g.
particularly supporting conceptions of numerosity as a set of modular processes
rather than as a unified concept. Various models have been developed to
describe how these different abilities interact (e.g.
Calculation is intuitively seen to be linked with language processes, because
it makes use of the ability to mentally manipulate sequences of symbols.
Consequently numeracy has sometimes been thought to be part of the language
faculty with no need to be seen as separate (e.g.
It would appear however that there are some numerical abilities that do
not use mental representations based on language or numerical notation (Dehaene,
1992). This approach should not be taken to mean that the role of linguistic
processes has been totally disregarded: as Dehaene's model describes, numeracy
involves a complex set of abilities, some of which are based on the mental
manipulation of symbols (e.g. mental arithmetic) and some of which are not
(e.g. representations of numerosity).
This debate forms part of a more general controversy over whether language
is a distinct system or the product of general learning or cognitive processes.
There is strong evidence that language is modular and that it can be further
divided into sub-modules (Clibbens,
One of the essential elements of numeracy is the ability to quantify sets
of data. It has been suggested that three main processes are used in quantification
which are counting, subitization and estimation (Klahr
and Wallace, 1973). Counting is the most well established concept, being
best defined by the five principles of
Gelman and Gallistel (1978). The
existence of these principles is well supported and a great amount of research
has focused on their development. Despite this a controversy remains over
whether a skeletal set of these principles is innate (Gelman
and Gallistel, 1978), guiding the acquisition of counting behaviours,
or whether the principles are progressively extracted after repeated practice
with rote learning procedures (e.g.
and Shipley, 1987).
Subitization is the process whereby very small sets (below 3 or 4) are quantified
immediately and estimation is used to quantify a large number of items.
The exact processes involved have not been precisely defined (Dehaene,
1992) and it is disputed whether they actually exist separately from
counting as defined by Gallistel and Gelman (1992).
A manifestation of the general ability view of numerical skills is the theory
proposed by Piaget (1952). Piaget argued that children will develop mathematical
concepts in their own time. Central to these concepts is number conservation,
the understanding of which is thought to precede all arithmetical operations.
This theory has been influential in the structuring of educational provision
but has been criticised by researchers such as
Donaldson (1978) and
and Gallistel (1978). The main criticism has been that children are able
to demonstrate understanding of conservation at younger ages than Piaget
claimed, if appropriate methodologies are used.
Alternative methodologies (e.g. the 'Magic' paradigm used by
Gelman and Gallistel, 1978) provide indirect evidence that three- or
four-year-old children can also understand addition and subtraction. For
Piaget understanding of these transformations does not occur until six or
seven years. Support for Gelman's claims comes from
Hughes (1981) who provided
more direct evidence of such understanding, as long as the situation presented
to the children was based in some sort of context.
The Cognitive Development of Children with Down Syndrome
Down syndrome is a chromosomal disorder, which causes a specific pattern
of physical and developmental characteristics. It is now generally accepted
that all children born with Down syndrome will have learning disabilities,
but the severity of the disability will vary between individuals (Carr,
One of the main debates surrounding cognitive development in Down syndrome
is whether it progresses through the same stages as typical development
but at a slower pace or whether it is qualitatively different (Lewis,
1987). It may be that some abilities develop in a different way to that
of typically developing children, while others emerge at a slower rate.
Support for this view comes from a review of the literature on language
skills in Down syndrome (Rondal,
1987). A further possibility is that milestones are reached in the same
order as typically developing children, but that the underlying developmental
processes are different (e.g. the 'developmental difference' theory proposed
by Morss, 1985).
Each of these positions has implications for educational provision and for
theoretical issues in the wider study of child development.
Numerical Ability in Children with Down Syndrome
Down syndrome has been the focus of much research, partly because the syndrome
is identifiable from birth and also because Down syndrome is the largest
sub-division of the learning disabilities (Lewis,
1987). One area of study that has been largely neglected in this population
is numerical ability (Bird
and Buckley, 1994). The following is a review of the only studies found
which investigate the numerical abilities of children with Down syndrome.
There is a basic understanding of the sorts of numerical abilities that
the children can be expected to achieve, with present educational programmes.
There are also some beginnings into investigating the numerical processes
that the children are using but it is clear that there is a need for further
investigation with this group.
Cornwell (1974) concluded that children with Down syndrome learn to count
using rote procedures and do not develop understanding of arithmetical concepts.
This finding was used to illustrate a general difficulty in concept formation
and abstraction. Gelman and Cohen (1988) made use of the conclusion that
children with Down syndrome only count using rote learning to investigate
the controversy about whether typically developing children are guided by
counting principles (Gelman
and Gallistel, 1978) or gradually extract the principles after using
rote counting procedures (e.g.
and Shipley, 1987). They found that generally the children with Down
syndrome were not as able as mental age matched pre-schoolers in solving
a novel counting problem. Due to the general difference between the two
populations the results supported the argument that typically developing
children make use of counting principles. However two of the children with
Down syndrome were found by Gelman and Cohen to be 'excellent counters'
who were able to make use of principled counting, but most of the children
did not display such ability. They stress that we cannot conclude from these
results that children with Down syndrome cannot take advantage of counting
principles and that with appropriate intervention this may be possible.
Caycho, Gunn and Siegal (1991) matched children with Down syndrome with
typically developing children of a similar 'developmental level' (using
the Peabody Picture Vocabulary Test - Revised), rather than using mental
age for matching. There was no significant difference between the two groups
in performance on tasks testing the counting principles. Therefore, it was
concluded that children with Down syndrome can make use of counting principles
and their counting competence is related to receptive language not to the
syndrome itself. Again it was stressed that the ability to take advantage
of the counting principles may be a function of the type of educational
programme the children are engaged in, and that it must not be assumed that
all children will develop such understanding, just that they are capable
of doing so.
Surveys of the children's abilities have also been carried out (e.g.
and Sacks, 1987; Carr,
1988; Irwin, 1989).
These have described the levels of attainment that the particular cohort
of children reached at the time but do not give detailed insight into the
processes that the children are using or techniques that are useful for
developing numerical skills.
Intervention studies that have been tried with children with Down syndrome
have also been reported. Irwin (1991) found that children with Down syndrome
who could 'count-all' were able to master 'counting-on' (counting-on from
the largest addend) for addition, using a structured teaching technique.
The children were able to generalise the skill to materials not used for
teaching. The Macquarie programme (Thorley
and Woods, 1979;
and Treloar, 1981) was a teaching programme based on the work of
and Gallistel (1978) which reported a variety of results with a small group
of children. Two children mastered all the objectives of the programme,
with one developing further skills. The remaining children mastered between
1 and 16 objectives, out of 38.
One issue that has been studied is how useful estimates of general ability
and labels such as Down syndrome or learning disability are in predicting
numerical ability. Two studies have provided contradictory evidence about
this (Baroody, 1986;
Sloper, Cunningham, Turner and Knussen, 1990).
Sloper et al (1990) found a significant correlation between mental age and
numerical ability (r = 0.73). They investigated the academic achievement,
including numerical ability, of 117 children with Down syndrome (aged 6
to 14 years). Academic attainment was measured by questionnaires (filled
in by the children's teachers) which assessed a range of numerical competencies
from "Discriminates between largest and smallest groups of objects" to "Does
simple division work" (see Table 2 for complete set of items). The children
displayed a wide range of abilities. Overall educational attainment was
found to correlate with mental age and type of school attended, with mainstream
schools being connected with higher educational attainments. This was supported
by Casey, Jones, Kugler and Watkins (1988) who found similar benefits for
children with Down syndrome attending mainstream schools, including in numerical
Baroody (1986) tested 100 'mentally handicapped' children (though not specifically
Down syndrome) using a series of games investigating numerical skills which
children are expected to have when entering school in the US. These include
tests of Gelman and Gallistel's five counting principles.
wide individual differences, with some children with lower IQs (33-49) in
the younger age group (6 to 10 years) performing at a higher level than
some of the children with higher IQs (51-80) in the older age group (11
to 14 years). Baroody concluded that labels such as IQ are not useful in
predicting levels of numerical ability. The conclusions that these two studies
make regarding the use of knowledge about general abilities for prediction
of numerical skills are starkly different, and the reasons for this difference
must be looked at more carefully.
One of the most prominent differences between the two studies is the population
examined. Although this may seem an obvious reason why the two studies have
different results, the predictions from either could be used to inform intervention
and theory about children with Down syndrome. A further (possible) difference
between the studies is the general ability measure used.
made use of IQ scores but the exact tests used to obtain these are not described.
Indeed it is not clear whether the children were even tested using the same
measure. To obtain mental age scores,
Sloper et al. (1990) used the McCarthy
Scales of Children's Abilities (McCarthy,
1972) and for children who did not attain a score on the McCarthy, the
Bayley Scale of Infant Development (Bayley,
1969). In addition the two studies cover a very different range of numerical
ability. The Baroody (1986) test only covers the skills that a typically
developing child would have on entering school in the US (aged six years)
Sloper et al (1990) checklist ranges from early skills up to arithmetic.
The current study was designed to test the hypothesis that the scores attained
by children with Down syndrome on the test used by
Baroody (1986) and the
scale used by
Sloper et al (1990) will positively correlate with each other,
both being claimed to be measures of numerical ability. Two standardised
tests of numerical ability from the British Ability Scales (Elliott,
1987) and the Kaufman Assessment Battery for Children (Kaufman
and Kaufman, 1983) were also used to test the validity of the two unstandardised
tests. It was predicted that if the Baroody and Sloper scales do indeed
measure numerical ability they should both positively correlate with existing
In addition an IQ measure (the British Ability Scales) was employed to test
the hypothesis that there would be a positive correlation between IQ and
numerical ability, as tested by the four different measures. From this the
previous conclusions that
Sloper et al (1990) and
Baroody (1986) reached
were investigated further using the same benchmark of general ability. Conclusions
about the relationship between numerical ability and 'general ability' may
not necessarily be taken further as debate surrounds what general cognitive
ability consists of and whether it is measured by IQ (Anderson,
To investigate the contributions of language to numerical ability, because
of the conclusions drawn by
Caycho, Gunn and Siegal (1991), the children
were also tested on the British Picture Vocabulary Test (Dunn,
Dunn, Whetton and Pintilie, 1982; the British standardised version of
the Peabody Picture Vocabulary Test). To extend the investigation of the
contribution of language skills further the Test for Reception of Grammar
was also used. Chronological age, schooling history and gender were also
collected for analysis. (Females with Down syndrome have been found to be
more advanced in general development and have higher IQ scores than males
(Carr, 1985) so it
may be a relevant variable for numerical skills.) The hypothesis that receptive
language, chronological age, schooling history and gender all significantly
correlate with numerical ability was tested (using the
Sloper et al (1990)
checklist and Baroody (1986) tasks for the measures of numerical ability).
The study was of a within subjects/correlational design, with all participants
being tested on the same battery of standardised and unstandardised tests.
The order of presentation of the battery of tests varied between subjects
but this was not manipulated systematically.
The 16 children who participated in the study all had Down syndrome. Eight
were female, eight male. They were aged between 7;0 and 12;6 (mean age:
8;9) at the time of testing. All attended mainstream primary schools.
The children involved in the current study were already being studied as
part of a research project conducted by staff at the Sarah Duffen Centre,
University of Portsmouth. The current research fell under the informed consent
that the parents and teachers had already given. Supplementary information
about the project was provided during individual visits. In addition it
was made clear that the child concerned did not have to take part in any
of the tests that the parents or teachers were not happy with (maintaining
the ethical standards for informed consent) and that all results would be
Each child was tested in a quiet area within their school, at the Sarah
Duffen Centre (a research centre with facilities for assessing children)
or at the child's home. The first author administered all the numerical
tests. The remaining tests were randomly administered either by this same
experimenter or one other experimenter as part of the wider study. The battery
of tests was administered to each child over several sessions, across one
or more days, varying with individual children's levels of fatigue and attention.
If any child displayed discomfort or unwillingness to perform any of the
tasks, testing on an individual task was discontinued.
Baroody Numerical Test
A series of tests based on those used by Baroody (1986) to identify basic
counting skills was used with the children. (From here on this group of
tests will be referred to as the 'Baroody Numerical Test'.) Rather than
using the criterion levels for each item that Baroody used to compare groups,
a continuous scale of scores was implemented to aid correlational analysis.
Therefore, each child could score between 0 and 136 points on this test.
Where necessary, test item materials were constructed from copyable teaching
workbooks. The set of tests were:
Oral Counting - the child is asked to count up to 40, both with and
Counting by Tens - counting 10 counters marked as 10p pieces.
Enumeration and Production of Objects - in the context of a shopping
game the child is asked to count sets of objects and count out specified
numbers of objects.
Cardinality Rule - a set of objects is counted in front of the child,
the display is covered and the child is asked how many objects have
Order-Irrelevance Principle - counting sets of blocks in various directions.
Finger Representations to 10 - ability to hold up a specified number
of fingers quickly.
Equivalence - this test involves the child identifying a picture from
a set of three, which matches a stimulus picture.
Class teachers and/or classroom assistants were asked to complete a rating
scale of the child's numerical ability, as constructed by
Sloper et al (1990)
which was based on a checklist devised by Lorenz (1985). This checklist
comprises 22 levels starting from "discriminates between largest and smallest
group of objects" to "does simple division sums" followed by "can do more
advanced number work. Please specify . . ." (see Table 2
for complete set
of items). Each item on the checklist is scored as "Can't do" (score = 1);
"Can do with help" (score = 2); or "Can do" (score = 3), with a possible
range of scores between 22 and 66. The checklist was presented to the teaching
staff with a front cover, which included instructions for completion of
the checklist, spaces for recording education details about the child and
who the form was completed by.
The chronological age of the child and amount of time they had spent in
mainstream schooling was collected from the class-teachers or classroom
assistants in conjunction with the 'Teacher Questionnaire'.
The following standardised tests were carried out with the children:
British Ability Scales (Elliott,
1987) - various sub-scales. These included the numerical ability
scale and the scales required to calculate the short-form IQ. For children
aged 5;0 to 7;11 years these are: recall of digits, similarities, matrices
and naming vocabulary. For children aged 8;0 to 17;5 years these are:
recall of digits, similarities, matrices and speed of information processing;
Kaufman Assessment Battery for Children (Kaufman
and Kaufman, 1983) arithmetic scale;
British Picture Vocabulary Scales (Dunn,
Dunn, Whelton & Pentillie, 1982) which measures receptive vocabulary;
Test for the Reception of Grammar or TROG (Bishop,
Table 1: Descriptive statistics of scores on the Teacher Questionnaire
from the current study and
Sloper et al (1990).
||Sloper et al study
Complete details of all the children's schooling histories were not readily
available but the information that was obtained from class teachers and
special needs assistants indicated that the children had highly individual
patterns of education. Because of the complex nature of this variable it
was not used in the statistical analyses, but a summary is given here to
describe the population studied.
Figure 1. Distribution of scores on Teacher Questionnaire.
Twelve of the children had attended mainstream schools for the whole of
their education so far, while two had transferred from special schools at
some point. Two of the children had attended mainstream school part-time
for a period, with one of these also attending a special school part-time.
Further information about where the second child spent the remainder of
the week, whether at home or at another school, was not available. The length
of time each child had spent in a particular type of education varied from
one to five years. One child had been kept back to repeat a year. Details
of pre-school groups or interventions were not readily available.
Figure 2. Distribution of scores on Baroody Numerical Test.
The distribution of the scores on the Teacher Questionnaire is presented
in Figure 1. Table 1 shows the mean, standard deviation, minimum and maximum
scores obtained. The table also shows the results found by
Turner and Knussen (1990) using the same measure in their study of 117 children
with Down syndrome. A higher mean score and a lower standard deviation can
be observed in the current study. (These differences have not been tested
for significance.) Differences between the two samples of children with
Down syndrome should be noted. The sample of 16 children in the current
study had a mean age of 8;9 years (range 7;0 to 12;6) and the sample of
117 children in
Sloper et al. (1990) had a mean age of 9;2 years (range
6;0 to 14;0). All the children in the current study attended mainstream
schools. In the
Sloper et al. (1990) study the children attended a variety
of schools, including mainstream and those for children with severe and
moderate learning disabilities.
Table 2. Results form the Teacher Questionnaire (Sloper et al., 1990)
To look at abilities of individual children the full results of the Teacher
Questionnaire are presented in Table 2, with the children in order of chronological
age. Wide variation can be seen in the pattern of abilities, with no simple
pattern of progression with age across the group. For the majority of the
children skills up to Item 8 (writes symbols, 0 to 9) have been acquired
or can be performed with help. From Item 9 (writes symbols, 10 to 20) the
majority of the children either cannot perform the skill or can only do
so with help. This is even more pronounced after Item 13 (subtracts from
written number up to 9, without materials).
This is a very general description of the performance of this group, and
it is clear from Table 2 that there are wide individual differences in performance.
The main exceptions to this description are children 1, 15 and 16; one of
the youngest and the two oldest children. Child 1 is able to perform the
first six skills on the questionnaire with help from an adult. The two oldest
children can do, or can do with help, items across the whole range of the
questionnaire, being unable to do only three or four of the items.
Table 3: Results from the Baroody Numerical Test.
A significant positive correlation was found between the Teacher Questionnaire
and the Baroody Numerical Test. This indicates that they measure overlapping
sets of numerical skills. They both significantly correlate (positively)
with the two standardised measures of numerical ability (the British Ability
Scales and the Kaufman) and this supports the construct validity of the
nonstandardised tests as measures of numerical ability. This conclusion
should be made with caution as the tests have not been tested for reliability
(a prerequisite for validity) and have only been tested on a limited sample
in this study.
As the children (apart from one) did not obtain IQ scores on the British
Ability Scales, the hypothesis that there would be a positive correlation
between the numerical test scores and the IQ scores could not be tested.
It would therefore be useful to replicate this study with a general ability
test that was more appropriate for the population studied, in an attempt
to resolve the conflicting conclusions of
Sloper et al (1990) and Baroody
(1984). Exactly what general ability consists of and how it can be measured
is a controversial issue (Anderson,
1992). Research by
Shipley and Shepperson (1990) suggests that the general
ability to process discrete objects guides children's counting behaviours
and they draw attention to literature that indicates that this ability also
aids vocabulary acquisition.
Table 4. Results form the Teacher Questionnaire (Sloper et al., 1990)
To investigate the relationships between numerical skills and other variables
further, the hypothesis that receptive vocabulary, receptive grammar, chronological
age, school history and gender would all significantly correlate with numerical
ability was tested. The details of the children's educational background
were complex and consequently were not used for analysis. All of the children
were attending mainstream school at the time of testing, but their school
histories leading up to this point were highly individual. The finding that
gender does not correlate with the numerical ability scores is inconsistent
with previous evidence (Carr,
Caycho, Gunn and Siegal (1991) studying children with and without Down syndrome,
found that use of the counting principles was related to receptive vocabulary
rather than mental age. In the current study small positive correlations
exist between the four numerical scales and the BPVS, but these are not
The numerical skills that
Caycho, Gunn and Siegal (1991) were investigating
were the use of Gelman and
Gallistel's (1978) counting principles, whereas
the numerical scales in the current study cover a wider range of numerical
skills. The counting principles are only explicitly tested by a sub-set
of items in the Baroody Numerical Test which has the smallest correlation
coefficient with the vocabulary scale out of the four numerical tests. The
current study, not having looked specifically at the counting principles,
cannot provide evidence to either support or refute the claims of
et al (1991).
In contrast to the relationships between receptive vocabulary and the other
variables, receptive grammar (as measured by the Test for Reception of Grammar;
Bishop, 1983) was found to correlate with all numerical measures, as well
as chronological age and receptive vocabulary. That receptive vocabulary
and grammar differ in their correlation with numerical skills, may be due
to different numerical skills being related to different aspects of language,
as suggested by Dehaene (1992). From the current study numerical ability
has a greater relationship with receptive grammar than vocabulary knowledge
in this group of children.
The nature of the relationships found between numerical ability and aspects
of language is relevant to the discussion about
Dehaene's (1992) model of
numerical processes, in which he supports the study of different populations
for the development of a comprehensive theory. Dehaene provides convincing
evidence that numerical processes need to be seen as separate from the language
faculty, which is the position supported by
Hurford (1987). Dehaene describes
how some numerical processes (those using verbal numerical notation) are
based on linguistic functioning but that some are completely independent
(e.g. representations of numerosity). The current research supports the
ideas implicit in Dehaene's work that different numerical competencies are
related to different sub-modules of language (e.g. lexical and syntactic
Baroody's description of the skills of typically developing children entering
school in the US at age 6 does not transfer to the British education system,
either for age or for policy of meeting children's special needs at school.
It would be informative to conduct a study of typically developing children
at infant schools in the UK to make more meaningful comparisons about the
skills achieved by the children with Down syndrome in the current study.
The group of children studied here do not show a steady progression of numerical
skills with chronological age (from the
Sloper et al. (1990) questionnaire).
It is likely that this is due to individual differences. A major factor
in this is likely to be the numeracy training that the children receive,
which up to now has been given little attention (Bird
and Buckley, 1994). Improvements in the teaching of reading to children
with Down syndrome have improved performance to levels that were once thought
and Buckley, 1994) suggesting that it is worth investigating intervention
techniques which could enhance numerical skills. Therefore further research
needs to carried out into teaching numerical ability, both in schools and
in pre-school programmes. In addition, research into the underlying numerical
processes and acquisition of these in this population is also required.
Using the label of Down syndrome in isolation has been thought to have little
use in determining educational provision including for numeracy and arithmetic
and Buckley, 1994). Preconceived ideas about limits on performance are
thought to be highly restrictive, causing self-fulfilling prophecies. Baroody's
research into children with learning disabilities supports this. As the
current study could not test the hypothesis that general ability is related
to numerical ability, nothing further can be added to the debate about the
usefulness of general ability for prediction. Although measures of receptive
language could be used to give some indication of the level that numerical
skills teaching should be aimed at, it would be far more appropriate to
use measures of numerical skills with the individual child for planning
Caution should be taken in interpreting the results found here because of
the small sample studied. With this population it is often necessary to
study small populations and for this reason it may be more appropriate for
future studies to focus on more detailed analysis of patterns of performance,
including errors. For example, Baroody (1986) was able to study the errors
that the children made on the numerical tasks.
Gelman and Cohen (1988) made
detailed analysis of the patterns of responses that the children gave to
novel counting problems. One of the problems with the current study was
the difficulty of recording all the children's responses accurately, while
administering all the test items. A solution to this would be to use video
recording equipment, enabling detailed analysis of error patterns. Such
analysis would also complement research using statistical analysis with
larger samples such as the one carried out by
Sloper et al
(1990) who studied
117 children with Down syndrome.
In children with Down syndrome all four measures of numerical ability are
useful for assessing numerical ability. The four tests have different characteristics
which could be taken advantage of in different situations. The two standardised
tests (from the British Ability Scales and the Kaufman Assessment Battery
for Children) can be used by psychologists where appropriate and for calculation
of standardised scores if they are required. For professionals, such as
teachers, who do not have training/access to use standardised tests both
the Sloper Questionnaire and the Baroody Numerical Test can be used to assess
a range of numerical skills. The Teacher Questionnaire is particularly useful
for accessing the knowledge that teachers and assistants have about a child
that may not perform to their best ability in a simple test situation. It
also enables the collection of a large amount of data, as in the
et al (1990) study. The Baroody Numerical Test is designed to make the test
situation child-centred, with most of the tasks being built into a game.
It is a useful test for a detailed assessment of counting skills. In addition
many of the tasks can be adapted to become training tasks (Baroody,
No conclusions could be drawn from the present study about the relationship
between numerical ability and general ability. Receptive grammar was found
to correlate significantly with numerical ability.
More work is required to determine how these skills and underlying processes
are developing and how remediation programmes can be best designed to aid
the development of numerical ability in children with Down syndrome.
This research was conducted as part of a B.Sc. Psychology (Hons.) degree
at the University of Plymouth.
Many thanks to all the children who took part in this study and to the head
teachers, class teachers and parents who allowed me the pleasure of working
with their children.
I am deeply indebted to the staff at the Sarah Duffen Centre, particularly
Sue Buckley, Gillian Bird and Angela Byrne for their support while setting
up the project and collecting my data.
- Anderson, M. (1992). Intelligence
and Development: A Cognitive Theory. Oxford: Blackwell.
- Baroody, A.J. (1986). Counting ability
of moderately and mildly handicapped children. Education and Training
of the Mentally Retarded, 21, 289-300.
- Baroody, A.J. (1992). Remedying common
counting difficulties. In J. Bideaud, C. Meljac and J.-P. Fischer (eds.),
Pathways to Number: Children's Developing Numerical Abilities.
Hillsdale, NJ: Lawrence Erlbaum Associates.
- Bayley, N. (1969). Manual for the
Bayley Scales of Infant Development. New York: The Psychological Corporation.
- Bird, G. and Buckley, S. (1994).
Meeting the Educational Needs of Children with Down Syndrome. Portsmouth:
University of Portsmouth. Available Online:
- Bishop, D.V.M. (1983). Test
for the Reception of Grammar (TROG). University of Newcastle-upon-Tyne.
- Buckley, S. and Sacks, B.
(1987). The Adolescent with Down Syndrome. Portsmouth: Portsmouth
- Carr, J. (1985). The development of intelligence.
In D. Lane and B. Stratford (eds.), Current Approaches to Down Syndrome.
London: Holt, Rinehart and Winston.
- Carr, J. (1988). Six weeks to twenty-one
years old: A longitudinal study of children with Down syndrome and their
families. Journal of Child Psychology and Psychiatry, 29 (4), 407-431.
W., Jones, D., Kugler, B. and Watkins, B. (1988). Integration of Down
syndrome children in the primary school: A longitudinal study of cognitive
development and academic attainments. British Journal of Educational
Psychology, 58, 279-286.
- Caycho, L., Gunn,
P. and Siegal, M. (1991). Counting by children with Down syndrome. American
Journal on Mental Retardation, 95 (5), 575-583.
- Clibbens, J. (1993). From theory to
practice in child language development. Down Syndrome: Research and
Practice, 1 (3), 101-106.
[Open Access Full
- Cornwell, A.C. (1974). Development
of language, abstraction, and numerical concept formation in Down syndrome
children. American Journal of Mental Deficiency, 79 (2), 179-190.
- Dehaene, S. (1992). Varieties of numerical
abilities. Cognition, 44, 1-42.
- Donaldson, M. (1978). Children's
Minds. London: Fontana.
L.M., Dunn, L.M., Whetton, C. and Pintilie, D. (1982). The British
Picture Vocabulary Scale. Windsor: NFER-Nelson.
- Elliott, C.D. (1987). British
Ability Scales. Windsor: NFER-Nelson.
- Fuson, K.C. (1988). Children's
Counting and Concepts of Number. New York: Springer-Verlag.
- Gallistel, C.R. and
Gelman, R. (1990). The what and how of counting. Cognition,
- Gallistel, C.R. and
Gelman, R. (1992). Preverbal and verbal counting and computation.
Cognition, 44, 43-74.
- Gelman, R. and Cohen, M. (1988).
Qualitative differences in the way Down syndrome and normal children solve
a novel counting problem. In L.Nadel (ed.), The Psychobiology of Down
Syndrome. Cambridge, MA: MIT Press.
- Gelman, R. and Gallistel,
C.R. (1978). The Child's Understanding of Number. Cambridge,
MA: Harvard University Press.
- Hughes, M. (1981). Can pre-school children
add and subtract? Educational Psychology, 1, 207-219.
- Hurford, J.R. (1987). Language
and Number. Oxford: Basil Blackwell.
- Irwin, K.C. (1989). The school achievement
of children with Down syndrome. New Zealand Medical Journal, 102,
- Irwin, K.C. (1991). Teaching children
with Down syndrome to add by counting-on. Education and Treatment of
Children, 14 (2), 128-141.
- Kaufman, A.S. and Kaufman,
N.L. (1983). Kaufman Assessment Battery for Children. Minnesota:
American Guidance Service.
- Klahr, D. and Wallace,
J.G. (1973). The role of quantification operators in the development
of conservation. Cognitive Psychology, 4, 301-327.
- Lewis, V. (1987). Development and
Handicap. Oxford: Blackwell.
- McCarthy, D. (1972). Manual for
the McCarthy Scales of Children's Abilities. New York: The Psychological
- McCloskey, M. (1992). Cognitive mechanisms
in numerical processing: Evidence from acquired dyscalculia. Cognition,
- Morss, J.R. (1985). Early Cognitive
Development: Difference or Delay? In D. Lane and B. Stratford (eds.),
Current Approaches to Down Syndrome. London: Holt, Rinehart and Winston.
- Piaget, J. (1952). The Child's Conception
of Number. London: Routledge & Kegan Paul.
- Pieterse, M. and Treloar,
R. (1981). The Macquarie Program for Developmentally Delayed Children:
Progress Report 1981. Macquarie University, Australia.
- Rondal, J.A. (1987). Language development
and mental retardation. In W. Yule and M. Rutter (eds.), Language Development
and Disorders. Oxford: Blackwell.
- Shipley, E.F. and
Shepperdson, B. (1990). Countable entities: Developmental changes.
Cognition, 34, 109-136.
- Siegler, R. and Shipley,
C. (1987). The role of learning in children's strategy choices. In L.
S Liben (ed.), Development and Learning: Conflict or Congruence?
Hillsdale, NJ: Lawrence Erlbaum Associates.
Sloper, P., Cunningham, C., Turner, S. and Knussen, C. (1990). Factors
relating to the academic attainments of children with Down syndrome. British Journal of Educational Psychology, 60, 284-298.
- Thorley, B.S. and Woods,
V.M. (1979). Early number experiences for pre-school Down syndrome children.
Australian Journal of Early Childhood, 4, 15-20.
See and Learn Numbers is designed to help parents and educators teach children with Down syndrome basic number skills and concepts.
See and Learn Numbers is designed to teach young children to count, to link numbers to quantity, to understand important concepts about the number system and to calculate with numbers up to 10.
Now available as teaching kits and apps. Find out more...