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Dyscalculia and Mathematical Difficulties: Implications
for Transition to Higher Education in the Republic of
Ireland
A paper presented at the Disability Service Symposium Trinity College Dublin,
June 2010
Alison Doyle
Disability Service
University of Dublin Trinity College
Disability Service,
Room 2054, Arts Building,
Trinity College,
Dublin 2, Ireland
Seirbhís do dhaoine faoí mhíchumas,
Seomra 2054,
Foígneamh na nEalaíon
Coláiste na Tríonóide,
Baile Átha Cliath 2, Éire
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Contents
Abstract ............................................................................................................ 3
Section 1: Literature review .............................................................................. 4
1.1 Introduction ............................................................................................. 4
1.2 Aetiology ................................................................................................. 6
Biological .................................................................................................. 8
Cognitive ................................................................................................. 11
Behavioural ............................................................................................. 15
Environmental ......................................................................................... 19
1.3 Assessment .......................................................................................... 19
Assessment instruments ......................................................................... 20
Working memory as an assessment device ............................................ 21
Computerized assessment ..................................................................... 23
1.4 Incidence .............................................................................................. 24
1.5 Intervention ........................................................................................... 25
Section 2: Accessing the curriculum .............................................................. 30
2.1 Primary schools programme ................................................................. 30
2.2 Secondary programme ......................................................................... 31
2.3 Intervention ........................................................................................... 33
Section 3: Transition to third level .................................................................. 35
3.1 Performance in State Examinations ..................................................... 35
3.2 Access through Disability Access Route to Education process ............ 42
3.3 Implications for transition to higher education ...................................... 44
3.4 Mathematics support in higher education ............................................. 46
Section 4: Summary ....................................................................................... 48
4.1 Discussion ............................................................................................ 48
4.2 Further research ................................................................................... 50
References and Bibliography ......................................................................... 52
APPENDICES ................................................................................................ 60
Leaving Certificate Statistics 2001 – 2009 .................................................. 60
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Abstract
This paper examines the neurological, cognitive and environmental features of
dyscalculia, which is a specific learning difficulty in the area of processing
numerical concepts. A review of the literature around the aetiology of
dyscalculia, methods for assessment and diagnosis, global incidence of this
condition and prevalence and type of intervention programmes is included.
In addition, the nature of dyscalculia is investigated within the Irish context,
with respect to:
the structure of the Mathematics curriculum
access to learning support
equality of access to the Mathematics curriculum
reasonable accommodations and state examinations
implications for transition to higher education
Finally, provision of Mathematics support in third level institutions is discussed
in order to highlight aspects of best practice which might usefully be applied to
other educational contexts.
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Section 1: Literature review
1.1 Introduction
Mathematical skills are fundamental to independent living in a numerate
society, affecting educational opportunities, employment opportunities and
thus socio-economic status. An understanding of how concepts of numeracy
develop, and the manifestation of difficulties in the acquisition of such
concepts and skills, is imperative. The term Dyscalculia is derived from the
Greek root ‘dys’ (difficulty) and Latin ‘calculia’ from the root word calculus - a
small stone or pebble used for calculation. Essentially it describes a difficulty
with numbers which can be a developmental cognitive condition, or an
acquired difficulty as a result of brain injury.
Dyscalculia is a specific learning difficulty that has also been referred to as
‘number blindness’, in much the same way as dyslexia was once described as
‘word blindness’. According to Butterworth (2003) a range of descriptive terms
have been used, such as ‘developmental dyscalculia’, ‘mathematical
disability’, ‘arithmetic learning disability’, ‘number fact disorder’ and
‘psychological difficulties in Mathematics’.
The Diagnostic and Statistical Manual of Mental Disorders, fourth
edition (DSM-IV) and the International Classification of Diseases (ICD)
describe the diagnostic criteria for difficulty with Mathematics as follows:
DSM-IV 315.1 ‘Mathematics Disorder’
Students with a Mathematics disorder have problems with their math
skills. Their math skills are significantly below normal considering the
student’s age, intelligence, and education.
As measured by a standardized test that is given individually, the
person's mathematical ability is substantially less than you would
expect considering age, intelligence and education. This deficiency
materially impedes academic achievement or daily living. If there is also
a sensory defect, the Mathematics deficiency is worse than you would
expect with it. Associated Features:
Conduct disorder
Attention deficit disorder
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Depression
Other Learning Disorders
Differential Diagnosis: Some disorders have similar or even the same
symptoms. The clinician, therefore, in his/her diagnostic attempt, has to
differentiate against the following disorders which need to be ruled out
to establish a precise diagnosis.
WHO ICD 10 F81.2 ‘Specific disorder of arithmetical skills’
Involves a specific impairment in arithmetical skills that is not solely
explicable on the basis of general mental retardation or of inadequate
schooling. The deficit concerns mastery of basic computational skills of
addition, subtraction, multiplication, and division rather than of the more
abstract mathematical skills involved in algebra, trigonometry,
geometry, or calculus.
However, it could be argued that the breadth of such a definition does not
account for differences in exposure to inadequate teaching methods and / or
disruptions in education as a consequence of changes in school, quality of
educational provision by geographical area, school attendance or continuity of
teaching staff. A more helpful definition is given by the Department for
Education and Skills (DfES, 2001):
‘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.’
Blackburn (2003) provides an intensely personal and detailed description of
the dyscalculic experience, beginning her article:
“For as long as I can remember, numbers have not been my friend.
Words are easy as there can be only so many permutations of letters to
make sense. Words do not suddenly divide, fractionalise, have
remainders or turn into complete gibberish because if they do, they are
gibberish. Even treating numbers like words doesn’t work because they
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make even less sense. Of course numbers have sequences and
patterns but I can’t see them. Numbers are slippery.”
Public understanding and acknowledgement of dyscalculia arguably is at a
level that is somewhat similar to views on dyslexia 20 years ago. Therefore,
the difference between being ‘not good at Mathematics’ or ‘Mathematics
anxiety’ and having a pervasive and lifelong difficulty with all aspects of
numeracy, needs to be more widely discussed. The term specific learning
difficulties describes a spectrum of ‘disorders’, of which dyscalculia is only
one. It is generally accepted that there is a significant overlap between
developmental disorders, with multiple difficulties being the rule rather than the
exception.
1.2 Aetiology
According to Shalev (2004):
Developmental dyscalculia is a specific learning disability affecting the
normal acquisition of arithmetic skills. Genetic, neurobiologic, and
epidemiologic evidence indicates that dyscalculia, like other learning
disabilities, is a brain-based disorder. However, poor teaching and
environmental deprivation have also been implicated in its etiology.
Because the neural network of both hemispheres comprises the
substrate of normal arithmetic skills, dyscalculia can result from
dysfunction of either hemisphere, although the left parietotemporal area
is of particular significance. Dyscalculia can occur as a consequence of
prematurity and low birth weight and is frequently encountered in a
variety of neurologic disorders, such as attention-deficit hyperactivity
disorder (ADHD), developmental language disorder, epilepsy, and
fragile X syndrome.
Arguably, all developmental disorders that are categorized within the spectrum
of specific learning difficulties have aspects of behavioural, cognitive and
neurological roots. Morton and Frith (1995) suggest a causal modelling
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framework (CM) which draws together behavioural, cognitive and neurological
dimensions, and contextualises them within the environment of the individual.
The underpinning rationale of this model is that no level should be considered
independently of the other, and it should include acknowledgement of the
impact of environmental influences. It is a neutral framework within which to
compare theories. Frith believes that the variation in behavioural or cognitive
explanations should not ignore possible common underlying factors at the
biological / neurological level. In addition, epidemiological findings identify
three major areas of environmental risk as socioeconomic disadvantage,
socio-cultural and gender differences. Equally, complex interaction between
biology and environment mean that neurological deficits will result in cognitive
and behavioural difficulties, particular to the individual. CM theory has been
extended by Krol et al (2004) in an attempt to explore its application to
conduct disorder (Figure 2). Therefore, discussion of the aetiology of
dyscalculia should include a review of the literature based on a CM
framework.
Whilst it could be argued that this approach sits uncomfortably close to the
‘medical’ rather than the ‘social’ model of disability, equally an understanding
of biological, cognitive and behavioural aspects of dyscalculia are fundamental
to the discussion of appropriate learning and teaching experiences.
Figure 1. Causal Modelling Framework (Krol et al., 2004)
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Biological
Brain imaging provides clear indicators with respect to the cortical networks
that are activated when individuals engage in mathematical tasks. Thioux,
Seron and Pesenti (1999) state that the semantic memory systems for
numerical and non-numerical information, are localised in different areas of
the brain. Rourke (1993) proposes that individuals with both a mathematical
and literacy disorder have deficits in the left hemisphere, whilst those
exhibiting only Mathematics disorder tend to have a right hemispherical deficit;
Evidence from neuroimaging and clinical studies in brain injury support the
argument that the parietal lobe, and in particular the intraparietal sulcus (IPS)
in both hemispheres, plays a dominant role in processing numerical data,
particularly related to a sense of the relative size and position of numbers.
Cohen Kadosh et al (2007) state that the parietal lobes are essential to
automatic magnitude processing, and thus there is a hemispherical locus for
developmental dyscalculia. Such difficulties are replicated in studies by
Ashcraft, Yamashita and Aram (1992) with children who have suffered from
early brain injury to the left hemisphere or associated sub-cortical regions.
However, Varma and Schwarz (2008) argue that, historically, educational
neuroscience has compartmentalized investigation into cognitive activity as
simply identification of brain tasks which are then mapped to specific areas of
the brain, in other words ‘….it seeks to identify the brain area that activates
most selectively for each task competency.’ They argue that research should
now progress from area focus to network focus, where competency in specific
tasks is the product of co-ordination between multiple brain areas. For
example, McCrone (2002) suggests a possibility where ‘the intraparietal
sulcus is of a normal size but the connectivity to the “number-name” area over
in Wernicke’s is poorly developed.’ Furthermore, he states that:
‘different brain networks are called into play for exact and approximate
calculations. Actually doing a sum stirs mostly the language-handling
areas while guessing a quick rough answer sees the intraparietal cortex
working in conjunction with the prefrontal cortex.’
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Deloche and Willmes (2000) conducted research on brain damaged patients
and claim to have provided evidence that there are two syntactical
components, one for spoken verbal and one for written verbal numbers, and
that retrieval of simple number facts, for example number bonds and
multiplication tables, depends upon format-specific routes and not unique
abstract representations.
Research also indicates that Working Memory difficulties are implicated in
specific Mathematics difficulties, for example Geary (1993) suggests that poor
working memory resources affect execution of calculation procedures and
learning arithmetical facts. Koontz and Berch (1996) found that dyscalculic
children under-performed on both forward and backward digit span tasks, and
whilst this difficulty is typically found in dyslexic individuals, for the dyscalculic
child it tends not to affect phonological skills but is specific to number
information (McLean and Hitch, 1999). Mabbott and Bisanz (2008) claim that
children with identifiable Mathematics learning disabilities are distinguished by
poor mastery of number facts, fluency in calculating and working memory,
together with a slower ability to use ‘backup procedures’, concluding that
overall dyscalculia may be a function of difficulties in computational skills and
working memory. However, it should be pointed out that this has not been
replicated across all studies (Temple and Sherwood, 2002).
In terms of genetic markers, studies demonstrate a similar heritability level as
with other specific learning difficulties (Kosc, 1974; Alarcon et al, 1997). In
addition, there appear to be abnormalities of the X chromosome apparent in
some disorders such as Turner’s Syndrome, where individuals functioning at
the average to superior level exhibit severe dysfunction in arithmetic
(Butterworth et al., 1999; Rovet, Szekely, & Hockenberry, 1994; Temple &
Carney, 1993; Temple & Marriott, 1998).
Geary (2004) describes three sub types of dyscalculia: procedural, semantic
memory and visuospatial. The Procedural Subtype is identified where the
individual exhibits developmentally immature procedures, frequent errors in
the execution of procedures, poor understanding of the concepts underlying
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procedural use, and difficulties sequencing multiple steps in complex
procedures, for example the continued use of fingers to solve addition and
subtraction problems. He argues that there is evidence that this is a left
hemisphere pre-frontal brain dysfunction, that can be ameliorated or improve
with age.
The Semantic Memory Subtype is identified where the individual exhibits
difficulties in retrieving mathematical facts together with a high error rate, For
example responses to simple arithmetic problems, and accuracy with number
bonds and tables. Dysfunction appears to be located in the left hemisphere
posterior region, is heritable, and is resistant to remediation. The Visuospatial
Subtype represents a difficulty with spatially representing numerical and other
forms of mathematical information and relationships, with frequent
misinterpretation or misunderstanding of such information, for example solving
geometric and word problems, or using a mental number line. Brain
differences appear to be located in the right hemisphere posterior region.
Geary also suggests a framework for further research and discussion of
dyscalculia (Figure 1) and argues that difficulties should be considered from
the perspective of deficits in cognitive mechanism, procedures and
processing, and reviews these in terms of performance, neuropsychological,
genetic and developmental features.
Figure 2. A Framework for Dyscalculia (Geary, 2004)
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Investigating brain asymmetry and information processing, Hugdahl and
Westerhausen (2009) claim that differences in spacing of neuronal columns
and a larger left planum temporal result in enhanced processing speed. They
also state that the evolution of an asymmetry favouring the left hand side of
the brain is a result of the need for lateral specialisation to avoid ‘shuffling’
information between hemispheres, in response to an increasing demand on
cognitive functions. Neuroimaging of dyslexic brains provides evidence of
hemispherical brain symmetry, and thus a lack of specialisation. McCrone
(2002) also argues that perhaps the development of arithmetical skills is as
artificial as learning to read, which may be problematic for some individuals
where the brain ‘evolved for more general purposes’.
Cognitive
Dehaene (1992) and Dehaene and Cohen (1995, 1997) suggest a ‘triple-code’
model of numerosity, each code being assigned to specific numerical tasks.
The analog magnitude code represents quantities along a number line which
requires the semantic knowledge that one number is sequentially closer to, or
larger or smaller than another; the auditory verbal code recognises the
representation of a number word and is used in retrieving and manipulating
number facts and rote learned sequences; the visual Arabic code describes
representation of numbers as written figures and is used in calculation.
Dehaene suggests that this is a triple processing model which is engaged in
mathematical tasks.
Historically, understanding of acquisition of numerical skills was based on
Piaget’s pre-operational stage in child development (2 – 7 years). Specifically,
Piaget argues that children understand conservation of number between the
ages of 5 – 6 years, and acquire conservation of volume or mass at age 7 – 8
years. Butterworth (2005) examined evidence from neurological studies with
respect to the development of arithmetical abilities in terms of numerosity –
the number of objects in a set. Research evidence suggests that numerosity
is innate from birth (Izard et al, 2009) and pre-school children are capable of
understanding simple numerical concepts allowing them to complete addition
and subtraction to 3. This has significant implications as “…. the capacity to
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learn arithmetic – dyscalculia – can be interpreted in many cases as a deficit
in the child’s concept of numerosity” (Butterworth, 2005). Butterworth
provides a summary of milestones for the early development of mathematical
ability based on research studies.
Figure 3. Milestones in Mathematical Development (Butterworth, 2005)
Geary and Hoard (2005) also outline the theoretical pattern of normal early
years development in number, counting, and arithmetic compared with
patterns of development seen in children with dyscalculia in the areas of
counting and arithmetic.
Counting
The process of ‘counting’ involves an understanding of five basic principles
proposed by Gelman and Gallistel (1978):
one to one correspondence - only one-word tag assigned to each
counted object
stable order - the order of word tags must not vary across counted
sets
cardinality - the value of the final word tag represents the quantity of items
counted
abstraction - objects of any kind can be counted
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order-irrelevance - items within a given set can be counted in any
sequence
In conjunction with learning these basic principles in the early stages of
numeracy, children additionally absorb representations of counting
‘behaviour’. Children with dyscalculia have a poor conceptual understanding
of some aspects of counting rules, specifically with order-irrelevance (Briars
and Siegler, 1984). This may affect the counting aspect of solving arithmetic
problems and competency in identifying and correcting errors.
Arithmetic
Early arithmetical skills, for example calculating the sum of 6 + 3, initially may
be computed verbally or physically using fingers or objects, and uses a
‘counting-on’ strategy. Typically, both individuals with dyscalculia and many
dyslexic adults continue to use this strategy when asked to articulate ‘times
tables’ where they have not been rote-learned and thus internalised.
Teaching of number bonds or number facts aid the development of
representations in long term memory, which can then be used to solve
arithmetical problems as a simple construct or as a part of more complex
calculation. That is to say the knowledge that 6 + 3 and 3 + 6 equal 9 is
automatized.
This is a crucial element in the process of decomposition where computation
of a sum is dependent upon a consolidated knowledge of number bonds. For
example, where 5 + 5 is equal to 10, 5 + 7 is equal to 10 plus 2 more.
However, this is dependent upon confidence in using these early strategies;
pupils who have failed to internalise such strategies and therefore lack
confidence tend to ‘guess’. As ability to use decomposition and the principles
of number facts or bonds becomes automatic, the ability to solve more
complex problems in a shorter space of time increases. Geary (2009)
describes two phases of mathematical competence: biologically primary
quantitative abilities which are inherent competencies in numerosity, ordinality,
counting, and simple arithmetic enriched through primary school experiences,
and biologically secondary quantitative abilities which are built on the
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foundations of the former, but are dependent upon the experience of
Mathematics instruction.
In the same way that it is impossible to describe a ‘typical’ dyslexic profile, in
that individuals may experience difficulties with reading, spelling, reading
comprehension, phonological processing or any combination thereof, similarly
a dyscalculic profile is more complex than ‘not being able to do Mathematics’.
Geary and Hoard (2005) describe a broad range of research findings which
support the claim that children with dyscalculia are unable to automatically
retrieve this type of mathematical process. Geary (1993) suggests three
possible sources of retrieval difficulties:
…. a deficit in the ability to represent phonetic/semantic information in
long-term memory……. and a deficit in the ability to inhibit irrelevant
associations from entering working memory during problem solving
(Barrouillet et al., 1997). A third potential source of the retrieval deficit is
a disruption in the development or functioning of a ……cognitive
system for the representation and retrieval of arithmetical knowledge,
including arithmetic facts. (Butterworth, 1999; Temple & Sherwood,
2002).
Additionally, responses tend to be slower and more inaccurate, and difficulty
at the most basic computational level will have a detrimental effect on higher
Mathematics skills, where skill in simple operations is built on to solve more
complex multi-step problem solving.
Emerson (2009) describes difficulties with number sense manifesting as
severely inaccurate guesses when estimating quantity, particularly with small
quantities without counting, and an inability to build on known facts. Such
difficulty means that the world of numbers is sufficiently foreign that learning
the ‘language of Mathematics’ in itself becomes akin to learning a foreign
language.
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Behavioural
Competence in numeracy is fundamental to basic life skills and the
consequences of poor numeracy are pervasive, ranging from inaccessibility of
further and higher education, to limited employment opportunities: few jobs are
completely devoid of the need to manipulate numbers. Thus developmental
dyscalculia will necessarily have a direct impact on socio-economic status, self
esteem and identity.
Research by Hanich et al (2001) and Jordan et al (2003) claim that children
with mathematical difficulties appear to lack an internal number line and are
less skilled at estimating magnitude. This is illustrated by McCrone (2002)
with reference to his daughter:
“A moment ago I asked her to add five and ten. It was like tossing a ball
to a blind man. “Umm, umm.” Well, roughly what would it be? “About
50…or 60”, she guesses, searching my face for clues. Add it up
properly, I say. “Umm, 25?” With a sigh she eventually counts out the
answer on her fingers. And this is a nine-year old.
The problem is a genuine lack of feel for the relative size of numbers. When
Alex hears the name of a number, it is not translated into a sense of being
larger or smaller, nearer or further, in a way that would make its handling
intuitive. Her visuospatial abilities seem fine in other ways, but she apparently
has hardly any capacity to imagine fives and tens as various distances along a
mental number line. There is no gutfelt difference between 15 and 50. Instead
their shared “fiveness” is more likely to make them seem confusingly similar.”
Newman (1998) states that difficulty may be described at three levels:
Quantitative dyscalculia - a deficit in the skills of counting and
calculating
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Qualitative dyscalculia - the result of difficulties in comprehension of
instructions or the failure to master the skills required for an operation.
When a student has not mastered the memorization of number facts,
he cannot benefit from this stored "verbalizable information about
numbers" that is used with prior associations to solve problems
involving addition, subtraction, multiplication, division, and square roots.
Intermediate dyscalculia – which involves the inability to operate with
symbols or numbers.
Trott and Beacham (2005) describe it as:
A low level of numerical or mathematical competence compared to
expectation. This expectation being based on unimpaired cognitive and
language abilities and occurring within the normal range. The deficit will
severely impede their academic progress or daily living. It may include
difficulties recognising, reading, writing or conceptualising numbers,
understanding numerical or mathematical concepts and their inter-
relationships.
It follows that dyscalculics may have difficulty with numerical operations, both
in terms of understanding the process of the operation and in carrying out the
procedure. Further difficulties may arise in understanding the systems that rely
on this fundamental understanding, such as time, money, direction and more
abstract mathematical, symbolic and graphical representations.”
Butterworth (2003) states that although such difficulties might be described at
the most basic level as a condition that affects the ability to acquire
arithmetical skills, other more complex abilities than counting and arithmetic
are involved which include the language of Mathematics:
understanding number words (one, two, twelve, twenty …), numerals
(1, 2, 12, 20) and the relationship between them;
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carrying out mental arithmetic using the four basic arithmetical
operations – addition, subtraction, multiplication and division;
completing written multi-digit arithmetic using basic operations;
solving ‘missing operand problems’ (6 + ? = 9);
solving arithmetical problems in context, for example handling money
and change.
Trott (2009) suggests the following mathematical difficulties which are also
experienced by dyslexic students in higher education:
Arithmetical
• Problems with place value
• Poor arithmetical skills
• Problems moving from concrete to abstract
Visual
• Visual perceptual problems reversals and substitutions e.g. 3/E or +/x
• Problems copying from a sheet, board, calculator or screen
• Problems copying from line to line
• Losing the place in multi-step calculations
• Substituting names that begin with the same letter, e.g.
integer/integral, diagram/diameter
• Problems following steps in a mathematical process
• Problems keeping track of what is being asked
• Problems remembering what different signs/symbols mean
• Problems remembering formulae or theorems
Memory
• Weak short term memory, forgetting names, dates, times, phone
numbers etc
• Problems remembering or following spoken instructions
• Difficulty listening and taking notes simultaneously
• Poor memory for names of symbols or operations, poor retrieval of
vocabulary
Reading
• Difficulties reading and understanding Mathematics books
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• Slow reading speed, compared with peers
• Need to keep re-reading sentences to understand
• Problems understanding questions embodied in text
Writing
• Scruffy presentation of work, poor positioning on the page,
changeable handwriting
• Neat but slow handwriting
• Incomplete or poor lecture notes
• Working entirely in pencil, or a reluctance to show work
General
• Fluctuations in concentration and ability
• Increased stress or fatigue
However, a distinction needs to be drawn between dyscalculia and maths
phobia or anxiety which is described by Cemen (1987) as ‘a state of
discomfort which occurs in response to situations involving mathematics tasks
which are perceived as threatening to self-esteem.’ Chinn (2008) summarizes
two types of anxiety which can be as a result of either a ’mental block’ or
rooted in socio-cultural factors.
Mental block anxiety may be triggered by a symbol or a concept that
creates a barrier for the person learning maths. This could be the
introduction of letters for numbers in algebra, the seemingly irrational
procedure for long division or failing to memorise the seven times
multiplication facts. [...] Socio-cultural maths anxiety is a consequence
of the common beliefs about maths such as only very clever (and
slightly strange) people can do maths or that there is only ever one right
answer to a problem or if you cannot learn the facts you will never be
any good at maths.
According to Hadfield and McNeil (1994) there are three reasons for
Mathematics anxiety: environmental (teaching methods, teacher attitudes and
classroom experience), intellectual (influence of learning style and insecurity
over ability) and personality (lack of self confidence and unwillingness to draw
19
attention to any lack of understanding). Findings by Chinn (2008) indicate
that anxiety was highest in Year 7 (1st year secondary) male pupils, which
arguably is reflective of general anxiety associated with transition to secondary
school.
Environmental
Environmental factors include stress and anxiety, which physiologically affect
blood pressure to memory formation. Social aspects include alcohol
consumption during pregnancy, and premature birth / low birth weight which
may affect brain development. Isaacs, Edmonds, Lucas, and Gadian (2001)
investigated low birth-weight adolescents with a deficit in numerical operations
and identified less grey matter in the left IPS.
Assel et al (2003) examined precursors to mathematical skills, specifically the
role of visual-spatial skills, executive processing but also the effect of
parenting skills as an environment influence. The research measured
cognitive and mathematical abilities together with observation of maternal
directive interactive style. Findings supported the importance of visual-spatial
skills as an important early foundation for both executive processing and
mathematical ability. Children aged 2 years whose mothers directed tasks as
opposed to encouraging exploratory and independent problem solving, were
more likely to score lower on visual–spatial tasks and measures of executive
processing. This indicates the importance of parenting environment and
approach as a contributory factor in later mathematical competence.
1.3 Assessment
Shalev (2004) makes the point that delay in acquiring cognitive or attainment
skills does not always mean a learning difficulty is present. As stated by
Geary (1993) some cognitive features of the procedural subtype can be
remediated and do not necessarily persist over time. Difficulties with
Mathematics in the primary school are not uncommon; it is the pervasiveness
into secondary education and beyond that most usefully identifies a
dyscalculic difficulty. A discrepancy definition stipulates a significant
discrepancy between intellectual functioning and arithmetical attainment or by
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a discrepancy of at least 2 years between chronologic age and attainment.
However, measuring attainment in age equivalencies may not be meaningful
in the early years of primary age range, or in the later years of secondary
education.
Wilson et al (2006) suggest that assessment of developmental symptoms
should examine number sense impairment. This would include:
reduced understanding of the meaning of numbers, and a low
performance on tasks which depend highly on number sense, including
non symbolic tasks (e.g. comparison, estimation or approximate
addition of dot arrays), as well as symbolic numerical comparison and
approximation.
They add that performance in simple arithmetical calculation such as
subtraction would be a more sensitive measure, as addition and multiplication
is more open to compensatory strategies such as adding or counting on, and
memorization of facts and sequences.
Assessment instruments
As yet there are few paper-based dyscalculia specific diagnostic. Existing
definitions state that the individuals must substantially underachieve on
standardised tests compared to expected levels of achievement based on
underlying ability, age and educational experience. Therefore, assessment of
mathematical difficulty tends to rely upon performance on both standardized
mathematical achievement and measurement of underlying cognitive ability.
Geary and Hoard (2005) warn that scoring systems in attainment tests blur the
identification of specific areas of difficulty:
Standardized achievement tests sample a broad range of arithmetical
and mathematical topics, whereas children with MD often have severe
deficits in some of these areas and average or better competencies in
others. The result of averaging across items that assess different
21
competencies is a level of performance […] that overestimates the
competencies in some areas and underestimates them in others.
Von Aster (2001) developed a standardized arithmetic test, the
Neuropsychological Test Battery for Number Processing and Calculation in
Children, which was designed to examine basic skills for calculation and
arithmetic and to identify dyscalculic profiles. In its initial form the test was
used in a European study aimed at identifying incidence levels (see section
1.4). It was subsequently revised and published in English, French,
Portuguese, Spanish, Greece, Chinese and Turkish as ZarekiR, This test is
suitable for use with children aged 7 to 13.6 years and is based on the
modular system of number processing proposed by Dehaene (1992).
Current practice for assessment of dyscalculia is referral to an Educational
Psychologist. Trott and Beacham (2005) claim that whilst this is an effective
assessment method where students present with both dyslexic and
dyscalculic indicators, it is ineffective for pure dyscalculia with no co-morbidity.
Whilst there is an arithmetical component in tests of cognitive ability such as
the Weschler Intelligence Scale for Children (WISC) and the Weschler Adult
Intelligence Scale (WAIS), only one subtest assesses mathematical ability.
Two things are needed then: an accurate and reliable screening test in the
first instance, and a standardized and valid test battery for diagnosis of
dyscalculia.
Working memory as an assessment device
Working Memory (WM) can be described as an area that acts as a storage
space for information whilst it is being processed. Information is typically
‘manipulated’ and processed during tasks such as reading and mental
calculation. However, the capacity of WM is finite and where information
overflows this capacity, information may be lost. In real terms this means that
some learning content delivered in the classroom is inaccessible to the pupil,
and therefore content knowledge is incomplete or ‘missing’. St Clair-
Thompson (2010) argues that these gaps in knowledge are ‘strongly
associated with attainment in key areas of the curriculum’.
22
Alloway (2001) conducted research with 200 children aged 5 years, and
claims that working memory is a more reliable indicator of academic success.
Alloway used the Automated Working Memory Assessment (AWMA) and then
re-tested the research group six years later. Within the battery of tests
including reading, spelling and Mathematics attainment, working memory was
the most reliable indicator. Similarly, recent findings with children with Specific
Language Impairment, Developmental Coordination Disorder (DCD),
Attention-Deficit/Hyperactivity Disorder, and Asperger’s Syndrome (AS) also
support these claims.
Alloway states that the predictive qualities of measuring WM are that it tests
the potential to learn and not what has already been learned. Alloway states
that ‘If a student struggles on a WM task it is not because they do not know
the answer, it is because their WM ‘space’ is not big enough to hold all the
information’. Typically, children exhibiting poor WM strategies under-perform
in the classroom and are more likely to be labelled ‘lazy’ or ‘stupid’. She also
suggests that assessment of WM is a more ‘culture fair’ method of assessing
cognitive ability, as it is resistant to environmental factors such as level of
education, and socio-economic background. The current version of AWMA
has an age range of 4 to 22 years.
In a review of the literature on dyscalculia, Swanson and Jerman (2006) draw
attention to evidence that deficits in cognitive functioning are primarily situated
in performance on verbal WM. Currently there is no pure WM assessment for
adult learners, however Zera and Lucian (2001) state that processing
difficulties should also form a part of a thorough assessment process. Rotzer
et al (2009) argue that neurological studies of functional brain activation in
individuals with dyscalculia have been limited to:
… number and counting related tasks, whereas studies on more
general cognitive domains that are involved in arithmetical
development, such as working memory are virtually absent.
23
This study examined spatial WM processes in a sample of 8 – 10-year-old
children, using functional MRI scans. Results identified weaker neural
activation in a spatial WM task and this was confirmed by impaired WM
performance on additional tests. They conclude that ‘poor spatial working
memory processes may inhibit the formation of spatial number representations
(mental numberline) as well as the storage and retrieval of arithmetical facts’.
Computerized assessment
The Dyscalculia Screener (Butterworth, 2003) is a computer-based
assessment for children aged 6 – 14 years, that claims to identify features of
dyscalculia by measuring response accuracy and response times to test
items. In addition, it claims to distinguish between poor Mathematics
attainment and a specific learning difficulty by evaluating an individual’s ability
and understanding in the areas of number size, simple addition and simple
multiplication. The screener has four elements which are item-timed tests:
1. Simple Reaction Time
Tests of Capacity:
2. Dot Enumeration
3. Number Comparison (also referred to as Numerical Stroop)
Test of Achievement:
4. Arithmetic Achievement test (addition and multiplication)
Speed of response is included to measure whether the individual is
responding slowly to questions, or is generally a slow responder.
The Mathematics Education Centre at Loughborough University began
developing a screening tool known as DyscalculiUM in 2005 and this is close
to publication. The most recent review of development was provided in 2006
and is available from
http://Mathematicstore.gla.ac.uk/headocs/6212dyscalculium.pdf The screener
is now in its fourth phase with researchers identifying features as:
24
Can effectively discriminate dyscalculia from other SpLDs such as
Asperger’s Syndrome and ADHD
Is easily manageable
Is effective in both HE and FE
Can be accommodated easily into various screening processes
Has a good correlation with other published data, although this data is
competency based and not for screening purposes
Can be used to screen large groups of students as well as used on an
individual basis
1.4 Incidence
The lack of consensus with respect to assessment and diagnosis of
dyscalculia, applies equally to incidence. As with dyslexia, worldwide studies
describe an incidence ranging from 3% to 11%, however as there is no
formalised method of assessment such figures may be open to interpretation.
Research by Desoete et al (2004) investigated the prevalence of dyscalculia in
children based on three criteria: discrepancy (significantly lower arithmetic
scores than expected based on general ability), performance at least 2 SD
below the norm, and difficulties resistant to intervention. Results indicated that
of 1, 336 pupils in 3rd grade (3rd class) incidence was 7.2% (boys) and 8.3%
(girls), and of 1, 319 4th grade (4th class) pupils, 6.9% of boys and 6.2% of
girls.
Koumoula et al. (2004) tested a sample population of 240 children in Greece
using the Neuropsychological Test Battery for Number Processing and
Calculation in Children, and a score of <1.5 SD was identified in 6.3% of the
sample. Findings by Von Aster and Shalev (2007) in a sample population of
337 Swiss children reported an incidence of 6.0 % using the same
assessment method and criterion. Mazzocco and Myers (2003) used multiple
tests of arithmetic skills (Key Math Subtests, Test of Early Math Ability, and
Woodcock-Johnson Revised Math Calculations) together with a criterion of
persistent diagnosis across more than one school year. Incidence rates for 3rd
grade children fell between 5% and to 21%.
25
Findings from cross-cultural studies indicate that incidence is more prevalent
in boys than girls, the risk ratio being 1.6 to 2.2. In terms of co-morbidity with
other specific learning difficulties, studies by Gross-Tsur et al (1996),
Barbaresi et al (2005) and Von Aster and Shalev (2007) provide evidence of a
coexisting reading difficulty, the percentages across all three studies falling at
17%, 56.7% and 64%. Additionally, a greater number of children with
dyscalculia exhibit clinical behaviour disorders than expected.
Barbaresi et al (2005) investigated the incidence of Mathematics learning
disorder among school-aged children, via a population-based, retrospective,
birth cohort study. The research study used a population sample of all
children born between 1976 and 1982. Data was extracted from individually
administered cognitive and achievement tests together with medical,
educational, and socioeconomic information. Findings identified a cumulative
incidence rate of Mathematics disorder by age 19 years within a range of 5.9%
to 13.8%. The results suggest that dyscalculia is common among school
children, and is significantly more frequent among boys than girls. This level
of incidence reflects a similar incidence of dyslexia, which is identified as
being between 4% and 10% of the population.
1.5 Intervention
At a neurological level, St Clair-Thompson (2010) states that remediation of
WM would enhance performance in academic progress. She suggests that
memory strategy training and practice in memory tasks are effective
intervention tools. This might include adjustments to the teaching environment
such as repetition of material in a variety of formats, breaking down tasks into
smaller units, and use of memory techniques. Research into the use of
computer programmes such as ‘Memory Booster’ (Leedale et al, 2004) whilst
demonstrating improved WM performance, does not confirm that they can
enhance or improve academic attainment (St Clair-Thompson et al, 2010;
Holmes et al, 2009).
26
Wilson et al (2006) developed and trialled software designed to remediate
dyscalculia, called ‘The Number Race’. The underlying rationale of this system
is the presence of a "core deficit" in both number sense and accessing such a
sense through visual symbolic representation. The programme claims to
remediate difficulties using mathematical problems which are adaptive to the
age and ability level of the child. The software was piloted with a small
sample of 7–9-year-old French children with mathematical difficulties, for 30
minutes a day over 5 weeks. Children were tested pre and post intervention
on tasks measuring counting, transcoding, base-10 comprehension,
enumeration, addition, subtraction, and symbolic and non-symbolic numerical
comparison. Whilst the sample exhibited increased performance on core
number sense tasks such as subtraction accuracy, there was no improvement
in addition and base-10 comprehension skills. However, this is the first step in
a series of clinical trials to build on this programme.
Sharma (1989) argues that Mathematics should be considered as a separate,
symbolic ‘language’ system and teaching should reflect this. Specifically, that
terminology, vocabulary and syntax of mathematical language must be taught
strategically to ensure understanding of mathematical concepts, to underpin
learning of mathematical methods. Sharma also makes the point that
consideration should be given to inclusive teaching principles, methods and
materials to address difficulties at every level. She suggests five critical
factors in delivering the Mathematics curriculum effectively:
1. Assessment of mathematical knowledge and strategies used by the
learner to determine teaching methodology.
2. Assessment and identification of learning style (whether quantitative or
qualitative) and recognition that this is unique to the individual. For example,
quantitative learners may favour learning the procedural aspect of
Mathematics, and to deduce answers from having learned general
mathematical principles. Qualitative learners are more dependent upon
seeing parallels and relationships between elements.
27
3. Assessment of seven ‘pre-Mathematics’ skills:
Sequencing
Direction and laterality
Pattern recognition
Visualisation
Estimation
Deductive reasoning
Inductive reasoning
4. Specific teaching of mathematical language and syntactical variations,
for example that 33 – 4 is the same as ‘subtract 4 from 33’ and 4 less than 33’.
5. A systematic approach to the introduction and teaching of new
mathematical concepts and models.
Principally, the consensus on guidelines for effective intervention can be
summarized as follows:
1. Enable visualization of Mathematics problems. Provide pictures,
graphs, charts and encourage drawing the problem.
2. Read questions / problems aloud to check comprehension. Discuss
how many parts / steps there may be to finding the solution.
3. Provide real life examples.
4. Ensure that squared / graph paper is used to keep number work and
calculation.
5. Avoid fussy and over-detailed worksheets, leave space between each
question so that pupils are not confused by questions that seem to
merge together.
28
6. Teach over-learning of facts and tables, using all senses and in
particular rhythm and music. Warning: meaningless repetition to learn
facts off by heart does not increase understanding.
7. Provide one-to-one instruction on difficult tasks. If a pupil does not
understand, re-frame and re-word the question / explanation
8. Use a sans serif font in minimum 12 point.
9. Provide immediate feedback and provide opportunities for the pupil to
work through the question again. Encourage opportunities to see where
an error has occurred.
10. In early stages of Mathematics teaching, check that the pupil has
understood the syntactical variations in Mathematics language.
Encourage the pupil to verbalize the problem stages, for example: ‘To do
this I have to first work out how many thingies there are and then I can
divide that number by the number of whatsits to find out how many each
one can have.’
11. Allow more time to complete Mathematics work.
12. Ask the pupil to re-teach the problem / function to you.
Whilst Sharma (1989) highlights the language of Mathematics as key in the
building of foundation skills, critically, in the NCCA Report (2005) only 17.2%
or primary teachers identified the use of Mathematics language as an effective
strategy in the teaching of Mathematics skills, and only 10.7% reported linking
Mathematics activities to real life situations. Butterworth (2009) suggests four
basic principles of intervention:
• Strengthen simple number concepts
• Start with manipulables and number words
29
• Only when learner reliably understands relationship between number
words and concrete exemplars, progress to numeral symbols
• Structured teaching programme designed for each learner
Technological aids tend to be limited to tool such as calculators, which include
talking calculators and enlarged display screens, buttons and keypads. There
are a plethora of computer programmes on the market which claim to improve
the underlying cognitive skills associated with reading, spelling and number.
However, caution should be exercised with regard to computerized training.
Owen et al (2010) researched the efficacy of brain training exercises
conducting an online study with more than 11,000 participants. Whilst
performance of all participants in improved over time on the experiment, re-
testing on the initial performance tests indicated that ‘these benefits had not
generalised, not even when the training tests and benchmark tests involved
similar cognitive processes’.
30
Section 2: Accessing the curriculum
2.1 Primary schools programme
The National Council for Curriculum and Assessment (NCCA) ‘Primary
Curriculum Review’ (2005) reported that 66.4% of teachers hardly ever or
never used diagnostic tests, and 77.2% hardly ever or never used
standardised tests as a means for assessing performance in Mathematics
(Figure 4). Whilst it is acknowledged that such tests do not play a role in
supporting the teaching and learning process, arguably they are necessary to
monitor the progress – or lack of – for pupils who are exhibiting difficulty in
accessing the Mathematics curriculum.
Figure 4. Use of Assessment Tools in Mathematics Education (NCCA, 2005)
The report identified 20.9% of teachers as stating that standardized tests were
unsuitable ‘to assess specific learning disability child in comparison to
mainstream’. In addition, they were of the opinion that there was an ‘over-
reliance on written assessment.’ Of the 459 teachers who responded to the
challenges of assessing Mathematics ability, over 80% of this number stated
that primary difficulties in assessment were time, the range of Mathematics
abilities amongst pupils, appropriate assessment tools and language.
With respect to time, teachers stated that large class sizes were a contributing
factor to difficulties particularly ‘time constraints for assessing children with
31
learning difficulties’. For classes with a wide range of ability level, difficulties
were expressed in assessing ‘how precisely each child coped with a new
concept’, ‘pinpointing [their] specific mathematical difficulty’, and ‘tailoring test
to individuals to pinpoint areas of weakness’. With respect to standardised
assessment tools, 20.9% of teachers felt they were inadequate for testing
performance against the revised curriculum and that they were inappropriate
to ‘assess specific learning disability in comparison to the mainstream.’
A critical point was made relating mathematical ability and language ability,
supporting Sharma’s (1989) assertions, with 7.1% of teachers observing
‘mathematical language itself to be problematic for certain children’ and that
‘lack of expressive language for Mathematics’ is a factor in difficulties. Clearly
then, there are practical constraints in assessing Mathematics performance,
which is a cause for concern. If a specific difficulty in Mathematics is not
identified during the early years of education where a solid mathematical
foundation is constructed, such difficulties will multiply exponentially.
2.2 Secondary programme
In September 2003 the NCCA introduced plans for the Project Mathematics
programme. This is a school-based initiative which aims to address issues
such as school completion targets, and access to and participation in third
level through changes to the Junior (JC) and Senior cycle (LC) Mathematics
curriculum. Objectives include a greater focus on the learner’s understanding
of key Mathematics skills, the role of Mathematics assessment, and the
contribution of such skills to Ireland as a knowledge economy. To achieve
these aims and objectives the project is committed to getting teachers
involved in changes to the curriculum, by encouraging lesson development,
adaptation and refinement that will feedback into the curriculum development
process. In terms of curriculum structure there will be incremental revisions to
syllabi, and an assessment approach which reinforces these changes.
In 2006 the NCCA conducted a review of Mathematics in Post-Primary
Education which included the following remarks:
32
The difficulties that students experience in Third Level are due to
mathematical under-preparedness in terms of mathematical knowledge
and skills as well as attitudes. The Leaving Certificate Ordinary Level
course is not working well for students in this regard and needs
attention.
The examination needs to be less predictable. At the moment it seems
easy for teachers / students / media to predict the format of the paper
and even the individual questions. The fact that ‘question 1 is always
about topic X’ reinforces the notion that rote learning is the way to score
highly.
If this is the case then we could assume that some pupils with dyscalculia may
be more successful in that they can revise to a set pattern, and anecdotally,
this appears to be a strategy that is widely used.
Project Mathematics aims to introduce a number of new initiatives: In the first
instance a bridging framework between primary and secondary level is
proposed. This will take the form of a common introductory course in first year
of secondary, with the purpose of building on the knowledge, understanding
and skills developed at primary school. For this reason, choice of syllabus
level at Foundation, Ordinary or Higher level will be delayed choice. With
respect to the JC years there will be two syllabus levels, ordinary and Higher
Level, with a Foundation Level examination based on the Ordinary syllabus.
The uptake targets are that at least 60% of the JC cohort will study at Higher
Level.
Planned syllabus changes at junior cycle and senior cycle includes 5 strands:
Statistics and probability
Geometry and trigonometry
Number
Algebra
Functions
33
The project began with the introduction of strands 1 and 2 into 24 schools in
September 2008. These schools continued to add two new strands (3 & 4) in
September 2009. The programme for all other schools will commence from
September 2010, and a programme of professional development for
Mathematics teachers will begin this autumn.
In April 2010, Mary Coughlan, Minister for Education, announced that a
scheme of bonus points would be introduced to encourage pupils to pursue
Higher Level Mathematics in the Leaving Certificate. John Power, Director
General of Engineers Ireland welcomed this suggestion, whilst acknowledging
that in isolation it would not provide a solution, but that specific training and
qualification in Mathematics for teachers is fundamental. The Royal Irish
Academy (2008) state that only 20% of teachers of Mathematics studied the
subject beyond the first year of their primary degree, and DES (2006) findings
indicate that 70% of school inspectors describe teachers’ knowledge of
methods of teaching Mathematics as ‘somewhat limited.’ Research by Ni
Riordain and Hannigan (2009) found that 48% of Mathematics teachers in
post-primary schools ‘have no qualification in Mathematics teaching.’
2.3 Intervention
Travers (2010) discusses inequitable access to the Mathematics curriculum
and the implications for provision of learning support within Irish primary
schools. Travers argues that the general allocation model of learning
support, and subtle changes to the wording of DES guidelines on the provision
of learning support, are constraining access to early intervention. He points
out that intervention is targeted at ‘pupils who are performing at or below the
10th percentile on nationally standardized tests’ with wording amended from
‘English and / or Mathematics’ to ‘English or Mathematics’, implying that
intervention is available for one but not both area of difficulty (DES, 2000).
Travers further points out that the 2005 inspection of literacy and numeracy
provision / achievement in disadvantaged schools, there was a significant
shortfall in provision of learning support in numeracy. Surgenor and Sheil
(2010) examined differences in learning support provision for English and
34
Mathematics across 172 Irish primary schools. Only 3% of schools provided
intervention support purely in Mathematics, compared to over 33% of schools
providing support in English.
Literature on quantity and quality of support for Mathematics in Irish schools
indicates that substantial increases in learning support staff, concrete
resources, quality of teacher training, curriculum structure and timetabling are
urgently required. The shortfall in provision is illustrated by the rising demand
for ‘grinds’ services in Mathematics at primary level, accessible only to those
parents with the requisite financial resources. It is clear that plugging holes in
the secondary curriculum is ineffective in the long term, and that a ‘bottom up’
rather than ‘top down’ approach is required, in that intervention schemes must
address Mathematics from the early years of education. In October 2009, the
NCCA published Aistear: The Early Childhood Curriculum Framework aimed
at primary age children from birth to 6 years. This scheme is targeted at
parents, teachers and other professional practitioners with an emphasis on
communication and learning through language in every subject area.
Engineers Ireland (2010) propose a 10-point action plan to address the
question of Mathematics and future performance at primary, secondary and
tertiary level. They also stress the advantages of a bottom up approach, and
provide suggestions for greater accessibility and flexibility in providing
Mathematics support, specifically:
The need to foster interest in Mathematics at both primary and
secondary level, and in particular within the Transition Year
programme.
Harnessing the power of ICT to contextualise the teaching of
Mathematics and Science at Primary and Second Level.
Construction of a Wiki-Solution web page to assist students with
problem solving in Mathematics and Applied Mathematics.
35
Section 3: Transition to third level
3.1 Performance in State Examinations
In order to matriculate to an Irish university, students must meet specific
minimum entry requirements for each institution of higher education. Currently
these are:
National University of Ireland: 6 subjects, including English, Irish and a
third language. Students must have achieved grade C at Higher Level.
in two of these subjects.
University of Limerick: 6 subjects, including English, Irish and a third
language. Students must have achieved grade C at Higher Level in two
of these subjects.
Trinity College Dublin: 6 subjects, with grade C on 3 Higher Level
papers and a pass in English, Mathematics and another language.
Dublin City University: 6 subjects, with a grade C on 2 Higher-Level
papers and a pass in Mathematics and either English or Irish.
Institutes of Technology
Honours Degree courses: grade C in 2 subjects at Higher Level and
grade D in 4 other subjects, including Mathematics and Irish/English.
Higher Certificate and Ordinary Degree courses: 5 grade Ds, including
Mathematics and Irish/English
.
Colleges of education: 3 grade Cs on Higher-Level papers, including
Irish, and three grade Ds, including Mathematics and English.
A pass means Grade D or above on Ordinary or Higher papers. A significant
number of pupils do not matriculate with a Leaving Certificate as a result of
failing the Mathematics examination. Oldham (2006) states that:
36
The percentage of students obtaining low scores (grade E, grade F, or
no grade) in Mathematics in the Ordinary-level Leaving Certificate
examination in particular means that some thousands of students leave
the school system each year without having achieved a grade regarded
as a `pass' in Mathematics. Such students are in general excluded from
third-level courses that require mathematical knowledge and skills.
Mac an Bhaird (2008) discusses factors associated with poor Mathematics
performance:
However, some of the main factors listed in [Lyons et al, 2003] and
elsewhere include bad publicity for Mathematics, negative attitudes
towards the subject, the high percentage of second-level students who
go onto third-level, the socio-economic background of the student,
increased competition for places, pressure on students and teachers to
achieve the highest possible points, little understanding of the context
or background of Mathematics, little appreciation of the applications of
Mathematics in everyday life, rote learning by heart, etc.
The State Examinations Commission provides annual and cumulative
statistics indicating performance in Leaving Certificate Mathematics on
Foundation, Ordinary and Higher level papers. Results are available in two
formats: as a percentage breakdown of candidates by grade awarded in each
subject, and percentage / number breakdown of results by gender across all
levels. Annual Leaving Certificate statistics were downloaded from
www.examinations.ie for the period 2001 - 2009.
These results were collated and re-tabulated, and a comparative analysis was
conducted across all grades for Foundation, Ordinary and Higher results
between 2001 and 2008 (statistics for 2009 are currently provisional and thus
were not included). A full analysis of these statistics is available in the
Appendices. Findings indicated that the number of pupils who failed to
matriculate in Mathematics between 2001 and 2008, and who were therefore
prevented from transitioning to college is 43, 892. In 2008 alone, 5,049
37
students failed to matriculate. It is worth noting that whilst there are
fluctuations in performance for A – D grades at all levels for the period 2001 –
2008, the number of students failing to matriculate having achieved E, F and
No Grade is fairly consistent (Figure 5).
2001 - 2008 All Levels
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
Year
No. of students
A
B
C
D
E
F
NG
Figure 5. Leaving Certificate Grade Comparison 2001 - 2009
This can be further illustrated by examining differences in performance for
each of the three levels of examination paper (Foundation, Ordinary and
Higher) separately, for the same period (Figure 6). Clearly, whilst there are
achievement fluctuations across the years for grades A to D, there is a
curiously ‘flat’ effect across E, F and No Grade results.
38
2001 - 2008 Foundation Level
0
500
1000
1500
2000
2500
2001 2002 2003 2004 2005 2006 2007 2008
year
No. of students
A
B
C
D
E
F
NG
2001 - 2008 Ordinary Level
0
2000
4000
6000
8000
10000
12000
2001 2002 2003 2004 2005 2006 2007 2008
Year
No. of students
A
B
C
D
E
F
NG
2001 - 2008 Higher Level
0
500
1000
1500
2000
2500
3000
3500
2001 2002 2003 2004 2005 2006 2007 2008
Year
No. of students
A
B
C
D
E
F
NG
Figure 6. Leaving Certificate Grade Comparison by exam Level 2001 - 2008
39
There are a number of questions that need to be asked in view of these
figures, not the least of which is, what do they mean? Why does the
percentage of pupils achieving E, F and NG remain reasonably constant
across the timeframe of 9 years, compared with pupils achieving higher
grades, and why is this the case across all three papers? Do they represent
pupils with mild general learning difficulties? Are they representative of pupils
who have not received adequate teaching and supports in Mathematics? Or
are they in fact pupils with an undiagnosed specific learning difficulty in
Mathematics? One further question to be investigated is how this compares
with the two remaining core curriculum subjects, Irish and English.
Comparative statistics were extracted from SEC results for the five-year period
between 2003 and 2008 in the three core curriculum subjects and across all
levels of papers. It is clear from the tables below that there is a discrepancy
between the percentages for the lowest grades achieved in both Irish and
English, compared with those in Mathematics, particularly at Foundation level
(Figure 8).
Irish LC results
0.0
5.0
10.0
Year
Percentage
E
F
NG
E
1.3
0.7
0.9
0.9
0.9
1.0
2.2
1.0
2.6
0.9
1.7
1.0
2.8
3.3
4.3
5.0
3.7
4.5
F
0.0
0.0
0.0
0.0
0.1
0.1
0.2
0.3
0.2
0.4
0.5
0.4
0.6
0.7
1.0
1.1
1.0
1.0
NG
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.0
0.0
0.0
0.0
0.0
2003
2004
2005
2006
2007
2008
2003
2004
2005
2006
2007
2008
2003
2004
2005
2006
2007
2008
Ordinary
Foundation
Figure 7. Leaving Certificate Results in Irish Language 2003 - 2008
40
English LC results
0.0
2.0
4.0
Year
Percentage
E
F
NG
E
1.2
1.1
1.6
1.3
1.6
1.1
1.8
2.3
2.2
2.0
2.6
2.7
F
0.1
0.1
0.1
0.1
0.2
0.1
0.3
0.5
0.5
0.4
0.6
0.6
NG
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.1
0.2
2003
2004
2005
2006
2007
2008
2003
2004
2005
2006
2007
2008
Ordinary
Figure 8. Leaving Certificate Results in English Language 2003 – 2008
Maths LC results
0.0
5.0
10.0
Year
Percentage
E
F
NG
E
3.5
3.2
3.2
2.5
2.9
3.5
4.3
4.9
5.7
4.8
4.5
3.7
8.1
7.7
7.9
8.2
7.8
8.0
F
0.6
0.9
0.8
0.7
0.7
0.7
1.2
1.4
1.8
1.6
1.9
1.8
3.1
3.3
3.5
3.1
3.3
3.7
NG
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2
0.1
0.3
0.2
0.3
0.5
0.5
0.3
0.4
0.5
2003
2004
2005
2006
2007
2008
2003
2004
2005
2006
2007
2008
2003
2004
2005
2006
2007
2008
Ordinary
Foundation
Figure 9. Leaving Certificate Results in Mathematics 2003 – 2008
The percentage ranges for all subjects can be summarized as below:
Subject (all
levels
Range of
percentages
No Grade
F
E
Irish
0.0 – 0.1
0.0 – 1.1
0.7 – 5.0
English
0.0 - 0.2
0.1 - 0.6
1.1 - 2.7
Mathematics
0.1 – 0.5
0.6 – 3.7
2.5 – 8.0
Table 1. Percentage ranges for Leaving Certificate subjects.
41
It could be suggested that this may be reflective of two factors: pupils who
have been granted an exemption from spelling and written punctuation
elements in English who might otherwise have failed to achieve a pass; pupils
who have been awarded an exemption from examination in Irish, who might
otherwise have failed to achieve a pass in that subject. As there is no
equivalent accommodation for Mathematics this may well be a reason for this
discrepancy, and thus an argument for equivalent recognition of the need for
reasonable accommodations in state examinations for specific learning
difficulty in Mathematics.
If the incidence of dyscalculia is taken as between 4% and 10% of a sample
population, then potentially the number of pupils with dyscalculia in this same
period might be estimated to be between 202 and 504.
Year
Total number of
students who failed
to matriculate
Potential number of
students at risk with
dyscalculia
4% incidence
10% incidence
2001
7402
296
740
2002
6409
256
640
2003
5211
208
521
2004
5096
204
509
2005
5270
210
527
2006
4697
188
469
2007
4758
190
475
2008
5049
202
504
Table 2. Potential number of Leaving Certificate candidates with Dyslexia
Of equal importance is what happens to these pupils who fail to matriculate?
What do they do and where do they go, and has this information been
compiled? Some pupils evidently repeat the Leaving Certificate to gain a
higher grade in Mathematics, some pupils feed into courses which do not
require matriculation, and it is likely that a number apply as mature students.
42
To investigate this latter circumstance, a sweep was conducted of mature
students registered with the Disability Service over the last 3 years.
A total of 144 mature students were registered on undergraduate courses,
with an age range of 24 to 68 years. Academic records were checked and
those who matriculated the Leaving Certificate and those for whom no archive
record was found, were removed. Of the remaining 95 students, 19 failed
Mathematics and thus did not matriculate. The remaining 76 students had no
Leaving Certificate result recorded, possibly because they left school after
completing Junior Cycle education. The 19 students failing to matriculate in
Mathematics are all registered with the Disability Service as having a specific
learning difficulty, and their performance in Mathematics LC is as follows:
Higher
level
Ordinary
level
Grade E
Grade F
NG
1
18
12
4
3
Table 3. Leaving Certificate Mathematics results for Mature students
This represents 20% of mature students registered with the service for whom
academic records were available. Investigation of academic performance in
senior cycle education for all mature students would illuminate whether these
findings can be generalised.
3.2 Access through Disability Access Route to Education process
Applicants to the Disability Access Route to Education (DARE) are invited to
submit a personal statement recoding their particular difficulties. The
incidence of dyscalculia is significantly under-represented in this cohort.
43
Year
Total no. of
applicants with
SpLD
Total no.
declaring
dycalculia
Total no.
providing
evidence
2008
1,223
3
1
2009
1,929
9
2
2010
1,398
3
2
Table 4. Number of students with Dyscalculia applying to DARE
In 2008 only 3 applications to DARE were formally submitted on the grounds
of dyscalculia, and of those only 1 provided documentation confirming
dyscalculia. In 2009 only 9 applications were formally submitted on the
grounds of dyscalculia, including one application which also stated a hearing
impairment, and of those only 2 provided documentation that included any
attainment scores; none specifically stated dyscalculia. In this same year only
three applicants in 2009 described dyscalculia as affecting performance:
[It] has affected my academic potential primarily in Mathematics, where
I have significant difficulties in a number of areas. I often struggle to
keep up with the class and maintain the levels required. Consequently,
I spend much longer than the average student studying Mathematics. I
also have great difficulty memorizing things such as ……. times tables.
The learning disability dyscalculia has had a huge impact on my
academic potential. Apart from Mathematics where the effect is severe
it also impacts on music theory and any subject that involves counting.
My concept of time is greatly inhibited also.
From a young age it was discovered that I had hearing problems. I
therefore missed hearing vital sounds needed for language and speech
development … My hearing problem was a huge factor in discovering I
had dyslexia. I had missed the basics in Language and Mathematics as
I either did not understand them or just did not hear it. This made my
school life difficult as the higher up in school I went the more
challenging it was which meant I missed out on more studies. In 4th
44
grade my work was below grade level and I had problems with
Mathematics.
In 2010 only 3 applicants formally applied for consideration under dycalculia,
and only one applicant specifically described their difficulties in Mathematics:
I have struggled with Mathematics throughout school. I have difficulty
knowing what to do with Mathematics problems but I find that when I
know implicitly I am fine. I can follow Mathematics methods step by
step but I need to practice them over and over again. I find it extremely
difficult to calculate in my head. I have difficulty learning multiplication
tables, common sequences, telephone numbers and number
sequences. I reverse numbers and symbols.
CAO applicants with a disability have the option of disclosing a disability prior
to entering college. Of the 546 who disclosed and applied to Trinity College
in 2008, 10 applicants failed to matriculate in Mathematics, only 1 of whom
had reported scores that identified them as having a difficulty in Mathematics,
reading and spelling. Of the 659 who disclosed and applied to Trinity College
in 2009, 10 applicants failed to matriculate in Mathematics, only 3 having
reported attainment scores that identified them as having a difficulty in
Mathematics, reading and spelling. There appears to be a lack of both
recognition and adequate assessment for a specific difficulty with
Mathematics.
3.3 Implications for transition to higher education
The Faculty of Engineering, Mathematics and Science in Trinity College
Dublin hosted a Mathematics symposium in March 2010 entitled ‘The Place of
Mathematics Education in Ireland’s Future’. The purpose of the symposium
was to review issues with regard to Mathematics curriculum at second level,
and review the proposal by some universities to re-introduce bonus points for
Higher Level (HL) Mathematics at Leaving Certificate. Presentations delivered
at this symposium are available at http://www.tcd.ie/ems/Mathematics-
symposium/presentations
45
Elizabeth Oldham (TCD) stated that less time than previously is given to
Mathematics in 1st and 2nd year, so time pressures in covering the syllabus,
limited resources and insufficient opportunities to do ‘up close’ work with
students results in pre-selection of Mathematics topics. She also noted that
some teachers of other subjects are asked to teach Mathematics. This is
problematic in that they may have limited specialised knowledge of the
subject, with the result that rules are taught ‘without reasons’. She added that
Foundation level is not recognised as providing a bedrock of mathematical
understanding and competence, and thus students opt for Ordinary level,
reinforcing a culture of rote learning and teaching. It is this method of
‘shortcut’ Mathematics that students bring with them to third level.
Maria Meehan (UCD) further emphasised the necessity for strong
mathematical foundations, stating that ‘Emphasis must be placed on the
understanding of mathematical concepts. “Teaching for understanding” and
“learning with understanding” takes time. Valuing understanding can result in
students’ development of mathematical skills’ Meehan also makes the point
that students / teachers need to recognize that Mathematics underpins many
Arts disciplines, and is not only necessary for achievement in academic
courses, but are an integral part of life skills.
Trinity Mathematics Waiver
Current admissions policy in Trinity stipulates that on the basis of information
contained in the evidence of a specific learning difficulty, the Disability Service
may recommend to that a matriculation requirement may be waived (the
modern language in the case of an applicant who has dyslexia or a hearing
impairment). Additionally, policy also states that in no circumstances will a
specific course requirement (for example Higher Level Leaving Certificate
Mathematics grade C3 for Engineering, or a language requirement specified for
a particular course) be waived.
However, having conducted an extensive review of Mathematics difficulties, the
Disability Service proposed that as part of the College Matriculation
requirements that Mathematics be open to a waiver under very specific
46
circumstances, such as students with dyscalculia or students who are blind and
who can demonstrate that they had limited access to the Mathematics
curriculum. It is not the intention that a Mathematics waiver be granted where
there is a mathematical requirement as a core component of a degree course,
for example within programmes such as Business, Sociology and Psychology.
In September 2009 Trinity introduced a Mathematics Waiver whereby students
may apply for a waiver of the Mathematics requirement if they function
intellectually at average or above average level, and have a specific learning
difficulty (dyscalculia) of such a degree of severity that they fail to achieve
expected levels of attainment in basic skills in Mathematics. Such evidence
must be provided by a fully qualified psychologist. It is hoped that this
initiative will enable students who might otherwise have been prevented from
participating in higher education.
3.4 Mathematics support in higher education
In 2009 Qualifax initiated an enquiry into the range of Mathematics support
provided to students in third level institutions. Colleges were asked to provide
details on access to specialized teaching and Mathematics assessment.
Responses which are summarized in the table below described initiatives that
include Mathematics programmes available to second level students, ‘second
chance’ admissions routes for students who did not matriculate in
Mathematics, and ongoing Mathematics support within college via a range of
strategies. Full text of these responses is available from
http://www.qualifax.ie/index.php?option=com_content&view=article&id=122&It
emid=183.
Of the 17 institutions surveyed all provided some form of Mathematics support
or advice, whether that be in the form of a dedicated centre, individual tutorials
or peer support. Only the NCI, IT Tallaght and IT Tralee provided outreach to
second level schools; the latter also engages in a pre-college Mathematics
skills course, Headstart. Many colleges permit the sitting of a special
Mathematics examination for prospective Engineering students, and there are
two interesting and unique approaches adopted by the American University
47
and Letterkenny IT. The American University will permit Business students
who did not matriculate in Mathematics to complete the first year of the
undergraduate course whilst preparing to re-take Math LC in the following
year. Letterkenny IT provide an Intensive Mathematics programme and
subsequent examination which permit students to apply for any vacant places.
Mac an Bhaird (2008) investigated the rationale and necessity for
Mathematics support in higher education. His paper reviews the information
collated by the Regional Centre for Excellence in Mathematics Teaching and
Learning (CEMTL) in the University of Limerick from all Mathematics support
facilities in Ireland. In addition he briefly discusses factors associated with
poor Mathematics performance at third level:
However, some of the main factors listed in [Lyons et al, 2003] and
elsewhere include bad publicity for Mathematics, negative attitudes
towards the subject, the high percentage of second-level students who
go onto third-level, the socio-economic background of the student,
increased competition for places, pressure on students and teachers to
achieve the highest possible points, little understanding of the context
or background of Mathematics, little appreciation of the applications of
Mathematics in everyday life, rote learning by heart, etc.
The Eureka Centre is hosted by the University of Loughborough
http://eureka.lboro.ac.uk.html and is specifically designed for students who are
not confident with Mathematics and statistics. The centre aims to help
students registered on any course through a series of events, resources and
information. These include automated Excel calculators for budgeting and
workshops targeted at mathematical tests as part of interview / employer
processes.
48
Section 4: Summary
4.1 Discussion
Public perception of dyscalculia is that this is a relatively ‘new’ addition to the
spectrum of specific learning difficulty however it is clear from a review of the
literature that identification of specific difficulty in the area of numerosity has
been investigated since the early 1990s. There is robust evidence for
hemispheric neurological deficits affecting numerical skills and reasoned
arguments for a hierarchical cognitive model for acquisition of mathematical
skills. Of particular interest is the emphasis on Mathematics as a language
system, and how this might affect mathematical understanding and
development. However, it also needs to be recognized that acquisition of
literacy and numeracy skills is not innate, and that perhaps the development of
arithmetical skills is as artificial as learning to read, which may be problematic
for some individuals where the brain ‘evolved for more general purposes’
McCrone (2002). Behaviourally, there is clear evidence of an inability to
visualise numbers and to represent them conceptually.
However, from an environmental perspective, consideration needs to be given
to the effects of ineffective teaching methods, lack of specialised support, the
time constraints of the curriculum and inappropriate assessment tools.
Evidence from teachers at primary level indicates that there are issues with
mathematical language, and assessment of achievement and identification of
difficulty using standard assessment methods. Proposed changes to early
years Mathematics programmes, and the delivery of the Mathematics
curriculum in secondary schools may address some of the problems in the
learning and teaching of Mathematics. Intervention programmes emphasise
the need for structured and staged approaches which require individualized
long term support. However, the shortfall in targeted support for those children
experiencing very real difficulties in Mathematics – compared to similar
provision for literacy - is inadequate.
Performance in state examinations in Mathematics over the period 2001 –
2008 indicates a lack of fluctuation in E, F and NG results, compared with A to
49
D grades. Additionally, results across the three core curriculum subjects of
Mathematics, English and Irish over the five-year period 2003 – 2008, suggest
a statistical difference which may be reflective of accommodations for dyslexia
in terms of the spelling and grammar waiver and exemption from Irish, and an
absence of accommodations for underlying Mathematics difficulties.
It is the view of the Royal Irish Academy that: “Mathematics is not perceived
simply as a service subject to be used in other disciplines and that
‘mathematical fluency’ is recognised as being particularly useful in a wide
range of professions (even when not explicitly required).” Whilst this is
undeniable, is it really the case that a qualification in Mathematics to Leaving
Certificate level is necessary for all pupils? What extra dimension does it
bring in terms of ‘real life’ skills? Is there an argument that, at its most basic
level, the content of the Foundation paper at Junior Certificate level is
sufficient for most people to function competently (mathematically) in everyday
life?
The Junior Certificate 2009 Foundation paper includes examination of
mathematical computation (long division, multiplication, square roots,
percentages); problem solving (calculating speed, time and distance, interest);
statistics (calculating the mean, histograms, constructing and interpreting
graphs); geometry (angles and areas) and probability. For those individuals
who have no desire to pursue higher level study with a mathematical
component, are basic skills in the above areas sufficient for competency in
everyday life skills such as managing a household budget and personal
finances? Arguably this has been sufficiently demonstrated in the UK system,
where Mathematics is compulsory only to GCSE level.
Johnson et al (2008) state that increasingly students transition to college and
only discover that they have a specific learning difficulty which was not
identified during second level education. Students with dyscalculia may still
achieve success in courses with mathematical components with the right
support and tutoring. However, reduced funding for supports means that
specific, individually tailored intervention is not always available. In addition,
50
such students need to be aware of the implications that an underlying difficulty
might have in terms of course and career choice. Whilst there are a number
of support strategies for students exhibiting difficulties with Mathematics,
students with dyscalculia require structured advice and guidance prior to
applying to the CAO, in terms of course content and course choice. Although
third level institutions strive to implement support programmes to address
difficulties with Mathematics, arguably such initiatives are a top down
approach aimed at ‘plugging the gap’ in mathematical knowledge.
Clearly there are courses where course content contains a mathematically
based core component (Psychology, Sociology, Science and Engineering, for
example), and thus competency is an expectation. However, issues that need
to be reflected upon include:
the relevance of a pass in Mathematics for arts courses which contain no
mathematical element, such as English, Classics or History
in addition to pure dyscalculia, consideration of a co-morbidity of several
disorders / conditions affecting acquisition of mathematical skills
acknowledgement that pupils with particular disabilities such as visual
impairment, have unequal access to the Mathematics curriculum
4.2 Further research
It is clear that any further discussion of the implications and incidence of
specific difficulty in Mathematics can only take place on the back of more in
depth statistical research and analysis. This might include:
Monitoring of Mathematics performance at primary level based on
models of acquisition of numerical concepts suggested by Geary and
Butterworth, against the new early years initiative Aistear.
Monitoring of Mathematics performance based at secondary level
measured against the new Project Maths curriculum.
51
Implementation of standardised assessment tools which are
appropriate for the assessment of specific difficulty in Mathematics, in
comparison to the mainstream.
Identification of students who do not matriculate on the basis of
Mathematics results, and their subsequent educational / work history.
Identification of the number of mature students registered on
undergraduate courses who did not matriculate in Mathematics.
Investigation of the psycho-educational profiles of students in second
level education who are struggling with both the Foundation and
Ordinary Level curriculum to determine either the presence of
dyscalculia, or poor Mathematics skills as a result of environmental
influences.
Pilot study using the Neuropsychological Test Battery for Number
Processing and Calculation in Children to determine incidence of
mathematical difficulty in primary school children.
52
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60
APPENDICES
Leaving Certificate Statistics 2001 – 2009
Foundation Level
Year
Total
number
of
students
A
grade
% of total
students
B
grade
% of total
students
C
grade
% of total
students
D
grade
% of total
students
E
grade
% of total
students
F
grade
% of total
students
NG
% of total
students
Total
number of
students
who failed
to
matriculate
2001
5227
412
7.8
1662
31.8
1741
33.3
1037
19.8
270
5.2
95
1.8
10
0.2
375
2002
5296
480
9.0
1678
25.2
1733
32.7
1028
19.4
260
4.9
103
1.9
14
0.3
377
2003
5702
696
12.2
1990
34.9
1739
30.4
952
16.6
245
4.3
70
1.2
10
0.2
325
2004
5832
580
10.0
1946
33.4
1863
32.0
1062
18.2
286
4.9
84
1.4
11
0.2
381
2005
5562
419
7.5
1733
31.1
1864
33.5
1115
20.0
319
5.7
102
1.8
10
0.2
431
2006
5104
400
7.9
1565
30.6
1775
34.7
1027
20.1
247
4.8
84
1.6
6
0.1
337
2007
5,580
545
9.7
1,908
34.2
1,742
31.3
1,008
18.1
252
4.5
106
1.9
19
0.3
377
2008
5,803
569
9.8
2,010
34.6
1,869
32.2
1,020
17.6
216
3.7
107
1.8
12
0.2
335
2009
6,212
10.9
36.0
30.9
17.1
3.7
1.2
0.2
2938
61
Ordinary Level
Year
Total
number
of
students
A
grade
% of
total
students
B
grade
% of
total
students
C
grade
% of
total
students
D
grade
% of
total
students
E
grade
% of
total
students
F
grade
% of
total
students
NG
% of
total
students
Total
number of
students
who failed
to
matriculate
2001
39984
5656
14.1
9974
24.9
9219
23.1
8515
21.3
4062
10.2
2228
5.6
330
0.8
6620
2002
38932
5281
13.6
9494
24.4
9575
25.0
8967
23.0
3675
9.4
1713
4.4
227
0.6
5615
2003
39101
4281
10.9
10384
26.5
10435
27.3
9520
24.4
3164
8.1
1198
3.1
119
0.3
4481
2004
37794
5937
15.7
10845
27.8
9390
24.9
7300
19.3
2893
7.7
1239
3.3
190
0.5
4332
2005
36773
4886
13.3
10001
27.3
9596
26.1
7872
21.4
2946
8.0
1290
3.5
182
0.5
4418
2006
35113
4018
11.4
8599
27.1
9774
27.2
7978
22.7
2872
8.3
1087
3.1
106
0.3
4065
2007
35077
4894
13.9
9738
27.7
9251
26.3
7137
20.4
2765
7.8
1143
3.3
147
0.4
4055
2008
35808
4483
12.5
10104
28.2
9507
26.6
7373
20.6
2857
8.0
1317
3.7
167
0.5
4341
2009
37273
12.7
27.2
27.4
22.3
7.5
2.5
0.3
TOTAL
37927
62
Higher Level
Year
Total
number
of
students
A
grade
% of total
students
B
grade
% of total
students
C
grade
% of total
students
D
grade
% of total
students
E
grade
% of total
students
F
grade
% of total
students
NG
% of total
students
Total
number of
students
who failed
to
matriculate
2001
9938
2099
21.2
2158
21.7
1886
21
1525
19.2
312
3.1
81
0.8
14
0.1
407
2002
9430
1245
13.2
2666
28.3
3154
33.5
1373
20.7
318
3.4
85
0.9
14
0.1
417
2003
9453
1257
13.3
2842
30.1
3106
32.9
1843
19.5
334
3.5
59
0.6
12
0.1
405
2004
9426
1534
16.2
2823
29.9
2940
31.2
1736
18.4
300
3.2
83
0.9
10
0.1
393
2005
9843
1525
15.5
3129
31.9
3029
30.7
1739
17.7
327
3.3
83
0.8
11
0.1
421
2006
9018
1280
14.2
3122
34.6
3014
33.4
1307
14.5
222
2.5
63
0.7
10
0.1
295
2007
8,388
1,287
15.4
2,836
33.8
2,595
30.9
1,344
16.0
253
3.0
60
0.7
13
0.2
326
2008
8,510
1,239
14.6
2,612
30.7
2,792
35.0
1,494
17.5
299
3.5
61
0.7
13
0.2
373
2009
8,420
15.0
33.1
32.5
16.0
3037
63
64
Comparison by Leaving Certificate level
2001
2002
2003
2004
2005
2006
2007
2008
A
8167
7006
6234
8051
6830
5698
6726
6291
B
13794
13838
15216
15614
14863
13293
14482
14726
C
12846
14462
15280
14193
14489
14563
13588
14168
D
11077
11368
12315
10098
10726
10312
9489
9887
E
4644
4253
3743
3479
3592
3341
3270
3372
F
2404
1901
1327
1406
1475
1234
1309
1485
NG
354
255
141
211
203
122
179
192
Total number of
pupils who
failed to
matriculate on
Mathematics
7402
6409
5211
5096
5270
4697
4758
5049
Comparison by year
Higher
2001
2002
2003
2004
2005
2006
2007
2008
A
2099
1245
1257
1534
1525
1280
1,287
1,239
B
2158
2666
2842
2823
3129
3122
2,836
2,612
C
1886
3154
3106
2940
3029
3014
2,595
2,792
D
1525
1373
1843
1736
1739
1307
1,344
1,494
E
312
318
334
300
327
222
253
299
F
81
85
59
83
83
63
60
61
NG
14
14
12
10
11
10
13
13
Non-
matriculation
407
417
405
393
421
295
326
373
Ordinary
A
5656
5281
4281
5937
4886
4018
4894
4483
B
9974
9494
10384
10845
10001
8599
9738
10104
C
9219
9575
10435
9390
9596
9774
9251
9507
D
8515
8967
9520
7300
7872
7978
7137
7373
E
4062
3675
3164
2893
2946
2872
2765
2857
F
2228
1713
1198
1239
1290
1087
1143
1317
NG
330
227
119
190
182
106
147
167
Non-
matriculation
6620
5615
4481
4322
4418
4065
4055
4341
65
Foundation
A
412
480
696
580
419
400
545
569
B
1662
1678
1990
1946
1733
1565
1,908
2,010
C
1741
1733
1739
1863
1864
1775
1,742
1,869
D
1037
1028
952
1062
1115
1027
1,008
1,020
E
270
260
245
286
319
247
252
216
F
95
103
70
84
102
84
106
107
NG
10
14
10
11
10
6
19
12
Non-
matriculation
375
377
325
381
431
337
377
335
2001
2002
2003
2004
2005
2006
2007
2008
Total number
of pupils who
failed to
matriculate
on
Mathematics
7402
6409
5211
5096
5270
4697
4758
5049
