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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