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A literature review is an objective, thorough summary and critical analysis of the relevant available research and non-research literature on the topic being studied (Hart, 1998). Its goal is to bring the reader up-to-date with current literature on a topic and form the basis for another goal, such as the justification for future research in the area. A good literature review gathers information about a particular subject from many sources. It is well written and contains few if any personal biases. It should contain a clear search and selection strategy (Rugg and Marian, 2007). Good structuring is essential to enhance the flow and readability of the review. Accurate use of terminology is important and jargon should be kept to a minimum. Referencing should be accurate throughout (Roberts, 2004).
According to Sharp and Howard (1996), there are two major reasons for reviewing the literature. The first is to provide the reader with a comprehensive background for understanding current knowledge and highlighting the significance of new research. It can inspire research ideas by identifying gaps or inconsistencies in a body of knowledge. The second is to critical review the existing literature.
The literature review therefore, should be a description and critical analysis of what other authors have written (Jankowicz, 2000). We will draw out the key points and trends, by providing readers with the necessary background knowledge to our research questions and objectives and establish the boundaries of our own research. For this reason, we will need to discuss critically the work that has already been undertaken on "children in primary school with Dyscalculia" and reference that work. The key to writing the critical literature review is to link together the different ideas we find in the literature, in order to form a coherent argument which set in context and justify our research.
Many researchers (such as Dees, 2000; Hart, 1998) suggest that in order to write our critical review, we will need to:
include the concepts and theories which are relevant to our chosen area,
show how our research relates to previous theories and published researches,
consider and discuss any significant supports and controversies to our ideas,
support our arguments with valid evidence in a logical manner,
justify our arguments by referencing previous research, by recognised experts in our research area
The literature sources that we are going to use can be divided into three categories: primary, secondary and tertiary (table 2.1). Primary literature sources include sources such as reports and articles. Secondary literature sources include books and Internet. Tertiary literature sources include dictionaries and abstracts, as well as encyclopaedias and bibliographies.
Table 2.1: Literature sources available (McMillan & Weyers, 2007)
Newspaper and magazine articles
2.2 UNDERSTANDING DYSCALCULIA
The identification of dyscalculia is not a simple task because the mathematical competence depends on effective functioning of many aspects of pupils. In order to decide the cause of a dyscalculic pupil each of these aspects will need to be explored so that the individual's weaknesses and strengths can be assessed. Only once such an assessment is made can appropriate and adequate approaches be given. (Jan Poustie, 2000).
2.2.1 UNDERLYING CAUSES OF DYSCALCULIA
As with other types of learning difficulties, dyscalculia is believed to involve the language and visual processing centers of the brain. (Butterworth, 2004). Henderson, et. al. (2003) have pointed out that dyscalculia may be inherited or can be caused by problems with brain development. Just as we find one family where parents and children are all very capable mathematicians, we often identify another in which mathematical difficulties are very common (Lerner 1993; Gaddes and Edgell, 1994).
Obviously, we must remember that environmental factors may also play a role in the transmission of a talent for, or a difficulty in, mathematics. A family where parents are mathematically competent may be a very different learning environment to one in which parents are uncertain about basic mathematics themselves (Gaddes and Edgell 1994).
However, it is clear that the biological influences do also play a significant role in dyscalculia. For instance, Shalev and Gross-Tur (2001) have pointed out that about 50% of the siblings of a pupil with dyscalculia can be expected to have it as well. Parents and siblings of a pupil with dyscalculia are ten times more likely to have dyscalculia than members of the general population (Henderson, et. al. 2003).
More specifically, according to Lansdown (1978), Reisman (1982), Kipriotakis (1989) and Trouli (1992), the factors that affect the performance of pupils in school lessons, particularly in mathematics could be classified into two broad categories: intrinsic or individual factors, which are associated with specific characteristics and peculiarities of each pupil and external or environmental factors that are associated with the pupil's family, school and the wider cultural environment. Surveys concluded that these factors are very important and can cause disturbances in pupils' learning or even lead to a high percentage of pupils in underperforming or general failure in mathematics (Henderson, et. al. 2003).
2.2.2 THE BIOLOGICAL BASIS OF LEARNING
Some pupils will be blessed with brains that seem purpose-built for mathematics. Other pupils may have dyscalculia and find mathematical thinking very difficult in comparison to other types of thinking and learning.
All learning and thinking develops through the evolution of specialised structures within the brain. Some of these structures are equipped to deal with the types of processes involved in mathematics (Bird, 2007). Glynis (2005) argues that between individuals there will naturally be differences in the efficiency with which these structures work. Furthermore, according to the National Research Council (2002), learning changes the physical structure of the brain. Also, these structural changes reconfigure the organisation and function of the brain. Finally, different parts of the brain may be ready to learn at different times.
How does the brain learn?
Brain cells are called neurons and there are at least 100 billion of them
in the human brain (Nolte, 1988). Each neuron has a cell body and a tree-like structure of branches called dendrites. The dendrites from one cell connect with those from other cells at the synapses, and exchange information.
Luria (1973) has pointed out that the synapses are a very important part of the learning process. Those that are frequently used become more complex and much more efficient at processing information. Those that are neglected fail to develop. Synapses develop in two main ways. In the first few years of life a child produces many, many more synapses than will ever be used. Incoming experiences activate some synapses which develop increased connections with other neurons. Synapses that are not used are selectively lost (Gaddes and Edgell, 1994). Furthermore, according to, Shalev, et.el. (1995). Furthermore, according to, Shalev, et.el. (1995), the second way of development is through the addition of new synapses throughout life, through an 'on demand' process. If a new synapse is needed the brain will, within limits create one!
According to the National Research Council (2000), four groups of rats were studied. The first group had a month of challenging acrobatic activities as they learnt to transverse an elevated obstacle course, the second group of rats had compulsory exercise on a treadmill, the third group had unlimited access to an activity wheel, and a control group of 'cage potatoes' had no exercise at all. At the end of the month the acrobats were the stand out winners when synapses were counted. Learning created synapses, exercise alone did not! But what, we might ask; do acrobatic rats have to do with dyscalculic pupils? The basic principles are the same. Learning new skills successfully develops brain connections.
Every individual learner has a brain circuitry that varies in its aptitudes when compared with others. Pupils with dyscalculia generally have more difficulty than most in building at least some of the neurological systems required for calculation and skilled mathematical thinking (Gaddes and Edgell, 1994).
Learning accumulates gradually as the brain structures develop
Henderson, et. al. (2003) have pointed out that learning is a gradual, accumulative biological process. It depends on the pupil having a range of experiences, of graded difficulty, which triggers the development of the underlying neurological connections. Like building up physical fitness, the process of building mathematical fitness is one of regular training, with gradual increments of difficulty and plenty of practice sessions. Although the brain is not a muscle, many of the same principles as for physical fitness apply when we are looking to optimise its learning (Butterworth, 1999).
The best learning occurs at the boundary between what the pupils can already do, and what they cannot yet do. Tasks that the pupils can do with just a little help will generate the best learning. The teacher will activate existing synapses and build up their strength and efficiency (Chinn, 2004). New synapses may also be developed through structured learning experiences (National Research Council, 2000).
What about right and left brain learners?
Nolte (1988), argues that the two halves of the brain are specialised, with the left hemisphere usually dealing primarily with language-related, sequenced information processing, and the right brain being more commonly involved in spatial reasoning and 'whole picture' thinking. However, the brain is an extraordinary complex organism, with many interconnecting and complementary systems. While the majority of us do have language located mainly in the left hemisphere and spatial reasoning in the right hemisphere, there are frequent exceptions to this pattern of 'wiring' (Socol, et.al. 1991). Gaddes (1994) has pointed out that although the brain is in two, quite distinct, halves there is a complex and intricate system of interconnection between the functions of both hemispheres. While some of us may be predominantly left- or right- brain thinkers, we can generally use either style of thinking quite efficiently, depending on the demands of a particular situation (Ashcraft, 1992). Sometimes the thinking is primarily right brain, sometimes left: but most often both thinking styles merge, each supplementing the other to maximise the efficiency of the thinking process.
Mathematics requires the integration of many aspects of thinking, sometimes making heavy demands on language skills, sometimes requiring spatial reasoning to take the lead. Many pupils with dyscalculia will have deficits in language or in spatial reasoning (Butterworth, 1999).
It is generally agreed that the mathematical competence depends on effective functioning of the pupil's
position and spatial conception
sequencing, laterality and direction
attitudinal and emotional factors
(Lerner, 1993; Haring and McCormick, 1990; Bley and Thornton, 1995; Miller and Mercer, 1998). Successful mathematical thinking therefore often requires the efficient integration of several different types of learning and thinking.
2.2.3 SYMPTOMS OF DYSCALCULIA
According to Miller and Mercer (1998), there is a lack of research on
that field of study. At the same time, other disorders, such as dyslexia, another learning disorder on the field of language acquirement, received considerably more attention of the scientific community.
However, after some researches which had been carried out and which we will refer later to; the most important factors associated to dyscalculia, may be linked to poor performance in arithmetic function and operation (addition, subtraction, multiplication, division) or linked to understanding the basic concepts which are prerequisites for the pupil's mathematical performance in school (Attwood, 2002; Butterworth, 2005 ). Lerner (1993) argues that, these factors are often identified in infants and young pupils. However, she states that each pupil is unique and not all pupils who have dyscalculia will present the same characteristics.
Difficulties regarding the position and spatial conception
The symptoms presented by pupils concerning the position and spatial conception are varied. According to Gifford (2005), they do not often complete their homework or copy symbols inaccurately. They, furthermore, tend to confuse the exercises and frequently face difficulties knowing what number comes after a particular number or what number comes before a particular number. In addition the use of a calculator may result in errors. Once dyscalculic learners have selected the appropriate calculation, they may then have difficulties between the stages of reading it on a page and transferring it to a calculator keyboard (Glynis, 2005). They face difficulties in identifying and using the desired button, especially with the symbols of the four operations. They may also face difficulties in auditory perception. Some dyscalculic pupils (and, indeed, others with auditory problems) have difficulty in distinguishing spoken numbers such as 9, 19 and 90, concluded to hearing problems in the classroom (confusing sounds). Moreover, many pupils with dyscalculia have difficulty in maintaining focus on a particular task. This will affect their attention to the teacher's instructions meaning that they begin tasks with incomplete information and that is usually why they are left behind from the rest of the classroom (Geary, 2004).
Difficulties regarding perceptual skills
Perceptual skills associated with visual disturbances may result to an incorrect reading or writing of numbers, symbols or operations and this leads to erroneous results in the execution of the four operations (Dowker, 2005).
According to Geary (2004), dyscalculic pupils with such difficulties may confuse certain written numerical symbols and letters. They may tend to reverse them or invert them (turn them upside down). Common errors are to confuse the numbers 6 and 9, 3 and 5, E and number 3. Additionally, and this may relate as much to a central processing deficit as to a visual perceptual deficit, many dyscalculic children have problems in discriminating the basic mathematical symbols ie: +, -, ÷, x (Dowker, 2004). Furthermore, Fuchs, et. al. (2004) point out that another common problem concerning the perceptual skills is the reversals of numbers with two-digits. Pupils who have the tendency to make reversals of individual digits, as mentioned previously, they can also reverse the digits of a two-digit number, during reading or writing (13-31 or 12-21, etc). Dyscalculic children also find telling the time on an analogue clock difficult and may have poor understanding relating to the passage of time, unlike their peers. Their reduced perception of time, affects directly on the overall ability to design and plan their activities (El-Naggar, 1996). Finally, they confront difficulties recognizing and using coins because they are unable to distinguish the differences in their sizes (Geary, 2005).
Difficulties regarding the sequencing, laterality and directional
The performance of pupils with dyscalculia may seriously be affected due to confusion in the sequencing, laterality and directional (Glynis, 2005). Many dyscalculic pupils have difficulty with concepts of leftness and rightness. They also have difficulty in remembering the correct sequence of the months of the year, the days of the week or the seasons. Levine (1999) argues that this is associated with letter and number reversals.
In our Western culture, reading is a left to right scanning process. Often, children are required to scan from left to right in maths, for instance:
6 + 2 + 1 =
23 - 6 =
4 x 2 =
If pupils have difficulties with directionality and sequencing, they may attempt to move from right to left rather than left to right. Fuchs and Fuchs (2005) have pointed out that in mathematics there is an additional problem over literacy which is that there are different start points of different types of sums.
In the above sum, pupils start at the right and work towards the left. However, Fuchs and Fuchs (2005) argue that in the following subtraction, the dyscalculic pupils start at the bottom right and have to remember to take the lower number from the top number, 'borrow' from the left upper number and move left:
In long division, in contrast, the pupils start at the left and move towards the right and move downwards whilst writing the answer at the top (Swanson and
Problems of remembering sequence, laterality and direction make life hard for a dyscalculic child who has to cope with all three of these issues - often at the same time.
Difficulties regarding insufficient memory (working, long-term memory)
It is clear that pupils with dyscalculia frequently have memory deficits (Wilson and Swanson, 2001; Gathercole and Pickering, 2000; Geary, 2004). The deficits may be in working memory (immediate memory that is used to store on-going, short-term information), or in long-term memory (where we store information for a period of time and then retrieve it).
Geary (2004) states that, "Many pupils with dyscalculia have difficulties in retrieving basic arithmetic facts from long term memory, a deficit that often does not improve". Wilson and Swanson (2001) showed that working memory deficits in both the verbal and visuospatial domains contribute to mathematical difficulties. Ericsson and Kintsch (1995) said that memory difficulties are particularly apparent when the pupils lack a solid conceptual understanding of the process they are attempting. If, for example, they understand the process of long multiplication, then they are far less dependent on remembering the steps in the process. If the pupils with dyscalculia forget how to do it, they can, with some time and application, work out what to do and proceed correctly, because now they understand what they have to do and are able to work out the correct procedure for themselves.
Glynis (2005) has pointed out that one element of working memory is the language-based 'phonological loop' where words are held temporarily while they are part of a process (such as calculating a total). Difficulties in working memory may mean that the dyscalculic pupils have to depend on counting on their fingers or making tally marks to keep a hold of where they are, and what they are doing when they are calculating (Dowker, 2004). Poor working memory will make it hard to recite multiplication tables because they forget where they are in the sequence as they recite (Glynis, 2005). Poor working memory in the visuospatial domain may mean that the
pupil's 'visuospatial sketchpad' is unreliable (Mazzocco, 2005). This in turn may mean that they import errors into their work, not because they do not understand what they are doing, but because they make mistakes when they copy their working out or transfer geometric forms from one place to another. (Chinn, 2000)
Ericsson and Kintsch (1995) have pointed out that long-term memory deficits are usually very obvious in dyscalculic pupils. Often pupils can be seen carefully counting their fingers to reconstitute a number fact that they have failed to retrieve from their memory (Dowker, 2004). To be successful in basic mathematics they have to remember written symbols (such as + and x) and the processes that they represent. They also have to remember a large number of basic number facts such as 3+7=10, 6x6= 36, double 12 is 24 (Chinn, 2004). Poor long-term memory also affects retention of and thus knowledge of mathematical formulae and methods.
Difficulties regarding the language of Mathematics
Butterworth (2004) argues that mathematics thinking is to a considerable extent dependant on the language used in mathematics. This language needs to be learned thoroughly in order for a child to be successful in mathematics. Unfortunately, many pupils with dyscalculia have significant difficulties with the language of mathematics.
Early language is the staring point for mathematical thinking
We know that language is a very important tool for the communication of information, questions and ideas. Much of a young child's comprehension of mathematical concepts will be strongly bound up with their language development (Bley and Thornton, 1995). They will learn about words and expressions such as more, less, bigger, longer, before, after, the same as, twice enough. They will learn to count and label shapes, generally before they start school. Dowker (2004) argues that pupils with insufficient language skills may have general difficulties in their language skills development or they may have specific difficulty with the language linked to mathematical concepts such as position, relationship, and size.
Mathematical thinking will be more difficult without language skills
According to Strawser and Miller (2001), language is a very important vehicle for thinking. It is very difficult to deal with new ideas, understand abstract concepts, handle information and ideas, solve problems and recall previous learning without using proper language (Glynis, 2005). If pupils do not have adequate language skills, their ability to handle some concepts and ideas will be reduced. El-Naggar (1996) argues that it is difficult to 'capture' concepts such as double, bigger than, twice
as much as, two triangles and a circle or two million people if they cannot use words as labels; but with adequate external or internal language dyscalculic pupils can represent not only objects but also actions and concepts, all essential elements in mathematical reasoning and learning. Language is important as a way of carrying thinking forward (Attwood, 2002). It also helps to link new ideas with ones already mastered (Poustie, J. (2000). Language is also very important in helping to handle sequences and maintain the order of information or a procedure. Sequenced language may be particularly difficult for pupils with dyscalculia (Attwood, 2002).
Children with Dyscalculia may not understand the language they recite
Henderson et. al. (2003) have pointed out that some pupils with dyscalculia have precise difficulty in linking the actions that they are supposed to carry out with significant language. All too often they have learned a script that is, mainly a pointless incantation-when they have no idea what all of this truly means (Levine, 1999).
Children with Dyscalculia may not use internal language to help with mathematics
According to Garnett (1998), pupils with dyscalculia often cannot, or do not, use their own internal language to manage the mathematical tasks they are attempting. They may imitate actions that they see their teachers or peers demonstrate, but fail to connect with the internal language that is supposed to accompany the process being taught (Wright et.al., 2006).
Mathematics has its own unique language
Mathematics has a language all of its own: sometimes it uses words or written symbols that are unique and need to be learned. For instance, tens, subtract, addend, multiplication and algorithms are seldom heard outside the mathematics classroom (Willis, 1990). Other words may be familiar to the dyscalculic pupils, but are used in
quite a different context: olden times/times tables, take away food/take away sums, stereo unit/tens and units. For pupils who may be having trouble with their first language, mathematical language can cause a lot of anxiety and frustration (DfES, 2005). What does equals mean to the dyscalculic child who thinks of it as eagles? What is the difference between: What is the time? I make it ten to two and What is ten times two?
Language helps to monitor thinking and learning
According to Hannel (2005) language is also very important in the way in which we monitor our own thinking and learning. Dyscalculic pupils may have difficulty in monitoring their own learning through internal, language-related thinking. As a result they may often not ask for help, because they cannot put into words what it is they do not know or understand (El-Naggar, 1996).
Difficulties regarding attitudinal and emotional factors
Levine (2002) argues that dyscalculia can have a dramatic impact on a pupil's mind set towards mathematics. Pupils who start off at school eager to learn may quickly become confused and frustrated by the difficulties that they encounter. It is generally true that motivation and teaching-approaches are two very good barometers of a pupil's confidence in any subject area. Unwilling and unco-operative learners are very frequently found to lack the foundation skills that they need to participate successfully in the task that is set (Glynis, 2005). According to Glynis (2005), mathematics does require clear thinking and steady concentration and we know that anxiety can seriously disrupt these important elements of mathematics thinking. Because anxiety, frustration, confusion and failure so often disrupt motivation, any pupil who shows disinterest, poor behaviour or lack of application to mathematics tasks should be assumed to be experiencing difficulties completing what they are being asked to do.
2.3 DYSCALCULIA AND DYSLEXIA
Chinn and Ashcroft (2006) argue that of the two disorders, dyslexia is usually more readily recognised and remedied, while dyscalculia often seems to go unnoticed. Perhaps this is because mathematics is seen as an intrinsically challenging subject, where it is more 'normal' and therefore acceptable to have difficulties. Butterworth and Yeo (2004) suggest that somewhere between 20% - 60% of pupils have both. Deficits in language and working memory may well create problems in the acquisition of both mathematics and literacy skills. However, the two disorders can also exist in isolation from each other.
Kosc (1974) argues: "It is obvious that the various symptoms concerning dyscalculia can occur mainly in conjunction with other symptoms related to reduce brain functions, especially with dyslexia and dysgrafia or other disorders of the central nervous system, due to brain damage". Porpodas (1993) notes that "the problem of a dyslexic child is mainly limited to the reading and writing language while learning other symbolic systems such as mathematical symbols mathematical and physical concepts, music elements, etc. may or may not be affected".
According to Stasinou (1999) "the syndrome of dyslexia is primarily a deficit in the individual's ability to handle symbolic material" and continues by wondering, "How many dyslexic children have dyscalculia and how many children with dyscalculia are or are not dyslexic". He notes that "there is a wide spread view which distinguishes the two distinct syndromes, the syndrome of dyslexia and that of dyscalculia". Furthermore, Doxa (1994) suggests that "some dyslexic pupils apart from the difficulties they face in literacy skills, they face as well difficulties in units of mathematics. Areas of the central nervous system used for the reading and writing, are also used for arithmetic calculations, designing of graphs, shapes, etc".
Agalioti (2000) argues that "the number of dyslexic pupils, who face severe mathematical difficulties, is the same or harsher than that concerning the reading difficulties and it is about 63%. A fairly high percentage of dyslexic pupils face serious difficulties also in mathematics". All the researches concede that this percentage is quite big (60%-70%). According to Chasty (1993), this percentage reaches the 70%, while according to Joffe (1990) it approximates the 60%.
Apart from these views, there are also views concerning dyscalculia which are referred to as not a distinct and autonomous entity in the area of specific learning difficulties (Agalioti 2000). Most of these researchers have dealt with the syndrome of dyslexia (Joffe 1990; Miles 1992; Chinn and Ashcroft, 1993). Miles (1992), considering also excessive the use of the term dyscalculia when all the learning difficulties can be included under the term of 'dyslexia'. However, Chinn and Ashcroft (1993) admit that there is a small percentage of pupils, who face dyscalculia as the specific problem.
2.4 RECOMMENDED TECHNIQUES FOR SPECIFIC DEFICITS ASSOCIATED WITH DYSCALCULIA
Arguably, the greatest challenge in order to create an efficient policy for meeting the needs of pupils with dyscalculia is the heterogeneous nature of the disability. As Mazzocco & Myers (2003) articulate: "Subtype differentiation is essential for guidelines on identification and intervention strategies, and it is important to recognize that these subtypes may not share one primary, core deficit". For this reason, McGlaughlin, et. al., (2005) call for interventions which focus both on precise math techniques and on other cognitive deficits in areas such as the working memory and the nonverbal-visual skills. Vogel, et.al., (1993) also mention the need for individual diagnoses and a deficit-based approach to intervention.
In the book Developmental Variation and Learning Disorders, Levine (1999) recommended sets of methods for working with dyscalculic pupils that correspond to specific cognitive deficits. For difficulties with reading perception, Levine (1999) suggests that pupils in order to check their work effectively, they can use visual patterns to pair with written explanations, design a personalised mathematics glossary of terms and concepts, use math-related computer software to increase conceptual understanding, and practice a lot in order to develop good estimating skills. For problems concerning working memory, Levine (1999) recommends that pupils concentrate on one task or sub task at a time, develop consistent approaches to problem solving, use self-monitoring procedures, practice extended arithmetical problems in their head to improve their working memory capacity as much as possible, and summarize complicated guidance or solution practices before attempting the solution. El-Naggar (1996) has pointed out that teachers help pupils create mnemonic aids and check over written work as often as possible to certify that when questioned about a mistake, these pupils recall the thought process at the time at which it was made. Levine (1999) argues that when teachers helping pupils who have deficits in nonverbal reasoning, they have to use manipulatives to emphasize mathematical terms and concepts. In addition, the researcher notes that real-world applications of these concepts are also effective.
According to Levine (1999), pupils with dyscalculia, should use flashcards to aid in the development of automaticity in essential arithmetic concepts, calculations, and manipulations. Moreover, mathematical computer software helps to keep pupils focused, and rehearsal of common calculation methods and multi-step procedures.
Nolting (2000) suggests a set of techniques to aid dyscalculic pupils with difficulties and indicates which assessment accommodations pupils may need to ensure that their mathematics knowledge is being tested fairly. These suggestions are the most extensive and the most closely suited to each potential deficit. Reisman, (1982) however, emphasizes that in order to clarify that the pupils who most need the learning skills will be successfully taught, teachers need to focus on direct intervention techniques such as "orientation courses, combination of curricula that focus on affective issues into developmental reading and writing classes, special study skills courses and teaching services with learning assistants trained to work with dyscalculic pupils".
Regarding pupils with visual deficits, Nolting (2000) advises note taking, student tape recorders, key word sheets, video records, enlarged graph paper for those with difficulty writing legibly, and copies of teach notes or teach notes online. In terms of considerations, dyscalculic pupils should be allowed twice the usual amount of time for test-taking, and also the tests ought to be 150 percent enlarged photocopies (Nolting, 2000).
For visual deficits, pupils who face difficulties with visual memory and/or visual discrimination, Nolting (2000) proposes the followings; teachers to help dyscalculic pupils read textbooks, enlarged photocopies, taking notes in different colored pens, and the use of talking calculators. Regarding tests, he points out that these pupils will need twice the usual amount of time and may benefit from enlarged tests, tests that are hand-written in felt-tip pen, the capitalization of all variables (such as A's and B's contrary to a's and b's), color-coded tests and finally these pupils should be accommodated within a private testing area. Pupils with auditory processing and short-term memory deficits have to use school-assigned note-takers, tape recorders, videotapes and books on tape. In "extreme" situations, Nolting (2000) recommends the use of Computer-Aided Real-time Translation (CART) devices.
Concerning dyscalculic pupils with deficits in the cluster of working memory, Lerner (1993) advices the use of calculators as often as possible, the creation and review of concept cards, videotapes that revise mathematical concepts and anything that may limit the requirements placed on long-term memory during problem solving. In terms of testing adjustments, he recommends the same adjustments listed for pupils with short-term memory deficits with the additional use of a calculator and a technique and/or fact sheet. At the end Nolting (2000) proposes that pupils with deficits in working memory will benefit from mathematics videotapes, mathematics study skills training, note cards, calculators and computer programs. During tests, he approves open-book testing (i.e., a testing situation in which the student's homework assignments, previous tests, and a calculator are at pupil's hand).
2.5 RESEARCH APPROACHES TO DYSCALCULIA
Slade and Russel (1971) presented a descriptive case study concerning four young pupils with difficulties in arithmetic function and operations. The researchers considered that the four pupils had dyscalculia and were based on the fact that while their mental condition was within normal limits, their performance on simple operations and problems concerning money on the other hand was very low. However, during their research they resulted to the fact that three of the four pupils were having psychiatric problems. These pupils are excluded from the category of people with learning difficulties because of the psychiatric problems.
Deluca et. al., (1991), Rourke and Finlayson (1978) researched dyscalculic pupils, trying to identify the cognitive strengths and weaknesses presented by them. The researchers tried to associate the performance in reading, writing and mathematics, with specific cognitive strengths and weaknesses.
At the beginning, the pupils were split into three groups according to their performance in reading, writing and mathematics, and in order to assess these skills, the researchers used some known tests, like Wide Range Achievement Test (WRAT). All pupils were having deficiencies in their performance in mathematics. In the first group, there were those pupils who were equally deficient in reading, writing and mathematics. In the second group, there were the pupils who had serious deficiencies in reading and writing and less in mathematics. Finally, in the third group, the pupils had deficiencies in mathematics, and they were at the average or above the average in reading and writing. During the research, pupils of both the first and the second group were found to have lower verbal than practical IQ in IQ tests and their performance in tests regarding their perceptual skills, their spatial conception and their skills concerning the sequencing, laterality and direction were better than pupils' of the third group. The pupils of the last group had lower performance in practical than verbal IQ and their performance on tests of verbal and auditory-perceptual skills were better than those of pupils from the other two groups. The researchers considered that the performance of pupils in the first and the second group was expected, since there were malfunctions of the left cerebral hemisphere, while, on the other hand, the performance of pupils in the third group was also expected due to malfunctions of the right hemisphere (Deluca et. al., 1991; Rourke and Finlayson, 1978). These results are not similar to those regarding acalculia which suggested that the skills concerning the sequencing, laterality and direction may also result from malfunctions of the left hemisphere (Dahmen et al., 1982; Grafman et al, 1982). The Swedish neurologist, Henschen (1926) used the condition acalculia for the first time, in order to mark the loss of computing capacity that occurs after brain damage (Henschen, 1926).
The research group of Professor Rourke (1993) proceeded also to a qualitative analysis concerning the errors of pupils in both the above groups. The pupils in the second group, who had better performance in mathematics rather than in reading and writing, used to avoid unfamiliar arithmetic operations and problems that required reading skills. They also had difficulties in remembering simple arithmetic function and operations or even algorithmic steps. They had reading difficulties and also memory deficiencies. The pupils in the third group who had normal performance in reading and writing skills and poor performance in mathematics, they made various mistakes which can be grouped into the following categories: mistakes concerning the sequencing, laterality and direction, the spatial conception, errors in the algorithmic procedure, memory deficits and reduced critical ability and logical thinking, which seems to be their main feature. According to Rourke (1993) and his colleagues, the pupils in the third group had non-verbal learning disabilities or had symptoms concerning the dysfunction of the right hemisphere, which affects the mobility, the spatial conception and the socio-emotional adjustment (Rourke and Conway 1998).
Geary (1994) made an effort in order to categorise the learning difficulties concerning mathematics reviewing relevant researches relying on diagnostic findings. He established three subtypes, having always concerns about the demarcation; the first subtype had as predominant characteristics the memory deficits. This type is associated with certain types of reading difficulties and is attached to the left cerebral hemisphere malfunctions. The second subtype with predominant features the difficulties in using mathematic procedures, strategies, and algorithms, is associated with reading difficulties and probably combined with dysfunctions of the left cerebral hemisphere. The third subtype with primary characteristic the disturbances in the spatial conception, is linked to malfunctions of the right cerebral hemisphere and especially of the posterior regions of the hemisphere. This type is not associated with reading difficulties and phonological deficits.
McCloskey et al., (1985) proposed a model concerning the type of errors made by pupils, while trying to write numbers. These pupils had the ability to write individual digits, instead they face serious difficulties in writing numbers with two-digits, (i.e., they used to write 874 instead of 8074 unlike their teacher's oral instructions).
Shalev et al., (1988) in order to interpret the performance of pupils with dyscalculia used the model proposed by McCloskey et al (1985). In the survey, took part pupils aged 8, 9 to 15 years old with dyscalculia and also some of their peers without dyscalculia called the control group. The researchers evaluated the comprehension of numbers and the execution of arithmetic function and operations. This survey concluded that there were no significant differences between the two groups in terms of the understanding and production of numbers. However, there were significant differences in the performance of arithmetic function and operations. Concerning the recovery of simple arithmetic function and operations (multiplication table, etc.) and algorithms, the researchers found significant differences between the two groups. In the tests concerning the cognitive functions, the pupils with dyscalculia presented significant deficiencies. All of them were dyslexics, ten had symptoms of dysgrafia (difficulty in writing) and nine pupils faced difficulties in gaining access to the desired word given to them orally. Those pupils were facing also language difficulties.
The relationship between problems in arithmetic function and operations and whether the pupils are right or left brain learners, was studied by Shalev et al., (1995) in a sample of 25 pupils with dyscalculia. For the evaluation of the children they used a mathematics test. The pupils were also underwent neurological and neuropsychological testing. The researchers diagnosed that 13 pupils had dysfunctions of the left hemisphere. That was based on mild neurological signs on the right side of their brain, also on the practical IQ which was higher than the verbal (P. IQ> V. IQ), on the emergence of dyslexia and finally, that there were no disturbances in spatial conception. For the remaining 12 children, the researchers diagnosed malfunction of the right hemisphere (Shalev et al., 1995). The criteria were; mild neurological signs on the left side of their brain, verbal IQ which was higher than the practical (V. IQ> P. IQ), disruptions in spatial conception and finally, language skills which were normal. These two groups of children were at the same age, gender and socioeconomic status of their families. The results of this study revealed that the malfunctions of either of the two cerebral hemispheres prevent the acquisition of mathematical skills. Nevertheless mathematical difficulties are more severe in the case of dysfunctions of the left hemisphere. More specifically, the results of the studies showed that the group with dysfunctions of the left hemisphere had significantly lower performance than the other group, in three areas: in the operations of addition and subtraction and also in the complex operations of multiplication and division. (Shalev et al., (1995)).