Ann Williams a a Deakin University, Melbourne, Australia

This article was downloaded by: [Ann Williams] On: 18 October 2012, At: 01:26 Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Australian Journal of Learning Difficulties Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/rald20 A teacher's perspective of dyscalculia: Who counts? An interdisciplinary overview Ann Williams a a Deakin University, Melbourne, Australia Version of record first published: 18 Oct 2012. To cite this article: Ann Williams (2012): A teacher's perspective of dyscalculia: Who counts? An interdisciplinary overview, Australian Journal of Learning Difficulties, DOI:10.1080/19404158.2012.727840 To link to this article: http://dx.doi.org/10.1080/19404158.2012.727840 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-andconditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

Australian Journal of Learning Difficulties 2012, 1 16, ifirst article A teacher s perspective of dyscalculia: Who counts? An interdisciplinary overview Ann Williams* Deakin University, Melbourne, Australia There are many reasons why children have difficulties with mathematics. Dyscalculia is one of them. Teachers need to know what dyscalculia is in order to effectively assist the child to overcome the challenges it presents. However, the literature on the remediation of dyscalculia is in its infancy. The potential for multidisciplinary research is great, but would require maths educators to become involved in this issue, where previously they have been silent. Successful teachers know their students and value knowledge that will help them to help their students. As well as knowing what dyscalculia is, teachers need to understand its causes and have effective strategies to deal with it. Teachers also need to know how dyscalculia affects a child s self-belief system in order to counter the potentially devastating effects of poor self-esteem. Reportedly, there is an incidence rate of 5% in Australian schools, yet there is a lack of recognition, identification and diagnosis of dyscalculia despite the fact that the behavioural characteristics of dyscalculia are well defined and generally agreed on. There are a number of reasons why the diagnosis of dyscalculia is masked. The existence of another (possibly previously diagnosed specific learning disability) is one of them; for example 50% of dyslexics have dyscalculia. This paper seeks to address the who, what, how and why of dyscalculia. Keywords: dyscalculia, math difficulties, learning difficulties, learning disabilities, maths anxiety, maths self-efficacy, resilience. Introduction Dyscalculia is a specific learning disability (SLD) that affects how a child learns arithmetic. In Australia, the issue with dyscalculia is one of equity. Dyscalculic children are not screened, assessed or given the remediation they need because of the poor recognition of dyscalculia and SLDs in Australia. This poor recognition flows from government, both state and federal, through our universities to teachers, parents and the community at large. This lack of recognition stems historically from a decision made by the Australian Senate in 1976, which concluded that SLDs did not exist (House of Representatives Select Committee on Specific Learning Difficulties, 1976). At the same time, in the USA, the UK and Canada, similar research concluded quite the opposite. Now SLDs are legally acknowledged by those governments, but not by the Australian government. This lack of recognition has led to a key gap in the research. On the one hand, there has been work performed by psychologists on dyscalculia, on the other, work *Email: awillia@deakin.edu.au ISSN 1940-4158 print/issn 1940-4166 online Ó 2012 Learning Difficulties Australia http://dx.doi.org/10.1080/19404158.2012.727840 http://www.tandfonline.com

2 A. Williams has been performed on remediation strategies. However, the research instigated has not been effectively coordinated. The study of dyscalculia would therefore benefit from inter-disciplinary research, and recently universities have seen the advent of a variety of Centres of Educational Neuroscience, which have such an aim. This could be a fruitful response to the difficulties mentioned. In the opinion of Bird (2007), with which this author concurs, the identification and diagnosis of any condition is only relevant if that diagnosis helps the teacher gain a better understanding of the child and also gain some insights into interventions that will help the teacher help the child. From a teacher s perspective, reading the international literature is very difficult because different paradigms have their own terminologies. Also confusing are the different names for, and definitions of, dyscalculia, that these international researchers have used. Although the definition of dyscalculia is contentious, there is consensus about its behavioural characteristics and consensus about its prevalence, which is about 5% worldwide (Goswami, 2008; Landerl, Bevan, & Butterworth, 2004; Shalev & von Aster, 2008). This prevalence could, however, be masked by the presence of other learning or behavioural difficulties (co-morbidity) (Pennington, 2006). Dyscalculia can also be overlooked in gifted students who have an SLD. Another important point is that a child s cognitive domain cannot be separated from their self-belief system and so their self-efficacy, maths anxiety, resilience and any behavioural/emotional problems all affect how they learn and this, in turn, complicates remediation. Different terminologies for dyscalculia Dyscalculia is an SLD. However, this term has a number of variations, for example, learning disability, learning difficulty, learning disorder and learning difference. Other terms in the specific learning disability field are similarly often poorly or inconsistently defined. Terminology differences make it confusing for a reader beginning to study this area. In the opinion of this author, the reasons for this are political rather than scientific for example some parents do not like their children to be labeled as disabled and so the term difficulties is often used in preference. Examples of this can be found internationally. The World Health Organisation (WHO) describes dyscalculia as a specific disorder of arithmetic skills (WHO, 2011), while the Diagnostic and Statistical Manual (DSM-IV) sees it as a Mathematics disorder (American Psychiatric Association, 1995). In the UK, the Department for Education and Skills (DfES) uses dyscalculia (Learning Support Services, 2010). Dowker (2005) notes that although the research in the area speaks of mathematical difficulties, it would be more strictly appropriate to speak of arithmetical difficulties (p. 324). Mathematics has many branches, arithmetic being only (a relatively unimportant) one of them. This proliferation of names makes for confusion. Additionally, the use of different criteria, tests and exclusions makes it difficult to make comparisons and to investigate dyscalculia. The definition of dyscalculia Despite the different nomenclatures used, a severe disability in learning arithmetic is common to all of the definitions. Nevertheless, there are many reasons why children have difficulties with mathematics. Dyscalculia is only one of them and greater

Australian Journal of Learning Difficulties 3 precision is therefore required. However, the behavioural characteristics of dyscalculia are generally agreed on. The following characteristics (by no means exhaustive), can be found in the booklet What is dyscalculia? (Learning Support Services, 2010). They are: An inability to subitise (see without counting) even very small quantities (p. 2). This reflects a basic deficit in a core aspect of number sense (Geary, Bailey, & Hoard, 2009, p. 277). Other characteristics stem from poor number sense: Number sense relates to whole numbers, number relations and number operations (Jordan, Glutting, Ramineni, & Watkins, 2010, p. 182). For example, an inability to estimate whether a numerical answer is reasonable, an inability to count backwards reliably, an inability to tell which of two numbers is larger. Another characteristic is confusing signs; þ and 6 or 7 and / and a reliance on counting on strategies using fingers rather than efficient methods of calculation. Dyscalculics also often have a problem managing money, as well as problems with different aspects of time. For example, delay in learning to tell the time from an analogue clock face and an inability to manage time in their daily lives (Burny, Vaicke, & Desoete, 2012). Other characteristics relate to memory problems. For example, weaknesses in both short-term and long-term memory, difficulties in learning multiplication tables (they are able to learn them but forget them overnight) and sequencing and directional confusion (difficulty in distinguishing between left and right or East and West). There are two characteristics of particular interest to the author. One is the issue of time because of its implications for teaching: the other, not mentioned in the booklet, is finger gnosia because of its possible use as a screener (Penner-Wilger et al., 2005). Different aspects of time are important when teaching alternative strategies to dyscalculics. Their brains need more time to process calculations (Butterworth, 2003; Jordan & Montani, 1997; Silverman, 2002), hence they need more time in examinations. They often have a poor sense of elapsed time and so are poorly organised (Silverman, 2002). They also have difficulty telling the time using an analogue clock, so need a digital clock. Neuro-scientific research now explains how finger knowledge could help children with dyscalculia (Fayol & Seron, 2005; Kaufmann, 2008). Butterworth (as cited in Kaufmann, 2008) states that young children throughout the world from different cultures use their fingers to help them to count before they attend school. Further research by Kaufmann (2008) has shown a neuro-functional link between fingers and numbers. Finger gnosis was found to be a specific predictor of numerical abilities by Noel (2005) (as cited in Kaufmann, 2008). Finger knowledge is not usually checked or taught. This is a gap in the literature and leads to a suggestion by the author that the use and knowledge of fingers should be taught explicitly to all children in their early years. There are three paradigms that have been used to explain the observed behavioural characteristics of dyscalculia and to differentiate between dyscalculia and other maths difficulties. Firstly, neuropsychologists believe that dyscalculia is a neurological disorder (Butterworth & Laurillard, 2010; Dehaene, Piazza, Pinel, & Cohen, 2005; Desoete, Ceulemans, Roeyers, & Huylebroeck, 2009; Fayol & Seron, 2005; Kaufmann, 2008; Landerl et al., 2004; Penner-Wilger et al., 2005; Piazza et al., 2010; Rousselle & Noel, 2007; Shalev & von Aster, 2008; Wilson & Waldie, 2010). Secondly, psychologists believe that it is a working memory deficit (Geary, 2010; Geary, Hamson, & Hoard, 2000; Keeler & Swanson, 2001). A third view shared by

4 A. Williams some other education researchers is that dyscalculia does not exist and the behavioural characteristics have emotional and/or experiential causes (Gifford, 2005b, 2006). There are two methods of defining dyscalculia using any of these three paradigms: qualitatively or quantitatively. Qualitative descriptions are given by the DSM-IV (American Psychiatric Association, 1995), DfES, UK (Learning Support Services, 2010) and WHO (2011). The quantitative methods used follow one or other of three broad categories of model. The discrepancy model is one in which there is a large difference between the achievement of the child in, for example, arithmetic, and the child s achievement in general. The second model is the severity model. This uses percentiles or other cut-off points to note the severity of the condition. Finally, the resistance to treatment model is one where a child does not respond to any remedial interventions and continues to use immature strategies such as, for example, finger counting instead of recalling number bonds (Shalev & von Aster, 2008). Some mathematics education researchers in the UK state that dyscalculia does not exist. There is, at the moment, a debate between some mathematics education researchers and others from non-educational paradigms (Gifford, 2005a, p. 1) concerning the nature and definition of dyscalculia. On the one hand, Gifford (2005b, 2006) states that the characteristics found in dyscalculics are due to delayed learning or attitudinal factors, not an innate disability. Gifford (2006) subscribes to the term mathematics difficulties, which has the implication that poor maths achievement is due to such external factors as inappropriate teaching, missing lessons, lack of motivation, poor attention and maths anxiety. Other researchers in neuroscience (Butterworth, 2010; Dehaene et al., 2005; Fayol & Seron, 2005; Landerl et al., 2004) have a different understanding. They suggest that innate representational systems play a key role in how we learn arithmetic. Moreover, they prefer the name dyscalculia or developmental dyscalculia. Geary (1993), Rourke (1993) and Siegler (1998) (as cited in Butterworth, 2005) prefer the name maths disabilities (MD). These psychologists think that the characteristics of dyscalculia are more likely to be due to cognitive disorders of memory, working memory, processing speed and delayed development. From this author s viewpoint, having taught dyscalculic children, dyscalculia does exist and its behavioural characteristics are beyond doubt. Self-efficacy Self-efficacy is a response of an individual to gain control over their lives (Bandura, 1995). In the framework of this paper, of particular interest is research on how selfefficacy beliefs affect children s motivation and achievement in maths (Zimmerman, 1999). In the general community, mathematics is seen as being a difficult subject. People who are good at mathematics are perceived as being brainy. This leaves the maths teacher with a major issue of how to motivate students. This is particularly so in the secondary years eight and nine, where students tend to lack academic motivation. Schunk (1991) discusses motivation in terms of self-efficacy and suggests that in order to increase academic motivation, one method that could be used would be increasing a student s self-efficacy. Schunk further suggests that students who have high selfefficacy in maths should work harder and persist longer in the face of difficulties than those students who are less self-efficacious. Pajares (1996) acknowledges the importance of high self-efficacy in maths problem solving and goes further to mention that self-efficacy to solve maths problems was more predictive of that performance

Australian Journal of Learning Difficulties 5 than were prior determinants such as sex or maths background or than variables such as maths anxiety, maths self-concept or perceived usefulness of maths (p. 326). Self-efficacy is also important in children with dyscalculia. It can be seen from the previous discussion that dyscalculia is primarily a deficit of counting and, hence, is arithmetic. Arithmetic is usually the first branch of mathematics taught in primary schools, so dyscalculics get their sense of self-efficacy from their failure in arithmetic from this early age. Further, most maths textbooks and good teachers try to extend all students, and particularly more able children, by giving them harder examples of the concept being taught. So, effectively, children practice a particular concept until they fail. This has a profound effect on their self-efficacy. Success engenders selfefficacy, whilst they are slow to recover their sense of efficacy following failure or setbacks (Bandura, 1994, p. 72). Another important influence on a dyscalculic s self-efficacy is that of their peers (Bandura, 1994). A common practice amongst primary teachers in motivating children to learn their multiplication tables is to place in a prominent position a list of all the children in the class with the multiplication tables they have learned. Dyscalculic children who cannot learn their tables can see quite clearly that all their peers have, for example, learnt the two times table easily, whilst they cannot learn it. The conclusion that they could easily reach is that they are dumb and stupid, which results in a plummeting of their self-efficacy in relation to arithmetics (Bandura, 1994). In order to increase the self-efficacy of dyscalculics, teachers must ensure that children are able to achieve success and limit the situations where they might fail. They should also explain that arithmetic is only one branch of mathematics. Other branches of mathematics require little or no arithmetic. All children with specific learning disabilities, including dyscalculics, are at risk of a low sense of self-worth. They are especially sensitive to their relative standing among the peers in activities that determine prestige and popularity (Bandura, 1994, p. 81), which can result in low social self-efficacy. A low sense of self-efficacy can lead to maths anxiety, however self-efficacy is more predictive of maths performance than is maths anxiety (Zimmerman, 1999). Maths anxiety Maths is a cognitively-based academic skill. As such it could present some issues when teaching all children, and particularly those who have maths anxiety. Maths anxiety was first described by Richardson and Suinn (1972), who said that: It is a feeling of tension and anxiety that interferes with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations (as cited in Ashcraft, Krause, & Hopko, 2007, p. 329). Ashcraft et al. (2007) define people with high maths anxiety as those who score more than one standard deviation above the mean on a Maths Anxiety Rating Scale. This is a Likert scale. It has proved to be remarkably reliable over the years but has been developed so that now there are a number of variations. With this definition, about 17% of the population have maths anxiety. In Australia there is an incidence of 4% of children with general anxiety (Siddons & Lancaster, 2004), but there is no research on the incidence of maths anxiety. Maths anxiety shows itself by negative self-talk and ruminating thoughts, which are thought to impinge on memory capacity and could be a reason why maths anxiety affects performance (Prevatt, Welles, Huijun, & Proctor, 2010). Prevatt goes further

6 A. Williams to suggest that anxiety-reduction techniques be used to improve the performance of students with maths anxiety. Maths is the only academic subject to have a phobia associated with it (Ashcraft & Ridley, 2005). Maths anxiety (MA) can affect parents and teachers. Parents can pass on their negative feelings to their children (Sparks, 2011). Interestingly, pre-service primary teachers averaged the highest on an American maths anxiety test (Ashcraft & Krause, 2007). The maths anxiety of teachers can also affect the maths anxiety of their students. The article quotes Ansari as stating that Teacher math anxiety is really an epidemic. I think a lot of people go into elementary teaching because they don t want to teach high school math or science (Sparks, 2011, p. 2). The traditional way that maths has been taught increases maths anxiety. Textbooks and worksheets all tend to have the harder sums at the end of the exercise. Teachers encourage students to do as many as they can. Teachers believe that the sign of a good mathematician is one who can do hard sums. The size effect (Fayol & Seron, 2005) shows that performance decreases as the size of numbers increases. The implication is that it is inherently the bigger numbers themselves that make the computation more difficult. Students tend to be taught drill rather than an understanding of the concept being taught. In order to allay children s anxiety and increase their confidence, a maths teacher should try to create a classroom where children feel able and confident to ask questions. An environment should be encouraged in which children feel safe and have no time or other pressures, and where they are able to arrive at their own understanding of the concept being taught. In Australia, the National Assessment Program Literacy and Numeracy (NAPLAN) is becoming increasingly important. The stress of timed tests and their concomitant pressure make maths anxiety a real issue for many children when having to do a NAPLAN test. The stress a child experiences during such tests causes an increase in anxiety, and maths anxiety compromises performance as working memory is involved (Ashcraft & Ridley, 2005). Despite the fact that maths anxiety is so prevalent in the community, it has not been studied widely in relation to dyscalculia. In fact, Rubinstein and Tannock (2010) were the first to study this relationship. Dyscalculics know from an early age that what other children find easy to do, they find either difficult or impossible (Rubinsten & Tannock, 2010). Because of this they tend to use avoidance strategies to ease the pressures they feel. Rubinstein and Tannock found that in dyscalculic children, arithmetic is related with fear but also For people with dyscalculia, childhood difficulties with numerical processes and poor math achievement intensify math anxiety, which further impedes math achievement (p. 10). Fear and maths anxiety are not the only negative feelings aroused by maths. Negative feelings lead some children to react by exhibiting inappropriate behaviour in the classroom. Emotional/behavioural problems There are a significant number of children who have both SLDs and emotional and/ or behavioural disorders (EBD), to the extent that the label SLD/EBD has been used (Rock, Fessler, & Church, 1997). This is not surprising as there is a likelihood that children s SLD and EBD share the same neurological substrates. What is surprising,

Australian Journal of Learning Difficulties 7 perhaps, is that this co-morbidity has not been studied previously. That emotions are the seat of learning has been known for a long time, but now, with the advent of neuro-imaging techniques, this connection can be seen at the neurological level (Hinton, Miyamoto, & Della-Chiesa, 2008). Furthermore, such research has found that learning is more effective if stress and fear are minimised and a positive and motivating environment is provided for students. Unfortunately, a positive approach has not been the experience for many students with SLD in Australia. People with SLD are over represented amongst the prison population and the unemployed and in mental health statistics. Juvenile delinquents have a significant incidence of SLDs (76%) and many students with SLDs experience emotional difficulties (Watson & Boman, 2005). Because of the lack of recognition of dyscalculia/slds in Australia, teachers are ill-equipped to help and support children with these issues. This is unfortunate, as research shows that the relationship between children and their teachers makes a difference to children s academic success and emotional wellbeing (Watson & Boman, 2005). School culture Family, school and community environments all affect how children grow, learn and develop. Research has shown that the culture of the school is particularly important. A culture in which there is a positive working environment, with strong leadership, a sense of community and a culture which fosters flexibility in organisation and a commitment to inclusive education is better able to support those experiencing learning difficulties (Watson & Boman, 2005). Given a supportive school community children with dyscalculia/slds may be given individual help. Because of its hierarchical nature this is harder to achieve in the secondary school context than in the primary school context. However, it should be possible to extract children from a classroom for individual help in both primary and secondary settings, although this depends on there being at least one person in every school who is knowledgeable of, and trained in, SLDs. The extraction of a child from a class needs the permission of the parents and a commitment from the principal of the school. In the experience of the author, special needs teachers are often seen as an extra resource to call on if there is a shortage of other teachers. Resilience In the affective domain, self-efficacy, maths anxiety and emotional/behaviour problems all affect the emotional well-being of dyscalculic children. They can lead to negative outcomes, as previously discussed. However, those children with dyscalculia/slds who overcome their problems and develop into successful adults are thought to be resilient. Risk and protective factors interact with the presence of a learning disability to facilitate or impede adjustment. However, there is evidence that attitude is the most important predictor of success in life, rather than the dyscalculia/sld (Firth, 2010). As Firth states, this is important because attitude can be changed (p. 26). These risk and protective factors may be internal characteristics of the individual or external characteristics. Characteristics identified by Firth are goal setting, self-awareness, perseverance, pro-activity, helpful support systems, and emotional coping strategies

8 A. Williams (p. 26). A resilience-coping programme to help all children, but particularly those with SLDs, is also mentioned by Firth. The characteristics mentioned by Firth are similar to those protective clusters of characteristics identified by Werner (1993), specifically:. Cluster 1: temperamental characteristics that enable the child to form a strong relationship with a caring adult.. Cluster 2: cognitive abilities which enable the child to plan, organise and reach his/her potential.. Cluster 3: care giving style of the parents, which reflects competence and foster self-esteem in the child.. Cluster 4: supportive adults who foster trust and act as gatekeepers for the future, for example Grandparents, youth leaders and members of church groups.. Cluster 5: a second chance at major life transitions, for example from high school to workplace, from single state to married state. Gardynik and McDonald (2005) have identified twice exceptional (2E) children as being particularly vulnerable. This research reveals that extrinsic factors are very relevant to a 2E child s success in life. If teachers teach to a child s strengths rather than concentrating on their weaknesses there is an increase in self-concept, motivation and task completion, while also improving basic skills. Screening instruments for dyscalculia In Australia, SLDs are not widely understood in the general community, nor are they an integral part of a mainstream teacher s education. The field of SLDs is highly specialised. By contrast, in the UK teachers are routinely given help, advice and support. An example of this is the booklet What is dyscalculia? (Learning Support Services, 2010), which is available to teachers. Silverman (2002) states that screening should be done early. It is more effective to help dyscalculic children if they are diagnosed in early primary school, as they can be taught alternative strategies to ameliorate the deleterious and cumulative effects of dyscalculia on both mathematical and personal development. Screening is time consuming, especially if an entire class needs to be screened to identify the dyscalculic children. Most instruments tend to only identify children as having dyscalculic tendencies and so a further, more comprehensive, assessment of cognition and intelligence is required to diagnose dyscalculia confidently. Some children may get a confirmation of dyscalculia, but false positives are possible because of the cut-off points on the sub-scales. These false positives may cause problems if not handled sensitively. Screening instruments are only a first step on the road to full assessment. There are a number of ways in which a child may be screened for dyscalculia. The types of screener that have been developed relate to the disciplines of the researchers. Butterworth (2003), a neuropsychologist, has developed a commercial screening instrument called the Dyscalculia Screener, which is widely used in the UK. It is consistent with his theory that dyscalculia is a core deficit in numerosity. This screener is computer-based and takes about 20 minutes to administer. It uses an

Australian Journal of Learning Difficulties 9 item-timed calculation to differentiate between, for example, the children who are finger counters and those who are fluent. It consists of four tests: (1) Simple reaction time: this test enables a more accurate calculation of response time. Slow response time is a characteristic of dyscalculia. (2) Dot enumeration: this is basically a measure of subitisation. The inability to subitise is a characteristic of dyscalculia. (3) Number comparison or numerical stroop: many dyscalculic children have difficulty comparing the magnitude of numbers. (4) Arithmetic achievement test (addition and multiplication): this test distinguishes those children who can recall the number facts from those who use other strategies. (p. 13) However, there have been reservations registered about this screening instrument (Maddux & Owens, 2003; Voutsina & Ismail, 2007). Other screeners are being developed by neuropsychologists who exploit one or more of the characteristics specific to dyscalculia. The screener being developed by Penner-Wilger et al. (2005) uses finger gnosia, fine motor ability and subitisation, while the one being developed by Piazza et al. (2010) uses number acuity. On the other hand, Jordan (2012), a psychologist, has developed a quantitatively different screener called the Number Sense Screener (NSS), to identify MD. This is consistent with her theory that MD is a problem with number sense (Jordan et al., 2010). Jordan s NSS tests for different aspects of number sense. The tests are of counting, number recognition, number comparison, nonverbal calculation, story problems and number combinations. The NSS is an untimed measure that takes approximately 15 minutes to administer. The items assess counting knowledge and principles (for example set enumeration, knowledge of the count sequence to at least 10, and principles of oneto-one correspondence, cardinality and stable order); number recognition (for example the ability to name written symbols such as 13, 37, and 82); number knowledge (for example what number comes right after seven or which number is bigger, five or four?); nonverbal addition/subtraction calculations (Jordan et al., 2010, p. 186). Another screener being developed is one by Geary et al. (2009) a cognitive developmental psychologist. The test is called the Number Sets Test. It was developed to assess the speed and accuracy with which children can identify and process quantities represented by Arabic numerals and object sets (p. 265). No screening instruments have been developed in Australia for teacher use. This is a clear gap. The other major consideration when using screening instruments is the co-morbidity or co-occurrence of other SLDs with dyscalculia, which could mask the identification of dyscalculia. The effect of co-morbidity and gifted/learning disabled on identification of dyscalculia Children with one SLD may also have other SLDs. Wilson and Waldie s (2010) research found dyscalculia can be co-morbid with:. dyslexia, which is a difficulty with literacy and has a co-morbidity prevalence of roughly 50%,

10 A. Williams. attention deficit/hyperactivity disorder (ADHD), difficulties with attention, concentration and behavioural control. This has a co-morbidity of roughly 40% and. central auditory processing difficulty (CAPD), difficulty with high level processing of auditory information, (no known co-morbidity prevalence). There is poor understanding amongst researchers of the reason for such high comorbidity rates. The reasons are complex and multi-factorial. Some factors involved are specific to dyscalculia, while others are shared with other SLDs (Landerl & Moll, 2010). It is quite possible for a child to have two or more SLDs, for example, dyslexia, dyscalculia and central auditory processing difficulty (CAPD). This can make the diagnosis of a particular SLD very difficult as the presence of one SLD may mask the presence of another. Furthermore, Jordan (2007) explains that children with a double deficit of both math and reading difficulties (MD/RD) do not respond as well to intervention as those children who only have a math difficulty (MD). The differentiating factor between the two groups is maths word problem solving ability. The former group (MD/RD) finding it much more difficult than the latter (RD). Both dyslexia and dyscalculia have been called the hidden SLDs as children do not usually exhibit early inappropriate or aberrant behaviour. In the case of ADHD, Asperger s syndrome and autism, however, early behaviour is often seen as different and is the predominant concern for a parent or teacher. Hence they may not look further for the co-morbidity of other SLDs. Another possible factor affecting the identification of dyscalculia is the comorbidity of giftedness and SLDs. These children are often called twice exceptional (2E) children. A full discussion of the problems of identification of gifted children and the difference between a gifted and a talented child is beyond the scope of this paper. The identification of a child as being gifted is fraught with difficulties (Coleman, 2003; West, 1997). However, in terms of 2E children the common practice of averaging the performance of subtests to give a Verbal IQ and a Performance IQ on Weschler Intelligence Scales for Children also tends to mask SLD in 2E children. Beckley (1998) acknowledges this and further specifies that a 2E child might belong to one of three neglected groups. Silverman (2002) acknowledges the challenges in assessing 2E children and the fact that their SLD may be undiagnosed. Alternatively, Lovett and Lewandowski (2006) feel that the issue in identifying 2E children is one of assessors unsound procedures, which need further evidence based research. Despite the contentious definition of dyscalculia and the gaps mentioned, there is hope that one area of interdisciplinary research might show promise. Interdisciplinary research and centres for educational neuroscience There is a key problem identified in current educational research, namely, how to make the research immediately relevant to the transformation of teaching practice and to bring about improvements in the classroom. A response to this is action research (Somekh, Stronach, Lewin, Nolan, & Stake, 2005). Another response could be interdisciplinary research conducted in centres for educational neuroscience. McCluskey and Studdert (2011) advocate interdisciplinary research in general, despite its inherent difficulties, because of the gaps in research that an

Australian Journal of Learning Difficulties 11 interdisciplinary study can identify and the different perspectives that can be used (as cited in Dropulich, 2011a, 2011b). The inherent difficulties are the different language and methodologies used by different disciplines, together with rigid organisational structures. McEwan (2010, pp. 3203 3205) states that: The demarcations between subjects were mere conveniences, or historical accidents, or the inertia of tradition. In particular, for dyscalculia, a fruitful interdisciplinary area for research is educational neuroscience, (Butterworth, Varma, & Laurillard, 2011; Tytler & Prain, 2009; van Nes, 2011). The interaction is a two-way field because teachers can incorporate the results of neuroscientific research into their practice and neuroscience can learn how brain disorders affect learning. Around the world there have been recent openings of centres of educational neuroscience, one of which is in Australia the Australian Research Council s Centre for Cognitive Disorders, at Macquarie University. There are a number of examples where educational neuroscience has either confirmed current thinking or elucidated it. For example multisensory teaching methods, which have long been believed to be beneficial for learning (Szucs & Goswami, 2007, p. 122), have recently been confirmed as an important intervention, there is evidence from functional magnetic resonance imaging to support this. Additionally, Thomson, Goswami and Baldeweg (2006) (as cited in Szucs & Goswami, 2007) state that there is promising research using electro-encephalograph techniques to find neural markers, which could identify children at risk of dyslexia or dyscalculia from a very early age (Butterworth et al., 2011). It can be seen from the previous discussion that interdisciplinary studies are at the forefront of SLD research. So this area of research should be particularly fruitful when it comes to improving how children with dyscalculia are taught. It may not be easy to find a way to use the results of neuroscientific research in maths education, but if such a research project can be done, it would enrich both disciplines (Szucs & Goswami, 2007; van Nes, 2011). Assessment of dyscalculia Assessment and remediation go hand in hand, as the accurate assessment of children will determine the form of remediation. Teaching mathematics is easy. It is a matter of analysing the topic to be taught and finding where the child is on firm ground and working forward from there (Emerson & Babtie, 2010; Poustie, 2000). However, although the teaching of mathematics is easy, the correct assessment of children who may have multiple disabilities is not (Isaksson, Lindqvist, & Bergstrom, 2010). A common ground is, however, that a careful assessment must be performed before any remediation processes commence. Remediation of dyscalculia The following remediation strategies could apply equally to children who have difficulties learning mathematics and to dyscalculic children. Pre-school counting skills are generally recognised as being an important prerequisite for an understanding of arithmetic (Howell & Kemp, 2010). Gersten et al. (2009) found that the use of heuristics and explicit instruction provided important increases in effect size (p. 1228). Explicit instruction would be familiar to many special needs teachers who use concrete materials such as Cuisenaire rods and paddle

12 A. Williams pop sticks (Bird, 2007; Wadlington & Wadlington, 2008). For many years, teachers have felt that multi-sensory instruction, together with the use of games, might be efficacious in helping children to learn. The use of games is felt to be particularly useful for children who have maths anxiety. Using games disguises the fact that the child is learning maths and also gives the child a chance to win, which gives them a positive experience and positive verbal feedback, thereby increasing self-efficacy. There are broadly two categories of games available: maths computer games and real games. Each has its own advantages and disadvantages. The advantage of a real game, for example dominoes, is the social interaction involved and the fact that it is multi-sensory. Dominoes can be played at a number of different levels. The most basic level helps dyscalculic children match the canonical pattern of, for example, five dots to the word five. The advantage of maths computer games is that they are highly motivating, providing they are not too easy and can focus on a particular skill. However, many math computer games are too hard for dyscalculic children and there are only a few maths computer games that focus on the understanding of a topic rather than just a skill. One such game which does both is The Number Bonds Game. This is an ipod app. that uses virtual Cuisenaire rods to teach the number bonds up to ten. This also has the advantage of actual multisensory rods could also be used. Another useful adjunct to classroom teaching is Number Shark. This is highly structured and was developed by special needs teachers. It also has reward games. In general, the use of games to help students with their maths has not yet been empirically proven to result in better performance; mixed success having been evident. A discussion of this is beyond the scope of this paper. However, the author suggests that a portfolio of games should be developed to complement the portfolio of lesson plans that teachers develop, use, adapt and refine over the years. A teacher should try many strategies to help an individual dyscalculic child in order to find the most effective one for that child. An effective strategy combining a heuristic and explicit approach is shown in Figure 1. Another strategy would be to ask the child to verbalise their thoughts as they work. Some children might find this helpful as it uses their auditory strengths (Ostad & Sorensen, 2007). One factor in helping dyscalculics is language. All words used should be familiar to the child and the correct mathematical terms should NOT be used until a concept is thoroughly understood. Also missing from the literature are alternative methods of learning multiplication tables. If a child has dyslexia and dyscalculia it is unlikely that they will be able to remember number facts because of the hypothesis that an impairment of rote verbal memory is partially responsible for dyscalculia in children with dyslexia (Dehaene et al., 2005, p. 449). So a strategy using stories should be considered (Silverman, 2002; Walker, 2000). None of the above interventions need necessarily take up too much time at the individual level but, as Shaddock, Giorcelli Figure 1. A successful strategy to help a dyscalculic child.

Australian Journal of Learning Difficulties 13 and Smith (2007) discuss, it may take more time for a teacher setting up an inclusive classroom. Conclusion In conclusion, it is clear from the literature that dyscalculic Australian children are not getting access to an equitable education. They have a poor number sense, have difficulties understanding basic arithmetic concepts and cannot learn multiplication tables. This leaves them at risk of developing maths anxiety, poor self-efficacy and behavioural problems. If dyscalculia were to be legally recognised by both the federal government and all the states, it would have far reaching implications in terms of policy and money. In Australia, research performed on SLDs has yet to be developed because of their historical lack of recognition. Increased recognition of dyscalculia in the community is also likely to increase teachers awareness as parents ask for help for their dyscalculic children. This, in turn, would add pressure for teachers and maths educators in particular to fill the gap between research conducted on dyscalculia by psychologists and the work of maths research educators and/or teachers. This would be challenging as it seems to be clear from the literature on dyscalculia that, for any person new to the subject, the proliferation of terminologies and definitions causes some confusion. However, there is consensus regarding the behavioural characteristics of dyscalculia. This provides a qualitative method of identification of dyscalculia. Other methods are screening instruments. However, researchers from different disciplines use different instruments, which makes it difficult to compare researchers results, and so makes it difficult to form an accurate view of the incidence of dyscalculia in the general population. This is further masked by the high prevalence of other co-occurring conditions. Assessment and remediation of dyscalculic children are closely allied as children s individual strengths and weaknesses must be ascertained before any remedial action is taken. Although research performed on dyscalculia is behind that of dyslexia, both in terms of time and money, there is hope for the future with the interest shown by the opening of a number of centres of educational neuroscience around the world. These inter-disciplinary organisations could provide a clearer understanding of the neural substrates that are believed to underpin dyscalculia and so offer teachers targeted and evidence based advice as to which strategies to try and which might be successful. References American Psychiatric Association. (1995). Diagnostic and statistical manual of mental disorders. Retrieved, August 20, 2011 from psychiatryonline.com.ezproxy-f. deakin.edu.au/content.aspx?aid¼7341 Ashcraft, M.H., & Krause, J.A. (2007). Working memory, math performance and math anxiety. Psychonomic Bulletin & Review, 14(2), 243 248. Ashcraft, M.H., Krause, J.A., & Hopko, D.R. (2007). Is math anxiety a mathematical learning disability? In D.B. Berch & M.M.M. Mazzocco (Eds.), Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (pp. 329 348). Baltimore, MD: Paul H. Brookes. Ashcraft, M.H., & Ridley, K.S. (2005). Math anxiety and its cognitive consequences. In J.I.D. Camobell (Ed.), Handbook of mathematical cognition (pp. 315 327). New York: Psychology Press.

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