Dyscalculia An Overview of Research on Learning Disability

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1 Dyscalculia An Overview of Research on Learning Disability Teresa Guillemot Teacher Education Programme, Mathematics and Computing ABSTRACT Dyscalculia is a mathematical learning disorder where the mathematical ability is far below expected for a person s age, intelligence and education. Researchers have found evidence that such a disability exists and due to their findings there is a need to address dyscalculia as an important educational issue in mathematics. Students with these mathematical learning disabilities in the Swedish school system today have the opportunity to be diagnosed through special tests. If these students are diagnosed with dyscalculia then they can receive special education corresponding to their needs. The concept of dyscalculia is being discussed, researched, tested and scrutinized in several institutions around the world. Test results indicate that somewhere between 3-6 % of the population suffers from severe mathematical learning disorders, even to the extent that it impairs the daily life. Recently there have been discoveries made suggesting that different mathematical operations and number handling are connected to specific areas in the brain. If these neuropsychological findings can be scientifically justified, researchers are one step closer to solve the puzzle of dyscalculia, also known as number blindness. The aim of this article is to discuss the different views of dyscalculia, both supporting and critical towards treating dyscalculia as an important educational issue, and illuminate the possible advantages of diagnosing mathematical learning disabilities. Keywords Dyscalculia, Learning disabilities, Mathematics education INTRODUCTION In cognitive science and the pedagogical and neuropsychological fields, different views exist of mathematical learning disorders. Dyscalculia has been defined as a specific mathematical learning disorder (MLD), where the mathematical ability is far below expected for a person s age, intelligence, and education. Researchers and scientists in the field have different opinions on whether dyscalculia as a diagnosis is possible to establish scientifically or not, or where it originates from in case it exists. This survey is focused on recent work from both perspectives, looking into possible explanations of the phenomenon. Teachers, parents and students are often aware of the fact that there are a number of persons with special difficulties in learning mathematics. Pupils with learning disorders may feel uncomfortable in learning situations, unless they are being treated in a way that facilitates their learning. In order to give the proper education to a pupil with a learning disorder, it is essential to understand the most suitable ways in which students with learning disabilities can acquire mathematical understanding, which is closely associated with abstract thinking. Findings pointing towards an area in the brain connected to abstract thinking were published by the French neurologist Broca already in 1861 (Butterworth 1, 1999). As early as in 1924 The German researcher Gerstmann examined patients that had lost their ability to perform mathematical operations due to damage to the left side of the brain. This disability was later called Gerstmann s syndrome (Butterworth, 1999). This article begins with the definitions of the topics discussed, followed by a survey of recent work of authors in the field of research. The research suggesting neurophysiologic basis for dyscalculia is important because of frequent opinion that learning difficulties are solely the result of lack of student s diligence and attentiveness. Also, even if a neurophysiological picture of the disorder can provide an explanation, the fact that the brain exhibits remarkable plasticity remains, which leaves us with the question: is it possible by exercise to remedy the neuropsychological state linked to dyscalculia? A discussion of the surveyed work follows in the end of the article. References in the text are given as endnotes. Definitions Dyscalculia is a broad term that includes many different kinds of difficulties in learning mathematics. Dyscalculia is also used for naming general difficulties in learning basic mathematics. Some of the different definitions include: Developmental dyscalculia: according to a number of researchers (Kosc, 1974, Shalev & Gross-Tsur, 1993, 2001, Wilson & Dehaene 2, 2007) this type of dyscalculia originates from a specific impairment in the brain function. However, this definition has not always been fully acknowledged by the research community. The neuroscience research field has though progressed, where studies are conducted concerning the brain (Wilson & Dehaene, 2007). Acalculia: where a person has lost all sense of meaning of numbers or where the person still understands numbers but is unable to perform basic calculations like addition and multiplication, due to a neurological damage. Pseudo-dyscalculia: finding math difficult, based on emotional blockage or a confidence problem. There are also several different theories of causes of dyscalculia: The Core deficit hypothesis: analogies are made between dyslexia 3 and dyscalculia. Even though the knowledge of its behavioural manifestations is incomplete (Wilson & Dehaene, 2007), neuroimaging results point to a specific region in the brain when it comes to numerical handling. Some types of dyscalculia are caused by an impairment of function in different regions in the brain (Butterworth, 1999; Gersten & Chard, 1999; Robinson, Menchetti & Torgesen, 2002). Working memory hypothesis; Wilson and Dehaene (2007) suggest a controversial explanation of dyscalculia, namely a 1

2 deficit in the components of the working memory, which is supported by Adler (2001). Research results on Dyscalculia Von Aster (2000) wrote about the state of dyscalculia, describing a study made at the Department of Child and Adolescent Psychiatry in Zürich, defining three subtypes of dyscalculia; The Verbal subtype was found in children with the largest difficulty when counting, especially when counting mentally, not using pen and paper. Subtraction was the most problematic operation but also remembering methods of counting. Children with the Verbal subtypes also had other learning disabilities. In nine of eleven children in the study, Von Aster found similar dyslexic conditions, and six of the children also had Attention-Deficit/Hyperactivity Disorder (ADHD). The Arabic subtype of dyscalculia included children with problems reading Arabic numbers out loud and writing them after hearing the numbers, but these children had no further learning disabilities. The third group of dyscalculia, called the Pervasive subtype, included disabilities with most mathematical thinking, writing, spelling and the children also possessed emotional and behavioural problems. The latest study found refuting the existence of dyscalculia as a phenomenon was made by Sjöberg in His opinion was that the results from previous studies were inconclusive when pointing towards dyscalculia as an indication of difficulties in mathematics. The idea that the pupils had not put enough effort and time into their work with math was a more likely scenario. Sjöberg claimed that the research results showing that 6 % of compulsory school pupils suffered from dyscalculia were incorrectly interpreted. Sjöberg presented his conclusions in a thesis where he studied 200 pupils from grade five in the Swedish compulsory school to the second year of education in the upper secondary school during a six year period. Of these students, 13 were having specific mathematical problems. Material was gathered regularly in which pupils filled in questionnaires. A total of 100 classroom observations were made and 40 of them were video recorded. On two occasions Sjöberg conducted in-depth interviews with the 13 pupils who had specific mathematical problems. Sjöberg noticed other components that were possibly affecting the pupil s low understanding of math like a low work rate during the lessons, disturbance in the classroom environment, large groups and also emotional stress in test situations. Some students claim to get blackouts during tests due to experiencing a high stress level. The results of the study did not refute the existence of dyscalculia as a phenomenon, but the findings made Sjöberg draw the conclusion that diagnosing dyscalculia should be exercised cautiously, if at all. All the students in his study finished math studies in upper secondary school with satisfying results. This led Sjöberg to the conclusion that in order to make a diagnose properly the whole environment has to be examined, not only the pupil with the assumed mathematical learning disorder. Unlike Sjöberg; Shalev, Gross-Tsur et al. (2000) suggested a prevalence of 3 6 % school children with dyscalculia. The range was interpreted as a consequence of different definitions of developmental dyscalculia. To determine prevalence we must develop a scientific and clinical consensus as to what constitutes a learning disability and which definition best describes the problem. (Shalev, Gross-Tsur et al., 2000). The definitions of dyscalculia were not precise and Shalev, Gross- Tsur et al. referred to different options like; a specific, genetically determined learning disability in a child with average intelligence and a learning disability in mathematics, the diagnosis of which is established when arithmetic performance is substantially below that expected for age, intelligence and education. The first study of dyscalculic prevalence was made by Kosc in Bratislava in Different mathematical assignments of basic character were used. Kosc regarded the children with results below the 10 th percentile as dyscalculic, which was 6.4 % of the 375 fifth-graders participating in the study. Several other studies have been made in other countries showing similar figures; In Germany by Badian (1983), Klauer (1992), in Switzerland by Von Aster (1997), in England by Lewis, Hitch and Walker (1994). The margin of errors in these studies were affected by reading difficulties, dyslexia, ADHD and other disabilities, since the children with these difficulties had a tendency to show poorer arithmetical skills than those only having dyscalculia. Shalev and Gross-Tsur (2000) acknowledged that other authors emphasized other reasons than brain dysfunctions for dyscalculia; lower social and economic status, mathematical anxiety, large classes and less well thought-through curricula and teaching. Kadosh 4 (2007) discovered through tests that a specific part of the brain was associated with automatic magnitude processing 5. Kadosh was using functional Magnetic Resonance Imaging (fmri) in order to show that the intraparietal sulcus (IPS), illustrated in figure A, had a role in the ability to recognize numbers. In order to cope with ambiguous results regarding left and right side of the brain, a transcranial magnetic stimulation (TMS) was used to block the energy activation of the brain on one side at a time. When blocking the right side of the brain, the test persons appeared to obtain dyscalculic behaviour, as opposed to the persons that were being tested without the blocking, where there was no appearance of dyscalculic behaviour. The experimenters also took into consideration that other areas of the brain that were communicating at the same time could be affected by the TMS. But according to careful studies of the fmri no other areas showed any activity during the automatic magnitude processing. However, the researchers did not claim to have found the cause of dyscalculia after this test, since it was not made clear whether developmental dyscalculia and the effects produced through TMS could be classified as identical deficits. The experimental procedures were carefully described in the report. It is important to notice that only five subjects were tested. This study confirmed the results of earlier studies, one of them made by Butterworth (2006), also using fmri and identifying the IPS as the centre responsible for the handling of number information. Figure A: Right side of the brain and the intraparietal sulcus (IPS) [6] Rubinstein and Henik (2009) regarded the IPS findings as surprising. The reasons they stated were heterogeneous behavioural deficits, comorbidity 7 and number processing represented in more than one brain area. Developmental disorders often generate multiple problems, according to Rubinstein and Henik. How is it possible to sort out the dyscalculia? Rubinstein and Henik focused on three different views; single restricted biological deficit, cognitive deficits due to instances of biological damage and neurocognitive damage causing developmental dyscalculia which might result in 2

3 several, not related, behavioural disorders. Rubinstein and Henik considered brain dysfunction as a possible reason for dyscalculia and stated that evidence has been found that IPS was involved in attention and related cognitive processes. Injury in the IPS area could thus result in dyscalculia. The authors recommended future researchers to examine the whole brain, since developmental disorders were considered heterogeneous. Focusing on single brain-behaviour deficits exclusively could prevent understanding of the variety of deficits connected to developmental dyscalculia and MLD, yet standardized tests of arithmetic computation could be a helpful screening tool to detect and separate dyscalculia from MLD, according to Rubinstein and Henik (2009). Geary et al. (2009) referred to other authors (Gross-Tsur, Manor, & Shalev, 1996; Kosc, 1974; Ostad, 1998; Shalev, Manor, & Gross-Tsur, 2005) when discussing predictions of the percentage of children diagnosed with a mathematical learning disorder, which was found to be in the range from 5 to 10 %. It was regarded critical on multiple levels that a diagnosis should be made at an early stage. The reason to that was for measures to be taken early in order to help the child develop their mathematical abilities in the future. The early stages of mathematical learning were essential for the later outcome in the education. The authors pointed out that the causes for mathematical learning disorders were still under investigation, even if some conclusions had been drawn about the main areas contributing to the disorder. Comparisons between normally achieving children and children with a MLD have been made. The latter group s counting strategies were less mature, their understanding of counting was not fully developed and they had continuous difficulties learning math and recalling basic arithmetic facts stored in the long-term memory. Working memory has come to develop a central role in the area of MLD. Children with MLD probably have a dysfunction in the basic recognition of numbers and their magnitude (Geary et al., 2009). The Number Sets Tests were designed by Geary et al. (2009) to measure the speed and correctness of the basic number and quantity recognition in children. The further developed test was meant to serve as a quick screening to find sensitivity for numbers and predict MLD. Participants in the test were 228 children from kindergarten, first, second and third grade. In the analysis were proficiency scores, IQ in the first year, working memory and mathematical cognition test scores taken into consideration. The IQ scores were measured using the test by Wechsler Abbreviated Scale of Intelligence (WASI; Wechsler, 1999). Examples of understanding numbers are given in figure B. Figure B: A part of the Number Sets Test (Geary et al., 2009) Number estimation was assessed through number lines; a blank line with two endpoints; 0 and 100. The assignment was to mark on the line where the number presented should be placed. The score was defined as the absolute difference between the marking of the number and the correct position of the number. The overall score was calculated as the mean of these differences across the trials. Measuring the children s counting ability was made by letting the child look at a puppet counting red and blue chips. The child had to tell if the puppet had made a correct calculation, and not double-calculated. Geary et al. (2009) noted that children with MLD made errors throughout the test when the first chip was counted twice like one one two three etc. The score for this part of the test was calculated as a percentage of the number of times the child successfully identified wrong calculations. To further asses the children s addition strategies, a mix of simple and complex addition problems was shown one at a time. Each problem should be solved as quickly and correctly as possible. The child could use any strategy to get the answer, but without pen and paper and the answer was to be told out loud. Geary et al. (2009) classified the trials into six different categories; specifically, counting fingers, fingers 8, verbal counting, retrieval, decomposition 9 or other/mixed strategy. The percentage of correct direct-retrieval trials for simple problems was correlated with the mathematics achievement scores, and used in the analysis by Geary et al. (2009). The Working Memory Test Battery for Children 10 was composed by nine subtests in order to assess the central executive 11, phonological loop and visuospatial sketchpad (Geary et al., 2009). The tests had six items from one to six to one to nine, where four were to be remembered to get to the next level. To get to the next level after that, the numbers remembered increased by one. The test ended when the child failed three times in a row. The central executive was investigated through three different tests; Listening Recall, Counting Recall and Backward Digit Recall. The idea of the first test was to let the child listen to a sentence, then determine if the sentence was true or false and then repeat the last word in a series of sentences. The second test then demanded of the child to count a set of dots on a card, maximum seven, and then remember the number of counted dots at the end of a series of cards. The third test, Backward Digit Recall is a standard format backward digit span 12 (Geary et al., 2009). The Phonological loop was examined with Digit Recall, Word List Recall, and Nonword List Recall, for instance by repeating words spoken by the experimenter in the same order. Series of words were presented to the child in the Word List Matching task. Then the words were presented again, maybe in another order, and the child had to decide whether the second list had the same order as the first list. The Visuospatial sketchpad was tried with other span tasks 13, like Block Recall. A board was set up with nine one-sided numbered blocks in a random arrangement; the numbers could only be seen by the experimenter. Then the experimenter touched series of blocks and the child should repeat the order. Mazes Memory task was conducted like this; the child got a picture of a maze with more than one solution. A picture of an identical maze with one solution drawn was presented. The child s task was to replicate the solution when the picture was removed. After a successful trial, the maze was increased with one wall. The tests were conducted every term for each age group, except for the children in kindergarten who were tested one time in spring. The location of the tests was at the children s schools, most of the time in a quiet place. Some of the children s tests were executed on the university campus or in a mobile testing van. A score below the 15th percentile on the mathematics achievement test characterized the child as having a MLD. Results between the 15th and 30th percentiles were being considered as low achieving (Geary et al., 2009). Pupils with results above the 30th percentile were considered normally achieving. No comments were made on high achieving children in the test. To find the most accurate measure to predict third grade mathematics achievement, all measures were compared in 3

4 independent regressions. The central executive and sensitivity measures predicted 25 % in the variation; the number line scores predicted 27 %. The number line measure appeared to somewhat assess the children s intuitive understanding of numerical quantity. The researchers claimed that their results provided initial support for their hypothesis. Competencies assessed by the sensitivity score were unrelated to reading achievement and achievement in general or to IQ or working memory (Geary et al., 2009). Of the non-mld children in third grade 96 % were correctly identified in first grade. According to Geary et al. (2009) the Number Sets Test was a potential screening tool for discovering children prone to MLD at an early stage. The results from the test in first grade could be used to detect 67 % of children risking MLD at the end of third grade and correctly identify nearly 90 % of non-mld children. Benefits of early identification of children at risk for MLD would be early remedial services and low costs. The authors emphasized that the Number Sets Test was not yet ready for use as a diagnostic tool since the test was not normed and it could not be used for making predictions later than the third grade. To enable further tests, the authors were willing to provide the test on request. Discussion Dyscalculia is a topic widely discussed in pedagogical and neuropsychological forums. Dyslexia has previously been established as a proper diagnose of the learning disorder concerning reading, spelling and writing but after years of controversies between the researchers in the fields of neuroscience and cognitive theory. Dyscalculia on the other hand has not yet reached the same status as dyslexia. In Sweden, Adler has been providing tests to diagnose dyscalculia for at least a couple of years. Sjöberg (2006) means that dyscalculia has not yet been defined and that there are other reasons for pupil s mathematical problems. Parameters pointed out are large groups, disturbing environments, bad teaching and mathematical anxiety (Sjöberg, 2006). The recommendations are to examine the whole environment and not only the student with the mathematical learning disorder. Sjöberg s test subjects later succeeded in finishing mathematics courses in upper secondary school, even with early indications of dyscalculia. Sjöberg's opinion is supported by Shalev and Gross-Tsur (2000) who propose lower social and economical status, mathematical anxiety, large classes and less well thought-through curricula and bad teaching as affecting parameters to difficulty in mathematical learning situations. Rubinstein and Henik (2009) are not satisfied with defining dyscalculia as being a separate disorder, and rather mean that dyscalculia is depending on a range of various disorders. Von Aster (2000) even defines three different subtypes of dyscalculia; Verbal, Arabic and Pervasive subtypes, which indicate that the disability is more complex than just a calculating issue. Language understanding and disorders like ADHD are factors contributing to difficulties in the learning process. Shalev, Gross-Tsur et al. (2000) disagree to that dyscalculia is perfectly defined due to the margin of errors in other researchers interpretations of test results. Kosc (1974) was one of the first researchers to make a large scale test on mathematical disorders. He came up with the result that 3-6 % of the tested children suffered from a mathematical learning disorder. Later performed studies by other researchers (Von Aster, 2000; Gross-Tsur V, Manor O, Shalev RS, 1996; Klauer KJ, 1992) show around the same percentage. Going deeper into the cognitive perception of mathematics, Geary et al. (2009) have developed tests to examine the working memory and its impact on the mathematical ability to discover and predict learning disabilities in young school children, up to second grade. This would lead to early remedial services to a low cost according to Geary et al. More recent research has although been made in this area, indicating that mathematical abilities can be connected to both sides of the brain, in the IPS. Kadosh et al. (2007) have performed clinical studies on the brain using fmri, with results confirming the IPS on the right side of the brain as a centre for number cognition. It is however not proven whether this mathematical ability is in need of other parts of the brain to fully function, even though the results are interesting and in need of further testing. Due to the small amount of gathered data, the statistics of the results are not reliable. Kadosh s study is confirming the findings made by Butterworth (2006), who also identified the IPS as the centre responsible for the handling of number information. Despite the research work, the fact remains that there still are pupils in school with difficulties when it comes to learning mathematics. These pupils are normally functioning, social, intelligent adolescents with seemingly no disorders. But when it comes to mathematics they no longer feel like intelligent well functioning persons that they really are and for some of them their whole world falls apart. In lectures I have learned that persons with dyslexia have felt excluded from learning reading, writing and spelling until they received a diagnosis of their disability which opened up a whole new world of understanding to them and also of them. A diagnosis may have the effect to make a person feel understood, and give an explanation to why it is hard to learn a specific matter. A diagnosis also prevents a person from being discarded as dumb. The main reason to why research on dyscalculia is conducted is hopefully to give a remedy to the persons with mathematical learning disabilities. A diagnosis of dyscalculia could help the affected children to find a way to not feel excluded from the possibility to understand a core subject like mathematics. To discover dyscalculia at an early stage may prevent these children from developing a negative self-image, leading to decreasing interest in mathematics, which in turn can affect the interest in other school subjects in a negative way. If the parents are involved and aware of the children s difficulties, then the extra support from home can help the children in their learning process. A diagnosis could give meaning to the children and their parents and give resources to teachers to better help students with disabilities. Waiting for the problems to solve themselves will probably have the effect of making the children dislike math even more in the future, since it is hard to understand why there is a problem learning math on your own. According to Adler (2001), the best way to teach math to children with dyscalculia is in individual sessions. Further, if trying to teach a child with dyscalculia math of the wrong kind, the situation for the child can even worsen. Today the classroom situation does not make it easy for children with dyscalculia. The classes have many pupils and there is not much time left for individual teaching. In many Swedish upper secondary schools a pupil must have a proper diagnosis to receive special education. Sometimes the school s economical situation still cannot allow extra individual teaching. We teachers see more and more students who are having severe problems learning mathematics. The scores on mathematical tests are getting lower each year, although the same tests are done every year in the first grade. That the student s mathematical abilities decrease with the years is an alarming trend that demands immediate attention from the authorities. There is a need to agree on criteria that will help recognizing and analysing dyscalculia. Finally, only with better understanding of the nature of developmental dyscalculia can we devise effective ways of helping the millions of our fellow citizens whose lives are blighted by it (Butterworth, 2006). Personally I look forward to the day when I as a teacher 4

5 in mathematics have better tools to help all of my students, including the ones with learning disabilities, succeed in mathematics. Acknowledgements I would like to thank my course leaders Jan Gustafsson and Gordana Dodig-Crnkovic for their support and enthusiasm, answering questions and helping their students managing the deadlines. Thank you! References Adler B. What is dyscalculia? NU-förlag, 2001 (See also Baddeley A. Working memory. Académie des sciences, Elsevier, Paris Butterworth B. The Mathematical Brain: Macmillan, London, Castelli F., Glaser D. and Butterworth B. Discrete and analogue quantity processing in the parietal lobe: A functional MRI study ( Cohen Kadosh R., Cohen Kadosh K., Schuhmann T., Kaas A., Goebel R., Henik A. and Sack AT. Virtual Dyscalculia Induced by Parietal-Lobe TMS Impairs Automatic Magnitude Processing Kosc L. Developmental Dyscalculia. Journal of Learning Disabilities, Vol DOI: / Osaka N., Logie R. and D Esposito M. The cognitive neuroscience of working memory. Oxford University Press. New York, 2007 Rubinstein O. and Henik A. Developmental Dyscalculia: heterogeneity might not mean different mechanisms Shalev R. S., Auerbach J., Manor O., Gross-Tsur V. Developmental dyscalculia: prevalence and prognosis. Steinkopff Verlag, Sjöberg G. If it isn t dyscalculia then what is it? A multimethod study of the pupil with mathematics problems from a longitudinal perspective. 2006, ISBN ( Von Aster M. Developmental cognitive neuropsychology of number processing and calculation: varieties of developmental dyscalculia. 9: II/41 II/57 Steinkopff Verlag, Wilson A., Dehaene S. Number sense and developmental dyscalculia. In Human behaviour, learning and the developing brain: Atypical development, pages: , 2007 Geary D., Bailey D. and Hoard M. Predicting Mathematical Achievement and Mathematical Learning Disability With a Simple Screening Tool - The Number Sets Test. University of Missouri. Printed in Journal of Psychoeducational Assessment, Volume 27, Number 3,

6 Endnotes 1 Brian Butterworth, professor of Cognitive Neuropsychology at University College, London 2 Stanislas Dehaene, Professor at Collège de France, has a Masters degree in applied mathematics and Computer science, PhD in Experimental Psychology, chair of Experimental Cognitive Psychology. 3 Dyslexia: learning disability involving difficulties in processing language in reading, spelling, and writing ( 4 Roi Cohen Kadosh, from the Institute of Cognitive Neuroscience, University College London, researcher in numerical cognition, parietal lobe functions and neurocognitive mechanisms amongst other areas. 5 Automatic magnitude processing: determining the magnitude of a number when compared to another number that also differs in appearance and physical size. 6 Figure taken from 7 The presence of one or more disorders. 8 Fingers means showing fingers by decomposing 8 into 5 and 3 and then adding = 10, = 15, (Geary et al., 2009) 10 WMTB-C; Pickering & Gathercole, 2001, (Geary et al., 2009) 11 The central executive; a neuropsychological explaining model of a central attentional control system in the brain (Baddeley, 1998) 12 Recalling numbers in reversed order (Wikipedia) 13 Span tasks are about repeating information directly after being presented to it. The span is about how long the list of remembered and repeated items is (Osaka, Logie & D Esposito, 2007, Wikipedia). 6