Theories of Mathematical Cognition and Their Influence of Developmental Dyscalculia - Essay Example

Modular and Integrated Theories of Mathematical Cognition and Their Influence of Developmental Dyscalculia. The problem of developmental dyscalculia became relevant in several decades, and the statistics data revealing a growing number of students suffering from disability in dealing with numbers and calculating, engage further researches in this sphere. At the same time, the necessity in the specialists of a higher level of mathematical skills increases each year. It becomes obvious, that a set of definite measures addressed towards restructuring the system of teaching and learning at schools, especially in the field of mathematics are to be forged and then integrated into the system of education to change the existing situation. A new system should be based upon different approaches to education, depending upon the differences in the level of students' development, their skills, and including the most effective methods that are suggested by various approaches to development and education. There are many ideas concerning the definition of dyscalculia, major characteristics of this condition, as well at the causes that make it appear. Some researchers speak of it as of a definite deficit of a fundamental ability of operating with numbers and understanding them. This deficit then results in learning process connected to arithmetical operations. This notion is supported by the proposal that a human is born with a definite ability that helps us understand and operate the numbers, with this ability integrated into special neural circuits. Different groups of psychologists during several decades observed the ontogeny of mathematical conceptions, and noticed the kid's growing recognition of the reasons of the fact that two set reveal identical numerosity, and understanding all kinds of operations that could or couldn't influence this numerosity. According to the recent findings, the babies in the first weeks of their lives reveal the capacity for distinguishing between visible objects, even in movement, and this capacity is functioning basing upon numerosity. They also may mentally represent and operate these objects when they are out of visibility. These capacities are considered to be a basis for a formation of a kit that is developed as a child grows, and the bad functioning of this capacity causes developmental dyscalculia. (1) The other group of scientists insists that difficulties in operating with numbers and connected to mathematics, appear as a result of destruction, or a series of destructions in cognitive systems, like memory or spatial thinking. This approach is an alternative to the one described above. According to their findings, a baby is born not with an embodied capacity for operating numerosities, but with ability for recognizing quantities and a knowledge of an integrated list of number words that helps in operating the numbers above four. According to this approach, the linguistic difficulties influence the development of the capacity for operating the numbers, inspite of the fact that these two spheres do not merge. These two approaches and some other ideas will be discussed further in this essay. Before giving a detailed view of these approaches, it is necessary to describe the nature of dyscalculia, its major characteristics, and the reasons for it to appear. At first, it is often difficult to recognize the dyscalculia, as there are many reasons for bad results in mathematics. For instance, a student may fail in math in case of bad teaching, or non-attending the lessons of math, or absence of interest to the subject. These circumstances may cause mistakes in diagnosing. Some people may find a cause of fail in the specific features of characters, such as laziness. Some authors combine developmental dyscalculia with dyslexia, difficulties in literacy, as they have similar symptoms. Developmental dyscalculia is mainly characterized by specific disability in learning mathematics. The major definitions claim that a kid usually fails in passing standard tests designed especially for child's age group, and respective educational level and level of development. Besides, these children face various problems in studying and even in everyday life. However, these standard tests are designed as to test skills that comprise capacities in spatial representation and verbal activities, thus touching not only specific abilities that refer to mathematical operations. Moreover, these tests may contain diverse notions of the level of necessary mathematical achievements, so they cannot stand for reliable source of diagnosing a kid's condition as the case of developmental dyscalculia. The U.K. Department for Education and Skills gives the following notion of dyscalculia: "A condition that affects the ability to acquire arithmetical skills. Dyscalculia learners may have difficulty understanding simple number concepts, lack an intuitive grasp of numbers, and have problems learning number facts and procedures. Even if they produce a correct answer or use a correct method, they may do so mechanically and without confidence." (3, p. 2) As far as the symptoms of developmental dyscalculia are concerned, there specialists agree that the kids suffering from developmental dyscalculia reveal problems in recognizing and remembering numbers and performing calculation operations. According to the research results, these children show bad results in subtracting, addition and multiplication. In addition, these kids often address to various strategies, such as counting with the help of fingers. Of course, these facts cannot be a single proof that a kid has dyscalculia. One of the features that are commonly recognized as the features specific to developmental dyscalculia is a problem with learning and recollecting arithmetical operations and numbers. These kids also have difficulties in processing various arithmetical operations, processing logical operations; they need much more time for solving the tasks, compared to average rate and have a high rate of errors done in processing and operating the numbers. Some authors include dyslexic problems, problems with spatial representation, memory and attention deficits into the list of possible factors that lead to dyscalculia. But there are the researchers that argue this idea, stating that these authors mixed the above-mentioned abilities with number processing capacities. The other facts that have been appeared within the researches and observations show that the kids with developmental dyscalculia have bad results when it is necessary to comprehend various concepts, for instance, the numerosity concept. This problem occurs even in very simple cases, like dealing with magnitudes. The first approach to the observation of developmental dyscalculia includes defining the cognitive and neuropsychological connections of developmental dyscalculia in order to find out the circumstances under which it appears or refer different kinds of developmental dyscalculia to different factors that cause this condition. The cognitive causes that were presented by the researches are connected to semantic and working memory, especially the problems that occur in functioning of these kinds of memory. Geary suggested that these are the problems with semantic memory, that may bring the further difficulties that the kids with developmental dyscalculia are faced with, and these problems may also cause troubles with reading, often associated with dyscalculia. This idea is supported by the fact that the kids with developmental dyscalculia have difficulties in recognizing and remembering data and mathematical operations. But there are authors that argue this opinion: "However, if this theory is correct, we should expect all dyslexic children to have number fact problems and vice versa. In addition, the argument confounds semantic memory with numerical processing. There is little empirical evidence for a non-numerical semantic deficit in dyscalculic children." (5, p.3) There was only one experiment that proofed this idea - Temple and Sherwood worked with a group of kids, showing problems in operating the numbers. As they have noticed, these children also had problems in defining the colour and naming the objects. But the authors didn't support the idea that there was a connection between speed of processing the data and capacity in arithmetic, insisting that a relatively small size of the group, which consisted of four kids, couldn't be a reliable evidence of the fact revealed. Another reason was the finding that semantic memory used for operating the numbers is supported by another system, not semantic memory in general. In his works Geary suggests, that the children with developmental dyscalculia "show two basic functional, or phenotypic, numerical deficits"(4, p. 346) 1. "The use of developmentally immature arithmetical procedures and a high frequency of procedural errors" (4, p. 346) 2. "Difficulty in the representation and retrieval of arithmetic facts from long-term semantic memory" (4, p. 346) The author states that the source of this condition is in problems that occur with long-term memory, dealing with semantics, as well as with the working memory, while operating the data. The importance of long-term memory is based on the idea that there is a developmental sequence from calculation procedures, like, for instance counting, to settling associations between questions and solving them. "Mastery of elementary arithmetic is achieved when all basic facts can be retrieved from long-term memory without error . . . [which in turn] appears to facilitate the acquisition of more complex mathematical skills" (4, p. 347) As the author claims, the settling down the connections between problems and solving them, is depending on observing the elements of the problem in working memory. Beside, the usage of ineffective strategies or operations in working memory may lead to the decrease of the volume of important information.(4) The kids with developmental dyscalculia count slower, making different calculation operations. The author supposes it happens, because the performance of such operation overloads the memory, thus prolonging the time needed for performing these operations, and making the time of keeping the facts shorter. The other researchers that refer to this approach, Koontz and Berch, worked with the kids that have and don't have difficulties in performing arithmetical operations, using the tests with digits and letters. Their findings show that the kids with the symptoms of developmental dyscalculia show worse results in both tests, though the memory dealing with phonology is not referred to basic numeric operations. During the studies in the field of this approach it became evident that the working memory is used in arithmetical operations. The researches tested phonological memory and came to a conclusion that the kids having developmental dyscalculia do not have problems with phonological memory. At the same time the tests showed that these children may have problems with their memory in the kits that are dealing with arithmetical operations and processes dealing with number and calculations. The studies in neuropsychology show that the notion of numbers is detached form semantic memory that deals with language. The systems of semantic memory for the information that includes number and don't include them, are referred to different systems of brain. So it is becomes evident that different brain systems cannot be responsible both for problems in math and reading. The problems with working memory are also referred to as the case of developmental dyscalculia. Geary supposes that the deficits in working memory may cause problems with processing the numerical operations, as well as influence the process of learning various mathematical facts. This aspect has been concentrated on the phonological loop, usually checked by a set of spoken points, like, for instance, digits, that should be remembered in a definite consequence. Problems in completing this task are regarded as the symptom of dyslexia and general difficulties in learning, and both these conditions are referred to problems in mathematical operations and, in particular, developmental dyscalculia. So, any kind of tests screening working memory should comprise the tasks for capacity in reading and IQ level. The other researches, Temple and Sherwood, also made a series of test, screening the working memory. They didn't find any kind of connection between the procedures of the working memory and capacities for numerical operations. Therefore, despite the problems with working memory may sometimes occur together with the problems in performing calculation operations and counting, they didn't find any facts that could proof that the difficulties in functioning of working memory lead to dyscalculia. Siegel and Ryan found out that the kids with developmental dyscalculia have worse results compared to average rates, in completing the tasks comprising counting and learning the numbers for their further performance, while they have the same results in the tests that hadn't digits in their contents. Thus the authors came to the conclusion that there was a part of working memory specialized in facts dealing with numbers, and this is the very system that leads to developmental dyscalculia. Bevan and Landerl observed the results of the tests with digit span and found out that the kids with developmental dyscalculia didn't show bad results, while kids with dyslexia had problems with doing the task. One more researcher, implementing this approach, Rourke, insisted that problems with calculation and operating numbers are connected to spatial thinking. Geary also observed this idea, adding that "a disruption of the ability to spatially represent numerical information . . . appears to affect both functional skills (e.g., columnar alignment in complex arithmetic problems) and the conceptual understanding of the representations (e.g., place value)" (3, p. 346) This idea found many supporters, stating that space and digits are connected in cognition. It lead to further researches in this direction and later a group of scientists (Piazza, Dehaene, Fias, Seron, Spalding and many others) suggested that the viewing mathematical magnitudes as a sort of a mental row of numbers. The results of their researches made evident that the problems with spatial thinking may influence a representation of mathematical magnitude. However, they didn't proof that the dysfunctions in spatial thinking may cause dyscalculia. One more group of scientists (Cohen, Dehaene, Angelergues, Hecaen, Houillier) regarded the ability of building up mental lines of numbers as a special ability of spatial thinking which is used in operating the numbers. They suggested defining a new kind of dyscalculia, which is connected to problems in spatial representation. This condition is seldom regarded as an evident symptom of developmental dyscalculia and it is not considered to be a factor that influences that notion of mathematical concepts. Koontz and Berch worked with the kids that have or have not developmental dyscalculia, testing them with the help of letter span and digit span, where letter span defined the god or bad functioning of the phonological memory which is not associated with numerical operations. These studies resulted in a conclusion that the kids with developmental dyscalculia show worse results compared to average rates in both kinds of tests, revealing deficits both in phonological and working memory. The other approach comprises a division of developmental dyscalculia into several types, depending upon the existence or absence of other deficits, trying to stress the processes that may influence these deficits. One of the important conditions associated with developmental dyscalculia is disability in reading. According to the estimations of the specialists, 40% of students having developmental dyscalculia, face dyslexia. One of the methods of dividing kids with developmental dyscalculia is dividing them into groups according to the presence or absence of reading disability in their cases. Rourke have studied kids with difficulties in mathematics operations and kids with better results in mathematics compared to their rates in reading. The kids that had only the problems in math also revealed deficits in spatial representation and psychomotor capacities, while kids with problems in reading encountered problems in fulfilling verbal tests. The author proposes that these researches make evident that problems in math and reading are the result of deficits of the left hemisphere of brain, while the problems with numerical operations occur due to disfunctions in the right hemisphere of brain. However, Rourke's notion of the symptoms related to dysfunctions in the right hemisphere is similar to the symptoms of Gerstmann's syndrome. It comprises the symptoms of developmental dyscalculia and the symptoms of disability in representing fingers in mind, dyslexia and dysgraphia. This syndrome causes harm to the left angular gyrus, and there are some reasons to consider developmental dyscalculia to be the result of the damage of left angular gyrus. Though, latest attempts to repeat Rourke's experiments didn't bring positive results, the researches didn't find any differences between children with problems in both reading and math and in math only. Fayol, Barrouillet, and Marinthe tried to check his thesis about a possible connection between mathematics difficulties and neuropsychological problems. They delivered a study where school children were given the tasks on graphisthesia, imultagnosia, and agnosia. At the same time they had to complete several simple tasks in arithmetic. The results revealed that a level of general intelligence is higher when shown in arithmetical tasks, then in neuropsychological tests. It means that association doesn't include causation. The other researches also paid attention to the studies of these dysfunctions: "Another set of deficits which are associated with developmental dyscalculia are finger agnosia, dysgraphia and difficulties with left-right discrimination. Taken together this symptom complex constitutes developmental Gerstmann's syndrome. However, since it appears that the four symptoms can appear individually and in any combination, and are frequently associated with other conditions such as reading disability it is unlikely that the symptoms are related in terms of a single underlying deficit."(5, p.103) Another deficit which has been related with developmental dyscalculia is ADHD, which is accompanied with bad eye and hand co-ordination, bad memory resources in cases when it is necessary to deal with non-verbal material and problems in social communication. Very few researches have observed differences between types of developmental dyscalculia, studying the tasks on mathematical processing. Shalev observed two groups of children, one with specific deficits in math and the other with the problems both in reading and math. Shalev found that those children who have problems in maths and reading had worse results than children with definite mathematical problems on division and subtraction. They also showed worse results in the majority of the WISC subtests and had a lower scores in IQ tests. However, the frame of mathematical impairment was similar for both groups. This study revealed no facts that could proof diversification between the two groups in mathematical processing. Jordan and Montani studied children with problems in maths operations and children who had maths incapability accompanied with more general problems in learning. Children having maths disability only showed better results in performing backup procedures in math, and completed the tasks without time restrictions, although they failed while performing time-restricted tests. Children having general problems in learning had difficulties in executing both tasks. The authors thought that children having problems in executing maths operations are able to succeed with tasks that have no time restrictions, due to better verbal skills. This research also shows that children with difficulties in general learning have more problems compared to the children with developmental dyscalculia. One more approach regards developmental dyscalculia as a problem that occurs in a special area of brain system. The researches shown above attempted to examine developmental dyscalculia by watching different kinds of capacities, not always related to operating the numbers, which are supposed to cause dyscalculia. This approach includes an assumption that the recognition and operating numerical information is a function of higher order, which depends upon the capacities shown. However, the facts in the sphere of neuropsychology and studying young children propose that operating the numbers is independent of other capacities, and is also presented at a basic level. Mathematical capacities, including arithmetic, are fulfilled by the parietal lobe. Neuropsychological facts showed that the capacity for recognizing numbers and calculating is not related to language; it is not connected to semantic memory for nonnumeric data; and is detached from working memory. The mathematical capacities are independent of other capacities. Besides, they also reveal in the first days of infant life. Thus numeral processing reveals to be a function that appears in infants, and is not dependent on other capabilities. This idea opposite to the notion of an important role of the capacities related to language, like, for instance, working memory or semantic memory, in developmental dyscalculia. It seems that major mathematical functions, such as understanding of arithmetical symbols, calculation, are based upon kits operating small numerosities. These kits also seem to provide a deficit causing dyscalculia. If developmental dyscalculia occurs due to basic difficulty with mathematical processing, children with developmental dyscalculia could encounter problems with even the simplest operations including numbers, such as counting small numbers of items, using numerals. Some proof for this reveals from works by Koontz and Berch, who showed that the children with developmental dyscalculia counted to 3 better than subitized. In conclusion, the data regarding math abilities as regards to developmental dyscalculia are rather conflicting, and they do not proof that problems with basic numeral processing may be a symptom of developmental dyscalculia. The steps delivered to find the cause of developmental dyscalculia by associating it with other deficits have not been successful. Neuropsychological research indicates a possible the presence of a 'number module' located in the parietal lobe for operating mathematical representations. Developmental dyscalculia is also sometimes regarded as a dysfunction in the sphere of numerosity and its processing. Although there has been no attempts of systematic examination the mathematical skills of children having developmental dyscalculia, the researchers have constantly found confirmation of deficits even with simple numeral tasks. Further inspection of the fundamental figure processing capacities of children with developmental dyscalculia is required for two reasons. First, a more accurate image of their deficits is necessary in to make theoretical base. If the deficit is mathematical in origin, all spheres of mathematical processing are influenced. If not, then more information about mathematical deficits shown by children with developmental dyscalculia will be useful in defying connection of arithmetic disabilities to other basic abilities. It is well-known that solving mathematical tasks may arouse anxiety in a kid. This fact concerns only mathematics and cannot be referred to any kind of challenging task. It is also known that anxiety influences a large set of cognitive functions, combining the functions that influence mathematical operations, like, for instance, working memory. At the same time, the emotional outcome of long term attempting of solving simple tasks that are easily done by the majority of students around, is unknown. The studying of kids having developmental dyscalculia shows that they feel emotional stress while trying to succeed in mathematics lessons. This effect should be also taken into consideration while trying to find out effective strategies that should reduce bad effects that are caused by developmental dyscalculia. (5) All experiments make evident that developmental dyscalculia is often accompanied by other dysfunctions. But it isn't still evident that these deficits influence developmental dyscalculia. There has been found no procedure or process that could be indicated as the forecasting the further developmental dyscalculia. Moreover, it hasn't been proofed that there are qualitatively diverse destruction frames that are common for the various types of developmental dyscalculia. The existence of such frames would be evident in case various types of developmental dyscalculia are related to different factors that cause this condition. There is no theoretical basis for the idea that any of these connections influence mathematical capacities. Moreover, there is no any rational theory which could describe and make clear such connections. At present, the most probable suggestions about the relations between different kinds of disabilities are based upon genetic or anatomical researches - that harm caused to a brain or dysfunction of a definite brain area may influence various cognitive functions, depending upon the degree of the damage.(5) There have been some authors that observed a condition of developmental dyscalculia form the point of view of the possible influence that the approaches described above, could have on it. One of this works, Integrated Versus Modular Theories of Number Skills and Acalculia, was written by Clark J.M. and Campbell J.I.. This work comprises the observation of two conceptions concerning the cognitive structure that modulates mathematical skills and developmental dyscalculia. They divide the approaches into two basic directions - the abstract-modular theory and specific-integrated theory. The authors give the general idea of these two approaches. Abstract-modular theory states that the number operations combine understanding, calculation and representing systems that are connected with the help of abstract quantity code. Specific-integrated theory suggests, that verbal, spatial and visual and other number codes are integrated into encoding system, and different points of operating the numbers include common processes and depend upon each other. The thesis about special number code is held up by structural inadequacy of a system that comprises nonfigurative codes, a phenomena of format in calculation operations, the distinction between developmental dyscalculia and individual deficits that may occur in processing the figures, and the understanding of the role of format codes that are integrated into the system of working memory. The alternative, associative theory of numeral processing is provided by the evident influence of modular views of abstract codes and other structural deficits, the facts that proof the existence of associative networks that reveal in calculation tasks, a condition of developmental dyscalculia, deficits in modular structures that cause problems in processing numbers, connections between those aspects of figures that concern semantic nature and verbal nature, and goes on into numeral and nonnumeric processing. These various logical and experimental observations are opposite to the abstract-modular theory and promote the notion of complex encoding, suggesting that numeral processing is influenced by embodied networks of special number codes. Summarizing all popular theories about the nature of dyscalculia, it is possible to define it as a condition similar to dyslexia in its nature and influencing the success of learning and working. The kids with developmental dyscalculia show bad results in fulfilling simple operations, like, for instance, a comparison of numbers. It is customary to regard it as the bad functioning of ability to operate numbers. The kids, showing the symptoms, that are concerned to be inherent to dyscalculia, complain that they have problems in recognizing number conceptions and easily lose the idea of the subject during the lessons of mathematics. The conception of seeing developmental dyscalculia as a problem with cognitive abilities, for instance, working memory or spatial representation or phonological memory, is not supported by the majority of the researches. It is believed, that there are neural circuits in brain that are specialized in operations with numbers, and these circuits are detached from the areas that are responsible other functions. Developmental dyscalculia seems to be heritable, which was proved in studying the twins and populations with genetic deficits, and many researches refer it to the X chromosome, but this doesn't mean that developmental dyscalculia is heritable in all cases. A role of heredity in case with developmental dyscalculia has been studied by various scientists. One of the most intriguing questions concerns the definition of what are the genes that may cause heredity of this condition. "A recent twin study showed that for DD probands, 58% of monozygotic co-twins and 39% of dizygotic co-twins were also DD and that the concordance rates were 0.73 and 0.56, respectively. In a family study, Shalev et al. (2001) found that approximately half of all siblings of children with DD are also dyscalculic, with a 5-10-times greater risk than for the general population."(1, p.463) Some anomaly of the X chromosome seems to influence abilities in performing mathematical operations more, than other cognitive capacities. This picture is evident in case with Turner's Syndrome, which shows good results in testing IQ, capacities in reading or learning language, but fail in fulfilling simple tasks in arithmetic. Though, it hasn't been proofed that developmental dyscalculia may reveal due to problems that occur during neural growth. A special screener has been worked out basing upon the most evident symptoms of developmental dyscalculia. It includes a number of time restricted tasks with counting and operations with magnitude. This screener allows making conclusions about the condition of major capacities in numerical operations. The screener proposes that there should be a special brain system which functions badly in cases of developmental dyscalculia. Some researches showed the results that allowed thinking that there should be a definite network in brain, which is responsible for operating numbers. Edmonds, Isaacs, Gadian, and Lucas, a group of scientists, observed a group of adults, who were born with a low weight. One group didn't reveal any problems in cognition, and the second showed definite problems in processing mathematical operations. The results showed that the people of the second group had less grey matter compared to the people from the first group. Although developmental dyscalculia is often regarded as a condition similar to dyslexia, dyscalculia was a subject of fewer researches, compared to dyslexia. As far as dyslexia is concerned, there is a definite set of evident symptoms, which helps diagnosing it, it is known that there are brain systems responsible for the process, and the genes that may cause bad functioning are also familiar to the scientists. In order to observe developmental dyscalculia, the scientists need to forge a common understanding of what it is, agree upon its symptoms and define what systems of brain influence the process. One more difference between dyslexia and dyscalculia is that the latter is not as widely recognized as the former. It is often mixed with the peculiarities of behaviour or features of a student's character that may lead to bad results in mathematics. Of course, forging common theory that would comprise the most probable suggestions and findings from various approaches would stimulate the process of further observation of this phenomenon and would be very helpful in finding the means of treating this special condition. References 1. Butterworth B. (2005) Handbook of Mathematical Cognition. Psychology Press 2. Clark, J. M., & Campbell, J. I. D. (1991). Integrated versus modular theories of number skills and acalculia. Brain and Cognition 3. DfES. (2001). Guidance to support pupils with dyslexia and dyscalculia. London: Department of Education and Skills. 4. Geary, D. C. (1993). Mathematical disabilities: Cognition, neuropsychological and genetic components. Psychological Bulletin, 114, 345-362 5. Landerla K., Bevana A., Butrworth B. Developmental dyscalculia and basic numerical capacities: a study of 8-9-year-old students. Cognition 93 (2004) 99-125 6. Newman R. (1998)Developmental Perspectives On Dyscalculia: Implications For Teaching In The Middle & Secondary School. R. M. Newman Communications 7. Redwood, F. (2003, February 4) When Sums Don't Add Up. The Daily Mail: Education Notebook Read More