Dyscalculia  is difficulty in learning or comprehending arithmetic, such as difficulty in understanding numbers, learning how to manipulate numbers, and learning facts in mathematics. It is generally seen as a specific developmental disorder.
Dyscalculia can occur in people from across the whole IQ range – often higher than average – along with difficulties with time, measurement, and spatial reasoning. Estimates of the prevalence of dyscalculia range between 3 and 6% of the population. In 2004, it was reported that a quarter of children with dyscalculia had ADHD. In 2015, it was established that 11% of children with dyscalculia also have ADHD. Dyscalculia has also been associated with people who have Turner syndrome and people who have spina bifida.
Mathematical disabilities can occur as the result of some types of brain injury, in which case the proper term, acalculia, is to distinguish it from dyscalculia which is of innate, genetic or developmental origin.
Signs and symptoms
The earliest appearance of dyscalculia is typically a deficit in the ability to know, from a brief glance and without counting, how many objects there are in a small group (see subitizing). Children as young as 5 can subitize 6 objects, especially looking at a die. However, children with dyscalculia can subitize fewer objects and even when correct take longer to identify the number than their age-matched peers.Dyscalculia often looks different at different ages. It tends to become more apparent as kids get older; however, symptoms can appear as early as preschool.
Although many researchers believe dyscalculia to be a persistent disorder, evidence on the persistence of dyscalculia remains mixed. For instance, in a study done by Mazzocco and Myers (2003), researchers evaluated children on a slew of measures and selected their most consistent measure as their best diagnostic criterion: a stringent 10th-percentile cut-off on the TEMA-2. Even with their best criterion, they found dyscalculia diagnoses for children longitudinally did not persist; only 65% of students who were ever diagnosed over the course of four years were diagnosed for at least two years. The percentage of children who were diagnosed in two consecutive years was further reduced. It is unclear whether this was the result of misdiagnosed children improving in mathematics and spatial awareness as they progressed as normal, or that the subjects who showed improvement were accurately diagnosed, but exhibited signs of a non-persistent learning disability.
Dyscalculia involves frequent difficulties with everyday arithmetic tasks such as the following:
- Difficulty reading analog clocks
- Difficulty stating which of two numbers is larger.
- Inability to comprehend financial planning or budgeting, sometimes even at a basic level; for example, estimating the cost of the items in a shopping basket or balancing a checkbook.
- Inconsistent results in addition, subtraction, multiplication and division.
- Visualizing numbers as meaningless or nonsensical symbols, rather than perceiving them as characters indicating a numerical value. (Hence the misnomer, "math dyslexia")
- Difficulty with multiplication, subtraction, addition, and division tables, mental arithmetic, etc.
- Problems with differentiating between left and right.
- A "warped" sense of spatial awareness, or an understanding of shapes, distance, or volume that seems more like guesswork than actual comprehension.
- Difficulty with time, directions, recalling schedules, sequences of events. Difficulty keeping track of time. Frequently late or early.
- Poor memory (retention & retrieval) of math concepts; may be able to perform math operations one day, but draw a blank the next. May be able to do book work but then fails tests.
- Difficulty reading musical notation.
- Difficulty with choreographed dance steps.
- Difficulty working backwards in time (e.g. What time to leave if needing to be somewhere at 'X' time).
- Having particular difficulty mentally estimating the measurement of an object or distance (e.g., whether something is 3 or 6 meters (10 or 20 feet) away).
- When writing, reading and recalling numbers, mistakes may occur in the areas such as: number additions, substitutions, transpositions, omissions, and reversals.
- Inability to grasp and remember mathematical concepts, rules, formulae, and sequences.
- Inability to concentrate on mentally intensive tasks.
- Mistaken recollection of names. Poor name/face retrieval. May substitute names beginning with same letter.
Both domain-general and domain-specific causes have been put forth. With respect to pure developmental dyscalculia, domain-general causes are unlikely as they should not impair one’s ability in the numerical domain without also affecting other domains such as reading.
Two competing domain-specific hypotheses about the causes of developmental dyscalculia have been proposed – the magnitude representation (or number module deficit hypothesis) and the access deficit hypothesis.
Magnitude representation deficit
Dehaene's "number sense" theory suggests that approximate numerosities are automatically ordered in an ascending manner on a mental number line. The mechanism to represent and process non-symbolic magnitude (e.g., number of dots) is often known as the "approximate number system" (ANS), and a core deficit in the precision of the ANS, known as the "magnitude representation hypothesis" or "number module deficit hypothesis", has been proposed as an underlying cause of developmental dyscalculia.
In particular, the structural features of the ANS is theoretically supported by a phenomenon called the "numerical distance effect", which has been robustly observed in numerical comparison tasks. Typically developing individuals are less accurate and slower in comparing pairs of numbers closer together (e.g., 7 and 8) than further apart (e.g., 2 and 9). A related "numerical ratio effect" (in which the ratio between two numbers varies but the distance is kept constant, e.g., 2 vs. 5 and 4 vs. 7) based on the Weber's law has also been used to further support the structure of the ANS. The numerical ratio effect is observed when individuals are less accurate and slower in comparing pairs of numbers that have a larger ratio (e.g., 8 and 9, ratio = 8/9) than a smaller ratio (2 and 3; ratio = 2/3). A larger numerical distance or ratio effect with comparison of sets of objects (i.e., non-symbolic) is thought to reflect a less precise ANS, and the ANS acuity has been found to correlate with math achievement in typically developing children  and also in adults.
More importantly, several behavioral studies have found that children with developmental dyscalculia show an attenuated distance/ratio effect than typically developing children. Moreover, neuroimaging studies have also provided additional insights even when behavioral difference in distance/ratio effect might not be clearly evident. For example, Price and colleagues found that children with developmental dyscalculia showed no differential distance effect on reaction time relative to typically developing children, but they did show a greater effect of distance on response accuracy. They also found that the right intraparietal sulcus in children with developmental dyscalculia was not modulated to the same extent in response to non-symbolic numerical processing as in typically developing children. With the robust implication of the intraparietal sulcus in magnitude representation, it is possible that children with developmental dyscalculia have a weak magnitude representation in the parietal region. Yet, it does not rule out an impaired ability to access and manipulate numerical quantities from their symbolic representations (e.g., Arabic digits).
Moreover, findings from a cross-sectional study suggest that children with developmental dyscalculia might have a delayed development in their numerical magnitude representation by as much as five years. However, the lack of longitudinal studies still leaves the question open as to whether the deficient numerical magnitude representation is a delayed development or impairment.
Access deficit hypothesis
Rousselle & Noël propose that dyscalculia is caused by the inability to map preexisting representations of numerical magnitude onto symbolic Arabic digits. Evidence for this hypothesis is based on research studies that have found that individuals with dyscalculia are proficient on tasks that measure knowledge of non-symbolic numerical magnitude (i.e., non-symbolic comparison tasks) but show an impaired ability to process symbolic representations of number (i.e., symbolic comparison tasks). Neuroimaging studies also report increased activation in the right intraparietal sulcus during tasks that measure symbolic but not non-symbolic processing of numerical magnitude. However, support for the access deficit hypothesis is not consistent across research studies.
At its most basic level, dyscalculia is a learning disability affecting the normal development of arithmetic skills.
A consensus has not yet been reached on appropriate diagnostic criteria for dyscalculia. Mathematics is a specific domain that is complex (i.e. includes many different processes, such as arithmetic, algebra, word problems, geometry, etc.) and cumulative (i.e. the processes build on each other such that mastery of an advanced skill requires mastery of many basic skills). Thus dyscalculia can be diagnosed using different criteria, and frequently is; this variety in diagnostic criteria leads to variability in identified samples, and thus variability in research findings regarding dyscalculia.
Other than using achievement tests as diagnostic criteria, researchers often rely on domain-specific tests (i.e. tests of working memory, executive function, inhibition, intelligence, etc.) and teacher evaluations to create a more comprehensive diagnosis. Alternatively, fMRI research has shown that the brains of the neurotypical children can be reliably distinguished from the brains of the dyscalculic children based on the activation in the prefrontal cortex. However, due to the cost and time limitations associated with brain and neural research, these methods will likely not be incorporated into diagnostic criteria despite their effectiveness.
Research on subtypes of dyscalculia has begun without consensus; preliminary research has focused on comorbid learning disorders as subtyping candidates. The most common comorbidity in individuals with dyscalculia is dyslexia. Most studies done with comorbid samples versus dyscalculic-only samples have shown different mechanisms at work and additive effects of comorbidity, indicating that such subtyping may not be helpful in diagnosing dyscalculia. But there is variability in results at present.
Due to high comorbidity with other disabilities such as dyslexia and ADHD, some researchers have suggested the possibility of subtypes of mathematical disabilities with different underlying profiles and causes. Whether a particular subtype is specifically termed "dyscalculia" as opposed to a more general mathematical learning disability is somewhat under debate in the scientific literature.
- Semantic memory: This subtype often coexists with reading disabilities such as dyslexia and is characterized by poor representation and retrieval from long-term memory. These processes share a common neural pathway in the left angular gyrus, which has been shown to be selective in arithmetic fact retrieval strategies and symbolic magnitude judgments. This region also shows low functional connectivity with language-related areas during phonological processing in adults with dyslexia. Thus, disruption to the left angular gyrus can cause both reading impairments and difficulties in calculation. This has been observed in individuals with Gerstmann syndrome, of which dyscalculia is one of constellation of symptoms.
- Procedural concepts: Research by Geary has shown that in addition to increased problems with fact retrieval, children with math disabilities may rely on immature computational strategies. Specifically, children with mathematical disabilities showed poor command of counting strategies unrelated to their ability to retrieve numeric facts. This research notes that it is difficult to discern whether poor conceptual knowledge is indicative of a qualitative deficit in number processing or simply a delay in typical mathematical development.
- Working memory: Studies have found that children with dyscalculia showed impaired performance on working memory tasks compared to neurotypical children. Furthermore, research has shown that children with dyscalculia have weaker activation of the intraparietal sulcus during visuospatial working memory tasks. Brain activity in this region during such tasks has been linked to overall arithmetic performance, indicating that numerical and working memory functions may converge in the intraparietal sulcus. However, working memory problems are confounded with domain-general learning difficulties, thus these deficits may not be specific to dyscalculia but rather may reflect a greater learning deficit. Dysfunction in prefrontal regions may also lead to deficits in working memory and other executive function, accounting for comorbidity with ADHD.
Studies have also shown indications of causes due to congenital or hereditary disorders, but evidence of this is not yet concrete.
To date, very few interventions have been developed specifically for individuals with dyscalculia. Concrete manipulation activities have been used for decades to train basic number concepts for remediation purposes. This method facilitates the intrinsic relationship between a goal, the learner’s action, and the informational feedback on the action. A one-to-one tutoring paradigm designed by Lynn Fuchs and colleagues which teaches concepts in arithmetic, number concepts, counting, and number families using games, flash cards, and manipulables has proven successful in children with generalized math learning difficulties, but intervention has yet to be tested specifically on children with dyscalculia. These methods require specially trained teachers working directly with small groups or individual students. As such, instruction time in the classroom is necessarily limited. For this reason, several research groups have developed computer adaptive training programs designed to target deficits unique to dyscalculic individuals.
Software intended to remediate dyscalculia has been developed. While computer adaptive training programs are modeled after one-to-one type interventions, they provide several advantages. Most notably, individuals are able to practice more with a digital intervention than is typically possible with a class or teacher. As with one-to-one interventions, several digital interventions have also proven successful in children with generalized math learning difficulties. Räsänen and colleagues have found that games such as The Number Race and Graphogame-math can improve performance on number comparison tasks in children with generalized math learning difficulties.
Several digital interventions have been developed for dyscalculics specifically. Each attempts to target basic processes that are associated with maths difficulties. Rescue Calcularis was one early computerized intervention that sought to improve the integrity of and access to the mental number line. Other digital interventions for dyscalculia adapt games, flash cards, and manipulables to function through technology.
While each intervention claims to improve basic numerosity skills, the authors of these interventions do admit that repetition and practice effects may be a factor involved in reported performance gains. An additional criticism is that these digital interventions lack the option to manipulate numerical quantities. While the previous two games provide the correct answer, the individual using the intervention cannot actively determine, through manipulation, what the correct answer should be. Butterworth and colleagues argued that games like The Number Bonds, which allows an individual to compare different sized rods, should be the direction that digital interventions move towards. Such games use manipulation activities to provide intrinsic motivation towards content guided by dyscalculia research. One of these serious games is Meister Cody – Talasia, an online training that includes the CODY Assessment – a diagnostic test for detecting dyscalculia. Based on these findings, Rescue Calcluaris was extended by adaptation algorithms and game forms allowing manipulation by the learners. It was found to improve addition, subtraction and number line tasks, and was made available as Dybuster Calcularis.
A study used transcranial direct current stimulation (TDCS) to the parietal lobe during numerical learning and demonstrated selective improvement of numerical abilities that was still present six months later in typically developing individuals. Improvement were achieved by applying anodal current to the right parietal lobe and cathodal current to the left parietal lobe and contrasting it with the reverse setup. When the same research group used tDCS in a training study with two dyscalculic individuals, the reverse setup (left anodal, right cathodal) demonstrated improvement of numerical abilities.
Dyscalculia is thought to be present in 3–6% of the general population, but estimates by country and sample vary somewhat. Many studies have found prevalence rates by gender to be equivalent. Those that find gender difference in prevalence rates often find dyscalculia higher in females, but some few studies have found prevalence rates higher in males.
Mental disabilities specific to mathematics were originally identified in case studies with patients who experienced specific arithmetic disabilities as a result of damage to specific regions of the brain. More commonly, dyscalculia occurs developmentally, as a genetically linked learning disability which affects a person's ability to understand, remember, or manipulate numbers or number facts (e.g., the multiplication tables). The term is often used to refer specifically to the inability to perform arithmetic operations, but it is also defined by some educational professionals and cognitive psychologists such as Stanislas Dehaene and Brian Butterworth as a more fundamental inability to conceptualize numbers as abstract concepts of comparative quantities (a deficit in "number sense"), which these researchers consider to be a foundational skill, upon which other mathematic abilities build. Symptoms of Dyscalculia include the delay of simple counting, inability to memorize simple arithmetic facts such as adding, subtracting, etc., There are very few known symptoms however, because there has been little research done on the topic.
The term dyscalculia dates back to at least 1949.
Dyscalculia comes from Greek and Latin and means "counting badly". The prefix "dys-" comes from Greek and means "badly". The root "calculia" comes from the Latin "calculare", which means "to count" and which is also related to "calculation" and "calculus".
- ^ American Heritage Dictionary
- ^ Collins Dictionary
- ^ Oxford Dictionaries Online
- ^ Random House Dictionary
- ^ a b c Butterworth, B (2010). "Foundational numerical capacities and the origins of dyscalculia". Trends in Cognitive Sciences. 14 (12): 534–541. doi:10.1016/j.tics.2010.09.007. PMID 20971676.
- ^ a b c d Butterworth, B; Varma, S; Laurillard, D (2011). "Dyscalculia: From brain to education". Science. 332 (6033): 1049–1053. Bibcode:2011Sci...332.1049B. doi:10.1126/science.1201536. PMID 21617068.
- ^ Shalev, Ruth. "Developmental Dyscalculia" (PDF).
- ^ Soares, MD, Neelkamal; Patel, MD, MBA, Dilip R (2015). "Dyscalculia". International Journal of Child and Adolescent Health. 8 (1): 15–26.
- ^ Klingberg, Torkel (2013), The Learning Brain: Memory and Brain Development in Children, Oxford University Press, p. 68, ISBN 9780199917105
- ^ Fischer, B; Gebhardt, C; Hartnegg,, K (2008). "Subitizing and visual counting in children with problems in acquiring basic arithmetic skills" (PDF). Optometry & Vision Development. 39 (1): 24–9.
- ^ Team, T. U. (n.d.). Understanding Dyscalculia. Retrieved September 02, 2017, from [www.understood.org]
- ^ Kucian; von Aster (2015). "Developmental Dyscalculia". European Journal of Pediatrics. 174 (1): 1–13. doi:10.1007/s00431-014-2455-7. PMID 25529864.
- ^ Mozzocco; Myers (2003). "Complexities in identifying and defining mathematics learning disability in the primary school-age years". Annals of Dyslexia. 53 (1): 218–253. doi:10.1007/s11881-003-0011-7. PMC 2742419 . PMID 19750132.
- ^ Posner, Tamar (2008). Dyscalculic in the Making: Mathematical Sovereignty, Neurological Citizenship, and the Realities of the Dyscalculic. ProQuest. ISBN 978-1-109-09629-3.
- ^ [www.as.wvu.edu]
- ^ Dehaene, S. (2001). "Precis of the number sense". Mind & Language. 16 (1): 16–36. doi:10.1111/1468-0017.00154.
- ^ Butterworth, B. (2005). Developmental dyscalculia. In J. I. D., Campbell (Ed.), Handbook of mathematical cognition (pp. 455–467). Hove, UK: Psychology Press.
- ^ Moyer, R. S.; Landauer, T. K. (1967). "Time required for judgements of numerical inequality". Nature. 215 (5109): 1519–1520. Bibcode:1967Natur.215.1519M. doi:10.1038/2151519a0. PMID 6052760.
- ^ a b Halberda, J.; Mazzocco, M. M. M.; Feigenson, L. (2008). "Individual differences in non-verbal number acuity correlate with maths achievement". Nature. 455 (7213): 665–668. Bibcode:2008Natur.455..665H. doi:10.1038/nature07246. PMID 18776888.
- ^ Halberda, J.; Ly, R.; Wilmer, J. B.; Naiman, D. Q.; Germine, L. (2012). "Number sense across the lifespan as revealed by a massive Internet-based sample". Proceedings of the National Academy of Sciences. 109 (28): 11116–11120. Bibcode:2012PNAS..10911116H. doi:10.1073/pnas.1200196109. PMC 3396479 . PMID 22733748.
- ^ Ashkenazi, S.; Mark-Zigdon, N.; Henik, A. (2009). "Numerical distance effect in developmental dyscalculia". Cognitive Development. 24 (4): 387–400. doi:10.1016/j.cogdev.2009.09.006.
- ^ Mussolin, C.; Mejias, S.; Noël, M. P. (2010). "Symbolic and nonsymbolic number comparison in children with and without dyscalculia". Cognition. 115 (1): 10–25. doi:10.1016/j.cognition.2009.10.006. PMID 20149355.
- ^ a b c Price, G. R.; Holloway, I.; Räsänen, P.; Vesterinen, M.; Ansari, D. (2007). "Impaired parietal magnitude processing in developmental dyscalculia". Current Biology. 17 (24): 1042–1043. doi:10.1016/j.cub.2007.10.013. PMID 18088583.
- ^ Piazza, M.; Facoetti, A.; Trussardi, A. N.; Berteletti, I.; Conte, S.; Lucangeli, D.; Dehaene, S.; Zorzi, M. (2010). "Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia". Cognition. 116 (1): 33–41. doi:10.1016/j.cognition.2010.03.012. PMID 20381023.
- ^ Rousselle, L.; Noel, M.P. (2007). "Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs. non-symbolic number magnitude". Cognition. 102: 361–395. doi:10.1016/j.cognition.2006.01.005. PMID 16488405.
- ^ De Smedt, B.; Gilmore, C.K. (2011). "Defective number module or impaired access? Numerical magnitude processing in first graders with mathematical difficulties". Journal of Experimental Child Psychology. 108 (2): 278–292. doi:10.1016/j.jecp.2010.09.003.
- ^ Mussolin, C.; De Volder, A.; Grandin, C.; Schlögel, X.; Nassogne, M.C.; Noël, M.P. (2010). "Neural correlates of symbolic number comparison in developmental dyscalculia". Journal of Cognitive Neuroscience. 22 (5): 860–874. doi:10.1162/jocn.2009.21237.
- ^ Shalev, Ruth (2004). "Developmental Dyscalculia". Journal of Child Neurology. 49 (11): 868–873. doi:10.1111/j.1469-8749.2007.00868.x.
- ^ a b Berch, Mozacco (2007). "Why Is Math So Hard for Some Children? The Nature and Origins of Mathematical Learning Difficulties and Disabilities". Brookes Publishing Company: 416.
- ^ Dinkel (2013). "Diagnosing Developmental Dyscalculia on the Basis of Reliable Single Case FMRI Methods: Promises and Limitations". PLOS ONE. 8 (12): e83722. Bibcode:2013PLoSO...883722D. doi:10.1371/journal.pone.0083722. PMC 3857322 . PMID 24349547.
- ^ Landerl; Bevan, A; Butterworth, B (2004). "Developmental dyscalculia and basic numerical capacities: a study of 8–9-year-old students". Cognition. 93 (2): 99–125. doi:10.1016/j.cognition.2003.11.004. PMID 15147931.
- ^ Landerl; Fussenegger, B; Moll, K; Willburger, E (2009). "Dyslexia and dyscalculia: Two learning disorders with different cognitive profiles". Journal of Experimental Child Psychology. 103 (3): 309–324. doi:10.1016/j.jecp.2009.03.006. PMID 19398112.
- ^ Rouselle; Noël (2007). "Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs non-symbolic number magnitude processing". Cognition. 102 (3): 361–395. doi:10.1016/j.cognition.2006.01.005. PMID 16488405.
- ^ Rosselli, Monica; Matute, Esmeralda; Pinto, Noemi; Ardila, Alfredo (2006). "Memory Abilities in Children With Subtypes of Dyscalculia". Developmental Neuropsychology. 30 (3): 801–818. doi:10.1207/s15326942dn3003_3. PMID 17083294.
- ^ Landerl, K; Bevan, A; Butterworth, B (2004). "Developmental dyscalculia and basic numerical capacities: a study of 8-9-year-old students". Cognition. 93 (2): 99–125. doi:10.1016/j.cognition.2003.11.004. PMID 15147931.
- ^ a b Shalev, R (2004). "Developmental Dyscalculia". Journal of Child Neurology. 19 (10): 765–771. doi:10.1177/08830738040190100601. PMID 15559892.
- ^ Geary, DC (1993). "Mathematical disabilities: Cognitive, neuropsychological, and genetic components". Psychological Bulletin. 114 (2): 345–362. doi:10.1037/0033-2909.114.2.345. PMID 8416036.
- ^ a b Rubinsten, O; Henik, A (February 2009). "Developmental dyscalculia: Heterogeneity might not mean different mechanisms". Trends Cogn. Sci. (Regul. Ed.). 13 (2): 92–9. doi:10.1016/j.tics.2008.11.002. PMID 19138550.
- ^ Grabner, RH; Ansari, D; Koschutnig, K; Reishofer, G; Ebner, F; Neuper, C (2009). "To retrieve or to calculate? Left angular gyrus mediates the retrieval of arithmetic facts during problem solving". Neuropsychologia. 47 (2): 604–608. doi:10.1016/j.neuropsychologia.2008.10.013. PMID 19007800.
- ^ Holloway, ID; Price, GR; Ansari, D (2010). "Common and segregated neural pathways for the processing of symbolic and nonsymbolic numerical magnitude: An fMRI study". NeuroImage. 49 (1): 1006–1017. doi:10.1016/j.neuroimage.2009.07.071. PMID 19666127.
- ^ Horwitz, B; Rumsey, JM; Donohue, BC (1998). "Functional connectivity of the angular gyrus in normal reading and dyslexia". PNAS. 95 (15): 8939–8944. Bibcode:1998PNAS...95.8939H. doi:10.1073/pnas.95.15.8939. PMC 21181 . PMID 9671783.
- ^ Pugh, KR; Mencl, WE; Shaywitz, BA; Shaywitz, SE; Fulbright, RK; Constable, RT; Skudlarski, P; Marchione, KE; Jenner, AR; Fletcher, JM; Liberman, AM; Shakweiler, DP; Katz, L; Lacadie, C; Gore, JC (2000). "The Angluar Gyrus in Developmental Dyslexia: Task-Specific Differences in Functional Connectivity With Posterior Cortex". Psychological Science. 11 (1): 51–56. doi:10.1111/1467-9280.00214. PMID 11228843.
- ^ Geary, DC (1990). "A componential analysis of an early learning deficit in mathematics". Journal of Experimental Child Psychology. 49 (3): 363383. doi:10.1016/0022-0965(90)90065-G.
- ^ McLean, JF; Hitch, GJ (1999). "Working Memory Impairments in Children with Specific Arithmetic Learning Difficulties". Journal of Experimental Child Psychology. 74 (3): 240–260. doi:10.1006/jecp.1999.2516. PMID 10527556.
- ^ Szucs, D; Devine, A; Soltesz, F; Nobes, A; Gabriel, F (2013). "Developmental dyscalculia is related to visuo-spatial memory and inhibition impairment". Cortex. 49 (10): 2674–2688. doi:10.1016/j.cortex.2013.06.007. PMC 3878850 . PMID 23890692.
- ^ Rotzer, S; Loenneker, T; Kucian, K; Martin, E; Klaver, P; von Aster, M (2009). "Dysfunctional neural network of spatial working memory contributes to developmental dyscalculia". Neuropsychologia. 47 (13): 2859–2865. doi:10.1016/j.neuropsychologia.2009.06.009. PMID 19540861.
- ^ Dumontheil, I; Klingberg, T (2012). "Brain Activity during a Visuospatial Working Memory Task Predicts Arithmetical Performance 2 Years Later". Cerebral Cortex. 22 (5): 1078–1085. doi:10.1093/cercor/bhr175. PMID 21768226.
- ^ Monuteaux, MC; Faraone, SV; Herzig, K; Navsaria, N; et al. (2005). "ADHD and dyscalculia: Evidence for independent familial transmission". J Learn Disabil. 38 (1): 86–93. doi:10.1177/00222194050380010701. PMID 15727331.
- ^ A. Anning, A. Edwards (1999). Promoting Children’s Learning from Birth to Five: Developing the New Early Years Professional. Maidenhead, UK: Open University Press.
- ^ S. Papert. Mindstorms: Children, Computers, and Powerful Ideas (Harvester Press, Brighton, UK, 1980).
- ^ a b Butterworth, B., Varma, S., & Laurillard, D. (2011). "Dyscalculia: from brain to education". Science. New York, N.Y. 332 (6033): 1049–53. doi:10.1126/science.1201536. PMID 21617068.
- ^ Fuchs LS; et al. (2008). "Remediating computational deficits at third grade: A randomized field trial". Journal of research on educational effectiveness. 1 (1): 2–32. doi:10.1080/19345740701692449. PMC 3121170 . PMID 21709759.
- ^ Fuchs LS; et al. (Jan 2013). "Effects of first-grade number knowledge tutoring with contrasting forms of practice". Journal of Educational Psychology. 105 (1): 58–77. doi:10.1037/a0030127. PMC 3779611 . PMID 24065865.
- ^ Powell SR, Fuchs LS, Fuchs D, Cirino PT, Fletcher JM (2009). "Effects of fact retrieval tutoring on third-grade students with math difficulties with and without reading difficulties". Learning Disabilities Research and Practice. 24 (1): 1–11. doi:10.1111/j.1540-5826.2008.01272.x. PMC 2682421 . PMID 19448840.
- ^ Wilson AJ, Revkin SK, Cohen D, Cohen L, Dehaene S; Revkin; Cohen; Cohen; Dehaene (2006). "An open trial assessment of "The Number Race", an adaptive computer game for remediation of dyscalculia". Behav Brain Funct. 2: 20. doi:10.1186/1744-9081-2-20. PMC 1523349 . PMID 16734906.
- ^ Hatton, Darla; Hatton, Kaila. "Apps to Help Students With Dyscalculia and Math Difficulties". National Center for Learning Disabilities and Math Difficulties. Retrieved Mar 26, 2014.
- ^ Callaway, Ewen (Jan 9, 2013). "Dyscalculia: Number games". Nature. Retrieved Mar 26, 2014.
- ^ a b c Butterworth, B., Laurillard, D. (2010). "Low numeracy and dyscalculia: identification and intervention". ZDM. 42 (6): 527–539. doi:10.1007/s11858-010-0267-4.
- ^ a b c Kucian, K., Grond, U., Rotzer, S., Henzi, B., Schönmann, C., Plangger, F., ... & von Aster, M. (2011). "Mental number line training in children with developmental dyscalculia". NeuroImage. 57 (3): 782–795. doi:10.1016/j.neuroimage.2011.01.070. PMID 21295145.
- ^ a b Räsänen, P., Salminen, J., Wilson, A. J., Aunio, P., & Dehaene, S. (2009). "Computer-assisted intervention for children with low numeracy skills". Cognitive Development. 24 (4): 450–472. doi:10.1016/j.cogdev.2009.09.003.
- ^ a b Käser, Tanja; Baschera, Gian-Marco; Kohn, Juliane; Kucian, Karin; Richtmann, Verena; Grond, Ursina; Gross, Markus; von Aster, Michael (2013-01-01). "Design and evaluation of the computer-based training program Calcularis for enhancing numerical cognition". Developmental Psychology. 4: 489. doi:10.3389/fpsyg.2013.00489. PMC 3733013 . PMID 23935586.
- ^ Rauscher, Larissa; Kohn, Juliane; Käser, Tanja; Mayer, Verena; Kucian, Karin; McCaskey, Ursina; Esser, Günter; von Aster, Michael (2016-01-01). "Evaluation of a Computer-Based Training Program for Enhancing Arithmetic Skills and Spatial Number Representation in Primary School Children". Developmental Psychology. 7: 913. doi:10.3389/fpsyg.2016.00913. PMC 4921479 . PMID 27445889.
- ^ Käser, T.; Busetto, A. G.; Solenthaler, B.; Baschera, G.-M.; Kohn, J.; Kucian, K.; von Aster, M.; Gross, M. (2013). "Modelling and Optimizing Mathematics Learning in Children". International Journal of Artificial Intelligence in Education. 23 (1-4): 115–135. doi:10.1007/s40593-013-0003-7.
- ^ Cohen Kadosh, R; Soskic, S; Iuculano, T; Kanai, R; Walsh, V (2010). "Modulating neuronal activity produces specific and long-lasting changes in numerical competence". Current Biology. 20 (22): 2016–2020. doi:10.1016/j.cub.2010.10.007. ISSN 0960-9822. PMC 2990865 . PMID 21055945.
- ^ Iuculano, T., & Cohen Kadosh, R. (2014). "Preliminary evidence for performance enhancement following parietal lobe stimulation in Developmental Dyscalculia". Frontiers in Human Neuroscience. 8: 38. doi:10.3389/fnhum.2014.00038. PMC 3916771 . PMID 24570659.
- ^ Shalev and Gross-Tsur; Gross-Tsur, V (2001). "Developmental dyscalculia". Pediatric Neurology. 24 (5): 337–342. doi:10.1016/s0887-8994(00)00258-7. PMID 11516606.
- ^ Gross-Tsur, Varda; Manor, Orly; Shalev, Ruth S. (1996). "DEVELOPMENTAL DYSCALCULIA: PREVALENCE AND DEMOGRAPHIC FEATURES". Developmental Medicine and Child Neurology. 38 (1): 25–33. doi:10.1111/j.1469-8749.1996.tb15029.x. PMID 8606013.
- ^ Kucian and von Aster, Karin; von Aster, Michael (2015). "2015". European Journal of Pediatrics. 174 (1): 1–13. doi:10.1007/s00431-014-2455-7. PMID 25529864.
- ^ Dehaene, S. (1997). The Number Sense: How the Mind Creates Mathematics. New York: Oxford University Press. ISBN 978-0-19-513240-3.
- ^ Trott, Clare (5 March 2009). "Dyscalculia". In Pollak, David. Neurodiversity in Higher Education: Positive Responses to Specific Learning Differences. John Wiley and Sons. ISBN 978-0-470-99753-6.
- ^ Kosc, Ladislav (1974). "Developmental dyscalculia". Journal of Learning Disabilities. 7 (3): 159–62. doi:10.1177/002221947400700309.
- Abeel, Samantha (2003). My thirteenth winter: a memoir. New York: Orchard Books. ISBN 0-439-33904-9. OCLC 51536704.
- Ardila A, Rosselli M (December 2002). "Acalculia and dyscalculia" (PDF). Neuropsychol Rev. 12 (4): 179–231. doi:10.1023/a:1021343508573. PMID 12539968.
- Tony Attwood (2002). Dyscalculia in Schools: What it is and What You Can Do. First & Best in Education Ltd. ISBN 1-86083-614-3. OCLC 54991398.
- Butterworth, Brian; Yeo, Dorian (2004). Dyscalculia Guidance: Helping Pupils with Specific Learning Difficulties in Maths. London: NferNelson. ISBN 0-7087-1152-9. OCLC 56974589.
- Campbell, Jamie I. D. (2004). Handbook Of Mathematical Cognition. Psychology Press (UK). ISBN 1-84169-411-8. OCLC 644354765.
- Brough, Mel; Henderson, Anne; Came, Fil (2003). Working with dyscalculia: recognising dyscalculia: overcoming barriers to learning in maths. Santa Barbara, Calif: Learning Works. ISBN 0-9531055-2-0. OCLC 56467270.
- Chinn, Stephen J. (2004). The Trouble with Maths: A Practical Guide to Helping Learners with Numeracy Difficulties. New York: RoutledgeFalmer. ISBN 0-415-32498-X. OCLC 53186668.
- Reeve R, Humberstone J (2011). "Five- to 7-year-olds' finger gnosia and calculation abilities". Front Psychol. 2: 359. doi:10.3389/fpsyg.2011.00359. PMC 3236444 . PMID 22171220.
- "Sharma: Publications". Dyscalculia.org.