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Dyscalculia: Definition, Symptoms, Types, Causes, “Big 5” of Intervention

William is eighteen years old, and he excels in drama and art. He’s an avid reader and writes poems and short stories in his spare time. However, he is unable to graduate with his senior class because he doesn’t have the required credits in mathematics.

Although he has normal intelligence, William has consistently done poorly in mathematics since primary school. After intensive tutoring and years of practice, he has finally become somewhat competent at basic facts and operations, but he has no idea how or when to apply them. When taking a math test, he simply takes numbers from each problem and inserts them in the algorithms that he memorized when studying for the test. He always carries his personal telephone book in his book bag because he can’t remember his own phone number or those of his friends.

Like William, many people have problems in learning mathematics. The nature of their problems vary. Some students can master basic facts but can’t do higher mathematics. Some can do higher math but can’t master basics. Some can follow math procedures one day but are unable to follow them the next day. Others may perform mathematical algorithms well in one situation but can’t apply them to new situations. Math disabilities can be very frustrating due to the complexity and variety of problems.

Definition

A 10-year-old with dyscalculia

Mathematics plays an important part in our lives, from basic trading at a market stall in Marrakesh or Beijing to the complex algorithms that guide international banking, from working out the time of a journey to see a friend in a nearby town to the time it takes a sub-atomic particle to travel around CERN’s Large Hadron Collider. But, not everyone is good at math.  is

Dyscalculia is defined as difficulty acquiring basic arithmetic skills that is not explained by inadequate schooling, emotional instability  or intellectual disability (IQ below 70). The term refers to a wide range of persistent and extreme difficulties in math, including weaknesses in understanding the meaning of numbers, and difficulty applying mathematical principles to solve problems. Coined in the mid-20th century, the word dyscalculia has both Greek and Latin origins: the Greek prefix ‘dys’ means ‘badly’, while ‘calculia’, from the Latin ‘calculare’, means to count. Literally, dyscalculia means to count badly; the reality is much more complex.

The term developmental dyscalculia may be used to distinguish the problem in children and youth from similar problems experienced by adults after severe head injuries. 

Sometimes the word acalculia is used to refer to complete inability to use mathematical symbols and the term dyscalculia is reserved for less severe problems in these areas.

An 11-year-old with acalculia

Students with pseudo-dyscalculia have severe math anxiety and may even develop math phobia (arithmophobia). Those who have been supported to overcome their math anxiety will potentially be able to function very well in math. Because of their difficulties with math, most dyscalculics have some anxiety about math (Hornigold, 2015).

Prevalence

As with other learning disabilities, reports of dyscalculia’s prevalence vary depending upon the definition and situation. However, research suggests that it has the same prevalence as dyslexia (about 6–8% of children) although it is far less widely recognized by parents and educators (Ardilla & Roselli, 2002).

Badian (1999) reports a prevalence rate of 6.9 percent with 3.9 percent of these students low in arithmetic only, and 3 percent of these students low in arithmetic and reading. She suggests that researchers differentiate between children with arithmetic difficulties and those with both arithmetic and reading problems, in order to prevent distorted interpretations of research. Peard (2010), however, contends that dyscalculia figures generally include a significant proportion of students who are better called “learned difficulties”, and that the incidence of a genuine learning disability in math — a permanent neurological disorder — is less than 2%. 

Attention deficits are also frequently comorbid with dyscalculia (Czamara et al., 2013; Shalev, Auerbach, & Gross-Tsur, 1995; Willcutt et al., 2013).

Symptoms

  • Poor number sense is a core deficit in dyscalculia. Number sense refers to a person’s ability to use and understand numbers.
  • How many stars are there in this diagram? If you can identify the number without counting you are subitizing! (Dilmen, 2006)

    Another core deficit is poor subitizing. The word ‘subitize’ comes from Latin meaning ‘sudden’. It refers to the ability to instantly identify the number of objects in a set without counting. Most people can subitize up to six or seven objects. A child with dyscalculia may find this very hard and may need to count even small numbers of objects. For example, if they are presented with two objects they may count the objects rather than just know that there are two.
  • Poor understanding of the signs +, -, ÷ and x, or may confuse these mathematical symbols.
  • Difficulty with addition, subtraction, multiplication and division or may find it difficult to understand the words “plus,” “add,” “add-together.”
  • Immature strategies such as counting all instead of counting on. The child may workout 137 + 78 by drawing 137 dots and then 78 dots and then counting them all.
  • Difficulty with times tables.
  • Poor mental arithmetic skills.
  • May have trouble even with a calculator due to difficulties in the process of feeding in variables.
  • Inability to notice patterns. The world of math is full of patterns and the ability to see, predict and continue patterns is a key math skill. Take the sequence of the 5 x table for example: 5, 10, 15, 20, 25 etc. This is a very clear pattern but a student with dyscalculia may not readily spot it.
  • Inability to generalize. Being able to generalize makes life so much simpler in math, but a dyscalculic student may find this very hard. They might not see that knowing that 3 + 4 = 7 means they also know that 30 + 40 = 70, or even that 3 inches + 4 inches = 7 inches.
  • May reverse or transpose numbers for example 63 for 36, or 785 for 875.
  • Difficulty with conceptualizing time and judging the passing of time.
  • Difficulty with everyday tasks like checking change.
  • Difficulty keeping score during games.
  • Inability to grasp and remember mathematical concepts, rules, formulae, and sequences.
  • Extreme cases may lead to a phobia of mathematics and mathematical devices.
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Symptoms by age

Toddlers and preschoolers

  • Difficulty learning to count
  • Difficulty sorting
  • Difficulty corresponding numbers to objects
  • Difficulty with auditory memory of numbers
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Kindergarten

  • Difficulty counting
  • Difficulty subitizing
  • Trouble with number recognition
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Early elementary

  • Difficulty with magnitude comparison
  • Trouble learning math facts
  • Difficulty with math problem-solving skills
  • Over reliance on finger counting for more than basic sums
  • Anxiety during math tasks
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Late elementary through middle

  • Difficulties with precision during math work
  • Difficulty remembering previously encountered patterns
  • Difficulty sequencing multiple steps of math problems
  • Difficulty understanding real-world representation of math formulae
  • Anxiety during math tasks
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High school

  • Struggle to apply math concepts to everyday life, including money matters, estimating speed and distance
  • Trouble with measurements
  • Difficulty grasping information from graphs or charts
  • Difficulty arriving at different approaches to same math problem
  • Anxiety during math tasks
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Types

According to the website Dyslexia in Ireland dyscalculia can be broken down into three subtypes:

  • Quantitative dyscalculia, a deficit in the skills of counting and calculating.
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  • Qualitative dyscalculia, a result of difficulties in comprehension of instructions or the failure to master the skills required for an operation. When a child has not mastered the memorization of number facts, he cannot benefit from this stored “verbalizable information about numbers” that is used with prior associations to solve problems involving addition, subtraction, multiplication, division, and square roots.
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  • Intermediate dyscalculia involves the inability to operate with symbols, or numbers.
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On the basis of his experience with arithmetic learning problems, Kosc described six types:

  • Verbal dyscalculia, which refers to problems in naming the amount of things.
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  • Practognostic dyscalculia, which refers to problems in manipulating things mathematically — for example, comparing objects to determine which is larger.
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  • Lexical dyscalculia, which refers to problems in reading mathematical symbols, including operation signs (+, – ) and numerals.
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  • Graphical dyscalculia, which refers to problems in writing mathematical symbols and numerals.
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  • Ideognostical dyscalculia, which refers to problems in understanding mathematical concepts and relationships.
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  • Operational dyscalculia, which refers to problems in performing arithmetic operations.
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Karagiannakis and Cooreman (2015) identified four areas or subtypes: core number, reasoning, memory and visual spatial. Dyscalculic students may have difficulty in all or maybe just one or two of these areas.

Dyscalculia can affect different aspects of math ability and can be split into four areas or subtypes leading to a variety of math profiles. Some students will have difficulty in all or maybe just one or two of these areas. (Hornigold, 2015) 

These types of dyscalculia have not been independently verified, and they are quite difficult to differentiate in students who have arithmetic learning disabilities. Nevertheless, these discussions illustrate the many problems students may have in arithmetic and mathematics.

Causes

Successful intervention is dependent on finding the cause or causes of a problem. Most problems can only be solved if one knows their causes. A disease such as scurvy claimed the lives of thousands of seamen during their long sea voyages. The disease was cured fairly quickly once the cause was discovered, viz. a vitamin C deficiency. A viable point of departure would therefore be to ask the question, “What causes dyscalculia?”

Although some causes of dyscalculia have a genetic origin (Kere, 2014), and environmental factors play an important role (Stein, 2018), cognition mediates brain-behavior relationships and therefore offers a sufficient level of explanation for the development of principled interventions. We thus need to understand the cognitive difficulties that underpin math failure, regardless whether their origin is constitutional or environmental (Elliott & Grigorenko, 2014).

It is important to understand that mathematics is a subject that consists of three aspects:

  • Foundational skills
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Research has shown that attention; visual perception; visual, visuospatial and working memory; and logical thinking (which makes problem solving possible) are important foundational skills of math.

Visual perception refers to the process of interpreting and organizing visual information. Visual perceptual skill is often subdivided into areas such as visual discrimination and visual memory. Visual discrimination involves the ability to attend to and identify a figure’s distinguishing features and details, such as shape, orientation, color and size. Visual memory refers to the ability to remember a visual image. 

One hundred seventy-one children with a mean age of 10.08 years participated in a study by Kulp et al. (2004). This study, conducted at the Ohio State University College of Optometry was designed to determine whether or not performance on tests of visual perception could predict the children with poor current achievement in mathematics. Kulp et al. concluded: “poor visual perceptual ability is significantly related to poor achievement in mathematics, even when controlling for verbal cognitive ability. Therefore, visual perceptual ability, and particularly visual memory, should be considered to be amongst the skills that are significantly related to mathematics achievement.”

Dr. Dénes Szűcs and team (2013) from the University of Cambridge, UK set out to compare various potential theories of dyscalculia in more than a thousand 9-year-old children. The researchers found that children with dyscalculia showed poor visuospatial memory performance. For example, they performed poorly when they had to remember the locations of items in a spatial grid. In addition, dyscalculic children’s ability to resist distraction from irrelevant information was also weak. On a task where they had to choose which of two animals was larger in real life they performed poorly when the real-life larger animal was smaller in its display size.

Working memory is the memory needed to carry out step-by-step procedures and to reason. Working memory enables you to hold information in mind while performing other actions. Siegel and Ryan (1989) found that children with dyscalculia did less well than controls on a working memory task involving counting and remembering digits, but not on a nonnumerical working memory task. This led them to speculate that there is a working memory system specialized for numerical information, and that children with dyscalculia have specific problems with this system.

Piaget and Szeminska (1941) suggested a relationship between ‘seriation’ (or the logical ability to sort objects based on differences while ignoring similarities) and ‘classification’ (or the logical ability to sort objects based on similarities, while ignoring differences) and the understanding of number. Although several neo-Piagetian researchers question the causality of seriation and classification for understanding number, recent studies revealed that children adequately solving seriation and classification tasks in kindergarten perform better in mathematical tasks in first and second grades (Desoete, 2015).
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  • Mathematical skills
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Mathematics comprise a wide variety of skills, including counting, adding and subtracting, multiplication and division, applying place value, fractions, reading time etc.

Landerla, Bevana and Butterworth (2004) concluded that dyscalculia is the result of specific disabilities in basic numerical processing, rather than the consequence of deficits in other cognitive abilities.
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  • Knowledge
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There is much in math that one simply has to know and therefore has to learn, for example many terms, definitions, symbols, theorems and axioms. These are all things that the learner must know, not things that he must know how to do.
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Intervention

Dyscalculia may have some serious implications for children if no intervention is provided. Primarily, dyscalculia may impinge on the emotional well-being of students. In the long-term, living with dyscalculia can be difficult. Difficulties vary from simply remembering important telephone numbers and dates, to paying the right amount to the cashier when going shopping and checking the change. Other tasks presenting difficulties could be cooking, planning appointments and being able to use the time available in a day appropriately. Adults who have dyscalculia often feel embarrassed when they are faced with everyday tasks which they cannot handle.

Bynner and Parsons (1997) indicate that students with poor numeracy skills tend to leave full-time education at their first chance. Butterworth and Yeo (2004) state that such persons are more likely to be unemployed, depressed, ill and arrested. All this illustrates that dyscalculia should be handled at a young age before it has irreversible effects.

Below is the “Big 5” of dyscalculia intervention: 
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  • Adhere to fundamental learning principles
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It should also be noted that learning is a stratified process. Certain skills have to be mastered first, before it becomes possible to master subsequent skills.

In order to be a football player, a person first has to master the foundational skills, e.g. passing, kicking and tackling. In the same way, in order to do math, a child first has to learn the foundational skills of math, like visual perception and visual memory. The child who confuses the signs +, -, ÷ and ×, may have a problem with visual discrimination of forms and/or visual discrimination of position in space. A child who has a poor sense of direction (i.e., north, south, east, and west), may have a problem with visual discrimination of position in space, etc.

The second step would be to master mathematical skills, which must be done in a sequential fashion. One has to learn to count before it becomes possible to learn to add and subtract. Suppose one tried to teach a child, who had not yet learned to count, to add and subtract. This would be quite impossible and no amount of effort would ever succeed in teaching the child these skills. The child must learn to count first, before it becomes possible for him to learn to add and subtract.

The third step would be to ensure that a learner catches up on the knowledge aspect of math.
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  • Minimize anxiety
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Math anxiety causes the release of cortisol into the bloodstream. Cortisol refocuses the frontal lobe to deal with the anxiety. Meanwhile, any unrelated learning in working memory is disrupted and lost (Sousa, 2015).

It is crucial to minimize levels of anxiety as much as possible, as anxiety may be a disability in its own right. Anxiety may fill up the working memory space of the brain without allowing the complete and effective processing of numerical tasks (Ashcraft et al., 1998).

Anxiety of any type causes the body to release the hormone cortisol into the bloodstream. Cortisol’s main function is to refocus the brain on the source of the anxiety and determine what action to take to relieve the stress. Heart rate increases, and other physical indicators of worry appear. Meanwhile, the frontal lobe is no longer interested in learning or processing mathematical operations because it has to deal with what may be a threat to the individual’s safety. As a result, the student cannot focus on the learning task at hand and has to cope with the frustration of inattention. Furthermore, the anxious feelings disrupt working memory’s ability to manipulate and retain numbers and numerical expressions (Sousa, 2015).
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  • Teach in a multisensory way
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Information is taken into the brain through three main channels: visual, auditory and kinesthetic. Many students with a learning disability have a weakness in one or more of these channels. Teaching in a multisensory way, using all three channels simultaneously, will help them as their weaker channels are supported by their stronger ones (Hornigold, 2015).
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  • Make the most of mistakes
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Everyone makes mistakes; they are vital to developing understanding. Unfortunately, constantly making mistakes in math can lead some children to give up. However, research by Jo Boaler and Carol Dweck at Stanford University has shown that synapses grow in the brain when mistakes are made, and that there is no growth when the answers are correct. Even if a mistake is not rectified, there will be growth. It is the struggle to get the right answers that fosters growth.

One of the most powerful moves a teacher or parent can make is in changing the messages they give about mistakes and wrong answers in mathematics. When we teach students that mistakes are positive, it has an incredibly liberating effect on them (Boaler, 2016).
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  • Repetition is not the enemy
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    In the 1920s and 1930s repetition and rote learning became to define bad teaching. Teachers were told that drill-and-practice dulls students’ creativity (Heward, 2003), and that rote learning in math classes is anti-right brain and therefore potentially criminal, as it robs all students of the opportunity to develop their human potential (Elliott, 1980). The phrase “drill and kill” is still used in educational circles, meaning that by drilling the student, you will kill his or her motivation to learn (Heffernan, 2010).
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When properly conducted, however, drill-and-practice is a consistently effective teaching method and should not be slighted as “low level,” and appears to be just as essential to complex and creative intellectual performance as they are to the performance of a virtuoso violinis (Brophy, 1986). In fact, a meta-analysis of 85 academic intervention studies with students with learning disabilities, carried out by Swanson and Sachse-Lee (2000), found that regardless of the practical or theoretical orientation of the study, the largest effect sizes were obtained by interventions that included systematic drill, repetition, practice, and review..
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Edublox 

Edublox’s dyscalculia help aims at

(1.) addressing the underlying shortcomings that interfere with math performance, such as poor visuospatial memory, working memory and logical thinking,

(2.) teaching math skills in a sequential fashion, which may include counting and skip-counting, adding and subtracting, multiplication and division, applying place value, fractions, using money, reading time etc., as well as

(3.) math knowledge.

We aim at minimizing anxiety, teach math in a multisensory way, make the most of mistakes, and repetition is not our enemy.

Case studies

Below is an example of a child’s progress after receiving math help from Edublox. She was diagnosed with dyscalculia as well as dyslexia and ADHD. Click here to follow her amazing journey to learning success.
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Below is another child’s progress after receiving math help from Edublox. She was diagnosed with dyslexia and acalculia. Click here to follow her journey to learning success..

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Below is the report of a third child whose math marks improved from 43% in the first term to 70% in the third term. Many of his other subject improved as well, as a result of the cognitive training he received:

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Visit the Edublox website
for more information or book a free consultation to discuss your child’s math learning needs.


Key takeaways


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Recommmended resources:

Boaler, J. (2016). Mathematical mindsets. San Francisco, CA: Jossey-Bass.
Hornigold, J. (2015). Dyscalculia pocketbook. Alresford, Hampshire: Teachers’ Pocketbooks.
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References:

Ardilla, A., & Rosselli, M. (2002). Acalculia and dyscalculia. Neuropsychology Review, 12(4), 179–231.

Ashcraft, M. H., Kirk, E. P., & Hopko, D. (1998). On the cognitive consequences of mathematics anxiety. In C. Donlan (Ed.), The development of mathematical skills. Hove: Psychology Press.

Badian, N. A. (1999). Persistent arithmetic, reading, or arithmetic and reading disability. Annals of Dyslexia49. https://doi.org/10.1007/s11881-999-0019-8

Boaler, J. (2016). Mathematical mindsets. San Francisco, CA: Jossey-Bass.

Brophy, J. (1986). Teacher influences on student achievement. American Psychologist, 41, 1069–1077.

Butterworth, B., & Yeo, D. (2004). Dyscalculia guidance: Helping pupils with specific learning difficulties in maths. London: Fulton Publishers.

Bynner, J., & Parsons, S. (1997). Does numeracy matter? London: Basic Skills Agency.

Czamara, D., Tiesler, C. M. T., Kohlböck, G., Berdel, D., Hoffmann, B., & Heinrich, J. (2013). Children with ADHD symptoms have a higher risk for reading, spelling and math difficulties in the GINIplus and LISAplus cohort studies. PLOS ONE, 8(5), e63859.

Desoete, A. (2015). Predictive indicators for mathematical learning disabilities/dyscalculia in kindergarten. In S. Chinn (Ed.), The Routledge international handbook of dyscalculia and mathematical learning difficulties (pp. 90-100). Abingdon, Oxon: Routledge.

Elliott, J. G., & Grigorenko, E. L. (2014). The dyslexia debate. Cambridge: Cambridge University Press.

Elliott, P. C. (1980). Going “back to basics” in mathematics won’t prove who’s “right”, but who’s “left” (brain duality and mathematics learning). International Journal of Mathematical Education in Science and Technology, 11(2), 213-219.

Franklin, D. (2018). Helping your child with language-based learning disabilities. Oakland, CA: New Harbinger Publications, Inc.

Hallahan, D. P., Kauffman, J., & Lloyd, J. (1985). Introduction to learning disabilities. Englewood Cliffs, NJ: Prentice Hall.

Heffernan, V. (2010, September 16). Drill, baby, drill. The New York Times Magazine. Retrieved February 10, 2020 from https://www.nytimes.com/2010/09/19/magazine/19fob-medium-heffernan-t.html

Heward, W. L. (2003). Ten faulty notions about teaching and learning that hinder the effectiveness of special education. Journal of Special Education, 36(4), 186-205.

Hornigold, J. (2015). Dyscalculia pocketbook. Alresford, Hampshire: Teachers’ Pocketbooks.

Karagiannakis, G. N, & Cooreman, A. (2015). Focused MLD intervention based on the classification of MLD subtypes. In S. Chinn (Ed.), The Routledge international handbook of dyscalculia and mathematical learning difficulties (pp. 265-276). Abingdon, Oxon: Routledge.

Kere, J. (2014). The molecular genetics and neurobiology of developmental dyslexia as model of a complex phenotype. Biochemical and Biophysical Research Communications452(2), 236-243. https://doi.org/10.1016/j.bbrc.2014.07.102

Kosc, L. (1974). Development of dyscalculia. Journal of Learning Disabilities, 7, 164–177.

Kulp, M. T. et al. (2004). Are visual perceptual skills related to mathematics ability in second through sixth grade children? Focus on Learning Problems in Mathematics, 26(4), 44-51.

Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic numerical capacities: A study of 8-9-year-old students. Cognition, 93, 99-125.

Mercer, C. D. (1997). Students with learning disabilities, 5th ed. Upper Saddle River, NJ: Merrill.

Peard, R. (2010). Dyscalculia: What is its prevalence? Research evidence from case studies. Procedia – Social and Behavioral Sciences, 8, 106–113.

Shalev, R. S., Auerbach, J., & Gross-Tsur, V. (1995). Developmental dyscalculia: Attentional and behavioral aspects. Journal of Child Psychology and Psychiatry, 36, 1261–1268.

Soares, N., Evans, T., & Patel, D. R. (2018). Specific learning disability in mathematics: a comprehensive review. Translational Pediatrics7(1), 48–62. https://doi.org/10.21037/tp.2017.08.03

Sousa, D. A. (2015). How the brain learns mathematics, 2nd ed. California: Corwin Press.

Stein, J. (2018). The magnocellular theory of developmental dyslexia. In T. Lachmann, & T. Weis (Eds.). Reading and dyslexia (pp. 97-128). Cham, Switzerland: Springer.

Swanson, H. L., & Sachse-Lee, C. (2000). A meta-analysis of single-subject design intervention research for students with LD. Journal of Learning Disabilities, 38(2), 114-136.

Szucs, D., Devine, A., Soltesz, F., Nobes, A., & Gabriel, F. (2013). Developmental dyscalculia is related to visuo-spatial memory and inhibition impairment. Cortex49(10), 2674-2688. https://doi.org/10.1016/j.cortex.2013.06.007

Willcutt, E. G., Petrill, S. A., Wu, S., Boada, R., DeFries, J. C., Olson, R. K., & Pennington, B. F. (2013). Comorbidity between reading disability and math disability: Concurrent psychopathology, functional impairment, and neuropsychological functioning. Journal of Learning Disabilities, 46, 500–516.

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