MATHEMATICS LEARNING DISABILITY
- Defining Dyscalculia
- Development of Mathematical Ability
- Dyscalculia and Dyslexia
- Dyscalculia and Math Anxiety
- Intervention Strategies
“Hi, my name is Van, and I have a learning disability. It’s called dyscalculia.”
So begins the vlog (video weblog) on Youtube (http://www.youtube.com/user /BlueforInsanity) titled Dyscalculia. The vlog was posted on May 3, 2010, and during that time, Van claimed that “I am looking at the end of my second year in college” and yet her mathematical ability goes only so far as basic math.
She went on to demonstrate how she does the four basic operations by counting on her fingers. “I don’t know what this is without counting it,” she said as she pointed at a card that reads “7+4”. Multiplying 7 by 4 took her over 20 seconds to do as she had to use counting again. Doing subtraction and division was even more confusing for her. She doubted herself even after she already got the right answers.
Van candidly enumerated other things that she was having difficulties with, such as telling the time using an analog clock; reading graphs, lists and maps; and following written instructions.
For people with dyscalculia, everything that has to do with numbers and mathematical skills is a constant struggle. Several studies have varied approximations of the prevalence of at least some form of dyscalculia among the population. However, the figures all tend to be between 3 and 8% (Geary, 2006; Butterworth, 2002; Kosc, 1974), which makes it about as common as dyslexia. Despite these, however, dyscalculia is much less recognized, studied and treated than its more popular cousin, dyslexia (Wilson and Dehaene, 2007).
Aside from that, or probably because of it, experts have varying definitions of dyscalculia, as well as its symptoms, causes, screening and treatment.
The United Kingdom (UK) Department for Education and Skills (DfES) defines dyscalculia as “A condition that affects the ability to acquire arithmetical skills. [Learners with dyscalculia] 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” (The British Dyslexia Association, 2009).
The UK Parliament Office of Science and Technology states that “people with dyscalculia have specific problems in learning arithmetical concepts and procedures, despite sufficient educational experience.
Mazzocco (2005) stated that “Dyscalculia is a genetic, neurological disorder that affects an individual’s ability to do mathematics” (as cited by Wadlington, 2008).
Butterworth (2002) cited the definition of “Mathematics Disorder” from The Diagnostic and Statistical Manual of Mental Disorders, fourth Edition (DSM-IV, Section 315.1) as applicable to define dyscalculia as well. The manual states the following criteria:
- Mathematical ability, as measured by individually administered standardized tests, is substantially below that expected given the person’s chronological age, measured intelligence, and age appropriate education.
- The disturbance in Criterion A significantly interferes with academic achievement or activities of daily living that require mathematical ability.
- If a sensory deficit is present, the difficulties in mathematical ability are in excess of those usually associated with it.
Furthermore, in the same study, Butterworth argues that “Dyscalculia appears to be a persistent congenital condition. Twin studies suggest that it is inherited, though little is known about which genes are involved. Any capacity specified in the genome is likely to be for simple concepts. The best candidate is for the concept of numerosity itself – that is, for a sense of the number of things in a collection.”
Rubinsten and Tannock (2010) defines dyscalculia as “a specific and severe deficit in the ability to process numerical information that cannot be ascribed to sensory difficulties, low IQ or inadequate education, and that results in a failure to develop fluent numerical computation skills.”
These definitions are mostly pertaining to what experts call “developmental dyscalculia” (as opposed to “acquired dyscalculia” or acalculia, which is not a pre-existing condition). However, Wilson and Dehaene (2007) contend that both developmental and acquired dyscalculia is caused by the same deficit in the brain, particularly in the horizontal intra-parietal sulcus (HIPS).
Although acalculia may be touched in some sections given its similarity, this paper primarily focuses on developmental dyscalculia, which will be referred to simply as dyscalculia.
In order to understand the deficits that a person with dyscalculia with regard to mathematical ability, we need to understand how mathematical ability normally develops as a child grows.
Jean Piaget had discussed the following key concepts to understanding numbers (as cited by Lee, 1995).
- Classification – the process of understanding relationships, such as similarity and differences.
- Ordering – the process of sequencing numbers.
- One-to-one correspondence – understanding that one object in a set is the same number as one object in another set, whether or not the objects are similar.
- Conservation – the quantity of an object in a set remains constant regardless of spatial arrangement.
Being able to master the above concepts is key for learning higher-level skills.
Adler (2001) contends that the development of mathematical ability starts at birth. Even after only a few days, an infant could already distinguish between two objects and just one object. This rudimentary understanding of numbers is already present at birth. Also at the moment of birth, the infant would already begin sorting things in his mind on the basis of categories and prototypes. Eventually, the child gains understanding of the relationships between things that he categorized.
At first, the infant would only recognize objects within the grasp of his senses; if it’s not there, it doesn’t exist. At about 1½ years, the child realizes that objects still exist even if he can’t see them. This is the first step in abstraction: concrete objects becoming ideas. At the age of 3-4 years, the child begins to determine numerical concepts of different sizes. The child learns how to count, at first learning just the sequence of numbers, and eventually learning to associate the number with the quantity of concrete objects.
Even before a normal child reaches school age, he already has developed a “number sense”, which would then be the foundation to build the basic arithmetic concepts when he starts school.
It is this “number sense” that people with dyscalculia has an initial deficit of (Wilson and Dehaene, 2007). Later on, learning new concepts of a higher order becomes problematic because of this deficit. This is also the reason why children with dyscalculia can’t be properly diagnosed as of yet until they start their formal schooling at the age of 6 or 7 years old.
According to Farmer, Riddick and Sterling (2002) as cited by Wadlington (2008), dyscalculia comes in many forms. Some people with dyscalculia have no problem understanding or performing the basic concepts, such as counting, number sense, basic arithmetic operations, but they would not know how to use these concepts to solve a higher level problem. Others would intuitively know the higher-level mathematical concepts, but have trouble performing or remembering basic arithmetic facts. And still some others would have a good understanding of basic and more complex mathematical concepts, but would have a hard time applying them to real-life or in similar situations. Some people would actually have challenges on all three areas!
An individual could actually have one, a few or all of the following types of dyscalculia (Wadlington, 2008; Adler, 2001):
- Verbal dyscalculia – refers to challenges in remembering and naming mathematical terms and symbols. A person with this type of dyscalculia would have challenges in recalling numbers, calculations or geometric shapes from memory.
- Practognostic dyscalculia – refers to challenges in using manipulatives or pictures when applying mathematical concepts. A person with this type often fails to understand and answer oral or written problems that are presented with words or in a text or picture. He or she is also unable to grasp the concepts of weight (which one is heavier or lighter), space (which one is larger or smaller), direction (left, right, up or down) and time (which one takes longer or faster). He or she also has challenges with the concepts of numbers as “much-more-most”, amount or quantity measures.
- Graphical dyscalculia – pertains to issues with writing mathematical terms, symbols, etc. Person with this type would write mathematical symbols, often numbers, in reverse or rotated. He or she has challenges in copying numbers, calculations or geometric figures. He or she is also unable to remember how mathematical symbols are written and how to write down numbers with more than one digit accurately such that zeroes may be lost or digits may be interchanged.
- Lexical dyscalculia – pertains to issues with reading mathematical symbols and vocabulary. A person with this type of dyscalculia would often mix up similar-looking numbers in reading, such as 6 and 9, 3 and 8 or 1 and 7. It would be difficult for him or her to recognize, and therefore use, calculation symbols such as +, –, × and ÷. He or she would also have difficulty in reading numbers with more than one digit, especially those containing zeroes. And he or she would often have confusion in reading directions, for instance 56 would be read as 65.
- Operational dyscalculia – pertains to challenges with basic arithmetic operations or failure to recognize or memorize basic arithmetic facts. A person with this type would have difficulties arranging numbers by size and have problems with number sequences. He or she would find the need to count on their fingers to manage basic calculation. He or she would have a bad memory on simple numerical facts, such as the multiplication tables. He or she would also have difficulties doing simple mental calculations and problems with counting backwards, and he or she would take a long time to solve simple mathematical tasks.
- Ideognostical dyscalculia – refers to challenges in grasping mathematical ideas or concepts. This usually is with regard to problems with complex thinking and mathematical abstraction.
Due to the varying nature and combinations of challenges that a person with dyscalculia might have, it is quite difficult to diagnose it. Standardized achievement tests would not be enough to accurately identify learners with dyscalculia because these tests do not differentiate dyscalculia from other possible causes of low achievement scores, such as poor mathematical background, ineffective instruction, low intelligence, lack of prerequisite skills, poor language skills, math anxiety, and several other factors that may cause mathematical deficits (Wadlington, 2008).
Lee (1995) espouses the use of noted child psychologist Dr. Ma. Lourdes “Honey” Carandang’s Rubic’s Cube approach, developed in 1981, in identifying learners with dyscalculia and helping them cope with it.
In Carandang’s interdisciplinary Rubic’s Cube approach, five aspects are considered:
- Physical – causes may be neurological, sensory impairment, motor deficiencies, etc.
- Intellectual – aside from IQ, a certain level of intellectual stimulation or motivation from family, school and the community is necessary.
- Emotional – the child’s feelings toward the subject, his teacher, or even himself may be the one hindering him to learn.
- Moral – the boundaries and implications of this disability should be clear. Will coddling the child and tolerating his transgressions be beneficial or more detrimental to his learning? How much special attention is enough and when is it too much?
- Social – How does the child’s classmates and society, in general, influence the child’s self-image?
Since dyscalculia is a neurological deficit, we should be able to rule out all other aspects that might hinder the child’s learning ability before we could conclude that the disability is in fact dyscalculia. This approach would require that we get to know the child and interview his parents, teachers and classmates, at the very least, in order to properly diagnose him.
Prof. Brian Butterworth (2002) developed an approach to screening possible candidates for dyscalculia using item-timed tests of capacity for “numerosity” or number sense. His method can be applied to any age group since the tests minimizes the effects of educational experience.
Butterworth’s Dyscalculia Screener is a computerized examination comprising of three computer-controlled, item-timed tests: number comparison, dot counting and item-timed arithmetic. The test also takes into consideration the test-takers’ simple reaction time and adjusts the timer accordingly based on this in order to minimize the possibility that the test-taker is simply a slow responder.
Low performance in arithmetic, but not on other tests, is attributed to poor learning or teaching, not dyscalculia. Also, since the Dyscalculia Screener takes note of the time it takes for a taker to answer each item, it is able to distinguish the takers who may have gotten an average number of correct answers but who was able to solve them in an abnormal way or abnormally slow manner.
The software is available for purchase and installation in personal computers.
It has been suggested that dyscalculia may just be a form or dyslexia, or may even be just a consequence of it. Studies have shown that about half of people with dyslexia also show signs of dyscalculia (The British Dyslexia Association, 2009). This is not surprising since dyslexia is a language processing problem, which could translate to difficulty in comprehending math language. For some, the difficulty actually lies in interpreting the math word problems instead of having difficulty performing math calculations or understanding math concepts per se.
People with dyslexia also often have trouble with directionality and sequencing. And since math also requires one to work in a specific order or sequence, then people with dyslexia may also find it difficult to solve math problems (Wadlington, 2008).
Butterworth (2002) was able to successfully conclude that dyscalculia is definitely not a form, nor is it a consequence, of dyslexia. The study compared groups with just dyscalculia, both dyslexia and dyscalculia, just dyslexia and a control group without the disabilities. The study was able to conclude that dyscalculia is not a matter of low IQ, it is not due to poor short-term memory, it is not due to poor language ability, it is not a consequence of slow reading, and yes, it is a deficit in basic numerical ability.
Math anxiety is feeling undue stress or tension whenever one is presented with a problem or activity that involves math. Rubinsten and Tannock (2010) cited that math anxiety may be due to environmental (past negative experience in math or math teachers), personal (lack of confidence or low self-esteem) or cognitive (low intelligence or poor cognitive abilities in math) factors.
Several studies have found out that math anxiety actually stems out from formal schooling in mathematics (Lazarus, 1974; Jackson and Leffingwell, 1999). As much as 16% of the students had been found out to have had their first negative experience in learning math as early as grades 3 or 4. Sadly, it seems that one of the major causes of math anxiety is the teachers who are supposed to guide the children’s learning in math.
This becomes an even more sensitive topic when it comes to children with dyscalculia. In a study by Rubinsten and Tannock (2010), they claim that “For people with [dyscalculia], childhood difficulties with numerical processes and poor math achievement intensify math anxiety, which further impedes math achievement. As educators come to appreciate the key role played by math anxiety, interventions that reduce it may become a key part of the math educational system. It might be that one of the most effective ways to reduce math anxiety is to improve math achievement from an early age through interventions focused on children with [dyscalculia] thus turning the cycle of failure-fear-failure to one of success-confidence-success. This is especially true if the assumption that [dyscalculia] is an innate condition is correct.”
The key is to pick out the child with possible dyscalculia early on and make the correct diagnosis, so that the child would be able to undergo immediate remedial sessions in math in order for him to cope. The longer it takes for the child to be diagnosed, the more he is left behind, the more anxious in math he becomes, and the less would be his chances in succeeding. For a lot of people with dyscalculia who didn’t get the help they needed early on, the disability causes them a life-long struggle.
Many curricula have been designed to suit the needs of learners with difficulties in Mathematics. Unfortunately, very few have been tested vigorously for efficacy, and the studies that do exist were not designed exclusively for learners with dyscalculia (Wilson, 2010).
According to Wilson (2010), the best approach to remediating dyscalculia would be to “(a) identify the areas where the child has a difficulty, and (b) try and target an intervention at these areas.” The learner’s difficulty might be due to basic mathematical abilities, such as understanding the meaning of numbers (which could be remedied by strategies emphasizing on understanding), or memorizing mathematical facts (which may be helped by drill-type interventions).
Chandler (2010), on the other hand, espouses the following intervention strategies:
- Repeated Reinforcement. One of the best ways to overcome dyscalculia is through repeated reinforcement. The student, along with a teacher, parent or tutor, should focus on the specific difficulty. This helps the student master the basics before moving on to new concepts. By removing the pressure of new material, a potentially negative feeling toward math can be changed to a positive experience.
- Use Graph Paper. Some students who suffer from dyscalculia will have problems with visual-spatial relationship. That means that they have difficulty relating one object to another, which in math translates to difficulty relating one number to another. This is most often noticed when the student cannot align numbers in columns or when calculations overlap on paper, according to ChildD.org. For these students, it may be helpful to use graph paper which provides clear columns to help in organizing the numbers.
- Apply Reasoning. Those who have good logic and verbal skills, but poor spatial skills can benefit from using reasoning skills instead of just memorizing numbers. For example, a child may be having difficulty remember their multiplication facts because they cannot see the relationship between the numbers. For example, if they are shown that 4 x 8 is equal to 4 x 4 doubled, as shown by Learning Disabilities Online, they can then see the relationship and understand the numbers instead of just trying to remember numbers that seem to have no meaning.
- Estimating. Utilizing estimating skills can also help students with dyscalculia. By estimating, the student is encouraged to think about the problem as a whole to obtain an answer. Specific skills can then build from this.
- Concrete Over Abstract. Students who have dyscalculia often have difficulty understanding abstract concepts, according to Teaching Expertise. Therefore, one strategy is to present concrete examples of problems before trying to explain the abstract concept. Use everyday examples with real objects to help the student visualize the math problem.
- Encourage Questions. As with any type of learning disability or any learning situation in general, always encourage the student to ask questions. This is especially helpful to those whose dyscalculia is caused by language difficulties. Asking questions can help reinforce ideas and provide additional explanation to help visualize the problem.
Wadlington (2008) listed down 25 useful strategies that would be effective especially when working with students with dyscalculia as well as other mathematical challenges.
- Prioritize mathematical goals for students. Differentiate between what they need to know and what would be nice to know.
- Seat students near the focal point of instruction and actively engage them in multisensory learning. This means that they are seeing concepts modeled, listening to concepts being explained, talking about concepts, and using movement and touch to learn.
- Teach students multisensory study skills to use for homework. For example, they can tape record your instructions about how to do a particular type of procedure (or record the procedure in their own words) and listen to the tape/repeat the procedure orally as they work the problem.
- Always present only a small amount of new material and make sure that new concepts build on old ones, using this sequence:
- Initially, use concrete objects or manipulatives (e.g., counters, shape blocks, geoboards, play money) to teach concepts and skills.
- Next, use pictures and diagrams to represent concepts and skills.
- Last, present concepts and skills abstractly. Keep in mind that students may need to go back to concrete objects or visuals for difficult or unusual tasks.
This concrete-pictorial-abstract (CPA) approach has been successfully implemented in Singapore, which is currently enjoying the distinction of having one of the best Mathematics basic education programs in the world. The concrete and pictorial representations are used to add meaning to the concept of numbers so that when learners are presented with the abstract concept of numbers, they would have a firm understanding of what those numbers stand for (Balano, 2011).
- Let students use manipulatives to work problems and demonstrate answers even when other more mathematically-abled students no longer need to do so.
- If students fail to master a skill or concept, find a new way to teach it. Do not repeat methods that did not work.
- On the other hand, know that students with dyscalculia will often need to drill on skills and over-practice them for the skills to become automatic.
- Teach mathematical vocabulary, signs, and symbols with concrete examples. Stress the meaning of mathematical language rather than just memorization.
- Make sure that students are talking and writing about mathematics. Set up cooperative groups in which students actually use mathematical language in real-life situations. Ask them to explain mathematical ideas or difficulties they may experience in math journals.
The importance of peer groups cannot be undermined. Van, for instance, claimed that she wouldn’t have passed high school math if it weren’t for her friend who would patiently discuss with her again after school what was discussed during class (http://www.youtube.com/user /BlueforInsanity).
- Seek a tutor who is an expert in principles of teaching/learning as well as mathematics and has excellent rapport with students.
- Make sure that the inability to learn basic math facts does not keep an individual from moving on to higher level mathematics. Provide fact charts, number lines, and calculators.
- Use error analysis to figure out why students get wrong answers. Do they miss basic facts, use incorrect operations, work in the wrong direction, or copy incorrectly? Then help students work on these specific errors.
- Do not allow students to practice errors. Monitor students as they work so errors can be caught quickly.
- Use memory tricks to help students remember. For example, “Please Excuse My Dear Aunt Sally” helps students remember the [hierarchy of operations] (i.e., parenthesis, exponents, multiplication, division, addition, and subtraction).
- If students have problems copying the problems from the text or board, give them photocopies on which to record answers.
- When students have difficulty understanding mathematical word problems, read them aloud and help them code the important parts. For example, students can underline the needed details, cross out irrelevant ideas, and circle the question.
- Teach students a variety of multisensory strategies to solve word problems. For example, they can create a picture/model or act out a problem.
- Include numerals and mathematical symbols for handwriting practice.
- Encourage students to work math problems on graph paper to make lining up numbers and symbols easier.
- Give students extra concrete experiences, so they will know if their answers make sense. Encourage students to always ask themselves “Does this make sense?” when they find a solution to a problem.
- Give tasks involving only one new concept or skill so students can experience success.
- Practicing test directions and taking self-timed tests can help prepare students for standardized math tests. Practice marking bubble sheets if this is a problem.
- There is no quick fix for dyscalculia. Help students stay motivated through helping them keep charts to track their progress and by rewarding small gains.
- Individuals with dyscalculia need self-understanding of their strengths and weaknesses. Always start by asking if they know what works best for them. Then help them develop additional strategies.
- Be an advocate for students with math disabilities. As students mature, teach them to be advocates for themselves.
Unfortunately, there are no specific effective strategies that can be implemented among all learners with dyscalculia in general. The key is getting to know the learner and finding out the root cause of his challenges. Intervention strategies should be applied depending on the root cause. Patience is also immensely important, for both the teacher and the learner, because dyscalculia is not something one overcomes easily. The support of the learner’s family, teachers and classmates would definitely help in boosting his confidence. It is imperative, however, that we recognize dyscalculia as a learning disability that can be overcome, and not something that describes a learner as dumb or stupid.
Dyscalculia may be inborn or innate, but it doesn’t mean there is no hope for people who have this disability. Just like any learning disability, dyscalculia can be overcome. As teachers, it is our job to watch out for signs of dyscalculia among our learners and help them overcome it.
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