Dyscalculia
Professor Mahesh Sharma, Professor of Education, Cambridge College
01-July-03

1.  What is dyscalculia?

Many students have difficulty learning mathematics for a variety of reasons. Not all of these students have dyscalculia. However, there are some basic areas of mathematical activity in everyday life that may indicate a dyscalculic tendency if persistently difficult and frustrating for a person. Such symptoms manifest as: mathematics anxiety and dyscalculia.

In very simple terms, analogous to dyslexia - which is dysfunction in the reception, comprehension, or production of linguistic information, dyscalculia can be defined as the dysfunction in the reception, comprehension, or production of quantitative and spatial information.

Dyscalculia is a collection of symptoms of learning disability involving the most basic aspect of arithmetical skills. On the surface, these relate to basic concepts such as: telling the time, calculating prices and handling change, and measuring and estimating things such as temperature and speed.

Dyscalculia is an individual's difficulty in conceptualizing numbers, number relationships, outcomes of numerical operations and estimation - what to expect as an outcome of an operation. Dyscalculia manifests in a person as having difficulty:

• Mastering arithmetic facts by the traditional methods of teaching, particularly the methods involving counting.
  
  
• Dealing with exchange of money-handling a bank account, giving and receiving change, and tipping.
  
  
• Learning abstract concepts of time and direction / schedules, telling and keeping track of time, and the sequence of past and future events.
  
  
• Acquiring spatial orientation/space organisation / direction, easily disoriented (including left/right orientation), trouble reading maps, and grappling with mechanical processes.
  
  
• Learning musical concepts, following directions in sports that demand sequencing or rules, and keeping track of scores and players during games such as cards and board games.
  
  
• Following sequential directions - sequencing (including reading numbers out of sequence, substitutions, reversals, omissions and doing operations backwards), organizing detailed information, remembering specific facts and formulas for completing their mathematical calculations.
  
  

Dyscalculia can be quantitative, which is a difficulty in counting and calculating; or qualitative, which is a difficulty in the conceptualizing of mathematics processes and spatial sense; or mixed, which is the inability to integrate quantity and space.

Schools have supported children who experience difficulties with mathematics, but dyscalculia has only recently been identified as a distinct condition for children and adults. It means that there are many adults and children who have never had their difficulties with mathematics formally identified. Furthermore, while there is currently a great deal of interest, little is known about the causes of dyscalculia. It is a complex phenomenon and may have several underlying causes.

Our work at Cambridge College in America with children and adults with learning problems in mathematics suggests that there seem to be several factors that may be implicated as the causes of mathematics learning problems:

• Cognitive factors.
  
  
• Inadequate and poor teaching - mismatch between the mathematics learning personality of a student and teaching style.
  
  
• Lack of prerequisite skills for mathematics learning.
  
  
• Delay in the development of mathematics language-vocabulary, syntax, and translation ability - from mathematics to English and English to mathematics.
  
  
• Inadequate mastery at levels of knowing: movement from intuitive to concrete, concrete to representational, representational to abstract, abstract to applications, and from applications to communication.

In all cases, it seems, the prerequisite skills for mathematics learning are affected. These prerequisite skills include: following sequential directions, spatial orientation/space organization, pattern recognition, visualization, estimation, inductive and deductive thinking. These prerequisite skills act as "anchors" for mathematics ideas. The degree to which these prerequisite skills are not developed or affected varies from learner to learner.

What is the incidence of dyscalculia?

Since people are just becoming aware of this condition, it is hard to quantify exactly how many people have dyscalculia. Although many people experience difficulty or disability in mathematics, recent studies show that approximately 4% of students show classical symptoms of dyscalculia. They may have normal abilities in other areas and appropriate or higher cognitive development. Each class may have one or two such students.

Is a dyslexic individual likely to be dyscalculic?

A link between dyslexia and dyscalculia hasn't been proved. The International Dyslexia Association has suggested that 60% of dyslexics have some difficulty with numbers or number relationships. Of the 40% of dyslexics who don't have mathematics difficulties, about 11% excelled in mathematics. The remaining 29% have the same mathematical abilities as those who don't have learning difficulties.

Since some of the same prerequisite skills are involved in both language acquisition and mathematics - at least in the early learning levels - coincidence of dyslexia and dyscalculia is not uncommon. Our observations show that about 40% of dyslexics also exhibit some symptoms of dyscalculia. However, the group of dyscalculic children/adults, like the group of dyslexics, is not a homogeneous one. Most people with dyscalculia don't necessarily suffer from any other learning difficulty. Indeed, they may well excel in non-mathematical areas.

Is dyscalculia widely understood?

All mathematics teachers have encountered children with mathematics learning difficulties and mathematics anxiety. Most of these teachers have some awareness of the nature of learning disabilities/problems in mathematics. However, few teachers are aware of the causes of these problems - learning disabilities, mathematics anxiety, and dyscalculia. In fact, very few of them are able to recognize and deal with the problems of dyscalculics.

American Academies of Neurology and Paediatrics have identified dyscalculia as one of the neurological conditions with a cluster of syndromes associated with it. Similarly, in 2001, as part of the national Numeracy Strategy, the government published guidance for teachers to help them support dyscalculic pupils. Dyscalculia is likely to be a more familiar condition to people who specialize in learning difficulties such as special needs coordinators and educational psychologists. In the U.S., many school psychologists, neurologists and neuro-psychologists have begun to diagnose this as a condition.

Dealing with dyscalculia?

Dyscalculia is a special need, and requires diagnosis, support and special methods of teaching. The support should give the learners an understanding of their condition, and equip them with coping and learning strategies that they can use in the classroom and in their day-to-day encounters with quantity and space. Since this is a heterogeneous group no general or single intervention can be recommended.

3.  What forms of instruction are most effective?

Dyscalculic learners lack an intuitive grasp of numbers and have problems learning number facts and procedures by the usual methods of teaching. Even when these learners produce a correct answer or use a correct method, they may do so mechanically and without confidence; they are anxious about it.

One objective of remedial instruction should be to improve learners' self-esteem by giving them real-life exposure to mathematics as a part of everyday life: ingredients needed in baking a cake, checking the change after purchasing something, or making estimations.

The milestone concepts for learning mathematics are: understanding number, place value, fractions, integers, spatial sense and variability. Because a person's mathematics difficulties generally originate from some dysfunction in one of these milestone concepts, one should begin instruction in these areas systematically.

Individuals with dyscalculia need help in organizing and processing information related to quantity and space. Since mathematics is a form of language, one should spend time on its vocabulary, syntax and translation - from mathematics to English and from English to mathematics.

These individuals can benefit from tutoring that can accomplish three objectives. First, to help them make-up the missing arithmetic concepts. Second, to help them connect these to their current mathematical needs. And, third, to help them develop the prerequisite skills for mathematics learning.

Effective teaching combines direct instruction (teacher-directed tasks, discussion, and concrete models) with strategy instruction (teaching ways to learn, such as memorization techniques for arithmetic facts, study skills and metacognition - learners identify strategies that help them to learn). Such teaching include:

• Sequencing and task-analysis (breaking down the task into parts and then synthesizing the parts into a whole, providing step-by-step prompts).
  
  
• Repetition and practice (automatizing arithmetic facts, daily testing, sequenced review).
  
  
• Socratic questioning and responses (structured questioning where teacher asks process or content questions to scaffold learning).
  
  
• Control of task difficulty (the teacher provides necessary assistance or tasks sequenced from easy to difficult).
  
  
• Use of technology.
  
  
• Teacher-modelled problem solving.
  
  
• Strategy cues (reminders to use strategies).

Every remedial/instructional session should have the following components:

• Developing the prerequisite and support skills.
  
  
• Learning arithmetic facts orally.
  
  
• Visualizing problems and information.
  
  
• Verbalizing and recording procedures and estimation.
  
  
• Helping the child form problems relating to the given concept.
  
  
• Counting forward beginning with a given number e.g. begin with 53 and count forward by two's, by three's, etc.
  
  
• Counting backwards beginning with a given number, e.g., begin with 97 and count backward, by 1's, 2's, 5's, and 10's.
  
  
• Showing patterns of number facts, e.g., 4 + 4 = 8, then 4 + 5 = 9; 8 + 10 = 18, then 18 + 10 = 28, 78 + 10 = 88, etc.

The three components of a mathematical idea: linguistic, conceptual and procedural should be carefully included in all mathematical instruction. Therefore, before the student performs any arithmetic operations he should be asked about the language, procedure, and the conceptual model of the problem first. When the student has completed the problem, they should be asked whether they know any other problems that are similar to the one just completed.

Based on our experiences with many dyscalculics, we are fairly certain that with the help of a competent tutor, with effort, discipline and structure, and appropriate concrete materials, dyscalculics can make a great deal of progress in mathematics learning and realize their potential.

Further readings and support:

• For Mahesh Sharma's UK courses and videos: Berkshire Mathematics,  http://www.berkshiremathematics.com
  
  
• For Mahesh Sharma's US courses and videos:
  
  • Center for Teaching/Learning of Mathematics, PO Box 3149, Framingham, MA 01701 Phone: 001 508 877 7895 Fax: 001 508 788 3600 Email:  ctlm@rcn.com
  • Mathematics Institute, Cambridge College, 1000 Massachusetts Ave., Cambridge, MA 02138 Phone: 001 617 873 0211 Fax: 001 617 876 2663 http://www.cambridgecollege.edu
    
    
• Dyscalculia, Mahesh Sharma and Eugene Loveless, CT/LM, 1988
  
  
• Dyslexia, Dyscalculia, and other Mathematical Problems, Mahesh Sharma, CT/LM, 1989
  
  
• How to Teach Arithmetic Facts Easily, Mahesh Sharma, CT/LM, 2000
  
  
• Mathematics for Dyslexics: A Teaching Handbook, Chinn and Ashcroft, Whurr, 1997
  
  
• The Mathematical Brain, Brian Butterworth, Macmillan, 1999