Dyscalculia in children: its characteristics and possible interventions
(Paper presented at OECD Literacy and Numeracy Network Meeting,
El Escorial, Spain, March 2004)
Ann Dowker, Department of Experimental Psychology, University of Oxford
It is well known that individual differences in arithmetical performance are very marked in both children and adults (Dowker, 1998). For example, British studies separated by 20 years, and by radical changes in mathematics education, have revealed a gap of about seven years in 'mathematics age' between the highest and lowest achievers in an average class of 10- or 11-year-olds (Cockcroft, 1982; Brown, Askew, Rhodes et al, 2002). Individual differences in arithmetic among children of the same age are consistently found to be large in most countries that have been studied. The average level of performance tends to be higher in Pacific Rim countries (TIMSS, 1996), though individual differences are high in these countries as well (Schmidt, McKnight, Cogan, Jackwerth and Huang, 1999). In all countries that have been studied, a significant number of children have real difficulty in mathematics (TIMSS, 1996).
Children's numeracy difficulties can take several forms. Some children have difficulties with many academic subjects, of which arithmetic is merely one; some have specific delays in arithmetic, which will eventually be resolved; and some have persisting, specific problems with arithmetic. It is the latter group for whom the term 'dyscalculia' may most appropriately be used.
It must be noted that there is continuous variation in arithmetical difficulties in the population; and that many people who would not be regarded as having severe and specific dyscalculia do have major and disabling problems with numeracy.
For example, Bynner and Parsons (1997) gave some Basic Skills Agency literacy and numeracy tests to a sample of 37-year-olds from the National Child Development Study cohort (which had included all individuals born in Britain in a single week in 1958). The numeracy tests included such tasks as working out change, calculating area, using charts and bus and train timetables, and working out percentages in practical contexts. According to the standards laid down by the Basic Skills Agency, nearly one-quarter of the cohort had 'very low' numeracy skills that would make everyday tasks difficult to complete successfully. This proportion was about four times as great as that classed as having very low literacy skills. Most of the adults with numeracy difficulties had already been experiencing difficulties with school mathematics at the age of 7.
The origins of these numeracy difficulties were presumably varied, though these were not examined. Presumably, only some of these adults would have been describable as 'dyscalculic': some would have had generally below-average IQs; some would have had limited or inappropriate instruction; and some would have had emotional and social problems affecting their performance in arithmetic. Nonetheless, the study shows the pervasiveness of numeracy difficulties and their importance in adult life.
Arithmetical ability is made up of many components
In order to study the nature of the arithmetical difficulties that children experience, and thus to understand the the best ways to intervene to help them, it is important to remember one crucial thing: arithmetic is no ta single entity: it is made up of many components, including knowledge of arithmetical facts; ability to carry out arithmetical procedures; understanding and using arithmetical principles such as commutativity and associativity; estimation; knowledge of mathematical knowledge; applying arithmetic to the solution of word problems and practical problems; etc.
Experimental and educational findings with typically developing children (Ginsburg, 1977; Dowker, 1998) and adults (Geary and Widaman, 1992) have shown that it is possible for individuals to show marked discrepancies between almost any two possible components of arithmetic. For example, Dowker (1998) studied calculation and arithmetical reasoning in 213 unselected children between the ages of 6 and 9. She reported (p. 300) that (1) individual differences in arithmetic are relatively marked; (2) that arithmetic is indeed not unitary and that it is relatively easy to find children with marked discrepancies [in either direction] between [almost any two] different components; and that (3) in particular it is risky to assume that a child does not understand maths” because he or she performs poorly in some calculation tasks”.
Studies of adults with acquired dyscalculia (Warrington, 1982; Dehaene, 1997; Butterworth, 1999; Delazer, 2003) show that almost any component of arithmetic can be selectively impaired: e.g. patients can show double dissociations between estimation and calculation; memory for facts and following procedures; written versus oral arithmetic; different arithmetical operations such as subtraction versus multiplication; etc.
It would thus be expected that at least some dyscalculic children might also show shown extreme discrepancies between different types of mathematical ability; and this has indeed been found when investigated. For example, Temple (1991) reports one child who could carry out arithmetical calculation procedures correctly but could not remember number facts, and another child who could remember the facts but not carry out the procedures.
Macaruso and Sokol (1998) studied 20 adolescents with both dyslexia and arithmetical difficulties, and found that the arithmetical difficulties were very heterogeneous, and that factual, procedural and conceptual difficulties were all represented.
Such findings are important, as they demonstrate that dyscalculic children need not have problems with all aspects of arithmetic, but may have strengths that could be used in intervention programs to compensate for and overcome their weaknesses.
JN. My own view is simple, any child clever enough to teach itself to speak is quite able to be taught perfectly in early arithmetic, by use of the abacus and using its fingers to perfect the concept of ten, by the time it is five years old. Teaching the child to chant, first from one to twenty, just as soon as it can speak properly, perfecting chanting the tens, by opening ten fingers ten times or pushing up the tens on the abacus, OR BOTH WAYS.
Before a child can physically speak it will understand the sound of many words, once it can speak the ability to make the sound of words simply by copying the sound far exceeds the childs ability to understand the meaning of many of those words, by using Abacus One and rapid pattern recognition with the fingers, every speaking child will perfect its understanding of numbers. ALL CHILDREN WILL VARY IN THE TIME TAKEN TO PERFECT THEIR AWARNESS AS TO THE MEANING OF NUMBERS, but every speaking child can perfect their understanding of numbers when they are taught systematically. MY WAY
Abacus One is a perfect Physical Copy in written numbers of the way we write numbers with numerals. We can use a flat printed copy as a substitute if an Abacus One is not available. I designed a flat counting board for copying and use with seven stones for children without any other resources, orphan Indian children.
THE ABACUS ONE WRITTEN MAP
Printed from an OECD website can become a world standard mathematic resource, Just as chess as become a world standard tool for developing spatial strategy teaching and metal development so can the Abacus One map teach Basic Arithmetic.
First the child can read in its own language, the words used in expressing arithmetic counting by starting to use it as a simple Abacus, counting and moving stones to add and subtract numbers.
Every mathematic principle can be explained by simple demonstration on the Abacus One Map, so I and any other concerned parties need to perfect processes and simple four child, or less games, to perfect these valuable mathematic concepts that will
PROVIDE EVERY CHILD ON EARTH WITH THE MENTAL STRUCTURES OF MATHEMATICS READING IN ENGLISH
Start as soon as the child can speak.
Perfecting the sound of the alphabet by chanting,
THERE IS NO OTHER WAY.
Cement a with the sound of a and b with the sound of b
Low case letters only in the rhythmic lay out, where the visual memory
Is assisted by the physical layout, where the easily remembered like o
Link the difficult p and q.
THREE DIMENSIONAL READING
Placing cards on letters to perfect the memory can be augmented by a
Daily POINT & PROVE exercise for one or 101 children at once.
Perfecting the sounds of letters as they are naturally used can become
Automatic by using small three dimensional objects to assist the
Memory of letters, an oxo cube, an apple for a, a potato for p, and so on.
ABACUS ONE provides every child with pictures of words that are
LEARNT NATURALLY as a meanings before they are recognized as
A PICTURE. There are twenty one different words on ABACUS ONE
ONE HUNDRED WORDS AS A PICTURE
and
one thousand or many less perfect pictures of low case letter groups,
WILL GIVE PERFECT EARLY READING ABILITY FOR MOST OF US.