Mathematics learning disabilities in Belgium
Running head: MATHEMATICS LEARNING DISABILITIES IN BELGIUM
CHILDREN WITH MATHEMATICS LEARNING DISABILITIES IN BELGIUM
Annemie Desoete1,2
Herbert Roeyers1
Armand De Clercq 3
1
Department of Clinical Psychology, Ghent University, Belgium
2
3
Artevelde College, Ghent, Belgium
Department of Applied Mathematics and computer science, Ghent University, Belgium
This study was supported by the the Artevelde College in Ghent and the Stichting Integratie
Gehandidapten (SIG) to whom the authors extend their thanks
Mathematics learning disabilities in Belgium
ABOUT THE AUTHORS
Annemie Desoete, PhD, is a researcher at Ghent University, in Ghent, Belgium. Her research
interests are in the area of mathematics learning disabilities. Herbert Roeyers, PhD, is assistant
professor in clinical psychology at Ghent University. His research interests are in the area of
developmental and behavioral disorders. Armand De Clercq, Ph D, is professor in applied
Mathematics. His research interests are in the area of computer science
Address: Annemie Desoete, Ghent University, .Department of Clinical Psychology Research
Group Developmental Disorders, Henri Dunantlaan 2, B-9000 Gent, Belgium.
Annemie Desoete
Ghent University, Department of Clinical Psychology
Research Group: Developmental Disorders
Address: Henri Dunantlaan 2
B 9000 Gent, Belgium
Email: Anne.Desoete@rug.ac.be
Herbert Roeyers
Ghent University, Department of Clinical Psychology
Research Group: Developmental Disorders
Address: Henri Dunantlaan 2
B 9000 Gent, Belgium
Tel: 32. (0) 9.264.64.48
Fax: 32. (0) 9.264.64.89
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Mathematics learning disabilities in Belgium
Armand De Clercq
Ghent University, Department of applied Mathematics and Computer Science
Address: Krijgslaan 281 S9
B 9000 Gent, Belgium
Tel : 32. (0). 9. 264 47 79
Fax : 32. (0) 9 264 49 95
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Mathematics learning disabilities in Belgium
Abstract
In Belgium between 3 and 8 percent of the children in the elementary schools
have mathematics learning disabilities. Many of these children have less developed
linguistic, procedural and mental representation skills. Moreover, a majority of the
children have been found to show inaccurate prediction and evaluation skills in grade 3.
Mathematics disabilities often become obvious in elementary school. While
children are retained, others are referred to special education.During the last 10 years the
number of children in special education as well as the number of children with learning
disabilities who are following a special guidance program in regular education (inclusive
education) have increased. Children in Belgium with mathematics learning disabilities
can get therapy for about two years. Nevertheless, many problems continue unresolved
even in high school and adulthood.
Key words: mathematics learning disabilities, Belgium, metacognition, prediction, evaluation,
Belgium, cognition, child.
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Mathematics learning disabilities in Belgium
CHILDREN WITH MATHEMATICS LEARNING DISABILITIES IN BELGIUM
Introduction
From 1974 to 1997, only 28 articles on mathematics learning disabilities were cited in
PsycInfo, whereas 747 articles could be found on reading disabilities (Noel, 2000). Therefore, we
agree with Ginsburg (1997) and Hanich, Jordan, Daplan and Dick (2001) that children with
mathematics difficulties have been understudied, and their problems have been underestimated.
We know that mathematics learning disabilities can seriously limit a student’s educational
opportunities (Rivera-Batiz, 1992). Moreover, the disabilities often persist into the college years
(Miller & Mercer, 1997) and many of these children continue to function below the arithmetical
level of a 13-year-old child (Cawley & Miller, 1989; Miller & Mercer, 1997).
In this article, we give first a general description of mathematics education in Belgium.
Second, we describe children with mathematics learning disabilities in our country. Third, we
discuss the problems with mathematical problem solving in children of average intelligence, both
with and without mathematics learning disabilities in Belgium. Finally, we inventory the
instructional interventions and the accommodations implemented to help students with
mathematics learning disabilities in Belgium.
Mathematics education in Belgium.
Regular education
Belgium is a small country in the center of Europe with a population of nearly 11 million.
Belgium has a French-speaking and a Flemish-speaking community, along with a very small
German-speaking community. This paper relates mainly to the Flemish-speaking community, or
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Mathematics learning disabilities in Belgium
Flanders, though some additional information is provided concerning the French-speaking
community. The situation in the French and German-speaking parts is similar to that of the
Flemish part.
In Belgium most children start kindergarten when they are two and a half years old. They
get three and a half years of kindergarten education. At the age of six, the children go to primary
school, where they learn to read, write and do mathematics. Primary classes include roughly 20 to
30 children in a single classroom. Children spend about seven hours at school every weekday,
except wednesday which has early dismissal. There is no school on Wednesday afternoon. After
primary school, students enter secondary education. This is followed either by vocational training
or by college or university.
All Belgian children have to continue their education (and in most cases remain at school)
at least until they are 18 years old. A small number of children are educated at home, because
since 1914 there has been compulsory ‘education’ in Belgium, but not compulsory ‘schooling’.
There are three Types of schools in Belgium: Catholic schools, schools organized by the federal
government, and schools organized by the cities or provinces. There are no major differences
among these three Types of schools, although each operationalizes the obligatory central
‘developmental aims’ (‘ontwikkelingsdoelen’) of the kindergarten and the ‘aims to be achieved at
the end of elementary education’ (‘eindtermen’) in a slightly different way with respect to the
‘mathematics curriculum’ (‘leerplan wiskunde’).
Schools are held accountable for student performance. Inspection is organized at a central
level. As in most European countries, several systematic and full inspections of the schools as a
whole are carried out in Belgium annually (Standaert, 2001). The CIPO model is used for this
inspection, and a school’s performance is evaluated in four key areas: Context, Input, Process,
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Mathematics learning disabilities in Belgium
and Output. This evaluation determines the results of each school with respect to student
performance (output) and the way in which these results have been achieved (process), taking
into consideration the context in which the school is operating (e.g., demographic, juridical,
administrative data) and the input data (e.g., information about pupils, school staff, material
resources). In Flanders, the evaluation of the quality of teachers is left to the local school or its
governing body, whereas the French-speaking community of Belgium has a central teacherassessment system (Standaert, 2001).
In the Flanders curriculum, 80 percent of the children must acquire 80 percent of the
minimum skills and knowledge by the end of their primary school education. Children who do
not meet these minimum aims and who read or solve mathematics problems on the level of third
graders without learning disabilities are referred to the “B-classes” of secondary education
(Hellinckx & Ghesquiere, 1999). If this lost ground is made up in the first year of secondary
education, then these children can go back to the “A-classes”, where they follow a general course
of studies aimed at preparing them for college or university. About 20% of the children with
learning disabilities in special primary education go to one of the A-classes in secondary
education (Hellinckx & Ghesquiere, 1999). The other children follow a specialized course of
vocational training.
Regular mathematics education in Flanders seems to be doing quite well for most
children. The PISA project (Program for International Student Assessment) recently compared
the learning performance of students in 30 different countries all over Europe, Asia, and America.
This project revealed that the 15-year-old Flemish student did best on mathematics (pisa
peco.htm). Flemish pupils obtained a second place on reading skills, just after youngsters from
Finland, and they scored third on science, after Korea and Japan. Especially, children in families
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Mathematics learning disabilities in Belgium
with a high social economic status did well in this study. Children in families with a lower social
economic status had results comparable with that of their peers in the other countries. Moreover,
students in the French speaking part of Belgium did worse on mathematics, reading and science
than their Flemish-speaking peers in the same country (Bormans, 2002; Galle, 2001).
The mathematics curriculum in Belgium changed in 1998 (Commissie wiskundeleerplan
van het gemeenschapsonderwijs, 1998; Vlaams Verbond van het Katholiek Basis Onderwijs,
1999). The former ‘mechanistic’ approach and even the ‘structuralistic’ approach (with ‘new
mathematics’) became history. Inspired by the Netherlands (Freudenthal, 1991; Treffers, 1991), a
more constructivistic ‘Realistic Mathematics Education’ (RME) was introducted, which
emphasizes the importance of students becoming active learners who construct their own
mathematics (Blöte, Van der Burg & Klein, 2001). This main paradigm of RME is that the
teaching of mathematics should build on the previous knowledge of children derived from reallife experiences with numbers. Dialectical ‘constructivists’ stress that the understanding of
arithmetic develops through active knowledge construction, transformation and discovery within
an approach involving scaffolding, and does not merely result from passive information
acquisition (Cobb, Yackel & Wood, 1992; Muthukrishna & Borkowski, 1996). They assume that
people actively construct their own knowledge by linking prior knowledge to new knowledge.
The emphasis is on a more functional use of basic mathematics concepts and skills and on the use
of schemata to assist abstract thinking. In addition, mathematics has become a tool for solving all
kinds of problems, for handling all kinds of tasks, and for communicating and working together
through a more (inter)active teaching process. In this RME approach, children in small groups are
given the opportunity to discover adequate strategies for themselves for a variety of tasks
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Mathematics learning disabilities in Belgium
presented in meaningful and rich ‘contexts’ (e.g., Milo, 2001; Van Luit, 1999; Verschaffel,
1999).
This conceptual change has had consequences on the content of mathematics education.
So-called structuralistic ‘new mathematics’ (AB …) has disappeared from the curriculum, and
some number algorithms, formulas and other domain-specific data (e.g., prime numbers, nine
control on multiplications, theory of collections) have disappeared with them. In contrast, the
accent is nowe on mental mathematics, on approximate arithmetic or number sense, on
mathematics using the calculator (from grade 4) and on problem solving and strategy use.
RME focuses on having students create and discuss their own solution procedures and on
using context problems to relate mathematics to real-life situations. RME resulted in a higher
level of flexibility and more conceptual understanding for average students without learning
disabilities (e.g., Blöte et al., 2001). However, children with mathematics learning disabilities did
not do as well (e.g., Woodward & Baxter, 1997). Several handbooks (e.g., ‘Pluspunt’) and
teaching methods based on this RME approach that have been used for Flemish children with
mathematics learning disabilities have resulted in ‘confused’ children who are unable to solve
math problems. This is presumably because RME places substantial demands on metacognitive
skills, which may not be adequately developed in children with mathematics learning disabilities
(Desoete, Roeyers, & De Clercq, 2003; Mercer, Jordan, & Miller, 1996). It appears that children
with learning disabilities profit less from RME than peers with age-adequate learning skills. In
clinical practice, children with mathematics learning disabilities seem to need a more structured
and direct instructional approach in which the teachers teach the conceptual understanding and
demonstrate and repeat the rules and procedural skills to be learned. The RME goal of flexible
use of procedures often resultes in children with mathematics learning disabilities lacking the
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Mathematics learning disabilities in Belgium
necessary structure and conceptual knowledge to deduct for themselves the procedural skills
needed to solve mathematical problems. For this reason, in Flanders there is considerable debate
as to the value of RME for these children (e.g., Feys, 1998; Milo, 2001; Kroesbergen, 2001; Van
Luit, 1992, 1994).
Special, inclusive or integrated education?
The regular school system has not met the needs of some children in Belgium. Since 1970
separate special schools have been created for categories of children. For young children (7-14
years) with reading, spelling or mathematics learning disabilities, Type 8 special primary
education has been set up. For children with learning disabilities, no special education program
has as yet been set up either in kindergarten or in secondary schools. In the 1970s about 3 percent
of school-age children went to special schools (Types l through 8). However, by the end of the
nineties, 3.91 percent were attending such special schools. This was the case for 0.73% of the
children in kindergarten, 5.92% of the primary school children, and 3.65 % of the secondary
school children. For Type 8 (children with learning disabilities), there has also been a large
increase in recent years. In 1990-1991 there were 6,015 children in Type 8, whereas in 20002001 10,049 (or +59.86%) of the children in special primary education in Belgium were being
educated in Type 8 schools. The number of children in Types l through 8 altogether represent
78.65% of the total number of children in special primary education (Mardulier, 2001). However,
there seems to be a large gender difference in the choice of regular versus special education. In
2000-2001 in Flanders boys went more often to special schools than girls.
In general, the budget for special primary and secondary education is about 414 million
euro, which means that 9% of the total education budget is spent on the 3.9% of the pupils who
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Mathematics learning disabilities in Belgium
are receiving this kind of education (Mardulier, 2001). With this extra budget, special education
is able to work with smaller classes (about 10 children in a class) using specially trained teachers
and a team of other specialists (speech and language therapists, physical therapists, ergonomic
therapists, psychologists, and sometimes also psychiatrists or other physicians) to assist in the
education of these children.
At the current time in Flanders there is a strong movement in favor of inclusive education
(Broekaert & Van Hove, 1995; Kennes, 2001; Van Hove & Roets, 2000; Wuyts, 2000). Some of
its proponents are even insisting on the “abolition” of special education for children with learning
disabilities. This inclusive approach to education is aimed at restructuring the schools in order to
meet the needs of the students who are ‘at risk’ and at promoting the value of diversity within the
human community. The guiding principle is that schools should accommodate to all children,
regardless of their physical, intellectual, social, emotional, linguistic or other characteristics. In
this spirit, the care-coordinator (‘zorgcoördinator’) will have an important task in regular schools
from september 2003 on.
An integrated education (GON Geintegreerd Onderwijs) experiment was started in 1980
on Flemish children who had a physical or sensorial disability (Types 4, 6 and 7). In 1983 it was
generalized to all Types of special education, and in 1986 it was put into law. School boards from
regular schools now have the possibility of cooperating with schools for special education where
students with learning or other disabilities are being educated. As a result, it is becoming possible
for some children with disabilities to remain in regular schools for one or two years, or in the case
of a severe disability unlimited time. The regular school then receives pedagogical-didactic
or/and therapeutic assistance from a school for special education. The school for special
education receives an integration allowance to assist the regular school in educating children with
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Mathematics learning disabilities in Belgium
disabilities. Although the percentage of children in special education has increased, the number of
GON pupils has also increased in ten years. In 1994-1995 there were 36 children in Type 8 with a
GON status, whereas in 2000-2001 there were already 190 children in such a GON education
system (Mardulier, 2001; Wuyts, 2000).
To summarise, regular mathematics education in Flanders seems to be doing quite well for
most children. In addition, although there is a strong movement in favor of inclusive and
integrated education, separate special schools exist for children with learning disabilities. This
leads us to the following part, namely the description of mathematical learning disabilities in our
country.
Mathematical learning disabilities in Flanders
Regular and special eduction
In Flanders most learning disabilities are not detected until in primary school, when
reading disabilities often become obvious. Likewise, many children with mathematics learning
disabilities in primary school did not encounter severe difficulties with preliminary mathematics
learning in kindergarten, although in some cases inefficient counting and not remembering the
numbernames was already present as a ‘marker’ Nevertheless, most children with mathematical
learning disabilies in Belgium are detected in Grade 1 when they have to master addition and
subtraction and, in some cases, even later (in Grade 3) when they have to learn to retrieve quickly
the time tables or to select and apply various problem-solving strategies in addition to the basic
mathematical operations.
In secondary education, college or university, students with mathematics learning
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Mathematics learning disabilities in Belgium
disabilities encounter numerous problems (with mathematics, physics) and very few teachers who
understand their learning problems. This often results in low self-concept and even in behavioral
problems. About 40 to 80 percent of the youngsters with learning disabilities develop behavioral
problems (Thiery, 2001). Parents are confronted with a very tumultuous period in the life of
these youngsters, with even suicidal behavior in some
Definition of learning disabilities in Flanders
Although various authors agree that an operational definition of learning disabilities is
meaningful (e.g., Kavale & Forness, 2000; Swanson, 2000), most studies are rather vague when it
comes to characterizing the children who fit in their category of ‘children with learning
disabilities’. In addition, several authors use different concepts for ‘disablement’ in mathematical
problem solving (‘mathematics learning difficulties’, ‘mathematics learning problem’,
‘mathematics learning disorder’, ‘mathematics learning disability’, ‘mathematics learning
retardation’, ‘mathematics learning deficiency’, ‘dyscalculia’) (e.g., APA, 1994; Dumont, 1994;
Fletcher & Morris, 1986; Hellinckx & Ghesquiere, 1999; Rourke & Conway, 1997; Swanson,
2000; Thiery & Anthonis, 2003; Van Luit, 1998, WHO, 1997).
In our studies, in accordance with the definition in the DSM IV (APA, 1994), we use
three criteria to determine whether children have mathematics learning disabilities. First, in
accordance with the 'discrepancy criterion', the child must perform significantly more poorly on
mathematics than one would expect on the basis of his or her general school results and/or
intelligence. Second, the 'severeness criterion’ is applied, which is based on the DSM IV (APA
1994, p. 46-51). So we only speak of a mathematics learning disability when the child’s
difficulties with mathematics, as measured by a valid test, are found to be minus two or more
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Mathematics learning disabilities in Belgium
standard deviations (SD) below the norm. The third criterion is the 'resistance criterion’, which
refers to the teacher’s assessments or the fact that the difficulties remain severe, even with the
usual remediation at school (remedial teaching or school therapist). Teachers' assessments are
used because, although some researchers question the trustworthiness of these data, reviews
indicate that these judgments can serve as worthy assessments of a student’s achievement-related
behavior (Meltzer, Roditi, Houser, & Perlman, 1998; Winne & Perry, 2000). Additionally, we
define mathematics learning problems as the unexplainable difficulties that a child has with
mathematics when validated by a test in which the child’s performance is between –2 SD and -1
SD below the norm (severeness criterion) (e.g., Ghesquière, Ruijssenaars, Grietens, & Luyckx,
1996). Moreover, these difficulties must be noticed by the teacher for them to be defined as a
‘mathematics learning problem’.
The three criteria (severeness, discrepancy and resistance) may seem very clear
parameters for ascertaining whether an individual child belongs to the group of subjects with
mathematics learning disabilities. However, in clinical practice the diagnosis also depends on the
test(s) chosen to measure the severeness criterion. The choice of the mathematics test(s) was
found to be crucial, in a study of 85 children in Flanders. No single test succeeded in identifying
all 37 children with a mathematics disability out of the 85 third graders, according to the
discrepancy and resistance criterion. Using a test on domain-specific mathematics knowledge
(measuring number knowledge and mental arithmetics), only 23 children with a mathematics
learning disability were detected. Using a test on word problems, only 16 children with a
mathematics learning disability were detected. On the basis of a test on mathematics number
facts, only 17 children with mathematics learning disabilities were detected. This study indicated
that a cocktail assessment - or test on number facts and at least a test on domain-specific
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Mathematics learning disabilities in Belgium
knowledge and a test on word problems - was needed to prevent the chosen test for determining
the diagnosis. The domain-specific test in combination with the test on word problems was able
to detect 27 out of the 37 children. The domain-specific test combined with the test on number
facts was able to find 34 out of the 37 children with mathematics learning disabilities. Only the
combination of a test on domain specific knowledge, a test on number facts, and a test on word
problems was found to be able to detect all 37 children with mathematics learning disabilities
(Desoete & Roeyers, 2000). It might be interesting to combine the assessment of those aspects
within one test (e.g. Grégoire, 1997) and to investigate if then all children with mathematics
learning disabilities can be detected. . Such studies are currently being prepared (Grégoire, Van
Nieuwenhoven, & Noel, 2003).
Prevalence of mathematics learning disabilities in Flanders.
In October 2001 a prevalence study was conducted among regular second, third and fourth
grade students in Flanders (n = 3978) (see Table 1).
<Table 1 here>
In the second grade, 2.27 % of the children (2.11% boys and 2.42% girls) were found to
have a mathematics learning disability according to the teacher and a mathematics test battery (2SD on one of the tests) (see Tot DIS in Table 1). About 1.51% of the disabilities had to do with
a memory deficit, translated in severe problems to retrieve number facts. About 0.75% of the
children had severe difficulties to deal with domain specific mathematics knowledge and
skills.Moreover, in the third grade 7.70 % of the children (7.15% boys and 8.32% girls) had a
mathematics learning disability (see Table 1). In 4.41% of the cases these children had severe
difficulties with number fact retrieval. In 3.29% of the cases these children had severe difficulties
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Mathematics learning disabilities in Belgium
with domain specific mathematics knowledge and skills. In addition, in the fourth grade 6.59 %
of the children (6.99% boys and 6.22% girls) had a mathematics learning disability in Flanders
(see Table 1). This was the case for number fact retrieval disabilities and domain specific
disabilities in 2.88% and 3.71% of the cases respectively. In addition, about 25.47% of the
second graders had mathematics problems according to the teacher and a mathematics test
battery (-1SD on one of the tests), in grade 2 (see Tot problem in Table 1). In grade 3 and grade 4
the prevalence rates of mathematics problems (-1SD) were 23.35% and 21.91% respectively. The
mean mathematics scores for boys and girls in second, third and fourth grade did not differ on
domain specific knowledge (DS) or on number fact knowledge (NF).
To conclude, several authors use different concepts for overlapping phenomena as
difficulties in mathematics.. Moreover, the prevalence rate of mathematics learning disabilities in
lower elementary schools in Belgium varied from 2.27 % till 7.70 %. In addition, we argued to be
careful as to the diagnosis of children with mathematics learning disabilities. More specifically,
we referred to the importance in young children to test number facts as well as domain specific
mathematics and general conceptual knowledge in order to prevent the chosen test to determine
the diagnosis. This leads us to the third part, namely the description of the problems with
mathematical problem solving in children with and without mathematics learning disabilities in
Belgium.
Problems with mathematics in children with and without learning disabilities
In the last decade, substantial progress has been made in characterizing the cognitive and
metacognitive skills that are important for success in mathematical problem solving (Boekaerts,
1999; Geary, 1993; Grégoire & Van Nieuwenhoven, 1999; Hacker, Dunlosky, & Graesser, 1998;
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Mathematics learning disabilities in Belgium
Lucangeli & Cornoldi, 1997; Montague, 1997; Simons, 1996; Wong 1996). On the basis of this
research and an analysis of 150 math protocols of third graders in our study, a conceptual model
of mathematical problem solving was developed (De Clercq, Desoete, & Roeyers, 2000; Desoete,
2002; Desoete, Roeyers, & De Clercq, 2002 a &b) (see Figure 1).
<Figure 1 here>
A variety of studies based on different theoretical approaches have provided
information regarding the cognitive skills that are important for young children in learning how
to solve mathematical problems. Numeral comprehension and production skills are the cognitive
skills that are necessary for the reading, writing, and comprehension of numbers composed of one
or more digits (e.g., read '5' or '14') (e.g., McCloskey & Macaruso, 1995; Van Borsel, 1998).
Problems with these cognitive skills lead to mistakes such as 15+9 = 18 (confusion between 5
and 2 and 9 and 6). Operation symbol comprehension and production skills are cognitive skills
that enable the reading, writing and comprehension of operation symbols (such as +, -, x, =, <, >)
(e.g., Veenman, 1998). Problems with these cognitive skills lead to mistakes such as 15x9 = 24.
Number system comprehension and production skills are the cognitive skills dealing with number
system knowledge and the position of decades and units (e.g., Veenman, 1998). Children making
these mistakes often have problems with the place of a number on a number line and do not know
how many decades and units there are for example in 15. Procedural skills are domain-specific
cognitive skills involved in calculating and solving mathematics tasks in number problem formats
(e.g., 15+9=__ or 81-5=__) (e.g., McCloskey & Macaruso, 1995; Noel, 2000; Veenman, 1998).
Problems with these cognitive skills lead to mistakes such as 15+9 = 105 or 114. Linguistic skills
are cognitive conceptual skills that enable children to understand and solve one-sentence word
problems (e.g. 9 more than 15 is __). Language holds a central place according to several authors
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Mathematics learning disabilities in Belgium
(e.g. , Campbell, 1998; McCloskey & Macaruso, 1995). Veenman (1998) stresses the importance
of general conceptual knowledge in mathematics. Van Borsel (1998) goes even further and
defends mathematics learning disabilities as a special kind of language disorder. We would not
go so far. However, we can see that if children do not know what 'more' means, then word
problems such as '9 more than 15 is what?' cannot be solved correctly. Mental representation
skills are cognitive skills that enable an adequate mental representation of the problem or task
(e.g., Geary, 1993; Montague, 1997; Vermeer, 1997; Verschaffel, 1999). A mental representation
is required in most word problems, since a simple 'translation' of key words in a problem (e.g.
‘more’) into calculation procedures (e.g. ‘addition’), without representation, leads to ‘blind
calculation’ or ‘number crunching’. This superficial approach leads to errors, such as answering
'24’ to tasks such as '15 is 9 more than __', ’27 is 3 less than __' and ‘48 is half of __’.
Contextual skills are cognitive skills that enable the solving of tasks in word problems consisting
of more than one sentence (e.g., Bert has 14 Digimon cards. Griet has 5 Digimon cards more than
Bert. How many cards does Griet have?__). In children with mathematics learning disabilities,
problems involving these tasks (cognitive complexity) were found to be related to problems with
working memory (and ‘cognitive overload’) and knowledge base (and ‘expertise’) (Baddeley,
1999; Logie & Gilhooly, 1998; McCloskey & Macaruso, 1995). Relevance skills are cognitive
skills enabling the solving of word problems with irrelevant information included in the
assignment (e.g., Bert has 14 Pokémon cards and 3 Digimon cards. Griet has 5 Pokémon cards
more than Bert. How many Pokémon cards does Griet have?__). Children can have difficulty
ignoring and not using information (e.g., 3 Digimon cards) in an assignment. Children with
mathematics learning disabilities often think that all the numbers have to be ‘used’ in order to
solve a mathematical problem, and will therefore answer ‘22’. Number sense skills is the the
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Mathematics learning disabilities in Belgium
ninth category of cognitive skill. These skills enable the solving of tasks such as 'the answer to 5
more than 14 is nearest to__. Choose between 5, 10, 15, 70 and 50'. The ability to estimate
without calculating the exact answer is called ‘number sense’. In clinical practice, almost all
children with mathematics learning disabilities have problems with number sense tasks (e.g.,
Sowder, 1992). In our study significant partial correlations were found between all cognitive
skills except between linguistic and procedural skills, linguistic and operation symbol
comprehension skills, linguistic and relevance and number sense skills and mental representation
and relevance and number sense skills (Desoete, Roeyers, & Buysse, 2000).
It is nowadays widely accepted that metacognition also influences mathematical problem
solving (Brown, 1987; Carr & Jessup, 1995; Desoete, Roeyers, & De Clercq, 2003 July; Hacker
et al., 1998; Lucangeli, Cornoldi, & Tellarini, 1998; Veenman, Kerseboom, & Imthorn, 2000;
Verschaffel, 1999). In recent studies, however, ‘metacognition’ has come to be used in multiple
and nearly unrelated senses to refer to a wide range of phenomena (Borkowski, 1992; Carr &
Biddlecomb, 1998; Schoenfeld, 1992; Wong, 1996). ‘Metacognitive knowledge’ has been
described as knowledge and deeper understanding of one’s own cognitive skills and products
(Flavell, 1976, 1979). Within metacognitive knowledge, Cross and Paris (1988) and Jacobs and
Paris (1987) distinguish between declarative knowledge, procedural knowledge and conditional
knowledge. Metacognitive skills relate to the voluntary control people have over their own
cognitive skills (Veenman, in press). The number of metacognitive skills being distinguished
varies from three to ten (Desoete, 2001). Substantial data have been accumulated on four
metacognitive skills: orientation, planning, monitoring and evaluation (Lucangeli & Cornoldi,
1997; Lucangeli et al., 1998). Simons (1996) described a third metacognitive component
('metacognitive beliefs') as the broader general ideas and theories (e.g. self-concept, self-efficacy,
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Mathematics learning disabilities in Belgium
motivation, attribution, conceptions of intelligence and learning - see Figure 4) that people have
about their own (and other people’s) cognition (McCombs, 1989; Vermeer, 1997; Wong, 1996).
A principal components analysis of metacognition revealed three metacognitive components that
explain 66 % to 67 % of the common variance (Desoete, Roeyers, & Buysse, 2001). Prediction
and evaluation were found to be interrelated as one of these components. As both these
metacognitive skills were measured either before or after the solving of exercises, we labeled this
metacognitive component ‘off-line (measured) metacognition’. In 165 third graders we were able
to differentiate between different mathematics ability groups on the basis of the off-line
metacognitive component (Desoete et al., 2001).
Recent studies have underlined the importance of several cognitive and metacognitive
skills for the purpose of distinguishing between children with mathematics learning disabilities,
children with mathematics learning problems and children with age-adequate mathematics
performance in Flanders (Desoete & Roeyers, 2002).
In terms of cognition, several children with a combined domain-specific and
automatization mathematics learning disabilities were found to have less developed language
comprehension skills (Desoete & Roeyers, 2001). In addition, several children with a domainspecific mathematics learning disability were found to have less developed mental representation
skills. However, children with merely automatization disabilities (and age-adequate domainspecific mathematics knowledge and skills) did not fail on these tasks (and some even had higher
scores) (Desoete & Roeyers, 2001). These findings support the idea that some children with
specific mathematics learning disabilities use blind calculation techniques depending on a simple
translation of key words in an instruction. Finally, several children with a domain-specific
20
Mathematics learning disabilities in Belgium
mathematics learning disability had problems with procedural skills, using several bugs (Desoete
& Roeyers, 2001; Van Lehn, 1990).
In terms of metacognition, a majority of children with mathematics learning disabilities
were found to have less accurate prediction skills than peers without learning disabilities.
Younger children outperformed all children with mathematics learning disabilities in the area of
prediction skills on tasks designed for first-grade students (so-called ‘easy tasks’). Furthermore,
children with specific mathematics learning disabilities had less accurate prediction skills relating
to number system knowledge and procedural calculation. In addition, children with combined
mathematics and reading learning disabilities were found to have less accurate prediction skills in
doing word problems involving language-related and mental representation tasks. Moreover, a
majority of the children with mathematics learning disabilities had less accurate evaluation skills
than peers without learning disabilities. In addition, children with mathematics learning
disabilities had problems especially in estimating their chances of success on the ‘easy tasks’.
Finally, children with mathematics disabilities did worse in the evaluation of number knowledge
and procedural calculation than did younger children who were their equals in the area of
mathematical problem solving (Desoete & Roeyers, 2002; Desoete, Roeyers, & De Clercq, 2001
August).
Studies at the group level certainly reveal interesting information. There is, however, a
certain risk of misinterpreting these studies, because they cannot be automatically applied to
individual children. Not all children with mathematics learning disabilities were found to have
the same inadequate cognitive or metacognitive skills. For example, children with a mathematics
automatization disability did not fail in linguistic, procedural or mental representation tasks
(Desoete & Roeyers, 2001). In addition, only a small majority of the third graders with
21
Mathematics learning disabilities in Belgium
mathematics disabilities had inaccurate off-line metacognitive skills. Furthermore, a minority of
the children without learning problems also had a severe deficit (-2 SD) on off-line metacognitive
skills (Desoete, 2001).
Taking all these findings together, what we seem to see is a broad spectrum of
mathematics learning disabilities that manifest themselves in a great variety of individual
cognitive and metacognitive profiles in young children. It might therefore be important to assess
cognitive and off-line metacognitive skills in children with mathematics learning disabilities.
Certainly procedural, linguistic, mental representation, prediction and evaluation skills have to be
tested in order to detect whether these skills are age-adequately developed. In addition, general
protocol cognitive or metacognitive intervention for all children with mathematics learning
disabilities might represent overconsumption of therapeutic resources, since not all individual
children were found to have below-average performance on tasks depending on these skills. This
leads us to the final part, namely the inventory of the instructional interventions and
accommodations implemented to help students with mathematics learning disabilities in Belgium.
Instruction for children with mathematics learning disabilities in Belgium.
When problems with mathematics are detected, the teacher or parent can ask the advice of
the school psychologist. In most cases, problems with mathematics are regarded as problems to
be solved within the classroom and within the school system. The school psychologist and the
care-coordinator (‘zorgcoördinator’) then look at exercises done in the classroom and, if
necessary, give an additional mathematics test. Special groups may be formed within the
classroom that for several hours during the week in order to differentiate between fast and slow
learners (‘broader care’ or ‘zorgverbreding’). When ‘ broader care’ is insufficient, then ‘remedial
22
Mathematics learning disabilities in Belgium
teaching’ (‘taakleerkracht’) is the usual next step. Here the children are given special exercises in
a special classroom for six months (on an individual basis or in small groups) in an attempt to
remedy their problems.
After six months to a year, children with resistant severe problems with mathematics in
primary school often go to one of the rehabilitation centers or centers specialized in handling
educational problems to get a more elaborated assessment. After an interview with the parents,
the child, the teacher and (sometimes) the school psychologist, and after an elaborated series of
tests, interviews and observations, the diagnosis and ‘label’ of learning disability is often given.
Some of the children with learning disabilities are also advised to repeat a year in school.
In Flanders 10% of all sixth graders have repeated one or more years of primary school
(Hellinckx & Ghesquiere, 1999). In other cases, children are referred to special education (Type
8), although, as previously mentioned, there is a strong movement toward inclusive education.
In Flanders, young children of average intelligence up to the age of 15 who have a
learning disability, a score lower than percentile 4 on a mathematics test and a need for a
multidisciplinary approach, can follow a course of therapy for 2 years (2 or 3 times a week) in a
rehabilitation center. Moreover, there are some situations in which psychologists and educators
and in some cases also remedial teachers and care coordinators set up a therapy plan for these
children. Then a special therapeutic program is set up. This program sometimes fits within the
regular school system, though for many children it is provided by psychologists and other
specialists outside of the school. In other cases, when a specific isolated mathematics retardation
of 1 year (for 7 to 9-year-olds) or 2 years (for 9 to 14-year-olds) is found, a mono-disciplinary
therapy is advised and children go to private mathematics therapists, (in Flanders, often
specialized speech and language therapists or specialized physiotherapists).
23
Mathematics learning disabilities in Belgium
The different approaches taken to therapy for children with mathematics learning
disabilities in Flanders can vary greatly. However, most therapists apply a rather (meta)cognitive
and task analytical therapy based on the learning of algorithms and heuristics. The instruction
principles are isolation (orientation, demonstration, practice, shortening, controle, working faster,
knowing when and where to use the principles), integration and generalization (Ruijssenaars,
2001). This is frequently combined with stimulating a realistic sense of self-esteem in young
children and a correct identification of the successes and failures in school performance. Often
therapists also use self-instructional techniques, for example to teach children how to handle
word problems (‘read the instructions’, ‘see if you understand all the words’, ‘make a diagram or
drawing of the situation’, etc.). A minority of therapists in Flanders use a more psychomotor
approach to dealing with mathematical problems in young children. However, in many cases a
combination of therapeutic approaches is used from a more eclectic perspective. The form that
the course of therapy actually assumes depends to a great extent on the assessment of the
problems of the individual child.
Summary
Flemish students seem to do quite well on mathematics compared with youngsters in
other countries. However, between 3 and 8 percent of students in Flanders have mathematics
learning disabilities in primary school. Moreover, mathematics learning disabilities seem to be
very persistent, leading to residual problems with mathematics in adolescence and even in
adulthood.
We have argued that although many authors agree that an operational definition of
learning disabilities is meaningful, most studies are rather vague when it comes to objectively
24
Mathematics learning disabilities in Belgium
defining the characteristics on the basis of which they identify these ‘children with learning
disabilities’. In addition, overlapping concepts and terms are frequently used to deal with a wide
range of phenomena relating to children who have difficulties with mathematical problem
solving. In an effort to be more explicit, we have adopted three criteria (discrepancy criterion,
severeness criterion and resistance criterion) for ascertaining whether an individual child belongs
to the group of subjects with mathematics learning disabilities.
In addition, we have stressed the need for care in the assessment and diagnosis
of children with ‘mathematics learning disabilities’. More specifically, we have referred to the
importance of using a cocktail of mathematical tests for young children, including at least one
test on number facts and a test on domain-specific mathematics knowledge or word problems, in
order to avoid drawing up an entire diagnosis on the basis of a single test. In addition, we have
found that teachers’ judgments seem to be an absolute requirement for confirming the test results.
Our research revealed the importance of also testing the reading skills of children with
mathematics learning disabilities in order to differentiate children with a specific mathematics
learning disability from children with a combined learning disability. Furthermore, quite a
number of children with mathematics learning disabilities were found to have less developed
linguistic, procedural and mental representation skills, and a majority of the children with
mathematics learning disabilities were found to show inaccurate off-line metacognitive skills. It
may be advisable also to assess the cognitive and metacognitive skills in young children with
mathematics learning disabilities.
We furthermore clarified that in Belgium most young children with learning
disabilities remain in regular kindergarten. Once the children are in primary school, the learning
disabilities often become obvious. With the new mathematics curriculum and the realistic
25
Mathematics learning disabilities in Belgium
approach (RME), additional research is certainly needed to determine their effect on children
with mathematics learning disabilities. Once the problems with mathematics are detected, the
school psychologist is consulted. This psychologist often suggests a differentiated approach
(broader care) or remedial teaching for six months. If this is not sufficient, a more elaborated
assessment is carried out and a therapy plan is established. This therapy is partially financed by
the government for a period of two years. After this period, in theory, the disabilities should be
‘remedied’. However, in practice, children in secondary or higher education with mathematics
learning disabilities encounter even more difficulties, which lead to a very low self-concept and
even to behavior problems. These findings indicate that more in-depth research is certainly
needed on the prevention and early detection of signals indicating potential mathematics learning
disabilities in preschool children. In addition, studies on adolescents and even adults with
mathematics learning disabilities would be useful in order to gain more insight into the emotional
and behavioral impact of the disabilities
The findings from current studies also provided evidence that educational interventions
for students with learning disabilities can produce positive effects of respectable magnitude
(Desoete, Roeyers, Buysse, & De Clercq, 2002; Efklides, Papadaki, Papantoiniou, &
Kiosseoglou, 1997; Masui, & De Corte, 1999; Swanson, Hoskyn, & Lee, 1999). However, a
number of problems that students with mathematics learning disabilities encounter continue to
make themselves felt even into adulthood. In Belgium, most therapists were found to apply a
rather (meta)cognitive therapy based on the learning of algorithms and heuristics. This approach
was often found to be combined with efforts to remedy the emotional aspects of the learning
disability. In most cases, a combination of therapeutic approaches is used for an individual child,
depending on the assessment results.
26
Mathematics learning disabilities in Belgium
It is important to keep in mind that we have restricted most of our studies to third graders
with mathematics learning disabilities in Flanders. Therefore on several aspects further research
can be recommended On the one hand, cognition and off-line metacognition needs to be studied
in younger and older children and in children with below or above average intelligence. In
addition, individual studies of children with mathematics learning disabilities are clearly needed
to help us translate the findings at the group level into remedial measures specifically suited to
individual children. We believe that the research data derived from such studies would improve
our understanding of persons with mathematics learning disabilities.
By way of conclusion, however, we have found that mathematics learning
disabilities can be very persistent. We have also found that a well elaborated assessment and
intervention procedure is of key importance, since there appears to be a broad spectrum of
mathematics learning disabilities that come to expression in a broad range of highly individual
cognitive and metacognitive profiles in young children
27
Mathematics learning disabilities in Belgium
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Mathematics learning disabilities in Belgium
Table 1. Prevalence of mathematics learning disabilities in Flanders
Gr2
Gr3
Gr4
Gr2 boys
Gr2 girls
Gr3 boys
Gr3 girls
Gr4 boys
Gr4 girls
n =662
n=661
n=699
n=637
n=644
n=675
2.11%
2.42%
7.15%
8.32%
6.99%
6.22%
NF disability 1.21%
1.81%
4.43%
4.39%
3.26%
2.52%
DS disability 0.91%
0.91%
3.29%
4.24%
4.66%
4.29%
Tot Problem 6.99%
6.22%
27.79%
23.15%
24.89%
25.90%
NF problem
19.33%
14.37%
16.16%
16.01%
14.91%
10.96%
DS problem
11.78%
13.31%
11.73%
13.81%
10.87%
15.26%
NF adequate 79.46%
83.81%
79.39%
79.59%
81.83%
86.66%
DS adequate 87.31%
85.93%
84.98%
81.95%
84.47%
80.44%
Tot DIS
Note. NF = number facts, DS = domain specific test, Tot DIS = total number of children with a
mathematics learning disability (-2 SD), Tot Problem=total number of children with a
mathematic learning problem (-1 SD), Gr2 = grade 2, Gr3 = grade 3, Gr 4 = grade 4
40
Mathematics learning disabilities in Belgium
Figure 1. Mathematical problem solving: a conceptual model
COGNITION
Numeral comprehension & production
Operation symbol comprehension &
production
Number system knowledge
Procedural calculation
Language comprehension
Context comprehension
Mental representation
Selecting relevant information
Number sense
METACOGNITION
Declarative Knowledge
METACOGNITIVE
Procedural Knowledge
KNOWLEDGE
Conditional Knowledge
Prediction
METACOGNITIVE
Planning
SKILLS
Monitoring
Evaluation
Self-concept
METACOGNITIVE
Self-efficacy
BELIEFS
41
Mathematics learning disabilities in Belgium
Motivation
Attribution
Conception of intelligence and learning
42