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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 Artevelde College, Ghent, Belgium

3 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


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: [email protected]

Herbert Roeyers

Ghent University, Department of Clinical Psychology

Research Group: Developmental Disorders

Address: Henri Dunantlaan 2


Tel: 32. (0) 9.264.64.48

Fax: 32. (0) 9.264.64.89

2

Armand De Clercq

Ghent University, Department of applied Mathematics and Computer Science

Address: Krijgslaan 281 S9


Tel : 32. (0). 9. 264 47 79

Fax : 32. (0) 9 264 49 95

3



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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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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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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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 teacher-

assessment 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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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 real-

life 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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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’ (AB …) 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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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 2000-

2001 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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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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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

12

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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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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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

15



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;

16

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

17



(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

18

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,

19



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 domain-

specific mathematics learning disability were found to have less developed mental representation

skills. However, children with merely automatization disabilities (and age-adequate domain-

specific 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 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 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

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



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

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



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

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



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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

Tot DIS 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%

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

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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

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Motivation

Attribution

Conception of intelligence and learning

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metacognition and mathematical problem solving in grade 3 · web viewdisabilities encounter...

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