Numerical competence in young children and in children with mathematics learning disabilities

Learning and Individual Differences 16 (2006) 351 – 367 www.elsevier.com/locate/lindif

Numerical competence in young children and in children with mathematics learning disabilities Annemie Desoete a,⁎, Jacques Grégoire b a

Department of experimental clinical and health psychology, Ghent University, Belgium b Catholique University of Louvain, Louvain-la-Neuve, Belgium Received 1 December 2006; accepted 14 December 2006

Abstract A longitudinal study was conducted on 82 children to investigate, firstly the numerical competence of young children and the predictive value of (pre)-numerical tests in kindergarten, and, secondly, whether children's knowledge of the numerical system and representation of the number size is related to their computation and logical knowledge and to their counting skills. In an additional cross-sectional study on 30 children with a clinical diagnosis of mathematical learning disability (MLD) of 8,5 years, age- and ability-matched with 2 × 30 children the same parameters of numerical competence were assessed. The longitudinal data showed individual differences in numerosity, as well as the relationship between a delay in arithmetics in grade l and problems on numerosity in kindergarten. In the cross-sectional results some evidence was found for the independence of numerical abilities in MLD-children. About 13% of them had still severe pre-numerical processing deficits (in number sequence production, cardinality skills and logical knowledge) in grade 3. About 67% had severe difficulties in executing calculation procedures and a lack of conceptual knowledge. A feature of 87% of the MLD-children was severe translation deficits, with a significantly worse knowledge of number words compared with the knowledge of Arab numerals. Finally a severe deficit in subitizing was found to be present in 33% of the MLD children. On a group level the processing deficits were linked to understanding numerosity, since the abilitymatched younger children and the MLD-children had the same pre-numerical and numerical profile. Implications for the assessment of mathematical disabilities and the value of TEDI-MATH® as an instrument in this process are discussed. © 2007 Elsevier Inc. All rights reserved. Keywords: Numerical system; Number size; Logical knowledge; Counting; Mathematics learning disabilities

1. Introduction 1.1. Components of numerical competence Several components (see Fig. 1) were found important for young children to develop numerical competence and to solve arithmetical problems adequately (Fuson et al., 1997; Gelman & Gallistel, 1978; McCloskey & Macaruso, 1995; Piaget & Szeminska, 1941; Sowder, 1992). We focus on five of these components underlying numerical competence of young children up to grade 3. ⁎ Corresponding author. E-mail address: [email protected] (A. Desoete). 1041-6080/$ - see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.lindif.2006.12.006

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Fig. 1. Theoretical framework integrating several components of mathematical problem solving.

Numerical competence depends on adequate logical operations on numbers. Piaget and Szeminska (1941) specified the logical abilities that children progressively acquire to master the concept of number. Co-ordination of classification (gathering items in a set) and seriation (ordering sets of items according to their size) are important for logical thinking on numbers. Although the work of Piaget remains an essential reference for practitioners working with children with mathematical problems, recent studies in the area of mathematics added insights on pre-numerical competence of young children (Donaldson, 1978; Grégoire, 2005; Grégoire, Van Nieuwenhoven, & Noël, 2004; Ruijssenaars, Van Luit, & Van Lieshout, 2004; Van Luit, 2002). In these studies the pragmatic context of studies, language and counting became more important. The second pre-numerical skill has to do with the counting procedures. There are two skills involved in these procedures: number word sequence production and cardinality skills. Firstly, counting depends on the pre-numerical competence and knowledge of the number word sequences (e.g., Gelman & Butterworth, 2005). In addition, counting can be seen as a tool to know the cardinal of a set (Briars & Siegler, 1984; Gelman & Meck, 1986; Kaye, 1986). According to Gelman and Gallistel (1978) counting is a procedure based on five principles: (1) the stable-order principle according to which the number words have to constitute a stable sequence; (2) the one–one principle according to which every items in a set must be assigned a unique tag; (3) the cardinal principle according to which the last number word pronounced represents the cardinal of the set; (4) the abstraction principle according to which any kind of object, taken as a unit, can be gathered to be counted; (5) the order-irrelevance principle according to which the elements of a set can be counted in any sequence as long as the other counting principles are respected. If these principles are acquired, children can count a linear or random pattern or heterogeneous set of items (e.g., Fuson, 1988). Furthermore, numerical competence depends on the insight in the number structure or on the knowledge of the position of decades and units and the ability to establish base-ten structure relationships (e.g., McCloskey & Macaruso, 1995). Knowledge of the numerical system is required to judge which of two Arab numerals or number words is the larger, to know how many decades and units are for example in 17 and to transcode (write a dictated number or read a number written in Arab code). The understanding and the production of numerical symbols is performed by two components, each one divided into a subsystem for Arab number processing and another one for verbal number processing. The transcoding procedure relates to the transformation for example of Arab representation into verbal representation, and vice-versa (Hüttemann, 1998; McCloskey & Macaruso, 1995). Numerical competence also depends on knowledge and computation skills to calculate and to solve arithmetical operations referring to objects, presented in a number problem format (e.g., 6 + 3=_ or 9 − 5=_ or 2 × 4=_) or in a verbal format (e.g., Denis had 2 marbles. He won two others. How many marbles had Denis in all?) (Carpenter, Franke, Jacobs, Fennema, & Empson, 1987; Riley, Greeno, & Heller, 1983; Siegler, 1987). The representation of number size (numerosity) is the fifth skill involved in numerical competence. This numerical skill is involved in subitizing (rapid apprehension of small numerosity) and in estimation of size (Gersten & Chard, 1999; Hannula & Lehtinnen, 2005; Sowder, 1992; Trick & Pylynshyn, 1994). There are even some arguments that problems encountered by pupils with mathematical learning disabilities may be due to a deficit in this skill (Butterworth, 2003).

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1.2. Mathematical learning disabilities Crucially when comparing studies is to take account of the fact that often different terms are used for the children with severe and persistent difficulties in mathematics but also sometimes the same terms are used for different children. In addition, findings about the prevalence, nature and outcomes of mathematical disabilities may vary considerably, depending on the used criteria to define the disability (Desoete, Roeyers, & De Clercq, 2004; Dowker, 2004; Geary, 2004; Mazzocco & Myers, 2003). 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 October 2001 a prevalence study was conducted among regular primary school children in Flanders (n = 3978). Between 3 and 8% of the children were found to have mathematical learning disabilities (Desoete et al., 2004). Similar prevalence rates have been found in other countries (e.g., Shalev, Manor, & Gros-Tsur, 2005). However, from 1999– 2004 only 29 articles on mathematics disabilities (math⁎disab⁎) and 206 articles on dyscalculia were cited in Web of Knowledge, whereas 1075 articles on reading disabilities (read⁎disab⁎) and 2782 articles on dyslexia could be found. 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. Difficulties in mathematics appear to be equally common in boys and girls, in contrast to language and literacy difficulties, which are more common in boys (Dowker, 2004). In 46% of the cases children with mathematics learning disabilities have severe and specific but isolated difficulties with mathematics and age adequate reading and writing skills (Ghesquière, Ruijssenaars, Grietens, & Luyckx, 1996). In 26% of the cases there are comorbide symptoms of ADHD (Gross-Tsur, Manor, & Shalev, 1996). The prevalence of a combined mathematics and reading disability varies from 17% (Gross-Tsur et al., 1996) to a little less than 50% (Badian, 1983). The prevalence of combined mathematics and writing disabilities is about 50% (Ostad, 1998). The chronicity of the disability is found to be associated with the severity of disability, the lower IQ, inattention and writing problems (Shalev, Manor, Auerbach, & Gross-Tsur, 1998). Some attempts to see mathematical learning disabilities (MLD) as a consequence of cognitive deficits that are not specific to understanding numerosity are made. Proposals include an abnormal representation in semantic memory, slow speed of processing, deficits of working memory, weak phonetic representations and a subitizing deficit (Butterworth, 2003). Another approach has involved subtyping MLD. Many authors (e.g., Geary, 2004) made a classification based on their observations and research findings. Since there is no single explanation for the cause of mathematical disabilities, researchers classified the syndromes they met in practice from different perspectives. This in turn led to different classifications of the wide range of observations in children with mathematical problems. Most mathematics learning disabilities are not detected until Grade 1 when children have to master addition and subtraction and, in some cases, even later (in Grade 3) when they have to learn to retrieve quickly the timetables or to select and apply various problem-solving strategies in addition to the basic mathematical operations. However the sooner an education program can focus on weak competences, understand the origin of the learning problem and give a sound foundation for remediation, the more efficient it can be. Early detection depends on good tests. Mathematic tests that focus on the assessment of performance are useful to situate individual scores among the distribution of the reference scores, but they cannot help us to understand observed problems (Grégoire, 1997). For that, we need tests based on a validated model of the specific learning we are assessing. Not many such tests are recently available. There are some instruments (Butterworth, 2003; Von Aster & Weinhold, 2002) that can provide a first screening of the child's learning problem. However, screeners often do not give us a sound foundation for remediation. There is also the Key Math explicitly (Connolly, 1988). Although designed with care the Key Math Revised has no support by a validated theoretical model of mathematics learning and no explanation is proposed for understanding the errors children make. An instrument that is validated by a combination of theoretical models and therefore can be used for an in-depth diagnostic assessment seems to be the TEDI-MATH (Van Nieuwenhoven, Grégoire, & Noël, 2001). This multi-componential instrument is based on a combination of neuropsychological (developmental) models of number processing and calculation. It has an age range from 4 to 8 years of age (kindergarten to 3rd grade). There is a German, Dutch and French version (Collet, 2003). 1.3. Aims of the study The short overview clearly shows that there is less research concerning the diagnostic assessment of mathematical disabilities in young children. The choice of a theoretical model is important as conceptual framework for the

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diagnostic assessment of learning disabilities. In the area of mathematical learning, new results are published every month. Therefore also in the assessment of mathematical disabilities it is important to integrate several models (see also Section 1.1.). The first study was set up to investigate in a normal population whether children with a delay in arithmetics in grade 1 already encountered problems on numerosity in kindergarten. If so an assessment of ‘markers’ in preschool can be very useful to start early intervention programs. Moreover a second study was set up to investigate whether the dissociation of numerical competence facets might have an additional value in the assessment of mathematical learning disabilities (MLD) in grade 3. The study aims to see if eventual processing deficits are specific to understand numerosity. In this case children with MLD and abilitymatched younger children will have the same profile. Furthermore it is investigated if the pre-numerical and numerical problems are representative of the majority of MLD-pupils and if subtyping MLD is needed. 2. Study 1 The aims of Study l were to examine children's numerical competence on the transition period where they change from kindergarten (preschool) to elementary school (grade 1). Our hypothesis was that young children vary in their (pre)numerical competence and on their focus on numerosity (see also Hannula & Lehtinnen, 2005). Within this study we focus on ‘markers’ that are related to the formation of early mathematical skills and may be are predictive if impaired for an eventual mathematical learning disabilities later on. 2.1. Method 2.1.1. Participants Participants were preschool children attending six kindergartens in the Dutch-speaking part of Belgium. All children were assessed at the end of kindergarten and in the first grade of elementary school. The sample included only white children, 45 girls (45.1%) and 37 boys (54.9%). All children followed regular preschool education. All parents were Dutch-speakers, with exception of one mother of Polish origin. This mother however also spoke very good Dutch. Permission for children to participate in this study was obtained from their parents. From the original dataset of 84 children in kindergarten only the data of 82 children were available after two years, resulting in 2 missing data. One child changed school and one parent did not accept that her child participated on the longitudinal study. 2.1.2. Measures TEDI-MATH (Grégoire et al., 2004 Flamish adaptation) is a test designed for the diagnostic assessment of mathematical disabilities. It was standardized on a sample of 550 Dutch speaking Belgian children from the beginning (November) of the 2nd grade of the nursery school to the end (May) of the 3rd grade of primary school. The test enlightens five facets of numerical competence: logical knowledge, counting, representation of numerosity, knowledge of the numerical system and computation. Appendix A shows the subtests of the TEDI-MATH. Twelve basic scores can be computed. The reliability coefficients vary from .70 (for computation with pictures) to .99 (for transcoding). A validation study showed that the TEDI-MATH could discriminate among pupils with different levels of mathematical knowledge according to the teachers. The raw score of the TEDI-MATH is converted into percentiles. It is suggested to consider possible disabilities under pc 10. However the authors state that these cut off scores are only indications and should be used with great caution. The diagnosis of learning disability can only be drawn from a global assessment of the child, including learning, intelligence, emotions, family and school context. Toeters (CLB Haacht, 1997) is a preschool assessment for language, and numeral competence, writing and working attitude. It was standardized on a sample of 498 Dutch-speaking children. The subtest mathematics includes items on counting, mathematical language and conservation. Kortrijk Arithmetic Test (Kortrijkse Rekentest Revision, KRT-R) (Baudonck et al., 2006) is a 60-item Belgian mathematics test on domain-specific knowledge and skills, resulting in a percentile on mental computation, number system knowledge and a total percentile. The psychometric value of the KRT-R (with norms January and June) has been demonstrated on a sample of 3246 children in total. A validity coefficient (correlation with school results) and reliability coefficient (Cronbach's alpha) of .50 and .92 respectively were found for grade 1. We used the standardized percentiles based on national norms.

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Arithmetic Number Facts test (Tempo Test Rekenen, TTR) (De Vos, 1992) is a test consisting of 200 arithmetic number fact problems (e.g., 5 × 9=_). Children have to solve as many number-fact problems as possible out of 200 in 5 min. The test has been standardized for Flanders on 220 third-graders (and on 10,059 children in total) (Ghesquière and Ruijssenaars, 1994). The teacher preschool-survey, which was created for this research line, is a rating scale (4-item) for teachers on the intelligence, pre-numerical skills, pre-reading skills and social skills of young children (e.g., very bad skills (1)/very good skills (7)). The teacher elementary school-survey is a similar rating scale on intelligence, arithmetics, reading and social skills of elementary school children. The teacher questionnaires were tested in previous studies in order to determine its construct validity. Test–retest correlations of .81 ( p b .01) and interrater reliabilities varying between .99 and 1.00 ( p b .01) were found (Desoete & Roeyers, 2002). 2.1.3. Design and procedure This study is part of a longer study were these children will be followed from the age of 4.9 years till 9.9 years. In this study we discuss the results of children's numerical competence at the end of preschool and one year later. The tasks were presented in the same order for all children. All preschool children were assessed individually, outside the classroom setting, where they completed TEDIMATH (Grégoire et al., 2004) and Toeters (CLB Haacht, 1997) in preschool. In addition regular kindergarten teachers completed a teacher survey in the same period. In grade 1 of elementary school the children completed the KRT-R (Baudonck et al., 2006) and the TTR (De Vos, 1992) on the same day for about 1 h in total. In addition regular elementary school teachers completed an elementary school teacher survey in the same period. The examiners, all psychologists or therapists skilled in learning disabilities, received practical and theoretical training in the assessment and interpretation of mathematics. The training took place two weeks before the start of the assessment. In addition, systematic, ongoing supervision and training was provided during the assessment of the first 10 children. The training included a review and discussion of the TEDI-MATH student profiles and involved one

Table 1 Preschool results Test item

Min–Max

Raw score (SD)

M pc (SD)

TEDI-MATH ® l. Knowledge of number word sequence 2. Counting set of items 3. Knowledge of the numerical system 3.1.1 Arab lexicon 3.1.2 Arab syntax 3.1 Arab numerical system 3.2 Oral numerical system 4.1 Seriation of numbers 4.2 Classification of numbers 4.3 Conservation of numbers 4.4 Inclusion of numbers 4. Logical operations on numbers 5.1 Arithmetical operations presented on pictures 5.2.1 Additions in arithmetical format 5.3 Wordproblems 6. Estimation of size: dot sets

1–8 1–13 4–8

4.04 (2.15) 7.60 (2.90) 7.12 (1.29)

52.76 (25.70) 40.22 (29.56) 74.74 (33.59)

1–17 7–24 6–12 0–2 0–2 0–4 0–3 0–11 0–6 0–10 0–8 0–6

9.22 (3.27) 16.21 (3.61) 9.60 (1.57) 1.40 (0.78) 0.73 (0.74) 0.91 (1.44) 1.88 (1.33) 4.94 (2.92) 3.88 (1.67) 2.15 (2.67) 2.70 (1.92) 4.90 (1.68)

75.85 (27.56) 60.94 (29.87) 53.94 (29.00) 77.88 (26.58) 64.29 (29.21) 77.13 (10.00) 66.59 (37.63) 58.56 (29.91) 55.61 (30.71) 70.18 (16.37) 62.87 (25.48) 55.35 (42.15)

TOETERS Visual scale Auditive scale Numerical competence Visual-motoric scale Total score

2.74 (0.75) 3.04 (0.83) 3.00 (1.00) 2.96 (0.88) 4.41 (1.67)

2.74 (0.75) 3.04 (0.83) 3.00 (1.00) 2.96 (0.88) 4.41 (1.67)

/ / / / /

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Table 2 Grade 1 results Test items KRT-R

M (pc)

SD

Mental arithmetics Number knowledge Total test

48.37 50.24 48.63

32.51 34.89 34.22

meeting during the assessment period. During and after the testing in each classroom, the first author visited the teacher of the class to supervise the ongoing study. 2.2. Results 2.2.1. Descriptive results At the time of pre-testing in kindergarten, the participants had a mean age of 5.9 year (SD = 4 months). For the standardized mean percentile score in kindergarten, we refer to Table 1. There were significant correlations between most subtests of TEDI-MATH and the subtest ‘arithmetics’ of TOETERS. The correlations with the number word sequences, counting and logical operations subtests of the TEDIMATH were .48 ( p b .05), .08 and .62 ( p b .01) respectively. The small correlation with counting can be explained by the difference in type of exercises. In addition, correlations were found with subtests knowledge of the numerical system and arithmetical operations of TEDI-MATH of .22 and .54 ( p b .01) respectively. Also problem-scores (bpc 10) on the TEDI-MATH were validated by the preschool teacher survey results. At the time of testing in elementary school, children were one year older and had a mean age of 6.9 year (SD = 4 months). For the standardized mean percentile scores on the KRT-R, we refer to Table 2. The mean Grade equivalent of the children was 6.96 years (SD = 6.40), meaning that they solved arithmetical facts as did children at the end of February and beginning of March. 2.2.2. Impact of pre-numerical skills in kindergarten on mathematical abilities in grade l 2.2.2.1. Arithmetical fact retrieval. In order to answer our research questions on pre-numerical skills a multivariate analysis of variance (MANOVA) was conducted with knowledge of number word sequence, counting and logical operations on numbers, as measured by TEDI-MATH, as dependent variables and belonging to one of the two arithmetical fact retrieval ability groups (less than 1/4 of the Grade equivalent, more than 1/4 of the Grade equivalent) as a factor. A power of .978 was observed. The MANOVA was significant (F(3, 39) = 7.57, p ≤ .0005) on the Table 3 Scores of children with different arithmetical fact retrieval skills Scores in pre-school

Pre-numerical skills 1. Number word sequence 2. Counting 4. Logical operations Numerical skills 3.1. Arab numerical system 3.2 Oral numerical system 5.1 Arithmetical operations presented on pictures 5.2 Arithmetical operations presented in arithmetical format 5.3 Arithmetical operations presented in verbal format 6. Estimation of size Note ⁎⁎p b .0005; ⁎p b .01.

More than 1/4 behind in grade 1

More than 1/4 in advance in grade 1

M (SD) (N = 21)

M (SD) (N = 22)

3.38 (2.202 6.14 (2.903) 3.43 (2.839)

5.23 (2.159) 9.68 (2.715) 5.91 (2.408)

15.95 (2.75) 9.19 (1.81) 3.38 (1.72) 0.90 (1.45) 1.86 (1.49) 0.81 (0.40)

16.55 (4.37) 10.05 (1.49) 4.68 (1.13) 3.09 (3.13) 2.91 (2.07) 0.95 (0.21)

F (1,41) 4.752⁎ 7.886⁎⁎ 6.901⁎ F (1,41) 0.28 2.87 8.69⁎ 8.50⁎ 3.63 2.21

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Table 4 Scores of children with different mental arithmetics skills Scores in pre-school

Pre-numerical skills 1. Number word sequence 2. Counting 4. Logical operations Numerical skills 3.1. Arab numerical system 3.2 Oral numerical system 5.1 Arithmetical operations presented on pictures 5.2 Arithmetical operations presented in arithmetical format 5.3 Arithmetical operations presented in verbal format 6 Estimation of size

pc ≤ 15 on mental arithmetics in grade 1

pc ≥ 50 on mental arithmetics in grade 1

M (SD) (N = 22)

M (SD) (N = 44)

2.45 (1.18) 5.91 (2.59) 2.41 (2.15)

4.95 (2.06) 8.68 (2.71) 6.52 (2.27)

F (1,64) 27.756⁎⁎ 15.778⁎⁎ 49.904⁎⁎

14.59 (3.14) 8.55 (1.68) 2.77 (1.69) 0.91 (1.31) 1.82 (1.50) 0.77 (0.43)

16.57 (3.66) 10.09 (1.31) 4.55 (1.32) 3.18 (3.06) 3.16 (1.79) 0.97 (0.15)

F (1,64) 4.69⁎ 16.84⁎⁎ 21.89⁎⁎ 11.06⁎⁎ 9.11⁎⁎ 8.11⁎

Note. ⁎p b .05, ⁎⁎p b .005.

multivariate level. On the univariate level, a significant difference was found between the groups for all dependent variables. For M and SD we refer to Table 3. From Table 3 we can conclude that children with a difficulty in learning and remembering arithmetical facts in grade 1 already struggled with pre-numerical competences in preschool. 2.2.2.2. Mental arithmetics. In addition we compared the pre-numerical skills of children average or good (≥ pc 50) in mental arithmetics in grade 1 with children with a delay (≤ pc 15) on mental arithmetics measured by a subtest of KRT-R (Baudonck et al., 2006). The MANOVA was significant on the multivariate level (F(3, 62) = 24.508; p ≤ .0005) with an observed Power of 1.00. On the univariate level both groups differed on all dependent variables. For M and SD we refer to Table 4. From Table 4 we can conclude that pupils with a delay in mental arithmetics in grade 1 already encountered problems with pre-numerical numerosity facets at preschool age. 2.2.2.3. Number knowledge. We also compared children average or good in number knowledge with children with below-average scores on number knowledge measured by a subtest of KRT-R (Baudonck et al., 2006). On the Table 5 Scores of children with different number knowledge Scores in pre-school

Pre-numerical skills 1.Number word sequence 2.Counting 4.Logical operations Numerical skills 3.1. Arab numerical system 3.2 Oral numerical system 5.1 Arithmetical operations presented on pictures 5.2 Arithmetical operations presented in arithmetical format 5.3 Arithmetical operations presented in verbal format 6. Estimation of size Note. ⁎p b .005, ⁎⁎p b .0005.

pc ≤ 15 on number knowledge in grade 1

pc ≥ 50 on number knowledge in grade 1

M (SD) (N = 21)

M (SD) (N = 45)

2.38 (1.53) 6.00 (2.86) 2.05 (2.29)

4.76 (1.92) 8.62 (2.68) 6.24 (2.44)

F (1,64) 24.692⁎⁎ 13.111⁎ 43.947⁎⁎

15.00 (3.41) 8.57 (1.57) 2.71 (1.93) 0.76 (1.18) 1.57 (1.75) 0.71 (0.46)

16.58 (3.78) 10.22 (1.31) 4.40 (1.29) 3.09 (3.09) 3.16 (1.95) 0.98 (0.14)

F (1,64) 2.65 19.99⁎⁎ 17.70⁎⁎ 11.09⁎ 10.04⁎ 12.09⁎

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multivariate level the MANOVA was found to be significant (F(3,62) = 22.786, p ≤ .0005) with an observed Power of 1.00. On the univariate level both groups differed on all dependent variables. For M and SD we refer to Table 5. From Table 5 we can conclude that children with a delay in number knowledge in grade l already had below-average pre-numerical competences at preschool age. 2.2.3. Impact of numerical skills in kindergarten on mathematical abilities in grade l 2.2.3.1. Arithmetical fact retrieval. In order to answer the research questions on the numerical skills in kindergarten, a multivariate analysis of variance (MANOVA) was conducted with knowledge of Arab numerical system, knowledge or oral numerical system, arithmetical operations presented on pictures, arithmetical operations presented in arithmetical format, arithmetical operations presented in verbal format as measured by TEDI-MATH, as dependent variables and belonging to one of the two fact retrieval ability groups (less than 1/4 of the Grade equivalent age, more than 1/4 of the Grade equivalent) as factor. A power of .780 was observed. The MANOVA was significant (F(6, 36) = 2.479, p ≤ .05) on the multivariate level. On the univariate level, a significant difference was found for arithmetical operations presented on pictures and in arithmetical format between the groups. For M and SD we refer to Table 3. From Table 3 we can conclude that children wit a delay in arithmetical facts retrieval in grade 1 already had significant lower scores on two of the six numerical subtests of TEDI-MATH in pre-school. 2.2.3.2. Mental arithmetics. In addition we compared the numerical skills of children average or good in mental arithmetics (≥ pc 50) with children with below-average (≤pc 15) on mental arithmetics measured by a subtest of KRT-R (Baudonck et al., 2006). The MANOVA was significant on the multivariate level (F(6, 59) = 7.567; p ≤ .0005) with an observed Power of 0.99. On the univariate level both groups differed on all dependent variables. For M and SD we refer to Table 4. From Table 4 we conclude that children with a delay in mental arithmetics in grade l already had significant lower scores on the numerical pretests of TEDI-MATH in pre-school. 2.2.3.3. Number knowledge. In addition we compared the numerical skills of children average or good (≥ pc 50) in number knowledge with children with below-average (≤ pc 15) on number knowledge measured by a subtest of KRT-R (Baudonck et al., 2006). The MANOVA was significant on the multivariate level (F(6, 59) = 8.664; p ≤ .0005) with an Observed Power of 0.99. On the univariate level both groups differed on all dependent variables. For M and SD we refer to Table 5. From Table 5 we can conclude that children with a delay in number knowledge in grade l already had significant lower scores on five of the six numerical subtests of TEDI-MATH in preschool.

Table 6 Intercorrelations of the subtests of the TEDI-MATH in preschoolers TM

2

3.1

3.2

4

5.1

5.2

5.3

5

6

1 2 3.1 3.2 4.1 4.2 4.3 4.4 4 5.1 5.2 5.3

0.49⁎⁎ /

0.16 −0.03 /

0.34⁎⁎ 0.17 0.07 /.

0.44⁎⁎ 0.18 0.24⁎ 0.39⁎⁎ 0.79⁎⁎ 0.49⁎⁎ 0.58⁎⁎ 0.72⁎⁎ /

0.36⁎⁎ 0.26⁎ 0.21 0.23⁎ 0.36⁎⁎ 0.27⁎ 0.19 0.49⁎⁎ 0.46⁎⁎ /

0.39⁎⁎ 0.22⁎ 0.31⁎⁎ 0.20 0.33⁎⁎ 0.13 0.23⁎ 0.49⁎⁎ 0.42⁎⁎ 0.45⁎⁎ /

0.35⁎⁎ 0.01 0.23⁎ 0.19 0.41⁎⁎ 0.14 0.30⁎⁎ 0.35⁎⁎ 0.45⁎⁎ 0.36⁎⁎ 0.46⁎⁎ /

0.12 0.18 0.32⁎ 0.18

0.58⁎⁎ 0.06 0.31 − 0.04 0.20⁎ 0.43⁎⁎ 0.19 0.14 0.29 0.10 0.22 0.16

0.46⁎⁎ 0.59 0.88⁎⁎ /

Note. Subtest 1 = Knowledge of the number word sequence, Subtest 2 = Counting sets of items, Subtest 3.1 = Knowledge of the numerical system in Arab numbers, Subtest 3.2. = Knowledge of the numerical system in number words, Subtest 4.1 = Seriation of numbers, Subtest 4.2 = Classification of numbers, Subtest 4.3.Conservation of numbers, Subtest 4.4. = Inclusion of Numbers, Subtest 4 = Logical operation on numbers total score, Subtest 5.1. = Arithmetical operations presented on pictures, Subtest 5.2 = Arithmetical operations presented in arithmetical format, Subtest 5.3 = Arithmetical operations presented in verbal format, Subtest 6 = Estimation of size (comparison of dot sets, subitizing). ⁎⁎p b 0.01, ⁎p b 0.05, two-tailed hypotheses.

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2.2.4. Relationship between knowledge of numerical system and representation of number size The correlations between the pre-numerical parameters (knowledge of number word sequences, counting and logical operations) and the numerical parameters (knowledge of the numerical system, arithmetical operations, estimation of size) were computed. The correlation matrix is presented in Table 6. A lot of facets are significantly correlated, indicating that kinds of competences or sub skills participate to the same general competence, which we call ‘the numerical competence’. However the correlations between the knowledge of the oral numerical system and representation of size are very low, indicating that there are numerical abilities that appear to be less dependent of other abilities. In addition, we computed how many children had a ‘clinical score’ or a b pc 10 on one of the TEDI-MATH subtests as preschool child. We found that 56 of the 82 children (or about 68%) scored above this critical cut off score on all subtests. However 12 children (or about 15%) had a clinical score on one of the TEDI-MATH subtests. Moreover 14 children (or about 17%) had a clinical score on two or three subtests of TEDI-MATH (see Table 7). In addition children without clinical score solved 15.43 (SD = 8.19) mathematical facts and had pc 59.12 (SD = 28.41) on KRT-R Mental arithmetics and pc 62.18 (SD = 29.69) on KRT-R Number knowledge. Children with a clinical score on one subtest solved 12.17 (SD = 6.78) mathematical facts and had pc 36.58 (SD = 29.99) on KRT-R Mental arithmetics and pc 32.92 (SD = 29.29) on KRT-R Number knowledge. Children with two clinical scores solved 10.18 (SD = 9.52) mathematical facts and had pc 18.55 (SD = 27.31) on KRT-R Mental arithmetics and pc 21.09 (SD = 36.36) on KRT-R Number knowledge. Children with three clinical scores solved 7.00 (SD = 4.00) mathematical facts and had pc 4.00 (SD = 3.46) on KRT-R Mental arithmetics and pc 3.67 (SD = 2.31) on KRT-R Number knowledge. 2.3. Conclusions There are a lot of individual differences in pre-numerical and numerical skills in kindergarten. Those differences in children's numerical competence even in preschool were found related to the mathematical skills of young children in the first grade of elementary schools. Table 7 Number of preschool children with below-average scores on the TEDI-MATH Subtests

Number of children

Prenumerical tasks 1. Number word sequence 2. Counting 4. Logical operations Numerical tasks Knowledge of the numerical system 3.1. Arab numerals 3.2. Number words Arithmetical operations 5.1. With pictures 5.2. Arithmetical format 5.3. Verbal format 6. Estimation of size Total number of problem subtests 0 1 2 3 4 5 6 7 8 9

bpc 10 preschool group 0 12 6 bpc 10 / 6 3 / 9 0 0 7 / 56 12 11 (total 23) 3 (total 26) 0 0 0 0 0 0 M = 0.52 SD = 0.86

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Children with average or above average mathematical skills in grade 1 already had higher pre-numerical skills in kindergarten compared with children with below-average mathematical skills in grade 1. Children that do not know the number word sequence and cannot count forward from a lower bound to an upper bound in preschool have delayed mathematical skills in grade 1 of elementary school. Not being able to count linear patterns and random pattern of items and understanding the cardinal in preschool is also a marker of potential problems in mathematics encountered by pupils in first grade. Also not being able to deal with logical operations on number often results in below-average mathematical results in grade l. Children with average or above average skills in mathematics in grade 1 already did not only have good prenumerical skills. They also had higher numerical skills in kindergarten compared with children with below-average mathematical skills in grade 1. Poor mathematics performers in grade 1 already had a delay in knowledge of the numerical system (Arab numerical system, Oral numerical system and knowledge of transcoding), problems with easy arithmetical operations (presented on pictures, presented in arithmetical format, presented in verbal format) and below-average skills to compare dot sets (subitizing) as preschoolers. Children with below average mathematics results in grade 1 appear to have lower knowledge of the oral numerical system and solve less arithmetical operations as preschooler than age-matched children with average or above-average number knowledge in grade 1. Several pre-numerical and numerical components are significantly correlated, indicating that they participate to the same general competence, which we call “the numerical competence”. However for example counting is correlated less significantly with other components than knowledge of number word sequences. Also there is no significant correlation between the knowledge of the Arab numerical system and the knowledge of the oral numerical system in preschool children. In addition, the correlations between the knowledge of the oral numerical system and representation of size are very low, meaning that the assessment of all these facets has to be done independently in order to get a picture of the numerosity of children. Finally, only 68% of preschool children in our dataset have a score of Npc 10 on all facets of numerical competence measured by TEDI-MATH. Fifteen percent of the preschoolers in our dataset fail on one of the subtests, 17% fail on two or three subtests. This means that only if children have 2 or 3 scores below pc 10, early intervention programs can be indicated focusing on numerosity and mathematical skills of young children. However, the more problems encountered as preschooler, the more difficulties children have in grade l to remember arithmetical facts, to execute mental calculation procedures and to acquire a good knowledge of number processing. 3. Study 2 The aims of Study 2 were to examine the features of pupils with a clinical diagnosis of mathematics learning disability. It is investigated if pupils with mathematics learning disabilities encounter as much problems as younger ability-matched children on the pre-numerical and numerical components of mathematical problem solving. It is investigated if the numerical deficits are specific to understand mathematics learning disabilities (MLD) and if all children with MLD have the same impaired amount of pre-numerical and numerical ‘markers’. 3.1. Method 3.1.1. Participants A combination of criteria (interview, testing, observation and teacher rating) were used to include the 30 children in the group of mathematics learning disabilities. All children had to attend a special school for young children with learning disabilities in Flanders. In addition, the children had below critical cut-off (− 2SD) scores on mathematical problem solving tests. In addition, the number of other causes of low mathematical functioning was restricted, since only white native Dutch-speaking children without histories of sensory impairment, brain damage, a chronic medical condition or insufficient instruction were included in this study (exclusion criterion). Moreover, each MLD child was screened for inclusion in the study, with the permission of the parents, based on the following criteria: be in grade 3 and demonstrate an unexplainable (exclusion criterion) but severe and resistant (severeness criterion) ability-achievement discrepancy, even with the usual remediation at school. The discrepancy was based on standardized arithmetical fact (TTR) and/or domain specific mathematics test (KRT3) scores below the 3rd percentile (− 2 SD) (severeness criterion). In addition, inefficient learning strategies (unsystematic planning behaviour or a lack of metacognitive regulating

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behaviour) had to be observed during mathematics testing by a school psychologist or a team of therapists (and confirmed by protocol analyses or videotapes afterwards). Finally, the MLD children had to be rated 1 on mathematics on a 7-point scale according to the teacher. The sample included 30 MLD children (20 boys and 10 girls). In addition 8 of them had comorbide symptoms of ADHD and 15 of them also had low reading or spelling scores. In our sample 17 of the children had a family member (brother, sister of parent) with learning disabilities. All the children were previously diagnosed with mathematics learning disabilities by the school psychologist. A profound understanding of the numerosity development of children with MLD requires age-related and abilityrelated comparisons (Torbeyns, Verschaffel, & Ghesquière, 2004). Since the immature strategy characteristics of MLD children at older ages might reflect their immature mathematical ability level, and thus can be explained by the delayhypothesis, such a combined design is used. The 30 age-matched children from grade 3 (GR3) and the 30 abilitymatched children from grade 2 (GR2) were gender-, SES- and birth order-matched with the MLD-group. The GR2 and GR3 pupils also had to be rated 4 on mathematics on a 7-point scale in the teacher questionnaire. In addition all the elementary school children used the same mathematics handbook in all groups. 3.1.2. Measures The TEDI-MATH (Grégoire et al., 2004) and the KRT-R (Baudonck et al., 2006) and TTR (De Vos, 1992) were used to measure (pre)numerical competence, as described in Study l. 3.1.3. Design and procedure This study is part of a longer study were a larger group of children will be assessed to look for subtypes in mathematical learning disabilities. In this study we discuss the results of children's numerical competence in the middle of elementary school. All subjects completed the KRT-R (Baudonck et al., 2006) and the TTR (De Vos, 1992) on the same day for about 1 h in total. In addition all children were also assessed with TEDI-MATH (Grégoire et al., 2004). 3.2. Results 3.2.1. Descriptive results The children with mathematical learning disabilities had a mean age of 8.9 year (SD = 4 months). The standardized mean percentile scores on the KRT-R Mental Arithmetics compared with grade 2 and KRT-R Number Knowledge compared with grade 2 were pc 36.33 (SD = 28.75; min1–max 93) and pc 27.07 (SD = 15.68; min 4–68). Children solved 48.07 (SD = 15.40; min 22–max79) arithmetical facts within 2 min. The children had a mean VIQ of 92.73 (SD = 10.41; min 77–max 124) and a mean PIQ of 92.20 (SD = 9.67; min 74–max 120) on the WISC-III. 3.2.2. Skills of children with mathematical learning disabilities In order to answer our research questions on the relation between pre-numerical skills and mathematics, a multivariate analysis of variance (MANOVA) was conducted with knowledge of number word sequence, counting and logical

Table 8 TEDI-MATH subtest scores Tasks

GR 2 M (SD)

GR 3 M (SD)

MLD M (SD)

Prenumerical tasks 1. Number word sequence 2. Counting 4. Logical operations

12.20b (1.09) 12.17b (1.44) 14.10b (2.86)

13.70a (0.65) 12.67a (0.71) 17.87a (0.35)

12.63b (0.93) 11.80b (1.27) 15.80b (2.62)

F (2, 87) 21.58⁎⁎ 4.06⁎ 21.16⁎⁎

Numerical tasks 3. Knowledge of the numerical system 5. Arithmetical operations 6. Estimation of size

116.67c (18.00) 55.30b (11.03) 16.97b (1.06)

166.20a (4.07) 76.90a (2.84) 17.90a (0.30)

135.77b (15.57) 60.40b (9.33) 16.93b (1,.01)

F (2,87) 96.33⁎⁎ 52.91⁎⁎ 11.99⁎⁎

Note. ⁎p b .05 ⁎⁎p b .0005. abc refer to post hoc differences on the .05 level.

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operations on numbers, as measured by TEDI-MATH, as dependent variables and belonging to mathematical ability groups (MLD, GR3, GR2) as a factor. A power of 1.00 was observed. The MANOVA was significant (F(6, 170) = 11.205, p ≤ .0005 on the multivariate level. On the univariate level, a significant difference was found between the groups for all dependent variables. For M and SD and posthoc indices we refer to Table 8. Children with mathematics learning disabilities (MLD) in grade 3 had the same pre-numerical skills as abilitymatched younger children. In addition MLD-children had the same calculation and estimation skills as ability-matched younger children. Furthermore the number of children with a ≤pc 10 score on number word sequences, counting and logical operations was 4, 4 and 6 respectively (see Table 9). In order to look for numerical differences a second multivariate analysis of variance (MANOVA) was conducted with knowledge of the numerical system, solving arithmetic operations and estimation of size, as measured by TEDIMATH, as dependent variables and belonging to mathematical ability groups (MLD, GR3, GR2) as a factor. A power of 1.00 was observed. The MANOVA was significant (F(6, 170) = 25.556, p ≤ .0005 on the multivariate level. On the univariate level, a significant difference was found between the groups for all dependent variables. For M and SD and posthoc indices we refer to Table 8. Children with mathematics learning disabilities in grade 3 had the same subitizing and estimation skills as ability-matched children. However they had more knowledge of the numerical system than ability-matched younger children. Children with mathematics learning disabilities also solved as many arithmetical operations than younger children with the same ability. For the number of children in the MLD-group with a ≤ pc 10 score we refer to Table 9.

Table 9 Number of MLD children in grade 3 with below-average scores on the TEDI-MATH Subtests

Number of children

Prenumerical tasks 1. Number word sequence 2. Counting 4. Logical operations Numerical tasks 3. Knowledge of the numerical system 3.1. Arab numerals 3.2. Number words 3.3. Base 10 system 3.4. Translation 5. Arithmetical operations 5.1. With pictures 5.2. Arithmetical format 5.3. Verbal format 5.4. Conceptual knowledge 6. Estimation of size Total number of problem subtests 0 1 2 3 4 5 6 7 8 9 10 11 12

bpc 10 MLD group 4 4 6 bpc 10 22 15 22 9 26 21 0 21 5 20 10 / 0 3 3 (total 6) 3 (total 9) 3 (total 12) 5 (total 17) 7 (total 24) 4 (total 28) 1 (total 29) 1 (total 30) 0 0 0 M = 4.73 (SD = 2.16)

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All MLD-children and none of the third graders without MLD had a below ≤ pc 10 score on at least one of the TEDI-MATH, meaning that there were no false positive or false negative scores in this dataset. In general the MLDchildren had a below ≤ pc 10 score on about five (M = 4.73; SD = 2.16; with a min of 1 and max of 9) of the twelve subtests of TEDI-MATH. More than half of the MLD children had ≤pc 10 scores on the knowledge of Number words, translation, calculations in formula format and on conceptual knowledge. Less than or about 25% of the MLD group had pre-numerical problems or problems in verbal formats. 3.3. Conclusions There are a lot of individual differences in pre-numerical and numerical skills between different age groups from kindergarten up to grade 3. In our dataset, children with mathematical learning disabilities (MLD) in grade 3 have similar pre-numerical skills than ability-matched children without learning disabilities. The MLD-pupils also have similar calculation and estimation skills than ability-matched children without learning disabilities. However, MLDchildren in our dataset have more knowledge of the numerical system than ability-matched younger children without learning disabilities. In addition, all MLD children fail in at least one of the subtests of the TEDI-MATH. Most children fail in about 5 basic mathematical skills. This might point to the fact that there is a broad spectrum of MLD disabilities, as already described by Fuchs and Fuchs (2002) and Kronenberger and Dunn (2003). 4. General discussion Most practitioners make a diagnosis of mathematical learning disabilities based on observational measures and criterion-based tests. Those tests (such as KRT-R (Baudonck et al., 2006)) check if the age-appropriate goals for mathematical education are reached. However, a good assessment of a mathematical disability has to be based on a validated model on mathematical problem solving and provide children, school, parents and practitioners with a solid base for remediation. In contrast to the criterion-based tests, an assessment with the TEDI-MATH (Grégoire et al., 2004) results in a profile of the pre-numerical and numerical strengths and weaknesses of the child, providing practitioners with a more solid base for remediation. In a longitudinal study 82 children were assessed in kindergarten and one year later in grade l. A feature of pupils encountering problems in the area of arithmetics in grade l was the below-average pre-numerical competence. Children ‘at risk’ in grade l struggled in preschool already with the knowledge of the number word sequence, encountered problems with counting as tool to know the cardinal of a set and had a delay in the logical abilities (classification, seriation etc.) compared with peers good in arithmetic. A second feature of pupils poor in arithmetics in grade l is the difficulty to execute simple calculation procedures even as pre-school child. Children with a delay in arithmetics in grade l were found to have a below-average score on calculation tasks in preschool, even when operations were presented with the aid of pictures. This might point to the fact that those children had processing difficulties even in preschool. Mc Lean and Hitch (1999) see this as a consequence of deficits of working memory. Geary (2003) attributes procedural problems to a lack of conceptual understanding and retrieval difficulties to a general semantic memory dysfunction. This is in line, with the belowaverage performance on calculation tasks presented in verbal format in preschool for children with a delay in mental arithmetics or number knowledge in grade l. No such pre-school differences were found between children with poor or good retrieval skills in grade l. Thirdly pupils poor in arithmetics in grade l already had below-average knowledge of the numerical system in preschool. There were more differences between good and poor pupils in grade 1 on the oral numerical system than on the Arab numerical system. In addition the knowledge of Arab and Oral numerals in pre-school did not differ between children in grade l that were fast or slow in remembering arithmetical facts. This might mean that number knowledge is not an underlying deficit in retrieval difficulties. Fourthly a generally well-recognised feature of pupils with mathematics learning disabilities is difficulty in learning and remembering arithmetical facts (Geary, 2004). In our dataset children with a slow speed of fact-retrieval in grade 1 had below average pre-numerical skills as preschooler and did worse on the calculation tasks with pictures and easy tasks presented in arithmetical format. Those children did not significantly worse on numerical system knowledge, calculations presented in verbal format and in subitizing tasks. This might point to the fact that the ‘markers’ for

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retrieval deficits are different than those for other problems in arithmetics and eventually to subtypes of children with mathematical learning problems. Fifthly, it is investigated if pupils poor in arithmetics in grade l already struggled as pre-school child with subitizing tasks. In our dataset children with a delay in mental arithmetics or number knowledge did significantly worse on subitizing and estimation of size than pupils that were better in arithmetics in grade 1. However, again no differences were found on those tasks in children with and without difficulties remembering arithmetical facts. Finally in study l only 68% of the preschool children had a score above pc 10 on all subtests of TEDIMATH. A lot of children still had problems with counting and logical operations compared with age-matched peers. Also there were a lot of children that had a very low score on the knowledge of Arab numerals and Number words in grade 3 or much problems with easy calculations on pictures. Only if children fail on 2 or more subtests without explanation this might point in the direction of mathematical learning disabilities later on. In accordance with the results in Study 1, the 30 children with mathematical learning disabilities (MLD) in Study 2 had remarkable differences in their pre-numerical and numerical facets of numerosity. A feature of about 66% of the pupils with mathematical learning disabilities is the difficulty to execute calculation procedures especially in arithmetical format. In line with Geary (2004), all most all of those children had also a lack of conceptual knowledge and understanding. In line with study 1, there were more children with mathematical learning disabilities in grade 3 with a below-critical cut off- score on the oral numerical system than on the Arab numerical system. In addition, 33% of the pupils with mathematical learning disabilities in grade 3 still struggled with subitizing tasks. However, again no below critical cut off-scores were found on those tasks in 67% of the MLD-children, meaning that this probably is not the unique explaining factor for MLD. Finally, all MLD-children in grade 3 had at least one score below pc 10 on the subtests of TEDI-MATH, where none of the age-matched children in grade 3 failed in those subtests. In average the MLD-children failed in about 5 out of the 12 subtests. The MLD-children in grade 3 however did not differ ‘as group’ in the knowledge of number word sequences and in the pre-numerical skill to count compared with ability-matched, but younger peers. The MLDchildren also did as well as the younger children in seriation, classification, conservation and inclusion skills. More than half of the MLD children had a below critical cut off-scores on the knowledge of number words, translation, calculations in formula format and on conceptual knowledge. Less than or about 25% of the MLD group had prenumerical problems or problems in verbal formats. These results should be interpreted with care since the numerosity skills might involve different cognitive skills and might be age-dependent and still maturing. In addition, the facets of numerical competence still need a full explanation from more applied longitudinal research on different age-, mathematics- and intelligence-groups. To exclude alternative possible explanations, our studies need to be replicated with larger samples. It would also be useful to compare the pre-numerical and numerical skills in children with specific mathematics learning disabilities (and intact reading skills) and in children with specific reading disabilities (and intact mathematical problem solving skills) and to investigate the modifiability of numerosity markers in preschool children ad risk. Such studies are currently being prepared. Summarizing, our studies suggest that children with a delay in arithmetics in grade l already encountered problems on numerosity in kindergarten. Moreover some evidence was found for the independence and dissociation of numerical abilities in children with a clinical diagnosis of mathematical learning disabilities (MLD) in grade 3. About 13% of the MLD-children still had pre-numerical processing deficits (in number sequence production, cardinality skills and logical knowledge) in grade 3. About 67% of the MLD-children in grade 3 had difficulties in executing calculation procedures and a lack of conceptual knowledge. A feature of 87% of the MLD-children was translation deficits, with a significantly worse knowledge of number words compared with the knowledge of Arab numerals. Finally a deficit in subitizing was found to be present in 33% of the MLD children. On a group level the processing deficits were linked to understanding numerosity, since the ability-matched younger children and the MLD-children had the same pre-numerical and numerical profile. Furthermore, findings from these studies support the use and importance of the TEDI-MATH assessment procedure. Taking into account the complex nature of mathematical problem solving, it may be useful to assess counting procedures, logical knowledge, knowledge of the numerical system, computation and representation of number size (numerosity) in young children at risk and in children with

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mathematical learning disabilities in order to focus on these factors and their role in mathematics learning and development.

Appendix A Subtests and examples of test items of the TEDI-MATH® Subtests

Items

1. Knowledge of the numer–word sequence Counting as far as possible Counting forward to an upper bound (e.g. “up to 9”) Counting forward from a lower bound (e.g. “from 7”) Counting forward from a lower bound to an upper bound (e.g. “from 4 up to 8”) Count backward Count by step (by 2 and by 10) 2. Counting sets of items Counting linear pattern of items Counting random pattern of items Counting a heterogeneous set of items Understanding of the cardinal 3. Knowledge of the numerical system 3.1. Arab numerical system Judge if a written symbol is a Number Which of two written numers is the larger 3.2. Oral numerical system Judge if a word is a number Judge if a number word is syntaxically correct Which of two numbers is the larger 3.3. Base-ten system Representation of numbers with Sticks Representation of numbers with Coins Recognition of hundreds, tens and units in written numbers 3.4. Transcoding Write in Arab code a dictated number Read a number written in Arab code 4. Logical operations on numbers 4.1. Seriation of numbers Sort the cards from the one with fewer trees to the one with the most trees 4.2. Classification of numbers Make groups with the cards that go together 4.3.Conservation of numbers Do you have more counters than me? Do I have more counters than you? Or do we have the same number of counters? Why? 4.4. Inclusion of numbers You put 6 counters in the envelope. Are there enough counters inside the envelope if you want to take out 8 of them? Why? 4.5. Additive decomposition of numbers A shepherd had 6 sheep. He put 4 sheep in the first prairie, and 2 in the other one. In what other way could he put his sheep in the two prairies? 5. Arithmetical operations 5.1. Presented on pictures There are 2 red balloons and 3 blue balloons. How many balloons are there in all? 5.2. Presented in arithmetical format Addition (e.g.: “6 + 3”; “5+.. = 9”, “..+3 = 6”) Subtraction (e.g.: “9 − 5”, “9−… = 1”, “…−2 = 3”) Multiplication (e.g.: “2 × 4” “10 × 2”) 5.3. Presented in verbal format E.g. “Denis had 2 marbles. He won two others. How many marbles had Denis in all?” 5.4. Understanding arithmetical operation properties (conditional knowledge) e.g.: addition commutativity “You know that 29 + 66 = 95. Would this information help you to compute 66 + 29? Why?” 6. Estimation of the size 6.1. Comparison of dot sets (subitizing) 6.2. Estimation of size Comparision of distance between numbers. e.g.: target number is 5. What number is closed to this (3 or 9)?

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