Off-line metacognition--a domain-specific retardation in young children with learning disabilities?

Abstract. Off-line metacognition (prediction and evaluation) was assessed in 437 normally intelligent children with or without learning disabilities in grades 2 and 3. Children with specific mathematics learning disabilities were compared with peers with specific reading disabilities, children with combined learning disabilities, age-matched peers and younger children matched at mathematical problem-solving level. Our results indicate that offline metacognition cannot be reduced to a demonstration of intelligence. Moreover, the off-line metacognitive scores of children with reading disabilities were comparable to those of age-matched peers without learning disabilities. Furthermore, significantly lower prediction and evaluation scores were found for children with specific or combined mathematics learning disabilities compared with age-matched peers. In addition, our data showed a different metacognitive profile for children with specific or combined mathematics learning disabilities, not comparable on all aspects to the profile of younger children, as suggested by the retardation or maturational-lag hypothesis. The educational implications of these results are discussed.

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Flavell introduced the concept of metacognition in 1976. He defined metacognition as the knowledge and active monitoring of one's own cognitive processes. Metacognition has become a general multidimensional construct enabling learners to adjust to varying tasks, demands and contexts (e.g., Greeno & Riley, 1987; Hutchinson, 1992; Montague, 1996, 1997; Wong, 1987). Moreover, metacognition is currently often used in an overinclusive way, including motivational and affective constructs (Boekaerts, 1999). Despite the emphasis on metacognition, many metacognitive concepts are interpreted differently by various researchers and include a wide range of phenomena. We will therefore start with a definition of the concepts, to avoid misunderstanding.

Metacognition has traditionally been differentiated into two central components, namely, metacognitive knowledge and metacognitive processes (Lucangeli, Galderisi, & Cornoldi, 1995). Metacognitive knowledge can be described as the knowledge, awareness, and deeper understanding of one's own cognitive processes and products (Flavell, 1976). In addition, metacognitive processes or "skills" can be seen as the voluntary' control people have of their cognitive processes (Brown, 1980).

One of the metacognitive skills is prediction. Prediction enables children to think about the learning objectives, proper learning characteristics and the available time. Moreover, children estimate or predict the difficulty of a task and use that prediction metacognitively to regulate their engagement related to outcome and efficacy expectation (Winne, 1997). A number of studies have dealt with the importance of prospective prediction skills in mathematics (e.g., Lucangeli & Cornoldi, 1997). Cornoldi (1998) showed that cognition is affected by predictions, which precede and are triggered by a specific task. The ability to predict enables children to foresee task difficulties and makes them work slowly on difficult tasks and more quickly on easier tasks. In addition, prediction makes children relate certain problems to other problems, develop intuition about the prerequisites required for doing a task and distinguish between apparent and real difficulties in mathematical problem solving (Lucangeli, Cornoldi, & Tellarini, 1998).

Another metacognitive skill, the evaluation skill, can be defined as the retrospective reflections that take place after an event has transpired (Brown, 1987), whereby children look at what strategies were used and whether or not they led to a desired result. Specifically, children reflect on the outcome and the understanding of the problem and the appropriateness of the plan, the execution of the solution method as well as on the adequacy of the answer within the context of the problem (Garofalo & Lester, 1985; Vermeer, 1997). Evaluation makes children evaluate their performance and compare task performance with people and use the final result in locating the error in the solution process (Lucangeli et al., 1998).

In this article we restrict prediction to predicting whether or not children are likely to solve a particular problem. Similarly, evaluation in this context is restricted to the outcome evaluation or to judgment of how well children did in the absence of feedback. Since prediction and evaluation are measured before or after the solving of exercises, we labeled them off-line (measured) metacognition, in contrast to on-line (measured) metacognitive skills, such as planning and monitoring. Off-line metacognition differentiated between average and above-average mathematical problem solvers and between students with a mathematics learning disability (Desoete, Roeyers, & Buysse, 2001).

Prediction and evaluation are related to concepts such as calibration, feeling of knowing, judgments of learning and the research on metacognitive knowledge monitoring assessment and the feelings of difficulty. Calibration can be defined in terms of whether the predicted value assigned to a single item is followed by the occurrence of that value on the criterion test. A comparison is made of whether the prediction before a task corresponds to the actual performance on the task (Nelson, 1996a). Some children know they know, others have the illusion of knowing, while yet others know they don't know and a last group does not know they don't know. The feeling of knowing (FOK) is "a rating made by people about the probability that they will be able to recognize an element of information" (Lories, Dardenne, & Yzerbyt, 1998, p. 7). Nhouyvanisvong and Reder (1998) reviewed various paradigms to clarify the FOK. They found that the judgments preceding execution of question-answering strategies (preretrieval FOK) were part of a more general process occurring automatically when a question is asked to help regulate strategy selection and operating. Judgments of learning (JOL) occur during or after acquisition and are predictors of future test performance on currently recallable items (Nelson, 1992, 1996b; Nelson & Narens, 1990; Reder & Ritter, 1992).

Tobias and Everson (1996) developed the Metacognitive Knowledge Monitoring Assessment (KMA) to assess what students think they know or do not know (what we call prediction) and what they really know and do not know. This relationship is analyzed in four scores (predicted score + real score +, predicted score + real score -, predicted score - real score -, predicted score - real score +). Correct knowledge monitoring is seen in the correspondence between the real scores and the predicted scores. This research design is very similar to the one we used. Furthermore, the study of Efklides, Papadaki, Papantoniou and Kiosseoglou (1997) on the "feelings of difficulty" is also related to our study on prediction and evaluation. Their feeling of difficulty is "the subjective experiences of task complexity" assessed on a 4-point rating scale (p. 233).

From a developmental point of view, metacognitive knowledge precedes metacognitive skills (Flavell, 1979). In school-aged children, metacognitive knowledge grows through the development of a strong conceptual knowledge base, domain-specific strategies and perturbation, resulting in the accommodation of schemes at higher levels of abstraction (Carr & Biddlecomb, 1998). Around the age of 9 to 10 years, metacognitive knowledge becomes a comprehensive theory and expands through reflection on one's own learning and the learning of others (Berk, 1997). In addition, metacognitive knowledge expands using efficient metacognitive skills (Carr, Alexander, & Folds-Bennett, 1994). Metacognitive skills have been found to be maturing until adolescence (Berk, 1997).

The metacognitive research on reading peaked in the 1980s (e.g., Jacobs & Paris, 1987) and has plateaued since (Wong, 1991, 1996). Metacognition has more recently also been applied to mathematics (e.g., Borkowski, 1992; Hacker, Dunlosky, & Graesser, 1998; Schoenfeld, 1992; Vermeer, 1997). Studies of problem-solving strategies in mathematically average-performing children have shown that metacognition is instrumental during the initial stage of mathematical problem solving, as well as in the final stage of interpretation and checking the outcome of the calculations (Verschaffel, 1999). Metacognition was also found to be important when the task demands challenge the child but do not overtax existing skills (Carr et al., 1994). Numerical and geometrical problem-solving abilities in particular were found to be strongly related to metacognitive skills, whereas this relation was only present for some children in arithmetic performance tasks (Lucangeli et al., 1998). Nevertheless, some authors remain skeptical about the importance of metacognition in young children (e.g., Siegler, 1989).

Children with mathematics learning disabilities were found to have less developed metacognitive knowledge or awareness and poorer metacognitive skills (Lucangeli & Cornoldi, 1997; Lucangeli et al., 1998). These children also verbalized fewer of those skills (Montague, 1998). In addition, it has recently been proposed that children with mathematics learning disabilities have different metacognitive beliefs than children with good mathematical performance (Lucangeli et al., 1998). Furthermore, children with reading learning disabilities were found to be weaker at integrating metacognition with on-line processing and problem solution than peers without disabilities (Swanson, 1993).

Although a certain consensus has been reached that metacognition has an important effect on students' achievement (Garcia & Pintrich, 1994; Metcalfe, 1998; Verschaffel, 1999; Wong, 1996), some questions remain. One of these questions considers the relationship between metacognition and intelligence. This relationship is hotly disputed. Brown (1978) and Sternberg (1979, 1985) conceptualized metacognitive skills as demonstrations of intelligence and as a part of the cognitive repertoire. Swanson's (1990) independency model, on the other hand, viewed intelligence and metacognition as two separate entities, where metacognitive skills could compensate for low intelligence scores. Furthermore, empirical evidence was found for the model, hypothesizing an interaction between metacognition and intelligence as well as an additional value of metacognition in the explanation of learning (e.g., Demetriou, Gustafsson, Efklides, & Platsidou, 1992; Sleife, Weiss, & Bell, 1985).

Another question is whether low metacognitive scores are to be considered as demonstrations of a maturational lag or retardation rather than as a deficit in children with learning disabilities. According to Wong (1996), the assumption that students with learning disabilities lack metacognitive skills is invalid. Instead these children appear to have less sophisticated metacognitive skills than peers without learning disabilities. Furthermore, low metacognitive scores in children with learning disabilities are considered by Borkowski and Thorpe (1994) to be the result of insufficient maturity in the development of the regulation of mathematical cognition. In this case metacognitive differences between children with and without learning disabilities can be explained according to the maturation lag or retardation hypothesis.

Another possible explanation is the deficit hypothesis, whereby metacognition is considered a deficit in children with learning disabilities (Geary, 1993). In the case of the deficit hypothesis, children with learning disabilities would have different or disharmonically developed metacognitive knowledge and skills, not comparable to the skills and knowledge of younger children matched on mathematical performance level. Davidson and Freebody (1986) found the deficit hypothesis incapable of explaining some of their research data.

Another question is whether metacognition is domain-specific or a more general phenomenon. Some authors regard metacognition as higher-order skills, affecting performance in a variety of academic areas and therefore as more general skills. In such cases, metacognitive components may seem pervasive across situations, and work interactively (Montague, 1996, 1997). The findings of Schraw, Dunkle, Bendixen, and De Backer Roedel (1995) support this domain-general hypothesis. On the other hand, much of the work on expert problem solving is consistent with the domain-specific hypothesis (Bereiter & Scardamalia, 1993). Expert problem solvers were found to be able to assess and update their mental representations in familiar domains, but to be no more able than novices at using these metacognitive skills in unfamiliar ones (Davidson & Sternberg, 1998). We refer to Perkins and Salomon (1989) for a comparison of the domain-specific and domain-general views.

In summary, much research on metacognition has yielded inconsistent results in younger children (e.g., Siegler, 1989). Furthermore, the debate on the relationship between metacognition and intelligence, the maturational-lag and domain-specificity hypothesis remains unresolved. In addition, although authors do agree that an operational definition of learning disabilities is meaningful in order to differentiate children with learning disabilities from children with mental retardation, and to make study more comparable (e.g., Kavale & Forness, 2000; Swanson, 2000), most studies do not differentiate between children with specific mathematics learning disabilities (MD), specific reading disabilities (RD) and children with combined reading and mathematics learning disabilities (MD+). Such differentiation nevertheless seems necessary, since over time a number of authors have shown that children with mathematics learning disabilities are a heterogeneous group (Ostad, 1998). For example, different neuropsychological profiles have even been found (e.g., McCloskey & Macaruso, 1995; Rourke, 1993).

THE PRESENT STUDY

Aim and Research Questions

The present study was designed to examine three differences between children without learning disabilities and children with specific or combined mathematics learning disabilities regarding off-line metacognition.

First, it was designed to show the relationship between mathematics, off-line metacognition and intelligence in young children. Specifically, we wanted to investigate Swanson's independency model in average-intelligent children in grade 3.

The second purpose of the study was to investigate the retardation or maturational-lag hypothesis, or to test the hypothesis that children with mathematics learning disabilities primarily show immature off-line metacognitive skills, comparable with mathematically average-performing younger children. Congruently with the retardation hypothesis, we could expect the same prediction and evaluation skills in children with specific mathematics learning disabilities, combined learning disabilities and in younger children matched at mathematical performance level.

Most studies end here; however, we wanted to perform two additional analyses. First, we wanted to investigate whether children with specific or combined mathematics learning disabilities in grade 3 also have more problems with prediction and evaluation on so-called easy tasks (or mathematical problem-solving tasks designed for children in grade 1 [[P.sub.1] and [E.sub.1]] or grade 2 [[P.sub.2] and [E.sub.2]]). We could hypothesize that since the recruited children with specific or combined mathematics learning disabilities have the same mathematical skills as children in grade 2, they would also have comparable prediction and evaluation skills. Second, we wanted to compare prediction and evaluation skills on different cognitive problem-solving tasks (numeral and operation symbol comprehension, number system knowledge, mental arithmetic, procedural calculation and word problems) in all children. According to the retardation or maturational-lag hypothesis, we could expect the same results for children with specific or combined mathematics learning disabilities and younger children on prediction about numeral and operational symbol comprehension ([P.sub.NR+0]), prediction about number system knowledge ([P.sub.K]), prediction about mental arithmetical problem solving ([P.sub.M]), prediction about procedural calculation ([P.sub.P]) and prediction about the solving of word problems ([P.sub.W]). Moreover, we could expect the same results for children with specific or combined mathematics learning disabilities and younger children on evaluation about numeral and operational symbol comprehension ([E.sub.NR+0]), evaluation about number system knowledge ([E.sub.K]), evaluation about mental arithmetical problem solving ([E.sub.M]), evaluation about procedural calculation ([E.sub.P]) and evaluation about the solving of word problems ([E.sub.W]). For the sake of completeness, we also compared predictions and evaluations on so-called difficult tasks ([P.sub.4] and [E.sub.4]), or tasks designed for fourth graders, and expected a similar pattern.

Furthermore, with Brown (1987, p. 107) we are interested in answering a critical question about metacognition, "Is it general or domain-specific?" In order to add some data to this debate, mathematically average-performing third graders (MA3) were compared with age-matched children with reading disabilities (RD) on off-line metacognition during mathematical problem solving. We hypothesized domain-specific metacognitive problems and low off-line metacognitive skills in children with specific mathematics learning disabilities (MD) and in children with combined mathematics and reading disabilities (MD+), but no such problems in children with reading disabilities (RD) solving mathematical tasks.

Method

Participants. The participants in this investigation consisted of third-grade (MD, RD, MD+) children referred by psychologists of multidisciplinary rehabilitation centers, teachers at schools for special education or paraprofessionals treating children with learning disabilities because of significantly below-grade-level mathematics and/or reading achievement.

Each referred child was screened for inclusion in the study with the permission of the parents, based on the following criteria: (a) The average intelligence had to be 90 < TIQ < 120. Furthermore, the participants had to demonstrate an ability-achievement discrepancy based on their total IQ and total standardized achievement test scores. Scores had to be below the 3rd percentile on frequently used tests on mathematics for the MD and MD+ children, and below the 3rd percentile on reading tests for the RD and MD+ group of children. The performance level of all children was at least one year below grade level according to the school psychologist. (b) To be accepted in our sample as children with learning disabilities (MD, RD, and MD+), the diagnosis had to be acknowledged and inefficient learning strategies had to be detected by a school psychologist or a team of therapists. (c) In addition, only white native Dutch-speaking children without histories of extreme hyperactivity, sensory impairment, brain damage, a chronic medical condition, insufficient instruction or serious emotional or behavioral disturbance were included. The final sample consisted of 62 MD children (29 boys and 33 girls), 53 RD children (30 boys and 23 girls) and 72 MD+ children (40 boys and 32 girls).

Two control groups (MA2, MA3) were included in the contrastive analysis in order to be able to investigate the domain-specificity hypothesis (and to compare RD with MA3) and the maturational-lag hypothesis (and to compare MD and MD+ with MA2).

The first control group (MA3) consisted of 130 (70 boys and 60 girls) average-intelligent third graders (ages 8-9) without a diagnosis of learning disability or other problems. Sixty of these children were matched with the children with mathematics learning disabilities (MD); 70 of these children were matched with the children with combined learning disabilities (MD+), based upon not more than a one-week difference in date of birth.

The second control group (MA2) consisted of 120 (52 boys and 68 girls) average-intelligent second-grade students (ages 7-8) without a diagnosis of learning disability or other problems. The sample was drawn at random, with the permission of the children's parents, from regular elementary classes. The matching was based on their mathematical problem-solving skills. For this purpose, children with mathematics learning disabilities in grade 3 (MD and MD+) and the group of young children in grade 2 (MA2) performed two tests on domain-specific mathematical knowledge for grades 2 and 3. Only children in grade 2 were accepted in this study if they could be matched with a child with mathematics learning disabilities and had less than 2 points of difference in performance scores on both tests (Kortrijk Arithmetic Test Grade 2 and Grade 3; Cracco, Baudonck, Debusschere, Dewulf, Samyn, & Vercaemst, 1995) compared with children with mathematics learning disabilities. Based upon these criteria, 55 children in grade 2 were matched with the children in grade 3 with specific mathematics learning disabilities (MA), and 65 children in grade 2 were matched with the children in grade 3 with combined mathematics and reading disabilities (MA+).

Participants in both control groups (MA2, MA3) were native Dutch-speaking Belgian children, with average intelligence (90 < TIQ < 120) and an overall school result of at least level B (60%). At the time of the testing, the third-grade subjects (MA3, MD, RD and MD+) had a mean age of 101.18 months (SD = 4.56 months), whereas the second graders had a mean age of 88.76 months (SD = 5.52). Furthermore, the final sample had a mean TIQ of 102.11 (SD = 6.86), a mean VIQ of 101.93 (SD = 6.77) and a mean PIQ of 101.74 (SD = 9.10).

Measures. The Kortrijk Arithmetic Test (Kortrijkse Rekentest, KRT) (Cracco et al. 1995) is a Belgian mathematics test of mental computation (e.g., 129 + 879 = --) and of number system knowledge (e.g., add three tens to 61 and you have --). Children have to read the instruction and write down the answer to 60 mathematical tasks within 45 minutes. The psychometric value has been demonstrated on a sample of 3,246 Dutch-speaking children. In all groups (MA2, MA3, MD, RD, MD+), the standardized total percentile based on Dutch norms was used. The version for grade 2 was used for MA2; the version for grade 3 was used for MA3, MD, RD, and MD+ children. In addition, the children in grade 2 also carried out the version for grade 3 and the children with mathematics learning disabilities (MD and MD+) also carried out the version for grade 2, in order to make matching possible.

The One Minute Test (Een Minuut Test, EMT; Brus & Voeten, 1999) is a test of reading fluency for Dutch-speaking people, validated for Flanders on 10,059 children (Ghesquiere & Ruijssenaars, 1994), measuring the capacity of children to read correctly as many words as possible. All children (MA2, MA3, MD, RD, MD+) were given one minute to read as many words as possible out of the same 116 words.

The intelligence of all children was measured, using Total IQ, since this seems to be the most reliable basis for documenting an ability-achievement discrepancy (Kavale & Forness, 2000). Furthermore, since the WISC-III was not available in Belgium, the WISC-R (Wechsler et al., 1986) was used. The psychometric value of the WISC-R is good and data for Flanders are available. In addition, since IQ is likely to be overestimated based on the WISC-R (Flynn, 1998; Gaskill, Frank, & Brantley, 1996; Lyon, 1995), a cutoff of 90 (pc 25) instead of 85 was used for normal intelligence.

The Evaluation and Prediction Assessment (EPA2000; De Clercq, Desoete, & Roeyers, 2000; Desoete, Roeyers, Buysse, & De Clercq, 2000, in print) consists of three parts (metacognitive prediction, mathematical problem solving, metacognitive evaluation). Children have to predict and evaluate on 80 mathematical problem-solving tasks, including tasks at grade 1, 2, 3 and 4. EPA2000 includes tasks on the comprehension of numbers and operation symbols (NR- and O-tasks) (e.g., put into the right order from low to high 39 37 38 40); number system knowledge (K-tasks) (e.g., complete this series 37 38 39 --); mental arithmetic (M-tasks) (e.g., 37 + 1 = --); procedural arithmetic (P tasks) (e.g., 37 + 653 = --); and word problems (W-tasks) (e.g., William wants to buy three cars. Two cars cost 1 euro. How long must William save? Choose between "till he has 6 euro," "till he has 3 euro," "till he has 2 euro," "till he has 1 euro"). In the measurement of prediction, children are asked to look at exercises without solving them and to predict on a 4-point rating scale whether they will be successful in this task. Children have to evaluate after solving the mathematical problem-solving task on the same 4-point rating scale. In EPA2000, children have to comprehend the instruction (with assistance for the reading aspect for RD and MD+ children) and to click on the answer with the mouse. All children (MA2, MA3, MD, RD, MD+) solved the same exercises. With EPA2000 both the accuracy in problem solving and the accuracy of predictions and evaluations are scored. Children can give four ratings (1 = absolutely sure I am wrong, 2 = sure I am wrong, 3 = sure I am correct, 4 = absolutely sure I am correct). Metacognitive predictions or evaluations are awarded 2 points whenever they correspond to the child's actual performance on the task (predicting or evaluating 1 and doing the exercise wrong and rating 4 and doing the exercise correctly). Predicting and evaluating, ratings 1 or 3, receive 1 point whenever they correspond. Other answers do not gain any points, as they are considered to represent a lack of off-line metacognition. For the mathematical problem solving, children obtain 1 point for every correct answer. The three scores (prediction, mathematical problem solving and evaluation) are unrelated. For instance, in theory a child can obtain maximum scores for prediction, a zero score for mathematics and a medium score for evaluation. The psychometric value has been demonstrated on a sample of 550 Dutch-speaking children (Desoete, Roeyers, & De Clercq, 2001, 2002). To examine the psychometric characteristics of the EPA2000 in this study, Cronbach's alpha reliability analyses were conducted. For prediction, mathematical cognition and evaluation, Cronbach's [alpha] of .74, .89 and .85, respectively, was found for the total test (80 items). For prediction and evaluation subscores for the different grades and for the different kinds of mathematical problem solving, tasks were computed on 100 points (see Table 1).

Data Collection

All subjects were assessed individually, outside the classroom setting, where they completed the KRT (Cracco et al., 1995), EMT (Brus & Voeten, 1999) and the EPA2000 (De Clercq et al., 2000) on two different days, for about two hours in total. The examiners, all psychologists or therapists skilled in learning disabilities, received practical and theoretical training in the assessment and interpretation of mathematics, reading and metacognition. 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 15 children with and without learning disabilities. The training included a review and discussion of the EPA2000 student profiles and involved several meetings during the assessment period.

RESULTS

Preliminary Comparisons

Preliminary comparisons revealed that the five mathematical ability groups (MA2, MA3, MD, RD, MD+) did not differ significantly in TIQ (F(4, 432) = 1.64, p = .16). Nevertheless, significant differences were found between the groups on VIQ (F(4, 432) = 2.96, p < .05) but not on PIQ (F(4, 432) = 0.67, p = .61). Children with combined mathematics and reading disabilities had lower VIQ scores than the other four groups. The groups did not, however, differ significantly in the socioeconomic level of the father (F(4, 432) = 2.19, p = .07) or the mother (F(4, 432) = 1.79, p = .13). Similarly, the four participant groups of grade 3 (MA3, MD, RD, MD+) did not differ significantly from each other in age (F(3,236) = 2.06, p =. 11). Finally, the five mathematical ability groups, as expected, differed significantly from each other on KRT (F(4, 432) = 123.30, p < .0005), EPA2000 cognition (F(4, 432) = 137.54, p < .0005) and EMT (F(4, 432) = 187.24, p < .0005). The average scores on the KRT (Cracco et al., 1995), EPA2000 (De Clercq et al., 2000) and the EMT (Brus & Voeten, 1999) as well as TIQ, VIQ and PIQ are presented in Table 2.

Post-hoc followup analyses (see abcd indexes in Table 2) revealed that children with a specific mathematics learning disability (MD) did not differ from children with a combined mathematics and reading disability (MD+) on the KRT (Cracco et al., 1995) or the EPA2000 (De Clercq et al., 2000). MD and MD+ children, as expected, had lower scores on the KRT than age-matched peers (MA3) and children with reading disabilities (RD). Furthermore, MD and MD+ children did worse on mathematical problem solving on the EPA2000 than mathematical problem-solving-matched children (MA2). Post-hoc analyses also revealed that RD children, as expected, performed worse than MA3 children on tests where they had to read assignments (KRT) but not on tests where they received assistance in reading the assignment (EPA2000). Furthermore, it can be concluded from Table 2 that RD children did not differ from MD+ children on the EMT. In addition, RD and MD+ children had lower scores on the EMT than MA2 children.

To summarize, children with mathematics learning disabilities (MD and MD+) did worse on mathematical problem solving than children with reading disabilities (RD) and age-matched peers (MA3), whereas children with reading disabilities (RD) and children with combined reading and mathematics disabilities (MD+) had lower reading scores than peers matched for mathematics learning disabilities (MD) and age (MA3).

Group Design Data Analyses

In order to investigate the relationship between mathematical learning, metacognition and intelligence, and given the high intercorrelations between the mathematical problem-solving tests (KRT and EPA2000 cognition), the internal structure of the mathematical problem-solving data was analyzed by principal-components analysis. This analysis was carried out to develop a mathematical problem-solving component empirically summarizing the correlations among the KRT and EPA2000 cognition variables. A one-component solution was extracted, explaining 76.41% of the common variance. The component matrix is presented in Table 3.

In order to investigate the relationship between the mathematics component, off-line metacognition and intelligence, Pearson correlations were computed between the mathematical problem-solving component score, prediction (P) and evaluation (E) and TIQ, VIQ and PIQ of all subjects (see Table 4).

Significant correlations were found between the mathematical problem-solving component and prediction (r = .71, p < .0005) and between the mathematical component and evaluation (r = .75, p < .0005). Furthermore, a significant correlation was found between the mathematics component and VIQ (r = .15, p < .005), but not between mathematics and PIQ. Significant correlations Were also not found between predictions and TIQ or between evaluations and TIQ. In addition, no significant correlations were found between prediction and VIQ, evaluation and VIQ, prediction and PIQ, evaluation and PIQ (see Table 4).

In order to further investigate the independency of intelligence and metacognition, partial correlations were computed between mathematical problem solving and prediction and evaluation, controlling for TIQ, VIQ and PIQ. Partial correlation coefficients between mathematical problem solving and prediction and between mathematical problem solving and evaluation of r = .71 (p < .0005) and r = .74 (p < .0005), respectively, were found. These results indicate that the relationship between metacognition and mathematics remains almost the same, controlling for the influence of intelligence.

In order to answer our research questions on the relation between off-line metacognition and mathematics, and in order to test the maturational-lag and domain-specificity hypothesis, a multivariate analysis of variance (MANOVA) was conducted with prediction (P) and evaluation (E) skills, as measured by EPA2000, as dependent variables and belonging to one of the five mathematical ability groups (MA2, MA3, MD, RD, MD+) as a factor. Post-hoc analyses were conducted using the Tukey procedure. With a medium effect size (f= .25), a power of > .91 was found.

The MANOVA revealed a significant main effect for the groups at the multivariate level (F(8, 862) = 40.21, p < .0005). Univariate significant between-subject effects were found for prediction (P) and for evaluation (E) (see Table 5).

Post-hoc analyses (see ab indexes in Table 5) demonstrated significantly lower prediction and evaluation scores for the children with specific or combined mathematics learning disabilities than age-matched children. No differences were found between children with a specific mathematics learning disability or combined mathematics learning disabilities and mathematical performance-matched younger children. In addition, children with reading disabilities did not have significantly lower prediction and evaluation scores than age-matched peers. These results might point in the direction of the maturational-lag and domain-specificity hypothesis.

In order to further analyze this maturational lag of children with specific and combined mathematics learning disabilities on off-line metacognition, we investigated whether third-grade students with mathematics learning disabilities also had problems with prediction on so-called easy tasks. By "easy tasks" we mean mathematical tasks designed for younger children (prediction on tasks grade 1 or P1 and prediction on tasks grade 2 or P2). For the sake of completeness, we also compared performance on "difficult tasks," or tasks designed for older children (prediction on tasks grade 4 or P4). We might expect no differences between children with mathematics learning disabilities and mathematical performance-matched children on prediction on tasks designed for grade 1, prediction on tasks designed for grade 2 and tasks designed for grade 4. A multivariate analysis of variance (MANOVA) was therefore conducted, with prediction on tasks designed for grade 1 (P1), prediction on tasks designed for grade 2 (P2), prediction on tasks designed for grade 3 (P3) and prediction on tasks designed for grade 4 (P4) as dependent variables and belonging to one of the five mathematical performance groups (MA2, MA3, MD, RD, MD+) as a factor. Post-hoc analyses were conducted using the Tukey procedure. With a medium effect size (f= .25), a power of .92 was found.

The MANOVA revealed a significant main effect for the mathematical performance groups at the multivariate level (F(16, 1311) = 26.32, p < .0005). Univariate significant between-subject effects were found for P1, P2, P3 and P4 (see Table 6).

Post-hoc Tukey analyses revealed that children with specific or combined mathematics learning disabilities (MD and MD+) did worse than age-matched children on the prediction tasks designed for grade 1. No difference was found between mathematical performance-matched younger children and children with specific or combined mathematics learning disabilities on the prediction tasks designed for grade 2, grade 3 and grade 4. Furthermore, the performance of children with reading disabilities was equal to that of age-matched children without learning disabilities on all prediction tasks. In addition, young children (MA2) and children with mathematics learning disabilities (MD and MD+) outperformed the children without learning disabilities in grade 3 (MA3) and the children with reading disabilities (RD) on prediction about tasks designed for grade 4.

We further investigated the evaluation skills on mathematical problem-solving tasks at grade 1, grade 2, grade 3 and grade 4. We expected no differences between children with specific mathematics learning disabilities (MD), children with a combined learning disability (MD+) and mathematical problem-solving-matched children (MA2) on evaluation tasks designed for grade 1 (E1), evaluation tasks designed for grade 2 (E2), evaluation tasks designed for grade 3 (E3) and evaluation tasks designed for grade 4 (E4). To test this hypothesis, a MANOVA was conducted with E1, E2, E3 and E4 as dependent variables and belonging to one of the five mathematical performance groups (MA2, MA3, MD, RD, and MD+) as a factor. Post-hoc analyses were conducted using the Tukey procedure. With a medium effect size (f = .25), a power of .92 was found.

The MANOVA revealed a significant main effect for the groups at the multivariate level (F(16, 1311) = 26.32, p < .0005). Univariate significant between-subject effects were found for E1, E2, E3 and E4 (see Table 7).

Post-hoc Tukey analyses (abc indexes in Table 7) revealed that children with specific or combined mathematics learning disabilities did worse than mathematical performance-matched younger children on evaluation tasks designed for grade 1. However, no significant differences were found on evaluation tasks designed for grade 2, evaluation tasks designed for grade 3 and evaluation tasks designed for grade 4, between the three groups of children (MA2, MD and MD). Furthermore, the performance of children with reading disabilities was equal to that of age-matched children on all evaluation tasks.

Since these results cannot be easily explained, we investigated whether the prediction and evaluation skills in children with specific or combined learning disabilities differed from those of younger children matched on mathematical performance on different aspects of mathematical problem solving, namely, numeral and operation symbol comprehension (NR+O), number system knowledge (K), mental arithmetic (M), procedural calculation (P) and word problems (W). In order to do so, a MANOVA was conducted with prediction on numeral and operation symbol comprehension ([P.sub.NR+O]), prediction on number system knowledge ([P.sub.K), prediction on mental arithmetic ([P.sub.M]), prediction on procedural calculation ([P.sub.P]) and prediction on word problems ([P.sub.W]) as dependent variables and belonging to the mathematical performance group of MA2, MD or MD+ as a factor. Post-hoc analyses were conducted using the Tukey procedure. With a medium effect size (f = .25), a power of .82 was found.

The MANOVA revealed a significant main effect for the mathematical performance groups at the multivariate level (F(10, 402) = 2.12, p < .05). Univariate significant between-subject effects were found for [P.sub.K], [P.sub.M], [P.sub.P] and [P.sub.W]. No significant between-subject effects were found for [P.sub.NR+O].

Post-hoc analyses (see ab indexes in Table 8) revealed better prediction performance for younger children matched on mathematical performance than for children with specific mathematics learning disabilities on number knowledge, mental arithmetic and procedural calculation tasks. Furthermore, young children matched on mathematical performance did better than children with combined learning disabilities on prediction about number knowledge and word problem tasks.

In order to investigate whether the evaluation skills of children with specific or combined mathematics learning disabilities differed from those of younger children on various aspects of mathematical problem solving, a MANOVA was conducted with evaluation on numeral and operation symbol comprehension tasks ([E.sub.NR+O]), evaluation on number system knowledge tasks ([E.sub.K]), evaluation on mental arithmetic' tasks ([E.sub.M]), evaluation on procedural calculation tasks ([E.sub.P]) and evaluation on word problem tasks ([E.sub.W]) as dependent variables, and belonging to the mathematical performance group of MA2, MD or MD+ as a factor. Post-hoc analyses were conducted using the Tukey procedure. With a medium effect size (f = .25), a power of .82 was found.

The MANOVA revealed a significant main effect for the mathematical performance groups at the multivariate level (F(10, 494) = 4.79, p < .0005). Univariate significant between-subject effects were found for [E.sub.K] and for [E.sub.P]. No significant between-subject effects were found for [E.sub.NR+O], [E.sub.M] and [E.sub.W].

Post-hoc analyses revealed significantly better evaluation scores for young children matched on mathematical performance on number knowledge and procedural calculation tasks compared with children with specific and combined mathematics learning disabilities (see ab indexes in Table 9).

DISCUSSION

Since metacognition is especially instrumental during the initial and final stage of mathematical problem solving (Verschaffel, 1999), this study focused on off-line metacognitive skills in young children, in grades 2 and 3. The differences between mathematically average-performing children and children with learning disabilities were investigated in order to add data on the independency, the maturational-lag and the domain-specificity hypotheses. Since different authors have stressed the importance of an operational definition of learning disabilities, children with specific mathematics disabilities were differentiated from children with specific reading disabilities and children with combined learning disabilities. Furthermore, all children had average intelligence and the socioeconomic level of both mother and father did not differ significantly.

We investigated the relationship between mathematical problem solving, off-line metacognition and intelligence. The data were in line with earlier investigations that have documented the relationship between mathematics and metacognition (e.g., Lucangeli and colleagues, 1997, 1998). In 437 children, a significant relationship between a mathematical component and off-line metacognition and between the mathematical component and verbal intelligence was found. Furthermore, no significant relationship was found between intelligence and off-line metacognition of children in grades 2 and 3. These results suggest that off-line metacognition cannot be seen as a demonstration of intelligence. Metacognition was nevertheless found to be important in the explanation of mathematical problem solving and had an additional value in the explanation of learning, as already pointed out by Swanson (1990).

We also investigated the retardation or maturational-lag hypothesis, meaning that children with specific or combined mathematics learning disabilities will perform worse on prediction and evaluation assignments than age-matched children without learning disabilities, but no such differences were expected compared with younger children matched on mathematical problem-solving skills. The data from the present study indicate a large discrepancy between off-line metacognition in children with mathematics learning disabilities compared with average-achieving peers. The pattern in these results could therefore be interpreted within the maturational-lag or retardation hypothesis. Young children with comparable mathematical performance scores on the EPA2000 (De Clercq et al., 2000) to children with mathematics learning disabilities (and even lower) had comparable prediction and evaluation scores on the EPA2000.

However, when we compared predictions and evaluations on the so-called easy tasks, or the tasks designed for younger (or older) children, subjects with specific or combined mathematics learning disabilities were expected to perform as well as younger children matched at mathematical performance level on prediction about tasks designed for the second or first grade and on evaluation about tasks designed for the second or first grade, according to the retardation or maturational-lag hypothesis.

On analyzing our results, however, a slightly different pattern was found. Children with mathematics learning disabilities had lower scores than younger children with comparable mathematical skills on prediction and evaluation on mathematics tasks designed for first graders. However, no such differences were found on prediction and evaluation about tasks designed for second, third or fourth graders. These results could not be totally explained by the maturational-lag hypothesis, but indicated a disharmonic metacognitive profile in children with mathematics learning disabilities.

Moreover, children in grade 2 and children with specific or combined mathematics learning disabilities outperformed the children in grade 3 without learning disabilities and the group of children with reading disabilities in grade 3 on prediction tasks related to mathematical problem-solving topics designed for grade 4. This may seem inconsistent, but interviews afterwards with some of the children taught us that children in grade 2 and the Children with mathematics learning disabilities were sure that they would not be able to solve such tasks, as they differed greatly from the ones they were used to solving. Therefore, these children correctly predicted being very sure about not being able to solve exercises of this kind. The children in grade 3 without mathematics learning disabilities might have had the illusion of being able to solve exercises of this kind, since the tasks appeared to be similar to the exercises they could solve in grade 3. This clarifies the finding, which at first glance otherwise appears strange.

In order to examine whether these results could be explained by analyzing the mathematical problem-solving tasks, off-line metacognition on numeral and operation symbol comprehension, number system knowledge, mental arithmetic, procedural calculation and word problems were compared in children with specific and combined mathematics learning disabilities and in younger children matched on mathematical performance. We found that subjects with a specific mathematics learning disability had significantly lower prediction scores than younger children on number system knowledge, mental arithmetic and procedural arithmetic. Moreover, children with a combined learning disability did worse than younger children on number system knowledge and word problem tasks. Furthermore, children with specific or combined mathematics learning disabilities did worse than younger subjects on the evaluation of number system knowledge and procedural calculation tasks. Again, these results cannot be explained by the maturational-lag hypothesis, but indicate that children with mathematics learning disabilities have a different off-line metacognitive profile than young children with comparable mathematical performance.

To sum up, at first glance children with specific or combined mathematics learning disabilities seem to have prediction and evaluation skills comparable to those of children one year younger, which could be interpreted according to the maturational-lag hypothesis. However, on analyzing this performance further, significant differences were found compared to children without learning disabilities, matched at the level of mathematical problem solving. Therefore, our data could not be interpreted according to the maturational-lag hypothesis. Further research seems to be indicated.

Finally, consistent with the domain-specificity hypothesis, we expected the performance of children with reading disabilities to equal that of children of the same age without learning disabilities on mathematically related prediction and evaluation tasks. In response to this research question, children with reading disabilities did not have significantly lower scores than peers without learning disabilities. Furthermore, the same pattern was found for all prediction and evaluation tasks in children with reading learning disabilities and peers without learning problems. These results are in line with earlier research on the domain specificity of off-line metacognitive skills (e.g., Schraw et al., 1995). Thus, it could be argued that children with reading disabilities might have domain-specific problems with off-line metacognition related to reading tasks, but not with prediction and evaluation related to mathematical problem-solving tasks. However, given that this study did not compare metacognitive skills across domains (e.g., reading and mathematics), additional research is needed to be able to draw conclusions on the domain specificity of metacognition per se and to draw links to the expert-novice literature in general.

The results of this study should be interpreted with care since metacognition might be age-dependent and still maturing until adolescence (Berk, 1997). In addition, depending on the particular nature of the mathematical task presented, metacognition may have a differential influence (Lucangeli et al., 1998). Furthermore, only off-line metacognitive skills were studied. Other answers may therefore be possible with on-line metacognitive skills or with metacognitive knowledge of beliefs. In addition, only children of average intelligence participated, and we were unable to match the five groups on VIQ, since VIQ was found to be lower for children with a combined learning disability compared with the four other groups. This could explain the lower scores on language-related items such as prediction about word problems, and certainly needs additional research. Moreover, since metacognitive skills were not compared across domains (e.g., reading and mathematics), additional research is needed to explore how domain-specific knowledge and experience interact in the production of proficient problem-solving performance with metacognition. It also remains unclear whether it is a question of metacognitive skills per se or whether the difference between non-experts and experts is a function of background and conceptual knowledge (the ability to represent problems, etc.) as a basis for effective metacognition (e.g., as a basis for making predictions or judgments about how well you can solve a problem or choosing a problem-solving strategy). Furthermore, the research on off-line metacognition in children with learning disabilities needs full explanation from more applied research on different age and intelligence groups.

Despite these limitations, this study may have important conceptual and educational implications. Since metacognition is important for mathematical problem solving and since metacognition cannot be reduced to demonstrations of intelligence, it has to be assessed separately, especially if things go wrong in mathematical problem solving. Furthermore, since we could not explain all our results according to the maturational-lag hypothesis, we cannot expect metacognition to develop spontaneously as children grow older and have more experience of mathematics. Metacognitive therapy should therefore focus on the metacognitive weaknesses and strengths of children with specific or combined mathematics learning disabilities, making them more aware of how they calculate or deal with word problems. Such programs seem to be helpful in addition to the more traditional mathematical training programs. Finally, training in off-line metacognition narrowly related to mathematical problem-solving tasks does not seem to be needed in children with specific reading disabilities. However, this study makes it clear that in all children with reading disabilities, mathematics also has to be assessed, since children with combined reading and mathematics disabilities (RD+ or MD+) have problems with off-line metacognition related to mathematical problem solving.

In summarizing, our studies support the use and importance of a metacognitive assessment procedure to differentiate between students with and without mathematics learning disabilities. Taking into account the complex nature of mathematical problem solving, it may be useful to assess off-line metacognition in young children with mathematics learning disabilities in order to focus on these factors and their role in mathematics learning and development. It might be possible that with more time allocated to off-line metacognitive instruction, especially during the initial stage and in the final stage of mathematical problem solving, some mathematics learning disabilities may become less pervasive.

Table 1
Cronbach's Alpha Analyses on EPA2000

                  Number of Items   Cronbach's [alpha]
                                        Prediction

NR- and O-tasks      27 items              0.90
K-tasks              10 items              0.88
M-tasks              10 items              0.87
P-tasks              10 items              0.95
W-tasks              23 items              0.94

Tasks grade 1        19 items              0.90
Tasks grade 2        37 items              0.94
Tasks grade 3        20 items              0.95
Tasks grade 4         4 items              0.86

                  Cronbach's [alpha]
                      Evaluation

NR- and O-tasks          .75
K-tasks                  .81
M-tasks                  .80
P-tasks                  .91
W-tasks                  .91

Tasks grade 1            .75
Tasks grade 2            .90
Tasks grade 3            .92
Tasks grade 4            .79

Note. NR and O = numeral and operation symbol comprehension,
K = number system knowledge, M = mental arithmetic, P = procedural
arithmetic, W = word problems.
Table 2
Children With and Without Learning Disabilities Compared

              MA2          MA3           MD          RD

               M            M            M            M
              (SD)         (SD)         (SD)        (SD)
            N = 120       N = 130       N = 62      N = 53

TIQ        102.50       102.71       101.47       102.77
            (7.71)       (5.46)       (7.37)       (6.37)
VIQ        101.99 (a)   102.68 (a)   102.18 (a)    02.89 (a)
            (7.30)       (5.35)       (6.50)       (4.96)
PIQ        102.27       101.64       100.29        02.75
            (7.44)      (10.06)      (10.41)       (8.23)
SES F       14.02        14.39        13.73        13.81
            (3.70)       (3.58)       (3.30)       (2.58)
SES M       14.02        14.19        13.97        13.66
            (2.78)       (2.53)       (2.65)       (3.01)
KRT         41.43 (b)    44.92 (a)    24.02 (c)    39.36 (b)
            (7.24)       (6.03)       (6.79)       (9.25)
EPA2000     53.86 (b)    67.54 (a)    50.37 (c)    66.09 (a)
            (7.52)       (4.59)       (9.02)       (5.39)
EMT         40.68 (c)    55.64 (a)    50.68 (b)    29.91 (d)
            (9.58)       (8.22)       (6.37)       (6.95)

              MD+

               M
             (SD)        F(4,432) =
             N=72

TIQ        100.47         1.64
            (7.39)
VIQ         99.56 (b)     2.96 *
            (8.84)
PIQ        101.57         0.67
            (9.26)
SES F       15.29         2.19
            (4.36)
SES M       14.89         1.79
            (3.25)
KRT         25.82 (c)   123.30 *
           (10.83)
EPA2000     49.39 (c)   137.54 *
            (8.16)
EMT         25.68 (d)   187.24 *
           (10.27)

Note. MA2 = age-matched young children in grade 2, MA3 = mathematical
performance-matched children in grade 3 without learning disabilities,
MD = children with specific mathematics learning disabilities,
RD = children with specific reading learning disabilities,
MD+ = children with combined mathematics and reading learning
disabilities.

* p [less than or equal to] .0005.

(a) (b) (c) (d) different indexes refer to significant between-group
differences with a significance level of .05.
Table 3
Component Matrix

                                 Mathematical Problem-Solving Component

KRT                                                 .87
EPA2000 mathematical cognition                      .87
Eigenvalue                                         1.53
% of variance                                     76.41
Table 4
Pearson Correlations Between the Mathematical Problem-Solving
Component, IQ and Off-Line Metacognition

       TIQ      VIQ         PIQ      Math. Comp.   Prediction

TIQ     -     .87 **      .75 **        .12 *         .03
             (p = .00)   (p = .00)    (p = .01)    (p = .53)
VIQ     -        -        .46 **        .15 *         .08
                         (p = .00)    (p = .00)    (p = .12)
PIQ     -        -           -          .03          -.04
                                      (p = .54)    (p = .45)
Math    -        -           -            -           .71 **
                                                   (p = .00)
Pred    -        -           -            -            -

       Evaluation

TIQ       .03
       (p = .51)
VIQ       .08
       (p = .10)
PIQ      -.04
       (p = .46)
Math      .75**
       (p = .00)
Pred      .79**
       (p = .00)

** p [less than or equal to] .0005.

* p [less than or equal to] .01.
Table 5
Metacognitive Prediction and Evaluation Skills

          MA2         MA3         MD          RD
           M           M           M          M
         (SD)        (SD)        (SD)        (SD)
        N = 120     N = 130     N = 62      N = 53

P **   64.79 (b)   79.27 (a)   61.90 (b)   76.30 (a)
        (9.62)      (8.16)      (11.59)     (9.14)
E **   64.12 (b)   79.77 (a)   62.90 (b)   77.17 (a)
        (9.92)      (6.79)      (12.34)     (7.54)

          MD+
           M
         (SD)      F(4, 432) =
         N=72

P **   61.21 (b)     74.79 *
        (8.80)
E **   60.63 (b)     79.79 *
        (11.17)

Note. P = prediction on all tasks in EPA2000, E = evaluation on all
tasks in EPA2000. MA2 = age-matched young children in grade 2,
MA3 = mathematical performance-matched children in grade 3 without
learning disabilities, MD = children with specific mathematics
learning disabilities, RD = children with specific reading learning
disabilities, MD+ = children with combined mathematics and reading
learning disabilities.

* p [less than or equal to] .0005.

** maximum score is reduced to 100 points.

(a) (b) different indexes refer to significant between-group
differences with a significance level of .05.
Table 6
Prediction on Tasks for Children in Grades 1 to 4

           MA2          MA3         MD          RD
            M            M           M           M
           (SD)        (SD)        (SD)        (SD)
         N = 120      N = 130     N = 62      N = 53

P1 **   83.25 (b)    92.97 (a)   72.76 (c)   88.88 (a)
          (9.17)      (7.88)      (17.73)     (9.79)
P2 **   65.27 (b)    81.16 (a)   60.96 (b)   78.15 (a)
         (11.61)      (8.68)      (11.99)     (9.36)
P3 **   49.10 (b)   69.79 (a)   47.04 (b)   65.32 (a)
         (15.25)     (11.62)      (11.47)    (13.76)
P4 **   52.96 (a)    38.83 (b)   47.43 (a)   35.78 (b)
         (22.57)     (21.69)      (19.95)    (23.27)

           MD+
            M
          (SD)      F(4, 432) =
          N = 72

P1 **   77.62 (c)     49.27*
         (9.99)
P2 **   61.81 (b)     78.99*
         (8.35)
P3 **   46.83 (b)     71.96*
         (9.14)
P4 **   49.02 (a)      9.18*
        (23.30)

Note. P1 = prediction on tasks level grade 1, P2 = prediction on
tasks level grade 2, P3 = prediction on tasks level grade 3,
P4 = prediction on tasks level grade 4. MA2 = age-matched young
children in grade 2, MA3 = mathematical performance-matched children
in grade 3 without learning disabilities, MD = children with specific
mathematics learning disabilities, RD = children with specific
reading learning disabilities, MD+ = children with combined
mathematics and reading learning disabilities.

* p < .0005.

** maximum score on P1, P2, P3 and P4 is 100.

(a) (b) (c) different indexes refer to significant between-group
differences with a significance level of .05.
Table 7
Evaluation on Tasks for Children in Grades 1 to 4

             MA2             MD             MD
              M              M               M
             (SD)           (SD)           (SD)
            N = 120        N = 130         N = 62

E1 **       84.87 (b)      94.08 (a)       79.20 (c)
            (9.08)         (5.64)         (10.11)
E2 **       66.50 (b)      82.21 (a)       65.20 (b)
           (11.51)         (6.63)         (14.06)
E3 **       45.50 (b)      73.94 (a)       48.98 (b)
           (15.15)        (10.97)         (14.79)
E4 **       38.57          37.86           38.13
           (22.39)        (21.36)         (25.64)

              RD            MD+
              M              M
             (SD)           (SD)         F(4, 432) =
             N = 53          N = 72

E1 **       91.73 (a)       80.91 (c)       57.84 *
            (7.39)         (7.92)
E2 **       79.80 (a)      64.64 (b)        67.01 *
            (8.22)         (9.74)
E3 **       69.17 (a)      44.83 (b)       110.78 *
           (12.16)        (11.80)
E4 **       34.46          41.60             0.85
           (19.23)        (19.12)

Note. E1 = evaluation on task level grade 1, E2 = evaluation on task
level grade 2, E3 = evaluation on task level grade 3, E4 = evaluation
on task level grade 4. MA2 = age-matched young children in
grade 2, MA3 = mathematical performance-matched children in grade
3 without learning disabilities, MD = children with specific mathematics
learning disabilities, RD = children with specific reading learning
disabilities, MD+ = children with combined mathematics and reading
learning disabilities.

* p < .0005.

** maximum score on E1, E2, E3 and E4 is 100.

(a) (b) (c) different indexes refer to significant between-group
differences with a significance level of .05.
Table 8
Prediction on Different Mathematical Problem-Solving Tasks Compared

                      MA2          MD            MD+

                       M            M             M
                     (SD)         (SD)          (SD)        F(2, 251) =
                    N = 120      N = 62        N = 72

[P.sub.NR+O] ***     69.65        66.08         65.11         1.77
                    (18.44)      (19.71)       (13.57)
[P.sub.K] ***        56.43 (a)    47.83 (b)     48.40 (b)     7.96 **
                    (17.07)      (17.76)       (14.54)
[P.sub.M] ***        64.56 (a)    58.33 (b)     59.02         6.05 **
                    (13.41)      (14.49)       (12.57)
[P.sub.P] ***        51.65 (a)    41.74 (b)     44.25         3.72 *
                    (27.62)      (24.71)       (22.44)
[P.sub.W] ***        52.14 (a)    48.37         47.51 (b)     4.07 *
                    (13.14)      (11.62)        (9.90)

Note. PNR+O = prediction on numeral and operation symbol
comprehension tasks, PK = prediction on number system knowledge
tasks, PM = prediction on mental arithmetic tasks, PP = prediction
on procedural calculation tasks, PW = prediction on word problem
tasks. MA2 = age-matched young children in grade 2, MD = children
with specific mathematics learning disabilities, MD+ = children with
combined mathematics and reading learning disabilities.

** p < .01.

* p < .05.

*** maximum score on PNR+O, PK, PM, PP, PW is 100.

(a) (b) different indexes refer to significant between-group
differences with a significance level of .05.
Table 9
Evaluation on Different Mathematical Problem-Solving Tasks Compared

                      MA2          MD           MD+

                       M            M            M
                     (SD)         (SD)         (SD)        F(2, 251) =
                   N = 120       N = 62        N = 72

[E.sub.NR+O] ***     69.04        69.56        65.74          0.94
                    (18.86)      (20.38)      (15.36)
[E.sub.K] ***        59.04 (a)    50.87 (b)    51.49 (b)      5.79 *
                    (18.51)      (20.75)      (15.70)
[E.sub.M] ***        59.25        61.11        64.11          2.74
                    (14.10)      (16.54)      (10.82)
[E.sub.P] ***        47.74 (a)    39.35 (b)    35.11 (b)      7.95 **
                    (22.41)      (22.06)      (19.59)
[E.sub.W] ***        51.54        50.50        47.53          2.27
                    (13.39)      (12.53)      (11.59)

Note. ENR+O = evaluation on numeral and operation symbol comprehension
tasks, EK = evaluation on number system knowledge tasks,
EM = evaluation on mental arithmetic tasks, EP = evaluation on
procedural calculation tasks, EW = evaluation on word problem tasks.
MA2 = age-matched young children in grade 2, MD = children with
specific mathematics learning disabilities, MD+ = children with
combined mathematics and reading learning disabilities.

* p < .01.

** p < .0005.

*** maximum score on ENR+O, EK, EM, EP, EW is 100.

(a) (b) different indexes refer to significant between-group
differences with a significance level of .05.

NOTES

This study was supported by the Stichting Integratie Gehandicapten (SIG), the Artevelde College Ghent and Centrum ter Bevordering van de Cognitieve Ontwikkeling (CeBCO), to whom the authors express their thanks. In addition, we thank an anonymous reviewer for constructive comments on an earlier draft of this paper.

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Requests for reprints should be addressed to: Annemie Desoete, Ghent University, Dept. of Clinical Psychology, B 9000 Ghent, Belgium.

ANNEMIE DESOETE, Ph.D, is a researcher, Ghent University and Artevelde College, Ghent, Belgium.

HERBERT ROEYERS, Ph.D., is assistant professor in clinical psychology, Ghent University, Ghent, Belgium.