EPA2000: assessing off-line metacognition in mathematical problem solving.

Introduction

Research from different theoretical approaches has provided information regarding processes that are important for young children to solve mathematical problems adequately (Donlan, 1998; Koriat, 1995; Lucangeli & Cornoldi, 1997; Metcalfe, 1998; Montague, 1998; Schunn, Reder, Nhouyvanisvong, Richards, & Stroffolino, 1997; Schwartz & Metcalfe, 1994). Our model of mathematical problem-solving integrates nine cognitive processes and two metacognitive parameters. To clarify our conceptual framework, we describe the cognitive processes included in mathematical problem-solving (see NR, S, K, P, L, C, V, R, N in Table 1).

Cognitive processes' enable the translation of numerical (NR-processes), symbolic (S-processes), simple linguistic (L-processes) or more complex contextual (C-processes) information into mental representations or visualizations (V-processes) of the problem or task. Furthermore, dealing with number system knowledge (K-processes), eliminating irrelevant information (R-processes) and estimating based on number sense (N-processes) typify mathematical problem solving and precede procedural calculation processes (P-processes), leading to the computing of the solution (Desoete, Roeyers, & Buysse, 2001).

In addition 'metacognition' seems to be involved in successful mathematical problem solving (see Pr and Ev in Table 1) (Desoete, Roeyers, & Buysse, 2000b; Lucangeli & Cornoldi, 1997; Montague, 1998; Tobias & Everson, 1996). Flavell (1976) defined metacognition as '...one's knowledge concerning one's own cognitive processes and products or anything related to them' (1976, p. 232). Studies concerned with problem- solving strategies in mathematically average-performing children have shown that inetacognition is instrumental during the initial stage ('Prediction', Pr) of mathematical problem solving, when subjects build an appropriate representation of the problem, as well as in the final stage ('Evaluation', Ev) of interpretation and checking the outcome of the calculations (Verschaffel, 1999). Prediction guarantees working slowly when exercises are new or complex and working fast with easy or familiar tasks. Evaluation refers to the retrospective verbalizations after the event has transpired (Brown, 1987), wher e children look at what strategies were used and whether they led to a desired result or not.

Children with mathematics learning disabilities show some typical shortcomings in different 'cognitive' processes (NR, S, K, P, L, C, V, R, N) of mathematical problem solving (e.g. Geary, 1993; McCloskey & Macaruso, 1995; Rourke & Conway, 1997; Verschaffel, 1999). Some of these children have problems in number (NR) and symbol (S) comprehension and production. They confuse 6 with 9, 'drie' (three in Dutch) with 'vier' (four in Dutch) or x with +. Other children with mathematics learning disabilities lack the needed number system knowledge (K) or make mistakes of a procedural (P) type. These children confuse digits and tens or forget, for example, a multidigit addition, to start in the right column. Language-dependent (L) and mental representation (V) related mistakes or problems dealing with linguistic or contextual (C) information as well as a lack of number sense (N) are also typical for some children with mathematics learning disabilities (Desoete et al., 2000; Desoete, Roeyers, Buysse, & De Clercq, 2000). Furthermore, children with mathematics learning disabilities often show below-average performances on the different metacognitive (Pr, Ev) parameters included in mathematical problem-solving (Desoete et al., 2001).

To focus on the problems of students with mathematics learning disabilities and to tailor a relevant instructional program, it is necessary to assess the 'cognitive' and 'metacognitive' strengths and weaknesses of these children. No test is currently available for a combined assessment of cognitive and metacognitive skills in grade 3 of the elementary school. The purpose of this article is to describe such assessment strategies for mathematics.

The Evaluation and Prediction Assessment (EPA2000) is a computerized assessment of cognitive and metacognitive skills. EPA2000 was adapted from a longer version of a semi-structured metacoguitive interview (Metacognitive Skills and Beliefs Assessment - MBA and MSA, Desoete & Roeyers, 1998) designed to assess processes, important for successful mathematical problem solving. A paper-and-pencil version was developed primarily to be used as a diagnostic-prescriptive tool, to assess primary school students' strengths and weaknesses in mathematical problem solving (Evaluation and Prediction Assessment, EPA, Desoete & Roeyers, 1999). Next, a less informal but highly motivating computer version was developed with the same items (Evaluation and Prediction Assessment 2000, EPA2000, De Clercq, Desoete, & Roeyers, 2000). EPA and EPA2000 were designed for normally intelligent children with or without mathematics learning disabilities in grade 3.

To provide background, the theoretical basis of EPA2000 is described first. The research findings that support the EPA2000 as a diagnostic-prescriptive tool are then presented. Finally an actual student protocol is used to illustrate the administration and interpretation of the EPA2000.

EPA and EPA2000 assesses nine cognitive (NR, S, K, P, L, C, V, R, N) and two metacognitive (Pr, Ev) processes found to be important in mathematical problem solving in grade 2 and 3 (see Table 1) (Desoete et al., 2000 & 2001). Exercises, in EPA and EPA2000, on Arabic Numeral comprehension and production or NR problems include the reading of single-digit and multiple-digit numerals as well as verbal numeral comprehension (e.g. Put into the right order from low to high: 39 37 38 40). The numeral comprehension additionally includes operation Symbol comprehension or S-problems (e.g. Which is correct? 38+1=39 or 38x1=39). Number system Knowledge or K-problems deal with insight into the number structure (e.g. Complete this series: 37, 38, 39, ?). Within the Procedural calculation items (or P-problems) the capacity to do additions, subtractions, multiplications and divisions is assessed (e.g. 37+1=?). Furthermore, exercises include items probing basic arithmetical facts and items with carry-over problems. Within the word problems of EPA and EPA2000, the L-problems demand a simple single-sentence Language analysis (e.g. 1 more than 37 is?). The C- type of word problems, however, depend upon Contextual language analysis in more than one sentence (e.g. baker problem in Figure 1). Another cognitive activity necessary to solve word problems is mental representation or Visualization of the problem (V-problems). '15 is 1 less than ?' is such a V-problem. Without visualization children answer 14, since they translate 'less' into 'minus', and answer '14' in a superficial number-crunching approach. In order to give correct answers, irrelevant information has to be eliminated in R-type word problem where Relevant information has to be selected. 'Lena has 24 Christmas balls, Grace has 15 Christmas stars and 8 Christmas balls. How many Christmas balls do they have altogether?' is such a R-problem. Here the number of stars is irrelevant. Furthermore, in EPA2000 some items on Number sense (N-problems) are included (e.g. 37 is nearest t o? Choose between 47, 40, 73 or 30).

As to 'metacognition', Verschaffel (1999) stressed its importance during the initial (prediction) and final (evaluation) stages of problem solving (see Table 1). Since these metacognitive skills are measured before or after the solving of exercises, we labeled them 'off-line (measured) metacognition'. In two studies (n=165) we found off-line metacognition capable of differentiating between good performers, moderate performers and children with mathematics learning disabilities (Desoete et al., 2001; Desoete, Roeyers, Buysse, et. al, 2000). To prevent floor or ceiling effects on children with and without mathematics learning disabilities in grade 3, exercises of different complexity (varying from grade 1 to 4) were introduced to measure mathematical problem solving in children of grade 3.

The EPA

The EPA paper and pencil version (EPA) (Desoete & Roeyers, 1999) has a three-part (metacognitive prediction - cognition - metacognitive evaluation) assessment. Children have to predict and evaluate with 80 mathematical problem solving tasks (e.g. NR-problems, S-problems, K-problems, P-problems, L-problems, C-problems, V-problems, R-problems, N-problems - see Table 1). In the assessment of prediction, children are asked to look at exercises without solving them and to predict if they will be successful in this task on a 4-point rating scale. Children have to evaluate after solving the same mathematical tasks on the same 4-point rating scale.

Metacognitive predictions or evaluations are awarded with two points, whenever they correspond to the child's actual performance on the task (doing the exercise correctly and rating 'absolutely sure I am correct', or doing the exercise wrong and rating 'absolutely sure I am wrong') (see Table 2). Predicting and evaluating, rating 'sure I am correct' or 'sure I am wrong' receive one point whenever they correspond. Other answers do not gain any points, as they are considered to represent a lack of off-line metacognition. As to the cognitive mathematical problem solving, children obtain 1 point for every correct answer.

The psychometric data of the EPA have been analyzed on 1336 third-grade children (Desoete et al., 2000b). Furthermore, mathematical processes (NR, S, K, P, L, C, V, R, N, Pr, Ev) were compared in normally intelligent children with mathematics learning disabilities (-2SD on mathematical performance tests), children with mathematics learning problems (-1SD on mathematical performance tests) and moderate achieving peers without learning disabilities on EPA (n=320) (Desoete, Roeyers, Buysse et al., 2000). In addition various experts on mathematics and on mathematics learning disabilities were consulted in order to increase the construct validity. As to the concurrent validity, Pearson product moment correlation coefficients were computed between the mathematical problem solving scores of the EPA and the scores of other mathematics tests for these children (n=145). A correlation of .56 (p<.0005) was found with another mathematical problem solving test frequently used in Belgium. In addition, a correlation of .79 (p<.0005) was found between the EPA mathematical problem solving scores and teacher ratings of mathematics skills. Furthermore, Cronbach's alpha reliability analyses were conducted. Reliability coefficients of .88 were found. As to metacognition, various authors were consulted to increase the construct validity. In addition, Cronbach's alphas of .79 and .73 respectively were found for the prediction and evaluation scores of the EPA in the same sample (n = 145). In another study with 30 third-grade students test-retest correlations of .81 (p<.0005) were found (De Clercq et al., 2000).

It became clear from these studies that the students and teachers were able to handle the instrument well. Findings support the use of this assessment procedure to differentiate between average (between -0.5 SD and +0.5 SD) and above-average (+2 SD) achievers on mathematical problem solving tests and peers with mathematics learning disabilities (-2SD on these tests) in the prediction and evaluation skills (Desoete et al., 2001).

However, this study revealed one restriction. There appeared to be an interference of cognitive and metacognitive mathematical solving processes with the paper and pencil assessment, even with teachers giving very explicit instructions to predict and not to calculate in the prediction phase. Because of these findings we decided to design an assessment without possible interferences between the cognitive and metacognitive processes. Since most studies suggest the equivalence of conventional and computerized instruments (Schulenberg & Yutrzenka, 1999), a computerized version was developed, which is easy to be modified and translated by a teacher without computer knowledge.

The EPA2000

The computerized assessment (EPA2000) is derived from the paper and pencil assessment (EPA) with exactly the same cognitive (NR, S, K, P, L, C, V, R, N) and metacognitive (Pr, Ev) tasks (De Clercq et al., 2000; Desoete, De Clercq, & Roeyers, 2000). With EPA2000 we are able to obtain a clear picture of and differentiate between cognitive and off-line metacognitive processes of third-graders. Since children have to click with the mouse while predicting, there is less time to calculate. In addition the prediction reaction time can be computed, in order to control for the interference between prediction and cognition. Furthermore children perform the cognitive tasks (NR, S, K, P, L, C, V, R, N) without seeing what they predicted and they evaluate without seeing their calculation results. The software is easily installed by teachers without much computer knowledge.

In the first part metacognitive prediction (Pr) skills are assessed (see Figure 1). Children have to predict on 80 mathematical problem solving tasks. Children are asked to look at the exercises without solving them and to predict whether they will be successful in this task on a color rating scale. In Figure 1 children have to predict on language related (Pr on L) tasks. Children might predict well and do the exercise wrong, or vice-versa (see Table 2).

In a second part, cognition (NR, S, K, P, L, C, V, R, N) is assessed. Children have to solve the same 80 mathematics problem solving tasks they predicted on before. In Figure 2 children are asked to solve a P-problem.

In a third part, children are asked to 'evaluate' (Ev) after solving the mathematical problem solving task, without seeing how they predicted or solved these tasks (see Figure 3). The same color rating scale as in prediction is used. The 80 prediction (Pr), cognition (NR, S, K, P, L, C, V, R, N) and evaluation (By) problems on the EPA2000 (Desoete, De Clercq et al., 2000) are exactly the same as those of the EPA (Desoete,& Roeyers, 1999).

The EPA2000 items are scored as in the EPA paper and pencil form (see Table 2). Results on the three subscales are the basis for developing cognitive and metacognitive profiles (see Appendix) for individual students. These profiles provide a graphic display (see Appendix) of a student's cognitive (NR, S, K, P, L, C, V, R, N) and metacognitive (Pr, Ev) mathematical problem solving strengths and weaknesses and can be used as a guide to tailor instruction by teachers for individual students.

The EPA2000 was tried out in one classroom with 30 children. The teacher installed the software and interpreted the results. It appeared that all children and the teacher were able to handle the instrument very well. In addition, the psychometric data were analyzed on 407 children (Desoete, Roeyers, & De Clercq, 2000). Cronbach's alphas were .89 for the cognitive scores, .74 for the metacognitive prediction skills, and .85 for metacognitive evaluation skills. In another study, with 30 third-grade children, test-retest correlations of .80 (p<.0005) for the EPA and EPA2000 were found (De Clercq et al., 2000).

The EPA2000 has recently been used in different studies focusing on children with mathematics learning disabilities (-2SD) in grade 3. An exploratory study (Desoete, Roeyers, & De Clercq, 2000 & 2001) was setup to investigate whether normally intelligent third graders with specific mathematics learning disabilities (n = 60) could be distinguished from children without learning disabilities (n = 60) in grade 3 on prediction and evaluation scores of EPA2000. In order to do so we compared two groups of normally intelligent children, controlling for differences in TIQ, reading skills and socio-economic level of both parents. Chi-square analyses revealed significant differences between the two groups ([chi square](2) = 68.05, p<.0005) (see Table 3). Eighty-three percent of the children could be classified into the correct diagnostic group on the basis of the two metacognitive scores.

Follow-up analyses (see Table 3) revealed that children with specific mathematics learning disabilities showed lower metacognitive prediction scores (F (1,118) = 76.18,p<.0005) and lower evaluation scores (F (1,118) = 82.55, p<.0005) than their age-mates without learning disabilities.

In another study (n = 407), our results indicate EPA2000 to be very useful in the assessment of normally intelligent (TIQ> 90) children with specific mathematics or combined reading and mathematics learning disabilities (Desoete & Roeyers, 2002). Children with ADHD had somehow more problems, since the assessment took too long for these young children.

A demo version of the EPA2000 can be downloaded free from http:// twiprofl.rug.ac.be/epa2000. For the complete version of EPA2000, the first author should be consulted at [e-mail]. In what follows we highlight the use of EPA2000 in the description of the cognitive and metacognitive strengths and weaknesses of Helmut, who was referred to us by a school psychologist because of significantly below grade-level mathematics achievement. EPA2000 was administered and interpreted with the teacher.

Administration and Interpretation of the EPA 2000

Helmut is a 9-year-old normally intelligent (WISC R TIQ 104, VIQ 109, PIQ 98) boy with mathematics learning disabilities. Helmut performs average in reading and poorly in mathematics at school. The intelligence subtests are presented in Figure 4. EPA2000 was administered by his regular teacher in collaboration with the school psychologist.

Helmut first made predictions (Pr) on his performance in the mathematical problem solving tasks. Then he solved the mathematical problem solving tasks (NF, S, K, P, L, C, V, R, N) and evaluated (Ev) his performance (see Table 1).

As to the prediction (Pr), he got a score of 96/160 or 60% (see Appendix).

Also the NR-tasks, the reading of single-digit exercises (9, 2, 7, 3, 4, 8, 5) was correct. The reading of multiple-digit exercises was good even when the digit name was not congruent with the number structure, with exception of the confusion of 71 and 37. Helmut read correctly 71, 41, 21, 40, 51, 82, 70, 91, 712 and 978. Furthermore, Helmut's verbal numeral comprehension was good. There was no confusion of written and oral number production. Helmut read 62, 81, 630, 311 and 407 without mistakes. Helmut did not have NR-problems. He got a score of 21/22 or 95% (see Appendix).

Also on operation symbol comprehension (S-problems), all items were solved correctly. Helmut knew <,>,x,+ and knew that the weight of a person is expressed in pounds. He got a score of 5/5 or 100% (see Appendix).

The number system knowledge (K-problems) was also assessed. Helmut could put 5 numbers (e.g. 19 28 37 46 or 105 150 5015 10) in the correct order, whereas he was mistaken with 10.1 11 15.1 51 and with the time structuration task. He got a score of 8/10 or 80%

As to the P-tasks, procedural additions to be solved by mental arithmetic (15+2=? And 42+51 =?) were correctly handled (see 2/2 addition Appendix). Subtraction (19-15=?) was solved correctly, with exception of 17-3=? (see 1/2 subtraction Appendix). Items to be solved with carry over (15+9=? And 17-9=?) were correct (see 2/2 carry over Appendix). Helmut knew simple arithmetical facts (3x7=?, 8x3=? and 8:2=?;35:7=?) (see 4/4 arithmetical facts). Procedural calculation tasks (15x7=? and 24x8=?) were incorrect whereas 210x30 was solved correctly (see 1/3 multiplication Appendix). The division task 98:7=? was incorrect, whereas 168:8=? was solved correctly (see 1/2 division Appendix). Procedural items to be solved by calculation procedures (27+653=?; 60+235=?; 210x30=?) were not correct (see 2/5 calculation procedures Appendix). In total he got 13/20 or 65% for P-tasks (see also graphic display in Appendix).

As to the language related word problems (L-problems), Helmut solved correctly 'twice 6 is ?', '1 less than 25 is?' and '1 more than 58 is?' (see 3/3 simply language factor in Appendix). Word problems involving an additional order factor (e.g. '? is half of 8' and'? is 2 less than 54') were correct (see 2/2 temporo-spatial or order factor Appendix). In total he got a score of 5/5 or 100% (see graphic display Appendix)

C-problems or word problems based on additional context information were correctly solved in the case of the postman problem but not in the case of the baker problem, key problem and the marbles problem (see 1/4 or 25% context factor Appendix).

As to the V-problems, the following word problems, where mental representation was essential in order to solve the problem, were incorrectly answered: '58 is 1 more than?' 16 is half of?' and '170 is 2 less than?' although '58 is 1 less than?' and '14 is twice?' were correct (see 2/5 or 40 % mental representation or visualization factor Appendix).

Furthermore, as to the R-problems, the word problems where Helmut had to eliminate irrelevant information (concert problem, km problem, Christmas stars problem, milk problem) were all incorrect (see 0/4 or 0% relevance factor Appendix).

In addition word problems based on number sense (N-problems) were correct in the case the flyer problem, but not in the case of the car problem, 27 near?, 99 near? and in the case of the bus problem (see 1/5 or 20% number sense Appendix).

Helmut often misjudged his own results and got a score of 85/160 or 53% on evaluation (Ev) (see Appendix). It took Helmut 40 minutes to complete the EPA2000. Helmut's cognitive and metacognitive profile was computed. Based upon the results of 550 third graders without learning problems (see in the graphic display in the Appendix) we were able to interpret Helmut's (see *) graphic display (Desoete, Roeyers, Buysse, De Clercq, 2000).

Summarizing the data, we found that Helmut's cognitive strengths were his numerical comprehension and production (NR), his symbol comprehension and production (S), his insight into the structure of the numbers (K) and his capacity for analyzing linguistic information (L). His cognitive weaknesses were dealing with addition contextual information (C), mental representation of the answer through visualization (V), selecting relevant information (R) and estimating in number sense tasks (N). As to the off-line metacognitive skills, we found Helmut retarded on prediction (Pr) skills but even more retarded on evaluation (By) skills. The following instructional recommendations could therefore be given: We recommended that Helmut receive comprehensive cognitive strategy instruction in coping with contextual cues (C), in problem representation strategies or visualization (V), in selecting relevant information (R) and in dealing with number sense (N). Furthermore, we recommended reflection moments after the mathematic al problem solving, to increase the prediction (Pr) but especially also the boy's evaluation skills (Ev). This intervention took place, in close collaboration with the teacher, in a rehabilitation center twice a week in two 30-mitt-sessions for one year.

Conclusion

The EPA2000 makes it possible for the teacher to obtain a fair intra-individual picture of the cognitive processes involved in mathematical problem solving of third grade children with or without mathematics learning disabilities, in order to analyze problem solving mistakes. The profile summarizes students' strengths and weaknesses and facilitates interpretation of the data, by graphing the scores from the scoring form. This allows instructional recommendations to be made. EPA2000 in this manner provides a picture of the number comprehension and production (NR), the operation symbol comprehension and production (S), the number system knowledge (K) and the capacities to calculate (P). We are furthermore able to note whether the problems with word problems are due to inadequate language-related strategies (L), problems to deal with context information (C) or whether they are due to inadequate mental problem representation and visualization (V). Furthermore, we obtain a picture of students' cognitive capacitie s to eliminate irrelevant information (R) as well as of the number sense skills (N) of third-graders. Furthermore, EPA2000 Student Profile facilitates the interpretation of the metacognitive prediction (Pr) and evaluation (Ev) skills, compared with same-age children.

Helmut's performance on the EPA2000 indicated that he was able to read single and multiple digits and comprehend operation symbols without problems. Furthermore, simple word problems based on single-sentence linguistic information without the need for mental representation of that information did not pose any problem for the boy. However, whenever number crunching was no longer adequate and the use of problem representation strategies was necessary, Helmut failed. In addition, he could not cope with contextual information nor could he eliminate irrelevant information or depend on a good number sense. In this way the EPA2000 provided the teacher with information about Helmut's cognitive problem-solving strategies and gave her cues as to a relevant cognitive instructional program for the boy. Furthermore, Helmut's prediction skills were better than his evaluation skills. However, evaluation is necessary to decrease one's impulsivity and to reflect upon one's actions in order to learn in the near future. Helmut should therefore be required to give a rationale for his decisions and answers to instill the notion that decisions and answers should be metacognitively guided.

To sum up, children with mathematics learning disabilities show shortcomings in different cognitive processes (NP, S, K, P, L, C, V, R, N) and in metacognition (Pr, Ev) associated with mathematical problem solving. To focus on the particular problems of students with mathematics learning disabilities and to tailor a relevant instructional program, it is necessary to assess the cognitive and metacognitive strengths and weaknesses of these children. This assessment can easily be done in the classroom, by a teacher in collaboration with a school psychologist. The assessment does not necessitate much computer knowledge. The EPA2000 is a motivating instrument., providing rich information about the processes involved in mathematical problem solving. The student's profile has several educational implications, enabling teachers and therapists in developing relevant instructional programs to optimize students' mathematical insights.

Appendix

Cognitive and metacognitive profile of Helmut

I. Cognitive profile                             51/80-63%
Numeral comprehension and
 production (NR-problems)                       21/22=95%
  Number reading Units                   7/7
  Number reading Tens Units              9/10
  Verbal numerical comprehension         5/5
Symbol comprehension and production
 (S-problems)                                    5/5=100%
Number system knowledge or insight
 into the number structure (K-problems)         8/1 0=80%
Word problems
 Language factor (L-problems)                    5/5=100%
   Simply language factor                3/3
   Language related to temporo-
spatial or order                         2/2
 Context factor (C-problems)                      1/4=25%
 Mental representation or visualization
 factor (V-problems)                              2/5=40%
 Relevance factor (R-problems)                     0/4=0%
 Number sense factor (N-problems)                 1/5=20%
Procedural calculation (P-problems)             13/20=65%
  Arithmetical facts (memory)
    Multiplication arithmetical facts    2/2
    Division arithmetical facts          2/2
  Calculation procedures (domain-
   specific skills)
    Addition                             2/2
    Subtraction                          1/2
    Carry over                           2/2
    Multiplication                       1/3
    Division                             1/2
    Calculation procedures >I00          2/5

II. Metacognitive profile
Prediction (Pr)                                96/160=60%
Evaluation (Ev)                                85/160=53%

Strengths: NR, S, K, L, compared with third graders; P is moderate compared with third graders.

Weaknesses: C, V, R, N, Pr and Ev compared with third grades

Recommendations: Therapy on C, V, R, N. Helping to develop prediction skills before starting mathematical problem-solving. Stimulating evaluating skills after mathematical problem solving tasks.

KEYS

NR Number comprehension and production

S Symbol comprehension and production

K Number System Knowledge

P Procedural calculation

L Dealing with linguistic information

C Dealing with contextual information

V Mental representation, visualization

R Selecting relevant information

N Number sense

Pr Prediction

Ev Evaluation

[TABLE 2 OMITTED]

Table 1

Cognitive and Metacognitive Strategies and Processes

COGNITION

Numeral comprehension and production
(NR)
e.g. Put into the right order from low to high 39 37
38 40
Operation symbol comprehension and production
(S)
e.g. Which is correct? 38+1=39 or 38x1=39
Number system knowledge
(K)
e.g. Complete this series 37 38 39?
Procedural calculation
(P)
e.g. 37+1=?
Language comprehension
(L)
e.g. 1 more than 37 is?
Context comprehension
(C)
e.g. William has 37 keys. James has I key more than
William. How many keys does James have?....
Mental representation visualization
(V)
e.g. 37 is 1 more than?
Selecting relevant information
(R)
e.g. William has 37 keys. James has I key more than
William and 2 keys less than Linda. How many keys
does James have?....
Number sense
(N)
e.g. 37 is nearest to? 47,40,73 or 30

METACOGNITION

Prediction
(Pr)
e.g. Do you think you can solve this exercise?
Evaluation
(Ev)
e.g. Are you sure about this answer?
Table 3

Discriminant Analysis of Off-Line Metacognition in Children With and
Without Mathematics Learning Disabilities in Grade 3

                                                       Group

                                                    Math. LD.

Scale            (max.)       Function
                              Coefficients  M                SD

Prediction       (160)        .48           99.34            18.78
Evaluation       (160)        .60           101.81           18.98

Group centroids                             -.88

Function 1       Eigen value  % variance    Canonical corr.
                 0.79         100           0.66

                           Group

                           No LD.

Scale
                 M               SD

Prediction       125.93          14.30
Evaluation       127.56          11.04

Group centroids  .88

Function 1       Wilks's Lambda
                 0.56

Note: Math. LD. = normally intelligent children with specific
mathematics learning disabilities in grade 3; No LD. = normally
intelligent children without learning disabilities in grade 3.
Figure 4

Intelligence profile of Helmut (d.o.b. 12.07.01)

Verbal Subtests of the WISC-R

I   Information          SS 7
S   Similarities         SS 15
A   Arithmetic           SS 9
V   Vocabulary           SS 12
C   Comprehension        SS 16
D   Digit span           SS 10

Performance subtests

PC  Picture Completion   SS 12
PA  Picture Arrangement  SS 12
BI  Blocks               SS 10
FC  Figure Completion    SS 8
SU  Substitution         SS 9
MA  Mazes                SS 8

Acknowledgements

This study was supported by the Stitching Integratie Gehandicapten (SIG), the Artevelde College Ghent and Centrum ter Bevordering van de Cognitieve Ontwikkeling (CeBCO), to whom the authors extend their thanks.

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