Can Offline Metacognition Enhance Mathematical Problem Solving? Annemie Desoete, Herbert Roeyers, and Armand De Clercq Ghent University The effectiveness of a short metacognitive intervention combined with algorithmic cognitive instruction was assessed in an elementary school setting. Two hundred thirty-seven 3rd-grade children were randomly assigned to a 5-session metacognitive strategy instruction, an algorithmic direct cognitive instruction, a motivational program, a quantitative-relational condition, or a spelling condition. Children in the metacognitive program achieved significant gains in trained metacognitive skills compared with the 4 other conditions. Moreover, the children in the metacognitive program performed better on trained cognitive skills than children in the algorithmic condition, with a follow-up effect on domain-specific mathematics problem-solving knowledge. Despite the consistency of findings, no generalization effects were found on transfer of cognitive learning. Several cognitive processes are required to answer problems like the following: “Kenny has 25 balls. This is 7 balls more than Mary, and 3 balls more than Kjell. How many balls does Mary have?”. First, children need to have adequate number reading. Children need to know that “25” is not 52 or 250 and that “7” is not 4 or 9. Next, children need to understand the language used in the problem. In addition, children need to build an adequate mental representation so that they will not translate “more” as “addition” and answer “32” (25 7 32). Also, children have to be able to execute adequate procedural calculations so that they will not answer “22” (25 – 7 repetition of 2, 7 – 5 2). Some children have difficulty with this task because of problems with the limited capacity of working memory (cognitive overload) and because of an insufficient knowledge base (or expertise) in math achievement (Baddeley, 1999; Keeler & Swanson, 2001; Logie & Gilhooly, 1998; McCloskey & Macaruso, 1995; Schneider & Pressley, 1997; Swanson, 1990). Moreover, some children fall behind in selecting relevant informa- tion to create an adequate mental representation of the problem (Feuerstein, Rand, & Hoffman, 1979; Greenberg, 1990). They can answer “35” (25 7 3) or “15” (25 – 7 – 3), even though the number of balls that Kjell has does not matter. In addition, the importance of number sense in mathematical problem solving has been clearly demonstrated by Sowder (1992) and Verschaffel (1999). In the last few years, various authors have described metacog- nition as essential to mathematical problem solving (e.g., Borkowski, 1992; Carr & Biddlecomb, 1998; De Corte, Ver- schaffel, & Op’t Eynde, 2000). Flavell (1976) defined metacogni- tion as “one’s knowledge concerning one’s own cognitive pro- cesses and products or anything related to them” (p. 232). Simons (1996) combined different metacognitive phenomena into three metacognitive components: namely, metacognitive knowledge, metacognitive skills (e.g., prediction, planning, monitoring, and evaluation), and metacognitive beliefs. Within those beliefs, mo- tivation drives and directs behavior (Heyman & Dweck, 1996) and can be seen as the vehicle for applying metacognitive knowledge and using metacognitive skills (Boekaerts, 1999). A principal-components analysis of metacognition revealed three metacognitive components, explaining 66%– 67% of the common variance (Desoete, Roeyers, & Buysse, 2001). Prediction and evaluation were interrelated as one of these components. Because both of these metacognitive skills were measured either before or after problem-solving exercises, we labeled this meta- cognitive component offline (measured) metacognition. In a sam- ple of 165 third graders, we were able to differentiate among various mathematic-ability groups on the basis of the offline metacognitive component (Desoete et al., 2001). Aim and Research Hypotheses Over the past few years, increasing attention has been paid to the idea of outcome measures (Swanson, Hoskyn, & Lee, 1999; Swan- son, O’Shaughnessy, McMahon, Hoskyn, & Sachse-Lee, 1998). It is interesting to hypothesize that metacognition embellishes the instruction of procedural knowledge. In this study, we tested this possibility using the five conditions described below (see Table 1). We expected positive outcomes because several metacognitive skills have been found to be trainable (Efklides, Papadaki, Papantoniou, & Kiosseoglou, 1997; Lucangeli, Cornoldi, & Tellarini, 1998). We chose five instruction variants that worked cumulatively (see Table 1). One potentially contributing aspect was added in each condition. By comparing the results of the children in the five conditions, we aimed to obtain an indication of whether any improvement in mathematical problem solving was due to one of the added components, namely, the metacognitive component. In addition, we investigated whether offline metacognitive skills need to be taught explicitly for the development of mathematical prob- lem solving. We took the option to train prediction (Pr), number reading (NR), procedural calculation (P), language-related (L), and Annemie Desoete, Herbert Roeyers, and Armand De Clercq, Depart- ment of Experimental Clinical and Health Psychology, Ghent University, Ghent, Belgium. This study was supported by the Stichting Integratie Gehandicapten, the Artevelde College Ghent, and Centrum ter Bevordering van de Cognitieve Ontwikkeling, to whom we express our thanks. Correspondence concerning this article should be addressed to Annemie Desoete, Department of Experimental Clinical and Health Psychology, Ghent University, Henri Dunantlaan 2, B 9000 Ghent, Belgium. E-mail: [email protected]Journal of Educational Psychology Copyright 2003 by the American Psychological Association, Inc. 2003, Vol. 95, No. 1, 188 –200 0022-0663/03/$12.00 DOI: 10.1037/0022-0663.95.1.188 188
Transcript
Can Offline Metacognition Enhance Mathematical Problem Solving?
Annemie Desoete, Herbert Roeyers, and Armand De ClercqGhent University
The effectiveness of a short metacognitive intervention combined with algorithmic cognitive instructionwas assessed in an elementary school setting. Two hundred thirty-seven 3rd-grade children wererandomly assigned to a 5-session metacognitive strategy instruction, an algorithmic direct cognitiveinstruction, a motivational program, a quantitative-relational condition, or a spelling condition. Childrenin the metacognitive program achieved significant gains in trained metacognitive skills compared withthe 4 other conditions. Moreover, the children in the metacognitive program performed better on trainedcognitive skills than children in the algorithmic condition, with a follow-up effect on domain-specificmathematics problem-solving knowledge. Despite the consistency of findings, no generalization effectswere found on transfer of cognitive learning.
Several cognitive processes are required to answer problemslike the following: “Kenny has 25 balls. This is 7 balls more thanMary, and 3 balls more than Kjell. How many balls does Maryhave?”. First, children need to have adequate number reading.Children need to know that “25” is not 52 or 250 and that “7” isnot 4 or 9. Next, children need to understand the language used inthe problem. In addition, children need to build an adequate mentalrepresentation so that they will not translate “more” as “addition”and answer “32” (25 � 7 � 32). Also, children have to be able toexecute adequate procedural calculations so that they will notanswer “22” (25 – 7 � repetition of 2, 7 – 5 � 2). Some childrenhave difficulty with this task because of problems with the limitedcapacity of working memory (cognitive overload) and because ofan insufficient knowledge base (or expertise) in math achievement(Baddeley, 1999; Keeler & Swanson, 2001; Logie & Gilhooly,1998; McCloskey & Macaruso, 1995; Schneider & Pressley, 1997;Swanson, 1990).
Moreover, some children fall behind in selecting relevant informa-tion to create an adequate mental representation of the problem(Feuerstein, Rand, & Hoffman, 1979; Greenberg, 1990). They cananswer “35” (25 � 7 � 3) or “15” (25 – 7 – 3), even though thenumber of balls that Kjell has does not matter. In addition, theimportance of number sense in mathematical problem solving hasbeen clearly demonstrated by Sowder (1992) and Verschaffel (1999).
In the last few years, various authors have described metacog-nition as essential to mathematical problem solving (e.g.,Borkowski, 1992; Carr & Biddlecomb, 1998; De Corte, Ver-schaffel, & Op’t Eynde, 2000). Flavell (1976) defined metacogni-tion as “one’s knowledge concerning one’s own cognitive pro-
cesses and products or anything related to them” (p. 232). Simons(1996) combined different metacognitive phenomena into threemetacognitive components: namely, metacognitive knowledge,metacognitive skills (e.g., prediction, planning, monitoring, andevaluation), and metacognitive beliefs. Within those beliefs, mo-tivation drives and directs behavior (Heyman & Dweck, 1996) andcan be seen as the vehicle for applying metacognitive knowledgeand using metacognitive skills (Boekaerts, 1999).
A principal-components analysis of metacognition revealedthree metacognitive components, explaining 66%–67% of thecommon variance (Desoete, Roeyers, & Buysse, 2001). Predictionand evaluation were interrelated as one of these components.Because both of these metacognitive skills were measured eitherbefore or after problem-solving exercises, we labeled this meta-cognitive component offline (measured) metacognition. In a sam-ple of 165 third graders, we were able to differentiate amongvarious mathematic-ability groups on the basis of the offlinemetacognitive component (Desoete et al., 2001).
Aim and Research Hypotheses
Over the past few years, increasing attention has been paid to theidea of outcome measures (Swanson, Hoskyn, & Lee, 1999; Swan-son, O’Shaughnessy, McMahon, Hoskyn, & Sachse-Lee, 1998). Itis interesting to hypothesize that metacognition embellishes theinstruction of procedural knowledge. In this study, we tested thispossibility using the five conditions described below (see Table 1).We expected positive outcomes because several metacognitive skillshave been found to be trainable (Efklides, Papadaki, Papantoniou, &Kiosseoglou, 1997; Lucangeli, Cornoldi, & Tellarini, 1998).
We chose five instruction variants that worked cumulatively(see Table 1). One potentially contributing aspect was added ineach condition. By comparing the results of the children in the fiveconditions, we aimed to obtain an indication of whether anyimprovement in mathematical problem solving was due to one ofthe added components, namely, the metacognitive component. Inaddition, we investigated whether offline metacognitive skills needto be taught explicitly for the development of mathematical prob-lem solving. We took the option to train prediction (Pr), numberreading (NR), procedural calculation (P), language-related (L), and
Annemie Desoete, Herbert Roeyers, and Armand De Clercq, Depart-ment of Experimental Clinical and Health Psychology, Ghent University,Ghent, Belgium.
This study was supported by the Stichting Integratie Gehandicapten, theArtevelde College Ghent, and Centrum ter Bevordering van de CognitieveOntwikkeling, to whom we express our thanks.
Correspondence concerning this article should be addressed to AnnemieDesoete, Department of Experimental Clinical and Health Psychology,Ghent University, Henri Dunantlaan 2, B 9000 Ghent, Belgium. E-mail:[email protected]
Journal of Educational Psychology Copyright 2003 by the American Psychological Association, Inc.2003, Vol. 95, No. 1, 188–200 0022-0663/03/$12.00 DOI: 10.1037/0022-0663.95.1.188
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mental-representation (M) skills because children with mathemat-ics learning disabilities are deficient in these skills (Campbell,1998; Desoete et al., 2001; Geary, 1993; McCloskey & Macaruso,1995; Montague, 1998; Noel, 2000; Rourke & Conway, 1997;Sowder, 1992; see Table 2 in this article).
We chose to investigate whether these skills could be trained innormal young children so that we could eventually set up a similarstudy of children with mathematics learning disabilities in thefuture. We chose evaluation (Ev), relevance (R), and number sense(N) tasks to measure transfer because these tasks are far-transfertasks that are in no way related to the trained content and, in thecase of the cognitive tasks (R and N), are not even partiallycorrelated with the trained content (Desoete, 2001; Desoete,Roeyers, & De Clercq, 2001).
The general aim of the present study was fourfold. First, weinvestigated the modifiability of offline metacognition in youngchildren. We hypothesized that a metacognitive strategy approach,combined with direct algorithmic cognitive instruction, couldmodify explicitly taught prediction skills in third graders (Hypoth-esis 1). We hypothesized no such improvement in prediction skillsfor the four other conditions or instruction variants.
Second, we explored the extent to which a metacognitive strat-egy approach, combined with direct algorithmic cognitive instruc-tion, adds value and is more effective than one of the four otherinstruction variants. We hypothesized that a metacognitive strategyapproach, combined with direct algorithmic cognitive instruction,could improve the NR, P, L, and M skills in young children(Hypothesis 2).
Third, we investigated the transfer-promoting function of ametacognitive strategy approach. We expected the metacognitivecondition to be the most effective intervention variant in promot-ing far transfer or in applying what is learned to uninstructedmetacognitive (Ev) and cognitive (R and N) mathematicalproblem-solving skills (Hypothesis 3).
Finally, we explored the extent to which a metacognitive strat-egy approach is the most effective intervention variant in promot-ing a follow-up effect 6 weeks later. We expected the metacogni-tive condition to be the most effective intervention variant inpromoting a follow-up effect on trained (P, L, and M) mathemat-ical problem-solving skills (Hypothesis 4).
Method
Participants
Participants were all third-grade children attending seven elementaryschools in the Dutch-speaking part of Belgium. The sample included 237White children: 114 girls and 123 boys. All children followed regularelementary education. Permission for the children to participate in thisstudy was obtained from their parents.
The children had an average intelligence, according to the teacher. Theirmeasured Full Scale IQ ranged between 79 and 135 on the CollectiveVerbal Intelligence Test for Grades 3 and 4 (CIT–34; Stinissen, Smolders,& Coppens-Declerck, 1974) in October of the third grade. The mean IQwas 104.80 (SD � 7.90), with the following raw subscores: GeneralDevelopment, M � 10.96, SD � 1.70; Contradictions, M � 13.08,SD � 2.20; Logical Relations, M � 15.29, SD � 2.68; AnalogicalReasoning, M � 12.42, SD � 2.92; Mathematics, M � 8.94, SD � 3.25;and Shifting, M � 13.63, SD � 3.55.
Participants were ranked on the basis of their pretest scores on theKortrijk Arithmetic Test (for second graders; KRT2; Cracco et al., 1995)and the Arithmetic Number Facts Test (TTR; De Vos, 1992) and were thenrandomly assigned to one of the conditions by the researchers. There weremultiple treatments tested at each school. The metacognitive group con-sisted of 49 children out of five classes. The algorithmic group consistedof 50 children out of five classes. The motivation, math, and control groupsincluded 38, 42, and 58 children, respectively, out of four, five, and fourclasses, respectively.
At the time of pretesting, the participants had a mean age of 99.59months (SD � 3.27 months). On the KRT2, children achieved a standard-ized mean percentile score of 39.78 (SD � 26.18). On the TTR, childrenachieved a standardized mean percentile score of 55.76 (SD � 31.99).Children’s mathematical skills on the Evaluation and Predication Assess-ment (EPA2000; De Clercq, Desoete, & Roeyers, 2000) were 57.63/80(SD � 8.18). The prediction score on EPA2000 was 102.31/160(SD � 16.46), whereas the evaluation score was 106.52/160 (SD � 18.43).In addition, the children read a mean of 39.93 (SD � 7.69) words correctlyin 1 min (Brus & Voeten, 1999).
Measures
Different mathematical problem-solving tests were used as pre- andposttest measures in this study. The KRT2 and the TTR were used to matchthe five groups of children. In addition, the KRT3 was used as an inde-pendent follow-up measure because it included only tasks we trained.
The KRT2 is a 60-item Belgian mathematics test on domain-specificknowledge and skills, resulting in a percentile on mental computation
Note. � � included compound. � � nonincluded compound.
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(especially tasks on P), number system knowledge (especially tasks onlanguage comprehension and visualization), and a total percentile. Thepsychometric value of the KRT2 (norms � June, Grade 2) and the KRT forthird graders (KRT3; norms � January, Grade 3) has been demonstrated ona sample of 381 and 523 Dutch-speaking students, respectively (andon 3,246 children total). In this study, because of different norm periods(October and January, Grade 3), two versions of the KRT were used onthese third graders. In October, the KRT2 was used as a pretest measure.The children were compared with second graders on this measure (percen-tile norms � June, Grade 2). Six weeks after the intervention (which tookplace in November), the KRT3 was used as posttest measure in January(here, percentile norms � January, Grade 3). To compare KRT2 andKRT3, percentiles were used. A validity coefficient (correlation withschool results) and a reliability coefficient (Cronbach’s alpha) of .65 and.90, respectively, for Grade 2. A validity coefficient and reliability coef-ficient of .64 and .91, respectively, were found for Grade 3. Because wefound performances on mental computation (e.g., 129 � 879 � ) andnumber system knowledge (e.g., “add three 10s to 61 and you get ”) onthe KRT2 and KRT3 to be strongly interrelated in our sample (Pearson’sr � .76, p � .01), we used the standardized total percentile based onnational norms. The KRT2 was used as pretest to match the five groups ofchildren. In addition, the KRT3 was used as an independent follow-upmeasure because it included only tasks we trained.
The TTR is a test consisting of 200 arithmetic number fact problems(e.g., 5 � 9 � ). Children have to solve as many number fact problemsas possible out of 200 in 5 min. The test has been standardized for Flanderson 220 third graders (and on 10,059 children total; Ghesquiere & Ruijs-senaars, 1994), but no further psychometric data are available. We used theTTR as a pretest to match the five groups of children, independent of theintervention model.
The One Minute Test (Een Minuut Test or EMT; Brus & Voeten, 1999)is a test of reading fluency for Dutch-speaking people, validated forFlanders on 361 third graders (and on 3,462 children total; Ghesquiere &Ruijssenaars, 1994), measuring the capacity of children to read correctly asmany words as possible out of 116 words (e.g., leg, car) in 1 min. Avalidity coefficient (correlation with teacher rating) and reliability coeffi-cient (test–retest correlation) of .77 and .91, respectively, were found forGrade 3.
The CIT–34 (Stinissen et al., 1974) is a verbal intelligence test forchildren that is made up of eight subtests, validated for Flanders on 622third graders (and on 3,701 children total). A validity coefficient (correla-tion with school results) and reliability coefficient (with the Kruder Rich-ardson [KR20] formula) of .67 and .95, respectively, were found.
The EPA2000 (De Clercq, Desoete, & Roeyers, 2000) is a computerizedprocedure for assessing various cognitive (e.g., NR, P, L, M, dealing withR and N; see Table 2) and metacognitive (Pr and Ev; see Figures 1 and 2)processes associated with mathematical problem solving in elementaryschool children. The psychometric value has been demonstrated on asample of 550 Dutch-speaking third graders (Desoete, Roeyers, & DeClercq, 2002). Moreover, Cronbach’s alpha reliability analyses revealedalphas of .74, .89, and .85 for Pr, mathematical cognition, and Ev, respec-tively, for the total test (80 items; Desoete & Roeyers, 2002). In addition,in a previous study on 1,336 third-grade children, partial correlations werecomputed between all cognitive skills. No partial correlations were foundbetween R and N tasks or between NR, L, and M tasks (Desoete, Roeyers,& Buysse, 2000). Moreover, P, L, and M tasks differentiated children withmathematics learning disabilities from children with mathematics learningproblems and children with age-adequate mathematical problem-solvingskills. Therefore, in this intervention study, two cognitive composite scoreswere computed, namely trained cognitive content (NR, P, L, and M skills;
Figure 1. Assessment sample of the trained metacognitive content (prediction). Metacognitive predictionswere awarded 2 points whenever they corresponded to the child’s actual performance on the task (e.g., doing theexercise correctly and rating “I am absolutely sure that I will solve the exercise correctly”). Predicting “I amquite sure that I will solve the exercise correctly” or “I am quite sure that I will solve the exercise wrong”received 1 point whenever the prediction corresponded with the child’s performance. Other answers did not gainany points, as they represented a lack of predictive metacognition. Trained metacognitive content had amaximum score of 160 points. Traffic light for “I am absolutely sure that I will solve the exercise correctly” �two green lights; traffic light for “I am quite sure that I will solve the exercise correctly” � one green light; trafficlight for “I am quite sure that I will solve the exercise wrong” � one red light; traffic light for “I am absolutelysure that I will solve the exercise wrong” � two red lights.
190 DESOETE, ROEYERS, AND DE CLERCQ
see Figure 3) and nontrained cognitive content (R and N skills; see Figure4). The two metacognitive scores (Pr and Ev skills) were also used.Cronbach’s alphas of .78, .74, .59, and .85 were found for trained cognitivecontent (NR, P, L, and M skills, with a maximum score of 38 points),trained metacognitive content (Pr skills, with a maximum score of 160points), nontrained cognitive content (R and N skills, with a maximumscore of 9 points), and nontrained metacognitive content (Ev skills, with amax score of 160 points), respectively (see Figure 2).
The motivation scale was made especially for this study and included aranking of 10 lessons on a 5-point rating scale (e.g., “I very much like amathematics lesson” vs. “I very much dislike a mathematics lesson”). Theinstrument was tested in a pilot study (N � 30) to determine the usefulness
for this age group and its sensitivity in measuring individual differences.Analyses showed that students in Grade 3 could handle the instrumentswell. Students were interviewed after the test about their reasons for theiranswers. The given answers all referred to the constructs in question.Furthermore, test–retest correlations of .81 ( p � .01) and interrater reli-abilities varying between .99 and 1.00 ( p � .01) were found.
Procedure
A pretest–intervention–posttest–follow-up design with control groupswas used in this study. The inclusion of five groups was important toensure that any treatment effect obtained by the metacognitive group could
Figure 2. Assessment sample of the nontrained metacognitive content (evaluation). Metacognitive evaluationswere awarded with 2 points whenever they corresponded to the child’s actual performance on the task (e.g.,doing the exercise correctly and rating “I am absolutely sure that I have done the exercise correctly”). Evaluating“I am quite sure that I have done the exercise correctly” or “I am quite sure that I have done the exercise wrong”received 1 point whenever the prediction corresponded with the child’s performance. Other answers did not gainany points, as they represented a lack of evaluation skills. Nontrained metacognition had a maximum score of160 points. Traffic light for “I am absolutely sure that I have done the exercise correctly” � two green lights;traffic light for “I am quite sure that I have done the exercise correctly” � one green light; traffic light for “Iam quite sure that I have done the exercise wrong” � one red light; traffic light for “I am absolutely sure thatI have done the exercise wrong” � two red lights.
Figure 3. Assessment sample of trained cognitive content (language task). For the trained cognitive mathe-matical problem solving, children obtained l point for every correct answer. Trained content had a maximumscore of 38 points.
191METACOGNITION AND MATHEMATICAL PROBLEM SOLVING
be attributed to the metacognitive strategy instruction rather than to otherfactors such as algorithmic direct instruction (as in Count City), motivationexperiences (as in the motivation group), quantitative-relation experience(as in the math group), or participation in a small group interventionprogram (as in the control group; see Table 1 and Figure 5). In addition,trainers and teachers were double blind with regard to the research ques-tions of this study.
The pretest battery consisted of a measurement of verbal intelligence(CIT–34), as well as an assessment of domain-specific mathematics knowl-edge (KRT2), mathematical number facts (TTR), and combined mathe-matical cognition and offline metacognition (EPA2000). Verbal intelli-gence was measured because in a previous study, in 437 third-gradechildren, a significant correlation was found between mathematical prob-lem solving and verbal intelligence (r � .15, p � .01) but not betweenmathematics and Performance IQ (r � .08, p � .12; Desoete & Roeyers,2002). In addition, we measured how many words could be read in 1 min(EMT) so that we knew whether additional help in reading the instructionswould be necessary. Finally, trained metacognitive content (Pr; see Figure1), trained cognitive content (NR, P, L, and M; see Figure 3), nontrainedmetacognitive content (Ev; see Figure 2), and nontrained cognitive content(R and N; see Figure 4) were measured with EPA2000.
The intervention took place in small groups (of about 10 children) inseparate classrooms five times in 2 weeks for 50 min each time. Eachsession consisted of the mathematics problems in accordance with theinstructions given in the program. The metacognitive experimental group(Number Town) was compared with four other instruction variants. In themetacognitive (Number Town) and cognitive (Count City) training, NR, P,M, and L skills were explicitly taught as trained cognitive content. In themotivation and math conditions, children also did exercises on these NR,P, M, and L tasks, without the tasks being in accordance with our concep-tual framework. Moreover, Pr was explicitly taught in the metacognitivegroup and is further referred to as trained metacognitive content. None ofthe five training sessions worked on tasks dealing with R or N, and thiscontent is therefore referred to below as nontrained cognitive content.Finally, none of the five types of training focused on Ev, and this contentis therefore referred to below as nontrained metacognitive content. Duringthe 2-week period that spanned the treatment manipulation, the children did
not receive any metacognitive strategy instruction from their ordinaryclassroom teacher.
The posttest outcome measure was administered to the participantsafter 5 hr of training. This posttest included EPA2000 measures of trainedmetacognitive content (Pr), trained cognitive content (NR, P, L, and M),nontrained metacognitive content (Ev), and nontrained cognitive content(R and N). In addition, because motivation can be seen as the vehicle formetacognition and a motivation group was included, we asked the childrento complete a motivation rating scale to evaluate how they experienced theintervention.
The follow-up test, 6 weeks after the last intervention, consisted of anindependent assessment of domain-specific mathematical problem solving(KRT3) that was neither created by us nor related to our model. The KRT3especially measured trained content (P, L, and M; see Table 2).
Teacher Training and Treatment Integrity
Four paraprofessionals were trained to teach all of the five instructionvariants (metacognitive intervention, cognitive intervention, computerizedmotivational intervention, math intervention, and spelling intervention).Each paraprofessional participated in three instruction variants. All para-professionals were skilled therapists and had experience with children withmathematics learning disabilities. Initial paraprofessional training tookplace 1 month prior to the start of the interventions. The paraprofessionalswere trained over 10 hr in total.
In addition, systematic ongoing supervision and training was providedduring the interventions. During initial training, the paraprofessionalslearned about current conceptions of mathematical problem solving andworked through the prepared training manual. Ongoing training includedreview and discussion of the next session plan and objectives and feedbackon the past session.
During and after the intervention, Annemie Desoete visited each class-room. Condition integrity was evaluated throughout the study by directobservation and semistructured interviews of the paraprofessionals before,during, and after each intervention session. The level of treatment integritywas obtained by calculating the percentage of treatment components im-plemented as designed over the 2 weeks of the study. Throughout inter-
Figure 4. Assessment sample of the nontrained cognitive content (number sense task). For the nontrainedcognitive mathematical problem solving, children obtained l point for every correct answer. Nontrainedcognitive content had a maximum score of 9 points.
192 DESOETE, ROEYERS, AND DE CLERCQ
ventions and across paraprofessionals, treatment integrity was very high,and a 97% fidelity to essential instructional practices was found. We alsoobserved that no metacognitive instruction was given by the regular schoolteacher during the 2-week training period.
Overview of Intervention Procedures
Each of the metacognitive (Number Town) sessions involved a direct Prstrategy as well as a direct cognitive (NR, P, L, and M) instruction (seeAppendix A). The tasks were especially created for the metacognitive and
cognitive group. This metacognitive training was verbal in nature andfocused on prediction of task difficulty (see Figure 1) as well as on thetasks and problem-solving procedures themselves (see Figures 3 and 5).Each session in the metacognitive condition started with an orientation orrehearsal phase. Then, the need for a metacognitive principle was experi-enced and brought about in small group sessions. We aimed to have 10children in each group. The children experienced the metacognitive train-ing as a motivating intervention because all children scored 4 or 5 on a5-point motivation rating scale for the intervention sessions.
Figure 5. Intervention sample procedural calculations tasks. The church is the metacognitive intervention, andthe house is the cognitive intervention.
193METACOGNITION AND MATHEMATICAL PROBLEM SOLVING
The algorithmic cognitive training (Count City) used exactly the sameexercises (house in Figure 5) as the metacognitive group (church in Figure5). There was direct cognitive instruction of NR, P, M, and L tasks (seeAppendix B) without Pr strategy teaching. A step-by-step presentation ofthe problems was used, without a prediction of task difficulty. The aim ofthe cognitive condition was to increase the mathematical problem-solvingskills in small group sessions through direct instruction without metacog-nitive strategy support. We aimed to have 10 children in each group. Thechildren experienced the cognitive training as a motivating interventionbecause all children scored 4 or 5 on a 5-point motivation rating scale forthe intervention sessions.
The computer-assisted training made use of most of the motivatingexercises in small group sessions on mathematical problem solving inGrade 3, without direct or strategy instruction. We aimed to have 10children in each group. Therefore, 100 mathematics therapists were con-sulted to select the five most attractive NR, P, L, and M exercises. Theirselections were five computerized math software programs: Multi (Dain-amic, 1992b), Top 100, Part 2 (De Winter & Witters, 1998a), Arithmic(Dainamic, 1992a), Top 100, Part 4 (De Winter & Witters, 1998b), andTempo (Dainamic, 1992c). The children worked with this software (oneprogram each session) in small group sessions (about 10 children in acomputer classroom, each child on a computer). The children experiencedthe computer training as a motivating intervention because all childrenscored 4 or 5 on a 5-point motivation rating scale for the interventionsessions.
With the math group, we investigated whether simple mathematicalproblem solving was sufficient to make children better problem solvers.Here, 100 mathematical therapists were consulted, and the most usedexercises for children in Grade 3 were selected and presented to thechildren in small groups. We aimed to have 10 children in each group. Theselection was five combinations of paper-and-pencil exercises. The mathtraining was not experienced as more motivating than ordinary mathsessions, as all children scored 2 or 3 on a 5-point motivation rating scalefor the intervention sessions.
Control subjects (control group) received the same amount of instruc-tional time as the children in the other four conditions. However, instead ofmath instruction, the control group received five sessions in small groupson the correct analysis of words in spelling and reading activities. Weaimed to have 10 children in each group. The control training was notexperienced as more motivating than ordinary math sessions, as all childrenscored 2 or 3 on a 5-point motivation rating scale for the interventionsessions. All participants received the same amount of instructional time(1st week � 3 � 50 min; 2nd week � 2 � 50 min).
Results
Preliminary Comparisons
Preliminary comparisons revealed that the children in the fiveconditions did not differ significantly in proportions of female andmale participants, �2(1, N � 237) � 0.34, p � .56. However, thechildren in the five conditions differed significantly in total intel-ligence quotient on the CIT–34, F(4, 232) � 3.21, p � .05, �2 �.05 (see Table 3). Tukey comparisons revealed that both computer-trained participants and the control group outperformed the meta-cognitive group on Full Scale IQ.
In addition, pretest scores and additional subscores were com-pared. The multivariate analysis of variance (MANOVA) with thetwo EPA2000 pretest mathematical problem-solving subscores(trained cognitive content and nontrained cognitive content) as thedependent variables and the condition (metacognitive condition,cognitive condition, motivation condition, math condition, andcontrol condition) as the independent variable was not significanton the multivariate level, F(8, 462) � 0.79, p � .61, �2 � .01.Moreover, the MANOVA with the two EPA2000 pretest metacog-nitive subscores (trained metacognitive content and nontrainedmetacognitive content) as the dependent variables and the condi-tion (metacognitive condition, cognitive condition, motivationcondition, math condition, and control condition) as the indepen-dent variable was not significant on the multivariate level, F(8,462) � 0.98, p � .45, �2 � .01. In addition, the analysis ofvariance with the KRT2 pretest percentile scores (to be used asfollow-up measure) as the dependent variable and the condition(metacognitive condition, cognitive condition, motivation condition,math condition, and control condition) as the independent variablewas not significant on the multivariate level, F(4, 232) � 0.81, p �.52, �2 � .03. The same results were found for the TTR pretest scores,F(4, 232) � 1.39, p � .24, �2 � .01, and for the EMT pretest scores,F(2, 232) � 1.65, p � .16, �2 � .03. Therefore, the five groups seemto be matched on mathematics and reading pretest scores.
Treatment Effects
To investigate the research hypotheses on the modifiability ofoffline metacognition (Hypothesis 1) as well as on the additional
Table 2Cognition and Metacognition
Item Training
Cognition
Numeral comprehension and production: “Read 5 or 29.” TProcedural calculation: “29 � 5 � .” TLanguage comprehension: “5 more than 29 is .” TMental representation: “29 is 5 more than .” TSelecting relevant information: “Wanda has 29 keys. Willy has 5 keys
more than Wanda and 2 keys less than Linda. How many keys doesWilly have?” N
Number sense: “29 is nearest to . Choose between 5, 20, 90, or 92.” N
Offline metacognition
Prediction: “Do you think you can solve this exercise?” TEvaluation: “Are you sure about this answer?” N
Note. T � trained; N � not trained.
194 DESOETE, ROEYERS, AND DE CLERCQ
Table 3Pre- and Posttest Characteristics of the Children in the Different Conditions
Note. Different subscripts refer to significant post hoc between-groups differences at p � .05. KRT2 �Kortrijk Arithmetic Test (for second graders); KRT3 � Kortrijk Arithmetic Test (for third graders); TTR �Arithmetic Number Facts Test.a Refers to the trained metacognitive content, with a maximum of 160. b Refers to the nontrained metacognitivecontent, transfer, with a maximum of 160. c Refers to the trained cognitive content, with a maximumof 38. d Refers to the nontrained cognitive content, transfer, with a maximum of 9. e Refers to the trainedcognitive content, follow-up, with a maximum of 100.* p � .05. ** p � .01.
195METACOGNITION AND MATHEMATICAL PROBLEM SOLVING
value of metacognitive instruction on the learning of cognitiveskills (Hypothesis 2), we measured trained content posttest scores(trained cognitive content, trained metacognitive content). Depen-dent measures were analyzed separately via a 5 (condition: meta-cognitive condition, cognitive condition, motivation condition,math condition, and control condition) � 2 (time: pretest, posttest)univariate analysis of covariance (ANCOVA), with repeated mea-sures on the second factor and intelligence as covariate. EachANCOVA determined whether significance existed among thefive conditions when compared on the dependent measure atpretesting and posttesting simultaneously. We were especiallyinterested in the Condition � Time interaction.
In addition, if there was a significant Condition � Time inter-action effect in the ANCOVAs, then we performed post hoc testson the posttest scores using an appropriate post hoc procedure(Tukey if equal variance could be assumed from the Levene testand Tamhane if equal variance could not be assumed from theLevene test). In addition, we calculated the observed power andeffect sizes.
It should be noted that preliminary analyses with the trainer inthe model as a second between-subject variable yielded no signif-icant main effects for the trainer ( p � .05) or the Trainer �Condition interaction ( p � .05) across all dependent posttestmeasures (trained cognitive content, trained metacognitive con-tent, nontrained cognitive content, and nontrained metacognitivecontent). Similarly, preliminary analyses with gender or school inthe model as a second between-subject variable yielded no signif-icant main effects or interactions across all dependent posttestmeasures ( p � .05). Thus, trainer, gender, and school were notconsidered further in the analyses.
In addition, all paper-and-pencil pretest, posttest, and follow-uptests were administered in the regular classroom (with an averageof 29 children in one class). In every classroom, there werechildren who participated in different treatment groups. The ex-perimenters were blind with respect to the treatment condition. Thecomputerized EPA2000 was administered on an individual basis.
Trained Metacognitive Content
To investigate the modifiability of offline metacognition, weanalyzed trained metacognitive content (or Pr) via a 5 (condition:metacognitive condition, cognitive condition, motivation condi-tion, math condition, and control condition) � 2 (time: pretest,posttest) univariate ANCOVA, with repeated measures on thesecond factor and intelligence as covariate. Moreover, post hocanalyses were conducted using the Tamhane procedure becauseequal variance could not be assumed: Levene F(4, 232) � 2.33,p � .05.
A significant interaction effect with a medium effect sizeemerged for Condition � Time, F(4, 231) � 160.38, p � .01,�2 � 0.74, power � 1.00, and a significant interaction effect witha small effect size emerged for Intelligence � Time, F(1,232) � 17.52, p � .01, �2 � 0.07, power � .99. In addition, asignificant main effect with a small magnitude emerged for con-dition, F(4, 231) � 9.12, p � .01, �2 � 0.14, power � .99; time,F(1, 231) � 23.82, p � .01, �2 � 0.09, power � .99; andintelligence, F(1, 231) � 58.82, p � .01, �2 � 0.20, power � 1.00.Means and standard deviations for the posttest are presented inTable 3.
The metacognitive trained children did better than the childrenin the other four conditions on this measure (see summary of posthoc analyses in Table 3). This measure indicated that the meta-cognitive group successfully learned the specific metacognitivecontent of their program, whereas the cognitive group did notspontaneously gain metacognitive insights while working on cog-nitive content.
Trained Cognitive Content
A second aim was to determine whether the metacognitivecondition had some value added, compared with the cognitivecondition, in promoting cognitive learning on NR and productiontasks, P tasks, L tasks, and M tasks. These tasks were includedin the training and considered as trained cognitive content (seeFigure 5).
To investigate the modifiability of cognitive skills, we analyzedtrained cognitive content via a 5 (condition: metacognitive condi-tion, cognitive condition, motivation condition, math condition,and control condition) � 2 (time: pretest, posttest) univariateANCOVA, with repeated measures on the second factor andintelligence as covariate. Moreover, post hoc analyses were con-ducted using the Tamhane procedure because equal variance couldnot be assumed: Levene F(4, 232) � 8.60, p � .01.
A significant interaction effect with a medium effect size wasfound for Time � Condition, F(4, 231) � 41.84, p � .01,�2 � 0.42, power � 1.00. Moreover, a significant interactioneffect with a small effect size was found for Intelligence � Time,F(1, 232) � 7.87, p � .01, �2 � 0.03, power � .80. In addition,a significant main effect with a small effect size emerged forcondition, F(4, 231) � 5.30, p � .01, �2 � 0.08, power � .97;time, F(1, 231) � 9.55, p � .05, �2 � 0.04, power � .87; andintelligence, F(1, 231) � 43.13, p � .01, �2 � 0.16, power � 1.00.Mean scores and standard deviations for the posttest are presentedin Table 3. The metacognitive trained children did better than thechildren in the other four conditions on this cognitive contentmeasure (see summary of post hoc analyses in Table 3). Nodifferences were found between children in the cognitive conditionand children in the motivation condition, math condition, or con-trol condition. The metacognitive group successfully learned thespecific cognitive content (see Figure 5) of their metacognitiveprogram. In addition, the cognitive group did not perform betterthan the children in the other three conditions on NR, P, L, and Mtasks, although these contents (see Figure 5) were taughtalgorithmically.
Generalization or Transfer
To investigate Hypothesis 3 on the generalization or metacog-nitive and cognitive transfer of mathematical problem-solvingskills, we measured nontrained content posttest scores (nontrainedcognitive content and nontrained metacognitive content). Depen-dent measures were analyzed separately via a 5 (condition: meta-cognitive condition, cognitive condition, motivation condition,math condition, and control condition) � 2 (time: pretest, posttest)univariate ANCOVA, with repeated measures on the second factorand intelligence as covariate. Each ANCOVA determined whethersignificance existed among the five conditions when compared onthe dependent measure at pretesting and posttesting simulta-
196 DESOETE, ROEYERS, AND DE CLERCQ
neously. In addition, if there was a significant Condition � Timeinteraction effect in the ANCOVA, then we performed post hoctests on the posttest scores using an appropriate post hoc procedure(Tukey if equal variance could be assumed and Tamhane if equalvariance could not be assumed). In addition, we calculated theobserved power and effect sizes.
Nontrained metacognitive content. One of the aims of thisinvestigation was also to evaluate the metacognitive transfer. Todo so, we investigated whether the metacognitive training, focus-ing on metacognitive Pr skills, also had a transfer effect onmetacognitive Ev skills. Therefore, nontrained content (or Evscores on EPA2000) was analyzed via a 5 (condition: metacogni-tive condition, cognitive condition, motivation condition, mathcondition, and control condition) � 2 (time: pretest, posttest)univariate ANCOVA, with repeated measures on the second factorand intelligence as covariate. Moreover, post hoc analyses wereconducted using the Tukey procedure: Levene F(4, 232) � 2.28,p � .06 (see Table 3).
A significant interaction effect with a small effect size wasfound for Time � Condition, F(4, 231) � 13.86, p � .01,�2 � 0.21, power � 1.00, but not for Time � Intelligence, F(1,232) � 2.86, p � .09. In addition, a significant main effect with asmall effect size emerged for time, F(1, 232) � 4.15, p � .05,�2 � 0.09, power � .99; condition, F(4, 231) � 3.34, p � .05,�2 � 0.06, power � .84; and intelligence, F(1, 231) � 40.13, p �.01, �2 � 0.15, power � 1.00. Mean scores and standard devia-tions for the posttest are presented in Table 3.
There was a significant difference between children in themetacognitive condition and children in the control condition (seesummary of post hoc analyses in Table 3). The metacognitivegroup learned the specific content of the sessions (trained meta-cognitive content and trained cognitive content), but only signifi-cantly more metacognitive (nontrained metacognitive content)generalization of learning took place in the metacognitive condi-tion compared with the control condition.
Nontrained cognitive content. We also addressed the criticalissue of cognitive transfer. To do so, we investigated whether themetacognitive training, focusing on NR and production, P, L, andM skills, had a cognitive transfer effect on mathematical problem-solving skills needed to deal with R and N tasks.
Therefore, nontrained cognitive content was analyzed via a 5(condition: metacognitive condition, cognitive condition, motiva-tion condition, math condition, and control condition) � 2 (time:pretest, posttest) univariate ANCOVA, with repeated measures onthe second factor and intelligence as covariate. We were especiallyinterested in the Condition � Time interaction.
No significant interaction effect was found for Time � Condi-tion, F(4, 231) � 0.91, p � .46, or Time � Intelligence, F(1,232) � 2.42, p � .12. In addition, no significant main effect wasfound for time, F(1, 231) � 3.40, p � .07, or condition, F(4,231) � 2.32, p � .06. However, a significant main effect with asmall effect size was found for intelligence, F(1, 231) � 48.18,p � .01, �2 � 0.17, power � 1.00. Mean scores and standarddeviations for the posttest are presented in Table 3.
As shown in Table 3, the metacognitive group learned thespecific content of the sessions (trained cognitive content andtrained metacognitive content), but significantly more cognitive(nontrained cognitive content) generalization of learning did nottake place than in the four other conditions.
Follow-Up Data, 6 Weeks After the Training
An important aim of the present study was to assess sustainedgrowth in mathematical problem-solving skills after the trainingtook place. Therefore, we used a nationally standardized measurethat was independent of our conceptual model, upon which themetacognitive and cognitive training were built (see Table 1). Thisassessment took place 6 weeks after the training and can beconsidered a measure of sustained mathematical problem-solvinggrowth.
To compare mathematical problem solving in the five condi-tions, we conducted a univariate ANCOVA, with condition (meta-cognitive condition, cognitive condition, motivation condition,math condition, and control condition) as the between-subjectfactor, posttest scores on the KRT3 as the dependent variable, andpretest scores on the KRT2 and intelligence as the covariates. Inthe ANCOVA, there was a significant main effect with a mediummagnitude (�2 � 0.69, power � 1.00) for condition, F(4, 230) �127.54, p � .01. Moreover, a significant effect with a high mag-nitude (�2 � 0.94, power � 1.00) was found for the mathematicspretest scores, F(1, 230) � 3,698.24, p � .01, but not for intelli-gence, F(1, 230) � 0.05, p � .82.
This significant main effect for condition was further analyzedusing Tamhane post hoc multiple comparisons, Levene F(4,232) � 3.00, p � .02. There were significant differences betweenchildren in the metacognitive group and children in the otherconditions on the posttest scores (see summary of post hoc anal-yses in Table 3). The children in the metacognitive conditionoutperformed the four other conditions.
Discussion
The central question underlying this study was whether offlinemetacognition could embellish instruction of procedural knowl-edge. Results show that it can.
First, children in the metacognitive group had significantlyhigher posttest mathematical problem-solving scores (trained cog-nitive content). The positive treatment outcomes were obtained byadding an aspect of offline metacognition onto mathematicalproblem-solving treatments.
Second, children in the metacognitive group had higher posttestprediction scores than children in the other four groups. Predictionseemed to be a modifiable metacognitive skill. Moreover, in ourstudy of the other (nonmetacognitive) groups, no such improve-ment was found. Evidently, offline metacognitive skills or strate-gies need to be taught explicitly to develop.
Third, the metacognitive group outperformed the spelling groupon metacognitive evaluation (nontrained metacognitive content),but no significant differences were found among the five groups onN and R problem-solving tasks (nontrained cognitive content).That is, very limited or absent generalization of learning tookplace. The lack of transfer or generalization of cognitive skillscould be due to the limited number of items (only nine items; seeTable 3) or to the lack of partial correlations between N, R, L, andM tasks (Desoete, Roeyers, & Buysse, 2000). It might also be thatthis lack of effect was due to the limited number of trainingsessions and to the fact that all metacognitive and cognitive skillshave to be taught explicitly and cannot be assumed to developfrom freely experiencing mathematics.
197METACOGNITION AND MATHEMATICAL PROBLEM SOLVING
Fourth, children in the metacognitive group performed betterthan children in the other four groups on the follow-up measure.Metacognitive instruction had a sustained effect on cognitive prob-lem solving 6 weeks after the training.
These results should be interpreted with care because there areseveral limitations to the present study. First, extrapolating ourconclusions and excluding possible alternative explanations, ourstudies need to be replicated with a sample of children withmathematics learning disabilities. Second, the interventions wereimplemented for a very brief period of time. The interventions tookplace for five sessions. We opted for this design because wefocused only on prediction skills—and did not want to train allmetacognitive skills—to determine what triggered the modifica-tion of skills. Finally, another limitation of this study was that theinterventions were implemented by paraprofessionals instead ofclassroom teachers. In reality, paraprofessionals are widely used toteach remedial instruction to students with learning disabilities.With adequate training and ongoing supervision, paraprofessionalsin this study could successfully modify metacognitive predictionskills in young children.
Some questions about metacognition and mathematical problemsolving remain unresolved. For example, metacognition might beage-dependent and still maturing until adolescence (Berk, 1997).The empirically demonstrated metacognitive components, there-fore, still need full explanation from more applied research ondifferent age groups. In addition, the other parameters included inmetacognition and the relationship between cognition, metacogni-tion, motivation, and emotion require additional research (Boek-aerts, 1999). Moreover, some questions about the cognitive com-ponents of mathematical problem solving remain unresolved. Forexample, additional research is necessary on the impact of workingmemory on the ability to hold and process highly contextualizedproblems. Similarly, more information is needed on the impact ofknowledge base on the ability to process a problem embeddedwithin a familiar or unfamiliar context (Keeler & Swanson, 2001).
References
Baddeley, A. (1999). Working memory. Oxford, England: Clarendon Press.Berk, L. E. (1997). Child development. Boston: Allyn & Bacon.Boekaerts, M. (1999). Metacognitive experiences and motivational state as
aspects of self-awareness. Review and discussion. European Journal ofPsychology of Education, 14, 571–584.
Borkowski, J. G. (1992). Metacognitive theory: A framework for teachingliteracy, writing, and math skills. Journal of Learning Disabilities, 25,253–257.
Brus, B. T., & Voeten, M. J. M. (1999). Een Minuut Test [One MinuteTest]. Lisse, the Netherlands: Swets & Zeitlinger.
Campbell, J. I. D. (1998). Linguistic influences in cognitive arithmetic:Comment on Noel, Fias, and Brysbaert (1997). Cognition, 67, 353–364.
Carr, M., & Biddlecomb, B. (1998). Metacognition in mathematics from aconstructivist perspective. In D. J. Hacker, J. Dunloksy, & A. C.Graesser (Eds.), Metacognition in educational theory and practice (pp.69–91). Mahwah, NJ: Erlbaum.
Cracco, J., Baudonck, M., Debusschere, A., Dewulf, B., Samyn, F., &Vercaemst, V. (1995). Kortrijkse Rekentest [Kortrijk Arithmetic Test].Kortrijk, Belgium: Revalidatiecentrum Overleie.
Dainamic. (1992a). Aritmix [Computer software]. Retrieved from http://www.dainamic.be
Dainamic. (1992b). Multi [Computer software]. Retrieved from http://www.dainamic.be
Dainamic. (1992c). Tempo [Computer software]. Retrieved from http://www.dainamic.be
De Clercq, A., Desoete, A., & Roeyers, H. (2000). EPA2000: A multilin-gual, programmable computer assessment of off-line metacognition inchildren with mathematical learning disabilities. Behavior ResearchMethods, Instruments, & Computers, 32, 304–311.
De Corte, E., Verschaffel, L., & Op’t Eynde, P. (2000). Self-regulation. Acharacteristic and a goal of mathematics education. In M. Boekaerts,P. R. Pintrich, & M. Zeidner (Eds.), Handbook of self-regulation (pp.687–726). San Diego, CA: Academic Press.
Desoete, A. (2001). Off-line metacognition in children with mathematicslearning disabilities. Unpublished doctoral dissertation, Rijks Univer-siteit Gent, Ghent, Belgium.
Desoete, A., & Roeyers, H. (2002). Off-line metacognition. A domain-specific retardation in young children with learning disabilities? Learn-ing Disabilities Quarterly, 25, 123–139.
Desoete, A., Roeyers, H., & Buysse, A. (2000). Achtjarigen, waarbijrekenen nooit routine wordt. Rekenstoornissen in Vlaanderen [Eight-year-olds for whom arithmetic never becomes routine. Mathematics andlearning disabilities in Flanders]. Tijdschrift Voor Orthopedagogiek, 10,430–441.
Desoete, A., Roeyers, H., & Buysse, A. (2001). Metacognition and math-ematical problem solving in Grade 3. Journal of Learning Disabilities,34, 435–449.
Desoete, A., Roeyers, H., & De Clercq, A. (2001, August 31). Themodifiability of cognition and metacognition in young children withmathematics learning disabilities. Paper presented at the European As-sociation for Research on Learning and Instruction Congress, Fribourg,Switzerland.
Desoete, A., Roeyers, H., & De Clercq, A. (2002). EPA2000. Assessingoff-line metacognition in mathematical problem solving. Focus onLearning Problems in Mathematics, 24, 53–69.
De Vos, T. (1992). Test voor het vaststellen van het rekenvaardigheids-niveau der elementaire bewerkingen (automatisering) voor het basis envoortgezet onderwijs: Handleiding [Test to determine the mathematicsability level for elementary operations (automatization) in primary andsecondary education: Manual]. Nijmegen, the Netherlands: Berkhout.
De Winter, W., & Witters, M. (1998a). Top 100 deel 2 getallenkennis(vervolg). Een programma voor het verkennen van getallen in hethonderdveld [Top 100 Part 2 number knowledge (continuation). Aprogram for the knowledge of numbers up to one hundred] [Computersoftware]. Dilsen-Stokkem, Belgium: OWG-Bureau.
De Winter, W., & Witters, M. (1998b). Top 100 deel 4 optellen enaftrekken tot 100. Een programma voor het verkennen van getallen in hethonderdveld [Top 100 Part 4 addition and subtraction up to 100. Aprogram for the knowledge of numbers up to one hundred] [Computersoftware]. Dilsen-Stokkem, Belgium: OWG-Bureau.
Efklides, A., Papadaki, M., Papantoniou, G., & Kiosseoglou, G. (1997).Effects of cognitive ability and affect on school mathematics perfor-mance and feelings of difficulty. American Journal of Psychology, 110,225–258.
Feuerstein, R., Rand, Y., & Hoffman, M. B. (1979). Theory, instruments,and techniques. Baltimore: University Press.
Flavell, J. H. (1976). Metacognitive aspects of problem solving. In L. B.Resnick (Ed.), The nature of intelligence (pp. 231–236). Hillsdale, NJ:Erlbaum.
Ghesquiere, P., & Ruijssenaars, A. (1994). Vlaamse normen voor studi-etoetsen Rekenen en technisch lezen lager onderwijs [Flemish standardsfor study evaluation or mathematics and technical reading in primaryschool]. Leuven, Belgium: Katholieke Universiteit Leuven [Universityof Leuven], Centrum voor studie en beroepsorientering.
198 DESOETE, ROEYERS, AND DE CLERCQ
Greenberg, K. H. (1990). Mediated learning in the classroom. InternationalJournal of Cognitive Education and Mediated Learning, 1, 33–44.
Heyman, G. D., & Dweck, C. S. (1996). Development of motivation. In E.De Corte & F. E. Weinert (Eds.), International encyclopedia of devel-opmental and instructional psychology (pp. 209–213). Oxford, England:Elsevier.
Keeler, M. L., & Swanson, H. L. (2001). Does strategy knowledge influ-ence working memory in children with mathematical disabilities? Jour-nal of Learning Disabilities, 34, 418–434.
Logie, R. H., & Gilhooly, K. (1998). Working memory and thinking.Current issues in thinking and reasoning. Hour, England: PsychologyPress.
Lucangeli, D., Cornoldi, C., & Tellarini, M. (1998). Metacognition andlearning disabilities in mathematics. In T. E. Scruggs & M. A. Mas-tropieri (Eds.), Advances in learning and behavioural disabilities (pp.219–244). Greenwich, CT: JAI Press.
McCloskey, M., & Macaruso, P. (1995). Representing and using numericalinformation. American Psychologist, 50, 351–363.
Montague, M. (1998). Research on metacognition in special education. InT. E. Scruggs & M. A. Mastropieri (Eds.), Advances in learning andbehavioural disabilities (pp. 151–183). Greenwich, CT: JAI Press.
Noel, M. P. (2000). La dyscalculie developpementale: Un etat de laquestion [Developmental dyscalculia: State of the art]. In M. Peseti & X.Seron (Eds.), Neuropsychologie des troubles du calcul et du traitementdes nombres (pp. 59–84). Marseille, France: Solal.
Rourke, B. P., & Conway, J. A. (1997). Disorders of arithmetics andmathematical reasoning: Perspectives from neurology and neuropsy-chology. Journal of Learning Disorders, 30, 34–45.
Schneider, W., & Pressley, M. (1997). Memory development between twoand twenty (2nd ed.). Mahwah, NJ: Erlbaum.
Simons, P. R. J. (1996). Metacognition. In E. De Corte & F. E. Weinert(Eds.), International encyclopedia of developmental and instructionalpsychology (pp. 436–444). Oxford, England: Elsevier.
Sowder, J. (1992). Estimation and number sense. In D. A. Grouws (Ed.),Handbook of research on arithmetics teaching and learning: A projectof the National Council of Teachers of Arithmetics (pp. 371–387). NewYork: Simon & Schuster.
Stinissen, J., Smolders, M., & Coppens-Declerck, L. (1974). CollectieveVerbale Intelligentietest voor derde en vierde leerjaar (CIT–34) [Col-lective Verbal Intelligence Test for Grades 3 and 4 (CIT–34)]. Brussels,Belgium: Centrum voor studie en beroepsorientering.
Swanson, H. L. (1990). Influence of metacognitive knowledge and aptitudeon problem solving. Journal of Educational Psychology, 82, 306–314.
Swanson, H. L., Hoskyn, M., & Lee, C. (1999). Interventions for studentswith learning disabilities: A meta-analysis of treatment outcomes. NewYork: Guilford.
Swanson, H. L., O’Shaughnessy, T. E., McMahon, C. M., Hoskyn, M., &Sachse-Lee, C. M. (1998). A selective synthesis of single subject inter-vention research on students with learning disabilities. In T. E. Scruggs& M. A. Mastropieri (Eds.), Advances in learning and behavioraldisabilities (pp. 79–126). Greenwich, CT: JAI Press.
Verschaffel, L. (1999). Realistic mathematical modeling and problemsolving in the upper elementary school: Analysis and improvement. InJ. H. M. Hamers, J. E. H. Van Luit, & B. Csapo (Eds.), Teaching andlearning thinking skills: Contexts of learning (pp. 215–240). Lisse, theNetherlands: Swets & Zeitlinger.
Appendix A
Sample Items From the Metacognitive Training (Number Town)
Session 1
The following story is told to the children:
In Number Town, there is a big market with a school and four biglanes (Question Lane with a cinema, Read Lane with the numberlibrary, Big Lane, and Bridge Lane with a baker and a swimmingpool) and four smaller streets (Add Street with a railway station,Remove Street, Times Street, and Division Street).
Three animals live in Number Town: a fast rabbit, a slow turtle, anda cat that estimates whether to be fast or slow according to thesituation. The rabbit lives in the market. The turtle lives on QuestionLane, and the cat lives on Big Lane.
The following questions are asked:If the three animals want to go to the baker while it is quiet in the town,
who would arrive at the bakery first?If the three animals want to go to the movie theater, who would arrive
first if there is a lot of traffic in the village?The principle of the first session is “taking time in advance avoids beingsorry afterward.” This principle is put on the first stage of the number stairsof Number Town.
Session 2
In a second session, the principle of the previous session is reviewed.The following story is presented:
The cat wants to walk in her street. She visits the church and fourstores. The church is full of additions and subtractions with big sizenumbers. The wine store is full of additions with big size numbers.The balloon store has lots of additions with big size numbers. Themarble store has additions and subtractions with small size numbers.The match store has additions with small size numbers.
Children are asked questions such as the following:Where does the cat have to walk slowly? Why?Where does the cat have to walk fast? Why?How will the turtle deal with the match store?How will the rabbit deal with the match store?What is the smartest way to deal with the match store?How will the turtle deal with the church?What is the smartest way to deal with the church?
The children are invited to reflect on where they can work fast and wherethey have to be more careful. They are also invited to do five exercises inwhich one can work fast or carefully.
The principle is experienced and then formulated: “Some exercises canbe solved quickly, whereas other exercises have to be solved very care-fully.” In addition, children have to solve the exercises reflecting on thisprinciple. Then children make their own exercises out of the match store,wine store, marble store, and balloon store and give these exercises to theirneighbor to solve. The second principle is written on the second stair of thenumber stairs.
(Appendixes continue)
199METACOGNITION AND MATHEMATICAL PROBLEM SOLVING
Appendix B
Sample Items From the Cognitive Training (Count City)
Session 1
The following story is told to the children:
Count City is a village where all houses contain mathematics exer-cises. There are red houses, blue houses, green houses, yellow houses,and orange houses. In every session, we will learn about one of thecolors of the houses. In every session, children learn a color of therainbow.
The children have to solve the questions in the red houses. They have toopen the doors and windows of the houses and solve the questions inside.In addition, the children are invited to follow the dots of a red house andto write “mathematics house” on the roof. Finally, the children play anumber reading game and color the red portion of the rainbow. Children inthis condition perform exactly the same exercises as the children in theNumber Town condition.
Session 2
In a second session, the children are asked what they learned in theprevious session. The following story is presented:
Tine walks in Count City and visits the blue houses. She visits the fiveblue houses. The first blue house is full of additions and subtractionswith big size numbers. The second blue house is full of additions withbig size numbers. The third blue house has lots of subtractions withbig size numbers. The fourth blue house has additions and subtrac-tions with small size numbers. The fifth blue house has additions in itwith small size numbers.
Children are asked questions such as the following:How did you solve the exercises? Why?Who can show us how to solve such an exercise?What are the steps to take?The children are invited to do five other exercises on the blackboard.The procedural algorithm is experienced and then formulated: “In an
addition, we start with the units and then add the tens . . . .” Then childrenmake their own exercises out of a blue page. The second portion of therainbow is colored. Children in this condition perform exactly the sameexercises as in the Number Town condition.
Received July 18, 2001Revision received February 26, 2002
