Learning and Individual Differences 20 (2010) 436–445

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Learning and Individual Differences

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Structural model of metacognition and knowledge of geometry

Utkun Aydın a, Behiye Ubuz b,⁎a Mugla University, Mugla, Turkeyb Middle East Technical University, Secondary Science and Mathematics Education, Ankara, Turkey

⁎ Corresponding author.E-mail addresses: utkun@metu.edu.tr (U. Aydın), ub

1041-6080/$ – see front matter © 2010 Elsevier Inc. Aldoi:10.1016/j.lindif.2010.06.002

a b s t r a c t

a r t i c l e i n f o

Article history:Received 21 October 2009Received in revised form 26 May 2010Accepted 13 June 2010

Keywords:Knowledge of cognitionRegulation of cognitionDeclarative knowledgeConditional knowledgeProcedural knowledgeGeometryStructural equation modelling

This structural equation modeling study aimed to investigate both direct and indirect relations betweenmetacognition and geometrical knowledge. The model was tested using data from tenth grade secondaryschool students (N=923). It was used to estimate and test the hypothesized effects of two metacognitiveconstructs (knowledge of cognition and regulation of cognition) on three knowledge constructs (declarative,conditional, and procedural knowledge) together with the interrelationships among these three knowledgeconstructs. Major findings from the model indicated: (a) a reciprocal relationship existed among declarative,conditional, and procedural knowledge; (b) knowledge of cognition had a positive direct effect onprocedural knowledge and a significant but negative direct effect on declarative knowledge; and (c)regulation of cognition had a positive direct effect on declarative knowledge and a significant but negativedirect effect on procedural knowledge.

uz@metu.edu.tr (B. Ubuz).

l rights reserved.

© 2010 Elsevier Inc. All rights reserved.

1. Introduction

A causal relationship between metacognition and students'mathematical knowledge has long been assumed to exist. Brown(1987) defined metacognition as “one's knowledge and control ofown cognitive system”. A considerable body of research has beendeveloped to explore this relationship using correlational analysis(Lucangeli, & Cornoldi, 1997; Sperling, Howard, Miller, & Murphy,2002; Sperling, Howard, Staley, & DuBois, 2004; Swanson, 1990;Tobias, & Everson, 2002; Veenman, Wilhelm, & Beishuizen, 2004;Veenman, Kok, & Blöte, 2005), crosstab analysis (Panaoura, Philippou,& Christou, 2003), latent variable modeling analysis (Panaoura &Philippou, 2005; Panaoura, 2007) and qualitative methods particu-larly interviews (Artzt & Armour-Thomas, 1992; Goos & Galbraith,1996; Maqsud, 1997; Pugalee, 2001, 2004; Stillman & Galbraith, 1998;Wilson & Clarke, 2004). The effect of metacognitive instruction onmathematical problem solving and reasoning has also been investi-gated in experimental settings (Schurter, 2002; Slife, Weiss, & Bell,1985; Kramarski, Mevarech, & Lieberman, 2001; Kramarski, Mevar-ech, & Arami, 2002; Kramarski, 2004; Mevarech, & Kramarski, 1997;Mevarech, 1999). Many of the afore-cited studies provide substantialevidence in favor of the positive unilateral interrelation amongcomponents of metacognition and student's mathematical knowl-edge. This, however, cannot explain to what extent these constructsinfluence one another, directly or indirectly. Veenman, Van Hout-

Wolters, and Afflerbach (2006) suggested the use of PCA and LISRELanalyses, which yield the best estimates among latent variables andmultiple indicators. Although knowledge of cognition and regulationof cognition were suggested as the two components of metacognition(Brown, 1987), previous research particularly focused on regulationof cognition. Besides that, context of the assessments mainly focusedon elementary school mathematics and rarely on secondary schoolmathematics, particularly concerning procedural knowledge.

The relationship between students' knowledge of concepts andprocedures has also long been an important issue in the mathematicseducation. The interrelation among different knowledge types wasparticularly investigated in the domains of counting (Gelman, Meck, &Merkin, 1986), single-digit addition (Baroody & Gannon, 1984),multi-digit addition (Fuson, 1990; Hiebert &Wearne, 1996), fractions(Byrnes & Wasik, 1991; Mack, 1990; Rittle-Johnson, Siegler, & Alibali,2001), decimal fractions (Moss & Case, 1999; Resnick et al., 1989),percent (Lembke & Reys, 1994), mathematical equivalence (Knuth,Stephens, McNeil, & Alibali, 2006; Perry, 1991; Rittle-Johnson &Alibali, 1999), linear equations (Star et al., 2005), calculus (Engel-elbrecht, Harding, & Potgieter, 2005), and algebra–geometry–analyticgeometry (Webb, 1979). In addressing the relationship, mostresearchers reported that types of knowledge are learned in tandemrather than independently (Rittle-Johnson & Alibali, 1999). The topicsstudied in this bulk of studies have been mainly limited to elementaryschool mathematics, particularly arithmetics.

Researchers assessed the conceptual knowledge through tasks thatinvolve “what” and “which” type of questions in the context ofprimary level of relationships (declarative knowledge), and/or thatinvolve “how” type of questions in the context of abstract level of

Fig. 1. The hypothesized model of metacognition and knowledge of geometry.

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relationships (conditional knowledge) since conceptual knowledgeinvolves building relationships between existing bits of knowledgethat is comprised of primary level of relationships and abstract level ofrelationships. These particular constructs, however, were not classi-fied as declarative and conditional knowledge rather introduced asconceptual knowledge. Additionally, knowledge of procedures wasassessed through tasks that involve the manipulation of algorithmsand procedures. Whether researchers are speaking of conceptualknowledge or procedural knowledge, they hold to the same premisethat any of these types of knowledge involve declarative, conditional,and procedural knowledge (Alexander, Pate, Kullikowich, Farrell, &Wright, 1989; Ryle, 1949).

Having established these facts mentioned above, the present studyaimed to test the hypothesized effects of metacognitive constructs(knowledge of cognition and regulation of cognition) on geometricalknowledge constructs (declarative, conditional, and procedural knowl-edge) together with the interrelationships among these knowledgeconstructs.We estimated themodel using structural equations to assessthe direct and indirect effects of the selected knowledge constructs oneach other, and metacognitive constructs on knowledge of geometry.The structural relationships among these constructswere interpreted asindices of effects of one construct on the other. Thus, the purpose of thisstudy was twofold: (a) to determine the effects of metacognitiveconstructs on knowledge of geometry, and (b) to determine therelationships among knowledge of geometry.

From a pedagogical point of view using knowledge of cognitionand regulation of cognition can effectively inform teaching andlearning. A major challenge for mathematics teachers is thus to fosterthe quality in thinking, assess more purposefully, and to better endshave students engaged in metacognitive processes. Students beingaware of what they know can portray their learning as a transition tosense-making. If so, such metacognitive processes may offer teachersmuch to alleviate the understanding of the reasons underpinningstudents' geometrical knowledge. It has been widely acknowledgedthat knowledge of mathematics is energized by declarative, condi-tional, and procedural knowledge. Students aligned with knowledgeof definitions, relational rules, and procedures are more apt to adoptwhat they know and do not know and use it effectively inmathematics. Similar issues of concern with corresponding inferencesin other subject areas can be evident when investigating students whoattach an elaborate action on their knowledge of physics, chemistry,etc. In this sense, the metacognition-knowledge model in the presentstudy offers relations specific for mathematics as well as holdsparallels and provides directions that can be specified to account forother subject areas in measures of both metacognition andknowledge.

1.1. The relationship among declarative, conditional, and proceduralknowledge

Declarative knowledge (DECKNOW) forms the ground on whichactions depend; conditional knowledge (CONKNOW) provides anoverview that supports the connection making and assists thereconstruction of actions; procedural knowledge (PROKNOW) pro-vides actions, changing and transforming the situations (Hiebert &Wearne, 1996; Mason & Spence, 1999). Hence, declarative knowledgerefers to factual information, procedural knowledge refers to thecompilation of declarative knowledge into algorithms, and condition-al knowledge demands the comprehension of accessing certain factsor employ particular procedures (Alexander & Judy, 1988). Anexample may help us to clarify the idea of these levels. When studentslearn about equilateral triangle, they learn a variety of knowledgeabout the properties of equilateral triangles. As a declarativeknowledge, they learn that an equilateral triangle has equal interiorangles. At this primary level, it is usually expected that students willrelate this fact to recognize the definition of an equilateral triangle. At

the abstract level, the student might advance this fact to theconditional knowledge of an “if-then statement” such as “If allinterior angles of a triangle are equal; then all side lengths of it areequal.”. This kind of connection between the fact of equal interiorangles and the fact of equal side lengths requires reflecting on the bitsof information. Taken together, conceptual knowledge needs to bedistinguished as declarative knowledge and conditional knowledge.To find the area of an equilateral triangle, students access proceduralknowledge, such as the application of the area formula algorithm.

The stream of research on the relationships between students'knowledge of concepts and procedures reported that students' gainsin procedural knowledge may lead to their gains in declarativeknowledge and/or conditional knowledge (Baroody & Gannon, 1984;Gelman et al., 1986; Pesek & Kirshner, 2000; Star et al., 2005) whilegains in declarative and/or conditional knowledge affect their gains inprocedural knowledge (Byrnes & Wasik, 1991; Engelbrecht et al.,2005; Hiebert &Wearne, 1996; Knuth et al., 2006;Mack, 1990;Moss &Case, 1999). Students' knowledge of facts and relational rules guidestheir attention to relevant features of the known and unknownvariables in the problem context. The organization of this knowledgeleads them to generate and select appropriate procedures for solvingproblems, choose among alternative procedures, and transform theknown procedure into a new problem situation. Students' knowledgeof procedures provides the extraction of key facts and principlesunderlying that procedure by making attentional algorithms avail-able. Systematic presentation of these algorithms further improvestheir explanations on the conceptual basis of facts and relational rulesthat they encounter. On the other hand, students' knowledge of factsto determine the core concepts in a problem strengthen their buildingrelations among these core concepts and facilitate their futureretrieval by the adaptation of existing links and procedures to thedemands of the problem. The evidence in this causal picture led us topredict that a reciprocal relationship exists among declarative,conditional and procedural knowledge (Fig. 1).

1.2. Metacognition and knowledge of mathematics

Metacognition was primarily introduced by Flavell (1971) andgenerally designated as ‘thinking about thinking’. Broadly speaking, itis “one's knowledge and control of own cognitive system”. Brown(1987, p. 66), including two main components: knowledge ofcognition (KNOOFCOG) and regulation of cognition (REGOFCOG).Flavell (1979) refers to metacognitive knowledge as person, task, andstrategy; while Brown (1978) classifies it into subcomponents asdeclarative, conditional, and procedural knowledge. While there isconsistent acknowledgement of the importance of awareness of tasknature and progress, researchers mark the conceptualization of theknowledge of the personal learning characteristics. Flavell (1979)

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attains a unified description of metacognitive regulation referred to asconscious use of strategies that accompany planning, monitoring, andcontrolling processes. In the same vein, Brown (1978) postulates thegeneral flow of these processes conveyed to planning, selecting,monitoring, evaluating, and debugging. Researchers commonlyregarded regulatory processes as strategic decisionswhich individualsengage during the execution of the task. Explanations used by the vastmajority of researchers relied primarily on considering Brown'sframework, which provided access to academic settings. A cursoryglance of the research emanating from metacognition, shows thatemphasis has been placed on Flavell's framework in concert with theemergence of problem solving as a means of understanding the effectof regulatory processes (Artzt & Armour-Thomas, 1992; Garofalo &Lester, 1985) and Brown's framework remains central to currentvisions of the consensus that self-report inventories are the leastproblematic technique to measure metacognitive ability (Schraw &Dennison, 1994; Sperling et al., 2002).

The causal relationship between metacognition and mathematicalknowledge exerts that students' regulation of cognition improvestheir gains in declarative (Veenman & Spaans, 2005; Wilson & Clarke,2004), conditional (Lucangeli & Cornoldi, 1997; Pugalee, 2001, 2004;Veenman & Spaans, 2005), and procedural knowledge (Artzt &Armour-Thomas, 1992; Goos & Galbraith, 1996; Lucangeli & Cornoldi,1997; Pugalee, 2001, 2004; Schurter, 2002; Stillman & Galbraith,1998). While students make use of setting goals and allocatingresources to activate their recognition of the key facts (DECKNOW),they reconstruct their thought processes such as debugging to explainwhy the condition in a relational rule is satisfied or not (CONKNOW).Throughout this planning and monitoring processes they direct theirverifications of the algorithms and selection of the procedures(PROKNOW). Their designation of regulatory processes such aschecking their comprehension of the problem, comparing the giveninformation in the problem context to information in memory, oremploying different strategies when they fail to draw a conclusionfacilitates their generation of correct procedures, evaluation of therelevance of known procedures and adaptation of alternativeprocedures.

The causal picture has also imposed knowledge of cognition tostudents' gains in declarative (Wilson & Clarke, 2004), conditional(Swanson, 1990; Wilson & Clarke, 2004) and procedural knowledge(Maqsud, 1997; Tobias & Everson, 2002; Wilson, & Clarke, 2004).Students' awareness of what information is given in the problem andwhat strategies to implement trigger them to identify the basic factsand recall the critical attributes of these facts (DECKNOW). Theknowledge of why certain strategies are more efficient leads them toexplain the relational rules and judge the links between these rules(CONKNOW). Aligned with the knowledge of how strategies can beintegrated into the problem solution students enrich the applicationof their procedures by the guidance of these strategies (PROKNOW)which would help them regulate their strategies toward implement-ing appropriate algorithms and checking whether the outcomes ofthese algorithms are satisfied by the underlying procedures.

The bulk of evidence in the research that suggested a unidirec-tional relationship between components of metacognition andknowledge types led us to hypothesize that knowledge of cognitionand regulation of cognition are positively related to declarative,conditional, and procedural knowledge (see Fig. 1).

2. Methods

2.1. Participants

923 tenth grade students (553 females and 370males) with an agerange from 17 to 18 from Ankara, Turkey participated in the presentstudy. Participants were from four Anatolian (n=410), three private(n=290), and two public (n=223) high schools. Anatolian and

private high schools are the schools that accept students according tothe Secondary School Entrance Examination (OKS) conducted byTurkishMinistry of Education. This exam includes 100multiple choicequestions in four domains: Turkish Literature, Mathematics, Science,and Social Sciences. Students attending to private high schools have topay a certain fee during the school year. To be accepted to public highschools students are required neither to take OKS nor to pay a fee tothe school administration.

2.2. Model and data analysis

To control the design effect in the sample resulting from the nestedstructure of the data, we assessed the intraclass correlation of all theinvolved constructs by performing analysis of variance (Raudenbush& Bryk, 2002). Following that we estimated the model, a latentvariable structural equation model, in several steps, using the LISREL8 program (Jöreskog & Sörbom, 1993). First, the measurement model(constructs and indicators) was specified and estimated; and thenstructural relationships in the model were specified and estimated;finally, different models were estimated to assess the direct, indirect,and total effects of the selected constructs. Structural equationmodeling is recommended to test and modify theoretical models(Anderson & Gerbing, 1988) for nonexperimental data. Beside theparameter estimates, the program provides fit indices to assess howwell the model fits the data. Such fit indices make it possible toevaluate the adequacy of the theoretical model in explaining the data.Fit indices recommended by Jöreskog and Sörbom (1993) were theratio of chi-square to the degrees of freedom (χ2/df), root meansquare error of approximation (RMSEA), root-mean-square residual(RMR), goodness-of-fit index (GFI), adjusted-goodness-of-fit index(AGFI), and comparative fit index (CFI). Schreiber, Stage, King, Nora,and Barlow (2006) suggested substantively interpretive models withchi-square ratios of three or less, a RMSEA from .06 to .08, a RMRbelow .08, a GFI above .90, an AGFI above .90, and a CFI above .90 asgood fitting.

After selecting items that logically seemed related to the five latentconstructs, we tested a measurement model to determine if theselected items had significant loadings on hypothesized constructs.Although reliabilities of some items were low, we considered themeasurement model reasonable with all significant factor loadings.Furthermore, to obtain conservative estimates of the interrelationsamong constructs we used the correlations to provide insight into thestrength of the relationships.

2.3. Latent constructs and measured indicators

The purpose of the study was to examine the relationships amongmetacognitive and geometrical knowledge constructs. Metacognitionwas measured by Jr. MAI items on a 5-point Likert scale developed bySperling et al. (2002). This instrument was adapted to Turkish by theresearchers. Geometrical knowledge on triangles was measured usingGeometry Knowledge Test (GKT) developed by the researchers.

To validate the Jr.MAI and GKT instruments, exploratory andconfirmatory factor analyses were conducted on two separatesamples (314 and 589 10th grade students respectively for Jr. MAI,and 260 and 297 10th grade students respectively for GKT) other thanthe sample of the present study.

As a result of these analyses, 17 Jr. MAI items were grouped in twometacognitive constructs: eight items on knowledge of cognition(KNOOFCOG) measuring students' knowledge about their owncapabilities and nine items on regulation of cognition (REGOFCOG)measuring knowledge about their own control processes during theexecution of the task. The Cronbach's alpha coefficient with 17 itemson the Jr. MAI was .85. The internal reliabilities of KNOOFCOG andREGOFCOG were .75 and .79, respectively.

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17 GKT open ended questions were grouped in three knowledgeconstructs: five questions on declarative knowledge (DECKNOW)measuring students' knowledge of definitions, facts, and symbols,six questions on conditional knowledge (CONKNOW) measuringtheir knowledge of relational rules, and six questions on proceduralknowledge (PROKNOW) measuring their knowledge of proceduresand algorithms. A focused holistic scoring scheme was developed bythe researchers reflecting the conceptual framework of declarative,conditional, and procedural knowledge with reference to Lane(1993). In developing the scoring scheme, criteria representing thethree constructs were specified for each of the five score levels (0–4). Based on the specified criteria at each score level, the highestscore of 4 was awarded for responses that the researchers regard asbeing entirely correct and satisfactory, while the lowest score of 0was reserved for no answer. The possible scores on GKT ranged from0 to 96. The Cronbach's alpha coefficient with 17 questions on theGKT was .88. The internal reliabilities of DECKNOW, CONKNOW, andPROKNOW were .62, .78, and .89, respectively. The reliabilitycoefficient for declarative knowledge part was low as this knowl-edge is related to “low-level learning” (Smith & Ragan, 1993) and iscomparable to recall levels of Bloom's taxonomy. As supported by

Fig. 2. Example of questions for declarative,

Kane (1986), reliability values greater than .50 are acceptable forthis kind of cases. Specimen questions of the GKT are presented inFig. 2.

To test the measurement model to determine if the selected itemsand questions in the Jr. MAI and GKT respectively had significantloadings on the latent constructs, a confirmatory factor analysis modelwas specified and estimated on the main sample.

3. Results

The intraclass correlation of all the involved constructs to controlthe design effect in the sample resulting from the nested structure ofthe data due to the school type and gender (Hosenfeld, Köller, &Baumert, 1999; Jürges, Schneider, & Büchel, 2005; Köller, Baumert,Clausen, & Hosenfeld, 1999) were calculated. The ICC values for thestudents within school types were calculated as .031, .035, .030, .01,and .001 for DECKNOW, CONKNOW, PROKNOW, KNOOFCOG, andREGOFCOG, respectively. The ICC values attached to gender werecalculated as .0001, .005, .015, .0006, and .0001for DECKNOW,CONKNOW, PROKNOW, KNOOFCOG, and REGOFCOG, respectively.As the ICC values are less than .25 (Heinrich & Lynn, 2001), much

conditional, and procedural knowledge.

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variation in the latent constructs is due to individuals rather thanschool types and gender. Hence, these ICC values did not signify theneed to use a multilevel model.

Table 1Description of latent constructs and descriptive statistics, reliabilities, and factorloadings of the corresponding observed variables.

Latent construct Coding M SD Rel.a Factorloading

Knowledge of cognitionI know when I understandsomething.

4.51 .78 .32 .47

I can make myself learnwhen I need to.

1=never 4.00 .98 .45 .67

I try to use ways of studyingthat have worked forme before.

2=seldom 4.02 1.23 .34 .58

I know what the teacherexpects me to learn.

3=sometimes 3.96 1.06 .35 .59

I learn best when I alreadyknow something aboutthe topic.

4=often 4.49 .86 .24 .49

I really pay attention toimportant information.

5=always 4.39 .84 .45 .67

I learn more when I aminterested in the topic.

4.65 .70 .22 .35

I use my learning strengths tomake up for my weaknesses.

3.72 1.07 .46 .68

Regulation of cognitionI draw pictures or diagrams to helpme understand while learning.

3.13 1.26 .30 .44

When I am done with myschoolwork, I ask myself ifI learned.

2.87 1.35 .42 .65

I think of several ways to solvea problem and then choosethe best one.

1=never 3.59 1.16 .36 .60

I think about what I needto learn before I start working.

2=seldom 3.74 1.16 .39 .62

I ask myself how well I amdoing while I amlearning something new.

3=sometimes 3.33 1.25 .33 .57

I use different learningstrategies depending on the task.

4=often 3.61 1.10 .46 .68

I occasionally check to makesure I 'll get my work doneon time.

5=always 3.62 1.09 .33 .58

I ask myself if there was aneasier way to do thingsafter I finish a task.

3.15 1.36 .35 .50

I decide what I need to getdone before I start a task.

3.95 1.08 .31 .56

Declarative knowledgeQuestion 1 3.70 .62 .38 .43Question 2 2.16 1.19 .35 .38Question 3 0–4 2.66 1.58 .86 .93Question 4 2.60 1.49 .59 .77Question 5 1.95 1.14 .64 .35

Conditional knowledgeQuestion 6 2.80 1.21 .30 .45Question 7 2.13 1.32 .36 .60Question 8 0–4 2.40 1.45 .39 .62Question 9 3.46 1.19 .59 .76Question 11 2.50 1.45 .34 .59Question 12 2.40 1.35 .43 .66

Procedural knowledgeQuestion 10 3.47 1.11 .67 .45Question 13 2.64 1.48 .73 .54Question 14 0–4 2.36 1.74 .85 .71Question 15 2.26 1.67 .87 .75Question 16 2.75 1.59 .83 .69Question 17 2.35 1.63 .81 .66

a Estimated in LISREL (Jöreskog, & Sörbom, 1993).

3.1. The measurement model

The measurement model was specified and tested before theestimation of structural relationships, as suggested by Anderson andGerbing (1988). Table 1 contains an exhaustive list of the observedvariables together with the descriptive statistics, reliabilities, andfactor loadings. Latent-variable models incorporate multiple observedvariables that provide an improvement in the construct validityreducing the measurement errors (Bentler, 1980).

The reliabilities listed indicate lower bound estimates of the truereliability (Jöreskog & Sörbom, 1993). As expected, PROKNOWindicators had the highest reliability. DECKNOW and CONKNOWindicators had moderate-to-high reliability. Indicators of KNOOFCOGand REGOFCOG had the lowest reliability estimates.

The factor loadings are standardized maximum likelihood coeffi-cients. The moderate-to-high factor loadings of the observed variableswere effective instruments in defining the latent variables. As shownin Table 1, all observed variables were significantly related to theirunderlying latent construct (pb .00). The relations yielded a fairlygood fitting measurement model, a χ2 (495)=1522.42, chi-squareratio=3.07, RMSEA=.047, RMR=.043, GFI=.91, AGFI=.90, andCFI=.96. Although the ratio of chi-square to degrees of freedom washigher than expected all other fit indices showed a good fit.

Table 2 presents the estimated correlations among latent con-structs, whichwere used as a basis in the subsequent estimation of thehypothesized model.

3.2. The structural model

We initially analyzed the hypothesized model (i.e., Model A) inFig. 1 for the structural relations. For the hypothesizedmodel, we usedthe structural equation modeling technique to assess (a) the directeffects of each of the two metacognitive constructs (KNOOFCOG andREGOFCOG) on three knowledge constructs (DECKNOW, CONKNOW,and PROKNOW) and (b) reciprocal relations among the threeknowledge constructs. We adopted pairwise deletion to constructthe covariance matrix among the variables at the pb .05 level. Theerror terms associated with the latent variables representing themetacognitive and knowledge constructs were hypothesized asuncorrelated. The hypothesized model covered the following nonsig-nificant paths (shown in parantheses are the statndardized estimate)from KNOOFCOG to CONKNOW (−.13) and from REGOFCOG toCONKNOW (.19). The goodness-of-fit indexes for the hypothesizedmodel follow: χ2(490; 1545.68)=3.15, RMSEA=.048, RMR=.068,GFI=.91, AGFI=.89, and CFI=.91. Although the ratio of chi-square todegrees of freedom ratio and the AGFI were less than satisfactory, allother fit indices showed a good fit. The squared multiple correlationsfor DECKNOW, CONKNOW, and PROKNOW indicated that thepredictors explained 20%, 21%, and 15% of the variance, respectively.

Next, following the recommendations of Jöreskog (1993), wetested a revised model to check which model was better. In thisrevised model (i.e., Model B), we deleted the nonsignificant pathsfrom KNOOFCOG and REGOFCOG to CONKNOWwithin considerationsabout the fact that high school mathematics engages students mostly

Table 2Correlations of the latent variables.

Latent Variable DECKNOW CONKNOW PROKNOW KNOOFCOG REGOFCOG

DECKNOW 1.00CONKNOW .42 1.00PROKNOW .40 .44 1.00KNOOFCOG −.11 .12 .02 1REGOFCOG −.04 .21 −.05 .58 1

Fig. 3. The alternative model C.

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on procedural questions rather than conditional questions. Studentsmay not rely on using different learning strategies or seeking foreasier ways to help their understanding of unfamiliar situations suchas “if-then” statements. We constrained two paths equal to zero thatwere freely estimated previously. The error terms associated with thelatent variables in Model B were hypothesized as uncorrelated. Theoutcomes of the reduced model yielded a good fit to the data, χ2(492;1542.91)=3.13, RMSEA= .047, RMR= .067, AGFI= .90, andCFI=.96. Because the model fit indexes improved, the reducedmodel seemed to fit the data better than did the hypothesized model.To decide which model was better, at least empirically, we performedmodel comparisons. According to the chi-square difference test,Δχ2=2.77, Δdf=2, Model B was significantly better than Model A.The summary of the goodness-of-fit statistics and the modelcomparisons are presented in Table 3.

We then tested an alternative model in which the error termsassociated with the latent variables were hypothesized as uncorrelated.In this alternative model presented in Fig. 3 below (i.e., Model C), weconsidered the possibility of two theoretical aspects. One assumptionconcerned the prevalence of knowledge constructs on metacognitiveconstructs. Therefore, we expected to find the main effects forknowledge constructs if KNOOFCOG was a mediator variable amongknowledge constructs and REGOFCOG. Based on the theory, weassumed that students' tendency to recognize definitions (DECKNOW),build mathematical relationships (CONKNOW), and apply algorithms(PROKNOW), would trigger them to be a good judge of their learning(KNOOFCOG) and motivate them to think about what they really needto learn before they begin amathematical task (REGOFCOG). The resultsof the Sobel test supported our assumption indicating that knowledge ofcognition functioned as amediator. Secondly, with the close associationbetween KNOOFCOG and REGOFCOG, the relations of the twometacognitive constructs were computed including the reciprocalpaths. The high correlation between KNOOFCOG and REGOFCOG hadbeen expected due to the fact that students who know when theyunderstand something tend to decidewhat they need to get donebeforethey start a task, and in turnmake themselves learn when they need to.

The alternative model attained a good fit to the data, χ2(503;1722.40)=3.46, RMSEA=.051, RMR=.045, GFI=.90, AGFI=.88,and CFI=.90. As expected, in this model reciprocal relations betweenKNOOFCOG and REGOFCOG moved very clearly into the foreground.The direct paths from CONKNOW to KNOOFCOG and from DECKNOWto REGOFCOG did not reach statistical significance. These paths werethen constrained to zero to detect whether the elimination from themodel would significantly reduce the model fit and the error terms ofthe latent variables were again hypothesized as uncorrelated.According to this additional calculation entered into Table 3, thedesignated paths did not make a significant contribution to the model(χ2(505; 1725.46)=3.41, RMSEA=.045, RMR=.042, GFI=.90,AGFI=.88, and CFI=.90). The chi-square difference test applied tocompare the initial alternative model with the nested alternativemodel (i.e., Model D) did not provide evidence of a significantdifference in fit Δχ2=3.06, Δdf=2.

To test whichmodel was better, at least empirically, we performednonnested model comparisons. Schreiber (2008) suggested the fitindex comparison of Akaike Information Criterion (AIC), Consistent

Table 3Goodness-of-fit statistics and comparisons for the metacognition-knowledge model of geom

Model χ2 df NC (χ2/df) RMSEA RMR

Model A 1545.68 490 3.15 .048 .068Model B 1542.91 492 3.13 .047 .067Difference of A and BModel C 1722.40 503 3.46 .051 .045Model D 1720.46 505 3.41 .045 .042Difference of C and D

Akaike Information Criterion (CAIC), and Expected Cross-ValidationIndex (ECVI) to test alternative models with the same data that arenot nested or hierarchically related. He recommended that smallervalues are good for comparisons. According to this nonnested modelcomparisonModel B was proved to be superior to Model C. Hence, thereduced model presented in Fig. 4 was considered to be the finalstructural model in the context of the present study.

Direct, indirect, and total effects of the structural model shown inTable 4 were examined for significance at the .05 level. A direct effectis the causal effect that is denoted by the directional relation from onevariable to the other. An indirect effect is the effect between twovariables that is mediated by one or more other variables. The sum ofdirect and indirect effects is the total effect.

KNOOFCOG had a significant positive direct effect on PROKNOW(.33). Students, who displayed metacognitive behaviors associatedwith KNOOFCOG such as judging of how well something isunderstood and motivating themselves when needed, tended to bemore successful on solving procedural knowledge questions. Incontrast, it had a significant but negative direct effect on DECKNOW(−.28), demonstrating that students who learn best when theyunderstand something about the topic do not tend to gear theirunderstandings on factual mathematics information. KNOOFCOG alsohad an indirect effect on PROKNOW (−.04) through influencingDECKNOW (.04). The indirect effects reaffirm the role of KNOOFCOGon DECKNOW and PROKNOW. These direct and indirect effects meanthat, for example, students who used their learning strengths to makeup their weaknesses were equipped with knowledge of concepts tomake justifications and in turn able to progress on algorithms.

REGOFCOG had a significant positive direct effect on DECKNOW(.24). Students who pay attention on important information tend tobe aware of the core knowledge leading them to make furthergeneralizations. In contrast, it had a significant but negative directeffect on PROKNOW (−.34). Students, who displayed metacognitivebehaviors associated with regulation of cognition, such as thinking ofseveral ways to solve a problem or deciding what to do before startinga task tended to be less successful in solving procedural knowledgequestions. One likely explanation for the unexpected results is due the

etry.

GFI AGFI CFI AIC CAIC ECVI Δχ2 Δdf

.910 .890 .910 1755.68 2367.58 1.90

.910 .900 .960 1748.91 2349.16 1.902.77 2

.900 .880 .900 1906.40 2442.54 2.07

.900 .880 .900 1905.46 2429.95 2.071.94 2

Fig. 4. The structural model of metacognition and knowledge of geometry.

442 U. Aydın, B. Ubuz / Learning and Individual Differences 20 (2010) 436–445

collinearity in data. As Maruyama (1998) illustrated collinearity givesrise to describe coefficients that change signs in the effects. This resultunderlined that as the level of awareness of how to regulateknowledge increases the level of performance on proceduralknowledge decreases. REGOFCOG had the indirect effect on DECK-NOW (−.04) through influencing PROKNOW (.02). Students whotend to monitor their solution procedures are able to unwrap the factswhile applying algorithms.

DECKNOW had a bilateral positive and significant direct effect onboth CONKNOW (.21) and PROKNOW (.12). One explanation for thereciprocal relationships is that, students who recall the attributes of aconcept are able to explain the relational rules relevant to the conceptand present these rules systematically in an algorithm. In other words,knowledge of relations and knowledge of procedures involve achallenge for students to analyze, interpret, and manipulate theirknowledge of concepts.

The two direct effects on CONKNOW were from DECKNOW (.21)and PROKNOW (.14). The strongest direct effect was that ofDECKNOW, while PROKNOW also exerted a substantial direct effecton CONKNOW. This finding indicates that students who are able toidentify the characteristics of a concept and correctly complete thesolution steps would be likely to state the relationships betweenconcepts and procedures.

PROKNOW was specified dependent of DECKNOW (.12) andCONKNOW (.14); and this specification was not disproved by thedata. This was in line with the conception that procedural knowledgedemands not only memorization of acquired knowledge such asdefinitions or symbols but also the links of these definitions andsymbols within the algorithms.

The total effects of DECKNOW on CONKNOW and CONKNOW onDECKNOW, DECKNOW on PROKNOW and PROKNOW on DECKNOW,CONKNOW on PROKNOW and PROKNOW on CONKNOW were .24,.17, and .19, respectively. These findings might indicate that allknowledge constructs were practically important contributors for oneanother.

Table 4Direct, indirect, and total effects.

Latent dependent variable DECKNOW CONKNOW P

Direct Indirect Total Direct Indirect Total D

DECKNOW .00 .07 .07 .21 .04 .25CONKNOW .21 .04 .25 .00 .08 .08PROKNOW .11 .04 .15 .14 .04 .18KNOOFCOG −.08 .02 −.06 .00 .03 .03REGOFCOG .06 −.04 .02 .04 .01 .05 −

4. Discussion

We investigated a structural model to explain the relationshipsamong metacognitive and knowledge constructs. There was strongsupport for the hypothesized relationships. All coefficients weresignificant and in the theoretically expected direction.

With respect to the structural model estimating the effects ofmetacognitive factors on geometrical knowledge, interesting findingsemerged.

KNOOFCOG had the strongest predictive power on PROKNOW;that as students become more aware of what they know and whatthey need to know, they show a higher performance in applyingalgorithms. That is, knowledge of how to use available information toreach a correct solution leads students to make effective progresswhen they encounter the procedural details such as computationaloperations in a geometry problem. This finding supports the findingsof previous studies (Maqsud, 1997; Tobias & Everson, 2002; Wilson &Clarke, 2004), which provided evidence that students' knowledge ofcognition positively contributes to their successful utilization ofalgorithms and procedures within relevant word problems. Thesignificant but negative effect of KNOOFCOG on DECKNOW providedsupport to the results of previous studies (Sperling et al., 2002, 2004;Swanson, 1990; Wilson & Clarke, 2004) indicating that students' self-awareness does not require a repertoire of declarative knowledge.This finding could be given the following explanation. Studentsequipped with an awareness of the subject might not attend to thestructural features of the problem, and yet were able to formmeaningful links between the given facts and the problem goal.Having established this relationship, these students would then nottend to learn more in a topic they already grasped its basic aspects.Overall, this result of the present study supports the argument thatthe accessing of simple facts during problem-solving does not requireconsiderable awareness of cognition.

The significantly positive direct effect of REGOFCOG on DECKNOWpresented in the model could throw some light on the relationshipraised by several researchers (Mevarech & Kramarski, 1997; Veenmanet al., 2004) where students who have the necessary regulatory skillsuse them appropriately during the search for the basic factualknowledge. From the metacognitive point of view the ability ofstudents to use available learning strategies can have them buildgeneralizations which draw out the links between their regulationprocesses and facts embedded in the problem. It appears that studentswho develop an understanding of when to deploy productivestrategies are able to structure the mathematical definitions duringproblem-solving. In contrast, themodel showed that REGOFCOG had asignificant but negative direct effect on students' PROKNOW.Although this result of the present study was inconsistent withprevious research (Kramarski et al., 2001, 2002; Kramarski, 2004;Mevarech & Kramarski, 1997; Mevarech, 1999), through the meta-cognitive lens it lends strong support for the claim that students usesubconsciously their regulatory skills and imply their plans when thetasks become automized (Lucangeli & Cornoldi, 1997). While solvingprocedural questions students may not need to plan their solutionstrategies, seek for an easier way to reach a correct solution, ormonitor their progress, as the given task requires the application of

ROKNOW KNOOFCOG REGOFCOG

irect Indirect Total Direct Indirect Total Direct Indirect Total

.11 .04 .15 −.08 .02 −.06 .06 −.04 .02

.14 .04 .18 .00 .03 .03 .04 .01 .05

.00 .06 .06 .09 −.06 .03 −.12 .03 −.09

.09 −.06 .03 .00 .53 .53 .59 .11 .70

.12 .03 −.09 .59 .11 .70 .00 .54 .54

443U. Aydın, B. Ubuz / Learning and Individual Differences 20 (2010) 436–445

straightforward algorithms and procedures. One favorable view ofthis finding might be that the Jr. MAI measures something other thanknowledge that supports previous research (Schraw & Dennison,1994; Sperling et al., 2002; Swanson, 1990) indicating the generallylow correlations between metacognitive and knowledge constructs.

Taken together those results might be partly attributable to thefocus of high school teaching of mathematics on proceduralknowledge. Therefore, students are better equipped to deal withprocedural knowledge in terms of linking within particular computa-tions (e.g operations) and algorithms (e.g., syntax of a problem).Students might tend to think about what they know about theproblem or check the outcomes of the problem solution whileapplying the syntax of the problem together with operations.However, the blend of the automatization may prevent studentsfrom productively regulating a suitable range of mathematicalstrategies.

The relationships emerged in the structural model have allowedus to dispense metacognitive training to make students aware oftheir knowledge and regulatory processes. The sense of thisperspective articulated the ways to designate the integration ofmetacognitive training together with the reconceptualization ofgeometrical knowledge delineated within procedural knowledge.For example, procedural knowledge might be introduced withquestions such as “Do you think of the algorithmic sequences beforeattempting a solution?”, or “Do you think of any other procedures toreach to the solution?”.

The significant effects of KNOOFCOG and REGOFCOG on DECK-NOW and PROKNOW could not be documented for CONKNOW. Ourstudy is in disagreement with those studies (Veenman & Spaans,2005; Wilson & Clarke, 2004) on the account that students' manner ofengagement in PROKNOW may affect their access to CONKNOWthereby hinder the activation of knowledge of the demands ofrelationships and regulation of the links among those relationships.

The pervasive indirect effects of KNOOFCOG and REGOFCOG onCONKNOW and to a lesser degree DECKNOW held PROKNOW as anessential mediator. The two metacognitive constructs influencedDECKNOW through paths involving CONKNOW and PROKNOW, afinding that indicates being aware of one's own capabilities andmonitoring of one's own strategies are dependent, in part onknowledge constructs. This elucidated that the high school mathe-matics teaching draws on PROKNOW based on factual knowledge(DECKNOW) and continued processing of relational knowledge(CONKNOW).

We contend that the interrelations among declarative, conditional,and procedural knowledge emerged in this study provided support infavor of the fact that the effects of one type of knowledge on the otherrepresent important features of the link between knowledge ofconcepts and knowledge of procedures (Pesek & Kirshner, 2000;Rittle-Johnson & Alibali, 1999; Rittle-Johnson et al., 2001; Rittle-Johnson & Koedinger, 2005).

Our study has the new perspective of a link among all studiesinvestigating the interrelationship among knowledge types. Coupledwith the reciprocal relationships, it holds the strongest promise inunfolding the relationship in which DECKNOW influenced the gains inCONKNOW as the gains in CONKNOW influenced DECKNOW. Inrelation to the questions (1, 7, 14) provided in Fig. 2, a studentknowing the definition of an equilateral triangle could provide anexplanation for the relationship between its sides and interior angles,and that such relationship satisfies the justification of a triangle withequal side lengths has equal interior angles relevant to the definition.This shows that the relationship between concepts or principlescannot function without accessing to definitions and facts or viceversa. From this perspective, a set of connections with evidence of arich web of linking nestle the properties which characterize amathematical concept; whereas this knowledge of properties alsorequire the relationships among these properties.

Similar line of evidence can be indicated in terms of the reciprocalrelationship between DECKN OW and PROKNOW. For instance Forinstance (see questions 1 and 14 in Fig. 2), a student who is able todefine an isosceles triangle knows that an isosceles triangle hasequal base interiors. Consequently, he/she would be able to use thisfeature in an isosceles triangle algorithm, which generalizes to partof the definition in relevant computations to find the unknownangle. This, in turn yields the application of the procedures to adoptthe fact that an isosceles triangle has two of its sides equal in length.These findings support the findings of some previous researchers(Byrnes &Wasik, 1991; Mack, 1990; Moss & Case, 1999; Perry, 1991;Star et al., 2005), who reported that students should have a rich storeof basic facts to adopt adequate procedures to the solution processwhile they should execute proper algorithms in which theyrecognize the correct facts.

A central result that framed the model is that CONKNOW andPROKNOW and the reciprocal relationships between the twomediate the links relative to the if-then statements and theprocesses. By becoming reflectively engaged in them, i.e., bybecoming involved in justifications and applications, knowledge-mediated practices offer the students a malleable array of geomet-rical knowledge. A student who is capable in procedures relevant toan equilateral triangle can evaluate the links between the attributesof the equilateral triangle in a given if-then statement or vice versa.That is, CONKNOW provides students to build links between therelational rules functioning within the procedures, while PROKNOWhelps students to improve their CONKNOW through strengtheningthe use of concept-algorithm sequences, reflecting on why algo-rithms work, and evaluating the correctness of possible answers tothe demands of problems. These findings were consistent with thefindings of previous research (Baroody & Gannon, 1984; Engelbrechtet al., 2005; Gelman et al., 1986; Hiebert & Wearne, 1996; Lembke &Reys, 1994), which reported that students who exhibited greatercomputational performance were likely to have a greater under-standing of relations among concepts. This implies that studentsextract relevant principles as they progress on a given proceduraltask and discover the relations within the practice process.Researchers' (Byrnes & Wasik, 1991; Engelbrecht et al., 2005;Hiebert & Wearne, 1996; Knuth et al., 2006; Mack, 1990; Moss &Case, 1999) consideration for the knowledge of concepts to becomepredominant on knowledge of procedures associated with compu-tational processing resonates with our results that students who areable to identify the characteristics of relationships correctly reflecton the algorithm sequences.

The reciprocal relationships among different types of geometricalknowledge in this study offers an integrated understanding thatpointed knowledge of concepts forms a basis for the acquisition ofnew procedures and fosters the learning of procedures (Byrnes &Wasik, 1991; Perry, 1991; Star et al., 2005). Accordingly, the currentinterrelations lend support to the previous contentions (Engelbrechtet al., 2005; Knuth et al., 2006), enabling us to present a compromisethat greater knowledge of concepts is associated with greaterknowledge of procedures or vice versa.

In our opinion, the distinction among declarative, conditional, andprocedural knowledge illuminates alternative ways in which usefulinstructional designs for mathematics courses could be implemented.One of the challenges that face education is to develop worthwhiletasks that require students to state concept definitions, makerelational explanations, and apply appropriate procedures. Theinstruction might begin with a series of activities that require thepresentation of the commands of basic definitions (DECKNOW) andcontinue with the incorporation of algorithms (PROKNOW) and theoutline of relational rules (CONKNOW). Thus, the combination ofknowledge types favors the synthesis of knowing that, knowing why,and knowing how and in turn could bring the accomplishment ingeometry.

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5. Limitations and recommendations

Caution is appropriate with respect to interpretations of the resultsof the present study that future research can address. First, knowledgeof geometry was modeled in terms of two metacognitive constructs.There are, however, additional variables that have potential effect onknowledge in geometry. Examples of such variables are anxiety,motivation, self-efficacy, self-concept and more. Therefore, moreresearch is needed in order to explore the effects that these variablesexert on different knowledge constructs. Second, metacognition is acomplex construct to measure with high reliability and validity.Although the measure of metacognition, including knowledge ofcognition and regulation of cognition constructs, was fairly reliableand valid, it would have been preferable to use measures includingmore items to assess the knowledge of cognition and regulation ofcognition to possibly strengthen these constructs. Third, the presentstudy employed a cross-sectional design, which only allowedinvestigation of relationships among metacognitive and knowledgeconstructs at one point in time. For example, the metacognitivedifferences could not be determined as developmental changes ofstudents. Last, caution is appropriate when interpreting the fit indexesthat are sensitive to sample size (Sivo, Fan, Witta, & Willse, 2006).

Despite these limitations, most of the hypothesized relationshipswere statistically significant and substantial in size, which support therobustness of the structural model related to metacognitive andknowledge constructs. In this study, we encountered some theoreticalissues that lend further credibility to the results are the use ofstructural equation modeling and specification of both direct andindirect effects of the constructs. Therefore, future research requirescross-validation and replication of this present study with anymodeling approach. Furthermore, future research may explorewhether the effect of metacognition differs in explaining knowledgein the content areas of operations, algebra, measurement, dataanalysis and probability, problem solving, and reasoning and proof.Closely related to the above effect of metacognition, the present studycan guide research in the overutilization of knowledge types ascenterpieces around which to craft different tasks (e.g. non-routineproblems).

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