Predicting Long-Term Growth in Students Mathematics … · 2016. 4. 25. · dent help seeking (Butler, 1998), and self-handicapping (Urdan & Midgley, 2001). As such, extrinsic motivation

Predicting Long-Term Growth in Students’ Mathematics Achievement:The Unique Contributions of Motivation and Cognitive Strategies

Kou Murayama, Reinhard Pekrun,and Stephanie Lichtenfeld

University of Munich

Rudolf vom HofeUniversity of Bielefeld

This research examined how motivation (perceived control, intrinsic motivation, and extrinsic motivation),cognitive learning strategies (deep and surface strategies), and intelligence jointly predict long-term growth instudents’ mathematics achievement over 5 years. Using longitudinal data from six annual waves (Grades 5through 10; Mage = 11.7 years at baseline; N = 3,530), latent growth curve modeling was employed to analyzegrowth in achievement. Results showed that the initial level of achievement was strongly related to intelli-gence, with motivation and cognitive strategies explaining additional variance. In contrast, intelligence had norelation with the growth of achievement over years, whereas motivation and learning strategies were predic-tors of growth. These findings highlight the importance of motivation and learning strategies in facilitatingadolescents’ development of mathematical competencies.

For decades, researchers in developmental and edu-cational psychology have been concerned with thedeterminants of academic achievement. Today,there seems to be agreement that both motivationaland strategy variables play an important role inexplaining academic achievement (for a review, seeRobbins et al., 2004). However, although a largeportion of previous research has focused on therelation between these variables and academicachievement assessed at a particular time point,fewer studies have investigated whether motiva-tional and strategy variables predict long-termgrowth in academic achievement. This is unfortunatebecause one of the ultimate goals in education is tofacilitate sustainable learning (long-term intraindi-vidual growth, i.e., change relative to currentachievement of the individual), rather than to focuson performance attainment at one point in time.Clearly, systematic investigation of the long-termdeterminants of growth in academic achievement isimperative. In the present research, using longitudi-nal data from six annual waves over the adolescentyears, we examined a variety of motivational vari-ables and cognitive strategies as predictors of both

concurrent level and long-term growth in mathachievement.

Motivation: Perceived Control and Intrinsic–ExtrinsicMotivation

Past studies have identified various motivationalfactors that affect academic achievement. The pres-ent research focused on three of these factors thatare regarded as especially important in motivationtheory and research: perceived control, intrinsicmotivation, and extrinsic motivation (Eccles &Wigfield, 2002; Pekrun, 2006). Perceived control isconceptualized as subjective appraisal of the causallink between one’s action and outcomes (Perry,Hladkyj, Pekrun, & Pelletier, 2001; Rotter, 1966).That is, perceived control reflects one’s expectancyto obtain a desired outcome through an action.Concepts of perceived control and competence (andrelated constructs of expectancies) play a prominentrole in motivation theories (Bandura, 1977; Eccles &Wigfield, 2002; Marsh & Shavelson, 1985; Pekrun,1993), and many studies have shown that perceivedcontrol and competence in academic domains arepositively related to achievement (Marsh, 1990;Meece, Wigfield, & Eccles, 1990). By implication, itis straightforward to expect that perceived control

This study was supported by an Alexander von HumboldtFoundation fellowship (to K. Murayama) and four grants fromthe German Research Foundation (DFG; to R. Pekrun, Project forthe Analysis of Learning and Achievement in Mathematics,PALMA; PE 320/11-1, PE 320/11-2, PE 320/11-3, PE 320/11-4).

Correspondence concerning this article should be addressed toKou Murayama, Department of Psychology, University ofMunich, Leopoldstrasse 13 (PF 67), Munich, Germany 80802.Electronic mail may be sent to [email protected].

© 2012 The AuthorsChild Development © 2012 Society for Research in Child Development, Inc.All rights reserved. 0009-3920/2012/xxxx-xxxxDOI: 10.1111/cdev.12036

Child Development, xxxx 2012, Volume 00, Number 0, Pages 1–16

is positively associated with concurrent level ofstudents’ achievement in domains such as mathe-matics.

Importantly, adolescents’ perceived control hasalso been shown to relate to their learning.Perceived control is linked to active and effortfulcommitment to learning (Skinner, Wellborn, &Connell, 1990), persistence when performing diffi-cult and challenging tasks (Cervone & Peake, 1986),and intrinsic engagement (Gottfried, 1990). Togetherthese findings suggest that perceived control shouldhelp adolescent students acquire new knowledge,and as such, should positively predict growth intheir achievement. Indeed, longitudinal research onstudents’ academic self-concepts has producedrobust evidence that competence-related self-appraisals can influence subsequent academicachievement (e.g., Marsh, 1990; for reviews, seeMarsh & Craven, 2006; Valentine, DuBois, & Cooper,2004). Although (past) self-concept research reliedon lagged analyses that do not consider absolutegrowth over time, the findings suggest thatcompetence-related appraisals such as perceivedcontrol can be a positive predictor of long-termchange in students’ academic achievement.

Intrinsic motivation is defined as motivation toengage in a task for the sake of interest in the taskitself and the inherent pleasure and satisfactionderived from the task, whereas extrinsic motivationis defined as motivation to engage in a task forexternal reasons (Deci & Ryan, 1985). Extrinsicmotivation is heterogeneous in that there are a vari-ety of different external rewards that produce moti-vated behavior. In the present research, we focuson extrinsic motivation driven by the desire to getgood grades, as this is one of the most prevalentforms of extrinsic motivation in academic settings(Lemos, 1996; Pekrun, 1993).

Intrinsic motivation is associated with variousvariables supporting learning, such as active andeffortful engagement (Ryan & Connell, 1989), per-sistence in the face of failure (Elliott & Dweck,1988), and positive emotional learning experiences(Pekrun, Goetz, Titz, & Perry, 2002). Extrinsic moti-vation, on the other hand, is typically driven byexpected short-term benefits of learning and islinked to instrumental learning independent ofinterest (Grolnick & Ryan, 1987), overuse of depen-dent help seeking (Butler, 1998), and self-handicapping(Urdan & Midgley, 2001). As such, extrinsicmotivation seems suited to benefit immediate, butrather ephemeral, academic achievement, whereasintrinsic motivation seems ideally suited to benefitenduring, long-term learning. Thus, we propose

that intrinsic motivation positively predicts growthin academic achievement (and possibly concurrentachievement as well; see also Marsh, Trautwein,L€udtke, K€oller, & Baumert, 2005), whereas extrinsicmotivation has short-term effects that are mani-fested in a link to current achievement, but doesnot promote long-term growth. Although thesetime-dependent relations (i.e., relations that emergeat a particular point in time) of intrinsic andextrinsic motivation are documented in a fewexperimental studies (Murayama & Elliot, 2011;Vansteenkiste, Simons, Lens, Soenens, & Matos,2005), little research has demonstrated such apattern of effects in the context of long-term growthof academic achievement.

Cognitive Strategies: Deep and Surface LearningStrategies

Learning strategies are strategies that areemployed by learners to study materials. Herein,we focus on two primary cognitive learning strate-gies that are widely recognized as having concep-tual and predictive utility: deep learning strategies(specifically, elaboration of learning material) andsurface learning strategies (rehearsal or memorization;Ramsden, 1988). Deep strategies involve challeng-ing the veracity of information encountered andattempting to integrate new information with priorknowledge, whereas surface learning strategiescharacterize the repetitive rehearsal and rote memo-rization of information (Entwistle & Ramsden,1983).

The extant data are mixed for the relationbetween these learning strategies and academicachievement, but tend to show null or positiverelations for deep strategies and null or negativerelations for surface strategies (Entwistle & Ramsden,1983; Meece, Blumenfeld, & Hoyle, 1988). However,most of the studies to date have focused on linkswith the level of academic achievement at one pointin time. A few studies have shown that learning-related strategy use has an impact on the growth ofacademic achievement during early childhood (e.g.,McClelland, Acock, & Morrison, 2006), but thelong-term relations of learning strategies withachievement during adolescence remain unclear.

Deep learning involves semantic understandingof study materials. Semantic understanding is anessential component in acquiring durable andmeaningful knowledge, and the knowledgeacquired through this semantic elaboration is morelikely to lead to later academic success. Therefore,we anticipated that deep strategies would have

2 Murayama, Pekrun, Lichtenfeld, and vom Hofe

positive relations with growth (as well as initiallevels) of academic achievement. In contrast, weexpected that surface strategies would not predictgrowth rates in achievement. Surface strategiesinvolve rote memorizing without deep elaboration,and studies have indicated that knowledgeacquired through rote memorizing fades quickly(Brown & Craik, 2000). Therefore, we anticipatedthat surface learning would not support furtherlearning, although these strategies might helpimmediate academic achievement.

Limitations in Previous Research on Long-Term Growthin Achievement

This study used latent growth curve modeling(McArdle, Anderson, Birren, & Schaie, 1990) toexamine predictors of growth in math achievement.One of the advantages of latent growth curvemodeling is that it can evaluate the predictors ofabsolute levels of growth. As noted earlier, previ-ous theorizing and evidence suggest that somemotivational variables and learning strategiesshould predict long-term growth in academicachievement. However, only a few studies have uti-lized latent growth curve modeling to directlyaddress the predictors of absolute levels of growthin achievement over time. Furthermore, the researchavailable that used latent growth curve modelinghas a number of important limitations that substan-tially reduce possibilities to draw firm inferences.

First, there is a significant lack of work that con-trolled for intelligence. Intelligence is a ubiquitous,strong predictor of academic achievement (Neisseret al., 1996), implying that it may be the mostimportant confounding variable when examininglinks between variables of learning, such as motiva-tion and cognitive strategies, and academic achieve-ment (Steinmayr & Spinath, 2009). It is easy toimagine students who have higher levels of intelli-gence to be more likely to be motivated and to useefficient strategies that result in their higher aca-demic achievement. Therefore, it is important toinclude intelligence as a predictor to evaluate thepredictive utility of motivation and cognitive strate-gies independently of basic cognitive abilities.

It should also be noted that the utility of intelli-gence to predict growth of achievement is by itselfof theoretical importance. Although researchersagree that intelligence is a strong predictor ofstudents’ level of academic achievement, they havenot reached consensus about its utility in predict-ing growth of achievement over time (Lohman,Ackerman, Kyllonen, & Roberts, 1999). Studies indi-

cated that intelligence scores did not (or onlyweakly) predict growth curves in performanceattainment (e.g., Gutman, Sameroff, & Cole, 2003;Underwood, Boruch, & Malmi, 1978; Woodrow,1946). These findings suggest that, unlike motivationand cognitive strategies, intelligence may not reflectthe capacity to learn new knowledge and skills.Importantly, however, the majority of these studiesfocused on short-term learning (sometimes learningwithin one experimental session). Studies investigat-ing the relations of intelligence with the long-termgrowth of academic achievement are still rare.

Second, much of the previous literature usinglatent growth curve modeling failed to establish acommon metric for assessments of achievementover time. For analyzing growth in academicachievement, if one simply uses achievement testscores or teacher-provided grades, it is not possibleto adequately study growth because these scoresare likely not comparable across years. To establisha common metric, item response theory (IRT) canbe employed, and anchor items can be includedacross adjacent time points to allow for verticalscaling (McDonald, 1999). Although IRT scaling isgenerally acknowledged today as an appropriateway to examine academic achievement over time,there still are many studies that have not used thismethodology, casting doubt on the validity of theirfindings on growth (e.g., Gutman et al., 2003; Hart,Hofmann, Edelstein, & Keller, 1997; Johnson,McGue, & Iacono, 2006).

Third, previous studies included relatively smallnumbers of variables, focusing on either motiva-tional or strategy constructs (e.g., McClelland et al.,2006; Shim, Ryan, & Anderson, 2008). However,given that motivational and strategy variables oftenshow substantial correlations, it is important toinclude multiple variables from both groups of con-structs to examine their unique contributions.

Finally, previous studies failed to consider thepossibility that the utility of motivational and strat-egy variables for predicting academic achievementmay change depending on students’ developmentalstage. Many studies have documented developmen-tal trajectories of motivation and strategies overtime (e.g., Fredricks & Eccles, 2002; Frenzel, Goetz,Pekrun, & Watt, 2010; Jacobs, Lanza, Osgood,Eccles, & Wigfield, 2002). It is possible, then, thatnot only the developmental trajectories but also thefunctions of motivation and learning strategies inshaping students’ achievement during early adoles-cence can change over time. However, few studieshave examined developmental change in the func-tions of these variables.

Long-Term Growth 3

Overview of the Present Research and Hypotheses

The current study aimed to broaden the extantknowledge about how motivation and cognitive strat-egies relate to growth in academic achievement, usingGerman longitudinal mathematics achievement datafrom six annual waves (Grades 5 through 10). We con-trolled for intelligence to evaluate the contributions ofmotivation and strategy variables, and utilized IRT toscale academic achievement over time. We includedall of the motivation variables (perceived control,intrinsic motivation, and extrinsic motivation) andlearning strategy variables (deep and surface strate-gies) together to investigate their unique contribu-tions. In addition, we compared the predictive powerof motivation and strategies at two different timepoints (Grades 5 and 7) to examine possible functionalchange in these variables. Grades 5 and 7 were chosenbecause they represent critical periods in the educa-tional careers of the German students who partici-pated in the study. Specifically, these studentsattended public schools in the German state of Bavariathat use ability tracking starting in Grade 5. Studentsare assigned to low-ability schools (Hauptschule), med-ium-ability schools (Realschule), and high-abilityschools (Gymnasium). In Grade 7, students’ assign-ments to low- versus medium-ability schools werereconsidered; thus, part of the student sample werereassigned to a different track. As such, for studentsin this study, Grades 5 and 7 represented criticalcareer and academic transitions. Furthermore, stu-dents in Grades 5 and 7 are in their early adolescence,a period characterized by a great deal of intellectual,social, physical, and emotional change (Lerner, 1993),which may imply that the functions of academic moti-vation and strategies also undergo change.

Succinctly stated, the study tested the followinghypotheses:

Hypothesis 1: Perceived control is a positive pre-dictor of the initial level as well as the growthrate of achievement.

Hypothesis 2: Extrinsic motivation is a positive pre-dictor of the initial level of achievement, whereasintrinsic motivation is a positive predictor of boththe initial level and the growth rate of achievement.

Hypothesis 3: Surface learning strategies are apositive predictor of the initial level of achieve-ment, whereas deep learning strategies are a posi-tive predictor of both the initial level and thegrowth rate of achievement.

Hypothesis 4: Intelligence is a positive predictorof the initial level of achievement, but has no rela-tion with growth rate.

Hypotheses 1–3 were tested while controlling forstudents’ intelligence. Furthermore, although wedid not have a specific hypothesis, we expected thatthe predictive power of motivation and strategiesmay change from Grade 5 to Grade 7.

Method

Participants and Design

The sample consisted of German students whoparticipated in the Project for the Analysis of Learn-ing and Achievement in Mathematics (PALMA;see Frenzel et al., 2010; Frenzel, Pekrun, Dicke, &Goetz, in press; Pekrun et al., 2007). This projectincluded a longitudinal study involving annualassessments during the secondary school years(Grades 5–10) to investigate the development ofmathematics achievement. At each grade level, thePALMA math achievement test was administeredtoward the end of the school year. Students’ intelli-gence and the self-reported motivational and strat-egy variables used in the current study wereassessed toward the end of the school year inGrades 5 and 7.

Samples were drawn from secondary schools inthe state of Bavaria, and were drawn so that theywere sufficiently representative of the student pop-ulation of Bavaria (Pekrun et al., 2007). The samplesincluded students from all three school types withinthe Bavarian public school system as describedearlier, including lower-track schools (Hauptschule),intermediate-track schools (Realschule), and higher-track schools (Gymnasium). These three school typesdiffer in academic demands and students’ entry-level academic ability. German students typicallyenter one of these schools in Grade 5 (i.e., the 1styear of the assessment), based largely on their prioracademic achievement.

At the first assessment (Grade 5), the samplecomprised 2,070 students from 42 schools (49.6%female, Mage = 11.7 years; 37.2% lower-track schoolstudents, 27.1% intermediate-track school students,and 35.7% higher-track school students). In eachsubsequent year, the study tracked the studentswho had participated in the previous assessment(s)and also included those students who had not yetparticipated in the study, but had become studentsof PALMA classrooms at the time of the assessment(see Pekrun et al., 2007). Overall, the study sampleconsisted of 3,530 students (49.7% female) who par-ticipated in at least one assessment (see SupportingInformation Appendix S1 available online for fur-ther details).


Measures

Socioeconomic Status (SES)

Family SES was assessed by parent report usingthe EGP classification (Erikson, Goldthorpe, &Portocarero, 1979) which consists of six orderedcategories of parental occupational status. In thecurrent analysis, the scores are coded so that highervalues represent higher family SES.

Mathematics Achievement

Mathematics achievement was assessed by thePALMA Mathematics Achievement Test (vom Hofe,Kleine, Pekrun, & Blum, 2005; vom Hofe, Pekrun,Kleine, & G€otz, 2002). Using both multiple-choiceand open-ended items, this test measures students’modeling competencies and algorithmic competen-cies in arithmetics, algebra, and geometry. It alsocomprises subscales pertaining to more specific con-tents (e.g., fractions, ratios, or functions).

The test was constructed using multimatrixsampling with a balanced incomplete block design(for details, see vom Hofe et al., 2002). Specifi-cally, for each measurement point, two differenttest versions were prepared that consisted ofapproximately 60–90 items each, and studentscompleted one of these two test booklets. Anchoritems were included to allow for the linkage ofthe two different test forms as well as the six dif-ferent measurement points. The obtained achieve-ment scores were scaled using one-parameterlogistic item-response theory (Rasch scaling; Wu,Adams, Wilson, & Haldane, 2007), with M = 100and SD = 15 at Grade 5 (i.e., the first measure-ment point). Additional analyses confirmed theunidimensionality and longitudinal invariance ofthe test scales (see Supporting Information Appen-dix S1 available online).

Motivation and Learning Strategies

Motivation and learning strategies were assessedby self-report scales developed for PALMA. Allscales pertained to the domain of mathematics, andreliability and validity of all the scales were ascer-tained in two large-sample pilot studies (Ns = 784and 1,613; Pekrun et al., 2007). Coefficient Omegawas used to estimate the reliability of the scalescores (see Table 1). Omega provides more fine-tuned estimates of scale reliability as comparedwith the commonly used Cronbach’s Alpha(McDonald, 1999).

Perceived control. Pekrun et al.’s (2007) PerceivedAcademic Control scale was used to assess stu-dents’ perceived control in mathematics (six items;e.g., “When doing math, the harder I try, the betterI perform”). Participants responded on a 1 (stronglydisagree) to 5 (strongly agree) scale.

Intrinsic motivation. Three items from Pekrun’s(1993) Intrinsic Academic Motivation scale were usedto assess students’ intrinsic motivation in mathemat-ics (e.g., “I invest a lot of effort in math, because I aminterested in the subject”). Participants responded ona 1 (strongly disagree) to 5 (strongly agree) scale.

Extrinsic motivation. Four items from Pekrun’s(1993) Extrinsic Academic Motivation scale wereused to assess students’ extrinsic motivation in math-ematics (sample item: “In math I work hard, becauseI want to get good grades”). Participants respondedon a 1 (strongly disagree) to 5 (strongly agree) scale.

Deep learning strategies. The PALMA elaborationstrategies scale (Pekrun et al., 2007) was used to assessstudents’ deep learning strategies (three items; e.g.,“When I study for exams, I try to make connectionswith other areas of math”). Participants responded ona 1 (strongly disagree) to 4 (strongly agree) scale.

Surface learning strategies. The PALMA rehearsalstrategies scale (Pekrun et al., 2007) was used to assessstudents’ surface learning strategies (three items; sam-ple item: “For some math problems I memorize thesteps to the correct solution”). Participants respondedon a 1 (strongly disagree) to 4 (strongly agree) scale.

Table 1Descriptive Statistics and Internal Consistencies of the Predictors

M SDObservedrange

Coefficientof reliability

Grade 5Perceived control 3.87 0.72 1.17–5.00 .75Intrinsic motivation 3.21 1.18 1.00–5.00 .87Extrinsicmotivation

3.37 0.99 1.00–5.00 .80

Deep learningstrategies

2.38 0.78 1.00–4.00 .67

Surface learningstrategies

2.49 0.75 1.00–4.00 .64

Intelligence 104.75 13.41 61–132 —

Grade 7Perceived control 3.43 0.83 1.00–5.00 .81Intrinsic motivation 2.57 1.07 1.00–5.00 .84Extrinsic motivation 2.87 0.97 1.00–5.00 .78Deep learningstrategies

2.03 0.70 1.00–4.00 .64

Surface learningstrategies

2.26 0.73 1.00–4.00 .65

Intelligence 99.61 14.56 55–145 —

Long-Term Growth 5

Intelligence. Intelligence was measured using the25-item nonverbal reasoning subtest of the Germanadaptation of Thorndike’s Cognitive Abilities Test(Kognitiver F€ahigkeitstest [KFT 4–12 + R]; Heller &Perleth, 2000).

Strategy of Data Analysis

Growth curve models can be fit in both struc-tural equation modeling (SEM) and multilevel mod-eling frameworks. We decided to use the SEMframework because it allows us to assess the overallfit between the empirical data and the growthcurve models we specified. Specifically, we evalu-ated the comparative fit index (CFI), the Tucker–Lewis index (TLI), the root mean square error ofapproximation (RMSEA), and the standardized rootmean square residual (SRMR). It should be notedthat nearly all current SEM packages compute CFIand TLI based on an inappropriate baseline modelwhen evaluating the fit of latent growth curvemodels (Wu, West & Taylor, 2009). In our analysisof latent growth curve models, we corrected theCFI and TLI by specifying the appropriate baselinemodel (see Wu et al., 2009). Following conventions,we regarded values higher than .90 for CFI andTLI, lower than .08 for the RMSEA, and lower than.10 for the SRMR as indicating an acceptable fit.

Prior to the analyses, all the predictors except fordemographic variables were standardized with zeromean and unit variance, and SES was mean cen-tered to facilitate the interpretation of the results.Based on Muth�en and Muth�en (2004), we adjustedthe standard errors and chi-square statistics to cor-rect for potential statistical biases resulting fromnon-normality of the data. Due to the longitudinaldesign of the study, there is a certain proportion ofmissing data. Accordingly, to make full use of thedata from students who only partially participatedin the investigation, we applied the full informationmaximum likelihood method to deal with missingdata (Enders, 2006). Additional analyses thataddressed issues of measurement error and thenested structure of the data (i.e., individuals nestedwithin schools) are reported in Supporting Informa-tion Appendix S1 available online.

Results

Longitudinal Analysis of Measurement Invariance forMotivation and Strategies

The descriptive statistics of the predictor vari-ables for Grades 5 and 7 are provided in Table 1.

Because we wanted to compare the predictiveutility of motivational and strategy variables acrosstwo different time points (i.e., Grades 5 and 7), weused confirmatory factor analysis to assess mea-surement equivalence across these time points. Spe-cifically, we evaluated the full scalar and errorvariance invariance model (see Steenkamp & Baum-gartner, 1998), in which the item intercept, factorloadings, factor variances and covariances, anderror variances were set to be equal across the timepoints. Support for this model would providestrong evidence for longitudinal invariance of themeasures. All of the motivation and learning strat-egy variables were analyzed together. The residualsassociated with indicators of the same items inGrades 5 and 7 were allowed to correlate. Theresults supported the longitudinal invariance of thescales, with good fit to the data, v2(663) = 2,548.98,p < .01, CFI = .92, TLI = .92, RMSEA = .031, SRMR= .060. These findings suggest that it is legitimateto compare the predictive utility of the motivationaland strategy variables across time points.

The disattenuated correlations of the latent fac-tors for the same constructs over time were .37 to.47 (ps < .01). The size of these coefficients indicatesthat the motivational and strategy variables were tosome extent stable across the 2 years but alsoshowed change, suggesting that it is possible thattheir functional roles for predicting academicachievement also changed over time. Comparisonof factor means indicated a decline in all of themotivation and strategy variables from Grades 5 to7 (ps < .01), which is largely consistent with previ-ous findings (e.g., Jacobs et al., 2002; Otis, Grouzet& Pelletier, 2005).

Latent Growth Curve Modeling

In the latent growth curve analysis, we firstconstructed a growth curve model without motiva-tional and strategy variables to specify the basicfunctional form of growth in mathematical achieve-ment. In this model, math achievement scoresacross six different time points (Grades 5–10) aremodeled by a set of common factors repre-senting the functional form of growth. Our initialexploration of the data indicated that a lineargrowth model (a model including an intercept anda linear slope) did not fit the data well, suggestingthat the growth curve includes nonlinear change.Accordingly, we decided to use an exponentialgrowth curve model (Grimm, Ram, & Hamagami,2011; for details about model selection, see Support-ing Information Appendix S1 available online).


Exponential growth curve modeling has severaladvantages over the commonly used quadraticgrowth curve model. Notably, exponential growthmodels provide interpretable parameters describinggrowth, such as the total amount of change overtime. This point is of particular importance for ourstudy, as we are interested in the factors thatinfluence the overall amount of growth over years.The model is represented by the following equa-tion.

yt ¼ g0 þ g1½1� expð�a � xtÞ� þ et;

t ¼ 5; 6; 7; 8; 9; 10ð1Þ

where yt is the math achievement score at grade t,and g0 and g1 are the latent variables representingthe intercept and total amount of (asymptotic)growth. a is a free parameter representing the rateof growth. ɛt is the error term for grade t, and weconstrained the variances of this term to be equalacross the time points [i.e., varðe5Þ ¼ varðe6Þ¼ varðe7Þ ¼ varðe8Þ ¼ varðe9Þ ¼ varðe10Þ; Hertzog,von Oertzen, Ghisletta, & Lindenberger, 2008]. xtrepresents the fixed time coding at grade tanchored at the initial time point [i.e., x5 = 0,x6 = 1, x7 = 2, x8 = 3, x9 = 4, x10 = 5]. By this partic-ular coding of time point, we can interpret g0 asthe predicted math achievement score at the initialtime point, and g1 as the asymptotic total amountof growth from the initial time point.

To control for participants’ demographic back-ground, variables representing gender, school type,and SES were included in the model. Gender wastreated as a dummy-coded variable (GENDER;0 = female, 1 = male), and school type was repre-sented by two orthogonal contrast variables. Thefirst school-type variable represents the differencebetween higher-track schools and the pooled inter-mediate- and lower-track schools (SCTYPE1; higher-track school = 2, intermediate-track school = �1,lower-track school = �1). The second variable rep-resents the difference between intermediate-trackschools and lower-track schools (SCTYPE2; higher-track school = 0, intermediate-track school = 1,lower-track school = �1). The following equationsshow the mathematical representation of this partof the model.

g0 ¼ lg0 þ c01GENDERþ c02SCTYPE1

þc03SCTYPE2 þ c04SESþ f0ð2Þ

g1 ¼ lg1 þ c11GENDERþ c12SCTYPE1

þc13SCTYPE2 þ c14SESþ f1ð3Þ

where c represents the path coefficients of the cor-responding variable. f is the error (or disturbance)term for each growth component.

The model showed an acceptable fit to the data,v2(36) = 757.48, p < .01, CFI = .96, TLI = .94, RMSEA= .075, SRMR = .090. The results (see Table 2)revealed that the mean total amount of growth wassignificantly positive (lg1 = 144.17, p < .01) with apositive growth rate (a = .06, p < .01). In addition,most of the path coefficients for gender, school type,and SES variables were significant, suggesting vari-ability in growth trajectories across demographicbackground variables. Figure 1 displays the esti-mated growth curves for male and female studentsfrom different types of schools.

An interesting observation in Figure 1 is thatthe difference between school tracks became largerwith increasing grade level, as indicated by thesignificantly positive relation of school type withtotal amount of change (c12 = 12.74, p < .01;c13 = 9.27, p < .01). This finding suggests that thetracking system used in Germany may be a sourceof a “Matthew effect” in math achievement—aphenomenon where the gap between high-achiev-ing and low-achieving students increases withtime (Bast & Reitsma, 1997; Baumert, Nagy, &Lehmann, 2012). To quantitatively evaluate thelink between tracking and the Matthew effect, we

Table 2Parameter Estimates for the Growth Curve Model Including Genderand School Type

Estimate SE

InterceptMean (lg0) 99.28** 0.31Gender (c01) 3.24** 0.45School type 1(high vs. mid or low track; c02)

4.51** 0.17

School type 2(mid vs. low track; c03)

6.24** 0.27

Family socioeconomic status (c03) 0.66** 0.13Total amount of growthMean (lg1) 144.17** 18.1Gender (c11) �1.08 2.37School type 1(high vs. mid or low track; c12)

9.27** 1.29


12.74** 2.02

Family socioeconomic status (c13) 0.85 0.67

**p < .01.

Long-Term Growth 7

computed the correlation between the intercept(g0) and the total amount of change (g1) beforeand after including the school-type variables.When the school-type variables were not included,this correlation was positive (r = .29, p < .01), indi-cating the presence of a Matthew effect. Control-ling for these variables dramatically decreased thecorrelation (r = .01, p = .79), suggesting that theMatthew effect observed in the data is linked totracking as used in the German secondary schoolsystem.

Predictive Relations of Motivation and StrategiesAssessed at Grade 5

To examine the relations of Grade 5 motivationaland strategy variables with the growth curve ofachievement, we included these variables as predic-tors of the growth components. We also includedintelligence as a predictor to control for students’basic cognitive abilities and to investigate the pre-dictive utility of intelligence for the growth compo-nents. Specifically, Equations 2 and 3 wereexpanded to take the following form:

g0 ¼lg0 þ c01GENDERþ c02SCTYPE1

þ c03SCTYPE2 þ c04SESþ c05CONTþ c06INTþ c07EXTþ c08DEEPþ c09SURFACEþ c10IQþ f0

ð4Þ

g1 ¼lg1 þ c11GENDERþ c12SCTYPE1

þ c13SCTYPE2 þ c14SESþ c15CONTþ c16INTþ c17EXTþ c18DEEPþ c19SURFACEþ c20IQþ f1

ð5Þ

where CONT, INT, EXT, DEEP, SURFACE, and IQrepresent the variables of perceived control, intrin-sic motivation, extrinsic motivation, deep learningstrategies, surface learning strategies, and intelli-gence assessed at Grade 5.

The model showed an acceptable fit to the data,v2(60) = 843.00, p < .01, CFI = .97, TLI = .93,RMSEA = .061, SRMR = .059. The parameter esti-mates of the model are presented in Table 3. Notsurprisingly, intelligence had a strong positive rela-tion with the intercept (c10 = 4.72, p < .01). In addi-tion, however, most of the motivational and

5 6 7 8 9 10

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160

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vem

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core

Gymnasium-maleGymnasium-femaleRealschule-maleRealschule-femaleHauptschule-maleHauptschule-female

Figure 1. Estimated growth curve of math achievement fromGrades 5 to 10 as a function of gender and school type.Note. Gymnasium = higher-track school; Realschule = intermedi-ate-track school; Hauptschule = lower-track school.

Table 3Parameter Estimates for the Growth Curve Model Including Motiva-tional and Strategy Variables Assessed at Grade 5

Estimate SE


3.36** 0.17


4.67** 0.27

Family socioeconomic status (c04) 0.45** 0.12Perceived control (c05) 2.21** 0.27Intrinsic motivation (c06) 1.53** 0.29Extrinsic motivation (c07) 0.74** 0.28Deep learning strategies (c08) �0.41 0.31Surface learning strategies (c09) �1.60** 0.27Intelligence (c10) 4.72** 0.27

Total amount of changeMean (lg1) 142.34** 17.6Gender (c11) �1.22 2.49School type 1(high vs. mid or low track; c12)

8.15** 1.26


11.69** 2.00

Family socioeconomic status (c14) 0.66 0.66Perceived control (c15) 3.78* 1.72Intrinsic motivation (c16) 1.04 1.84Extrinsic motivation (c17) 1.56 1.72Deep learning strategies (c18) �2.52 1.88Surface learning strategies (c19) �5.99** 1.71Intelligence (c20) 0.24 1.61

*p < .05. **p < .01.


strategy factors yielded significant incremental rela-tions with the initial achievement score, over andabove intelligence. Specifically, perceived control,intrinsic motivation, and extrinsic motivation weresignificant predictors of the intercept factor(c05 = 2.21, c06 = 1.53, and c07 = 0.74, ps < .01),whereas surface learning strategies were a negativepredictor (c09 = �1.60, p < .01). Deep learning strat-egies did not show any significant relation (p = .18).

Intriguingly, in contrast to the relations with theintercept factor, intelligence did not predict the totalamount of growth of math achievement over time(c20 = 0.24, p = .88). However, two of the motiva-tional and strategy variables turned out to be signif-icant predictors of growth of math achievementover time. Specifically, perceived control was a sig-nificantly positive predictor of the slope factor(c15 = 3.78, p < .05), whereas surface learning strate-gies were a significantly negative predictor(c19 = �5.99, p < .01). This result underscores theunique importance of motivational and strategyvariables in affecting not only current achievementbut also the long-term growth of achievement.

To facilitate the interpretation of the findings ongrowth, we estimated the growth curves ofstudents with scores of 1.5 SD above the mean forperceived control and 1.5 SD below the meanfor surface learning strategies (“high-growth stu-dents”), and students with scores of 1.5 SD belowthe mean for perceived control and 1.5 SD abovethe mean for surface learning strategies (“low-growth students”). To avoid redundancy, Figure 2presents the estimated growth curves only for malehigher-track (Gymnasium) students. The shape ofthe growth curves for females and for intermediate-or lower-track students was basically the same.Figure 2 visually confirms that the slope is steeperfor high-growth than for low-growth students.

Predictive Relations of Motivation and StrategiesAssessed at Grade 7

The previous analysis partially supported ourhypotheses in that (a) intelligence had strongpositive relations with initial level of math achieve-ment but showed null relations with growth rateand (b) perceived control had positive relations notonly with initial level but also with the growth rate.To further explore our hypotheses, we next focusedon students’ growth in math achievement fromGrades 7–10. Specifically, we investigated whetherthe growth curve estimated in the Grades 7–10 datacan be predicted by motivation and learning strate-gies assessed at the Grade 7. The model was the

same as in the previous analysis, Equations 1, 4,and 5, with the following exceptions. First, Equa-tions 4 and 5 included the motivational and strat-egy variables and intelligence assessed at Grade 7instead of Grade 5, as our focus was to examine thepredictive utility of the variables assessed at Grade7. Participants’ school type was also coded basedon the information at Grade 7. Second, we used themath achievement scores assessed across Grades7–10, that is, t = 7, 8, 9, 10 in Equation 1 andx7 = 0, x8 = 1, x9 = 2, x10 = 3. Third, we fixed thegrowth rate parameter to the value obtained in theprevious analysis (i.e., a = .06) to make the Grade 5and Grade 7 analyses comparable. This constraintimproved model fit in terms of the Akaike informa-tion criterion and the Bayesian informationcriterion, and did not significantly increase the chi-square fit statistics, Dv2(1) = 1.09, p = .30.

The model showed an acceptable fit to the data,v2(28) = 477.79, p < .01, CFI = .97, TLI = .91, RMSEA= .073, SRMR = .060. The parameter estimates of themodel are presented in Table 4. For the prediction ofthe intercept, the pattern of results was almost thesame as in the Grade 5 analysis; in addition to thestrong positive explanatory power of intelligence(c10 = 6.10, p < .01), perceived control and extrinsicmotivation were significantly positive predictors ofthe initial level of math achievement (c05 = 3.35,p < .01; c07 = 1.18, p < .01), whereas surface learningstrategies were a significantly negative predictor

5 6 7 8 9 10

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High-growth StudentsLow-growth Students

Figure 2. Estimated math achievement growth curve from 5th to10th grades for high-growth and low-growth students.Note. To avoid redundancy, only male Gymnasium students areplotted. High-growth students = 1.5 SD above the mean for per-ceived control and 1.5 SD below the mean for surface learningstrategies. Low-growth students = 1.5 SD below the mean forperceived control and 1.5 SD above the mean for surface learn-ing strategies.

Long-Term Growth 9

(c09 = �1.49, p < .01). These results again corrobo-rate the importance of motivational and strategy fac-tors in accounting for current math achievement overand above intelligence. On the other hand, intrinsicmotivation and deep learning strategies were not sig-nificant predictors of the intercept factor.

Intriguingly, for the prediction of total amount ofgrowth, the Grade 7 analysis showed a pattern ofresults that differed substantially from the Grade 5findings. Specifically, whereas perceived control,extrinsic motivation, and surface learning strategiesdid not predict growth of math achievement, intrin-sic motivation and deep learning strategies weresignificantly positive predictors of the total amountof growth (c16 = 4.51, p < .05; c18 = 4.64, p < .05).Again, intelligence had null relations with theamount of growth (c20 = 0.37, p = .87).

As in the Grade 5 analysis, these findings indi-cate that motivational and strategy variablesuniquely contribute to growth of math achieve-

ment. Furthermore, these findings also indicate thatthere is developmental change in the predictivepower of these variables. As assessed at Grade 7,intrinsic motivation and deep learning strategiesdid not have any significant relations with the inter-cept factor but related to subsequent growth, sug-gesting that they may not have immediate utilitybut may provide long-term benefits in facilitatingmath achievement at this age. As an illustration forthe nature of this finding, Figure 3 displays the esti-mated growth curves of male higher-track (Gymna-sium) students with scores 1.5 SD above the meanfor intrinsic motivation and deep learning strategies(“high-growth students”) and students with scores1.5 SD below the mean for these variables (“low-growth students”). The shape of the growth curvesfor females and for intermediate- or lower-trackstudents was basically the same. Figure 3 visuallyconfirms that high-growth and low-growth studentsdid not differ in terms of the initial status at Grade7, but developed differently across the subsequentgrade levels.

Discussion

This research examined the predictive power ofmotivation, cognitive strategies, and intelligence forexplaining the long-term growth of adolescents’

7 8 9 10

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core

High-growth StudentsLow-growth Students

Figure 3. Estimated math achievement growth curve from 7th to10th grades for high-growth and low-growth students.Note. To avoid redundancy, only male Gymnasium students areplotted. High-growth students = 1.5 SD above the mean forintrinsic motivation and deep learning strategies. Low-growthstudents = 1.5 SD below the mean for intrinsic motivation anddeep learning strategies.

Table 4Parameter Estimates for the Growth Curve Model Including Motiva-tional and Strategy Variables Assessed at Grade 7

Estimates SE


4.47** 0.17


6.88** 0.28

Family socioeconomic status (c04) 0.33** 0.12Perceived control (c05) 3.35** 0.27Intrinsic motivation (c06) 0.37 0.29Extrinsic motivation (c07) 1.18** 0.28Deep learning strategies (c08) �0.49 0.28Surface learning strategies (c09) �1.49** 0.25Intelligence (c10) 6.10** 0.27

Total amount of changeMean (lg1) 134.39** 2.41Gender (c11) 6.41 3.40School type 1(high vs. mid or low track; c12)

3.80** 1.42


6.23* 2.76

Family socioeconomic status (c14) 1.69 0.96Perceived control (c15) �2.76 2.18Intrinsic motivation (c16) 4.51* 2.25Extrinsic motivation (c17) �0.55 2.10Deep learning strategies (c18) 4.64* 2.23Surface learning strategies (c19) �0.81 2.02Intelligence (c20) 0.37 2.28

*p < .05. **p < .01.


achievement in mathematics from Grades 5 to 10.The results showed that motivation and strategiespredicted current math achievement over andabove intelligence. Furthermore, our findingsrevealed that motivation and strategies alsoexplained the growth of achievement. Growth waspositively predicted by perceived control, intrinsicmotivation, and deep learning strategies, and it wasnegatively predicted by surface learning strategies.In contrast, intelligence turned out not to be a pre-dictor of growth (after controlling for demographicvariables), despite being a strong predictor ofcurrent math achievement. Overall, these findingslargely support our hypotheses.

Relations of Motivation and Strategies With Initial Leveland Growth of Achievement

Among the motivational factors, both perceivedcontrol and intrinsic motivation predicted long-term growth in math achievement. Specifically,perceived control (assessed at Grade 5) and intrin-sic motivation (assessed at Grade 7) positively pre-dicted the total amount of growth in mathachievement. This lends support to the notion thatthese constructs do not merely reflect subjectiveperceptions of current ability (perceived control) ortransient, ephemeral positive emotion (intrinsicmotivation), but motivational tendencies that shapefuture learning and resulting achievement (Deci &Ryan, 1985; Marsh & Craven, 2006). In contrast,extrinsic motivation predicted only initial levels butnot growth in achievement, illustrating an interest-ing contrast with the relations observed for intrinsicmotivation. The short-lived nature of extrinsic moti-vation and long-term benefit of intrinsic motivationare in accordance with previous experimental find-ings (Murayama & Elliot, 2011; Vansteenkiste et al.,2005), and our results provide the first evidencethat this pattern is evident even over long periodsof time.

With regard to learning strategies, both deepstrategies and surface strategies were related tolong-term growth in math achievement, buttheir relations had opposite signs (deep learningstrategies were positively linked, whereas surfacelearning strategies were negatively linked togrowth). Furthermore, surface learning strategieswere also negatively linked to initial level ofachievement. Although we had not expected thatsurface learning strategies would lead to lowerinitial levels and smaller growth rates of mathachievement, it is plausible that relying on surface,rehearsal-based strategies can interfere with the use

of more efficient strategies and thus be deleteriousfor achievement outcomes. It should be noted,however, that the present findings do not indicatethat surface learning strategies are maladaptivein any situations, as the effectiveness of learningstrategies may largely depend on learning materialsand phases of learning (Alexander & Murphy,1998). Surface learning strategies may be useful forsome tasks (e.g., skill tasks) or at initial stages oflearning.

A seemingly counterintuitive finding is thatintrinsic motivation (assessed at Grade 7) and deeplearning strategies (assessed at Grades 5 and 7)were unrelated to concurrent achievement.Although unexpected, this is actually consistentwith the extant literature suggesting that the rela-tion between these constructs and performance isunclear (Meece et al., 1988; Murphy & Alexander,2002; Utman, 1997). Students with high intrinsicmotivation are less concerned about how well theyperform on upcoming achievement tests. Accord-ingly, although intrinsic motivation should providelong-term benefits, such a noninstrumentalapproach to learning may not add much to currentperformance. As for deep, elaborative learningstrategies, previous studies indicated that elabora-tive learning may not be an efficient means of deal-ing with an upcoming achievement test becausesemantic elaboration is a relatively slow learningprocess and therefore costly if time is limited (Gar-ner, 1990). Given the somewhat counterintuitivenature of this phenomenon, further research isneeded to replicate the present findings and toexamine possible conditions under which these con-structs are positively related to (concurrent)achievement scores.

Developmental Differences in the Functions ofMotivation and Strategies

It is noteworthy that the motivational and strat-egy variables predicted growth at different develop-mental stages. For motivational factors, the relationbetween perceived control and growth in mathachievement was observed when perceived con-trol was assessed at Grade 5, whereas differentialrelations for intrinsic versus extrinsic motivationemerged only when motivation was assessed atGrade 7.

One possible explanation is that our findingsreflect developmental differences in students’understanding of different dimensions of motiva-tion. Perceived control (i.e., cognitive representa-tions of action–outcome contingencies) seems to be

Long-Term Growth 11

a straightforward notion that may require littlecognitive capacity to understand. Thus, it is likelythat students develop individual appraisals of per-sonal control over achievement and that theseappraisals influence their academic achievementfrom early on (e.g., in the first grades of elemen-tary school). In contrast, understanding the intrinsicversus extrinsic reasons underlying one’s behavioris quite complicated and may require higher orderthinking. As a result, comprehension of the intrin-sic–extrinsic distinction to the extent that intrinsicmotivation can clearly relate to long-term achieve-ment growth may occur only later (e.g., in earlyadolescence). In fact, previous findings indicate thatchildren’s perceptions of control (expectancybeliefs) differentiate into subject-specific, distinctconstructs at a very early stage in elementaryschool. In contrast, subcomponents of task value(such as intrinsic value vs. utility value) are lessdifferentiated among younger children, but becomemore distinct during early adolescence (Wigfield &Eccles, 1992).

We also observed that perceived control atGrade 5 predicted growth in math achievement,whereas perceived control at Grade 7 failed to doso. One possible explanation is that, as mathemat-ics content becomes more difficult, students’ judg-ments of personal control over achievement areanchored more to concrete experiences with currentmath problems rather than to judgments of per-sonal math ability and possible future trajectoriesof developing math competence (see Vallacher &Wegner, 1987), resulting in less predictive powerfor long-term growth. In accord with this interpre-tation, contrary to the predictive relations withgrowth, the link between perceived control andconcurrent achievement was stronger at Grade 7than at Grade 5.

With regard to strategy variables, growth inmath achievement was positively predicted by deeplearning strategies at Grade 7, but not yet at Grade5. Previous developmental studies on strategy usehave shown that students exhibit an increasingreadiness across adolescence to deploy strategiesthat are elaborative and generative (Christopoulos,Rohwer, & Thomas, 1987; Pressley, Borkowski, &Johnson, 1987). In addition, these studies alsorevealed a “utilization deficiency,” a phenomenonwhereby children, up until a particular develop-mental stage, are able to use a learning strategy,but are unable to reap its performance benefits(Bjorklund, Coyle, & Gaultney, 1992; Miller & Seier,1994). Our findings shed light on these findings,suggesting that an effective use of deep learning

strategies may occur only later in students’ aca-demic career.

The Role of Intelligence and the Matthew Effect

One of the features of the current investigation isthat we controlled for intelligence when examiningthe predictive relations of motivation and cognitivestrategies. This is by itself of considerable impor-tance, as discussed at the outset. In addition, theinclusion of intelligence as a predictor producedinteresting findings: Long-term growth in mathachievement was predicted by motivational andstrategy factors, but not by students’ intelligence(after controlling for demographic variables). Thisstands in marked contrast to the commonlyobserved finding that intelligence explains a muchlarger proportion of the variance in current achieve-ment scores, as compared to motivational and strat-egy variables (e.g., Spinath, Spinath, Harlaar, &Plomin, 2006). We should be aware that this studyfocused on the development of achievement in oneacademic domain only. Nonetheless, our findingsclearly underscore the importance of paying atten-tion to adolescents’ motivation and learning strate-gies when wanting to understand the developmentof their academic achievement. Thus, an intriguingmessage from this study is that the critical determi-nant of growth in achievement is not how smartyou are, but how motivated you are and how youstudy.

Another interesting finding is the Matthew effect,a phenomenon that over time more able individualsbecome even better and less able individualsbecome even worse, thus widening the gapbetween haves and have-nots. The Matthew effecthas been investigated in a number of differentdomains, such as reading ability (Stanovich, 1986)and job payment (Tang, 1996). The current studyobserved the Matthew effect in the domain of mathachievement during adolescence, and the findingssuggest that the effect may have been linked to theGerman school tracking system. It should be keptin mind, however, that the current analysisprovides little information about the backgroundcharacteristics of students attending schools fromdifferent tracks (e.g., in terms of learning environ-ments at home) or the educational quality of theseschools that may have produced this Mattheweffect. These varied explanations would have differ-ent educational implications (see also Baumertet al., 2012). Further research is needed to investi-gate such mechanisms that widen the gap betweenstudents differing in initial achievement.


Limitations and Directions for Future Research

Our findings should be interpreted in the contextof several limitations that suggest directions forfuture research. First, although the IRT scaling hasthe great advantage of establishing a common met-ric across different grade levels—an issue that wasnot addressed in many previous studies—it rests onthe critical assumption that all math achievementtest items measure a single unidimensional con-struct. By testing the dimensionality and longitudi-nal invariance of the PALMA Math AchievementTest, we were able to show that there was no majorviolation of unidimensionality assumptions in thisstudy (see Supporting Information Appendix S1available online). However, unidimensionalityassumptions cannot be completely met in longitudi-nal studies of academic achievement, given theneed to include items measuring various contentsthat match the diversity of the curriculum bothwithin and across grade levels. Future researchwould do well to investigate the impact of suchlocal heterogeneity of items on growth trajectories.

Second, although the present research producedevidence about the predictive power of motiva-tional and strategy variables on growth (aftercontrolling for demographic variables and intelli-gence), these relations are still correlational innature. Thus, one should exercise caution wheninterpreting these links in terms of causality.Although this was not the primary concern of ourresearch, alternative approaches such as a cross-lagged analysis (e.g., Marsh & Craven, 2006)or dual change score analysis (see McArdle &Hamagami, 2001) may be better suited to examinecausal ordering and possible reciprocal effects. Pre-vious research has shown that students’ academicself-concepts and their achievement can be linkedby reciprocal causation (Marsh, 1990). It seemslikely that students’ intrinsic and extrinsic motiva-tion and surface versus deep strategy use also arereciprocally linked with their academic achieve-ment. Future research should examine the natureof these reciprocal associations.

Conclusion

Numerous studies have been conducted to betterunderstand students’ motivation and learning strat-egies as promoting the academic development ofcompetence and knowledge. Researchers definedmotivation as the process whereby goal-directedactivity is instigated and sustained (Pintrich &Schunk, 2002). Learning strategies are described as

planned sets of coordinated study tactics that aredirected by a learning goal and aim to acquire anew skill or gain understanding (Alexander & Mur-phy, 1998). According to these views, motivationand learning strategies should, by their nature,facilitate long-term learning processes. The presentresearch documents that these variables are indeedimportant for students’ academic growth over theschool years.

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Supporting Information

Additional supporting information may be found inthe online version of this article at the publisher’swebsite:

Appendix S1. Additional Analyses.


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Predicting Long-Term Growth in Students Mathematics … · 2016. 4. 25. · dent help seeking (Butler, 1998), and self-handicapping (Urdan & Midgley, 2001). As such, extrinsic motivation