2009 university students’ achievement goals and approaches to learning in mathematics.pdf

8/12/2019 2009 University students achievement goals and approaches to learning in mathematics.pdf

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Copyright The British Psychological SocietyReproduction in any form (including the internet) is prohibited without prior permission from the Society

University students achievement goals andapproaches to learning in mathematics

Francisco Cano* and A. B. G. BerbenDepartment of Educational Psychology, University of Granada, Granada, Spain

Background. Achievement goals (AG) and students approaches to learning (SAL)

are two research perspectives on student motivation and learning in higher education

that have until now been pursued quite independently.

Aims. This study sets out: (a) to explore the relationship between the most

representative variables of SAL and AG; (b) to identify subgroups (clusters) of studentswith multiple AG; and (c) to examine the differences between these clusters with

respect to various SAL and AG characteristics.

Sample. The participants were 680 male and female 1st year university students

studying different subjects (e.g. mathematics, physics, economics) but all enrolled on

mathematics courses (e.g. algebra, calculus).

Methods. Participants completed a series of questionnaires that measured their

conceptions of mathematics, approaches to learning, course experience, personal

2 2 AG, and perceived AG.

Results. SAL and AG variables were moderately associated and related to both the

way students perceived their academic environment and the way they conceived of the

nature of mathematics (i.e. the perceptual-cognitive framework). Four clusters of

students with distinctive multiple AG were identified and when the differences between

clusters were analysed, we were able to attribute them to various constructs includingperceptual-cognitive framework, learning approaches, and academic performance.

Conclusion. This study reveals a consistent pattern of relationships between SAL

and AG perspectives across different methods of analysis, supports the relevance of the

2 2 AG framework in a mathematics learning context and suggests that AG and SAL

may be intertwined aspects of students experience of learning mathematics at

university.

Achievement goals (AG) and students approaches to learning (SAL) are two prominent

perspectives in contemporary research on student motivation and learning in highereducation. The former originated in the United States and refers to the reasons orpurposes that students adopt while engaged in academic work (Elliot & McGregor, 2001),

* Correspondence should be addressed to Dr Francisco Cano, Facultad de Psicologa, Granada, 18071, Spain(e-mail: [email protected]).

The

British

Psychological

Society

131

British Journal of Educational Psychology (2009), 79, 131153

q 2009 The British Psychological Society

www.bpsjournals.co.uk

DOI:10.1348/000709908X314928


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whereas the latter originated in Europe and Australia and refers to the way students goabout the task of learning: their intentions and learning/study processes (Biggs, 2001;

Entwistle & McCune, 2004).Although both AG and SAL seem to be associated with students motivation and

perceptions of the teachinglearning environment, there is a dearth of researchexamining the relationship between the two perspectives. This information is crucial toexpand researchers understanding of students learning experience in higher educationand to advance their work on the AG perspective. The purpose of this study, therefore, is

to explore the interplay between AG and SAL perspectives.

AG perspective

According to Pintrich (2003), a substantive question for researchers with a motivationalscience perspective is what motivates students. Of the various constructs proposed in

answer to this question, such as adaptive self-efficacy, control beliefs, and AG, the lasthas been one of the most active areas of motivation research in classroom contextsover the last 15 years (Pintrich, 2003, p. 676). AG are conceived of as cognitiverepresentations of what individuals are trying to do or what they want to achieve : : :

and represent the individuals orientation to the task or situation, their general focusor purpose for achievement Pintrich, Conley, and Kempler (2003, p. 321). Briefly, AG

refers to the reasons or purposes that individuals adopt while engaged in competence-

relevant settings (Elliot, McGregor, & Gable, 1999; Midgley, Kaplan, & Middleton, 2001).The type of goal endorsed leads to differential patterns of cognition, affect, and

behaviour (Ames, 1992, Dweck, 1999; Elliot & McGregor, 2001).The AG perspective is based on achievement motivation theory and social cognitive

theoretical models (Pintrich, 2003), but researchers differ on the nature of AG, as therecent review by Kaplan and Maehr (2007) reveals. They found different theoreticalmodels such as self-schemas, needs, and situation-schemas, but acknowledged that mostof the research would fit the last model, in which AG emerge from the characteristics ofa specific situation. Researchers also differ in their methodology: from experimentalmanipulations and questionnaires to qualitative investigations.

Research on AG has varied from a simplistic two-goal model to a trichotomous model

and recently to a multiple goals model (Pintrich, 2003).Early work in this area such as Dweck and Legget (1988) tended to use a simple

mastery (good) versus performance (bad) dichotomy, known as the normativeperspective, Pintrich (2000). A mastery goal, also labelled task-involved or learninggoal, orients learners to mastery of the task and increasing ones competence, whereas aperformance goal, also labelled ego-involved or ability goal, orients learners to the selfand to demonstrating ones competence (Ames, 1992). Elliot and his colleagues gave thefirst impetus to an initial revised goal theory perspective (Pintrich, 2000) when theyproposed a trichotomous achievement goal framework by dividing performance goalsinto approach and avoidance (Elliot & Church, 1997; Elliot & Harackiewicz, 1996).

Some theorists (e.g. Elliot, 1999; Elliot & McGregor, 2001) have revised and extendedthe trichotomous framework by accepting (a) four distinct AG and (b) different patternsof combination among them, this being known as the multiple goal perspective(Pintrich, 2003). The former supports a traditional, variable-centred analysis of students

adoption of single separate goals, whereas the latter allows for a person-centred analysisof patterns of multiple AG. These theorists considered that competence, the conceptualcore of the achievement goal construct, may be differentiated in two dimensions:

132 Francisco Cano and A. B. G. Berben


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definition and valence. The former refers to standards used in evaluating competence:absolute/intra-personal (mastery) and normative (performance), whereas the latter

refers to the ways in which competence can be valenced: positive (approachingsuccess) and negative (avoiding failure).

This 2 2 framework comprises four main AG: mastery-approach, in which theindividual seeks to increase his/her level of competence by gaining understandingand learning as much as possible; mastery-avoidance, in which he/she strives to avoidmisunderstanding or the failure to learn as much as possible; performance-approach,

in which the learners purpose is to demonstrate his/her competence and receivepublic recognition for ability and superiority; and performance-avoidance, in whichhe/she seeks to avoid looking incompetent, less able, or doing badly relative toothers. These AG are measured by the four subscales of the achievement goal

questionnaire. Students responses to them have shown high reliability and constructvalidity (Elliot & McGregor, 2001).

Besides describing personal goals, that is goals that are pursued, AG theorists havepaid attention to goal structures, that is goals that are perceived, which are defined asthe type of achievement goal emphasized by the prevailing instructional practices andpolicies within a classroom, school, or other learning environment (Wolters, 2004,p. 236). As with early work on personal goals, theorists have distinguished between amastery goal structure, with an emphasis on learning, understanding, effort andimprovement and a performance goal structure, with an emphasis on performing well,

high ability relative to others, grades and test scores (Ames, 1992; Ames & Archer, 1988;Maehr & Midgley, 1996). Goal structures are not generally characterized in terms of theapproachavoidance dimension (Linnenbrink, 2005, p. 199) as evidenced by Urdans

(2004) scales, one of the most widely used instruments for measuring classroom goalstructures, which include scales only for mastery and performance goal structures.

Antecedents and consequences of AG

Much of the research on the normative two-goal perspective has found that for most

cognitive (Ames, 1992; Dweck & Legget, 1988) and achievement outcomes (e.g. Church,Elliot, & Gable, 2001; Kaplan & Maeher, 1999), mastery goals have a beneficial effectwhile performance goals are detrimental. By contrast, research on the multiple goalperspective suggests that a combined emphasis on mastery- and performance-approachAG is more beneficial for cognitive (e.g. Harackiewicz, Barron, Tauer, Carter, & Elliot,2000; Midgleyet al., 2001; Pintrich, 2003) and achievement outcomes (e.g. Bouffard,Boisvert, Vezeau, & Larouche, 1995; Wentzel, 1991, 1993; see Linnenbrink, 2005 for areview).

The 2 2 AG framework was examined by Elliot and McGregor (2001), who looked

at several antecedents (e.g. need for achievement, self-determination, fear of failure, andperceived class engagement) and consequences (e.g. anticipatory test anxiety, deep vs.surface processing, exam performance). These authors found that the pattern forperformance-avoidance goals was more negative than for mastery-avoidance goalswhich,in-turn, were more negative than those for mastery-approach goals. Previous researchsuggests that mastery goals are intimately connected with intrinsic motivation, whereasperformance goals are considered to be an index of extrinsic motivation (Heyman &

Dweck, 1992). Further studies have shown, however, that while performance-avoidancegoals are negatively associated with intrinsic motivation (Elliot, 1999), performance-approach goals are positively associated (Elliot & Harackiewicz, 1996).

Achievement goals and approaches to learning 133


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Generally, students endorsement of multiple goals has been examined by looking atinteractions between AG variables in regression analyses or by using median splits or

cluster analysis (Midgleyet al., 2001). The latter was recently recommended by Pastor,Barron, Miller, and Davis (2007), as regression techniques do not allow researchers tocharacterize the common goal orientation profiles in their sample (p. 12) and mediansplit techniques are highly dependent on the procedure on the sample median beingused (p. 13).

In the 2 2 AG framework, two recent studies have examined the antecedents and

consequences of students patterns (i.e. profiles) of multiple AG. Wang, Biddle, andElliot (2007), using cluster analysis (Wards method), confirmed the four-factor structureof the 2 2 AG framework in the physical education context and identified four AGclusters: moderate AG; low AG; high AG; and mastery AG. While the low AG group was

the least motivationally adaptive, as the multivariate analysis of variance (MANOVA) andpost hoc tests showed, the group with high AG demonstrated the most positivepsychological characteristics (e.g. perception of competence) and outcomes (e.g.enjoyment). However, the authors suggest that more research using cluster analysis ofAG is needed to explore the effects of combinations of mastery and performance AG onbehaviour. Pastoret al.(2007), using latent profile analysis (LPA), a complex techniquederived from cluster analysis, found that six different profiles were needed and thatstudents assigned to these profiles showed statistically significant differences on somevariables, such as competitiveness and academic achievement. However, the authors

acknowledge some of the limitations of this technique and encourage researchers todelve into the consistency and validity of cluster solutions, suggesting they examinestudents goals for a specific course rather than their goals for a given semester (Pastor

et al., 2007, p. 39).The research provides substantial evidence that students AG are determined by their

perceptions of the classroom learning environment (goal structure) (see Wolters, 2004for a review). For example, the more students perceived their learning environment asmastery-structured, the more likely they were to endorse a mastery-goal orientation(e.g. Kaplan & Maehr, 1999; Roeser, Midgley, & Urdan, 1996). Although our understandingof the link between goal structures and achievement is only tenuous (Wolters, 2004), itwouldappear that this link is usuallynegative in the case of performance goal structure (e.g.Anderman& Anderman, 1999; Anderman & Midgley, 1997; Midgley & Urdan, 2001; Urdan,

Midgley, & Anderman, 1998) and often fails to emerge in the case of mastery goal structure(Anderman & Anderman, 1999; Urdanet al., 1998).

The relationship between personal AG and learning strategies (e.g. cognitive, deep,metacognitive), widely reviewed by Elliot et al. (1999) and Wolters (2004), is clearlypositive in the case of mastery goals (e.g. Archer, 1994; Pintrich, 2000; Valle et al., 2003;Wolters, 2004), but more ambiguous regarding performance goals (e.g. Archer, 1994;Pintrich & Garcia, 1991). When the strategy under examination is deep/surfaceprocessing, research shows that performance and performance-avoidance goals arepositively related to surface processing and unrelated to deep processing (e.g. Elliot

et al., 1999; Pintrich & Garcia, 1991).Research into the relationships between perceived goals and both learning strategies

and achievement has indicated (a) that a mastery goal structure generally predicts theuse of learning strategies, whereas a performance structure does not (Ames & Archer,

1988; Wolters, 2004) and (b) that a performance goal structure correlates negativelywith achievement (Midgley & Urdan, 2001), whereas the correlation between master ygoal structure and achievement fails to emerge (Urdan et al., 1998; Wolters, 2004).



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Perceptions of the teachinglearning environment (academic environmenthenceforth), motivation for engaging in learning and learning strategies are revealed

as major factors in another research perspective focused on understanding howstudents set about learning in higher education.

SAL perspective

This perspective, often used in Europe and Australia, is one of the two most significant

lines of research on student learning in higher education, along with self-regulatedlearning (SRL, Pintrich, 2004), most often used in North America. Pintrich (2004) hascriticized SAL researchers because they tend to include fewer psychological constructsand theories and most assess student learning at a more generic level than SRLresearchers, which makes these two perspectives incommensurable in some ways.Although our study does not seek to relate these different perspectives, it suggests somepossible answers to Pintrichs criticisms by describing and explaining the SALperspective and linking it to the AG perspective.

The SAL perspective is usually characterized as being based on bottom-up models oflearning generated from the perspective of the students learning experience and

grounded in different research procedures. Early studies in this area were carried out bytwo Swedish researchers, Marton and Saljo (1976a, 1976b), who used naturalisticexperiments, such as reading an academic article, and in-depth interviews. Students

generally tackled the academic task in one of two main ways, called deep and surfaceSAL, which related to their manner of interpreting the instructions and the learning task.This interpretation created an intention, that is motivation to learn, which led to adistinctive process, that is strategy of learning.

According to Watkins (2001, p. 167), at this point the SAL literature proceeded intwo contrasting but not incompatible directions. The first direction was generated

by these Swedish researchers and focused on a qualitative analysis of how studentsperceived the content and process of learning, which they called phenomenography(Marton, 1981). The second direction was developed by Biggs in Australia and

Entwistle in the United Kingdom and focused on quantitative methods such as self-report surveys and questionnaires (Entwistle & McCune, 2004) to assess whatstudents usually do while learning and studying. Research in this direction led to thedevelopment of two learning process inventories: the study process questionnaire(SPQ; Biggs, 1987) and the approaches to studying inventory (ASI; Entwistle &Ramsden, 1983).

There is some diversity in the views of the most representative SAL researchers abouthow many dimensions and subscales are really needed. While Entwistle and Ramsden(1983), Entwistle, McCune, and Walker (2001) and even Biggs in his 1987 research

established three SAL dimensions: deep; surface; and achieving, other authors such asRichardson (2000), reviewing the psychometric research on the SPQ and ASI, suggestedthat only the first two could be considered valid, which was implicitly accepted byBiggs, Kember, and Leung (2001).

Regarding subscales, while for Entwistle et al. (2001) SAL are defined by a compositeof several subscales, for example, deep approach is defined by subscales such as seekingmeaning, relating ideas, use of evidence; and interest in ideas, for other authors such

as Biggs (1987) and Biggs et al. (2001), each learning approach is composed of onlytwo subscales, one assessing intention (motivation) and another assessing strategy.SAL are not stable traits of individuals, but processes adopted during learning, which



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Biggs has always described as multidimensional because they comprise congruentmotive-strategy packages which are only meaningful in context (Biggs, 1987, 1993;

Biggset al., 2001).The surface approach involves an extrinsic motivation (reproducing the material

being studied, fulfilling course requirements and avoiding failure with the least personaleffort and involvement) and a repetitive strategy (memorizing specific facts andaccurately reproducing them). By contrast, the deep approach is characterized by anintrinsic motivation (trying to understand the authors intentions and seeking self-

fulfilment from the material) and the use of a more meaningful strategy (searching formeaning, integrating formal knowledge with personal experience and relating facts toconclusions) (Biggs, 2001; Entwistle et al., 2001).

SAL have been shown to depend on students conceptions of learning. Students who

conceive of learning as the extraction of meaning and transformation of information arelikely to adopt a deep learning approach while students who conceive of learning asincreasing and reproducing knowledge are likely to adopt a surface approach (VanRossum & Schenk, 1984). Further, SAL have been shown to be linked to studentsperceptions of the teachinglearning environment. Students who perceive a well-planned and well-resourced academic environment that includes clear goals andstandards, good quality teaching and appropriate assessment and workload tend todeploy a deep approach to learning, while students who perceive that the quality ofteaching is poor, assessment is focused on memorizing and workload is too high tend to

adopt a surface learning approach (e.g. Trigwell & Prosser, 1991a, 1991b). Theseperceptions, measured through the course experience questionnaire (Ramsden, 1991)in two recent investigations (Lizzio, Wilson, & Simons, 2002; Richardson, 2006), were

found to influence students learning outcomes both directly and mediated throughtheir SAL, with positive perceptions being positively related to deep approach, overallmarks, and general satisfaction. In a meta-analysis based on 55 independent sampleswith 27,078 respondents from 15 countries, Watkins (2001) found that academicachievement had average correlations of2 .11 and .16 with surface and deep learningapproaches, respectively.

Some researchers have focused on analysing students experience of the subjectmatter they are studying rather than their experience of learning in general. Forexample, Crawford, Gordon, Nicholas, and Prosser (1994, 1998a) developed a

questionnaire on how university students viewed the nature of mathematics. Thisassessed fragmented versus cohesive conceptions of mathematics (mathematics as afragmented body of knowledge versus mathematics as a way of thinking and ofunderstanding the world). They also modified the SPQ (Biggs, 1987) in order tospecifically assess SAL mathematics. In a later study, Crawford, Gordon, Nicholas, andProsser (1998b) extended the previous research by showing (a) that fragmentedconceptions of mathematics were associated with surface learning approaches tomathematics and perceptions of both workload and assessment as inappropriate (theformer too high, the latter focused on memorizing) and (b) that cohesive conceptions of

mathematics were associated with deep learning approaches to this discipline andperceptions of clear goals and good teaching. Further, by using cluster analysis, theauthors detected two subgroups of students whose average subscale scores weredifferent for all the variables analysed. Students allocated to the first subgroup

held fragmented conceptions of mathematics, unsatisfactory perceptions of theirlearning environment, surface learning approaches and a low level of achievement.Students allocated to the second subgroup reported cohesive conceptions



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of mathematics, satisfactory perceptions of their learning environment, deep learningapproaches, and a high level of achievement.

SAL and AG

Very little attention has been paid to the interface between SAL and AG. It could beargued that this was explored in two studies by Elliot et al. (1999) and Elliot andMcGregor (2001) but an analysis of the instruments they used suggests this was not the

case. Their results showed that mastery goals were significant predictors of self-reporteddeep processing, while performance-avoidance goals predicted surface processing.Elliot et al. (1999) also reported that some motivational dispositions were significantpredictors of AG: self-determination (a continuum between intrinsic and extrinsicmotivation) positively predicted mastery approach and negatively predicted masteryavoidance and performance avoidance, and fear of failure positively predicted masteryavoidance, performance approach, and performance avoidance. However, theirresearch does not appear to be in line with the SAL perspective, at least not withBiggs proposal, because when the items defining deep and surface processing scales areexamined, only learning strategies are found, not the congruent motive-strategy

package characteristic of learning approaches (Biggs, 2001).Since Pintrichs (2003) claim that it will probably be more useful for future

motivational science research to examine how different constructs from different

theoretical models relate to one another rather than attempting to discover newconstructs or create new theories (p. 677), three aspects in which SAL and AG seem tocoincide have been identified.

First, SAL and AG perspectives both include some forms of motivation: intrinsic(deep approach) versus fear of failure and extrinsic (surface approach) in the caseof SAL (e.g. Entwistle et al., 2001), and increasing ones competence (mastery-goal

orientation) versus demonstrating ones competence (performance goal orientation)in the case of AG (e.g. Elliot & McGregor, 2001). SAL forms of motivation werereferred to in a recent work as intrinsic and extrinsic goals (Pintrich, 2004, p. 388),

which together with meanings and strategies, are actively constructed by learnersduring the learning process. SAL and AG might be assumed to be related, giventhe links between (a) mastery goals, deep processing, and intrinsic motivation(Elliot et al., 1999; Heyman & Dweck, 1992) and (b) performance goals, surfaceprocessing, fear of failure, and extrinsic motivation (Elliot et al., 1999; Heyman &Dweck, 1992).

Although a reference to students goals is included in Pintrichs (2004) model of SRL,it was not considered appropriate to analyse this model in relation to the SALperspective for three reasons: (a) goal orientation adoption only appears in 1 of the 16

boxes (four phases by four areas of regulation) in which this model is organized; (b) thequestionnaire used to assess self-regulation was developed in 1991 and as Pintrich(2004) admits, does not represent an instrument designed to assess allcomponents ofthe current conceptual model (p. 392); and last but not least, (c) this instrument onlyassesses the simplistic early mastery and performance goals.

Second, SAL and AG seem to be grounded in the context or situation, specificallyin students perceptions of their academic environment: SAL are related to their

perceptions of their learning environment in general, for example good teaching,assessment practices (e.g. Crawfordet al., 1998b; Richardson, 2006) while personal AGare related particularly to perceived goals (goal structures) for example motives for



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effort, grading practices (e.g. Ames, 1992; Wolters, 2004). Accordingly, it could bespeculated that while the mastery structure generally correlates with positive

perceptions of the academic environment (and deep learning approach), theperformance structure correlates with negative perceptions of the academicenvironment (and surface learning approach).

Third, some studies following the SAL and AG research perspectives seemto focus on specific subject matter or educational context (e.g. mathematics,physical education) (e.g. Crawford et al., 1998b; Wang et al., 2007), rather than on

a given semester or general educational level. This focus on domain-specificitysomewhat refutes Pintrichs (2004) criticisms that learning is operationalised ata generic level in the SAL perspective, this being in line with his belief that thecourse level is a good compromise between an overly global level focused on

college learning in general and a more micro-analytic level focused on different taskswithin a course (Pintrich, 2004, p. 394). Taking advantage of the empirical dataon 1st-year mathematics students gathered by SAL researchers, an analysis of the AGof these students would help both to extend the 2 2 AG model to this subjectand to examine the relationships between SAL and AG variables in this specificcontext.

It could be argued that in these researches, SAL were assessed by means of Biggssquestionnaire, which has led to some concerns being raised about its psychometriccharacteristics (e.g. Richardson, 2000). Recent research has also found that the latest

version of this questionnaire, the R-SPQ-2F, which has fewer subscales and items, hasan underlying structure that is apparently non-hierarchical and that SAL might bedefined as a co-variation between a motive and its intended strategy (Justicia,

Pichardo, Cano, Berben, & De la Fuente, in press). These authors consider itreasonable to continue using this questionnaire, but computing only the main scalescores, that is, deep and surface SAL, rather than subscale scores for the motive andstrategy subscales. Therefore, it seems appropriate to use the R-SPQ-2F in the presentstudy for the sake of consistency with other research carried out in the mathematicsdomain and because its simple first-order two-factor structure allows for a quick andeasy administration and assessment of deep and surface SAL.

An exploration of the relationship between SAL and AG perspectives couldbe undertaken by two distinct methods, suggested by the 2 2 AG framework:

(a) a variable-centred method based on the multivariate correlations (canonical variates)between the set of SAL variables and the set of AG variables and (b) a person-centredmethod based on clusters of AG, which according to Wang et al.(2007) is valuable yetunder-utilized at present (p. 147). We agree with their suggestion but we would includea split-sample cross-validation procedure (Everitt, Landau, & Leese, 2001), which theyomit to do. Further, the characteristics of these clusters could be examined using theSAL and AG variables not used to define them (e.g. conceptions of and approaches tolearning, academic performance).

Therefore, on the basis of the literature review and bearing in mind the

aforementioned limitations of previous research, the present study was designed toaddress the following research questions:

(1) Do any relationships (canonical variates) exist between the set of SAL variables

and the set of AG?(2) What types of clusters of multiple AG can be detected in 1st-year mathematicsstudents?



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(3) and (4) What are the differences, in terms of the set of variables not used todetermine cluster membership, between students allocated to each of these

clusters and how can these differences be characterized?

Method

Participants

A total of 680 1st-year university students (382 females and 298 males) with an averageage of 19.3 years (SD 2:4) participated in this study. They were recruited from 24randomly selected mathematics classes (e.g. algebra, calculus), these courses beingoffered in diverse academic areas (e.g. mathematics, physics, economics) at a majorstate-funded university in Spain.

Measures

Five instruments were used and all the items on their subscales were rated on a 5-pointLikert-type scale, from 1 (never or rarely true of me) to 5 (always or almost always trueof me). These instruments were translated into Spanish, adapted to take cultural

differences into account, then independently translated back and further modifiedwhere necessary. Their subscales and corresponding Cronbachs alpha (Cronbach,1951) for this study were as follows.

Conception of mathematicsThe conception of mathematics questionnaire (Crawford et al., 1998a) was used toassess participants conceptions of mathematics. This questionnaire comprises 18 items

grouped into two subscales: fragmented conception (Cronbachs alpha :84) andcohesive conception (Cronbachs alpha :86).

Approaches to learningA modified version of the revised two-factor-SPQ (Biggs et al., 2001) assessed

SAL. This questionnaire was modified to specifically assess SAL mathematics. Themodified questionnaire is composed of 20 items grouped into four subscales: deep

motive (Cronbachs alpha :

74); deep strategy (Cronbachs alpha :

58); surfacemotive (Cronbachsalpha :67); and surface strategy (Cronbachs alpha :63). Eachmotive/strategy combination defines a distinct learning approach: deep approach (deepmotive and deep strategy) and surface approach (surface motive and surface strategy).

Course experience

Information about participants experiences of studying mathematics waselicited using theshort form of the course experience questionnaire (Ramsden, 1991). This questionnaire iscomprised of five subscales: good teaching (six items) (Cronbachs alpha :81);appropriate assessment (three items) (Cronbachs alpha :48); clear goals (four items)(Cronbachs alpha :53); appropriate workload (four items) (Cronbachsalpha :64);and generic skills (six items). The last subscale was not included due to time constraints in

administering the questionnaires and because is assessed competencies such as teamworkand written communication which could be considered general rather than specific to thelearning of mathematics.



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Achievement goalsThe achievement goal questionnaire (Elliot & McGregor, 2001) was used to assess

participants AG. Thisquestionnaireincludes 12 items grouped into four subscales:masteryapproach (Cronbachs alpha :72); mastery avoidance (Cronbachs alpha :69);performance approach (Cronbachs alpha :88); and performance avoidance(Cronbachs alpha :65). Although Elliot and McGregor (2001) used a 7-point scale forthe assessment of AG, a 5-point scale was used in the current study to maintainconsistencywith the scaling of the other items in the questionnaires.

Classroom goal structures

Urdans (2004) scales of classroom goal structures assessed participants perceptions ofthe mastery and performance classroom goal structures. Urdans scales comprise eightitems measuring a mastery goal structure (Cronbachs alpha :63) and seven itemsmeasuring a performance goal structure (Cronbachs alpha :83).

Procedure

Participants received information about the study and specific instructions about how tofill in the questionnaires, which they had to respond to in relation to mathematics,during lecture time. They were told that their participation in the study was voluntary,

were assured of the confidentiality of their responses and that their answers would notaffect their grades. In addition, they were asked to provide demographic information,such as name, age, and sex, and to give written consent to their examination marks inmathematics being accessed for the purposes of this study.

Results

Variable-centred analysis

Canonical correlations between the set of SAL variables and the set of AG variables are

presented in Table 1. The analysis detected six pairs of canonical variates of which thefirst four were statistically significant (x235 433:84;p , :001;x

224 29:61;p , :001;

x215 68:57; p , :001; and x28 17:23; p , :02). The four canonical correlations

were: .67; .52; .44; and .27, representing the 45, 27, 20, and 7% overlapping variance forthe first, second, third, and fourth pairs of canonical variates, respectively. With sixcanonical correlations included, x2

48 847:46; p , :001, and with the first fiveremoved, x23 1:02; p :79. Thus, the first four canonical correlations werestatistically significant but on the basis of variance accounted for (e.g. whether..10),only the first three accounted for a salient amount of variation and the subsequent

account should be restricted to these.The results of canonical correlation analysis showed that students scores on the

eight SAL scales and on the six AG scales were somewhat related, as indicated by:(a) canonical correlation, which measures how strongly the two sets of variates in eachpair relate to each other; (b) eigenvalues, which represent the correspondingoverlapping variance; (c) the total percentage of within-set variance; and (d) the totalredundancy (i.e. the amount of variance there is in the canonical variates from one

set extract to the other in both directions). However, Richardson recently(2007) demonstrated that the majority of measures of multivariate association aresystematically conservative, in that they ignore the multivariate properties of the data,



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and commended the complement of the statistic known as Wilks L (lambda), (1 2 L)(Wilks, 1932, 1935), as the only appropriate measure of strength of association. In our

study, the data yielded a significant value of Wilks L of .283, F48; 3;281:06 19:98,p , :001. Hence, 1 2L :717, which implies that the two sets of variables shared71.7% of their variance.

With a cut-off correlation of .30, those variables in the SAL set that correlatedclosely with their first canonical variate were good teaching (.68), deep approach(2.71), negative surface approach (2.65), clear goals (.42), cohesive conceptions ofmathematics (.37) appropriate workload (.36) and appropriate assessment (.37), andthe variables in the AG set that correlated closely with their first canonical variate wereclass mastery (.84), mastery approach (.63), and negative performance avoidance (2.43).

Taken together, these variables suggest that those who perceive their academicenvironment as mastery-structured and high-quality-orientated tend to deploy a mastery-approach goal and a deep learning approach (accompanied by an avoidance of surfaceapproach).

Thesecond canonical variate in the SAL set was composed of deep approach, negativegood teaching, appropriate workload, and cohesive conception of mathematics

while the corresponding canonical variate from the AG set was composed ofclass performance, mastery approach, mastery avoidance, and performance approach.Taken together, these variables indicate that an academic environment perceived

Table 1. Results of canonical correlation analysis between approaches to learning and achievement

goals variables

Canonical variates

First Second Third

Canonical correlations (rc) .67 .52 .44

Eigenvalue (r2c) .45 .27 .20

Structure coefficientsStudent approaches set

Fragmented 2 .15 .10 .39

Cohesive .37 .30 .00

Clear goals .42 2 .17 .13

Good teaching .68 2 .47 .52

A. workload .36 2 .62 2 .24

A. assessment .37 2 .17 2 .12

Deep approach .71 .60 2 .02

Surface approach 2 .65 2 .04 .64

Percent of variance .25 .14 .11

Redundancy .11 .03 .02

Achievement goals set

Class mastery .84 2 .23 .46

Class performance 2 .04 .65 .42Mastery approach .63 .62 2 .21

Mastery avoidance .19 .69 .09

Performance approach .04 .33 .13

Performance avoidance 2 .43 .25 .63

Percent of variance .22 .25 .14

Redundancy .10 .06 .02



12/23


as unsatisfactory in quality (performance structured, inappropriate assessment, andnegative teaching) may be associated with a cohesive conception of mathematics, a deep

learning approach and a combination of mastery-goals (approach and avoidance) andperformance-approach goals.

The third pair of canonical variates has high loadings (correlations) on surfacelearning approach, good teaching, and fragmented conception of mathematics in theSAL set, and class mastery, class performance, and performance avoidance in the AG set.In other words, the perception of an academic environment where although teaching is

good, classroom goal structure is both mastery- and performance-oriented and studentshold a fragmented conception of mathematics is linked to performance-avoidance goalsand surface learning approaches.

Person-centred analysis

To identify AG patterns, a split-sample cross-validation clustering procedure (Everittet al., 2001) was used. The total sample was randomly split into two halves (Sample Aand Sample B) and hierarchical cluster analysis (Wards method of linkage and squaredEuclidean distance), followed by non-hierarchical cluster analysis (K-means procedure

and squared Euclidean distance) was applied to each half. Prior to these analyses, all thevariables were standardized usingzscores (mean of 0 and standard deviation of 1). TheWard method was preferred because it optimizes the minimum variance within clusters,

avoids the tendency to form long, elongated clusters, found in some methods as singlelinkage, and outperforms most other clustering methods in conditions of cluster overlap(Aldenderfer & Blashfield, 1984). In addition, the squared Euclidean distance waschosen because these authors recommend using it with Wards method (Aldenderfer &Blashfield, 1984). The best number of clusters to use was determined by following twoprocedures (see Wishart, 2005, for a review) implemented in the ClustanGraphics

program (Wishart, 2006): tree/validate and Mojenas upper tail rule (Mojena & Wishart,1980). The former suggested that a partition of four clusters was optimal for bothsamples and the latter confirmed that this solution was statistically significant for Sample

A (t 112:71, realizeddeviate 6:12, df 338, p , :05) as well as for Sample B(t 104:36, realized deviate 5:67, df 338, p , :05). This solution was replicatedby theK-means procedure in which the cluster centres (i.e. means) resulting from thehierarchical analysis were used as seed points. The next step was to reclassify Sample B(K-means procedure) according to the cluster centres obtained for Sample A (K-meansprocedure). The agreement between this clustering (case location) and the initial

K-means clustering of Sample B, measured by means of Cohens k (Cohen, 1960), was.80. Similarly, Sample A was reclassified according to the cluster centres obtained fromSample B and the agreement coefficient between this clustering and the initial clustering

of Sample A was .76.Finally, the samples were combined and subjected to a non-hierarchical K-means

clustering procedure (focal point; Wishart, 2006) by grouping the cases into fourclusters. The means, standard deviations, andzscores describing the four clusters arepresented in Table 2, which also includes the univariate tests that followed a MANOVAon the four AG scores that defined the clusters. The multivariate cluster effect wasstatistically significant (WilksL :10, F12; 1;780 197:86, p , :001, effect size

measured by means of tau squared t2

:

52).Clusters were interpreted as being higher or lower on the four AG, using a criterion zscore of^.5 (Wang & Biddle, 2001). Accordingly, Cluster 1 (labelled low AG, specifically



13/23


on mastery goals) was characterized by low mastery approach and mastery avoidancescores, and moderately low performance approach scores; Cluster 2 (labelled low AG, butmoderately high mastery approach) consisted of low performance-avoidance goals andmoderately high mastery approach; Cluster 3 (labelled high AG, but low performanceapproach) contained a combination of moderately high mastery approach, high mastery

avoidance, and low performance approach and Cluster 4 (labelled high AG, specifically onperformance approach) was characterized by high scores in practically all AG.

Relationship between AG clusters and outcome variables

This was examined by means of a one-factor multivariate analysis of variance (MANOVA)followed by a descriptive discriminant analysis (DDA). MANOVA/DDA involves theanalysis and description of the effects of a grouping variable (predictor) on a set ofoutcome variables (criteria) (Huberty & Olejnick, 2006). The assessment of the effectsof cluster membership (predictor) on SAL, AG, and academic performance (criteria)involved three main steps.

(1) Thefirst stepwas to examine all cluster differences simultaneously by carrying out aMANOVA omnibus test. This test showed a statistically significant effect for the predictor,first when only AG variables were used as criteria (Wilks L :89, F6; 1;350 12:50,

p , :001, t2 :05) and then after the addition of SAL and academic performance variables(WilksL :66,F33; 1;962:86 8:91,p , :001,t

2 :12). This set of 11 criteria variableswas used in subsequent analyses. Descriptive information for the fourclustersis reportedinTable 3, including F-to-remove values. The latter indicate that the two variables that

contribute most to overall cluster differences are Deep and Surface SAL, followed byperformance goal structure, fragmented conception of mathematics and academicperformance. Clear goals and good teaching contributed the least to group differences.

Table 2. Cluster analysis of scores in the four achievement goals

Clusters

Achievement goals

Cluster 1

(N 185)

Cluster 2

(N 139)

Cluster 3

(N 191)

Cluster 4

(N 165) F

Mastery approach

M 2.82 3.96 4.02 4.02 207.57*

SD .56 .52 .51 .60z 21.13 .36 .45 .44

Mastery avoidance

M 2.49 3.08 3.78 3.76 175.68*

SD .55 .77 .54 .66

z .95 2 .24 .60 .57

Performance approach

M 1.92 1.97 1.60 3.40 276.15*

SD .73 .74 .50 .65

z .31 2 .25 2 .63 1.28

Performance avoidance

M 3.43 2.25 3.65 3.77 193.16*

SD .72 .55 .50 .62

z .12 21.31 .38 .53

Note. df 3; 676;*p , :001.



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Table

3.

Resu

ltso

fMANOVAan

ddescr

iptive

discr

iminantana

lysis

(DDA)

Descr

iptivein

formationan

dstructurers

forthe

four-c

lusters

Pa

irw

isecompar

isons:F-t

o-removean

dstructurers

Outcomevar

iables

C1

C2

C

3

C4

Fu

a

rs

(1)

rs(2

)

rs(

3)

Fu

b

C2

C4

rs

Fu

c

C3

C4

rs

Classro

om

goa

ls

Masterygoa

l

2.8

1

3.1

8

2.9

9

3.0

0

2.1

9

.44*

2.0

9

.00

1.9

6

.29

.04

2

.05

Performancegoa

l

2.7

3

2.6

9

2.9

2

3.1

3

5.3

1

.07

.6

2*

.00

13

.57

2

.51

6.1

3

2

.42

Conceptiono

fmathematics

Fragmented

3.2

8

2.9

5

3.3

2

3.2

4

4.9

1

2

.27

.3

5

2

.46*

7.5

3

2

.40

.68

.17

Cohes

ive

3.3

5

3.7

1

3.6

1

3.7

4

2.4

9

.41*

.2

7

.03

1.1

0

2

.04

1.5

5

2

.32

Course

exper

ience

Clea

rgoa

ls

2.9

9

3.1

3

3.0

4

3.1

2

.61

.15

.0

6

.17*

1.6

8

.00

.13

2

.20

Goo

dteac

hing

2.5

3

2.8

0

2.5

6

2.6

2

.47

.20

2.1

4

.23*

.26

.21

.00

2

.12

A.w

or

kloa

d

2.7

1

2.9

1

2.5

4

2.6

2

2.9

1

.10

2.3

7

.56*

.20

.33

4.1

9

2

.18

A.a

ssessment

3.4

4

3.7

7

3.6

1

3.5

0

1.7

6

.25*

2.2

1

2

.24

.61

.32

2.4

4

.23

Learnin

gapproac

hes

Deepapproac

h

2.3

0

2.9

7

2.7

3

2.8

9

15

.36

.83*

.2

9

.03

.15

.13

3.7

3

2

.45

Surfaceapproac

h

2.8

3

2.1

1

2.5

3

2.6

2

12

.89

2

.69*

.4

7

.16

30

.54

.13

5.4

1

2

.22

Aca

dem

icPer

formance

5.5

5

6.2

4

5.8

0

6.4

4

4.7

1

.25

.2

0

.41*

4.2

7

2

.09

9.0

7

2

.50

Can.C

orre

lation

(rc

)

.48

.3

3

.17

Eigenva

lue

(r2 c)

.30

.1

2

.03

Note

.C

1

C4

meansonoutcomevar

iables

for

Cluster

1,

Cluster

2,

Cluster

3,

an

dCluster

4.

rs

corre

lations

betweeneac

honeo

ftheresponsevar

iables

scores

an

dtherespective

linear

discr

iminantfunctions

(LDFs).

*Largesta

bso

lutecorre

lation

betweeneac

houtc

omevar

iablean

danyo

fthe

discr

iminant

functions.

Fu,

Un

ivar

iateF

-to-remove.

adf

1

3,

df

2

676;

b,cdf

1

1,

df

2

666

.



15/23


(2) The second step was to describe the nature of the MANOVA omnibus effect byexamining the linear composites of the criteria variables, called linear discriminant

functions (LDFs), which maximize the separation of all clusters. This multivariateanalysis is preferable to univariate analyses because it takes into account thecorrelational structure of SAL and AG variables detected earlier. The dimensionality testindicated three statistically significant LDFs to describe cluster differences (LDF1:L :66,x233 276:30;p , :001,F33; 1;962:86 8:91,p , :001; LDF2:L :86,x

220

100:04; p , :001, F18; 1;334:00 5:15, p , :001; LDF3: L :96, x29 21:57; p , :01,

F9; 668 2:42,p , :02). While the first two LDFs reflect high percentages of variance,65.71 and 27.14, the third reflects only 7.15%.

A plot of the corresponding cluster centroids is reported in Figures 1 and 2.

From this plot, it appears that: (a) the separation among the four clusters may beattributed to LDF1(Cluster 1 is separated from Clusters 3 and 4 but even more clearlyfrom Cluster 2, which is at the greatest distance from it) and (b) the separation ofClusters 1 and 2 from Clusters 3 and 4 may be attributed to LDF2. Third, the separationof Cluster 3 from Clusters 4, 1 and 2 may be attributed to LDF3.

To determine what constructs (i.e. latent variables) these LDFs represent, it isnecessary to examine the correlations between the 11 criteria variables andtheir three respective LDF scores. These are the structure rs, which are reportedin the central columns of Table 3. According to Comrey (1973), correlations of.32 (10% overlapping variance) are considered poor, .45 (20% overlapping variance)

fair, .55 (30% overlapping variance) good, .63 (40% overlapping variance) verygood, and .71 (50% overlapping variance) excellent.

The first construct could be labelled constructive learning process and suitableperceptual-cognitive framework because it is primarily defined by a deep way ofgoing about learning accompanied by a cohesive conception of mathematics and a

Figure 1. LDF plot of cluster centroids (LDF1vs. LDF2).



16/23


mastery-structured perception of the learning environment. The second constructcould be labelled performance-structured perception of the learning environmentbecause this attribute shows the highest absolute correlation. Finally, the third constructcould be labelled appropriate in terms of workload, conception of mathematics and

academic performance because the first, the second (negative fragmented conceptionof mathematics) and the third variables show the highest structure rs.

(3) The third step consists of examining the separation between two pairs of clusters

of particular interest in this study: Clusters 2 and 4 and Clusters 3 and 4, for two mainreasons partly related to students academic performance. First, because students fromClusters 2 and 4, who are clearly separated by their different perceptions of theirlearning environment (LDF2), seemed to show similar mean scores on academicperformance (6.24 and 6.44, respectively) and deep approach (2.97 and 2.89,respectively). Second, because the variable academic performance seemed to be asignificant indicator of LDF3, which separated Cluster 3 from the other clusters andespecially from Cluster 4. With each of these two contrasts there is only one LDF toconsider. Their corresponding cluster-difference structures and univariate F-to-remove

values are given on the right-hand side of Table 3.The construct underlying the difference between Clusters 2 and 4 is a reproductive

learning process and unsuitable perceptual-cognitive framework because it consistsprimarily of a surface approach to learning accompanied by a fragmented conception ofmathematics and a performance-structured perception of the learning environment.These unsuitable characteristics appear characteristic of students allocated to Cluster 4,whereas it seems they do not emerge in students allocated to Cluster 2.

The construct underlying the difference between Clusters 3 and 4 is a mixedpathway to learning. This construct, which appears to be defined by a mix of indicators,

some of them suitable (high academic performance and deep learning approach) andsome unsuitable (performance goal structure), seem to characterize students allocatedto Cluster 4 but not those allocated to Cluster 3.

Figure 2.LDF plot of cluster centroids (LDF2 vs. LDF3).



17/23


Discussion

The findings of the present study taken as a whole seem to indicate a generallyconsistent pattern of relationships between SAL and AG perspectives across differentmethods of analysis, support the relevance of the 2 2 AG framework in a mathematicslearning context and shed light on students experience of learning this discipline atuniversity.

One main result of this study was that a moderate association between SAL andAG perspectives emerged when the variable-centred method was used. This yielded

three main pairs of canonical variates that suggest a wide variation in studentslearning experience. Although the first and second pairs of canonical variates wereboth characterized by an emphasis on deep approaches to learning and a cohesiveconception of mathematics, students learning experiences seemed to differ in each.While in the first, the adoption of mastery goals appeared to be associated with apositive perception of the academic environment, in the second, the simultaneousadoption of mastery-approach and mastery-avoidance goals appeared to be

associated with an unsatisfactory perception of the academic environment. In thethird pair of canonical variates, a different way of learning emerged: surfaceapproach, which seemed to be associated with three characteristics: a fragmentedconception of mathematics, perception of the academic environment as oriented inopposite directions (both performance-structured and mastery goal structured; bothpositive and negative quality of teaching), and the adoption of performance-avoidance goals.

Taken together, these pairs of canonical variates seem to suggest (a) that AG and SALare linked to both the way students conceive of the nature of mathematics and the way

they perceive their academic environment (i.e. perceptual-cognitive framework);(b) that variations in both AG and this perceptual-cognitive framework are related toa differential pattern of processes (e.g. learning approaches); (c) that the two aspectsof the academic environment (i.e. perceived teaching quality and perceived goals)might be related; and (d) that different aspects of the SAL perspective are interrelated, asare different aspects of the AG perspective.

These findings advance the research on student motivation and learning in higher

education by providing an insight into the relationship between AG and SAL, which

appear to be intertwined aspects of students experience of learning mathematics.

Further, they confirm previous results regarding the relationships among differentaspects of the SAL perspective (perceptions of the academic environment, conception

of mathematics, and learning approaches) (e.g. Crawford et al., 1998a, 1998b) and of

the AG perspective (AG and goal structures) (e.g. Ames & Archer, 1988; Wolters, 2004).

With regard to the latter, these results also extend previous studies by providing

evidence on the applicability of the 2 2 AG perspective to a mathematics learning

context and by showing that some combinations of AG appear to be consistent with the

multiple goal perspective (Midgleyet al., 2001; Pintrich, 2003).

The second main result of the present study is that, using a person-centred method,four different types of AG clusters are identified among mathematics students. They canbe summarized as follows: low AG, specifically on mastery goals (Cluster 1); low AG butmoderately high mastery approach (Cluster 2); high AG but low performance approach

(Cluster 3); and high AG, specifically on performance approach (Cluster 4). Thesefindings are generally consistent with prior research in sport and exercise settings (Elliot& Conroy, 2005; Wang et al., 2007), seem to confirm that students can combine



18/23


different AG according to the multiple goal perspective (Pintrich, 2003) and extendprevious research to the academic context of mathematics learning.

Having identified these clusters, we were able to use them as predictors and examinetheir relationship with some of the criteria variables. A statistically significant maineffect for cluster membership emerged, first when only AG variables were used and thenafter the addition of SAL and academic performance variables, making a total of11 variables. Two of these criteria variables, deep and surface SAL, when consideredsimultaneously, contributed most to all the cluster differences. The highest percentages

of variance in these differences may be attributed to two constructs: constructivelearning process and suitable perceptual-cognitive framework and performance-structured perception of the learning environment. We also found that differencesbetween Clusters 2 and 4 may be attributed to a construct labelled reproductive

learning process and unsuitable perceptual-cognitive framework, which appearsto be the exact opposite of the construct which separated Cluster 1 from the rest.Further, the resulting differences between Clusters 3 and 4 students may be attributedto the construct mixed pathway to learning. All these results provide sufficientevidence to validate the cluster solution and constitute the third main result of thepresent study.

The results were consistent with previous research in that students in someclusters were more adaptive than in others (Wang et al., 2007). Thus, the first dimensionto reflect cluster differences seemed to show that students in Clusters 2, 3, and 4, but

especially in Cluster 2, appear to be the most adaptive because, compared with studentsfrom Cluster 1, they approached their learning in a deep way and held a matureconception of mathematics (SAL variables), and perceived their learning environment

as mastery-structured (AG variable). The second dimension to reflect cluster differencesseemed to indicate that students from Clusters 1 and 2 were more adaptive thanthose from Clusters 3 and 4 because the former appeared to perceive their learningenvironment as moremastery-structuredthan the latter. However, students fromCluster 4showed some suitable characteristics compared to students from Cluster 3. The pairwisecontrast detected that the former seemed to follow a mixed but apparently more adaptivepathway to learning which resulted in a better academic performance than the latter.Moreover, it was interesting to note that students allocated to Cluster 4 appeared to showsimilar mean scores on academic performance and deep approach to those of students

from Cluster 2.These findings confirm previous results regarding the relationship between some

aspects of the SAL perspective (conception of mathematics and learning approaches)(e.g. Crawfordet al., 1998b; Entwistleet al., 2001) and the AG perspective (AG and goalstructures) (e.g. Ames & Archer, 1988; Wolters, 2004). Furthermore, they are somewhatin line with earlier work linking goal structures and grades (Kaplan & Maehr; 1999;Roeseret al., 1996; Wolters, 2004). However, they conflict with other research that hasfailed to find any relation between mastery goal structure and grades (Anderman &Anderman, 1999; Urdan et al., 1998) and that has found a negative relation between

performance goal structure and grades (e.g. Anderman & Anderman, 1999; Anderman &Midgley, 1997; Midgley & Urdan, 2001; Urdan et al., 1998). It is difficult to find a precisereason for these results because as Wolters (2004) recognized, these relations are lessthan straightforward (p. 238). The fact that the final mark for a specific mathematics

course was used instead of overall GPA might be one of the factors implicated, butadditional research is need to address this issue empirically, including the way in whichachievement is assessed (Linnennbrink, 2005; Wolters, 2004).



19/23


These findings on multiple goals seem to confirm our previous innovative results thatlink SAL and AG perspectives when analysing single separate goals. SAL and AG

characteristics appear to vary within each cluster and to emerge as distinctive aspects ofthe different ways of experiencing mathematics learning defined by each of the clusters.Thus, once again, SAL and AG are revealed as apparently intertwined aspects ofstudents experience of learning mathematics at university.

It is worth examining the finding that students exhibiting high AG, particularlyon performance approach (Cluster 4), evidenced some positive characteristics

(e.g. academic performance, deep learning approach) as did students exhibiting lowAG but moderately high mastery approach (Cluster 2). At first sight, this finding mightlead to the conclusion that the adoption of both mastery and performance AGrepresents the ideal motivational profile, at least in terms of achievement, as did some

authors (e.g. Bouffard et al., 1995; Wentzel, 1993). However, this study serves tochallenge such a conclusion, if we bear in mind that there are substantial differencesbetween Clusters 2 and 4 that may be attributed to the reproductive learning processand unsuitable perceptual-cognitive framework characteristic of students who adoptedboth mastery and performance AG (Cluster 4). These students perceived the academicenvironment as performance-structured, held a more fragmented conception ofmathematics and adopted the highest levels of surface learning approach blended withmoderate amounts of deep learning approach. Thus, students in Clusters 2 and 4appeared to be following different routes to reach a similar level of academic

performance, which is in line with the principle of equifinality (Ford, 1992). However,as the analysis of SAL and AG characteristics suggests, these routes seem to be associatedwith different costs in term of learning experiences.

The findings of this study are subject to some limitations in that firstly, it used self-report instruments to measure most of the variables and the Cronbach a for somesubscales were low; secondly, the only measure of learning outcomes collected was thefinal mark for the mathematics course; and thirdly, participants came from a ratherheterogeneous population, due to the great diversity of academic areas in whichmathematics courses are taught. The magnitude of the reliability coefficient for somesubscales is below the level required for academic decisions about an individual student,although in line with those reported in the literature (see Watkins, 2001 for a review onSAL subscales) and acceptable for a research instrument used for group comparisons

(Watkins, 2001, p. 171). The other limitations could be overcome in future studies byusing (a) observations and interviews, (b) different measures of success in college,including quantitative and qualitative learning outcomes, and (c) a more homogeneouspopulation (e.g. mathematics degree students).

Notwithstanding these limitations and bearing in mind the need for caution, ourresults suggest three tentative inferences, the first two being of interest to mathematicsteachers and teacher trainers and the third to researchers.

Firstly, it might be worthwhile for mathematics teachers to pay attention not onlyto the mathematical content itself but to how their students are experiencing

mathematics learning. Informed reflection on this issue might lead them to re-examinesome aspects of their work. Our results are correlational, hence no causality linkscould be inferred. However, they seem to suggest that the approaches adopted bystudents to learn mathematics are apparently related not only to their ways of

conceiving this subject matter and to their personal goals but also to their perceptionsof teaching quality in general and to the type of goal emphasized by instructionalpractices in particular.



20/23


Secondly, an informed view about mathematics teaching and learning would helpteachers become aware that despite the appearance of homogeneity among students in

a particular class, there are distinct subgroups of students defined by their endorsementof different multiple personal goals. These goals seem to be predicted by studentsconceptions of mathematics and by their perceptions of the goal structures.

Analysis of these subgroups of students reveals two important aspects to take intoaccount: (a) their members show different pathways to learning and substantialvariations in their experience of learning mathematics and (b) though some of these

pathways lead to similar levels of academic performance, the most appropriate in termsof constructive experience of learning mathematics is apparently that of students whoadopt low AG but a moderately high mastery approach. The pathway adopted by thesestudents seems to be related to teaching practices that foster a mastery-structured

perception of the learning environment such as an emphasis on learningand understanding, on trying hard and on valuing all students (Linnenbrink, 2005;Wolters, 2004).

Thirdly, the results, taken in their totality, emphasize the need to shift attention awayfrom considering AG and SAL as independent constructs towards a more systematicview of students experience of learning mathematics as a whole. At a broad level, thisexperience seems to be characterized by three main aspects: (a) perceptual-cognitiveframework (i.e. antecedents), defined by conceptions about the subject matter andperceptions of the academic environment (quality of teaching in general, with a

particular emphasis on goal structure); (b) processes adopted in achievement settings,consisting of personal goals and related learning approaches (AG and SAL); and(c) learning outcomes (i.e. consequences). The relationships among these aspects seem

to suggest that students processes could mediate the links between antecedents andconsequences, a topic that would merit investigation in future studies.

Acknowledgements

The authors acknowledge the helpful comments of both the Editor and two anonymous referees

on a draft of this article, and the support and assistance of Barbara Lamplugh in drafting the

manuscript in English.

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