Paper presented at the Annual Conference of Australian Association for Research in Education

Adelaide

November/ December 1998

Self-schema is a term coined by Markus Hazel (1977), which is defined as a cognitive generalisation of one's self-knowledge in a specific domain from the past experiences. This type of self-knowledge is dynamic and situational. Self-schema serves as an organizer that mediates and regulates behaviours. In addition, self-schema also provides incentives, standards, plans, rules and scripts for behaviours (Alexander, 1997; Cross & Markus, 1994; Oyserman & Markus, 1993). It has been found that self-schema have bearing on information processing about the self (Markus, 1977), forming perceptions about others (Lewicki, 1983; Markus & Smith, 1981) and drawing inferences from ambiguous social information (Catrambone & Markus, 1987). As such, self-schemas will be relevant for motivation research. This paper reported two studies that look into how students' domain specific self-schemas in learning mathematics affect their learning engagement patterns. Learning engagement patterns are defined in terms of what learning goals and learning approaches students are having in learning mathematics.

Diagram 1 shows the theoretical model proposed in this paper. Self-schema is taken as an independent variable and its effects on performance is mediated through goal orientations and learning approaches. Students' self-schema in a subject domain and perceived learning purposes will explain why students learn a specific subject, in this case, mathematics. Whereas, learning approaches will explain how students learn it.

The link between self-schema and goal orientations can be justified empirically. Ng (1997) explored the relationship among self-schema, perceived teacher's teaching goals, perceived relationship with teacher, and goal orientations. It was found that self-schema outweighed the other two social variables in predicting students' goal orientations. In particular, self-schema was the most important predictor for mastery goal. Similar results were found in a subsequent study (Ng, 1998). These empirical findings substantiate the postulation that self-schema is causally tied to students' goal orientations.

Little research to the author's knowledge has been done to establish the link among goal orientations, learning approaches and performance. However, the relationships among these variables can be derived from the empirical studies in the respective fields.

The research on goal orientations has successfully contrasted the effects of these two goals, mastery and performance goal, on students' learning strategies. Students' learning strategies have been represented through the use of cognitive strategies (e.g. Meece, Blumenfeld & Hoyle, 1988; Nolen, 1988; Pintrich, 1989; Pintrich & Garcia, 1991;), the use of self-regulatory strategies (e.g. Pintrich, 1989; Meece, 1991; Pintrich & Garcia, 1991), the manipulation of learning resources (e.g. Pintrich & Garcia, 1991; Pintrich & De Groot, 1990), and different types of cognitive engagement (e.g. Greene & Miler, 1996; Meece et al., 1988).

It has been found that mastery-oriented students employ more frequently elaboration and organisation strategies that enable them to have a deeper understanding of the learning materials. Performance-oriented students rely more frequently on rehearsal strategies and as a result are confined to a surface-level of understanding (Pintrich 1989; Pintrich & DeGroot, 1990; Pintrich & Garcia, 1991). Similarly, Nolen (1988) found that mastery oriented students reported more use of deep-processing strategies including discriminating important from unimportant information, integrating new information to the existing knowledge and monitoring comprehension. A performance orientation correlated with the use of surface-level strategies like repeated reading, memorising new words and rehearsing information. In addition, mastery-oriented students always valued the use of the deep processing strategies.

Likewise, a similar pattern prevails when students' goal orientations are associated with self-regulatory strategies and the management of learning resources. Mastery-oriented students are usually more planful. They will set realistic goals and always monitor their process of learning. In addition, they also show greater concern for their management of study time, study environment as well as seeking help appropriately (Pintrich, 1989; Pintrich & DeGroot, 1990; Pintrich & Garcia, 1991).

Given the benefits of cognitive and self-regulatory strategies, it is not surprising to find that mastery oriented students reveal a deeper processing of information and hence a better performance than students stressing performance goals (Graham & Golan, 1991; Nolen, 1988; Pintrich, 1989; Pintrich & DeGroot, 1990; Pintrich & Garcia, 1991). Mastery goals are therefore described as adaptive and performance goals are labelled as maladaptive to learning. However, recent studies (Harackiewicz, Barron, & Elliot, 1989) have questioned the maladaptive nature of performance goals.

It is important to point out that goal orientations are not directly related to performance. It is the use of various forms of learning strategies that predicts performance level. In other words, the effects of goal orientations on performance are actually mediated through the appropriate employment of learning strategies (Wentzel, 1989; Pintrich & Garcia, 1991). Motivation per se will not guarantee an improvement of performance. Learning strategies are necessary.

Biggs (1987) factor analysed a list of variables by which he derived three higher order factors called approaches to learning. Each factor is a congruent motive-strategy mix. These three high order factors are deep, surface and achieving approach. Studies in these approaches found that they will lead to different levels of performance as well as different kinds performance (Biggs, 1987, 1993; Entwistle & Kozeki, 1985).

Deep approach and surface approach are orthogonal to each other. Deep approach is characterised by an interest in the task and the employment of deep strategies, like relating new ideas to prior knowledge, that maximise understanding. In contrast, surface approach is a work minimising orientation by which students learn with reproductive strategies, like rote learning, that yield a superficial understanding of the learning materials. As a result, deep approach links with a deep understanding and better results. The reverse is true for surface approach. The last factor, achieving approach, is characterised by a focus on performance. Students will utilise regulatory strategies like organising time, working space, planning ahead in order to secure a high performance. Achieving approach is therefore associated with high levels of performance. (Biggs, 1979, 1987, 1988, 1993; Kember & Gow, 1989; Liu, 1997) These three approaches and their relationship with performance have been found among high school students as well as college students (Biggs, 1989; Kember & Gow, 1989; Ramsden, Martin, & Bowden, 1989; Trigwell & Prosser, 1991). The findings in learning approaches have also been validated in cross-cultural settings (Entwistle & Kozeki, 1985; Kember & Gow, 1990).

Drawing together the findings from the research on goal orientations, learning approaches and performance, we can make the following assumptions:

a/ mastery goal would be positively related to deep approach and achieving approach and negatively related to surface approach; the same association with learning approaches would be held for functional goal and social solidarity goal.

b/ performance goal would be positively related to achieving approach. Given the maladaptive nature of performance goal, they would be negatively related to deep approach but positively related to surface approach

c/ deep and achieving approach would lead to a high performance while surface approach would result in a low performance.

The aim of the survey study was to establish the relationship between students' domain specific self-schemas and their learning engagement patterns, expoused in terms of goal orientations, learning approaches and anticipated achievement levels. A group of Chinese students in Hong Kong was asked to complete a questionnaire that probed their self-schemas, goal orientations, learning approaches and anticipated performance in learning mathematics. The study was administered in late 1997 by the author in collaboration with teachers in several secondary schools in Hong Kong.

The participants were nine secondary 4 classes (equivalent to year 10 in Australia) from five secondary schools in Hong Kong. In total, 329 valid cases were gathered. These students constituted a mixed-achievement sample. They came from schools of varying achievement bands. Secondary schools in Hong Kong are classified according to students' collective performance into five different achievement bands. Band 1 schools are mainly made up of high achievers while band 5 schools are populated mainly by low achieving students. The participants in this study came from two band 2, one band 3, one band 4 and one band 5 secondary school. The age of the students ranged between 14 and 18 with a mean of 15.36.

Self-schema in this study is a composite variable formed by collapsing four variables together, affect, efficacy, importance and future self, which are the four dimensions of self-schema suggested by Garcia and Pintrich (1993, 1994). Students rated items measuring these variables on a 5-point Likert scale, ranging from strongly disagree (1) to strongly agree (5).

'Affect' was assessed by two questions: "I enjoy learning mathematics", and "solving mathematical tasks is fun". The Cronbach alpha was 0.63.

'Efficacy' was measured by four items. The Cronbach alpha was 0.76. Sample items are: "I keep pushing myself in learning difficult things in mathematics for I believe that with persistent effort I will master them later on", and "I'm as smart as other in doing mathematics".

'Importance' was measured by four items. The Cronbach alpha was 0.81. Sample items are: "A good result in mathematics is important for my future success", "I consider mathematics as 'my subject'", "It is important for me to do well in mathematics".

'Future self' was measured by three items. The Cronbach alpha was 0.74. Sample items are: "I'll choose to study mathematics or other related subjects in my future studies", and "The job I would like to do in the future requires a lot of mathematical skills and is strongly related to mathematics".

Factor analysis extracted one factor from these four variables. The scores of these four variables were then collapsed together forming a composite variable called self-schema in mathematics learning. This composite variable had a Cronbach alpha value of 0.89.

All items measuring students' goal orientations were preceded by a stem, "I study mathematics because ...". Students rated each item on a 5-point Likert scale ranging from strongly disagree (1) to strongly agree (5).

Students' mastery goal was measured by three items. The Cronbach alpha was 0.81. Sample items are: "I enjoy solving problems" and "I want to master different mathematical skills".

Students' functional goal was measured by three items. The Cronbach alpha was 0.66. Sample items are: "I need to do well in mathematics in order to get into the university program that I want" and "I need to do well in maths to get the job I want".

Students' performance goal was measured by six items. The Cronbach alpha was 0.82. Sample items are: "I want to get better results", "I want to show that I am smart", "I do not want to look stupid".

Students' social solidarity goal was measured by three items. The Cronbach alpha was 0.67. Sample items are: "I want to be useful to the society", and "I want to help my friends to learn mathematics".

Students' learning approaches in mathematics were assessed by 36 items taken from Mathematics Learning Process Questionnaire (MLPQ) developed by Liu (1997). Students rated themselves on a 5-point Likert scale (1=very true of me, 5=very untrue of me) about their learning approaches in mathematics. The MLPQ items were adapted from the Learning Process Questionnaire (LPQ)(Biggs, 1978). While Biggs' LPQ measures students' general approaches to learning, Liu's MLPQ measures students' subject specific learning approaches (mathematics). In a series of studies, Liu (1997) found three recurring learning approaches in mathematics similar to Biggs framework. The Cronbach alpha values of these three learning approaches in Liu's studies ranged from .55 to .84, which were comparable to Biggs samples (1987). In this study, the Cronbach alpha values for learning approaches derived from the MLPQ ranged between .71 and .86. A forced factor analysis with varimax rotation of the MLPQ resulted in three factors, which were congruent to Biggs framework of learning approaches. Items with loading value greater than .40 were selected to form the following learning approach constructs.

The achieving approach was composed of 14 items with a Cronbach alpha value of .86. This approach is related to a desire for performance and strategies that secure a high performance. Sample items are "I have a strong desire to do best in maths", "I'll work for top mark in maths whether or not I like the subject", "I find that doing well in maths can give me a deep personal satisfaction".

The deep approach was composed of 7 items. The Cronbach alpha value was .73. These items are associated with a desire and strategies to gain an deep understanding of the learning materials. Sample items are "I spend a great deal of my free time working on maths problems and puzzles that I found in books and magazines", "Soon after a maths class, I'll attempt the exercises to make sure that I understand the materials", "In studying a new topic in maths, I often recall materials I have learned and see if there is a relationship between them".

The surface approach was composed of 9 items with a Cronbach alpha value of .71. By this approach, students are concerned with expending minimum effort in learning mathematics. Sample items are "I prefer maths topics in which I have to learn just formulas rather than solve problems", "In maths, I only do enough to get a pass and no more", "I think maths teacher shouldn't expect us to work on topics outside the set course".

Students' anticipated performance was measured by one item. Students were asked to rate their possible mark at the end of the academic year in a 6-point scale (A, B, C, D, E and No target grade).

The data collected was first checked for outliers and normality. The screened data was then subjected to correlation analysis. It was followd by a path analysis.

Table 1 shows the correlation matrix of the variables in this study. Self-schema showed strong correlation with mastery goal and moderate correlations with other goal orientations. These relationships were consistent with previous findings (Ng, 1997, 1998). Self-schema also correlated positively with achieving and deep learning approach but negatively with surface approach.

All goal orientations were positively related to achieving and deep approach but negatively related to surface approach. These correlations between goal orientations and learning approaches generally reflected the hypotheses. However, some exceptions were found with performance goal. The positive relationship between performance goal and deep approach and at the same time a negative non-significant relationship between performance goal and surface approach were unexpected. These

findings may suggest that performance goal might not be totally maladaptive. Aside from linking with learning approaches, goal orientations were moderately related to each other.

Anticipated performance was positively related to achieving and deep learning approach. It was, as expected, negatively related to surface approach. These findings were consistent with previous studies in these variables.

The path analysis in this study was conducted through EQS 5, using maximum likelihood estimation. After pairwise deletion of the data, 289 cases were available for analysis. The raw data was then transformed into a covariance matrix. All the paths in the initial model and the revised model were assessed simultaneously, which means that each path coefficient represents the unique association of the two variables linked by the path without interference from other paths. The Wald test was used to exclude non-significant paths. Lagrange Multiplier test was used to include paths that may contribute significantly to the model.

Fit indices. The EQS programme provides different indices to ascertain the model fit. The Chi-square value, c ², is among the most common indices that has been widely used to determine the goodness of fit of a model. However, as c ² is based on restrictive assumptions and sensitive to sample size, it may not be a good estimate of overall model fit (Tabachnick & Fidell, 1996). Herein, other indices, the Comparative Fit Index (CFI), the Bentler-Bonett Normed Fit Index (NFI), the Bentler-Bonett NonNormed Fit Index (NNFI), the Lisrel GFI/AGFI Index and the Root Mean Squared Residual (RMR) were emphasised as indicators of the goodness of fit. CFI, NFI, NNFI, GFI/AGFI values of .90 or above indicate a good model fit to the sample data. RMR value of .05 or below corroborates a model fit (Bentler, 1990; Bentler & Bonett, 1980).

The hypothesised model followed closely the theoretical model depicted in Diagram 1. Effects of self-schema on anticipated performance was assumed to be mediated through various goal orientations and learning approaches.

This hypothesised model tested:

a/ The direct effect of self-schema on four goal orientations: It was assumed that self-schema would link positively with these goal orientations and a relatively strong link was expected to be found with mastery goal.

b/ The relationship between goal orientations and learning approaches: It was assumed that mastery goal would be positively related to achieving and deep approach but negatively related to surface approach. These relationships between goal orientations and learning approaches were expected to be repeated in functional goal and social solidarity goal. Performance goal was expected to link positively with achieving and surface approach but to be negatively related to deep approach.

c/ The relationship between learning approaches and anticipated performance: Achieving and deep approach would be positively related to anticipated performance while negative relationship was expected to be found between surface approach and anticipated performance.

Table 2a and 2b show the significant standardised path coefficients for this hypothesised model. In general, the model supported the hypothesised relationships among the variables. Self-schema was causally linked to all goal orientations. The strongest tie was found with mastery goal (b =.67, p<.001). The learning approaches were also significantly related to anticipated performance as expected. The relationship between goal orientations and learning approaches generally fitted the hypotheses, however, with some exceptions. These exceptions include:

a/ Functional goal was not significantly related to surface approach;

b/ Performance goal was positively related to deep approach;

c/ Social solidarity goal was not related to any learning approach.

Overall, the relationship specified in the hypothesised model was found, suggesting that there is a linear relationship between self-schema and anticipated performance mediated through goal orientations and learning approaches. However the significant Chi-square value (c ² <15, N=289> =209.36, p<.001) suggested that this model did not account for all the data. Other goodness of fit indices (CFI=.79; NFI=.78; NNFI=.61; GFI=.84; AGFI=.64; RMR=.08) also indicated that the model fitted the data only moderately. Modification was necessary in order to attain a better fitting and more parsimonious model.

With reference to the Wald test, the Lagrange multiplier test and related empirical findings, additional paths were added. It has been argued that a path, or the causality implied by it in a covariance structure model, is an assumption between the variables linked by the path (Brannick, 1995). In this study, therefore, the decision of which path would be included was driven by a theoretical understanding and empirical findings.

Path set 1: Paths from self-schema to various learning approaches.

Path set 3: Paths among different goal orientations.

Path set 4: A path from deep approach to achieving approach.

The revised model was then subjected to analysis. Non-significant paths were deleted. The Chi-square value (c ² <17, N=289> = 37.21, p=.003) of the modified model has greatly reduced, suggesting a substantial improvement in the goodness of fit. Though the Chi-square value was still significant, all the other goodness of fit indices indicated that the revised model fitted the data well (NFI=.96; NNFI=.96; CFI=.98; IFI=.98; MFI=.97; GFI=.97; AGFI=.92; RMR=.02). As mentioned earlier, Chi-square value, as a goodness of fit index, has its limitation. Bentler (1995) explained that c ² is sensitive to sample size; with large samples, trivial differences between the sample and the estimated population covariance matrix will often lead to significant findings; with small samples, the computed c ² may not be distributed as c ², causing inaccurate probability levels. Therefore, a significant c ² may not necessarily mean that the model does not fit the data. Given that c ² has substantially reduced and all the other indices consistently and distinctively signalled that this modified model fitted the data well, the revised model was then accepted as the best solution. Wald test and Lagrange multiplier test suggested that no path was needed to be included or deleted. Diagram 2 shows the final model with significant standardised path coefficients.

Table 3 shows the decomposition of effects of the final model. Self-schema was the most important variable in the model. It demonstrated strong total effect on mastery goal (b =.67, p<.001). Its total effect on other goals was moderate (b =.54, p<.001 for functional goal; b =.43, p<.001 for performance goal; b =.43, p<.001 for social solidarity goal). It is interesting to note that self-schema's effects on mastery goal and performance goal were mediated through other goal orientations. In the case of mastery goal, social solidarity goal mediated the effect of self-schema (b =.11, p<.001). As for performance goal, both functional goal and social solidarity goal mediated the effect of self-schema (b =.29, p<.001). The presence of these two mediating variables has weakened the direct effect of self-schema on performance goal (b =.14, p<.001).

Self-schema also predicted the employment of learning approaches. As expected, self-schema was positively related to achieving approach (b =.39, p<.001) and deep approach (b =.32, p<.001). A negative relationship was recorded between self-schema and surface approach (b =.-41, p<.001). Indirect effects of self-schema on learning approaches were mediated through mastery goal and performance goal.

Finally self-schema demonstrated a distinct direct effect on anticipated performance (b =.32, p<.001). Indirect effects of self-schema on anticipated performance mediated through achieving and surface approach were also significant.

In short, self-schema revealed its importance in this model in two ways: a/ its direct effects on all the dependent variables; b/ its indirect effects mediated through goal orientations and learning approaches on anticipated performance.

Mastery goal linked positively with deep approach (b =.18, p<.001) and negatively with surface approach (b =-.26, p<.001). However, it was not significantly related to achieving approach. These relationships highlight the strong task focus nature of mastery goal. Students learning with a mastery orientation will concentrate on understanding and learning. This concentration will draw students away from concerns about marks and performance levels. This explains why mastery goal did not link with achieving approach, which has its focus on performance.

Performance goal was the only significant goal orientation linked with achieving approach (b =.29, p<.001). This is expected as both performance goal and achieving approach have their focus directed onto high achievement. However, its positive link with surface approach (b =.27, p<.001) and deep approach (b =.12, p<.001) in the same model warrant further elaboration.

Social solidarity goal and functional goal were not significantly related to any learning approach. Their effects were mediated through mastery and performance goal.

It is important to highlight that all goal orientations were not significant in predicting anticipated performance. Anticipated performance was predicted positively by achieving approach (b =.18, p<.001) and negatively by surface approach (b =-.24, p<.001). In other words, goal orientations appeared not to influence how students perceived their future performance. It is how they approach learning that is more important in this particular prediction. Nevertheless, note that deep approach did not predict anticipated performance. This can be explained by the strong task focus of deep approach that may have overshadowed the importance of achievement levels.

In general, the final model depicts a linear relationship among these four categories of variables. This empirical model is consistent with the theoretical model explained earlier in this paper. Specifically, the model highlights the overwhelming importance of self-schema in influencing why and how one approaches learning as well as how one anticipates future performance.

Self-schema is the most important variables in the model. It is statistically significant in explaining the variance in goal orientations, learning approaches and anticipated performance. In other words, a positive self-schema in learning mathematics will probably open up a variety of learning options for students. Of course, with a positive self-schema, students will be more likely to adopt a mastery goal, though the other three goal orientations are still within one's choice spectrum. Likewise, a positive self-schema will also allow student to choose to learn through either deep or achieving approach. In contrast, a negative self-schema will be more likely to direct students to a surface approach which will have adverse effects on their perceived performance and probably on their actual performance. In addition, self-schema will also help students to anticipate their future performance. This may be critical, as low expectations for future success are usually associated with a low level of persistence, devaluing the task, and a withdrawal of effort expenditure.

The significance of self-schema lies not just on its strong direct effects on the goal orientations, learning approaches and anticipated performance, its effects are also mediated through goal orientations onto learning approaches and subsequently on anticipated performance. In light of these findings, the original theoretical model was revised. Diagram 3 shows this revised theoretical model in which the linear relationship among these key variables are still held. The major alteration is the addition of paths linking self-schema with learning approach and performance, reflecting the direct effects of self-schema on these variables.

This revised model opens up for consideration two quite different paths for successful learning of mathematics:

a/ A mastery focused path. This path starts from self-schema, mediating through mastery goal to deep approach. The mastery-focused path implies that learning is overtly concentrated on the understanding and mastering of the knowledge. Achievement or performance is probably seen as a by-product that comes necessarily as students engage in such deep learning.

b/ A performance focused path. This path starts with self-schema through performance goal and various learning approaches to anticipated performance. This path can further be divided into two streams : an adaptive stream and a maladaptive stream.

The adaptive stream links performance goal to either deep or achieving approach and ends with a positive anticipated performance. The maladaptive stream links performance goal with surface approach and a negative expectation of future performance.

The adaptive performance path suggests that the performance goal may not be detrimental to learning, as it has been depicted in the early studies of goal orientations. In the light of the current model, the performance goal would lead to an adaptive pattern of learning engagement through achieving approach as well as deep approach. Recent studies (Greene & Miller, 1996; Harackiewicz et al., 1998; Wolters, Yu, & Pintrich, 1996; Young, 1997) have shown that certain types of performance goals are adaptive and will have positive effects on learning. These adaptive performance goals have been labelled as relative ability goal (Greene & Miller, 1996) or approach performance goals (Elliott & Dweck, 1988). It is only when students try to do the minimum or to avoid work that their performance and learning will be undermined.

Nevertheless, the presence of a maladaptive stream alongside with the above adaptive stream has created interpretation difficulties for performance goals. For it is hard to imagine that the same performance goal will lead to an adaptive deep or achieving approach and simultaneously link positively with a maladaptive surface approach. This confusion might have been caused by the structure of the performance goal construct in this study. Performance goal construct in this study was formed by both positive statements like "I study mathematics because I want to get a better result" and negative statements like " I study mathematics because I don't want to look stupid.". "Getting a good result" and "not looking stupid" certainly represent two contrasting orientations toward learning. Harackiewicz and et al. (1998) argued that it is crucial for researchers to distinguish these two orientations in order to ascertain the effects of performance goals. The collapse of these two types of performance goals into the same construct may have been the cause of the present confusion in the model. The questionnaire items, therefore, need to be reworded in order to distinguish these two performance orientations.

Another interesting point about the model and the learning paths discussed above is the interrelationship among goal orientations. The significant paths that go from social solidarity goal to performance goal and to mastery goal as well as that from functional goal to performance goal may be interpreted as the existence of immediate achievement goals and enduring goals. This interpretation imposes a temporal frame onto goal orientations. In this line of thinking, mastery goals and performance goals can be considered as more immediate goals that relate directly to learning and performance of a task or a subject. These two goals provide the immediate reasons for learning engagement. Functional goals and social solidarity goals on the other hand represent enduring goals for learning behaviours. They represent an individual's long term pursuits; as such, these enduring goals will motivate students to learn for either a mastery or a performance focus as long as the learning ties with their enduring endeavours.

In the case of functional goal, "getting a desired job" and "gaining an entry to a desired university programme" can be understood as enduring goals that fuel the endeavour for performance. In the same way, social solidarity goals like assisting others to advance academically can be considered as long term goals that provide supports for mastery effort and striving for high performance. Following this interpretation, the model here suggests that students' mastery and performance goals can be explained by other long-term endeavours. The enduring goals can be seen as an incentive or a justification for learning engagement with either a mastery or a performance focus.

The aims of the interview study is to illuminate the survey findings with qualitative data. The interviewees were selected by their teachers. Eight year 10 students from a band 3 and a band 2 secondary school that participated in the earlier survey were interviewed. Each student was interviewed for 30 minutes to 45 minutes in a study centre close to their school. Because of the time constraint, the last four students were group-interviewed, over a two-hour period.

Students were interviewed in a semi-structured format. They were asked to respond freely to several open ended questions, for example, 'what do you think of mathematics?'. These open-ended questions serve as a prompt to stimulate students to speak. At the end of the interview, students were asked to respond to several hypothetical classroom situations which depicted a mathematics teacher taught with different goal emphases. Students' responses to these cases clarified their learning goals and revealed their desired learning environments.

For the purpose of this paper, two interviewees' accounts of their learning experiences in mathematics were reported. These two students were chosen because they demonstrated how students with extreme self-schemas studied mathematics.

Wai experienced a change of her self-schemas in learning mathematics from positive to negative in high school. Originally, she had a positive self-schema in learning mathematics during her first year in high school. She was good at maths, which was reflected in high marks. In addition, she looked forward to do more maths in the future and thought she had the ability to do it. These positive schematic characteristics led her to a mastery pattern of learning, which could be shown by the amount of time she spent on learning mathematics.

However, she attributed her positive schema to her good teacher. To Wai, a good teacher is one who can teach well, takes time to help students and is approachable. In addition to the teacher factor, she also claimed that her good performance in primary school has helped her maintain the positive schema. The following excerpt demonstrates these points.

Teachers' classroom behaviours had a major influence on how Wai dealt with maths. She explained that her failures in form 2 and 3 (year 8 and 9) were the result of the bad teaching and classroom mismanagement practice of her teachers.

Therefore, Wai did not concentrate in the class. Teacher was not the only factor. She explained that an institutional factor has also come into play. She was assigned to a low-achieving class, which the teacher could not control. In addition, the poor teaching practice afforded Wai and others a chance for mucking around during the mathematics lessons.

At one stage, Wai did try to get a grip on herself. However, the poor teaching practice of her form 3 (year 9) teacher again has frustrated her. She accused the teacher of lacking of assertiveness in classroom discipline and attributed her inattention to his poor teaching strategies.

The experience in these two years has lasting detrimental impact on Wai's learning of mathematics. A negative self-schematic view about mathematics emerged as a result. In form 4 (year 10), She dreaded learning maths and thought that she would not be able to do well in it.

The following excerpt shows that Wai developed a maladaptive pattern of learning as a result of her negative self-schematic view on learning mathematics. The maladaptive learning pattern is characterised by:

• surface learning strategy: memorisation and cramming before a test.
• a reluctance to expend effort;
• a low self-efficacy belief;
• a negative perception of mathematics

The next excerpt shows Wai's current negative self-schema. Note that the excerpt also reveals Wai's struggles and contradictions in learning mathematics. She expounded at the beginning of the excerpt that she hated memorisation and yet she employed it to learn mathematics. Wai was not happy with an average result, however she found it hard to forsake her handicapping learning approach. She did acknowledge that learning required effort and time. However, she explained that she would not expend effort as she found the subject hard and uninteresting.

She also delayed expending effort in getting a good result. She justified that she could wait until the next year. It was not yet the 'crunch time' for a serious study.

The following excerpt shows how she crammed before a test or an examination.

Wai did try to revive the situation. However, her negative schema in learning mathematics led her to laziness, which prevented her from expending effort and time into the subject.

When responding to a speculative question about her performance in maths, she believed that she could only do well if she had a good teacher. She did have a good teacher in year 10. Notwithstanding that her good teacher can drive her to pay attention in the class, her handicapping negative self-schema on learning mathematics and the associated maladaptive learning styles cannot be changed in a short period of time. She claimed that her goal in learning mathematics was to get a good result. Ironically, a good result to her means only a 'clean pass'.

Bing maintained a positive self-schema in learning mathematics. She has always been doing well in mathematics. She did maths because she liked it and found it interesting. She explained that her interest in the subject had been the result of the favourable learning environment at home.

She enjoyed doing mathematics task and solving difficult questions. This interest in mathematics led her to a learning pattern that was characterised by mastery and self-regulation. Note that at the end of the following excerpt, she explained the causal links between a positive schemas in learning mathematics and the learning approaches.

In addition, she would try to master the task by herself. She was reluctant to copy from others. With difficult questions, she would make sure that she understood them and practised similar ones.

She stated clearly that she was always confident in doing mathematics since primary school. In talking about her ability and confidence in doing the subject, she related the following story about a test she had in primary school

However she loved to compete with other and like to beat other to be the first in the class.

The focus on relative ability and competition led her to some particular learning preferences and perceptions about her classmates and teachers. First, she preferred to study alone and rejected learning in a group.

In addition, she preferred not to ask help from other classmates which she probably considered as an indication of a lack of ability.

However, she enjoyed giving help. To Bing, giving help was a moment of triumph and a chance for consolidating her superiority over her classmates.

She treated her classmates with suspicion. She had the impression that when she asked people how they revised for a test, most of them will go 'I don't know, I didn't pay much attention in the class. I don't understand something here, something there'. The following excerpt demonstrates this point. Bing was talking to Nagi, her classmates, during the interview.

Bing saw no problem for teachers to publish students' marks. Most of her past teachers published students' marks. She reckoned that this would help her know where she was positioned in the class.

In addition, she thought that performance and relationship with teacher were tied. She believed that teachers would treat high achieving students in a better way.

Aside from learning with a focus on performance, she also leant with a mastery goal. She defined success in terms of both mastery and outperforming other. In order to outperform other, she believed that she needed to do her best, which means efforts.

However, she held a fixed notion of ability. She related how her father explained to her that individual's ability was limited.

Because of this fixed notion of ability, she tended to rationalised her effort. She told two examples to support her view.

The concept of fixed ability caused her to refrain from putting extra effort into the revision for examination as she thought she might not be getting a proportional reward from her effort and time expended.

Nevertheless, her positive self-schematic view on learning mathematics was very stable. She would hold onto her love of the subject even with a bad teacher. 'I won't be affected by other. Like if I have got into something, it can't be changed so easily. It's not like a decision, you can change it easily. It is something you have been nurturing for years.'

The two cases revealed the effects of self-schemas on students' learning engagement patterns. Wai represents students with a negative self-schema, which has led her to a very low level of learning engagement. Wai learnt with a performance avoidance goal, as she was very reluctant in expending effort and satisfied with a passing grade in mathematics. In addition, she employed a surface learning approach by which she used strategies like memorisation to study mathematics. The surface engagement pattern resulted in a poor performance.

In a stark contrast, Bing learnt with a positive self-schema. The positive self-schema has brought Bing into a deep engagement mode in learning mathematics by which she studied with a combination of a mastery and a relative ability goal. The relative ability goal might have driven her to a crippling belief, a fixed notion of ability. However, it seems that her interest in the subject and the competitive spirit have offset the detrimental effects of the fixed notion of ability depicted in the literature (Dweck, 1986). She maintained a deep-achieving approach to learn mathematics, which was characterised by a focus on performance and a stress on effort expenditure. A deep-achieving approach has been hailed as the most adaptive and rewarding mode of learning (Biggs, 1987). Predictably, she did well in mathematics.

In general, these two cases confirm the patterns of findings derived from the final model of the survey study. The presence of the two paths of learning has been demonstrated. Bing represents students learning with the adaptive path, which goes from a positive self-schema, through mastery and relative ability goals, via a deep-achieving approach and lands successfully on a high performance. Wai, in contrast, belongs to students learning with a negative self-schema, which leads her to performance avoidance goal and surface learning approach, and inevitably ends with a poor performance. In addition, Bing's case shows that performance goal will not be totally detrimental as suggested in the early studies in goal orientations (Harackiewicz & et al., 1998).

Concerning the development of self-schemas, Wai's case demonstrates that students' self-schemas can be affected by teachers and their teaching. Teachers' teaching strategies and classroom management skills will help to create a learning environment that favours learning and the development of positive self-schemas. In Wai's case, as her teachers had failed to produce such a facilitating learning environment, the further development of her initial positive self-schema has been stifled. Ironically, a negative self-schema emerged and came to maturation two years later when Wai dreaded learning mathematics in year 10.