Issues In Educational Research, Vol 17, 2007
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Students' and teachers' efficacy in use of learning strategies and achievement in mathematics

Deepa Marat
Unitec Institute of Technology, New Zealand

In the context of the critical role of self-efficacy in educational achievement, this present research examined students' and teachers' efficacy in use of learning strategies in mathematics, and the relationship with achievement. The second phase of a multi-method doctoral study, ninety-two students and ten teachers from a diverse secondary school were surveyed on efficacy in use of learning strategies, and the factors they perceived as facilitating or inhibiting the use of learning strategies in the classroom context. The findings are contextualised within a multi-method framework comprising two research projects. The cumulative findings bring to light illusory-efficacy in a sizeable number of student participants who did not achieve, highlighting the importance of true efficacy and learning strategies to reduce disparities and enhance achievement.

Introduction

The construct of self-efficacy has been researched extensively in the field of psychology and education. Introduced by Bandura (1977), self-efficacy refers to people's beliefs in their capabilities to organise and execute courses of action to attain goals (Bandura, 1991). A multi-dimensional construct, self-efficacy influences human functioning directly and indirectly through its effects on determinants of behaviour (Bandura, 1997) such as motivation, self-regulation, attribution, and emotion. Self-efficacy is not merely a self-recognition of being good in school, but judgments of capability in task specific skills to accomplish goals related to learning and performance (Pintrich & Schunk, 2002). For instance, self-efficacy in mathematics include the self-belief in strategies such as learning computation and problem-solving, working on unfamiliar problems, metacognitive strategies, and working in small groups (Schoenfeld, 1992; Schunk, 2000, 2001).

Self-efficacy differs from concepts such as self-concept and self-esteem. Self-concept is a composite view of oneself based on past environmental experiences, and oriented to the past. Self-efficacy is future-oriented and represents an individual's use of self-perceptions in successfully accomplishing a particular task with the target behaviour clearly specified (Skaalvik & Bong, 2003). Self-esteem involves emotional reactions to actual accomplishments (Linnenbrink & Pintrich, 2003). The "dynamic and malleable nature of self-efficacy perceptions render them more amenable to experimental procedures aiming at efficacy enhancement" (Bong & Skaalvik, 2003, p.30), and thereby enhancing the level of achievement. This distinction between self-efficacy and self-concept is significant in the education context. Research on self-related constructs for advancing achievement in New Zealand schools have focussed on the assessment of students' self-concept (Chamberlain & Caygill, 2002; Education Review Office, 2002; Nash & Harker, 1997; Hughes et al., 2000), and thereby undervalued by omission the role of self-efficacy in learning and achievement of students.

Self-efficacy, use of learning strategies, and achievement

Self-efficacy and its role in students' achievement have been researched extensively, with various studies highlighting positive correlation between self-efficacy and performance attainment (Bandura, 1986; Bempechat & Drago-Severson, 1999; Goddard, Hoy, & Hoy, 2000; Pajares, 1996; Pajares & Valiante, 1997; Patrick, Hicks, & Ryan, 1997; Schunk, 1995). Factors which impact on academic self-efficacy include goal setting, information processing, models, parental support, skills in self-regulation, resource management, teacher feedback, and rewards (Pintrich & Schunk, 2002; Schunk, 2000; 2001). Crucial for the development of self-efficacy and enhancing achievement in students is teacher efficacy. Teacher efficacy has been defined as "teacher's belief in his or her capability to organise and execute course of actions required to successfully accomplish a specific teaching task in a particular context" (Tschannen-Moran, Hoy, & Hoy, 1998, p.233). Teacher efficacy includes teacher beliefs in maintaining an orderly classroom conducive to learning, self-belief in instructional practices in different knowledge domains, enlisting resources and parental involvement, and counteracting social influences that subvert students' academic pursuits (Bandura, 1997). Teacher efficacy also influences instructional practices, and can predict students' performance and self-efficacy (Pajares, 2002). Teachers with high self-efficacy help create mastery experiences for their students, teachers with low instructional-efficacy can undermine students' cognitive development and self-efficacy. Statements about the value of a task in the classroom by the teacher influence self-efficacy and motivation (Pintrich, & Schunk, 2002). Efficacy also plays a significant positive role in teachers' adoption of innovations in educational processes (Evers, Brouwers, & Tomic, 2002).

Research spanning two decades on use of learning strategies and self-efficacy, has been examined by Schunk (2001). He reports that for students who scored below grade-level in mathematics, use of cognitive modelling enhanced achievement. Elementary students with difficulty in subtraction, when provided exposure to mastery or coping models, performed well. Modelling, followed by monitoring and training in self-monitoring produced better performance, as compared to modelling without monitoring. Strategies underlying teacher-led motivated learning in students include (Pintrich & Schunk, 2002):

Ensuring that students have the capability to learn what is being taught.
Providing information on the relevance of learning in students' lives.
Providing information on strategy use and feedback on how using strategies has helped improve performance.
Content tailored such that students understand it, and instructional presentations catering to individual differences.
Setting learning goals and guiding students to monitor daily progress.
Feedback on progress in learning and rewards linked to progress.
Using models who help develop self-efficacy and increase motivation.
In-group situations, designing tasks which can be successfully attained if members work industriously.

Some of these learning strategies are implicit in Akonga (Maori pedagogy), advocated by Bishop, Berryman, Tiakiwai, and Richardson (2003) in the New Zealand educational context, and encompass:

Narrative pedagogy to promote committed, partnerships between teachers and students.
Co-operative learning providing opportunities to students for acquiring academic skills and working towards achieving common goals.
Formative assessment which involves students actively engaged in self-evaluation of having achieved learning intentions, celebrating achievement and enhancing self-esteem.
Student generated questioning which facilitates culturally appropriate discourse based on student realities.
Oral language/ literacy across the curriculum to help students process their own learning based on their own realities.
Integrated curriculum collaboratively co-constructed by teachers and students based oncultural realities, empowering students to be life-long learners.
Critical reflection of teaching-learning practices.
A reciprocal partnership (Ako) of teachers and students being learners and teachers.

While research on student achievement in New Zealand reports the creditable comparative standing of New Zealand students among OECD countries (Education Review Office, 1994; Statistics New Zealand, Undated), there are also reports which highlight concerns about New Zealand students' achievement (Maxim Institute, 2003; Minister of Education, 2002), particularly in mathematics. Forty-four percent of New Zealand 14 year olds have been reported to score below fixed international benchmarks in mathematics (Maxim Institute, 2003). A report on ways in which mathematics teachers of Year 1 to Year 8 students developed teaching programmes from the achievement objectives of the mathematics curriculum (Education Review Office ERO, 2002) highlights that in most of these schools there was low incidence of in-depth investigative problem solving in mathematics. While teachers introduced the mathematics session by recapitulating prior learning, there was no focus on clarification of the objectives of the teaching-learning session or setting goals or expectations for learning. Further, for a significant number of lessons there was no conclusion. Reasons were essentially time-related. There were no formal procedures to obtain feedback on the effectiveness of teaching strategies. The report recommends more research on group teaching in New Zealand, on the advantages of whole-class teaching, and on teaching strategies which facilitate teachers to make informed decisions on choice of appropriate strategies.

The present study explores the role of self-efficacy and learning strategies in students' achievement in mathematics. The learning strategies examined include: (a) strategies for self-regulatory learning, (b) resource management; and (c) motivation. These strategies were derived from factor analysis and multiple discriminant analysis of phase I of the research (Marat, 2003; 2005b), and developed into two survey questionnaires (Appendix I and II). Phase I of the research comprised 137 student participants from two culturally diverse schools in Auckland. Participants were assessed for self efficacy in achievement in mathematics. The major findings were: (a) Students are reporting relatively high levels of self-efficacy for achievement in mathematics which are not reflected in actual achievement; (b) Three major factors emerge as independent variables which impact on achievement in mathematics; (c) These factors are also seen to have predictive value in determining students' level of achievement in mathematics.

Methodological approach and methods

The present research was based on a unified approach of research paradigms (Keeves, 1997), leading "to the adoption of a common approach to educational inquiry in which researchers and practitioners work together to solve educational problems" (p.5). Since the two main paradigms of educational research, that is, the naturalistic and the scientific are "not independent of the cultural and social context in which they operate' (Husen, 1997, p.21), the principles underpinning a New Zealand research approach, the Kaupapa Maori approach of respect, consultation, sharing processes and knowledge was also an integral part of the research.

Kaupapa Maori approaches to research are based on the assumption that research that involves Maori people, as individuals or communities, should set out to make a positive difference for the researched. This does not have to be an immediate or direct benefit...The research approach also has to address seriously the cultural ground of respect, of working with communities, of sharing processes and knowledge. (Smith, 1999, p.191)

From the initial stages of the research process, which involved formulation of the research problem, consultancy with practitioners and experts from the field of educational research led the present project. The emerging results of the first phase of the research was presented at the Joint New Zealand and Australian Association for Research in Education Conference (Marat, 2003), and audience feedback was integrated into the research-in-progress.

The choice of methodology and methods in the present study resulted from a synthesis of quantitative and qualitative approaches. The essential test was "[t]o ensure that the evidence obtained enables us to answer the initial question as unambiguously as possible" (de Vaus, 2002, p.9). While the framework of the research method was survey-based, correlational methods, and content analysis were used to analyse the data. Thus, this research comprising two phases was based on a framework comprising: (a) mixed method design, and (b) multimethod design. Morse (2003) explains mixed method design as incorporating both quantitative and qualitative strategies, and multimethod design as one inclusive project comprising two or more research projects, each complete in itself. What emerges is methodological triangulation described as "[t]wo or more subprojects, each of which exhibits methodological integrity. While complete in themselves, these projects fit to complement or enable attainment of the overall programmatic research goals" (p.190).

Ninety-two students from Form VII and ten teachers from a culturally diverse secondary school participated in this research. Purposive-theoretical sampling was used to invite participants, after lack of response from a probability based random sample of secondary schools invited to participate in the research. In the present study, culture is considered as a contextualised, dynamic, and evolving construct, and much more diverse than dichotomous categorisation of individuals into individualistic or collectivistic cultures. This is to accord for the multicultural nature of New Zealand society, and the intra-ethnic and intra-individual diversity within the cultural groupings. Culture although a significant variable, has not been considered in analytical interpretation in the present research. While no comparisons are drawn between cultural groups, the distinct cultural identities of participants are valued and acknowledged. The decision to desist from using culture as a variable for analyses was based on an underlying ideology of considering culture as seamless construct, and the inappropriateness to compartmentalise participants and their responses into narrow categories based on cultural affiliations. As stated by Bandura (2002):

[A] selected cultural factor that yields a small difference in group averages is generalized to all individuals in the cultural grouping as though they all believed and behaved alike as dichotomously classified. ... Human behaviour is socially situated, richly contextualized and conditionally expressed. (p.276)

However, since there is substantial body of work which highlights disparities in achievement grounded in the history and culture of New Zealand (Simon, 2000; Jenkins & Jones, 2000, Ministry, 2001), and initiatives which target achievement of specific cultural groups and 'achievement of all' (Ministry, 1998; Minister of Education, 2003), the literature reviewed encompasses culture as a construct having important bearing on self-efficacy and achievement. The purpose is two-fold: (a) to ensure that the present research has relevance for the diverse learning context in New Zealand schools, and (b) to highlight the links, if any, between historically grounded disparities, self-efficacy and achievement.

The conceptualisation of teacher efficacy in the present research is narrow, since the purpose of this research was to explore student efficacy comprehensively. Although teacher beliefs have been investigated, it is suggested that in-depth investigation on the role of teacher efficacy as a construct which impacts on student achievement will have to be considered in future research.

The students' survey tool (Appendix I) sought information on efficacy in the use of thirteen learning strategies in mathematics (Alpha 0.77). The term self-belief was used to explain the construct of self-efficacy to aid easy comprehension by participants. An open-ended statement at the end of each item sought information on how students used the learning strategy, and factors perceived as facilitating or inhibiting the use of these strategies in the classroom. To assess the relationship between students' efficacy and achievement, consent to access achievement scores was sought from students. The achievement results of only 40 students could be used for inferential analysis. Some of the students did not clearly write their names in the survey precluding use of their achievement grades. Teachers' survey included seven items on efficacy in use of learning strategies in the classroom context (Appendix II). An open-ended statement at the end of each item sought recall of instances in the past year when the teacher guided students in use of specific strategies, and factors which facilitated or inhibited use of strategies.

Data analyses

Quantitative data were analysed using both descriptive and inferential measures. Correlations were computed between self-efficacy in learning strategies and achievement in the examinations. Qualitative data were analysed based on content analysis, categorising participant responses in the open-ended survey items into emerging themes within the framework of theories of self-efficacy, self-regulation and motivation. Based on Miles and Huberman's method of Qualitative Data Analysis (Rouse & Dick, 1994), the three-step process comprised: (a) data reduction (synthesising responses from the students' and teachers' surveys separately, the term 'selective responses' is used to denote this synthesis in Tables 2 & 6); (b) data display (Tables 2 & 6); and (c) conclusion drawing and verification (section on discussion).

Results

Demographics

Student participation rate for the survey was 92% (n=92). The gender breakdown for students show 40 females, and 51 males. One did not report gender. Ten teachers from the mathematics department volunteered to participate.

Analysis of student self-efficacy scores in use of learning strategies

Participants reported self-efficacy in use of most of the learning strategies (Table 1). Efficacy in use of online and school resources, practising mathematics to learn, monitoring progress, setting goals, and seeking guidance were reported by a sizeable number of participants. Seventy-four percent of participants reported lack of efficacy in making time-plans; and sixty percent of students reported lack of efficacy in reviewing set goals. The reported efficacy in a number of strategies was not congruent with students' comments to the open-ended survey items (Table 2). Correlation analysis to assess the relationship between efficacy in use of learning strategies and achievement in mathematics showed positive significant correlations between students' practising mathematics to learn, using school resources, increasing confidence to perform successfully in the examination, and achievement in the examination. See Table 3 for summary of participants' achievement scores, and Table 4 for correlation analysis, and Figures 2, 3, and 4 for cross-tabulated data on efficacy in learning strategies and achievement.

Analysis of teacher efficacy in use of learning strategies

Most of the teacher participants reported efficacy in use of some strategies. The strategies included guiding students to set goals, assisting students to override negative thinking, and guidance in time management. Some teachers reported lack of efficacy in use of models, guiding students to revisit goals, mastery learning, and in-class opportunities to use resources. See Table 5 and Table 6 for details of teachers' responses.

Summative analyses

Students are reporting self-efficacy in use of most of the learning strategies. The results show highest positive correlations between beliefs in use of strategies for practising mathematics to learn, using school resources, confidence in one's capability to perform successfully in the forthcoming examinations, and achievement in mathematics.
Teachers report that they believe in the use of learning strategies in the classroom context. Notable exceptions were mastery learning and provision of school and online resources. Time, workload, shortage of resources, increase in number of students, and lack of awareness were attributed by teachers as some of the factors which inhibited the use of learning strategies.
The use of learning strategies in the classroom is mostly incidental, and not a deliberately planned process.

Table 1: Students' self-efficacy in the use of learning strategies in mathematics

Questions Response (%)

Yes No

Deliberately set goals 63 37

Deliberately made time-plan 26 74

Deliberately took control over any negative thinking 51 48

Deliberately praised oneself when successful 62 38

Felt motivated when others succeeded 66 34

Practised mathematics to learn 79 21

Regulated the environment to facilitate learning 51 49

Monitored progress 71 29

Reviewed set goals 39 60

Sought guidance 65 32

Used resources provided by the school 82 17

Used online resources 72 27

Felt increase in confidence in capability to perform successfully in mathematics 62 36

n= 92 Missing data ranged from 1.1% to 3.3%

Table 2: Thematic summary of qualitative responses on learning strategies by student participants

Emerging themes Response
(Frequency count)

a) Goal setting

Goals set to achieve better grades
Day-to-day goals set
No goals
13
5
6

Lead Statement:
When I worked in mathematics this year I deliberately set goals.
Comments from a selective number of participants:
To pass everything - every internal; I really wanted to pass my internals, especially the first one so I made sure I worked on it; I set goals but they change constantly and only kept in my head; Set goals to get high marks in assessment; A goal to achieve well in maths; Goals are mostly short-term; I wanted to achieve excellence and merit; I don't set goals.

b) Time plan

No plan 10

Lead statement:
When I worked in mathematics this year I deliberately made a time plan
Comments from a selective number of participants:
I do not plan I do; I didn't really set a certain timetable of study; I don't have a plan; I try, but don't follow it; Do not make time-plan; I made a rough plan but did not necessarily keep to it; Because I can do no more; Too restrictive; No I did not place any time deliberately aside for maths; I just did the work when I had to do it until it was done.

c) Deliberately praised oneself when successful

No 7

Lead statement: When I worked in mathematics this year I deliberately praised myself when successful. Comments from a selective number of participants: Don't want to get big-headed; No this leads to overconfidence; Do not praise myself; It's something I should do; Because if I did I would get overconfident; I am not like that

d) Felt motivated on seeing others succeed

Wanted to succeed like them 9

Lead statement:
When I worked in mathematics this year I felt motivated when I saw others like myself succeed.
Comments from a selective number of participants:
I felt that if I can do it so can I and worked towards it; Made me realise I could achieve too; I wanted to achieve just as well others did.

e) Practised mathematics to learn

To do well in examinations 18

Lead statement:
When I worked in mathematics this year I made myself practice so that I could learn.
Comments from a selective number of participants:
It's the only way to improve in mathematics; Before exams I studied; I bought study guides and got help from my dad; Every time I have time I would practice and revise my work; Did homework; I usually concentrate my effort on solving merit and excellence questions; Going over and over problems to ensure I knew the material (yet I failed damn it!!!)

f) Monitoring progress

Checking grades
No, don't care 8
8

Lead statement:
When I worked in mathematics this year I made sure I was aware of my progress.
Comments from a selective number of participants:
Mark down what grades I have got so far; I also like to see how I can do in relation to the class; Keep tabs on my achievement; I don't really keep up-to-date on where I am; No I don't; Was not aware of my progress.

g) Reviewed goals

Did not review, no goals set
Went over results, keeping in mind what was done 11
4

Lead statement:
When I worked in mathematics this year I reviewed my mathematics goals after I had set them.
Comments from a selective number of participants:
I didn't set any goals; Don't have goals; Have not changed them; No I never review; To see what I have to work on; I always keep my goals in mind; Day to day goals so they don't really need reviewing rather I just tried to complete the goal; I never had any as from starting the subject I was bored stiff.

h) Sought guidance / help

From text books
Maths tutor
Maths teacher 3
4
6

Lead statement:
When I worked in mathematics this year I sought guidance and help.
Comments from a selective number of participants:
Revision books; Maths tutoring both in school and out of school; From parents and my tutor; Get all the help I need from teacher and friends; From my dad; Asked the teacher questions and used text books; I sought help from teachers but haven't really received it; As no one ever explains it properly it may well be Arabic.

i) Resources provided by the school

Text books 10

Lead statement:
When I worked in mathematics this year I used resources provided by the school.
Comments from a selective number of participants:
Text book; Teacher and text books; My maths text book; Bought extra text books; Sigma mathematics; Old exam papers and text books; Test papers and library books; Work books and text books.

j) Online Ministry resources

Download exemplars from NZQA, TKI websites 17

Lead statement:
When I worked in mathematics this year I used online Ministry resources.
Comments from a selective number of participants:
It is very useful; Yes, I use online exemplars; Downloaded practice exams; Visited TKI for studying; NZQA website; NZQA website exemplars.

k) Confidence to succeed in examinations

Feel increase in confidence
Not confident/ worried 6
5

Lead statement:
When I worked in mathematics this year I feel increasing confidence in my capability to succeed in mathematics
Comments from a selective number of participants:
It's quite difficult; No I feel less confident than ever at this point in time; I feel I am getting better; Because I don't succeed; Or so I thought until I found out my marks, then life seemed pointless...; As the system is corrupt and obsessed with making the subject as boring as possible. God if you really want to help maths students, get rid of the subject completely as it is demoralising; When I got a good mark; I feel I know how to study well for a maths exam now; I got 1 E and 2 M in recent exams. I believe I can do better in external exams; I'll get there....

Table 3: Achievement results in mathematics in the mid-year exams

Grade Frequency (%)

Excellence 5

Merit 25

Achieved 37.5

Not achieved 30

Did not sit 2.5

n=40

Table 4: Correlation scores between achievement grades and use of learning strategies

Questions Correlation Approx.
significance

Practised mathematics to learn .296 .027

Used resources provided by the school .347 .009

Felt increase in confidence in capability to perform successfully in mathematics .341 .010

n = 40

Discussion

The aims of the present research were two-fold: (a) to assess students' and teachers' self-efficacy in learning strategies, and (b) to assess the relationship between students' self-efficacy in learning strategies and achievement in mathematics. An important point to consider in the discussion is the very low acceptance rate by schools invited to participate in phase two of the research which precluded a probability based random sample, and an adequate sample size. Further, the results of only a sub-sample of forty student participants could be considered in the inferential analysis. However, the combined summary of phase one and phase two of the research is an essential consideration in the discussion. This helps position the findings within the theory of self-efficacy which underpinned the study, and also enhances the validity of the study.

Figure 1: Cross tabulation: Practised mathematics with achievement

Figure 2: Cross tabulation: Used school resources with achievement

Figure 3: Cross tabulation: Increase in confidence in capability to succeed mathematics with achievement

Table 5: Teachers' response to use of learning strategies in the classroom context during the year

Questions Response
(Frequency count)

Yes No

Guided to work towards setting goals 8 2

Assisted to override negative thinking 9 1

Provided guidance on time management 9 1

Provided models on successful use of learning strategies 6 4

Provided opportunities for mastery learning 4 6

Guided students to revisit their goals 8 2

Provided in-class opportunities to use school and Ministry online resources 1 9

n= 10

Table 6: Qualitative teacher responses to use of learning strategies in the classroom context during the year

Items and Comments

a) Guided to work towards setting goals
Comments of teacher participants:
Encourage to use a study time-table when preparing for exams;
Students were given practice papers for revision;
Students were updated on the credits they have or not obtained and what they should now direct themselves to achieve;
What does this mean?

b) Assisted to override negative thinking
Comments of teacher participants:
I do this all the time;
Always mention about how mathematics is used in real life;
Positive posters all around, positive words like "You can do it" It's not all bad;
Explained the alternative methods in TRIG to overcome the application of Algebra in Year 11 mathematics alternative.

c) Provided guidance on time management
Comments of teacher participants:
Showed students how to revise;
Too busy teaching mathematics;
Devised a revision programme for students to achieve especially in Year 11;
Diary/time management skills.

d) Provided models on successful use of learning strategies
Comments of teacher participants:
Probably; Most learning styles are covered in our schemes;
What, like me? I model this on many occasions;
Time constraints;
Not aware of any models, used examples in text books to develop students thinking in order to solve problems.

e) Provided opportunities for mastery learning
Comments of teacher participants:
Going over questions not done well in class;
What does this mean? It is a long time since I was at training College;
Not common practice; Class-time one-on-one tutoring; Gave a number of repeated or similar examples of problems in order to achieve mastery of a concept in mathematics; Varied learning techniques used.

f) Guided students to revisit their goals
Comments of teacher participants:
Topic review at the end of topic for tests and exams; Students need to set their own goals where a majority of students did not realise the importance of this; Constantly updated students with their credits and asked them to evaluate where they were and how many credits they need etc. to complete their course; Termly reassessment.

g) Provided in-class opportunities to use school and Ministry online resources
Comments of teacher participants:
Big class, not enough computers in the lab; Too busy teaching; No computer times available; Computer rooms are always booked so it is not feasible as some other schools do; No IT resources or hardware available in the classroom to use online resources; Referred students to appropriate website which they could access from the school library or home.

Summary of phase I and phase II

Students are reporting self-efficacy for performing successfully in mathematics, and in use of learning strategies in mathematics.
The major factors emerging as independent variables which impact on achievement in mathematics include self-efficacy for self-regulated learning, resource management, use of cognitive and motivational strategies, practicing mathematics to learn, confidence to perform successfully in the forthcoming examination, and meeting others' expectations.
Teachers report self-efficacy in use of most learning strategies.
The reported level of students' self-efficacy is not reflected in achievement in mathematics.
The use of learning strategies is not a deliberate or planned process.

Use of learning strategies, implications for teaching-learning and achievement

Student participants, in Phase II of the study, report self-efficacy in self-regulatory learning, resource management, motivational strategies, and confidence to succeed in examinations (Table 1). However many of the students' comments to goal setting, reviewing goals, and monitoring progress reveal lack of use of these strategies for learning (Table 2). Students who report using goal setting as a strategy, set broad goals for "passing everything, getting high marks, to achieve well". Half of the students who chose to comment on monitoring progress stated that they did not monitor their progress and didn't care about the process, and a sizeable number of students did not review set goals. According to Zimmerman (2001) setting general goals, distal, absolute and non-hierarchical goals have proved to be causal factors for lack of motivation, and failure to achieve among learners. Lack of specificity in goals lead to lack of focus on what to do next, distal goals lead to delayed feedback, absolute goals lead to slow progress, and non-hierarchical goals lead to failure in discriminating between strategy processes from performance outcomes in a hierarchical manner which impact adversely on quality of performance. Setting proximal goals is very important, as are situational factors such as rewards and teacher feedback. Reviewing set goals was not a strategy in use by over 60% of participants in the present study. Kitstantas, Reiser, and Doster (2004), researching on goal settings, self-evaluation, and organisational skills in ninth and tenth graders, report that use of process goals led to higher self-efficacy, better performance, more performance satisfaction, and strategic attributions, as compared to outcome goals. Braten, Samuelston, and Stomso (2004) examined the mediating role of self-efficacy in performance goal orientated contexts. They state that students who used performance-avoidance goals and reported high self-efficacy maybe vulnerable to maladaptive patterns of learning. Here, self-efficacy acts as a buffer against negative self-image. Over 60% of student participants in the present study reported that they set goals when they worked in mathematics, however only 39% reviewed set goals.

Analyses of teachers' reported use of learning strategies and comments reveal that while teachers use the strategies, it is not a deliberate and planned process (Tables 5 & 6). Eight out of ten teachers surveyed reported providing guidance in goal setting, however it emerges that the guidance were not specifically tailored for the topic or task. Comments included "encouraging use of a study-timetable, students given practice papers for revision". Four teachers reported providing guidance on mastery learning, and six teachers stated using modelling as a learning strategy. Comments on the mastery learning process revealed that it was usually teacher modelled problem solving, or going over questions in which students had not performed well. Cognitive modelling based on mastery learning has been recommended as an effective learning strategy in mathematics. Schunk (2001) provides a summary of research interventions in the use of learning strategies and self-efficacy spanning over two decades which aimed at enhancing student achievement in mathematics. In a study targeting students who scored below grade-level in mathematics, cognitive modelling produced enhancement in achievement. Elementary students with difficulty in subtraction, when provided exposure to mastery or coping modelling, performed well. In another study using multiple modelling strategies, in the initial stages, students observing coping-emotive models reported highest learning self-efficacy. With instructions and practice, all groups reported similar levels of self-efficacy and achievement. Modelling, followed by monitoring and training in self-monitoring produced better performance, as compared to modelling without monitoring. "Mastery learning enables 75 to 90 percent of the students to achieve the same high level as the top 25 percent learning under typical group-based instructional methods" (Block, 1971, p.3).In Phase I of this research project 42.7 percent of students did not achieve successfully, and 11 percent of students did not attempt the examinations, in Phase II 30% did not achieve, and 2.5% did not attempt the examinations.

Time and lack of access to resources such as computers and the Internet within the classroom were factors highlighted by teachers as impeding use of some of the learning strategies. While only one out of ten teachers reported providing in class guidance to students in use of resources, 72% of students reported efficacy in using online resources and 82% of students reported efficacy in using school resources. Efficacy in practising mathematics to learn was reported by 79% of student participants and 62% reported increase in confidence to perform successfully in mathematics. Inferential analysis reveals positive significant correlations between efficacy in use of three strategies, viz: practising mathematics to learn, using school resources, increase in confidence to perform successfully in examination and level of achievement (Table 4). Some of the participants who were in the achieved and the merit categories also reported lack of efficacy in use of learning strategies (Figures 1, 2, &3). While assessing the relationship between efficacy and achievement, it is important to consider the learning and performance self-efficacy distinction (Bandura, 2003). A high level of learning self-efficacy facilitates acquisition of skills, self-doubts about performance in challenging situations is conducive to achievement. A number of student participants who did not achieve also reported efficacy in use of learning strategies, and a sizeable number of students (30%) did not achieve in the examinations (Table 3).

Bandura (1997) and Schunk (1994) attribute discrepancies between reported self-efficacy and achievement to the source of self-efficacy information, value of the task undertaken, and the presence or absence of skills required to accomplish the task. While efficacy can enhance motivation, students cannot produce 'new fangled performances' if the subskills for the exercise of personal agency are absent (Bandura, 1997). In a similar vein, Schunk (1994) states that high self-efficacy will not produce competent performances in the absence of requisite knowledge and skills. The explanation by Cramer (1998) on positive illusions is significant in this context. Reporting "positive illusions" about the self and giving responses which might be socially desirable are indicative of denial. These are coping strategies. Reviewing studies on the use of overly positive self-evaluations, and self-deceptive coping in college students, Cramer states that when the outcome measure is an external criterion and not self-reports, use of coping mechanisms such as these do not result in positive benefits.

It is arguable that students in the present study who report self-efficacy in use of learning strategies which were not reflected in achievement could be reporting illusory self-efficacy, rather than true self-efficacy. The nature of individual self-belief, if not based on self-awareness (Goleman, 1998) of true capability to undertake action, could be illusory, functioning only as a coping strategy or as a defense mechanism, leading to inflated levels of self-efficacy. Goleman defines self-awareness as "having a realistic assessment of our own abilities and a well-grounded sense of self-confidence" (p.318). According to Bandura (1991) "Perceived self-efficacy is concerned with people's beliefs about their capabilities to organise and execute courses of action" (p.158). The concerns voiced by Donnolley (2002), and Hames (2002) about the New Zealand educational system are significant in this context. Flexibility in the curriculum, while not providing assistance for teachers to effectively plan instructional strategies, and the stress on self-esteem by acceptance of low achievement levels appear to cost students the required knowledge and skills. Hames (2002), queries whether the emphasis on self-esteem, is not impacted adversely by poor performance, and might perhaps be fuelling egotism in students. The attribution of causes for failure, the role of culture, and perceptions of positioning the self in the wider socio-cultural context are other significant variables in this interplay between self-efficacy and achievement.

A way forward

The findings from phase one and phase two of this research, and the recommendations by ERO (2002) suggest that there is an imminent need to integrate learning strategies in the secondary classroom context. Introduction of statements of intent on learning strategies to be used by students are worth considering. Within the self-efficacy theoretical framework, these strategies have to provide coverage for each of the curriculum strands in terms of the achievement aims and achievement objectives. For instance, learning strategies for goal setting which are an important aspect of motivation, self-efficacy and achievement should accord for clarity and specificity in stating the goals, setting proximal, distal and challenging goals, review of goals within the achievement aims and achievement objectives specific to each strand. The use of motivational principles by teachers in the classroom could include, among others: clarity in expectations, realistic expectations, comparing students' achievement to their past achievements, focus on cooperation rather than competition, and provision of immediate, task-specific feedback (Margolis & McCabe, 2004). Teachers need to assist students to develop internal standards in self-evaluation of efforts (Pajares, 2002). "The teacher's challenge is to ensure that their students' internal standards are rigorous without being debilitating, realistic without being self-limiting, fluid without being wishy-washy, consistent without being static" (p.121). Highlighting the need for teachers to focus on students' self-efficacy beliefs and self-regulatory strategies in professional practice, Pajares states "Teachers have the responsibility to increase their students' competence and confidence as students progress through school. Educational programs that seek to empower students must not only cultivate the knowledge to succeed but must endeavour to maximise the potential for success by nurturing the belief that one can indeed succeed and the self-regulatory strategies required to help bring about that success" (2002, p.121).

Strategies to promote mastery learning were not integral to the teaching-learning process as reported by teacher participants in the present research. Incorporating a cognitive-mediational approach to learning in the classroom requires teachers to have knowledge of instructional strategies that facilitate focus on students' thinking, understanding and self-regulated learning (Putnam & Borko, 1997). The authors suggest that teachers consider the concept of distributed cognition, the understanding that cognition is not solely an individual- oriented construct, but is distributed over the individual, others, and the environment, with all activities in the classroom having to be considered in the out-of-school context where activities are collaborative, and require resources which are beyond the capacity of individual members, but distributed among many individuals and requires a collectivistic orientation to be accomplished successfully. In the bicultural and multicultural New Zealand educational context, incorporating the concept of Akonga (Bishop et al, 2003) could be considered a parallel to this concept of distributed cognition. Finally, it is important to consider the perspective that fundamental changes in teacher beliefs and practices about students (and student learning and performance) can take place slowly, and requires considerable support mechanisms to guide teachers to the linkages between particular beliefs and practices in specific teaching learning contexts (Calderhead, 1996). "Teachers are expected to function in ways that acknowledge, and are in accordance with, the cultural practices, circumstances, and beliefs of their students. Their role as teachers in multicultural settings requires them to act as personal agents, proxy cultural agents, and collective cultural agents" (Gibbs, 2005, p.104). Teacher professional development practices in New Zealand should consider the role of teacher efficacy, agency, and use of learning strategies to facilitate positive outcomes in learning and performance for students and teachers.

Concluding comments: A prologue for future consideration

Reform-led efforts emphasise providing New Zealanders with strong foundations for future learning, and high levels of achievements by all school leavers (Minister of Education, 2003). The statement of intent 2005-2010 of the Ministry of Education (Ministry of Education, 2006) aims to raise student achievement and reduce disparity through effective teaching, strengthening family and community engagement to nurture and encourage learning, and providing quality providers. Whilst this focus on active engagement of all stakeholders in teaching-learning and achievement is noteworthy, in the New Zealand educational context research on the role of individual-level variables which impact on student achievement have focused among others, on assessment of students' self-concept. These studies have informed practice and initiatives aimed at enhancing student achievement. This research demonstrates that it is crucial to supplement measures of self-concept with assessment of students' and teachers' efficacy which inform curricular praxis, and incorporate learning strategies across all educational sectors.

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Appendix I

Appendix II

Author: Deepa Marat works as a Research Consultant at the Postgraduate Centre Matai Kahurangi Unitec Institute of Technology in Auckland. Her research interests include self-efficacy, self-related constructs, and agency. She is at present engaged in two collaborative research projects with a focus on student engagement and success in the tertiary sector. Underpinning each of these projects is a methodological framework led by indigenous research methodologies from Australasia. Email: dmarat@unitec.ac.nz

Please cite as: Marat, D. (2007). Students' and teachers' efficacy in use of learning strategies and achievement in mathematics. Issues In Educational Research, 17(2), 207-231. http://www.iier.org.au/iier17/marat.html

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Questions	Response (%)
Questions	Yes	No
Deliberately set goals	63	37
Deliberately made time-plan	26	74
Deliberately took control over any negative thinking	51	48
Deliberately praised oneself when successful	62	38
Felt motivated when others succeeded	66	34
Practised mathematics to learn	79	21
Regulated the environment to facilitate learning	51	49
Monitored progress	71	29
Reviewed set goals	39	60
Sought guidance	65	32
Used resources provided by the school	82	17
Used online resources	72	27
Felt increase in confidence in capability to perform successfully in mathematics	62	36
n= 92 Missing data ranged from 1.1% to 3.3%

Emerging themes	Response (Frequency count)
a) Goal setting
Goals set to achieve better grades Day-to-day goals set No goals	13 5 6
Lead Statement: When I worked in mathematics this year I deliberately set goals. Comments from a selective number of participants: To pass everything - every internal; I really wanted to pass my internals, especially the first one so I made sure I worked on it; I set goals but they change constantly and only kept in my head; Set goals to get high marks in assessment; A goal to achieve well in maths; Goals are mostly short-term; I wanted to achieve excellence and merit; I don't set goals.
b) Time plan
No plan	10
Lead statement: When I worked in mathematics this year I deliberately made a time plan Comments from a selective number of participants: I do not plan I do; I didn't really set a certain timetable of study; I don't have a plan; I try, but don't follow it; Do not make time-plan; I made a rough plan but did not necessarily keep to it; Because I can do no more; Too restrictive; No I did not place any time deliberately aside for maths; I just did the work when I had to do it until it was done.
c) Deliberately praised oneself when successful
No	7
Lead statement: When I worked in mathematics this year I deliberately praised myself when successful. Comments from a selective number of participants: Don't want to get big-headed; No this leads to overconfidence; Do not praise myself; It's something I should do; Because if I did I would get overconfident; I am not like that
d) Felt motivated on seeing others succeed
Wanted to succeed like them	9
Lead statement: When I worked in mathematics this year I felt motivated when I saw others like myself succeed. Comments from a selective number of participants: I felt that if I can do it so can I and worked towards it; Made me realise I could achieve too; I wanted to achieve just as well others did.
e) Practised mathematics to learn
To do well in examinations	18
Lead statement: When I worked in mathematics this year I made myself practice so that I could learn. Comments from a selective number of participants: It's the only way to improve in mathematics; Before exams I studied; I bought study guides and got help from my dad; Every time I have time I would practice and revise my work; Did homework; I usually concentrate my effort on solving merit and excellence questions; Going over and over problems to ensure I knew the material (yet I failed damn it!!!)
f) Monitoring progress
Checking grades No, don't care	8 8
Lead statement: When I worked in mathematics this year I made sure I was aware of my progress. Comments from a selective number of participants: Mark down what grades I have got so far; I also like to see how I can do in relation to the class; Keep tabs on my achievement; I don't really keep up-to-date on where I am; No I don't; Was not aware of my progress.
g) Reviewed goals
Did not review, no goals set Went over results, keeping in mind what was done	11 4
Lead statement: When I worked in mathematics this year I reviewed my mathematics goals after I had set them. Comments from a selective number of participants: I didn't set any goals; Don't have goals; Have not changed them; No I never review; To see what I have to work on; I always keep my goals in mind; Day to day goals so they don't really need reviewing rather I just tried to complete the goal; I never had any as from starting the subject I was bored stiff.
h) Sought guidance / help
From text books Maths tutor Maths teacher	3 4 6
Lead statement: When I worked in mathematics this year I sought guidance and help. Comments from a selective number of participants: Revision books; Maths tutoring both in school and out of school; From parents and my tutor; Get all the help I need from teacher and friends; From my dad; Asked the teacher questions and used text books; I sought help from teachers but haven't really received it; As no one ever explains it properly it may well be Arabic.
i) Resources provided by the school
Text books	10
Lead statement: When I worked in mathematics this year I used resources provided by the school. Comments from a selective number of participants: Text book; Teacher and text books; My maths text book; Bought extra text books; Sigma mathematics; Old exam papers and text books; Test papers and library books; Work books and text books.
j) Online Ministry resources
Download exemplars from NZQA, TKI websites	17
Lead statement: When I worked in mathematics this year I used online Ministry resources. Comments from a selective number of participants: It is very useful; Yes, I use online exemplars; Downloaded practice exams; Visited TKI for studying; NZQA website; NZQA website exemplars.
k) Confidence to succeed in examinations
Feel increase in confidence Not confident/ worried	6 5
Lead statement: When I worked in mathematics this year I feel increasing confidence in my capability to succeed in mathematics Comments from a selective number of participants: It's quite difficult; No I feel less confident than ever at this point in time; I feel I am getting better; Because I don't succeed; Or so I thought until I found out my marks, then life seemed pointless...; As the system is corrupt and obsessed with making the subject as boring as possible. God if you really want to help maths students, get rid of the subject completely as it is demoralising; When I got a good mark; I feel I know how to study well for a maths exam now; I got 1 E and 2 M in recent exams. I believe I can do better in external exams; I'll get there....

Grade	Frequency (%)
Excellence	5
Merit	25
Achieved	37.5
Not achieved	30
Did not sit	2.5
n=40

Questions	Correlation	Approx. significance
Practised mathematics to learn	.296	.027
Used resources provided by the school	.347	.009
Felt increase in confidence in capability to perform successfully in mathematics	.341	.010
n = 40

Items and Comments
a)	Guided to work towards setting goals Comments of teacher participants: Encourage to use a study time-table when preparing for exams; Students were given practice papers for revision; Students were updated on the credits they have or not obtained and what they should now direct themselves to achieve; What does this mean?
b)	Assisted to override negative thinking Comments of teacher participants: I do this all the time; Always mention about how mathematics is used in real life; Positive posters all around, positive words like "You can do it" It's not all bad; Explained the alternative methods in TRIG to overcome the application of Algebra in Year 11 mathematics alternative.
c)	Provided guidance on time management Comments of teacher participants: Showed students how to revise; Too busy teaching mathematics; Devised a revision programme for students to achieve especially in Year 11; Diary/time management skills.
d)	Provided models on successful use of learning strategies Comments of teacher participants: Probably; Most learning styles are covered in our schemes; What, like me? I model this on many occasions; Time constraints; Not aware of any models, used examples in text books to develop students thinking in order to solve problems.
e)	Provided opportunities for mastery learning Comments of teacher participants: Going over questions not done well in class; What does this mean? It is a long time since I was at training College; Not common practice; Class-time one-on-one tutoring; Gave a number of repeated or similar examples of problems in order to achieve mastery of a concept in mathematics; Varied learning techniques used.
f)	Guided students to revisit their goals Comments of teacher participants: Topic review at the end of topic for tests and exams; Students need to set their own goals where a majority of students did not realise the importance of this; Constantly updated students with their credits and asked them to evaluate where they were and how many credits they need etc. to complete their course; Termly reassessment.
g)	Provided in-class opportunities to use school and Ministry online resources Comments of teacher participants: Big class, not enough computers in the lab; Too busy teaching; No computer times available; Computer rooms are always booked so it is not feasible as some other schools do; No IT resources or hardware available in the classroom to use online resources; Referred students to appropriate website which they could access from the school library or home.