The issue of professional competences is closely linked to educational advances and development, and from this perspective the value given to the teacher's competence is an asset necessary in the reformulation of the teacher's training and, more particularly, of the teacher's performance. Some research about the teacher's classroom performance has been conducted (Rico, 1990; Ball, 1988; Llinares and Sanchez, 1990). Other studies focus on the different teaching approaches given to the teaching of mathematics: student-centred, content-based related to comprehension, content-based focused on routine actions or classroom-based (Kuhs and Ball, 1986).
Llinares (1988) describes certain features in the learning and changing processes that the mathematics teacher goes through, trying to balance the difficulties between theory and practice. From a 'professional perspective', there is knowledge that originates from professional performanceand from the knowledge that supports and justifies decisions and actions in the work context of mathematics teaching. However, the teacher's training and the teacher's performance are two different issues. Several research studies relate these two issues. Eisenhart et al. (1993) identify the teacher trainees' difficulties in teaching in actual practice and analyse the way in which they contextualise the content learned at the university. Other researchers describe difficulty variables found in the students while attempting to introduce problem solving, among these are the meaning of the concepts involved, the regulation of their beliefs and the distinction between problem and exercise.
Thus, researchers talk about subordinate competences in problem solving or about a hierarchy of competences regarding contents (cultural and practical aspects plus those connected with the training), mathematical methods (modelling principles used), didactic procedures (strategies developed), and psychological procedures (level 2 of conscious operation) (Cazzaro, Noel, Pourbaix and Tilleuil, 2001). Blanco (1991) states that faced with difficulties in their perceptions and interpretations of information, students give more emphasis to algorithmic than to conceptual processes. This suggests a difference between the 'professional knowledge' of the teacher of mathematics and the analysis of the epistemological nature of such knowledge. Professional knowledge, according to Bromme and Tillema (1996), is knowledge oriented towards professional activity. Such knowledge involves not only specific training in data and problem solving methods, but also information necessary to define and understand the problems faced by the professional.
The emergence of Reforms in the last decade has made the teacher a main concern for trainers and researchers in Mathematics Education, making the issue of the competences needed by such teacher to adapt to and face those reform processes increasingly popular in international congresses. The current Chilean Reform is aimed at improving teaching quality in education as a whole. In the context of school teaching, the reform has made considerable effort to implement educational practices in a way that can secure the development of feasible, long lasting transformations that enrich and renew teaching practices. This led us to develop an experimental assessment proposal of the competences of the mathematics teacher in Chile. The focus of the assessment was the setting of criteria that consider the aims of the Reform, its contents and conceptions, the teaching methods and assessment procedures employed by the teacher of mathematics.
Competences, Frameworks and Quality
For the purpose of this research, the competences of the mathematics teacher are defined as the skills effectively and efficiently acquired when teaching mathematics. Competences must necessarily be associated to quality, since the aim is to teach but to teach well. The assessment proposed here includes specific and general competences, competence context frameworks, and qualitative dimensions (Poblete, Diaz, 2001).
We define general competences as:
Ability to innovate, inquire and create during the mathematics teaching and learning process.
Capacity to encourage a favorable atmosphere for the process of mathematics learning
Capacity to face socio-cultural diversity during the process of mathematics teaching
Team work capacity in the professional work of the teacher
Capacity to self-criticise their role as a trainer and as teacher of mathematics
Skill to apply mathematics knowledge
Capacity to adapt, update and project as a teacher of mathematics
Capacity to foster and encourage ethical development in the student
Regarding specialized competences, we see these as:
Skill to plan didactic activities in mathematics
Capacity to face curriculum, methodological and technological demands
Skill in using varied teaching strategies
Capacity to understand, identify and apply mathematics learning theories
We have related these competences to context frameworks of the mathematics teacher regarding knowledge of and about mathematics content; didactic know-how of and about teaching, of and about the teaching and learning process, and of and about assessment. We have also related these to the capacity of knowing about and being transversal in terms of values, and about knowing how to be evolutionary regarding adaptability (Poblete, Diaz, 2001).
The intersection of frameworks and the way they connect and represent each other allow the teacher of mathematics to perform educational actions in which s/he can demonstrate that competence. What is interesting in this interaction is that the actions performed by the teacher consider a quality conception (a conception of a subjective nature depending on how we define and accept quality). Thus, we have outlined specific features typical of the mathematics teacher that relate this quality conception to varied dimensions, among them:
Relevance: the educational aims that the teacher of mathematics wants to achieve
Efficiency: the optimisation in the use of educational resources that the teacher of mathematics makes to help the learners
Effectivity: concordance between the mathematics teacher's plans and the achievements obtained in the context in which s/he teaches.
Efficacy: congruence between the educational results achieved by the teacher of mathematics and the selection, distribution and organisation of resources.
Processes: relationship between the performance of the mathematics teacher and the results achieved.
While many studies of effective teaching highlight the context specific nature of effective teaching. This study showed that teachers make use of situated possibilities afforded by the specific context in the best interests of the students. This study found that the development of professional competence is characterized by constant engagement in reflection and in responding to challenges, thereby engaging in the kind of learning that extends one's competence. This study contributes to the body of research literature on teacher cases, research on secondary teacher's subject matter knowledge (Cooney, Shealy, & Arvold, 1998; Cooney & Wilson, 1995), and the influence of teachers' conceptions on instruction (e.g., Ball, 1993; Eisenhart et al., 1993; Raymond, 1997; Schifter & Simon, 1992; Shulman, 1987; Thompson, 1992) and on students' dispositions in mathematics. The findings of this study provide additional support to the finding from the Third International Mathematics and Science Study involving teaching at the eighth grade: 'teachers' beliefs about mathematics learning and instruction were to some extent related to their preparation' (Mullis et al., 2000, p. 191). The teachers from the Indian setting demonstrated multicultural competence. The findings from the Indian setting suggest that effective mathematics teaching demands mathematical enculturation of students by developing an understanding of the diverse cultural and academic backgrounds of the students. Cultural traditions prevalent in the teachers' work context interact with teachers' pedagogical and mathematical conceptions, and with the instructional context to influence student outcomes. Many researchers (Banks, 2002; Ladson-Billings, 2001) recommend teacher multicultural education in the wake of changing ethnic composition of the United States.
'Teacher-education research lacks a common theoretical basis, which prevents a convincing development of instruments and makes it difficult to connect studies to each other' (Bl??meke, Felbrich, M??ller, Kaiser & Lehmann, 2008). Since then, not only the research on prospective but also the research on practicing mathematics teachers' knowledge has continued to develop. Two research groups were particularly productive by assessing teacher knowledge with direct measures: one from Michigan State University in the context of 'Mathematics Teaching in the 21st Century (MT21; see e.g. Schmidt, Bl??meke & Tatto, 2011)' and the 'Teacher Education and Development Study: Learning to Teach Mathematics (TEDS-M; see e.g. Tatto, Schwille, Senk, Rodriguez, Bankov, & Reckase, in press; Bl??meke, Kaiser & Lehmann, 2010)', the second one from the University of Michigan in the context of 'Learning mathematics for teaching' (LMT; see e.g. Delaney, Ball, Hill, Schilling & Zopf, 2008; Hill, Ball & Schilling, 2008)'. This pioneering work has paved the way for the present special issue.
The results of comparative studies also provide benchmarks of what level and quality of teacher knowledge can be achieved and they point at country-specific strengths and weaknesses. Efforts to fill existing research gaps have been made since the late 1990s. Several comparative small-scale studies on mathematics teachers and mathematics teacher training are available (e.g., An, Kulm and Wu, 2004; Ma, 1999; Burghes, 2008). Much of the teacher research, however, neglected the content domain, focused on other subdomains of mathematics teachers' competencies like beliefs (Bramald, Hardman, & Leat, 1995; Calderhead, 1996) or intended to capture knowledge by self reports. Studies including direct measures of teacher knowledge and cross-country studies are still needed (Brouwer, 2010; Wilson, Floden, & Ferrini-Mundy, 2001).
There are numerous factors that influence achievement in mathematics. Among other factors pupil-teacher relationship and school disciplinary climate (Shin, Lee & Kim, 2011), teacher competence and classroom atmosphere (Lamb, 2001) and assessment methods (Ellerton & Clements, 2008) were found to influence academic achievement. Affective factors such as pupils' values, beliefs, attitudes and emotions are said to play significant roles in learner achievement in mathematics (Grootenboer & Hemmings, 2007). One factor that is highly associated with pupil achievement is the pupils' attitudes towards mathematics. For instance, in Zimbabwe during summative assessment results analysis in teacher development program, mathematics teachers are largely blamed for not fostering positive attitudes towards the subject during instruction (Mandebu, 1996). This blame prompted the present study in order to identify the types of attitudes that influence academic achievement in 'O' Level mathematics in order to provide insight to similar discussions as those in teacher development sessions.
Performance in mathematics attracts attention from all walks of life creating constant search for ways for improving mathematics education (Grevholm, 2000: Julie 2004: van der Sandt & Nieuwoudt, 2005). Learner achievements are receiving great attention because mathematics is a critical filter for school leavers' employment opportunities and full participation in society (Brumbaugh & Rock, 2001). Thus the utility of the subject is high but one wonders whether pupils studying 'O' Level realize the importance of mathematics in their future lives. Hence an analysis of pupils' attitudes towards mathematics may shed light into the dispositions that pupils have towards mathematics.
A review of school'based educational research has revealed that the majority of secondary school pupils find mathematics as the most difficult, abstract, deadly and boring subject (Amirali, 2010). Other research studies have shown that students in primary school enjoy mathematics but when they move to secondary school their interest towards the subject declines, (Larzim, Abu & Wan 2003; Chambers, 1998). Some societal views about mathematics such as mathematics problems have one and only one answer and can be solved in a particular way, mathematics is a solitary activity, done by individuals in isolation, mathematics requires
Study of the past researches shows that teachers competencies plays an important role over the performance of students in mathematics. Professional competency develops confidence among the teachers. Also teachers personality traits, attitudes and beliefs about teaching of mathematics at secondary school level mean a lot in teaching and learning process of mathematics.
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