3. Enterprise Architecture
3.1. La bases teóricas para una Enterprise Architecture
According to Pellerey (1985, p.3246), “Mathematics instruction has been a main item of any educational curriculum since the most ancient times.” From a historical point of
38 view, the origin of Mathematics could be traced back to ancient Egypt (Burton, 1991; Pellerey, 1985). This means that Mathematics as a discipline or its instruction is very ancient and as such it has gone through a lot of transformation. Pellerey (1985, p.3247) further mentions that the teaching of Mathematics at Secondary schools, around the 19th century was characterized by the rhythm of explanation, study, exercise and interrogation. This situation persisted until the 1960s and 1970s when the New Mathematics movement gained momentum (Pellerey, 1985). However, as Pellerey (1985, p.3248) states: “The support given to the renewal of the teaching of Mathematics (and science), sometimes in an incoherent way, by the psychology of Piaget, lost force, due to new studies and research in the psychology of Mathematics.”
A report from a survey prepared by the Schools Council (1977) in London on Mixed- ability Teaching in Mathematics states that the aims and objectives of Mathematics teaching have changed over the period of time due to the changing needs of the society (p.19). The Schools Council (1977, p.20) also proposes what they call characteristics of good Mathematics teaching as follows:
• Quality: The mathematical content given and variety of tasks should be of great quality in order to ensure that concepts or relationships are formed with strategies in developing mathematical activities. The tasks should be appropriate for the learners in terms of level of difficulty, interest and relevance.
• Continuity: In order to ensure continuity teachers should be aware of the structure of the Mathematics course and of the progress of individual learners.
• Autonomy: In order to develop learners’ autonomy in class, the teacher should try to encourage learners to organize their own materials, assessment or tasks. Learners’ independence can also be encouraged by using a variety of teaching
39 approaches where learner autonomy is enhanced. For example, providing a workshop scheme with a choice of tasks for learners to pursue their own interests. • Discussion: Teacher-pupil as well as pupil-pupil discussions should be used as
powerful agents in the classroom in order to promote good teaching and learning. Several authors (Ashlock, Johnson, Wilson and Jones, 1983; Goulding, 1997; Nickson, 2000 and Selinger, 1994) highlight the importance of social, affective and cognitive domains in the teaching and learning of Mathematics. For example, Goulding (1997, p.144) states: “If learning is influenced by social, affective, and cognitive dimensions then teachers clearly have to attend to all these factors in the classroom in creating learning opportunities for pupils.” The teaching and learning of school Mathematics, in particular, is complex because it is influenced by several factors. For example, Sanders (1994, p.29) mentions that teaching of Mathematics can take a variety of forms because of the different views about how to learn Mathematics, as well as the diverse nature within Mathematics education circles. The integration of teaching and learning Mathematics, therefore, becomes an important feature in this discussion. Jaworski’s model (1994) of investigative Mathematics teaching (see Fig. 2.2) referred to as a ‘Teaching Triad’ gives a good explanation of the relationship between teaching and learning Mathematics. The model was designed after a lengthy observation in Mathematics classrooms “as a device to aid characterization of an investigative approach to Mathematics teaching” (Jaworski, 1994, p.183), with teachers for whom:
40 Fig 2.2 The Teaching Triad [from Jaworski (1994, p.107)]
• Management of learning is manifested in a set of teaching strategies and beliefs about teaching which influence the prevailing classroom atmosphere and the way in which lessons are conducted.
• Sensitivity to students is inherent in the teacher-student relationship and the teacher’s knowledge of individual students and influence ways in which the teacher interacts with, and challenges, students.
• Mathematical challenge arises from teachers’ own epistemological standpoints and the way in which they offer Mathematics to their students depending on students’ individual needs and level of progress.
Jaworski (1994, p.107) does not provide a detailed account of the inter-linkage and/or inter-relationships of the three domains but her description of the characteristics of observed Mathematics classrooms sum up the following features:
• Type of tasks which teachers set for students to work on; • Introduction of a task by the teacher to the students; • Emphasis on mathematical thinking processes; • Organization of classroom -groups – and discussions; • Use of apparatus or equipments: practical work; • Mode of operation of teacher;
41 • Student activity and behaviour and
• Teacher evaluation of learning, feedback for planning (p.171).
Although Jaworski (1994a) supports constructivist teaching in Mathematics, she further claims that the social dimension contributes significantly to individual student’s construction of meaning (p.218). The social aspect in Mathematics teaching is one of the major components of this study in terms of social constructivism and Learner-Centred Education. Moreover, Jaworski (1994a, p.218) states that constructivism (social) is a philosophy that underpins much of what is regarded as good practice in Mathematics teaching and learning. However, the diverse and obscure nature of Mathematics (Goulding, 1997) could influence ways in which both teachers and educators approach the teaching of Mathematics. According to Jarworski (1994) and Ernest (1989), Mathematics as a subject allows learners the opportunity to construct their own knowledge and understanding. However, “Teaching Mathematics is difficult, particularly if it is based on a constructivist perspective” (Jaworski, 1994a, p.230).
Summary
This chapter describes the historical background of Learner-Centred Education. It describes the implementation of curriculum as discussed by Fullan (1991), Orstein and Hunkins (1993) and others. Review of related literature addressing curriculum implementation in other countries as well as in Namibia specifically is given. The chapter also addresses curriculum reform and policies globally, then nationally. The last section discusses the approaches to Mathematics teaching and learning in relation to the LCE concept. In summary, a theoretical conclusion that social constructivist theory supports
42 and forms the basic foundation of the concept Learner-Centred approach is drawn from the literature review. The next chapter addresses the methodology and methods used to collect and analyse the data.
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