Improvement of mathematics learning is basically linked to development in teaching, and that teaching develops through a learning process that involves teachers and the ones who receive knowledge. Teachers and educators can make use of inquiry as a tool to promote the involvement critically with key questions and issues in question. This method involves addressing mathematical tasks in classrooms, developing approaches to mathematics teaching or creating methods of working with teachers for promoting mathematical teaching (Jaworski, 2004)

Teaching Mathematics as Learning in PracticeIt has been noticed that mathematics education has addressed philosophical and epistemological perspectives with respect to mathematics and to mathematics learning over the past few decades. Theoretical considerations that involve knowing nature of mathematical knowledge, what it means to know mathematics and to come to know it, how mathematics knowledge is related to knowing in social settings more widely, has been a topic of consideration and debate .

It has been seen that constructivism and sociocultural theories have been highly influential in addressing mathematical knowledge and its learning. But unfortunately the position of mathematics research, teaching and learning has always been seriously anomalous and underdeveloped.

Theories lead to analysis, explanation, but they fail to provide the plan for action; rarely do they provide direct guidance for practice. We can promote mathematics learning from theoretical perspectives. Research has shown that the sociocultural setup of the society in which mathematical learning and teaching takes place are too complex. It is always reasonable that teaching mathematics should drive us to the better gain of knowledge of mathematical learning, but to theorise teaching is a big issue that most teachers seem to be dealing with. (Cooney, 1994)

The process of growth in knowledge especially in the field of mathematics continues throughout practice. E. Wenger also conceptualises learning as developing identity (as a teacher, for example) through participation in practice and learning is a “process of becoming”. To learn the science of Mathematics one must engage themselves in the ideas through involvement in communicative practice, polish those ideas through exercising his/ her thinking and align it to the proper thoughts. Alignment of imagination in a community of practice leads to the individual members aligning themselves with characteristics of the practice. With the exercise of broader thinking during engagement, it can be a vital process in which one questions the reason and consequences of aligning with norms of normal practice.

If the community of practice is the school, it is seen as a process of becoming a teacher within this community, hence, looking at any school community, one would identify teachers as aligning themselves critically with respect to a broad and rich picture of the world.

On analysing the work of Brown and McIntyre’s in social practice theory, we will find that the process of legitimate peripheral participation, for any teacher being drawn into this community, might be seen as one of learning to develop the normal desirable state and maintain it for the well-being of all those involved as ‘normally’ perceived in the school setting. Thus participation here looks more like a perpetuation of the practice – an alignment that lacks a critical dimension – although I recognize that we might debate the meaning of these terms.

A main reason of mathematics method courses is to help pre-service teachers develop the means to facilitate the knowledge of mathematics. These method courses differ from each other in institutions in structure and philosophy; however, mathematics teacher education programs tend to emphasise the development of both mathematics content knowledge and pedagogical knowledge. (Shulman,1986)

The research in the field of – Learning Communities in Mathematics is structured to develop inquiry communities amongst teachers and didacticians to promote inquiry at all levels. It puts into practice, and simultaneously the research, the ideas expressed in this article. Analysis of data of the didacticians, focuses on tensions, and brings to light the considerable learning opportunities in creating inquiry communities with teachers.

Analysis of mathematical thinking in schools:

If we analyse the mathematical study culture in school, ‘How do we know whether a student in class actually understands what’s taught in mathematics class? This cognitive state if could be judged there would have been ‘understanding’ of mathematics within the classroom.

The ‘understanding’ of mathematics can be judged by observing behaviour of particular kinds – written, verbal, diagrammatic or symbolic forms of maths..

It can be said through the result of studies that a competent participant in the school mathematics who understands will display the behaviour that will lead to recognition of that understanding.

This leads to many further questions about what it means to be a‘competent participant’, what might determine ‘appropriate circumstances’.

In assessing maths education the main factors needs to be considered-

One is categorized as ‘psychological’ that stresses on the design of tests to be used as research, diagnosis or evaluation instruments to measure or describe student’s achievement or understanding within the mathematical domain of interest.

No matter how mainstream research has become but a ‘curriculum reform’ focus, analysing traditional forms of assessment from the point of view of validity and often proposing new forms of assessment more aligned with curriculum objectives.

Both these factors aim to focus at what students produce and to attempt to devise instruments that will cause them to produce fair representations of their knowledge.

It should be taken care that in classroom education – assessment practices and its consequences should become more transparent that allows opportunities for teachers and students to resist them and take control. We should look at the ways in which ‘being mathematical’ is constituted in the school education by various evaluations made by teachers of their students or any contestatations.

STATUS OF MATHEMATICS EDUCATION AS KNOWLEDGE FIELD

We need several answers to critical points mentioned below–

Basis of mathematics education in the field of knowledge

Mathematics education a discipline, a field of enquiry or interdisciplinary area

Relationship with other disciplines such as philosophy, sociology, psychology or linguistics

Research methods and methodologies are employed and what is their philosophical basis and status

Mathematics education research community judge knowledge claims

The role and function of the researcher in mathematics education

The status of theories in mathematics education

Impact of modern developments in philosophy (post-structuralism, post-modernism)

These abovementioned factors are most important for the philosophy of mathematics education which needs to considered and explored.

Controversies in the Philosophy of Mathematics Education

The philosophy of mathematics education by certain set of people can be considered to be a dry and overly academic domain. Other issues could be as follows:

1. Mathematical Philosophy:

Mathematics is one of the oldest sciences and it is also a field of certain and cumulative knowledge, mathematics and its philosophy seems an unlikely area for controversy. Recently issues are emerging about philosophical views of science and mathematics. Foundationalists and absolutists are in support of the view that mathematics is certain, cumulative and untouched by social interests or developments beyond the normal patterns of historical growth whereas Fallibilists, humanists, relativists and social constructivists, have been arguing that mathematics is through and through historical and social, and that there are cultural limitations to its claims of certainty, universality and absoluteness.

2. Aims of Mathematics learning:

Five aims of Mathematical education identified are as follows:

Industrial Trainer aims – ‘back-to-basics’: numeracy and social training in obedience

Technological Pragmatist aims – useful mathematics to the appropriate level and knowledge and skill certification,

Old Humanist aims – transmission of the body of mathematical knowledge

Progressive Educator aims – creativity, self- realisation through mathematics

Public Educator aims – critical awareness and democratic citizenship via mathematics

These aims are best understood as part of an overall ideological framework that includes views of knowledge, values, society, human nature as well as education.

3. Theories of Learning Mathematics:

The Constructivist theories of learning are considered very old, a number of distinguished speakers in the 80s tried radical constructivism, most notably the strong version due to Ernst von Glasersfeld (1995).

It was seen that the attacks on radical constructivism at that conference, that were intended to expose the weaknesses of the position fatally, served instead as a platform from which it was launched to widespread international acceptance and approbation. This is not without continuing strong critiques of constructivism from mathematicians and others (e.g., Barnard and Saunders 1994).

It led to further controversy between different versions of constructivism, most notably radical constructivism versus social constructivism (Ernest, 1994)

4. Mathematics Teaching

The teaching of mathematics is also an area in which there can be substantial clashes of philosophy or ideologies. Some of the areas of issues are as following:

1. Mathematical pedagogy – problem solving and investigational approaches to mathematics versus traditional, routine or expository approaches

2. Technology in mathematics teaching – whether electronic calculators be permitted or they interfere with the learning of number and the rules of computation, computers be used as electronic skills tutors or as the basis of open learning?

3. Mathematics and symbolisation – whether mathematics should be taught as a formal symbolic system or emphasis be put on oral, cognitive

4. Mathematics and culture – whether traditional mathematics with its formal tasks and problems be the basis of the curriculum, or it is to be presented in realistic, authentic contexts.

5. Research Methodologies in Mathematics Education

Traditionally research in mathematics education used the methodologies of

Psychological,

Agro-biological and

Scientific research paradigm.

Linking Philosophies of Mathematics and Mathematical Practice:

The link between philosophies of mathematics and mathematical practices is a central issue in philosophy of mathematics. It has been established that there is a strong and complex link between philosophy and pedagogy.

2 philosophical views of mathematics have been described. It is far from uncommon for teachers and others and the experience of learning itself to confirm this view. Such an image is often, but not always, associated with negative attitudes to mathematics.

We see a counterexample arose in a study on student teacher’s attitudes and beliefs about mathematics. A subgroup of mathematics specialists was identified who combined absolutist conceptions of the subject with very positive attitudes to mathematics and its teaching. However amongst non-mathematics-specialist future primary school teachers there was weak correlation between fallibilist conceptions and positive attitudes to mathematics and its teaching (Ernest 1988, 1989b).

A research on children’s attitudes towards mathematics in the past two decades shows strong liking of the school subject, mainly in the years of elementary schooling .But in the later years of academic career attitudes is disconnected to negative are relatively rare. This change in attitudes is due to many factors as adolescence, peer-attitudes, the impact of competitive examinations, and most importantly the image of mathematics conveyed in and out of school.

The most important issue for mathematical education is the relationship between mathematics and values, as the popular image of mathematics is undoubtedly value laden.

Finally, it is possible that the social context of schooling to be so powerful that a teacher with connected values and a humanistic view of school mathematics is forced into ‘strategic compliance’ that results in an evolved modern mathematics classroom practice. This practice may originate with either an absolutist philosophy, or a fallibilist philosophy, but in both cases ‘crosses over’. The research has confirmed that teachers with very distinct personal philosophies of mathematics (absolutist and fallibilist) have been constrained by the social context of schooling to teach in a traditional, separated way (Lerman 1986).

It is suggested that values as well as beliefs and philosophies play a vital role in determining the core fundamentals and philosophies of mathematics classroom practice. This is not surprising as these values are typical of teacher-pupil relations, the degree of competitiveness, the extent of negative weight placed on errors, the degree of public humiliation experienced in consequence of failure.

A lot of work on the philosophy of mathematics education is related to exploring the link between the philosophies of mathematics implicit in teachers’ beliefs, in texts and the mathematics curriculum, in systems and practices of mathematical assessment and in mathematics classroom practices. A lot of progress has been made in impacting such influences, and it is clear that the relationships are complex and non-deterministic.

Also, light has been thrown on a number of dimensions of the philosophy of mathematics education, what it is, and its implications for the classroom. Major concern that still lies is that of inquiring into and questioning some of the presuppositions underlying theories and research in mathematics education and the practices of teaching and learning mathematics.

REFERENCESAssociation of Teachers of Mathematics (1987). Teacher is/as Researcher. Derby: ATM.

Banwell, C. S., Saunders, K. D., & Tahta, D. S. (1972). Starting Points. Oxford: Oxford University Press.

Bassey, M. (1995). Creating Education Through Research. Edinburgh: British Educational Research Association.

Bartolini-Bussi, M. G. (1994). ‘Theoretical and Empirical Approaches to Classroom Interaction.’ In R. Biehler, R. W. Scholtz, R. Strässer and B. Winkelmann (Eds.), The Didactics of Mathematics as a Scientific Discipline (pp. 121-132). Dordrecht: Kluwer.

Bauersfeld, H. (1995). The Structuring of the Structures: Development and Function of Mathematizing as a Social Practice. In L. P. Steffe & J. Gale (Eds.), Constructivism in Education. Hillslade, NJ: Lawrence Erlbaum Associates.

Bauersfeld, H. (1988). Interaction, Construction, and Knowledge: Alternative Perspectives for Mathematics Education. In D. A. Grouws & T. J. Cooney (Eds.), Perspectives on Research on Effective Mathematics Teaching. Reston, Va: National Council of Teachers of Mathematics.