In the later part of the 20th Century, there have been intensive efforts to improve mathematics education in schools. Educationists, subject experts, teacher-educators, teachers, parents, psychologists, policy makers, text book and resource material producers and many more are showing their concern about the state of mathematics education at the school level. Many attempts were made in this regard. After more than a quarter century of attempted reforms, there appears to have been little improvement. Where are we still going wrong? Why is mathematics still a problem subject for so many? The problem is widespread. If we cannot find at least a partial answer to this question, there is no reason to expect that future efforts will be any more successful than past. I don’t think that there is a single answer to the problem, nor that any one person knows all the answers. The mathematical culture that is adapted may be one of the causes and mathematical enculturation is one of the solutions.

 
The Present Mathematical Culture in Secondary Schools

The importance of teaching mathematics as an integrated subject is recognised every where. But still we are sticking on to the idea of ‘teaching mathematics’. It is better to move from ‘teaching mathematics’ towards ‘mathematics education' (Bishop, 1988), though educating the pupils mathematically is more difficult, challenging and complex than teaching them some mathematics.

In most of the countries including India the mathematical culture adapted in schools has the following characteristics

• Curriculum of procedures, methods, skills, rules and algorithms which insist on ‘doing’ mathematics rather than ‘thinking’ mathematics.
• Quantum of content of mathematics subject matter, completion of stipulated syllabus in the given rigid time, examinations by giving less or no importance to the abilities, interests and cognitive level of the learners.
• Text books written by authors mostly who have less or no chance of knowing about the pupils, teachers and mathematics classroom.
• The assumptions made such as ‘mathematical knowledge flows from higher level to lower level’. Here the teacher is considered as at ‘higher level’ and student as at ‘lower level’ ignoring the interpersonal relationship. Mathematics is considered as optional, pupils can drop themselves whenever they find it difficult, meaningless, or of less useful to them.
• Normally ‘what to teach’, ‘how to teach’, ‘how much to teach’, ‘how long to teach’ etc., are dictated to the teachers.
• Teachers generalise the learner ability i.e., teacher plans the lesson, teaches the lesson aiming at the average generalised ability of the learner considering it as every student’s ability, resulting that it is not suitable to anybody because the average represents no body in the group.
• Mathematics teaching is dominated by dehumanisation, depersonalisation and decontextualisation. School pupils many times confused why are they learning about algebra, trigonometry theoretical proofs of theorems. They mean that if Apollonius theorem or Pythagoras theorem is true every where, so what? Why should they learn about universal truths? If the teaching is not contextual, the aims are not realised by the pupils, mathematics learning is meaningless to them.
• The focus is on ‘how many sums the pupil did’, ’how many topics are learnt’, ‘how much portion the teacher completed’ and not on what is going on in the minds of the pupils. If the pupil gets the correct answer there ends learning. It is rarely attempted to know if the pupil does it correctly, how could he arrive to the correct answer, and if a child fails, where has he gone wrong, what misconceptions led him to arrive a wrong conclusion. There is no scope given to get multiple answers to the same task. Emphasis is more on the product rather than on the process.

The Values of Mathematical culture

The values of mathematical culture are providing different dimensions and directions to mathematics teaching at school level. Greenfield and Bruner(1966), offer the idea that "some environments push cognitive growth better, earlier and longer than others. What does not seem to happen is that different cultures produce completely divergent and unrelated modes of thought".

• Since the time of the Egyptian and Hellenistic civilisations Rationalism. logic and reason have become basis for mathematics education. Rationalism, with its focus on deductive reasoning challenged the trial-error pragmatism, and inductive reasoning. The main aim of mathematics teaching was to develop logical, rational, abstract and theoretical thinking. The mathematical ideas are developed by proofs, extensions, examples, counter-examples, generalisations and abstractions.
• In the western culture it could be observed that Objectism as the driving force in the development of mathematics. The focus is on the origin of ideas which form with the interaction of the environment and it is material objects which provide the intuitive and imaginative bases for these ideas (Newman 1959). Pythagoreans developed the abstract idea of numbers starting from concrete objects like stones and pebbles and proceeded to particles and points. Therefore, to the Pythagoreans numbers were objects, literally.
• While Rationalism insists on logic and Objectivism on materialism, the third dimension of mathematical culture focuses on the aesthetic pleasure and satisfaction that a learner experiences when a pattern is suddenly revealed by organisation or structuring a messy collection of number facts, or a set of random shapes.
• Another aspect is the feelings of growth, of development, of progress, and of change which led to the alternativism - the recognition and valuing of alternatives. It means finding multiple solutions to the same task or problem. Multiple solutions or ways out of a problem definitely leads to progress. Horton (1967) wrote: "in traditional cultures there is no developed awareness of alternatives to the established body of theoretical tenets; whereas in scientifically oriented cultures, such an awareness is highly developed". This may be the reason that the ‘Investigations’ and ‘Investigatory approach’ are picking up the attention even in mathematics teaching.
• The sociological component is concerned about the examination of mathematical truths, propositions and ideas, where from the mathematical ideas come from and who generate them.
• The technological value imposed on mathematics due to the introduction of calculators, computers and ICT

Mathematical Enulturation and Mathematics Education

The idea of enculturation entered interestingly in the field of mathematics education. Enculturation, is a creative, interactive process among those who live in it with those who born in it. Though the ideas, norms and values may be similar to the previous generation, it would be inevitably different due to the re-creation role of the next generation. The mathematics education of the child is affected at three levels, informal, formal and technical.

As cultural transmission is certain, the mathematics teaching is affected informally. Many times this interaction is informal and so mathematical culture can play an important role in informal enculturation. Initially before the children go to formal schooling many mathematical concepts like many and few (number concept), size (big and small), shape (regular, irregular, plane, curved), area, volume, capacity (more, less and measurement), distance (far, near, length, breadth, height, depth, perimeter), time (tense and measurement), weight (heavy, light and measurement), are formed and attained through informal inter action with the elder members of the society. Adults transmit the mathematical culture by approving and disapproving the actions based on mathematical logic. They also develop symbolic ideas through stories, music, discussion and examples from day to day life. These reactions with social experience develop among the pupils the abilities of rational thinking, perceiving the logic, finding the relation between cause and effect and reason for every action. By insisting on regularity, punctuality, following the rules with reason, rational thinking, pupils develop mathematical attitude. The social interactions also vary according to the different personality and interests of the children. No two children are identical and therefore the ways they interact will differ. As a result, no two people will develop an identical conception of their shared mathematical culture. So the adults who share the mathematical culture play an important role in informal mathematical enculturation.

At the formal level the mathematical culture is to be filtered and the best of the mathematical cultural heritage is tried to be transmitted. Formal mathematical enculturation is possible through ‘formal education’. Therefore it should help to explain and understand various aspects of informal mathematical culture and should make it more structured by refining it. The formal enculturation should take place at an appropriate level intentionally and explicitly. Though this responsibility is entrusted to the schools, the situation is far beyond satisfaction. Mathematics is a universal language of symbols and concepts. So the main goal of formal mathematical enculturation is to induct the children in to the symbolisation, conceptualisation and values of mathematical culture. The child and culture should be given equal importance. One cannot be over emphasised at the cost of the other. But what is observed in the developing countries like India, the mathematical culture is given more importance ignoring the individual and the suitability of the culture to the individual. The curriculum and subject content planned for a different society with different social, cultural and technological environment is adapted and at the same time the process is not adapted. We cannot confine to the process oriented because of the culture’s frame of knowledge, nor we can just concentrate to that knowledge, since education is more than just imparting knowledge. So the teacher education Institutions should take the role of liaison-fare and guide the future teachers. There should not be any conflict between informal and formal mathematical enculturation, they should supplement each other.

Due to modernization and technological developments school mathematics is affected at the technical level. The advent of Computers and calculators and other technological developments broadened the mathematical applications in society. As such the mathematics education should be modified according to the industrial, technological and societal requirements. It should no longer remain as a product of pure mathematics. Here also the under developed countries and developing countries, which cannot afford heavy investment and without laying a sound suitable environment, the syllabuses and curricula of more-technological societies are considered as models and facing utter failure. It is because these changes are in-appropriate to the social and cultural environment of the child. Here again there is a conflict between mathematical enculturation at informal level and at technical level. We cannot neglect the individual personalities. A cultural perspective on Mathematics education must surely recognise the existence of individual differences and we can no more consider ‘children’ as ‘child’. Cultural learning is a creative and re-creative act on the part of every person. Cultural learning is thus no simple one-way process from teacher to learner. Therefore it is necessary to think about the type of mathematical enculturation we would like to bring. Then the questions rise how should it be done? and at what level? Should it be done at an informal level? or at the formal level? or at the technical level? The teacher who is the key person and the Mathematical Enculturator in the formal enculturation process should establish a proper rapport among these three levels of enculturation by eliminating the opposing forces. Teacher education Institutions have to take the major responsibility in inculcating these abilities in mathematics teachers.

The next question is what should be the approach in the curriculum? Should we follow the Behaviourist Approach aiming at improving learning by a ‘task analysis’, or the ‘New-Math approach insisting on a ‘systematic description of mathematics’, or the structural approach based on the theories of Bruner and Diene, or the Formative approach that focuses on cognitive abilities and affective and motivational attitudes of the pupils, or the Integrated-Teaching Approach which is based on problem-solving process.

The Behaviourist approach insists on sequential learning and the main objective is mastery of specific mathematical content. In this approach only cultural transmission is possible but not enculturation. The New-Math approach is just like ‘an old wine in a new bottle’. It reorganises and describes the mathematics content with common uniform and precise language. The structuralist Approach also gives importance to the mathematics subject content. The Kilpatrick’s Formative approach goes beyond the subject matter and aims at the development of cognitive abilities and motivational attitudes which describe in terms of personality traits. The Integrated approach insists on the flexibility of the curriculum and problem solving processes. The combination of Formative and the Integrated teaching approach focusing on the process may be the best solution for mathematical enculturation since it provides flexible curricular units and open-ended processes for the learner according the psychology of the individual. (the suggestions offered are not final, subjected for discussion).

What cultural components should be considered for enculturation? Is it the symbolic component that is based on ‘rationalism’ and ‘Objectivism’? or the societal component that insists on the uses of mathematical explanations or the cultural component that insists on alternativism and openness? What activities are to be planned accordingly? In case if we would like to have the integration of all these, how to balance them in the curriculum? A combination of all these three components namely symbolic, societal, and cultural may supplement each other and can bring mathematical enculturation. The symbolic component helps in developing the intellectual (cognitive) abilities and Objectism in explicit exploration, the societal component takes care of the applicative value and the cultural component will look into the technical and alternatives of the existing phenomenon.

How to bring the Enculturation process in action? Should it be interpersonal and interactional? Should it be formal, institutionalised, intentional and accountable?

Should it be concerned with mathematical concepts, meanings, processes and values? or should it be suitable to the social context? or should it be for all?

Above all, according to Bishop, J (1988), the ‘enculturation’ was focused on values with an insist of moving away from a ‘transmission’ image of mathematics education. According to Bishop enculturation can not be done by one person to another, culture is not a ‘thing’ which is transmitted from one person to another, nor is the learner merely a passive recipient of culture from the Enculturator. Enculturation is an interpersonal process and therefore it is an interactive process between the teacher and the taught. There should be a strong relationship between teacher and Mathematics Teacher Educators.

Since mathematical enculturation is an intentional, shaping process, the teacher’s task is to create a particular kind of social environment for the learner and it is the learner’s task to construct ideas and modify them in interaction with that environment. The psychologists, Educationists, the subject experts and curriculum framers should provide the supporting system.
 

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