A FEW THOUGHTS & MANY QUERIES ABOUT MATHEMATICAL
CULTURE AND MATHEMATICS EDUCATION
Chilakamarri Vijayalakshmi
King’s College, London, Visiting Academic from INDIA
Abstract
In this paper an attempt is made to discuss the impact of culture on teaching and
learning mathematics at the school level. All over the world research findings
reveal examples of students rejecting mathematics, fearing it, disliking it, given a
chance preferring it as last resort, even if they learn resorting to rote and
instrumental methods. To what extent is culture the cause of it? How can
enculturation help students in enjoying mathematics and in transferring the
problem solving skills to new situations? Many more queries and thoughts about
the influence and impact of culture on mathematics education at school level are
put before you for discussion.
In the later part of the 20th Century, there have been intensive efforts to improve
mathematics education in schools. Educationists, subject experts, teachereducators, 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 inappropriate 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.
References
Berger, P.L. and Luckman, T (1979). The Social Construction Of Reality : a
treatise on the sociology of knowledge. Harmondsworth: Penguin.
Bishop, A.J (1988). Mathematical Enculturation: A Cultural Perspective on
Mathematics Education. Netherlands: Kulwer Academic Publishers.
Burkhardt, H(1981). The Real World and Mathematics. Glasgow: Blackie.
Cole, M. and Scribner, S (1974). Culture and Thought. New York: Wiley
Gay, J and Cole, M (1967). The New Mathematics in an Old Culture. New
York: Holt, Rinehart and Winston.
Griffiths, H.B. and Howson, A.G (1974). Mathematics, Society and
Curricula. Cambridge: University Press.
Howson, A.G., Keitel, C. and Kilpatrick, J (1981). Curriculum Development
in Mathematics. Cambridge University Press.
Mann, W.J.A & Tall, D (1992). Computers in Mathematics Curriculum..
Leicester: The Mathematics Association.
Mottershead, L (1985). Investigations in Mathematics. Oxford: Blackwell
Orton, A and Frobisher, L (1996). Insights into Teaching Mathematics.
London: Cassell.
Pimm, D (1987). Speaking Mathematically. London: Routledge and Kegan
Paul
Resnikoff, H.L. and Wells, R.O (1984). Mathematics in Civilization. New
York: Dover.
Selinger, M. (Ed.). (1995). Teaching Mathematics. London: Routledge
Stenhouse, L (1967). Culture and Education. New York: Weybright and
Talley.
Vijayalakshmi, Ch. (1993). ‘Development of Computer Assisted Instruction
in Mathematics’, Innovative work of Teacher Educators, 1-19.
Vijayalakshmi, Ch. (1994). ‘Implications of Brain-Research for Teachers &
Teacher Educators’, Edu-Vision, 18-20.
Wilder, R.L. (1981). Mathematics as a Cultural System , Oxford: Pergamon
Press.