Chapter
4
Ethnomathematics and Mathematical Enculturation
Ethnomathematics is a
programme which looks into the generation, transmission, institutionalisation
and diffusion of knowledge with emphasis on the socio-cultural environment. By
drawing on the cultural experiences and practices of individuals and of
communities, ethnomathematics allows for an easier flow of scientific ideas
with children, reducing the effects of cultural blocks.
[D’Ambrosio, 1990, p. 369]
The above description of
ethnomathematics directly contrasts with the current views held by many of the
modern and traditional philosophers and mathematicians, particularly those who
strongly advocate the absolutist view of mathematical knowledge noted in
chapter 3. However, despite these strongly held views, the notion of
ethnomathematics, like that of modern technology (i.e. calculators and
computers), is inevitably here to stay, particularly within the community of
mathematics educators. In particular today, the need to produce great mathematicians
at the expense of the majority of the citizens is strongly challenged by the
need to produce mathematically literate citizens who are able to manage their
affairs more successfully in a so-called ‘modern technological’ society. This
subsequently leads to another concern which is of even greater importance to
this particular literature survey. This relates to the question of how best to
run the business of education, in particular, the teaching-learning process
within mathematics as a means of enabling young people to not only become
mathematically literate citizens, but also become successful everyday problem
solvers. Thus in educational terms, this notion is well-expressed in the
comments made by Bishop (1991b, p. 3) that:
Educating
people mathematically
consists of much more than just teaching them some mathematics. It is much more
difficult to do, and the problems and issues are much more challenging. It
requires a fundamental awareness of the values
which underlie mathematics and a recognition of the complexity of educating
children about those values. It is not enough merely to teach them mathematics,
we need also to educate them about
mathematics, to educate them through mathematics,
and to educate them with mathematics.
Seen
in the above context, ethnomathematics is therefore a notion generated as a
result of what others might call a ‘self-defeating prophecy’ of the current
practices of mathematics education. This simply means that not only have these
practices been unsuccessful in fulfilling the educational needs of the majority
of the learners, but in doing so, they have also not fulfilled their societal
obligations in producing a mathematically literate society.
It is in the light of the above
existing problems concerning the teaching-learning process that this particular
chapter explores the notion of ethnomathematics in anticipation of
incorporating the type of educational environment it advocates into the
mathematics classrooms of today. This is envisaged not only as means to make
mathematics learning more meaningful to the learners, but in doing so enable
these learners to realise and to apply the power of mathematics in solving many
of the everyday unfamiliar problem situations in their environment. This is
above all, in this author’s view, the prime objective of the whole exercise of
education in general and mathematics education in particular.
4.1 Background and definition
4.1.1 Background
For many thousands of years, all
societies one way or another have developed mathematical practices that are
considered most appropriate to their daily lives and cultures, an area of
mathematics known as ethnomathematics
(Bishop, 1991a; Borba, 1990; D’Ambrosio, 1990a; 1991). In contrast to the more
commonly known ‘academic mathematics’, that is mathematics taught and learned
in schools, ethnomathematics is the “mathematics which is practised among
identifiable cultural groups, such as national-tribal societies, labour groups,
children of a certain age bracket, professional classes and so on, with their
jargons, codes, symbols, myths and even specific ways of reasoning and
inferring” (D’Ambrosio, 1991, p.18). Furthermore, ethnomathematics can also be
seen as a very broad range of human activities, which throughout history have
been replaced by equivalent practices through formalising and codifying and
eventually incorporated into the so-called ‘academic’ mathematics found within
the formal schooling systems (Bishop, 1991a; Borba, 1990; D’Ambrosio, 1990a,
1991; Gerdes, 1994). On the other hand, ethnomathematics is still alive in many
of these culturally identified groups, subsequently forming an integral part of
many of the routines in their practices (Lancy, 1983; Saxe, 1981). In this
context, the notion of ethnomathematics therefore lies on the borderline
between the history of mathematics and cultural anthropology (D’Ambrosio,
1990a, 1991; Gerdes, 1994; Pinxten, 1994).
4.1.2 Mathematics as a neutral and value-free
knowledge
It was not until very recently
that the notion of ethnomathematics started gaining popularity, probably
because of the commonly held view of the “universality of mathematics”
(D’Ambrosio, 1991, p. 15) which implied that mathematics is both culture- and
value-free (Bishop, 1991a; Gerdes, 1994; Julie, 1993). As noted in the National
Statement on Mathematics for Australian Schools:
The universal nature of many
mathematical ideas and the extensive use of symbolic notation to portray
abstract ideas ... lead many people to the mistaken view that mathematics is
culturally neutral and value-free. (Australian Educational Council, 1991, p.
15)
The absolutist view of mathematical
knowledge (i.e. formalism, logicism) has for the most part of the last two
thousand years dominated much of mathematics. This view of mathematical
knowledge is still the one currently held by many of the modern and traditional
philosophers (Ernest, 1991; Pinxten, 1994). According to this view, mathematics
is a “body of infallible and objective truth, far removed from the affairs and
values of humanity” (Ernest, 1991, p. xi).
As Bloor (1973), in support of this view further notes:
Mathematics and logic are seen as
being about a body of truth, which exist in their own right independently of
whether anyone believes them or knows about them. On this view even if there
were no human beings mathematical truths would still be true. (cited in Stigler
& Baranes, 1988, p. 257)
Hence, the implication for mathematics
education according to this view is that teaching and learning mathematics
consist of discovering the already existing truths of formal logic, thereby
logically initiating the pupil in the deductive and basically decontextualised
knowledge of the mathematical field (Pinxten, 1994). This view suggests that
mathematics teaching involves teaching the students to view the quantities and
their manipulations as contextless and divorced from meaning, that is to say,
independent of human activities (Ascher & D’Ambrosio, 1994).
4.1.3 Mathematics as a socially constructed
knowledge
The above view of mathematical
knowledge is however challenged not only by mathematics educators because of
the growing dissatisfaction with the current teaching programs based on the
absolutist philosophy of mathematics (Ernest, 1991; Pinxten, 1994; Julie,
1993), but also by a growing number of philosophers and mathematics educators
such as Abraham and Bibby (1988), Ascher and D’Ambrosio (1994), D’Ambrosio
(1990a), Ernest (1991) and Bishop (1991a, 1991b) who affirm that mathematics is
fallible, changing, and like any other body of knowledge, the product of human
inventiveness. As argued further by Ascher and D’Ambrosio (1994):
The absolute suppression of
context, and the quantification of values for comparison, in order to value
something more than the other, is ... a sign of philosophical damage done to
modern thought; it leads to a world deprived of human values, even of human
feelings. (p. 40)
In addition, the absolutist view of
mathematical knowledge is also challenged by recent research evidence from both
anthropological and cross-cultural studies (e.g. Bishop, 1979; Gerdes, 1985;
Lancy, 1983; Saxe, 1981, 1982; Zaslavsky, 1994) which not only support the idea
that mathematics has a cultural history, but also that from these cultural
histories have come what can only be described as different mathematics
(Bishop, 1991a, 1991b). As D’Ambrosio (1991) further noted:
... recent research ... shows
evidence of practices which are typically mathematical, such as counting,
ordering, sorting, measuring and weighing, done in radically different ways to
those which are commonly taught in the school system. (p. 15)
Mathematical practices were developed as a
result of every single culture, tribe, community and individual trying to cope
with everyday needs, problems and challenges for their survival in direct relationship
with the environment and fellow human beings. It can be viewed as a means of
trying to understand what is going on, to explain what is seen and felt,
thereby contributing either directly or indirectly to the building up of
knowledge (Ascher & D’Ambrosio; Bishop, 1979, 1991a; D’Ambrosio, 1990a,
1991; Stigler & Baranes, 1988). In other words, mathematical practices and
ideas arose out of the real needs and interests of human beings (Zaslavsky,
1994). This view is also shared by Abraham and Bibby (1988) who, in their
effort to generate discussion about the potential role of a Mathematics and Society curriculum in
formal education, contend that “mathematics cannot be understood without some
understanding of the social institution of mathematics - that is socially
organised mathematical activity” (p. 4). This view implies that in order to
have some understanding of the role mathematics plays in structuring our
experiences and judgements, one needs to have some understanding of the human
actions that give rise to such major developments in mathematics.
In the light of the above views,
it is obvious that various cultural mathematics, referred to in this paper as
‘ethnomathematics’, existed in all societies in one form or another from the
most simple everyday practical activities to more complex abstractions.
Moreover, these views also suggest that new developments in mathematics can
only occur when previously established knowledge is challenged by the need to
solve unfamiliar problem situations occurring in everyday contexts. Seen in
this context, ethnomathematics is therefore best summarised by D’Ambrosio
(1990a, p. 369) that:
mathema
[are] the
actions of explaining and understanding in order to transcend and of managing
and coping with reality in order to survive. Throughout all our own life
histories and throughout the history of mankind, technes (or tics) of mathema have
been developed in very different and diversified cultural environments, i.e. in
the diverse ethnos. So, in order to
satisfy the drives towards survival and transcendence, human beings have
developed and continue to develop, in every new experience and in diverse
cultural environments, their ethno-mathema-tics. These are communicated
vertically and horizontally in time and for the reason of being more or less
effective, more or less potent and sometimes even for political reasons, these
various tics have either lasted and
spread (e.g. counting, measuring) or confined themselves to restricted groups
and even disappeared.
In other words, ethnomathematics refers to what Nunes
(1992) called “forms of mathematics that vary as a consequence of being
embedded in cultural activities whose purpose is other than doing mathematics” (p. 557). These
include among other things, building houses, exchanging money, weighing
products, and calculating proportions for a recipe which involve numbers,
calculations, and precise geometrical patterns.
4.2 Ethnomathematics and the Development of
Mathematics.
4.2.1 Mathematics as a cultural knowledge
In his philosophy of mathematics,
Wittgenstein (cited in Stigler & Baranes, 1988), while simultaneously
questioning the notion that mathematics is a system that exists outside the
practice of doing mathematics, argued that mathematics is “social in nature,
and inseparable from the social realm in which it is used” (p. 257). Thus, from
Wittgenstein’s perspective, the formalisation of mathematics was not a
discovery of the foundations of mathematics, but rather a theoretical
enterprise that came after the development of mathematical practices. For
Wittgenstein, mathematics is not a system of unique formal logic that exists on
its own but rather a collection of techniques or games, each with its own
consistent rules that are not necessarily connected to other games (Stigler
& Baranes, 1988). As Bishop (1991a, p. 30) further elaborates: “The thesis
is therefore that mathematics must now be understood as a kind of cultural
knowledge, which all cultures generate but which need not necessarily ‘look’
the same from one cultural group to another”. Accordingly, he further argues
that just as all human cultures generate language, religious beliefs, rituals,
food-techniques etc., so it seems do all human cultures generate mathematics.
Thus, in his view, mathematics is a pan-human phenomenon. Moreover, if the
above perspective is to be taken as it is, it means that just as each cultural
group generates its own language, religious beliefs etc., so it logically seems
that each cultural group is also capable of generating its own mathematics.
Obviously this kind of thinking has some fundamental implications for teaching
and learning of mathematics in terms of re-examining many of our traditional
beliefs about the theory and practice of mathematics education (Bishop, 1991a).
4.2.2 The universal mathematical activities
In the light of the description
in the above section, it is worthwhile to point out that while there are
specific mathematical practices or activities that differ within and among
different cultural groups, in general, there are some practices that Bishop
(1991a) called “universal” across different cultural groups. According to his
analyses, Bishop (1991a) argued that there are six fundamental activities which
are both universal, in that they appear to be carried out by every culture
group ever studied, and also necessary and sufficient for the development of
mathematical knowledge. These, as he described them, are as follows:
Counting
: the use of a
systematic way to compare and order discrete objects. It may involve body or finger
counting, tallying or using objects or strings to record, or special number
names. Calculation can also be done with the numbers, with magical and
predictive properties associated with some of them.
Locating
: exploring
one’s spatial environment and conceptualising and symbolising that environment,
with models, maps, drawing and other devices. This is the aspect of geometry
where orientation, navigation, astronomy and geography play a strong role.
Measuring
: quantifying
qualities like length and weight for the purposes of comparing and ordering
objects. Usually measuring is used where phenomena cannot be counted (e.g.
water, rice) but money is also a unit of economic growth.
Designing
: creating a
shape or a design for an object or for any part of one’s spatial environment.
It may involve making the object, as a copyable ‘template’, or drawing it in
some conventional way. The object can be for technological or spiritual use and
‘shape’ is a fundamental geometrical concept.
Playing
: devising and
engaging in games and pastimes with more or less formalised rules that all
players must abide by. Games frequently model a significant aspect of social
reality, and often involve hypothetical reasoning.
Explaining : finding ways to represent the
relationship between phenomena. In particular, exploring the ‘patterns’ of
number, location, measure and design, which create an ‘inner world’ or
mathematical relationships which model and thereby explain the outer world of
reality. (Bishop, 1991a, p. 32-33)
Thus, mathematics as a cultural knowledge
derives from humans engaging in the above six fundamental activities, as Bishop
(1991a, p. 33) put it, “in a sustained and conscious manner”. Accordingly, he
argued that the performance of these activities by members of the respective
cultural groups can either be in a mutually exclusive way or perhaps, more
significantly, by interacting together as in ‘playing with numbers’, examples
of which are number patterns and magic squares which arguably contributed to
the development of algebra. Taken in this context, Bishop (1991a) is also
supported by many other writers (D’Ambrosio, 1991; Ernest, 1991; Gerdes, 1994;
Pinxten, 1994) who argue that many of the highly significant ideas found in the
current so-called ‘Western mathematics’ taught in schools are themselves the
results of human engagement in activities such as those described above. As
Bishop (1991a, p. 34), on the basis of the above six fundamental activities,
further elaborated:
From these basic notions, the
rest of ‘Western’ mathematical knowledge can be derived, while in this
structure can also be located the evidence of the ‘other mathematics’ developed
by other cultures. Indeed we ought to re-examine labels such as ‘Western
Mathematics’ since we know that many different cultures contributed to the
knowledge encapsulated by that particular label.
For example, Nelson (1993) in
this respect provides some significant historical examples of mathematical
developments such as the use of 360 degrees for complete revolution tracing it
back to the Babylonian sexagesimal (60-based) number system, the notation for
the angle to the Greeks, the sign for equality to Robert Recorde (1557), the 
notation to Descartes
(1637), and both the decimal notation and the concept of zero to the Indians
and Chinese, just to name a few. These examples illustrate the fact that the
study of historical developments in mathematics can be of rich learning
experience for the learners through the use of projects which can be investigation-based
thereby simultaneously providing the opportunity for the learners to involve
parents or other members of the local community who are the bearers of various
mathematical practices embedded in cultural activities.
4.2.3 Mathematics as a cultural phenomenon
If the current experiences are
taken to be any indication, there is another aspect of ethnomathematics which
is usually left out by many of the school systems during the teaching of formal
mathematics. Unlike the more obvious observable mathematical practices
described so far, this aspect of ethnomathematics relates to the notion of what
Bishop (1991a) called “mathematics as a cultural phenomenon” (p. 32). In other
words, if ethnomathematics is to be taken as a cultural product, according to
White (cited in Bishop, 1991a) this simply means that the functions of culture,
more than anything else, are to relate man to his environment on the one hand,
and to relate man to man, on the other. Thus, ethnomathematics is also a notion
that is ideological, sociological and sentimental in nature, in that it is used
by members of specific cultural group to represent and to explain cultural
phenomena that could not be adequately defined in any practical terms.
In this respect,
White (cited in Bishop, 1991a) subsequently went further to divide the
components of culture into four categories, namely:
·
ideological: composed of beliefs, dependent on symbols,
philosophies;
·
sociological: the customs, institutions, rules and
patterns of interpersonal behaviour;
·
sentimental: attitudes, feelings concerning people,
behaviour;
·
technological: manufacture and use of tools and implements.
(cited
in Bishop, 1991a, p. 32)
Moreover, it has to be pointed out that
while the above four components are inter-related, White (1959) strongly argues
that the technological factor is the most basic one with all other factors
dependent upon it. In other words, for White the technological factor
determines in a general way at least the form and content of the social,
philosophic and sentimental factors.
The significance of written
language in terms of mathematical symbolism as one of its particular conceptual
‘tools’ is also highlighted by Bishop (1991a). Based on White’s schema, he
argued that mathematics as an example of both a cultural phenomenon and a
symbolic technology, offers an opportunity to explore ideology, sentiment and
sociology, and therefore simultaneously attend to the notion of ‘values’ as
well.
In this respect,
Ascher and D’Ambrosio (1994) contend that in the education of children, values
are being taught through an emphasis on contextless numbers whereby everyday
examples are phrased in terms of numerical equivalents. It is therefore not
uncommon to find that for those learners who are alienated by mathematics,
there is a feeling that mathematics is both emotionless and lacks the necessary
human feelings that initially created it. According to Ascher and D’Ambrosio
(1994), this therefore means that the most “important point that ethnomathematics
adds, however, is that, nonetheless, expressions of mathematical ideas do have content and cultural context”
(p. 39), a notion that has been a large source of misunderstanding in the past.
The above argument is strongly
supported by the evidence from Buxton’s research (cited in Bishop, 1991c)
concerning adults’ fear of mathematics, which suggested that mathematics is
largely seen as:
1.
Fixed,
immutable, external, intractable and uncreative.
2.
Abstract
and unrelated to reality.
3.
A
mystique accessible to few.
4.
A
collection of rules and facts to be remembered.
5.
An
affront to common sense in some of the things it asserts.
6.
A
time test.
7.
An
area in which judgements not only on one’s intellect but on one’s personal
worth will be made.
8.
Concerned
largely with computation.
(Cited in Bishop, 1991c, pp. 195-196)
If the above evidence is of any
significance, this paper therefore strongly shares the view that the practices
of mathematics and mathematicians are not fundamentally different from human
thought as embedded in other domains (Stigler & Baranes, 1988).
Subsequently, it therefore views the practice of mathematics as not the
discovery of truths existing outside the realm of human activity. Rather it
considers mathematics to be, as Stigler and Baranes (1988) put it:
“domain-specific, context-bound, and procedurally rooted as are other forms of
knowledge” (p. 258). Taken in this context, it therefore views mathematics
learning to be not uniquely immune to the influences of culture, but rather as culturally dependent as learning in other
domains.
4.3 Views of Culture and Mathematics
The analysis of the discussion so
far in this chapter generates two distinct approaches to the study of cultural
influences on mathematical knowledge. According to Nunes (1992), in the first
approach which is consistent with the view suggested by Stigler and Baranes
(1988), the definition of mathematical knowledge is somewhat implicit, in that,
it appears to be based on the content of knowledge. Thus, according to Stigler
and Baranes (1988), “mathematics is not a universal, formal domain of knowledge
waiting to be discovered, but rather an assemblage of culturally constructed
symbolic representations and procedures for manipulating these representations”
(p. 258). They therefore argue that the incorporation of representations and
procedures by children into their cognitive systems is a process that occurs in
the context of socially constructed activities. In this context, they further
argue that mathematical skills that children learn in school are the results of
a combination of previously acquired (or inherited) knowledge and skills, and
new cultural input, rather than logically constructed on the basis of abstract
cognitive structures. According to this view, culture therefore functions not
as an independent variable that can promote or retard the development of
mathematical abilities, but is considered as an integral part of the
mathematical knowledge itself (Stigler & Baranes, 1988). In this approach,
the variations between cultures are
therefore treated as a reflection of the differences in language and numeration
systems (Nunes, 1992).
The second perspective, as noted
by Nunes (1992) and advocated by Ascher and D’Ambrosio (1994), Bishop (1991a,
1991b) and D’Ambrosio (1990a, 1991), contends that the analysis of cultural
influences on mathematical knowledge can demonstrate both the differences and
invariance in mathematical knowledge across cultures. According to this view,
‘mathematizing’ reality is therefore seen as representing reality in such a way
that more knowledge about the represented reality can be generated via
inferences using mental representations as contended by Bishop (1991a) in his
analyses of the six fundamental activities described in the previous section.
Moreover, there is also no need to manipulate such reality any further for the
purpose of verifying this new knowledge. In using this view, Nunes (1992)
further argues that the invariant logical structures are therefore part and
parcel of mathematical knowledge, regardless of the location where this
knowledge is developed, be it in school or out of school. However the most
important consideration is the ability to make mathematical inferences on the
basis of these logical structures, rather than the content of knowledge, that
distinguishes mathematical knowledge. As Nunes (1992, p. 558) clearly put it:
Merely reciting count words ...
would not be considered mathematical knowledge in this perspective. Despite the
fact that numbers are the content of knowledge ... no inferences result from
mere recitation, thus no mathematical knowledge is involved.
If count words are used to represent sets
and make inferences about relationships between sets, the count words bear on
mathematical knowledge. Taken in this context, it therefore means that
different cultures have found distinct solutions to the organisation of their
count words. In educational terms, these differences, like those of other
cultural knowledge have important effects on how quickly children learn the count
words within the context of formal classroom situations (Nunes, 1992).
4.4 Mathematical Enculturation
Mathematics education, like those
of other ‘academic’ educational disciplines, is so often characterised by what
Bishop (1991b, p. 89) described as the “formal mathematical enculturation” into
the so-called Mathematical culture. Because of this, it has as its goal, the
induction of children (learners) into the symbolisations, conceptualisations
and values of the Mathematical culture. Such characterisation clearly indicates
the involvement of both the process
and content of mathematics. Because of the culture’s frame of
knowledge, the formal mathematical enculturation cannot be just
process-oriented, but nor should it just attend to content knowledge, since
education is more than mere transmission (Bishop, 1991b). Moreover, this
equally means that the ‘enculturation process’ has the responsibility to both
the child and culture, respecting the individuality and personality of the
children at the same time the characteristics of the culture. As Bishop (1991b)
explains further:
To ignore the [first] would lead to indoctrination,
while to ignore the [second] would
lead to anarchy. Mathematical enculturation needs to be conceptualised as a
social interactive process carried out within a certain knowledge frame but
with the goal of recreating and redefining that frame (p. 89).
It is with such a background in
mind that mathematics, like any other academic teaching subject, must be guided
by some form of curriculum. In short, it is the ‘objectified’ representation of
mathematical culture for the purposes of the educational process. This not only
helps in thinking about the kinds of activities to be used for enculturing the
learners into the mathematics culture, but also provides a suitable structure
for organising those activities into one coherent whole (Bishop, 1991b). For
the purpose of this literature survey in particular, this logically means that
the ‘culturally-oriented’ mathematics curriculum must therefore seek a way to define the knowledge frame which will
not only allow personality to flourish, but most importantly, create a suitable
learning environment necessary for effective social interaction to take place
within the context of learners’ socio-cultural environment.
In view of the above suggestion,
it is argued by Lester (1989) that the knowledge gained in out-of-school
situations, is often characterised by five key features developing out of
everyday activities which, as he puts it:
1.
occur
in a familiar situation;
2.
are
dilemma driven;
3.
are
goal directed;
4.
use
the learner’s own natural language; and
5.
often
occur in an apprenticeship situation allowing for observation of the skill and
thinking involved in expert performance. (cited in Masingila, 1993, p.18)
On the other hand, knowledge
acquired in formal classroom situation all too often grows out of a
transmission paradigm of instruction and is therefore viewed to be lacking in
meaning, that is to say, lack of context, relevance and specific goal (Ascher
& D’Ambrosio, 1994; Boaler, 1993a; Masingila, 1993). Accordingly, it is
argued by Resnick (cited in Masingila, 1993) that schools place too much
emphasis on the transmission of syntax (procedures) rather than on the teaching
of semantics (meaning) thereby discouraging children from bringing their own
intuitions to at least have some bearing on the school learning tasks.
Students need in-school
mathematical experiences to build on and formalise their mathematical knowledge
gained in out-of-school situations. The mathematical experience in school in
terms of the guidance and structure provided by teachers can be of significant
benefit to students in that it can help them to make connections among
different mathematical ideas. In this context, some mathematics educators (e.g.
Bishop, 1991b; Julie, 1993; Masingila, 1993; Zaslavsky, 1994) have suggested
that mathematics teachers should establish master-apprenticeship relationships
with their students in order to help initiate them into the mathematics
community. Working with others toward common goals, being actively involved in
doing mathematics, and discussing and refining mathematical ideas are all
characteristics of being part of the mathematics community. At present,
however, according to Masingila (1993) many students become isolated from the
mathematics community rather than becoming a part of it. A key reason for this
isolation is the wide gap that exists between mathematical practices in school
and in out-of-school situations which also includes ethnomathematics defined in
chapter 4 of this presentation representing the ‘cultural practices of
mathematics’.
Thus, in terms of the objective
of this study, the implication is that this wide gap is also seen to be mainly
responsible for greater mathematical learning difficulties experienced by many
of the Papua New Guinean students. As noted in chapter 3, this is because of
the current approaches to mathematics education in PNG which do not take into
account the learner’s cultural and social environment in which many of these
ethnomathematical practices are embedded. For one thing, these learners should
not be characterised as empty mugs, but rather, as equal partners who, in their
own small ways, contribute to shaping the educational process, thus leading to
meaningful and realistic educational experiences.
With such a view in mind, the
prime objective of this study is to close this gap between doing mathematics in
school and doing mathematics in out-of-school situations. The implication of
this objective is that the alternative mathematics education curriculum should,
as one of its main objectives, incorporate the social history of mathematics
not only to reflect on the historical developments of mathematics, but to use
it as the basis to generate further mathematical inferences and knowledge in
benefiting everyone in society. Moreover, such an approach will not only become
a rich educational experience for the learners, but also preserve the rich
cultural heritage of mathematics to benefit future generations.
4.5 Conclusion
The definition of
‘ethnomathematics’ as generated by the discussion in this chapter can be seen
as the mathematics that is practised by members of different cultural groups
embedded in a very broad range of human social activities which include their
jargons, codes, symbols, myths and even specific ways of reasoning and
inferring. Moreover, the analysis of these cultural practices also reveals that
ethnomathematics can also be characterised as both a content and process of
mathematics. The ‘process’ of mathematics notably involves the six universal
mathematical activities namely, counting, locating, measuring, designing,
playing and explaining. These fundamental activities are considered to be not
only necessary, but are also sufficient for the generation and development of
mathematical inferences and knowledge. Thus, throughout the history of a
specific cultural group these fundamental activities have both individually and
in interaction been instrumental in developing the complex symbolisations and
conceptualisations of mathematics which also include the so-called ‘Western’
mathematics taught in schools.
Because of its significant
educational implication, the ‘process’ aspect of ethnomathematics is therefore
consistent with both the objectives of this study and the underlying educational assumption of the current
philosophy of education for PNG which both aim to generate, transmit,
institutionalise and diffuse mathematical knowledge to learners with a strong
emphasis on their socio-cultural environment. Consequently, such an emphasis is
therefore in direct conflict with the educational assumptions of the current
approach to mathematics education in PNG, because of its view of mathematics as
being independent of the learners’ socio-cultural environment. This approach to
mathematics education is seen to be largely responsible for the dislike of
mathematics by many learners in PNG, a major concern of many of the recent
research studies. In this respect, ethnomathematics therefore aims to draw from
the cultural experiences and practices of the individual learners, the
communities, and the society at large in using them as vehicles to not only
make mathematics learning more meaningful, but more importantly, to provide the
learners with the insights of mathematical knowledge as embedded in their
social and cultural environment. In doing so, it not only allows for an easier
flow of mathematical ideas in reducing the effects of many of the social and
cultural factors, but also creates a more meaningful learning environment that
is conducive to any human social process of which mathematics is its highest
symbolic technological product.
It is with such views in mind
that the next chapter will further consider some specific ethnomathematical
practices within the cultural context of PNG which have the potential to
generate mathematical ideas or inferences that could be incorporated into the
formal mathematics classrooms in PNG. In particular, the aim is to use them as
the basis for making the process of teaching and learning mathematics noted in
the previous section as ‘mathematical enculturation’ more meaningful to the
learners. Most importantly though is for the learners to ‘relationally’
understand mathematical knowledge such that it can be applied in any unfamiliar
everyday context of the learners’ own environment as determined by the overall
societal needs and problems.