**The Role
of Ethnomathematics in Mathematics Education**** in Papua
New Guinea****:** **Implications for Mathematics Curriculum**

It is
not uncommon to hear many Papua New Guinean students at all levels of the
national education system commenting in saying that they have always found
mathematics, compared with other prescribed school subjects, a difficult one to
learn and understand. Depending on the type of cultural background that one
comes from, and how well one attaches to mathematics as a formal school
subject, there will surely be a variety of responses on this comment given by
different individuals. For some, such comment is expected and they would wholeheartedly
welcome it, because it further increases the knowledge gap between what the few
privileged individuals know about the subject as an academic discipline and
those who are not, particularly in terms of its everyday use as an important
cultural tool. For others who show no interest in the comment just ignore it
probably regarding it as one of the day to day slogans of the experiences that
children go through in schools. For those who seriously call themselves
mathematics educators, such comment is unacceptable, because it directly brings
into question their professional integrity.

This
paper aims to bring to the attention of everyone involved in the business of
education services of the need to address this situation head on because
ignoring it would mean several scenarios. Firstly, this can be easily seen and
interpreted by many concerned individuals as evidence of professional
negligence. Secondly on a more serious note, the far-reaching implications it
would have on the society where its everyday existence and survival depended
heavily on the utility of the subject as an important cultural tool. While
there are many other factors that may have contributed towards students making
such comments, this paper argues that it is the type of classroom practices
currently employed by teachers of mathematics that contribute greatly towards
students’ learning difficulties in mathematics though much of it are external
to ordinary classroom teachers.

In
the light of the above background and the current educational-reform in PNG,
this paper strongly suggests for a change in the current culture of Mathematics
Instruction in PNG, which is fundamental to the overall success of the reform
in PNG’s National Education System (NES). In reflecting on the research conducted
on the carpet layers including other research examining the mathematics
practices in everyday situations, Masingila (1993) suggested three areas of
mathematics education that require some attention for possible improvement by
everybody involved in the teaching of school mathematics. These are (a) the
school mathematics curriculum, (b) the methods used in teaching school
mathematics, and (c) research in mathematics education. In addressing the
existing problems within the current school mathematics curriculum and the
methods used in teaching school mathematics in the context of the education
system in PNG, the paper aims to provide some suggestions for consideration
under the following four sub-headings.

i.
Re-defining the Role of a Teacher;

ii.
Defining the Role of Ethnomathematics in Mathematics
Education;

iii.
Shift in teaching emphasis from Procedural to Conceptual
understanding;

iv.
Implications for Mathematics Curriculum

In
general, it is common knowledge in PNG that many students, including parents,
see teachers both as an authority and the only source of all knowledge and
information in the formal classroom. In many ways teachers themselves in their
own ways either consciously or unconsciously encourage the development of this
view during their formal and informal interactions with students. While such a
view may have advantages in other educational settings it can also become a
barrier to effective learning in the formal classroom situation in the sense
that it fails to acknowledge students as equal partners of the
teaching-learning process. In other words, those teachers who strongly advocate
this view somewhat treat students, as *empty
mugs* that need to be filled up with information that are only available
within the boundaries of the formal classroom. Thus, in the actual
teaching-learning situation, it only requires the teacher to reinforce this
view by simply not giving any opportunity for students to either ask questions
or express an opinion on what is being taught. From the teacher’s point of
view, there is also an aspect of insecurity in that it may at times directly
bring into question their professional integrity as a teacher in terms of the
content knowledge of the subject as an academic discipline.

The
redefining of the role of a teacher within the classroom context would
basically require a teacher to see himself or herself as a *facilitator* of the teaching-learning process rather than an *authority *and *transmitter *of the mathematical knowledge (Matang, 1996; 1998; 1999). In practice however, this approach
would require the teacher to acknowledge students as equal partners of the
teaching-learning process who in turn are also
required to contribute meaningfully to educational activities that occur in the
classroom. In other words, students are made to be *active participants
of information sharing **process* rather
than *passive recipients of information
presentation.* Weissglass (1992) has suggested that if mathematics is to
become vibrant and vital subject then student-to-student and
teacher-to-students interactions, or vice versa, must be encouraged and
promoted by teachers. A significant feature of a mathematics-learning
environment under such an approach is that both the teacher and students build
the mathematics together in developing special pride in learning activities
facilitated by the spirit of free and open investigations. The learning climate
in the classroom should provide an atmosphere of open communication between
students and their teacher through cooperation and collaboration during which
the teacher is expected to encourage students
to ask questions at the same time accept variety of problems from students. Mathematics instructional material should be relevant to the pupil’s
interests and needs allowing for student experimentation. On the other hand
mathematical problems should be such that they allow students to investigate
and develop variety of problem-solving strategies with the teacher facilitating
teacher-student discussions on the validity of these strategies through
reflective teaching and learning (Matang, 1998).

A
number of research results [e.g. Bishop, 1991a; Masingila, 1993; Pinxten, 1994; Saxe, 1982; Zaslavsky, 1994]
have shown that there are significant contrasting situations that exist between
the type of mathematics practices carried out in the everyday situations within
cultures and the way school mathematics are taught in schools. Masingila (1993)
in particular has highlighted
that “knowledge gained in out-of-school situations often develops out of
activities which: (a) occur in a familiar setting, (b) are dilemma driven, (c)
are goal directed, (d) use the learner’s own natural language, and (e) often
occur in an apprenticeship situation allowing for observation of the skill and
thinking involved in expert performance” (p. 18).

If
ethnomathematics is defined both as the cultural or everyday practices of
mathematics of a particular cultural group and a program that looks into the
generation, transmission, institutionalisation and diffusion of knowledge with
emphasis on the socio-cultural environment (Bishop, 1991b; D’Ambrosio, 1990; 1991), then it has a role to play in the context of the
teaching-learning process in the formal classroom. This is because
ethnomathematics is both context-relevant and problem-specific thus it provides
the necessary linkage between the everyday cultural practices of mathematics
and the teaching of the related but abstract concepts found in school mathematics
(Boaler, 1993; Kaleva, 1992, Pinxten, 1994). Moreover, it also has the
potential to enable students to not only make important connections between
them but also find the relevant meaning to many abstract mathematical ideas
taught in schools at the same time legitimises
the relevance for learning school mathematics (Boaler, 1993; Masingila, 1993). This approach
if taken on board will also be in line with the rationale for one of the most fundamental principles
of education, that is, teaching from *known
*to *unknown*. As argued by Resnick
(in Masingila, 1993) that “schools place too much emphasis on the *transmission of syntax* (procedures)
rather than on the *teaching of semantics*
(meaning) and this discourages children from bringing their intuitions to bear
on school learning tasks” (p. 18). Providing the necessary link between the
students’ ethnomathematical knowledge gained in out-of-school situations and
the formal mathematics learnt in school is where the role ethnomathematics
becomes fundamentally important particularly if students are to make any
meaning of the abstract mathematical concepts taught in school mathematics. In
other words, ethnomathematics complements the efforts of both the teacher and
students in the learning of formal school mathematics in providing the relevant
*contextual meaning* to many abstract
mathematical ideas which otherwise would be difficult for students to learn and
understand (Boaler, 1993; D’Ambrosio, 1991). What is required of the classroom teacher is to
build upon the ethnomathematical knowledge that students bring into mathematics
classroom from their everyday experiences. One way to achieve this is to
encourage students to make relevant
connections between these two worlds whereby the role of the teacher is to
facilitate mathematical activities in which students identify and use relevant
everyday cultural practices of mathematics to explain the conceptual meanings
associated with the abstract mathematical ideas found in school mathematics.
Such a teaching approach will also formalise the students’ ethnomathematical
knowledge gained through practical experiences in which students also develop sense of ownership to that
knowledge. This way it encourages the students to learn
mathematics in a more meaningful and relevant way. Teaching of school mathematics, where ethnomathematics plays a very
central role, can be more effective because it has the potential to
“yield more equal opportunities provided it starts from and feeds on the
cultural knowledge or cognitive background of students” (Pinxten, in Masingila,
1993, p. 20). An example of one such common ethnomathematical practices that
are familiar to many Papua New Guinean students and can be used in teaching
geometrical concepts such as properties of shapes, lines of symmetry, etc, is
given below in Figure 1.

Figure 1

(Source: Matang, 1996:67)

The above pattern
is part of one of the common geometrical patterns found on the walls of houses
usually woven using either sago stalks or bamboo. While the finished product is
not mathematically important, it is the planning, the structure, the imagined
shape, the perceived spatial relationship between object and purpose, the
abstracted form and the abstracting process that are of significant importance
to mathematics education (Kaleva, 1992; Matang, 1996). It is one of the many
familiar examples that is not only educationally rich in teaching a number of
formal geometrical concepts in school mathematics but can also be investigated
further for its higher mathematical insights commonly pursued in academic
mathematics. However, in the context of mathematics instruction in schools, it
makes the learning of mathematics more meaningful and relevant in providing the
students with even more reasons to learn mathematics.

** **

**Shift in mathematics teaching emphasis**

Based
on current observations of what is happening in the actual teaching-learning
situations in the mathematics classrooms in PNG, it would seem at least to this
author, that for a long time much of the teaching emphasis has been on what
Hiebert and Lefevre (1986) called *procedural
knowledge*. The disadvantage of this approach is that while it promotes the
development of procedures for obtaining correct answers in either an
arithmetical or mathematical problem, it pays little to no attention at all to
the many important relationships that exist between different parts of the
problem which Hiebert and Lefevre (1986) called *conceptual knowledge*. For instance, if a group of PNG secondary
school students were asked to give an answer to the fraction problem ½ x ¼, it
would take them less than one minute to give the answer _{}. On the other hand, it would take a long time for the same
group of students to explain the respective meanings of both the operational
and conceptual relationships between the operand fractions of ½ and ¼ and the
final answer of _{}. In many instances, if the author’s own experiences is any
indication, the students would simply give up on providing the explanations to
similar problems hence forcing the teacher to do all the explanation. The
situation is no different even at the secondary teacher training level
particularly at the University of Goroka in PNG. During a third year
mathematics course taught by this author at the University of Goroka, students
were asked to either explain or illustrate both the operational and conceptual
relationships between ½, ¼, and _{} to the arithmetical
problem ½ x ¼. It turned out that only less than half of the students in the
group provided any form of response with less than a third of the respondents
providing responses that were somewhat closer to the author’s expectations.

According
to Hiebert and Lefevre (1986), *conceptual
knowledge* is strongly characterised by the fact that it is rich in
relationship, that is, it is a knowledge network whereby linking relationships
are as prominent as the discrete pieces of information. The development of
conceptual knowledge is highly dependent on the construction of the important
relationships between different pieces of information. On the other hand,
procedural knowledge is made up of two distinct parts – the formal language, or
symbol representation system of mathematics and the algorithms, or rules, for
completing the mathematical tasks. While each knowledge type do complement each
other at some stage within the classroom, the problem arises when students are
faced with the real life problem situation that is unfamiliar to what they
learn in schools given the fact that problems encountered in real life are
never standard.

Advocating
strong teaching emphasis on conceptual knowledge development is based on the
observations that if one analyses the strategies used in solving everyday
problems in real life, one would find that there are many different ways to
solving the same problem. Making a choice on which of the strategies to use is
basically an individual’s choice. Thus, the current teaching emphasis on
procedural knowledge development in mathematics does not provide the students
with the flexibility to develop other problem solving strategies that are
equally important and necessary in solving problems in everyday life. Instead,
the current approach gives the students the impression that there is one and
only one standard method for solving all types of problems in everyday life.

In
accommodating the potential role of ethnomathematics in the teaching of school
mathematics, one of the first implications for the current mathematics
curriculum is that it should be flexible enough to accommodate the
ethnomathematical knowledge gained from everyday practices of mathematics that
students bring into the mathematics classroom. To achieve this, it is strongly
suggested that mathematics curriculum should include as (Masingila, 1993) has
pointed out a “wide variety of rich problems that: (a) build upon the
mathematical understanding students have from their everyday experiences, and
(b) engage students in doing mathematics in ways that are similar to doing
mathematics in out-of-school situations” (p. 19). In comparing the school-based
knowledge of general mathematics students with the experienced-based knowledge
of carpet layers, Masingila (1993) found that some of the difficulties the
students had in solving the floor covering problems were very much related to
their lack of exposure to rich, constraint-filled problems. Currently students
are not encouraged to, and in
some cases, even discouraged from making connections between how they do
mathematics in school and out-of-school situations. As a result, students do not see any relevance for the learning of school mathematics in relation to what they do and encounter outside of the boundaries of the formal mathematics
classroom. Integration of ethnomathematics into the mathematics curriculum not
only enables students to develop a wide variety of problem-solving strategies
but also legitimises their ownership of such knowledge. This in turn adds more
meaning to many abstract mathematical ideas, a subject which many students in
PNG come to perceive it to be boring, meaningless and non-reflective subject
(Matang, 1998).

Based
on the above discussions, it is obvious that ethnomathematics or indigenous
(cultural) mathematical knowledge has a role in the teaching of formal school
mathematics in that it is context-relevant and constraint-filled problem solving strategies
providing the necessary contextual meaning to many abstract mathematical
concepts found in school mathematics. In
order to accommodate the role of
ethnomathematics in mathematics teaching, the mathematics classroom teacher need to see herself/himself as the
facilitator of the teaching-learning process rather than an authority and
transmitter of knowledge in the formal mathematics
classroom. This requires the teacher to acknowledge students as equal partners
of the teaching-learning process in that they are seen as active participants
of information-sharing process
rather than passive recipients of information presentation. Utilising students’ rich ethnomathematical
knowledge in the classroom encourages the development of conceptual knowledge
base amongst students. Moreover it also enables students to develop wide
ranging problem-solving strategies that require both the teacher and students to further verify their validity
in a variety of both familiar
and unfamiliar situations thereby making
mathematics to be both meaningful and reflective subject.

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