THE
ROLE OF ETHNOMATHEMATICS AND REFLECTIVE LEARNING IN MATHEMATICS EDUCATION IN
PAPUA NEW GUINEA
University of Goroka
Introduction
It is argued by a number of mathematics educators
around the world that the mathematical knowledge, ideas and intuitions
developed by a specific cultural group are largely ignored by the current
mathematics curriculum found within their respective education systems. This
situation is particularly apparent within the teaching/learning process whereby
such mathematical knowledge developed by children and derived from pre-school
experiences within their own environment is most often ignored during the
teaching of formal mathematical concepts. This conduct is based on the dominant
assumption that the only mathematics that children come to know is what they
learn in school. In particular, Masingila (1993) has argued that the
acquisition of knowledge in school all too often grows out of a ‘transmission’
paradigm of instruction and is largely devoid of meaning, that is to say that
there is a lack of context, relevance and specific goal during the process of
acquiring such knowledge. In other words, mathematics teaching in schools
places too much emphasis on the transmission of procedures rather than on the
teaching of meaning, thereby discouraging children from bringing their
intuitions to bear on school learning tasks (Resnick, in Masingila 1993).
The above situation causes far-reaching negative educational effects for mathematics education in PNG in terms of having a ripple effect on all facets of the teaching-learning process, starting from teacher education activities down to actual classroom activities at the elementary school level. These negative educational outcomes have also been highlighted by research evidence from a number of studies (e.g. Adetula 1990; Clarkson 1992; Lancy 1983; Lean, Clements and del Campo 1990; Saxe 1982; Souviney 1983) which have cited language, culture and learning modality as the three key factors contributing to the mathematics learning difficulties experienced by many students who come from diverse cultural backgrounds. In particular, the research evidence not only shows that much mathematical misunderstanding has occurred as a result of the kind of teaching practices adopted, but also shows how little understanding is achieved, even in situations where correct answers are given (Matang 1996). This situation applies with force to the case of PNG, where past and present mathematical learning difficulties experienced by many students reflect underlying problems existing in many mathematics classrooms.
Ethnomathematical Practices
Mathematics is a product of culturally based human
social activities (bishop 1991b; D’Ambrosio 1990, 1991) and all societies have
developed mathematical practices that are considered most appropriate to their
daily lives and cultures. These systems are referred to as ethnomathematics.
These cultural practices include jargons, codes, symbols, myths and specific
ways of reasoning and inferring (D’Ambrosio 1990, 1991) and may involve one or
more of the ‘universal’ mathematical activities of counting, locating,
measuring, designing, playing and explaining (Bishop 1991a). PNG, with a
staggering cultural diversity of more than 800 different languages, is no
exception, since such a rich cultural environment has the potential to not only
enhance further knowledge development, but at the same time form contrasting
situations for meaningful and reflective learning to occur. It is, however,
impor4tant to note that one of the key features of these cultural practices is
that, in nearly all cases, they depend on the contextual reality to provide the
necessary conceptual meaning of many of the mathematical concepts formally
taught in schools, rather than depending on abstraction which tends to
characterise many of the ‘western’ thoughts.
An example which highlights the importance of
contextual meaning can be found in the lack of representation of the concept of
zero in many of PNG’s numeration systems (Matang 1996). There is a logical appeal
in this, in that if a person fails to at least touch an object, then no further
discussion takes place on it. Thus it is no surprise to find that many PNG
students experience difficulties understanding the presentation, within the
formal classroom situation, of abstract mathematical knowledge, such as the
concept of zero, in the absence of its relevant contextual meaning.
The
Role of Reflective Learning in Mathematics’ Education
According to Wheatley (1992), reflection plays a
critically important role in mathematics learning; just completing tasks is
insufficient. Sigel’s view (cited in Wheatley 1992) is that reflection in
mathematics learning involves distancing oneself from the action of doing
mathematics, whereby second-order schemes are constructed. It is one thing to
ask a person to solve a problem, but it is quite another thing to ask the same
person to take their own action as an object of reflection. Persons who reflect
have greater control over their thinking and can decide which of several paths
to take, rather than simply being in action.
Moreover, there are also serious limitations of the currently used ‘explain-practice’ method of instruction in that students do not easily give meaning to abstract formulations which are not in their experience, mainly because of the way in which mathematics is viewed by many as a formal system to be learned in the abstract and then ‘applied to the real world’ (Masingila 1993; Sigurdson and Olson 1992; Wheatley 1992). To learners, new experiences become more meaningful only if they are related to previous experiences. Thus, in order to enable students to construct a mathematical world which is coherent and useful, mathematics instruction should begin with situations from which generalisations can be made. Mathematical formulations should flow from meaningful experiences rather than beginning with abstract formulations and then applying them to ‘real world’ problems.
An
Alternative Approach to Mathematics Teaching
It seems logical to suggest that there is a strong
connection between what is found in the formal school system in terms of the
current teaching practices in mathematics and the learning difficulties
experienced by many PNG students. This calls for an alternative approach to
mathematics education in PNG in terms of underlying philosophy, the type of
curriculum, and the type of teaching practices employed within the context of
the classroom. One way of addressing the above problem is to adopt a more
interactive and reflective mathematics teaching approach, based on the
learner’s own sociocultural background. This will not only provide the students
with the necessary contextual meaning for understanding the conceptual meaning
of many abstract mathematical concepts, but at the same time portray mathematics
as a meaningful, attractive and reflective subject, thereby developing critical
thinking in mathematics (Matang 1996). In other words, it is strongly suggested
that reflective teacher-student interactions should characterise all formal
mathematics classroom activities in PNG. This can be achieved by using
examples, illustrations, experiences and problem-certered learning situations
found within the student’s own sociocultural environment. According to Wheatley
(1992), establishing an environment which encourages reflection is a complex
exercise because, apart from selecting relevant tasks, the teacher must also
negotiate social norms in which students can become a community of inquirers.
This will apparently require both the students and teacher to learn to talk
mathematics and to listen to each other. As argued by Wheatley (1992:539)
effective mathematics instruction:
·
promotes
autonomy and commitment;
·
is
based on theoretical models of children’s mathematics learning;
·
has
‘sense-making’ as a goal;
·
involves
the negotiation of social norms; and
·
encourages
and facilitates reflection.
The above prescription can also be seen as one way
of providing accountability to the general, particularly considering the time
and other resource allocations required by mathematics within the formal
education system.
Implications for Teacher Education
There are basically three key areas at the teacher
level where such an approach will have implications: the role of the teacher;
teacher beliefs and values; and teacher background knowledge (Matang 1996).
Of the three areas mentioned, defining the role of a
teacher is probably the most difficult one, but at the same time it is
considered one of the most crucial factors in the successful implementation of
any curriculum change.
In the light of the reflective and interactive
teaching approach envisaged, this is a fundamental issue, I that such an
approach to mathematics education will not only require a change in one’s
underlying views concerning mathematics teaching, but also requires a
redefining of the role of a teacher within the classroom environment. In other
words, it requires a teacher to change from being an ‘authority’ and
‘transmitter’ of knowledge to being a ‘facilitator’ in the teaching/learning
process.
Secondly, teacher beliefs and values concerning the
role and nature of mathematics in society are also an important factor. The
current mathematics teaching approach, because of its emphasis on the
acquisition of skills involving learning of rules and procedures, most often
results in teachers adopting the ‘expository’ teaching method.
The final implication relates to teacher content
knowledge in mathematics. Though this may not appear to be consistent with what
is being envisaged, if teachers are to be successful in effectively employing
the reflective and interactive teaching approach, it is necessary that they be
given every opportunity to do an in-depth and reflective investigation of
mathematical content knowledge during their initial teaching training. This will
inevitably require mathematics teacher educators also to change their current
view and teaching approach. This will enable them to adopt and design
mathematics teacher training programs that are characterised by reflective
teacher/student interactions with emphasis on critical analysis of mathematical
knowledge through the use of projects and critical investigative work (Matang
1996). It is only then that mathematics education in PNG will be seen to be not
only fulfilling its educational obligation to the learners, but also
simultaneously achieving the objective of producing a mathematically literate
PNG society.
Conclusion
The current ‘explain-practice’ method of instruction
has negative educational outcomes for mathematics education in PNG in that it does
not effectively enable learners to meaningfully construct mathematical
relationships. This situation is mainly a result of mathematics teaching done
in the absence of the contextual meaning associated with many of the formal
mathematical topics taught in schools. An alternative teaching approach,
underpinned by reflective learning as its central theme, is therefore
suggested, using examples, illustrations and experiences from the learner’s own
socio-cultural
environment. Such an approach should not only promote self-esteem on the part
of the learners, but also at the same time portray mathematics as a meaningful,
attractive and above all, a reflective subject.
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