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.
(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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