Ethnomathematics and the Age-old Debate of Conceptual Knowledge versus Procedural Knowledge in Mathematics: Implications for Mathematics Education in Papua New Guinea

Rex A. Matang

The difficulties experienced by many school students in Papua New Guinea (PNG) in learning school mathematics under the current mathematics curriculum based heavily on Western, mainly Australian models, is well known. Furthermore, the debate on whether conceptual knowledge or procedural knowledge is more important does not even make this situation any better. Using the literature on research in ethnomathematics and mathematics education, this discussion paper proposes for an integration of ethnomathematics into the formal mathematics curriculum as one way to address student-learning difficulties where mathematics teaching emphasis is more concept-oriented. This is based on the rationale for the most basic educational learning theories of teaching from known to unknown that the learning of mathematics is more effective and meaningful if the teaching of mathematics progressively begins with the more familiar learning situations embedded in the everyday mathematical practices of the learner’s own socio-cultural environment.

Introduction

The complexity involving the question of what should be taught, or for that matter, considered relevant to be included in the formal school curriculum, is a well-known age-old educational debate particularly in mathematics [e.g. Carpenter, 1986; Hiebert and Lefevre, 1986; Davis, 1986]. The debate, while remains healthy, it somewhat stems from speculations made about which type of knowledge is more important in terms of its relevancy, or what might be an appropriate balance between them to be considered a prerequisite for their inclusion in the prescribed school curriculum. Further questions on how students learn mathematics, or how they should be taught, also opens up further floodgates to other general questions relating to knowledge acquisition by children. More particularly in mathematics, the discussion of what constitutes conceptual and procedural knowledge, and the distinction between the two notions of ‘concept’ and ‘procedure’ also play an important role in providing some insight into how children acquire knowledge, not only in the formal school system but also informally in out-of-school situations.

While attempting to identify some of the obvious differences between the conceptual and procedural knowledge in mathematics, the main thrust of this discussion paper aims to provide some insight into both the advantages and the disadvantages of the two types of knowledge as one way to address those concerns raised by some sections of the PNG society as to the direction which mathematics education should take in the face of the ever-increasing presence of computers and calculator-related technology in schools (Matang, 1996; 1998). These concerns are addressed in the context of defining the role of ethnomathematics in the mathematics classroom in terms of teaching emphasis that is more concept-oriented and its educational benefits to students in their struggle to not only learn but to make sense of the mathematical concepts embedded in school mathematics in relation to everyday practices of mathematics.

Conceptual and Procedural Knowledge in Mathematics

According to Hiebert and Lefevre (1986), conceptual knowledge in mathematics is strongly characterized as the knowledge that is rich in relationships. In other words, it is a knowledge-based 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. Relationships interconnect individual facts and propositions so that all pieces of information are linked to some kind of knowledge network. In essence, a unit of conceptual knowledge cannot be treated as an isolated piece of information; by definition it is part and parcel of conceptual knowledge only if an individual recognizes its relationship to other pieces of information.

On the other hand, procedural knowledge is made up of two distinct parts – first, the formal language, or symbol representation system, of mathematics and the other part comprising of the algorithms, or rules, for completing specific mathematical tasks. The first part includes familiarizing with the symbols used to represent mathematical ideas and an awareness of the syntactic rules for writing those symbols in an acceptable form. It is important to note that, in general, knowledge of the symbols and syntax of mathematics implies only an awareness of surface features, not knowledge of meaning. The second part of procedural knowledge consists of rules, algorithms, or procedures used to solve mathematical tasks. In other words these are specific step-by-step instructions that prescribe how one needs to go about in completing the tasks at hand. A key feature of procedural knowledge that differentiates it from conceptual knowledge is that they are executed in a predetermined linear sequence where it can be thought of as production systems that require some kind of input (Hiebert and Lefevre, 1986; Davis, 1986; Carpenter, 1986).

While each knowledge type do complement each other in almost all teaching-learning situations within the confines of the formal classroom, it is however argued here that there are significant practical difficulties associated with the teaching approaches that strongly emphasise procedural knowledge in the context of actual use of such knowledge by students in everyday situations. These difficulties become particularly more obvious when students are faced with the need to solve everyday problems that are particularly unfamiliar to the type of mathematical problems they were asked to solve in the formal classroom setting. Masingila (1993) in particular argued that because of the students’ limited exposure (or lack of it) to a variety of real life problems, majority of them are unable to solve these problems. It is also argued here that the difficulties faced by students in solving many practical problems is indicative of one of the weaknesses of the current teaching emphasis that promotes procedural knowledge whereby teachers so often do not encourage students to utilize their out-of-school mathematical experiences to complement the learning of formal school mathematics. Thus in reality, many students do not see the need to learn school mathematics further adding to barriers of meaningful learning of mathematics as many of these formal mathematical methods are viewed by students to be inappropriate in solving many everyday practical problems at hand.

Current mathematics teaching emphasis

Current observations on actual teaching-learning situations in the mathematics classrooms in PNG indicate that for a long time much of the mathematics teaching emphasis has been mostly biased towards procedural knowledge (Clarkson & Kaleva, 1993; Kaleva, 1995; Matang, 1996). The emphasis of such teaching approach is based on the dominant assumption that “mathematics can be learned in school, embedded within any particular learning structures, and then lifted out of school to be applied to any situation in the real world” (Boaler, 1993, p. 12). The evidence of such teaching emphasis is mainly reflected through the expository teaching methods employed by the teachers hence dominance by classroom practices such as rote learning and performance of rules and algorithms are common features of many mathematics classrooms. The teacher is basically seen as the authority and transmitter of knowledge shared within the classroom. The extent to which this happens at the various levels of the National Education System differs from one level to another level and between each individual teacher and their lesson presentations.

The above observation is further supported by the preliminary results from a small research conducted by the author through a questionnaire given to a group of third and fourth year trainee mathematics teachers at the University of Goroka. The questionnaire simply asked the students to represent using the diagram the fraction operation and most importantly to explain in words the relationships between the three fractions namely, , and . Out of a total of 20 students who responded, only 2 students, representing 10%, responded correctly. This result is of great concern particularly given the fact that these students are training to become mathematics teachers. While it would take them less than one minute to give the answer , it would however take a long time for the same group of students to explain the relationships between different parts of the problem to the answer obtained including the interpretation of the respective meanings. In almost all instances, half the students would simply give up providing the explanations hence indirectly forcing the teacher to explain. The result while not conclusive it is indicative of one of the fundamental problems of the current mathematics teaching emphasis in Papua New Guinea that strongly emphasizes the teaching of procedural knowledge whereby mathematics classrooms are most often dominated by the more familiar teaching pattern of explain-example-exercise method (Matang, 1996). The main disadvantage of this approach is that it does not enable the students to realise and appreciate the important conceptual relationships that exist among the different pieces of information found in many mathematical operations. It is however acknowledged here that though many of the mathematics classrooms in PNG are dominated by the 3-e teaching pattern, teachers are so often unaware of its long-term implications in terms of student benefits probably due to their ignorance of the underlying philosophical base of such an approach to mathematics teaching.

The role of ethnomathematics in mathematics education

A number of research results [e.g. Bishop, 1991a; Boaler, 1993; 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). Furthermore, a number of mathematics educators [e.g. Bishop, 1991a; Carpenter, 1986; D’Ambrosio, 1990; 1991; Kaleva, 1995] have strongly acknowledged that before children enter any formal school system they posses highly developed informal systems of arithmetic. Carpenter (1986) in particular further asserts that contrary to popular notions, most young children are relatively successful at analyzing and solving basic addition and subtraction problems using their own invented informal modeling and counting strategies. He argues that the solution procedures used by children appear to be linked to conceptual knowledge reflecting better problem-solving models than the ones taught in schools which most often appear to be superficial.

If ethnomathematics is considered both as the cultural or everyday practices of mathematics of a particular cultural group including those of school children, and a program that looks into the generation, transmission, institutionalization and diffusion of knowledge with emphasis on the socio-cultural environment (Bishop, 1991b; Borba, 1990; D’Ambrosio, 1990; 1991; Pinxten, 1994), then it has a role to play in the context of the teaching-learning process in the formal classroom. This is because ethnomathematics, unlike the school mathematics, is both context-relevant and problem-specific thus provides the necessary linkage between the everyday cultural practices of mathematics and the teaching of school mathematics (Boaler, 1993; Masingila, 1993; Saxe, 1982; 1983). As Borba (1990) puts it “the ethnomathematics developed by different groups are likely to be more efficient at solving problems related to their cultures than academic mathematics is because the ethnomathematics developed by a given cultural group is linked to the obstacles which have emerged in this group. Thus formalizing the role of ethnomathematics not only enables students to make important connections between in-school and out-of-school mathematics but also helps them to find relevant meanings to many abstract mathematical ideas taught in schools at the same time legitimizes the reasons for learning school mathematics (Boaler, 1993; Masingila, 1993). Recognition of students’ ethnomathematical knowledge also increases their self-esteem, which in turn increases their performance on school mathematics. In comparing the final grades of students in Intermediate Algebra at the Orange Coast College in California, Arismendi-Pardi (2001) found that there was statistically significant difference between the mean scores for students who were taught with ethnomathematical pedagogy and those who were taught without an ethnomathematical pedagogy. The mean score for students who were taught with ethnomathematical pedagogy was higher than the mean score for those students who were taught without ethnomathematical pedagogy. This result further support the argument that if the role of ethnomathematics is properly utilized in the classroom then it has the potential to improve the mathematics performance of students in schools at the same time narrow the gap between the way mathematics practices are carried out in everyday situations and the way school mathematics are taught in schools.

If the role of ethnomathematics is taken on board it will also be in line with the rationale for one of the most fundamental educational learning theories, that is, to teach from known to unknown. In the case of mathematics teaching, this means teaching from concrete to abstract. In supporting this approach, Resnick (in Masingila, 1993) argued 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 of ethnomathematics becomes fundamentally important. Accommodating the role ethnomathematics at this stage is necessary particularly if students are to establish any meaningful connections between the school mathematics and their own everyday mathematical experiences in real-life situations. In other words, ethnomathematics complements the efforts of both the teacher and students in the learning of formal school mathematics in terms of meaning-making relating to somewhat abstract mathematical ideas which are otherwise difficult for students to learn and understand (D’Ambrosio, 1991; Matang, 1996). Therefore what is required of the mathematics classroom teacher to do is to basically build upon the students’ ethnomathematical knowledge that they bring to school from their everyday experiences. This can be achieved firstly by the teacher recognising students’ ethnomathematical knowledge, and then utilising them to teach school mathematics whereby students are further encouraged to make important meaningful connections between the two mathematical worlds. This approach to mathematics teaching will also formalise the students’ informal mathematical knowledge where students also develop sense of ownership to that knowledge thus contributing to their self-esteem. This teaching approach also encourages the students to learn mathematics in a more meaningful and relevant way in which both the teacher and students are seen as equal partners of the teaching-learning process (Matang, 2001). Teaching school mathematics where ethnomathematics plays a very central role has every chance to be more effective and successful because it has the potential to generate more equal opportunities for all provided it starts from and feeds on the cultural knowledge or cognitive background of students (Pinxten, in Masingila, 1993).

Implications for Mathematics Education

The most likely implications for accommodating the role of ethnomathematics will be basically in two main areas namely mathematics curriculum and pedagogy of mathematics. Owing much to its rigidity the current mathematics curriculum does not give prominence to the experienced-based ethnomathematical knowledge of students that they bring into the formal classroom to give meaning to somewhat abstract concepts embedded in school mathematics. In many ways the current classroom practices, which are governed by the existing mathematics curriculum, indirectly discourages students from making important meaningful connections between how they do mathematics in school and out-of-school situations though it may not be the initial intentions of the current curriculum. As a result, students do not see any relevance for the learning of school mathematics in terms of its usage in everyday problems of survival. It is therefore strongly suggested that the mathematics curriculum must be re-designed in such a way that it is flexible enough to accommodate the ethnomathematical knowledge of students gained from everyday practices of mathematics during the teaching of school mathematics. To achieve this, Masingila (1993) has suggested that the current mathematics curriculum should be reviewed to include 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). 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 found in school mathematics, a subject which many students in PNG come to view it as boring, meaningless and non-reflective subject (Matang, 1998; 1999).

In terms of pedagogical implications, because the current mathematics classroom practices place too much emphasis on the procedural knowledge which basically involve remembering rules and performing formulas and algorithms, it is therefore suggested that the teaching emphasis must now be redirected to focus more on the conceptual knowledge development (Clarkson & Kaleva, 1993; Matang, 1999). This means that the mathematics classroom practices including the teaching methods employed by teachers must be seen to emphasise mathematics teaching that is both concept-oriented and context-based. Undoubtedly these can both be facilitated by the ethnomathematical knowledge of students. In practical terms, what is required of the mathematics teacher is to plan his her lessons in such a way that students’ ethnomathematical knowledge is incorporated into the planning of the individual lessons. To successfully accommodate the role of ethnomathematics in the mathematics classroom, it is particularly important to realise that it is a prerequisite requirement for teachers to see themselves as the facilitators of the teaching-learning process rather than authorities and transmitters of knowledge (Matang, 1998; 2001; Weissglass, 1992).

One obvious advantage of such an approach to teaching mathematics is that apart from equipping the students with the relevant skills and knowledge necessary for survival in everyday life, it also enables them to make meaningful connections of many abstract mathematical ideas in solving many everyday real-life problems. Moreover, given the fact that everyday real-life problems are never standard it will be to students’ disadvantage if teaching emphasis is only aimed at procedural knowledge. This is because such an approach to teaching only limits students’ mathematical abilities to comprehend and analyse everyday problem, which so often may require a number different ways to solve.

Conclusion

Based on the above discussions, it seems obvious that ethnomathematics or indigenous (cultural) mathematical knowledge has a role to play in the teaching of formal school mathematics in that it is context-relevant and constraint-filled problems providing the necessary contextual meaning to many abstract mathematical concepts found in school mathematics. To accommodate the role of ethnomathematics, it is suggested that 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 whereby students’ out-of-school mathematical experiences are utilized to complement the efforts of both the teacher and students in learning school mathematics. 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 not only encourages the development of conceptual knowledge base amongst students but is also an attempt to bring mathematics classroom practices more in line with the practices of everyday context-based problem solving strategies. Such teaching approach will enable students to develop wide ranging problem-solving strategies that require both the teacher and the students to further verify their validity in a variety of both familiar and unfamiliar teaching situations. Thus, in the long term this will not only make mathematics to be meaningful and reflective subject but relevant to solving everyday problems found in a complex and an evolving technologically-oriented society.

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