STAKEHOLDERS' RECEPTIVENESS TO AN ETHNOMATHEMATICS CURRICULUM FOUNDATION: THE CASE OF CAMEROON by HENRY KANG D.I.P.E.S. 1, University of Yaounde 1,1992 M . E d . , University of Regina, 1996 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E F A C U L T Y O F G R A D U A T E STUDIES (Curriculum and Instruction) T H E U N I V E R S I T Y O F BRITISH C O L U M B I A November 2004 © Henry Kang, 2004 ABSTRACT The purpose of this study is to assess the curriculum stakeholders' receptiveness to a curriculum built on an ethnomathematics foundation. The stakeholders w h o participated i n this study were secondary school students, secondary mathematics teachers, pedagogic personnel and teacher educator. The students and the teachers were from two secondary schools i n Cameroon, one of w h i c h was public and the other private. Data collected for analysis include audiotaped interview transcripts, questionnaires and field notes from classroom observations of the teaching of an ethnomathematics unit. From an analysis and interpretation of the data, a picture emerged of the stakeholders' level of interests and concern about adopting an ethnomathematics curriculum foundation. Findings from this study indicate that the stakeholders are generally receptive to an ethnomathematics curriculum but are also concerned about the demands such a curriculum w o u l d have on the cultural knowledge background of those i n the mathematics classroom. The study also indicates that the stakeholders' encounter w i t h an ethnomathematics approach can help them develop a broader view of mathematics and raise awareness of the presence of mathematical processes i n cultural practices. The study notes that the stakeholders demonstrated both situational and actualized interests that were complex and not fixed. W h e n a particular cultural activity facilitated mathematics teaching and learning, the stakeholders exhibited actualized interest to an ethnomathematics curriculum. W h e n the lesson activities demanded m u c h from the stakeholders i n terms of cultural background knowledge and the teaching and learning implements, the stakeholders showed situational interest. The study also suggests that stakeholders' interests i n an ethnomathematics curriculum are complex and interrelated, and are influenced more by external factors than by a given phenomenon. ii The findings also suggest that stakeholders had some concerns regarding an ethnomathematics curriculum and that these concerns were more complex and varied w i t h each stakeholder according to h o w each viewed her/his role i n the education process. The study's analysis of the stakeholders' receptiveness provides useful and important implications for relevant mathematics education, teacher education and above all, curriculum reform. It also highlights the importance of involving all those concerned w i t h the education process to play major roles in the curriculum development process. iii TABLE OF CONTENTS Acceptance Page i Abstract ii Table of Contents iv Acknowledgements vii Dedication viii List of Tables ix List of Figures x C H A P T E R 1 T H E N A T U R E A N D SIGNIFICANCE OF THE STUDY 1 1.1 Introduction 1 1.2 Purpose of the Study 2 1.3 Rationale of the Study 3 1.3.1 Performance i n Mathematics -. 5 1.3.2 Cultural Relevance 8 1.3.3 Education Reforms 12 1.4 Significance of the Study 14 1.5 Conclusion 15 C H A P T E R 2 T H E O R E T I C A L PERSPECTIVES 17 2.1 Introduction 17 2.2 Conceptual Components of Receptiveness 17 2.3 Philosophical and Theoretical Foundations of E t h n o m a t h e m a t i c s . . . 24 2.4 2.3.1 Some Current Definitions of Ethnomathematics 25 2.3.2 Etymology and Operational Definition 27 Conclusion 40 C H A P T E R 3 REVIEW OF R E L A T E D LITERATURE 41 3.1 Introduction 41 3.2 Evolution of Formal Education 41 3.2.1 Pre-colonial Education and the Birth of Formal Education. 44 3.2.2 Colonial and Post-colonial Education 51 iv 3.3 The Current State of Education 66 3.4 Studies on Ethnomathematics 69 3.5 Conclusion 81 CHAPTER 4 RESEARCH METHODOLOGY 83 4.1 Introduction 83 4.2 Statement of Research Overview 83 4.3 Methodology Used i n this Study 85 4.4 The Research Site 91 4.5 The Schools 93 4.6 The Stakeholders 96 4.7 4.6.1 The Pedagogic A d v i s e r 97 4.6.2 The University Faculty 98 4.6.3 The Teachers 99 4.6.4 The Students 99 Data Collection 101 4.7.1 Step One: Questionnaires 102 4.7.2 Step Two: The Ethnomathematics Unit 104 4.7.3 Step Three: Questionnaires and Interviews 107 4.8 Data Analysis 109 4.9 Validity and Reliability of Data Collection and Analysis 110 4.10 Conclusion 112 C H A P T E R 5 PRESENTATION OF FINDINGS 113 5.1 Introduction 113 5.2 Findings 113 5.2.1 The Nature of Mathematics 114 5.2.2 Mathematics and Culture 137 5.2.3 School Mathematics C u r r i c u l u m 145 5.2.4 C u r r i c u l u m Reform Process 169 5.2.5 Importance of Formal Education 178 5.2.6 Possibilities of an Ethnomathematics C u r r i c u l u m 187 5.3 Summary 193 5.4 Conclusion 195 v C H A P T E R 6 A N A L Y S I S OF FINDINGS 197 6.1 Introduction 197 6.2 Stakeholders' Perception of an Ethnomathematics C u r r i c u l u m . . . . 198 6.3 Stakeholders' Levels of Interests towards an Ethnomathematics Curriculum 6.4 6.5 206 Stakeholders' Levels of Concern towards an Ethnomathematics Curriculum 221 Conclusion 229 C H A P T E R 7 CONCLUSION, IMPLICATIONS, & R E C O M M E N D A T I O N S . 234 7.1 Conclusion 234 7.2 Implications 235 7.3 Recommendations 239 REFERENCES 243 APPENDICES 261 Appendix A 262 (i) 263 Letters of Information and Request for Consent (ii) Interview Protocol 282 (iii) Questionnaires 284 Appendix B 317 Ethnomathematics Unit 318 vi ACKNOWLEDGEMENT I want to express sincere and grateful appreciation to the following people for making the completion of this dissertation a reality: First of all, I would like to acknowledge my debt of gratitude to the professors and graduate students of the University of British Columbia's Faculty of Education for their kind and thoughtful support during my doctoral studies. In particular, I would like to thank my supervisor Dr. Susan Pirie first for agreeing to cheer my dissertation committee, guiding me throughout my course work and raising the right questions at each stage to bring about internal consistency in the dissertation. I am deeply indebted to her and to the other members of my dissertation committee namely Dr. Cynthia Nicol and Dr. Lyndon Martin for their generosity, emotional support, intellectual advice and patient flexibility at various stages during my course work and the writing of this dissertation. I am also heavily indebted to Dr. John Willinsky for supporting me financially through research assistantship positions and for initially chairing my dissertation committee. His mentorship during my course work was very instrumental in the conceptualization of this project. I would also like to gratefully extend my acknowledgement to the Centre for Cross Faculty Inquiry in Education, formerly Centre for the Study of Curriculum and Instruction in the Faculty of Education at tbe University of British Columbia, and in particular, Dr. Karen Meyer for the financial support in the form of graduate assistantships during her tenure as director. This eased the pain of doctoral studies. v Sincere appreciation is extended to the students and teachers of Cameroon College of Arts, Science and Technology (CCAST) Bambili and City College of Commerce (CCC) Mankon, the North West provincial pedagogic adviser for mathematics, the Teachers' Resource Centre in Bamenda, the professors at Ecole Normale Superieure (ENS) of the University of Yaounde 1, Cameroon. I am deeply indebted to their cooperation and participation in this project. I also am indebted to a variety of friends and colleagues in Canada and Cameroon whose encouragement kept me going: the M U Group at UBC, Dr. Lynn Fels, Ndoh Martin Akwo, Ewi Alfred Zuoh, and to P. N . Eba for his teachings during my years in ENS Bambili and for the kind support during my data collection. Thank you. I would equally like to extend my appreciation to Dr. Leke Tambo of the University of Buea, Cameroon for his mentorship going way back when I was an undergraduate student at ENS Bambili. His believe in my abilities and potential, and his steadfast support is far more appreciated than he knows. Finally, I would like to express my heartfelt thanks to my wife Francoise and our son Dylan-Michel for bearing with me throughout this project. Without the moral support and continuing encouragement from you and our extended families in Cameroon and Canada, this dissertation would not have been possible. I sincerely apologize for anyone who is not properly acknowledged here. vii DEDICATION This dissertation is dedicated w i t h profound love to m y parents E w i Michael Inah and N s i h Lamenda E w i w h o i n 2001, both slipped the sturdy bonds of this earth to touch the face of G o d while I followed m y dream i n Canada. M a y their souls rest i n peace! I equally dedicate this work to m y son and father-incarnate Dylan-Michel, and m y lovely wife Francoise, for their love, k i n d support and patience throughout the course of this work. viii LIST OF TABLES TABLE Page 5.1 Experience i n C u r r i c u l u m Reform by Stakeholders 170 6.1 Stakeholders' Views of Mathematics and Ethnomathematics 201 ix LIST OF FIGURES FIGURE Page 2.1 Ethnomathematics: interaction between culture and mathematics . . . 39 3.1 Dimensions of Ethnomathematics Research 71 x Chapter 1 NATURE AND SIGNIFICANCE OF THE STUDY 1.1 Introduction There is burgeoning literature (Bockaire, 1988; Carraher, Carraher & Schliemann, 1987; Gay & Cole, 1967; Gerdes, 1988 & 1994; Knijnik, 1997a, 1997b; Lakoff & N u n e z , 2000; Masingila, 1994 & 1995; Saxe, 1985, 1988 & 1991; Zaslavsky, 1973 & 1994) to demonstrate that the mathematics, w h i c h most people learn i n contemporary schools, is not the only mathematics that exists. This demonstration of other forms of mathematizing has called into question commonly observed approaches to mathematical learning and teaching such as rote memorization, d r i l l and practice, skills being taught out of context, a lack of connectedness w i t h i n mathematics and responsiveness to the learner. In addition to the contributions d r a w n from the growing literature cited above, other research has also indicated considerations for enhancing the learning of mathematics to embrace the concept of cultural relevant teaching (Gilliland, 1995; Taylor & Stevens, 2002; Zaslavsky, 1994). This teaching perspective has been shown to be a promising aspect of education reform by seeking to promote academic success centred on students' cultural identities or cultural backgrounds. These revelations and suggestions seem to have been embraced by the Cameroon government and a growing cadre of Cameroonian educators w h o have begun to promote a new vision of mathematics education similar to what many multicultural societies ascribe to. In 1995 the Cameroon government sponsored a one-week national forum on education involving Cameroonians of a l l walks of life w i t h the aim of formulating a new educational policy for the country. This largest 1 gathering of Cameroonians since independence came at an unprecedented time i n the nation's history of educational development and took place w i t h i n a background of popular demands for democracy, decentralization, effective management, accountability, pedagogic reform and above all, relevance i n education ( M I N E D U C , 1995). The last issue i.e. relevance i n education was regarded as the most critical. Yet, government efforts at addressing this issue and many others have been slow and sporadic. Mathematics education is one of those areas that have received little attention. A n d while many educators i n Cameroon believe that a relevant cultural perspective to the school curriculum may be the solution, there is little research work undertaken to understand h o w the curriculum stakeholders especially the teachers i n the field w i l l receive such a perspective. A n ethnomathematics curriculum foundation may provide the relevant cultural perspective and knowledge of how the stakeholders w i l l respond to such a curriculum and this is essential before such a proposal is submitted to the Cameroonian government for consideration. 1.2 Purpose of the Study The purpose of this project is to examine the stakeholders' (students, teachers, pedagogic advisers/inspectors, and university faculty) interests towards a proposed ethnomathematics and response foundation to the school mathematics curriculum i n Cameroon. In other words: How receptive are the curriculum stake- holders in Cameroon to a proposed ethnomathematics foundation to the school mathematics curriculum? To design a relevant and responsive school mathematics curriculum, it is important to understand the characteristics of the people w h o w i l l engage i n it; and to assess the people's interest, it must be done w i t h respect to meaningful mathematical activities or objects, however broadly these activities or objects are defined. Furthermore, a thorough analysis of interest requires even more than a consideration of the person and the mathematics activity, because a personactivity interaction occurs within a social context, and the context can have a 2 considerable influence on that interaction, and thus on the person's interest. Thus, to assess the stakeholders' receptiveness to this proposed new vision, the study (1) developed a sample unit on a mathematics topic taught i n secondary schools i n Cameroon. This sample unit incorporated illustrations of mathematical concepts d r a w n from cultural practices i n Cameroon. The sample unit was then tested w i t h secondary mathematics teachers for its efficacy i n the teaching and learning of mathematics; and (2) examined the response of the stakeholders towards the proposed alternative curriculum foundation by having them talk about their interests, concerns, potential problems and the possibility of implementing such a curriculum i n Cameroonian secondary schools. 1.3 Rationale of the Study It is generally agreed that mathematics knowledge is created and needed by all humans (Ascher, 1991; Bishop, 1988a, 1990; Burton, 1988; D'Ambrosio, 1985; Gerdes, 1997), but the process of imparting the formalized aspects of that knowledge i.e. school mathematics, has not yielded great results i n many a society. Numerous studies have documented the poor level of students' performance and the declining enrolment numbers i n mathematics and mathematics related fields of studies i n high schools and universities i n many countries around the w o r l d (see for example, Eshiwani, 1985; Hogbe-Nlend, 1985; Masingila et al., 1996). This situation is even more deplorable i n developing or less developed regions of the w o r l d like Cameroon that are still struggling to emerge from the grip of their colonial past. Inappropriateness of the school curricula has been blamed for this situation, w h i c h has prompted the need for reforms, by the Cameroonian government. This is because school mathematics i n Cameroon is continuously being taught as a set of rules and formulas that students have to memorize so as to be able to solve series of problems d r a w n out of life situations alien to the students. Yet, many reforms have been carried out over the last twenty years to remedy this situation w i t h dismal 3 success (Ngwana, 2003). This is because most of the educational reforms, especially i n the area of mathematics education, have for the most part focused on coating school mathematics problems w i t h a thin veneer of 'real w o r l d ' associations. The continuous failure of these reforms has also been attributed to the poor process of reform implementation (Ndongko & Tambo, 2000). This is because the major stakeholders usually affected by these reforms such as students and teachers are often not involved i n the development and field-testing phase of these reforms. While it is encouraging to have an official policy statement on education stating that "the main aim of education i n Cameroon is the reviving of the Cameroonian culture and the reestablishment of a Cameroonian personality" ( M I N E D U C , 1995, p. 2), what exactly those Cameroonian cultural elements are that should be incorporated into school curricula, and what sort of Cameroonian personality we expect to re-establish is not clear. The transplantation of curricula from the highly industrialized capitalist nations to formerly colonized regions like Cameroon i n the 1960s (e.g., the so-called 'African Mathematics Program') fanned the negation of indigenous Cameroonian mathematics. This has been cited as one of the fundamental causes for the verified l o w levels of attainments i n mathematics (Fasheh, 1982; Bishop, 1990), students having to learn mathematics d r a w n from cultures completely alien to theirs. This project's focus is developed partly i n response to the existing disconnection between school and indigenous mathematics, partly to the failure of past reforms, and partly i n response to the ever-decreasing number of students enrolled in mathematics and mathematics related courses at the h i g h schools and universities i n Cameroon. The declining situation has necessitated the development of a curriculum that is more i n line w i t h the present realities i n Cameroon by incorporating cherished cultural and indigenous knowledge practices familiar to Cameroonian children. While many educators i n Cameroon (Eba, 1993; Nji, 1994; N d o n g k o & Nyamnjoy, 2000; N g w a n a , 2003) believe that a cultural perspective to the school curricula may be just what is 4 required, there is little research work undertaken to understand how the curriculum stakeholders w i l l receive such a perspective. The foregoing discussion draws its impetus from three main issues: the problem of poor performance i n mathematics education; the issue of cultural relevance i n mathematics i n terms of content, process and methodology; and finally the issue of educational (curriculum) reforms. These issues are discussed at length i n the succeeding sections. 1.3.1 Performance in Mathematics Mathematics i n most developing countries such as Cameroon is continuously being taught as a set of rules and formulas that students have to memorize i n order to apply these i n solving textbook problems d r a w n out of life situations alien to the students'. Appropriateness has hardly been a guiding principle w h e n developing a school mathematics curriculum. For most students and teachers i n Cameroon, studying mathematics is basically to enable them to pass the many public examinations they engage i n at various levels of education. After close to a quarter century since school mathematics was made compulsory to all junior secondary schools, the data o n students performances has not been encouraging, to say the least. Reforms over the years have focused on identifying clear objectives for mathematics education without a contextual philosophy. A l t h o u g h the cited objectives of teaching mathematics usually assert k n o w i n g certain mathematical facts and being able "to think correctly, logically, and scientifically" ( M I N E D U C , 1995) this has not been the outcome. The reality is that Cameroonian children are failing to learn mathematics effectively i n school. Ekane (2000) has documented the degree of failure through the analysis of test scores and students' performances i n mathematics certificate examinations. However, research findings based on testing procedures do not fully reveal the nature of the difficulties experienced by Cameroonian children when faced w i t h the 5 task of learning "Western" mathematics i n the school environment. N o r do such findings reveal the degree to which present approaches to mathematics education result i n Cameroonian "children perceiving school mathematics more i n terms of a meaningless ritual than as a purposeful pursuit" (Hogbe-Nlend, 1994, p. 13). The result is that after several years of schooling many children have only learned the answers to mental arithmetic that have little relationship w i t h the w o r l d as they k n o w it. M u c h of this unusable mathematical knowledge is soon forgotten, both between periods of school attendance, and once school days are left behind. If this situation is to be remedied, Cameroonian people and educators must together negotiate the what, how and why of mathematics education for Cameroonian children. U n l i k e the other subject areas, there is a steady decline i n the number of students taking u p Advanced Level (A-Level) and higher level mathematics i n universities i n Cameroon. Ekane (2000) points out that the number of students w h o took u p A - L e v e l mathematics i n 1998 was similar to what it was 10 years ago. However, since 1963, there has been a substantial increase i n number of students participating i n high school and university education. The number of students taking A - L e v e l mathematics has been rising since then and seems to have reached its peak i n the early 1990s. However, since 1992, the rate of students taking A - L e v e l mathematics has been falling steadily by almost 15% ( C G C E Board, 1998). O n the other hand, the National Research Council's (NRC) report Everybody Counts ( N R C , 1989) asserted that "mathematics is the worst curricular villain i n driving students to failure i n school. W h e n mathematics acts as a filter, it not only filters students out of careers, but frequently out of school itself" (p. 7). Beside this assertion, there is little hard evidence to support the claim that poor performance i n mathematics is possibly one of the many factors that leads to the problem of l o w intake i n mathematics and mathematics related programmes i n high schools and other higher education institutions. Past literatures (Dick & Rallis, 1991) have shown 6 that w h e n students choose a subject to specialize i n at A - L e v e l or university, they are very much influenced by the perceived relevance and usefulness of the subject areas in their future careers and their self confidence i n that subject. If students perceived mathematics as a difficult, alien and irrelevant subject and they lack self-confidence i n the subject, then they w i l l avoid taking up this subject at higher level. Furthermore, Dick and Rallis (1991) found that the effect of socializers, including parents, teachers and peers i n influencing both subject choice and career choices could be subtle but powerful. Thus, I w i s h to argue that if the teachers' and pedagogic specialists' understanding of the relationship between mathematics and culture is poor, then we can expect that these socializers or significant others i n the lives of students are likely not to encourage the students from taking u p mathematics, at higher level. Consequently this problem of l o w enrolment of mathematics students at higher level may be perpetuated. Yet, there are still a number of issues that need to be addressed here. Firstly, before further effort is given to promote the incorporation of relevant cultural practices into school mathematics curricula (content, process and methodology), and to change the widespread views about the cultural neutrality of the content and process of mathematics, we need to have a better understanding of the stakeholders interest and response to this ethnomathematical approach. Unfortunately, reviews of related literature indicate that relatively few systematic research studies have been conducted on the stakeholders' receptiveness to this ethnomathematical approach. Most studies have dealt w i t h demonstrating the possibility of applying such an approach i n the teaching of mathematics. Other studies have focused on demonstrating the existence of this cultural knowledge and strategies w i t h the hope that the revelation of the existence of this k i n d of knowledge w i l l get those involved i n curriculum reform and the process of teaching to incorporate these into their curricular prescriptions and teaching strategies. 7 In arguing that the students' poor performances and continuous decline i n enrolment i n mathematics and mathematics related fields of study and the failure of reforms i n mathematics education is about responsiveness and relevance, I draw u p o n cultural and sociocultural theories of learning to aid m y analysis of the level of stakeholders' receptiveness to an ethnomathematical foundation to the school mathematics curriculum. In this analysis, I asked: h o w receptive are the curriculum stakeholders to a proposed ethnomathematical foundation to the school mathematics curriculum and, consequently, what are their m a i n concerns and what changes w o u l d such a curriculum necessitate? The treatment of the findings w i l l provide foundational guidelines i n the successful adoption and implementation of this curriculum. The study is thus premised on the understanding that change i n mathematics education is an interconnected process that depends on political developments i n a country and cannot be sustained without the active participation of all stakeholders concerned. It begins w i t h an analysis of the context of reform. This is done because it provides a practical account of how "reform and developments i n mathematics education cannot be separated from the context i n w h i c h these take place" (Mwakapenda, 2002) and the knowledge and assumptions influencing the formulation of reforms. 1.3.2 Cultural Relevance A m o n g the many problems facing education i n general, and mathematics education i n particular i n Cameroon today, the appropriateness and responsiveness of the curriculum content is perhaps the most critical (Samoff, Metzler, & Salie, 1993). The mathematics curricula i n Cameroonian secondary schools are governed by structures like the University of L o n d o n General Certificate of Education and the French Baccalaureate, which are culturally laden to a very high degree. The transplantation (Bishop, 1990) of this alien colonial curriculum was "part of a deliberate" (Bishop, 1990, p. 55) and long-standing strategy of acculturation and assimilation by 8 the colonial governments (French and British) - "intentional i n their efforts to instruct i n 'the best of the West', and convinced of their superiority to any indigenous mathematical systems and culture" (Bishop, 1990, p. 55). The adopted mathematics curricula ignored the fact that mathematical developments i n other cultures, however different, followed different paths of intellectual inquiry, h o l d different concepts of truth, different sets of values, different visions of the self, of the other, of mankind, of nature and the planet, and of the cosmos. A s a result, students are seldom taught that several of the ancient Greek mathematicians, Pythagoras and Thales for instance, travelled and studied i n places such as Northern Africa where they acquired m u c h of their mathematical knowledge (Joseph, 1991). Cameroonian students k n o w little of the mathematical inventions or applications of such ancient non-European cultures as the Egyptians, the Babylonians, the M a y a n , etc. The students spend most of their time memorizing mathematical formulae and relationships of western cultures as w o u l d be demanded of them i n the many national examinations they w i l l be writing. The students are not aware of non-western contributions because the teachers themselves have not been taught that many cultures, including theirs, have contributed to the development of mathematics; cultures whose members were, and are, certainly intelligent, resourceful, and creative (Barta, 1995). The implication of this is that many of the students are educated away from their culture and away from their society. Mathematics, a subject which, i n fact, "could so easily have made connections w i t h indigenous cultures and environment and w h i c h could have been relevant to the needs of the indigenous society" (Bishop, 1990, p. 56) is not thought of i n those terms by proponents of a culturally neutral conception of mathematics, despite the current literature on the relationship between culture and mathematics. It is reasonable to suggest here that, as a consequence of the inappropriate curriculum content, large numbers of students fail their examinations i n mathematics every year i n Cameroon. For example, between 1994 and 1999, 9 the failure rate i n the O-level mathematics examination has been running as h i g h as 54% while the rate of attrition continues to grow (Ekane, 2000). This i n turn is affecting the number of students w h o proceed to university to read mathematics or mathematics-related subjects like engineering. Textbooks constitute the base of school knowledge, particularly i n third w o r l d countries where there is a chronic shortage of qualified teachers. In many instances, teachers adhere closely to texts, using them as the sole source of school knowledge, assigning students lessons contained i n the text and testing students only on the knowledge contained i n the texts. "[Yet] most third w o r l d countries have been so immersed i n the problems of p r o v i d i n g schooling to children ... that they have paid little attention to curriculum development and even less to the content of school textbooks" (Altbach & Kelly, 1988, pp. 3,10). T w o examples to contextualise Altbach and Kelly's (1988) argument for the case of mathematics textbooks content i n the Cameroon education system are given below. In an introduction to the topic of 'probability' i n one mathematics text being used, students are urged to note the following: Students (10-12 years old) should be able to list/display the possible outcomes w h e n a die is tossed twice; Students should be able to answer a question such as 'Is it as easy to become a millionaire in Cameroon as i n England or in Tanzania?' w i t h explanations. (Berinyuy, 1998, p. 59) The content of the above examples reveal m u c h about the forms of mathematics being valued and ways of operating i n mathematics practice. In the first example, w h y use the term 'die' w h i c h the students may not be familiar with? The question could be made culturally relevant by couching it i n terms of cowries w h i c h are common and familiar to the students. In the second example, an assumption is 10 made that students are i n possession of economic knowledge of England or Tanzania which may not be the case. O n the issue of making real world contexts accessible to students i n a mathematics classroom, Silver et al. (1995) conclude that "increasing the relevance of school mathematics to the lives of children involves more than merely p r o v i d i n g 'real-world' contexts for mathematics problems; real-world solutions for these problems must also be considered." (p. 41). The idea of building on children's informal mathematical developed i n part from the foregoing theory of how understanding knowledge develops, especially i n mathematics. If we see children's informal mathematical knowledge as part of a well-connected network of ideas and concepts, then using that knowledge as a starting point from which to base instruction does not only make sense but is also relevant. Evidence confirms that helping all those involved i n the process of mathematics education build on children's informal knowledge i n mathematics classrooms helps children use their intellect well, make meaning out of mathematical situations, learn mathematics w i t h understanding, and connect their informal knowledge to school mathematics (Carpenter et al., 1989; Fennema et al., 1996; Mack, 1990). This viewpoint is supported by Matthews (2003) who states that teaching from a student's cultural background is "more than merely presenting cultural information through 'cultured' examples and illustrations, it must be understood that the very act of b u i l d i n g on students' cultural knowledge is essentially an active one, and a dialogical endeavour." (pp. 79-80) One of the methods that certainly lends itself w e l l to such an approach and to this project is that of ethnomathematics research. Since D ' A m b r o s i o coined the term "ethnomathematics" over twenty years ago, this research field has been recognized as a valid educational tool i n mathematics and mathematics education w i t h already recorded levels of success i n uncovering and understanding mathematical processes embedded i n cultural practices and enhancing the teaching and learning of school 11 mathematics (See for example, Ascher & Ascher, 1981; Boaler, 1993; Gerdes, 1988; Masingila, 1994). M a n y mathematics educators and researchers alike are already employing this approach to mathematics education and research. For example, Gerdes (1988) has used an ethnomathematical approach to 'uncover' hidden moments of geometrical thinking i n A n g o l a n sand drawings; to develop a better understanding of mathematics practice i n everyday situations, Masingila (1994) has used this approach to identify the mathematics concepts and processes used i n the context of carpet laying. These brief examples of recorded successes (Gerdes, 1988; Masingila, 1994) i n ethnomathematics research demonstrate the wealth of mathematics knowledge not being tapped into. A s long as the view of mathematics as a culturally neutral subject continues to be promoted, students' performances i n school mathematics w i l l continue to be poor. Hence, school mathematics will continue to be meaningless, elitist, incomprehensible and unpopular to the vast majority of students not only i n Cameroon, but the w o r l d over. This study thus proposes that one of the ways Cameroonians can break away from this cycle of inappropriate education is by identifying individual characteristics, practices and behaviours that are valued i n the society, analysing them, and according to priority, developing a curriculum that w o u l d develop such individuals. To do this, the Cameroon government needs to get all those involved i n the education of the child to participate i n crucial decision making sessions regarding the curriculum. The role of the teachers and students needs to be emphasized, and most importantly input from the students should be seriously considered since they are the ones w h o are going to be affected most. 1.3.3 Education Reforms Reforms i n mathematics education i n Cameroon have not so far solved the inherent problems precisely because of their reformist nature. Reforms i n the last two decades have merely rearranged the same content into a "new and i m p r o v e d " 12 format. To make education i n mathematics more relevant, responsive and appropriate, new instructional methods and materials, and appropriate curriculum content must be sought, examined, and incorporated. Apart from claiming that school mathematics i n Cameroon is "devoid of everyday life" (Nji, 1994, p. 9), very little or no attempt is being made to consider many instances i n mathematics textbooks i n Cameroon, i n w h i c h the meaning and image of mathematics and connections between school mathematics and everyday reality are i n fact inappropriate, distorted and inadequately addressed. Teachers, researchers, mathematicians, and policymakers have all argued about what curricula should be used i n classrooms. A l t h o u g h opponents and proponents of different curricula have disagreed about the importance of contextualized mathematics instructions, they have rarely considered the ways i n which foreign and transplanted curricula and mathematics content can or do impede students liking the learning of mathematics and claim of ownership i n the development of mathematical knowledge. Yet, to be successful, the constant changes to school curricula call for different kinds of learning approaches and changes i n the roles of each of the curriculum stakeholders i.e. students, teachers, pedagogic inspectors, university faculty and parents. I contend that the curriculum reform process could have been enhanced if it had considered what it means to provide an education that is shaped by an understanding of mathematics education as a process of migrating from acculturation to "enculturation" (Bishop, 1988a) together w i t h an understanding that reform cannot be considered separately from contextual factors that influence its functioning i n practice. I w i l l argue that success i n any future reform policies w i l l be determined by the degree of involvement of all the curriculum stakeholders. In line w i t h Cameroon's aims of education i n incorporating Cameroonian cultural elements, school mathematics seems a fertile ground for the incorporation of such cultural elements. This view is being supported by a growing number of mathematics 13 educators w h o see a culturally relevant approach as germane to success i n mathematics education for all. But before we jump into implementing culturally relevant teaching approaches, it is important to understand h o w such approaches w o u l d be received by those w h o stand to be affected the most. This study thus aims to make a systematic enquiry into the receptiveness of the stakeholders (students, teachers, pedagogic inspector/adviser, university faculty) to a proposed ethnomathematical foundation to the school mathematics curriculum i n Cameroon. Knowledge of the interests and response of the stakeholders w i l l be germane to success of future curriculum reforms. 1.4 Significance of the Study There is a growing call for appropriateness and relevance i n school mathe- matics curriculum i n Cameroon (Eba, 1993; Nji, 1994; N W - M T A , 2002), but very little systematic enquiry into h o w such a responsive curriculum w o u l d be received and implemented by the main curriculum stakeholders. Therefore the findings of this study w i l l provide systematic and empirical data on the stakeholders' receptiveness to an ethnomathematical foundation to the school mathematics curriculum, w h i c h w i l l be very useful i n developing relevant curricula for other school subjects or knowledge domains. Firstly, by w o r k i n g w i t h those involved i n the education process, the results of this study might inform us of the role each of the stakeholders play i n effecting curriculum change. Secondly, by demonstrating the presence of mathematics i n cultural practices in Cameroon, this revelation might help develop i n Cameroonian students, a greater awareness of the presence of mathematics i n their everyday life. The findings might also promote i n the students, the feeling and rightful claim of ownership to a vast amount of the mathematics knowledge taught i n today's schools. 14 Lastly, the findings w i l l suggest possible implications for mathematics education and mathematics teacher education. This knowledge may help to enhance better curriculum planning and teacher development programmes i n the future. The findings might also encourage the local production of textbooks and educational resources that are more i n line w i t h the everyday realities of the learners and the teachers. 1.5 Conclusion Research i n educational reform must attend to the constituencies, their interests, and the dynamics among them, to understand h o w school curriculum reforms effect change or fail to. Efforts to reform school curricula must be examined i n terms of underlying assumptions, social and historical contexts, and the values, ideologies, and goals of vested constituent groups. H a v i n g briefly described the current state of mathematics education i n Cameroon, the importance and significance of cultural relevance i n mathematics education, and the importance of involving all the stakeholders of the curriculum, I argue that there is a need to carry out this study. Chapter 2 is a presentation of some theoretical perspectives o n receptivity (with respect to the school curriculum) and some philosophical and theoretical foundations of ethnomathematics. Chapter 3 is a presentation of a brief contextual chronological overview of the evolution of the school curriculum i n Cameroon as a backdrop in developing the overall argument driving the project. This presentation is necessary because the evolution of the school curriculum i n Cameroon is inextricably linked to its social and political history, and above all, to it history of colonial subjugation. This presentation is followed by a critical review of some related studies i n the field of mathematics education research, w i t h particular attention to research studies focus- 15 ing on the cultural connections to mathematics and mathematics teaching and learning. Chapter 4 provides some background to the research setting, the research participants and the methodology employed i n the collection of data for the study and an explanation of h o w the data was analysed. Chapter 5 presents the findings related to the research question i n the form of a comparative interpretive commentary. Chapter 6 summarizes the findings i n Chapter 5 i n the form of an interpretive analysis, and indicates some considerations. Chapter 7 summarizes the conclusions, recommendations and implications highlighted i n Chapter 6. These conclusions, recommendations and implications are presented i n the form of curriculum guidelines. The appendices contain letters of information and requests for consent, research questionnaires and interview protocol as w e l l as the unit. 16 ethnomathematics Chapter 2 THEORETICAL PERSPECTIVES 2.1 Introduction This chapter begins w i t h an operational definition of the term receptiveness based on the rationale outlined i n Chapter One. Some conceptual components of receptiveness are explored to delineate it from other closely related concepts. Next, some philosophical and theoretical foundations of ethnomathematics are analysed i n an attempt to develop a framework for the methodology and analysis used i n this study. 2.2 Conceptual Components of Receptiveness The term receptiveness is used in this study to refer to the level of willingness or readiness to favourably consider a new suggestion or proposal. A person's willingness to favourably consider an idea is often influenced by their level of interests i n that idea. A person's readiness to consider an idea can sometimes be gleaned from his/her response to the introduction of that idea. Hence determining h o w receptive the stakeholders are to a proposed ethnomathematical curriculum foundation w i l l require determining their level of interests and their response to the proposal. This is because interest is a powerful motivator for any action or change i n behaviour, and as Deci (1992) asserts, "people typically pursue avocations primarily because of interest i n the activities" (p. 43). However, he continues, "interest can also be an important motivator i n one's vocation ... because people w h o are interested i n their work are typically committed to doing it w e l l " (pp. 43-44). Deci's assertions are i n line w i t h a view furthered by Piaget (1958) that each individual constructs new knowledge on the basis of actions w h i c h are of interest to the individual. Therefore 17 an understanding of the concept of interest i n relation to receptiveness, especially w i t h respect to curriculum change is germane. The concept of interest has often been associated w i t h intrinsically motivated behaviours because people seem to adopt those behaviours out of interest. Intrinsically motivated behaviours towards an activity are those done simply for personal reward of enjoying the activity itself i.e. they are usually freely undertaken. Deci (1992) argues that, "the quality of one's motivated behaviour differs as a function of the extent to which the person is interested i n an activity" (p. 46). Even i n self-determination theory, interest is also closely linked to intrinsic motivation by being "conceptualised as the core affect of the self - the affect that relates one's self to activities that provide the type of novelty, challenge, or aesthetic appeal that one desires at that time" (Deci, 1992, p. 45). Extrinsically motivated behaviours towards an activity are those instrumental for some other reward, such as money, praise, or grades on a report card. That is, these are behaviours that are undertaken as a means to some end - as an instrument for achieving some outcome other than the spontaneous satisfaction that accompanies the activity. W h e n interest is defined as an affect that occurs i n the interaction between a person and an activity (person-object interaction), one can then move to a focus either on the activity or on the person. For example, one can explore the characteristics of activities that tend, on average, to make them interesting to a group of people (e.g. curriculum stakeholders). By k n o w i n g what characteristics of tasks (e.g. ethnomathematics) are interesting to the stakeholders, it w i l l be possible and easier to design educational materials (e.g. ethnomathematics curriculum) that, on average, w i l l be more interesting and thus more intrinsically motivating for the stakeholders especially the students and teachers. W h e n interest is treated as a person variable by focusing on the degree to w h i c h a person is interested i n a particular activity or class of activities e.g. ethno18 mathematics, then one is concerned w i t h the person's enduring interests i n an activity over an extended period of time. Such an approach allows interest to be viewed as an i n d i v i d u a l difference or "dispositional variable" (Deci, 1992, p. 46). Thus, the main nature of "interest" is that it is constantly changing in the web of the individual's personal culture, sometimes disappearing, at other times leading to new forms of interest. "Interests" then can be viewed as an intermediate by-product of the imagination processes that guide the person's mental/affective and actional systems towards some object. Krapp, H i d i , and Renninger (1992) identify three related concepts of interest (1) interest as a characteristic of the person (individual interest), (2) interest as a characteristic of the learning environment (interestingness), and (3) interest as a psychological state. Both individual interest and interestingness can bring about experiences and psychological states i n an i n d i v i d u a l that are generally referred to as interests. Typical characteristics of this state might include increased attention, greater concentration, pleasant feelings of applied effort, and increased willingness to learn. Interests can pertain to objects i n the physical or natural environment, to symbolic representations, or to activities (Rheinberg, 1989). Individual interests are conceived of as dispositions that are based on mental schemata associating the objects of interest w i t h positive emotional experiences and the personal value system. T w o forms of interest can be identified: actualised and situational (Krapp, H i d i , & Renninger, 1992, p. 6). Actualised interest is a state phenomenon while situational interest is generated by external stimuli. In other words, situational interest has more to do w i t h the "interestingness" of the situation or object under consideration while actualised interest has to do w i t h the intended actions as a result of the "interesting" situation. For example, when one says that a historical approach to the teaching of mathematics is an interesting approach but does not necessarily buy into the whole notion of using history i n the teaching and learning of mathematics then 19 one is said to be expressing situational interest. O n the other hand, if one says that a historical approach i n the teaching and learning of mathematics is not only interesting but also essential, and one is quite w i l l i n g to adopt this approach i n their mathematics instructions, then one is demonstrating actualised interest. While the above illustration suggests that situational interest is often a precursor to actualised interest, a series of actions eliciting situational interest may lead to the development of dispositional interest w h i c h is different from actualised interest. In fact, H i d i (1990) argued that situational interest, triggered by environmental factors, may evoke or contribute to the development of long-lasting individual (dispositional) interests. Efforts to measure interest usually lead to the operationalization of the concept on the basis of the person's encounters w i t h specific external object domains of the w o r l d . The common-sense use of "interest" can be viewed as somehow linked w i t h "motivation," w h i c h i n turn is somehow linked to "effort," and "novelty." Valsiner (1992) views interest as an invented concept to refer to a certain psychological phenomenon. For example, if one says, "I am interested i n a cultural approach to the teaching of mathematics" then one's actions and reasoning/feeling about a cultural approach to the teaching of mathematics all become parts of the phenomenon to be explained. Educational research studies i n the area of interest theory have tended to be on trying to understand the relationship between interest and academic outcomes. In such cases, interest is treated as a dispositional variable. W i t h many of the studies correlational i n nature, some of them, treating interest as a dispositional variable, have revealed a causal relation between academic interest and achievement i n the school setting while others have found very little of a causal relation. But gauging one's interest i n a process is a very difficult task especially if that interest has to do w i t h one's profession. This is because interest when conceptualised as a personobject relation (Krapp, H i d i , & Renninger, 1992; Prenzel, 1988; Prenzel, Krapp, & H . 20 Schiefele, 1986; U . Schiefele, 1992, 1996) is characterized by value commitment and positive emotional valences. Interest-driven actions are characterized b y experience of competence and personal control, the feeling of autonomy and self-determination, and a positive emotional state. These characteristics are often necessary for one to be receptive (express a favourable willingness to consider) new ideas or class of activities such as ethnomathematics. Theories of instructions have suggested that classroom instruction and materials that are interesting play a large role i n determining learning achievement (Hofer, 1986; Todt, 1985). This suggestion is based on an assumption that there is a relation between the interestingness of texts, its connectedness to the material covered, and the grades and later success of the student. K u b l i (1987) conducted a study on the factors that contributed to the interestingness of physics, and concluded that girls could become more interested i n physics if physics problems were linked to social or everyday situations. Similarly, Tobias (1990) came to the same conclusion w i t h respect to students learning both mathematics and science. The interestingness of an activity or class of activities does not usually explain w h y people respond positively or show a higher degree of interest i n getting involved or participating. Deci (1992) contends that people's willingness to participate i n uninteresting activities depends on their experiencing the activities as having personal importance or instrumental value to them. If people understand the importance of the activities, social contexts that are characterized by involvement and autonomy support w i l l tend to foster the internalisation of regulations for these uninteresting though important activities. A n d through internalisation and integration, the activities may gradually become more interesting for the person. For example, students and teachers alike w i l l be more motivated i n exploring and understanding the production techniques of cultural objects such as fish traps i n an attempt to glean the mathematics embedded i n them even when the act of weaving fish traps is not an interesting activity. This motivation to understanding the 21 production techniques and consequently the mathematics embedded i n the fish traps has a more personal meaning and importance to the students and teachers because they associate w i t h the art of weaving fish traps and the rewards associated w i t h producing a 'good' fish trap such as a cage full of fish. Interest does not always occur when one is involved i n a familiar activity, setting or object. Deci (1992) maintains that one can equally experience interest w h e n one encounters novel, challenging, or aesthetically pleasing activities or objects i n a context that allows satisfaction of the basic psychological needs and thus promotes development. W h e n interest is experienced in this situation, it is often viewed as one's preferences or 'enduring interest'. While people tend to have stronger preferences for activities at w h i c h they are more competent or have greater potential, Deci (1992) also contends "regardless of people's level of proficiency, they are more interested i n activities that provide optimal challenge" (p. 51). These preferences are influenced by the activities available i n people's environment and by the social context, as people are more likely to develop preferences for optimal challenges that are available to them than for those that are not. Deci (1992) believes that this is because when people engage i n activities w i t h i n a social context that allows satisfaction of their fundamental psychological needs for competence, autonomy, and relatedness, they w i l l be likely to maintain or develop enduring interests i n those activities, whereas when they engage i n activities w i t h i n a social context that thwarts the satisfaction of the three psychological needs, they w i l l lose or fail to develop enduring interests. A study conducted by Koller, Baumert, and Schnabel (2001) to investigate the relationships between academic interest and achievement i n mathematics among Grade 7, Grade 10 and Grade 12 students revealed that the relationship between academic interest and mathematics achievement is moderated by changes i n the instructional setting and not wholly by the curriculum alone. They found that when instructions were not primarily driven by extrinsic values such as written examina22 tions and the associated positive and negative consequences, interest became a more important antecedent of mathematics achievement. Students' interests i n learning new content (e.g. ethnomathematics) can often be explained i n experiential terms. For example, a student w h o is interested i n ethnomathematics content means he or she is experiencing a certain quality of attention and a certain sense of delight. Deci (1992) concurs w i t h this view by stating that "the experiential quality of interest has a positive hedonic valence and is related to the feelings of excitement and enjoyment..." (p. 49). When a learner is able to identify a cultural connection w i t h a learning situation, they tend to become interested i n the learning activity. A study conducted by Pallascio et al. (2002) on Inuit students to identify the sociocultural elements that teachers and curriculum developers should consider along w i t h students' interest and motivation w h e n designing curricula and w h e n planning instruction observed that the students manifested the weakest interest i n the most decontextualized activities—that is, the activities having the fewest cultural connotations, whereas the more contextualized activities, w h i c h had greater cultural or everyday connotations, drew a more positive response. Numerous studies have shown that competence-promoting information enhances interest (Deci, 1971; Harackiewicz, 1979; Ryan, 1982), whereas information that signifies or ensures incompetence diminishes interest (Deci & Cascio, 1972; Vallerand & Reid, 1984). Studies using path analytic procedures (Harackiewicz, Abrahams, & Wageman, 1987) have further confirmed that competence-promoting feedback and structure are important for enhancing people's sense of competence and i n turn their intrinsic motivation and interest, whereas competence-diminishing feedback and structure undermine intrinsic motivation and interest. The distinction between perceived self-competence and intrinsic motivation is also central to Deci and Ryan's (1985, 1991) cognitive evaluation theory. F r o m this theoretical perspective, these authors posit that "we w o u l d expect a close relation23 ship between perceived competence and intrinsic motivation such that the more competent a person perceives him/herself to be at some activity, the more intrinsically motivated he or she w i l l be at that activity" (1985, p. 58). For example, teachers are more w i l l i n g to consider curricula changes if they feel that they are competent to implement such changes and they believe that implementing those changes w i l l result i n greater student learning. The foregoing discussion on the concept of interest thus establishes a necessary aspect of the theoretical grounding of this study - the operationalization of receptiveness as a state phenomenon i.e. the degree of willingness to consider a new idea or class of activities - and h o w it w i l l be used i n both the data collection and analysis. The stakeholders' interests and response to an ethnomathematical foundation to the school mathematics curriculum w i l l be examined by considering their expressed actualised (intended action as a result of w o r k i n g on an ethnomathematics unit i n the classroom) and situational (the interestingness of the ethnomathematics foundation proposal) interests during their participation i n the study. H a v i n g outlining some conceptual aspects of the theoretical grounding of receptiveness guiding this study, I w i l l n o w lay out i n the ensuing section, some philosophical and theoretical perspectives on ethnomathematics. These perspectives are germane to the collection and analysis of the data i n this study. 2.3 Philosophical and Theoretical Foundations of Ethnomathematics One of the subjects of the ethnomathematical programme can be illustrated by the work of G a y and Cole (1967) i n w h i c h they investigated the mathematics of the Kpelle society i n the West African nation of Liberia. The purpose of their study was to find a way to b u i l d a bridge that w o u l d help the Kpelle use their mathematical experiences to learn the Western conceptualisations of mathematics. Gay and Cole d i d not expressly use the term 'ethnomathematics'; rather, they referred to this form of knowledge as 'indigenous mathematics'. The entree of ethnomathematics 24 into mathematics education discourse can be traced back to two decades ago, specifically i n 1984, when D ' A m b r o s i o used the w o r d at the Fifth International Congress on Mathematical Education (D'Ambrosio, 1984). Over the years, notions related to ethnomathematics have been frequently used by the literature and by popular writings on culture and mathematics. It is sometimes located within major fields comprising Cultural Anthropology, Anthropology and Archaeology, and Social Sciences. From the root, mathematics, and the prefix, ethno- from ethnography, it can be presumed that ethnomathematics refers to the study of mathematics i n relation to culture. However, despite its seeming popularity as a theoretical concept, it is still ill-defined. A l t h o u g h its importance as a research construct is well recognized b y scholars, any reference to it i n the academic literature is often fleeting and, at best, tangential. A s a result, ethnomathematics does not permit rigid measurements and fine-grained analysis of its attributes. A respectable body of research literature on the topic is still being generated and only a handful of books on the subject are presently available. Providing an acceptable definition of ethnomathematics and some philosophical orientations are, therefore, the first steps toward a systematic study of the subject. 2.3.1 Some Current Definitions of Ethnomathematics Before proposing a definition, it is worth examining the current definitions of ethnomathematics w i t h i n anthropological literature. D ' A m b r o s i o (1985) is perhaps considered as the forerunner when examining the evolution of the term ethnomathematics and its varied definitions. Accredited for having coined the term ethnomathematics, D ' A m b r o s i o defined ethnomathematics as the "mathematics practiced among identifiable cultural groups, such as national-tribal societies, labour groups, children of certain age bracket, professional classes, and so on. Its identity depends largely on focuses of interest, on motivation, and on certain codes and 25 jargons which do not belong to the realm of academic mathematics. We may even go further i n this concept of ethnomathematics to include much of the mathematics w h i c h is currently practiced by engineers, mainly calculus, w h i c h does not respond to the concept of rigor and formalism developed i n academic courses of calculus." (1985, pp. 45-46). This definition therefore suggests that "even the form of mathematics produced by professional mathematicians can be seen as a form of ethnomathematics" (Borba, 1990, p. 40). However, the definition is ethnocentric because it does not involve the standard study of mathematics (i.e. academic mathematics), i m p l y i n g that the term only suggests mathematics studied by other cultures. Another definition is provided by Paulus Gerdes w h o is considered as one of the main contributors to the field of ethnomathematics. Most of Gerdes' work is on the elaboration of academic mathematics inspired by traditional practices. H e defines ethnomathematics as "the cultural anthropology of mathematics and mathematical education" (Gerdes, 1997, p. 332). H e views ethnomathematics as an active reclaiming of a mathematical point of view as part of indigenous culture. Gerdes' view concurs w i t h D'Ambrosio's that ethnomathematics is a research field aimed at contributing to the mathematical awareness of colonised people and drawing attention to mathematics as a cultural product. Ascher and Ascher (1986), two researchers of African counting cultures, defined ethnomathematics as "the study of mathematical ideas of non-literate peoples" (p. 125). This definition is too restrictive to permit a generalizable investigation of the topic. It implies that mathematics contains a cultural component only when discussing the mathematics of non-literate peoples (Borba, 1990). Further, it implies that a people can have a culture only if they are non-literate (or i n some alternate way, an Other to the examiner of culture). This interpretation of ethnomathematics is a concrete example of ethnocentrism and an encouragement of the idea that proper mathematics is a notion defined only by the literate peoples. M o r e importantly, w i t h anthropology's acceptance of Boas' theory of cultural relativity i n 26 the early 1900's, this definition also seems grossly antiquated. Boas (cited i n Rosaldo, 1993) argued for the integrity of separate cultures which were equal w i t h respect to their values. Differences between cultures w i t h respect to technological or other development conferred them w i t h neither moral superiority nor moral inferiority, including differences w h e n compared to one's o w n culture (Rosaldo, 1993). A somewhat concise definition of the concept is provided by Borba (1997) i n his statements that ethnomathematics is the "mathematical knowledge expressed i n the language code of a given sociocultural group" (p. 265). This definition clearly characterizes ethnomathematics as the study of mathematics w h i c h takes into consideration the culture i n w h i c h mathematics arises. While this definition relates culture to mathematics and opens the door for testing hypothesized relationships between the two, it too seems inadequate to permit a more eclectic investigation of the topic. A broader definition of the concept that emphatically links its roots to the mores and values of groups of people is thus warranted. 2.3.2 Etymology and Operational Definition A n ideal starting point for defining a term is by borrowing its meaning from the dictionary. However, as mentioned earlier, the w o r d ethnomathematics is not found i n a standard dictionary. To the point, the definition of ethnomathematics has not been standardized at all. Nonetheless, few w o u l d disagree that etymologically ethnomathematics is the concatenation of the prefix ethno- onto the word mathematics. Thus, what is obvious is that there are two different literatures that examine ethnomathematics: Anthropology and Mathematics. From this, one can gather that ethnomathematics is at the crossroads of culture and mathematics. But, because these two subjects are so divergent, it is unclear exactly h o w they interrelate and give birth to ethnomathematics. A fitting definition can, however, be created by examining the w o r d itself and the definition of the prefix ethno- and the root mathematics. The prefix ethno comes from the w o r d ethnology. The American Heritage 27 College Dictionary (1993) defines ethnology as "the science that analyzes and compares h u m a n cultures; cultural anthropology." The same dictionary also defines mathematics as "the study of the measurement, properties, and relationships of quantities, using numbers and symbols." U p o n examination of these etymologies and u p o n examination of the conceptual differences i n the mathematics of different cultures, it becomes apparent exactly h o w large a topic ethnomathematics is. Ethnomathematics does not only include the meekly interesting facts about h o w cultures count on their toes, fingers, or ears. It also includes a m y r i a d of other topics that can be analysed and studied: > What is the function of mathematics w i t h i n a culture? > H o w does mathematics affect one's culture (leading also to h o w technology affects ones culture)? > W h y is there a cultural feeling that mathematics is a universal subject? > What conceptual differences are found i n the mathematics of different cultures? > H o w do different cultures count? D o these methods suggest something about the values of the underlying society? > What mathematical area of study does a society stress, and what about the culture that dictated that those topics be studied? > H o w do social hierarchies w i t h i n a culture affect the development of mathematics w i t h i n that culture? To accommodate the myriad of topics above, the definition of ethno- mathematics itself must not be a restrictive one. It must be simple and yet provide a basis to study divergent topics that emerge because of variations i n h u m a n cultures. To accomplish this, it is important to look at some contemporary understandings of culture and mathematics. Culture according to Hollins (1996), is: 28 "the essence of w h o we are and h o w we exist i n the w o r l d . It is derived from understandings acquired by people through experience and observation (at times speculation) about h o w to live together as a community, h o w to interact w i t h the physical environment, and knowledge or beliefs about their relationships or positions w i t h i n the universe." (p. 6) While H o l l i n s ' definition focuses on what culture is, the meaning of culture can also be gleaned from definitions that look at what culture does. H a l l (1977) quoted i n Hollins (1996) concisely described the function of culture thus: Culture is man's medium; there is not one aspect of human life that is not touched and altered by culture. This means personality, h o w people express themselves (including shows of emotion), the way they think, how they move, how problems are solved, h o w their cities are planned and laid out, h o w transportation systems function and are organized, as w e l l as h o w economic and government systems are put together and function. ..." (p. 5) Discernible from the above definition is the suggestion that culture has a pervading influence on h o w a group of people lives and learns. The following definitions of the term, culled from different sources, are equally relevant for the purpose of this study. The Oxford English Dictionary (OED) defines culture as: a. "The training, development, and refinement of m i n d , tastes, and manners; the conditions of being thus trained and refined; the intellectual side of civilization, b. A particular form or type of intellectual development. Also, the civilization, customs, artistic achievements, etc., of a people, especially at a certain stage of its development or history" (The Oxford English Dictionary, 2 n d Edition, 1991). A Cultural Anthropology (Kottak, 1994) text defines culture as a concept distinctly pertaining to humans. "Cultures are traditions and customs, transmitted 29 through learning, that govern the beliefs and behaviour of the people exposed to them. Children learn these traditions by growing u p i n a particular society" (Kottak, 1994, p. 13). The concept of culture can be problematic since the w o r d has numerous definitions and elaborations. "What most have i n common, and what is significant for us, is that i n any culture, the people share a language; a place; traditions; and ways of organizing, interpreting, conceptualising, and giving meaning to their physical and social worlds" (Ascher, 1991). Even w i t h i n this definition, defining a group of people and their cultural aspects can also be problematic. "Because of the spread of a few dominant cultures, there is no culture that is completely selfcontained or unmodified" (Ascher, 1991). Culture as used i n this study refers to a set of norms, beliefs, values, and practices that are common to a group of people. Culture here is taken i n its broadest sense. Thus one can talk of Cameroonian culture, w o r k i n g class culture, and also the culture of university mathematicians as a group i.e. it is not limited to race or place. Mathematics refers to the study and use of numbers and symbols i n relational terms. The focus is not only on the evolutionary aspect of its contents but also on h o w they are learned and used. The Oxford English Dictionary defines mathematics as follows: "Originally, the collective name for geometry, arithmetic, and certain physical science (as astronomy and optics) involving geometrical reasoning. In modern use applied, (a) i n a strict sense, to the abstract science w h i c h investigates deductively the conclusions implicit i n the elementary conceptions of spatial and numerical relations, and w h i c h includes as its m a i n divisions geometry, arithmetic, and algebra; and (b) i n a wider sense, so as to include those branches of physical or other research w h i c h consist i n the application of this abstract science in concrete data. W h e n the w o r d is used i n its wider sense, the abstract science is distinguished as pure mathematics, and its concrete applications (e.g. i n astronomy, various branches of physics, the theory of probabilities) as applied or mixed mathematics" (The Oxford English Dictionary, 2 n d Edition, 1991). 30 The emergence of the field of ethnomathematics is forcing a radical rethinking of our definition of mathematics. W i t h the uncovering, unfreezing of mathematical practices hidden i n many cultural practices (Gerdes, 1988), and the constant growth and development of the mathematics practiced by mathematicians (Tymoczko, 1986) around the world, there is the need for an ongoing adaptation of what mathematics is. For the purpose of this study, the nature of mathematics that w i l l be used is one rooted i n a falliblist philosophy that, according to Ernest (1985), "gives a realistic account of the nature of mathematics as it is actually practiced" (pp. 608-609) rather than as it ought to be. According to this view, mathematics is seen as a dialogue between people tackling mathematical problems. That is, mathematics is a h u m a n creation born of and nurtured from practical experience, always g r o w i n g and changing, open to revision and challenge, and whose claims of truth depend on "guessing by speculation and criticism, by the logic of proofs and refutations..." (Lakatos, 1976, p. 5). Mathematical methods therefore are not perfect and cannot claim absolute truth. Mathematical truth is not absolute but relative because i n fact truth is time dependent (Grabiner, 1986) and space dependent (Wilder, 1986). Time dependent because what is scientifically true today, might be a falsehood i n the future as theoretical assumptions change, as occurred w i t h the theories of Euclid and Ptolomeus. Mathematical methods are also space dependent because different peoples and different cultures have different ways of doing and validating their mathematical knowledge (Ascher, 1991). A s such, it can be characterized as a h u m a n activity that cannot be viewed i n isolation from its history, its sociology and its applications i n life. A l t h o u g h less able to offer a tidy overall philosophical picture of mathematics like other schools of thought on the subject such as: logicism, formalism, constructivism and Platonism, falliblism is perhaps the one school of thought on the nature of mathematics that offers to me, useful and interesting educational parallels and implications. For example, its denial of the externalised objectivity of mathe31 matics emphasizes i n d i v i d u a l experience, w h i c h i n educational terms is a childcentred view. Falliblism, I w i l l argue, is the only school of thought w h i c h sees history as germane to, and, beyond this, as an essential part of mathematics. From an educational point of view, a philosophy of mathematics which admits its history, sociology, applications i n life and hence a consideration of the gradual evolution of mathematical concepts, is not only a valuable but also a commendable one. Studies o n contextualization have indicated the desirability of considering an individual's cultural environment. According to Bruner (1996), "all mental activity is culturally situated. Indeed, it is impossible to understand mental activity if one fails to account for the cultural environment and the resources it makes available - i n other words, the m y r i a d details w h i c h shape the m i n d and determine its scope. Learning, remembering, speaking, imagining - all of this becomes possible only because we participate i n a culture" (p. 7). A brief examination of the history of mathematics displays some of the relationships between culture and mathematics. Mathematical ideas and concepts as defined by Western culture, including arithmetic and geometry, were developed at various historical periods across the w o r l d , and different strains of mathematics were pursued i n each culture. Different cultures stressed different aspects of mathematics and treated mathematics differently. For instance, many cultures classify mathematics differently and do not have a strong dividing line between Mathematics or Physical Sciences and the Social Sciences. In these cultures, mathematics is taught integrated w i t h i n the humanities. Culture also greatly affects our view of mathematics; differential consideration of facts concerning the evolution of certain mathematical concepts and relations has distorted the history of mathematics itself. Hersh discusses mathematics as a h u m a n activity: "Mathematics is human. It's part of and fits into human culture. Mathematical knowledge isn't infallible. Like science, mathematics can 3 2 advance by making mistakes, correcting and recorrecting them.... There are different versions of proof or rigor, depending on time, place, and other things.... Mathematical objects are a distinct variety of social-historic objects. They're a special part of culture. Literature, religion, and banking are also special parts of culture. Each is radically different from the others. M u s i c is an instructive example. It isn't a biological or physical entity. Yet it can't exist apart from some biological or physical realization—a tune i n your head, a page of sheet music, a high C produced by a soprano, a recording, or a radio broadcast. M u s i c exists by some biological or physical manifestation, but it makes sense only as a mental and cultural entity. What confusion w o u l d exist if philosophers could conceive only two possibilities for music—either a thought i n the m i n d of an Ideal Musician, or a noise like the roar of a vacuum cleaner.... Mathematics is a social-historic reality.... There's no need to look for a hidden meaning or definition of mathematics beyond its social-historic-cultural meaning. Socialhistoric is all it needs to be.... forget immaterial, inhuman 'reality'." (Hersh, 1997, pp. 13-16) W h e n discussing the origin of mathematics, we cannot help but think about the usefulness of it and that it originated because of its use i n society. Perhaps, however, it emerged because of its aesthetic quality and the enjoyment of creating order out of chaos through rational thinking. If you ask virtually any mathematician, he/she w o u l d agree to the statement, "mathematics, like music, is worth doing for its o w n sake" (Guillberg, 1997, p. 25). The usefulness of mathematics is what tends to conceal and disguise the cultural aspect of mathematics. Guillberg (1997) notes that no one ever asks about the usefulness of music: "The role of music suffers no such [cultural] distortion, for it is clearly an art whose exercise enriches composer, performer and audience; music does not need to be justified by its contribution to some other aspect of human existence". Mathematics, like music can exist without its usefulness, and can be appreciated as an exercise that enriches those who come into contact w i t h it. 33 Different aspects of mathematics were developed at various historical periods by different cultures across the w o r l d . Proof that each culture developed its o w n mathematics is presented u p o n examination of the different methods developed for solving algebraic systems such as quadratic equations. Each culture stressed a different aspect of mathematics i n its development. Babylonians invented a place value number system, knew different methods of solving quadratic equations (which w o u l d not be improved u p o n until the sixteenth century A.D.) and knew the relationship between the sides of a right-angles triangle, w h i c h came to be k n o w n as the "Pythagorean theorem (Joseph, 1997). Egypt pursued geometry to aid i n the creation of complicated architectural structures. Egyptian fractions and the heightened accuracy of p i were developed as a tool for the development of these structures. India developed the number system and pursued more theoretical aspects of mathematics. We can examine the differences i n mathematics from culture to culture and notice a culture's effect on the development of mathematics. Greeks have been credited w i t h the development of a more sophisticated form of mathematics that serves as the basis of what we use today. Despite the common perception that Greeks were the founding fathers of mathematics, Greeks learned most of their mathematics from Egyptians. Egyptian mathematics was superior to the Greeks, and the latter often went to be schooled i n Egypt. Aristotle's teacher, Eudoxus, one of the notable mathematicians of the time, had studied i n Egypt before teaching in Greece. Thales (d. 546 B.C.) was reported to have travelled w i d e l y i n Egypt and Mesopotamia and learned m u c h of his mathematics from these areas. "Some sources even credit Pythagoras (fl.500 B.C.) w i t h having travelled as far as India i n search of knowledge, w h i c h may explain some of the close parallels between Indian and Pythagorean philosophy and religion." (Joseph, 1997) Documented evidence is found w i t h i n even the Greek mathematical literature itself of the intellectual debt they owed to the Egyptians and Babylonians (a generic term that is often used to describe all inhabitants of ancient Mesopotamia), 34 and fulsome acknowledgement is given w i t h i n many of the texts. There are scattered references of the knowledge acquired from Egyptians i n fields such as astronomy, mathematics, and surveying, w i t h sources varying from Herodotus (fl. 450 B.C.) to Proclus (fl. A . D . 400). Some Grecian commentators even considered the priests of M e m p h i s to be the true founders of science. Aristotle (fl. 350 B.C.) considered Egypt to be the cradle of mathematics. The Greeks are usually given credit for the determination of p i despite Egypt's more accurate estimate of p i . This is not surprising as the advancements of Africa are often attributed to others due to cultural misconceptions. To explain Egypt's responsibility for the development of p i , we must first examine their representation of fractions. Egyptians used a different representation of fractions i n place of the common Western fraction format (which they had no knowledge of). The Egyptian representation of fractions remained the common technique of fraction representation and computation until the 19th century. The Egyptians represented a fraction by a sum of unit fractions, e.g. 1/a + 1/b + 1/c + ... where a b c are increasing integers. For example, according to the Egyptians, the fraction 5/6 can be represented by as 1/2 +1/3. In fact, every rational number can take on the Egyptian representation of fraction format. A famous "mysterious, so called, meaningless" triple, 13, 17, 160, was found throughout Egyptian architecture and manuscripts. W h e n translated into Egyptian form of fractions representation, we notice that 3 + 1/13 + 1/17 + 1/160 approximates p i to 4 significant digits w h i c h is m u c h better than 3.16 w h i c h is usually attributed to the Egyptians. We must keep i n m i n d that mathematics is a cultural product. Other cultures, although they do have the ideas or concepts that we deem as mathematical, do not distinguish them and class them together as we do (Ascher, 1991). The definitions of mathematics are based solely on the Western experience, even though they are often phrased universally. Even w i t h i n the Western culture, the definition of mathematics is fluid, and is generally defined to include whatever the Western professional class 35 called mathematicians do. A n example of the fluidity of the definition of mathematics is cited by Barton (1996) concerning the traditional Konigsberg bridge problem that was considered a recreational problem for centuries before becoming legitimate branch of mathematics as network theory. In the ensuing sections, I describe h o w mathematics developed and the role of culture i n its evolution to set the stage for understanding ethnomathematics. " N o t much study has been done i n ethnomathematics, perhaps because people believe i n the universality of mathematics. This seems to be harder to sustain, for recent research, mainly carried on by anthropologists, shows evidences of practices w h i c h are typically mathematical, such as counting, ordering, sorting, measuring and weighing, done i n radically different ways than those w h i c h are commonly taught i n the school system" (D'Ambrosio, 1997). There is a societal belief that mathematics is a universal and standard concept across ethnological boundaries. Barton (1996) characterizes this belief as borne of a philosophical difficulty and states that "there is little agreement on the extent to w h i c h mathematics is universal, and on h o w mathematical ideas can transcend cultures" (Barton, 1996, p. 201). Its theorems and laws are viewed as generalizable and universally applicable. This belief stems from mathematics' axiomatic principle that its premises and assumptions must be held as constant despite the variations i n the usage environment. This constancy principle has endowed mathematics w i t h an ideal platform, sought by less precise disciplines, to explain varying phenomena i n comparative terms. There is a perception that mathematics is an effective tool for analysing, examining, and verifying truth. It has provided mathematics w i t h an aura of objectivity amidst a predominantly subjective, chaotic, and nebulous w o r l d . While its assumptions and theorems are universal, their application, usage, and even the methods used to learn them seem to be culturally influenced. Thus, just as a language (e.g., English) is spoken or written differently by people of different cultures, mathematics-related communication appears to be punctuated b y cultural 36 differences and similarities. Some obvious examples are the following: Many mathematics languages are base-20, based on the number of fingers and toes. Nahuatl, a language of Central Mexico, is one of these, as is Choi, a M a y a n language spoken i n northern Chiapas, Mexico. The French language also expresses its numbers i n a base-20 format after the number sixty. A number system of base ten may seem to be obvious to the reader because it matches the number of fingers on the hand. However, the Y u k i of California think their system based on eight is the most logical for a similar reason. The Yuki's base eight system is based on the number of interfinger spaces. Knuckles are used i n yet other cultures. M a n y cultures use different words for the same number depending on what they are counting. For instance, the D i o i language has fifty-five numeral classifiers. Gilbertese, spoken on the Gilbert Islands, w h i c h is n o w part of the Republic of Kiribati, has 18 numerical classifiers. Some of these are animate objects and ghosts, groups of humans, days, years, generations, coconut thatch, rows of thatch, rows of things (other than thatch), customs, modes of transportation, etc (Ascher, 1991). One study showed h o w diversely number counting can be done on fingers (Zaslavsky, 1991, 1994). Ten children were asked to count to eight on their fingers secretly. Then all at one, they were asked to display h o w they represented the number eight on their fingers. The children had a multitude of different ways of representing the number eight. It is thus clear that despite its universality paradigm, aspects of mathematics have significant cultural overtones. By examining these cultural attributes, factors contributing to teaching and learning-effectiveness i n mathematics can be analysed and understood. For the purposes of this study, therefore, arguments by Powell and Frankenstein (1997a) provide a starting point for a refined and concise definition on ethnomathematics as including "the mathematical ideas of peoples manifested i n written or nonwritten, oral or non-oral forms, many of w h i c h have been either 37 ignored or otherwise distorted by conventional histories of mathematics" (p. 9). The above definition makes clear the subject of ethnomathematics. However, settling on a particular definition is a little futile because of the existing levels of contradictions within the ethnomathematics literature which according to Barton (1996), are "about the meaning of the term 'ethnomathematics' in particular, and also about its relationship to mathematics as an international discipline" (p. 201). Hence, rather than provide another definition of ethnomathematics, I have elected to offer statements about what ethnomathematics involves. For this study, ethnomathematics will be seen both as a subject and process of the social construction of knowledge at a cultural level i.e.: the study of the culturally-related aspects of mathematics; it deals with the comparative study of mathematics of different human cultures, especially in regard to how mathematics has shaped, and in turn been shaped by, the values and beliefs of groups of people. The above definition describes ethnomathematics as a legitimate offspring of the interaction between culture and mathematics. It suggests that the study and use of mathematics has cultural overtones and must be viewed as such. It offers a framework to discuss and explain evolutionary issues in mathematics as due to differences in human subcultures. At the same time, it suggests that the economic and technological disparities of societies can be explained by the influence mathematics has had on the thinking and behaviour of people of those societies. Figure 2.1 diagrammatically provides a framework and describes the relationships of variables specified in the above definition. Relationships emphasize reciprocity between culture and mathematics. Culture affects mathematics, as does mathematics affect culture. The interplay within culture and mathematics is ethnomathematics. 38 Fig. 2.1: Ethnomathematics: interaction between culture and mathematics Ethnomathematical research according to D ' A m b r o s i o is mathematics education research aimed at "analyzing the w a y i n w h i c h mathematical knowledge is colonized and h o w this knowledge rationalizes social divisions within society and between societies" (1990, 1994). In trying to make sense of ethnomathematics, Barton (1996) identifies three dimensions on which ethnomathematical studies can be classified. These are time, culture and mathematics and each lies on a continuum w i t h ethnomathematics studies located at any point i n the space. According to Barton's classification, studies that focus on understanding the mathematical practices and conceptions of ancient societies are attending to the time dimension; studies that focus on analysing cultural practices for the presence of mathematics are adhering to the culture dimension; and studies that focus o n demonstrating the presence of mathematics i n every culture and advocating for unbiased historical accreditation. Hence, ethnomathematics research w i l l be characterized as "the cultural anthropology of mathematics and mathematical education" (Gerdes, 1997, p. 343) i.e. 39 a cultural w i n d o w for examining any form of knowledge or social activity characteristic of a social and/or cultural group, that can be recognized by other groups such as anthropologists, but not necessarily by the group of origin, as mathematical knowledge or mathematical activity. 2.4 Conclusion The foregoing supports the proposition that culture has occupied a central role i n the development of mathematics. While economic nature seems to have given birth to mathematics, environmental factors unique to different societies have impacted its growth. Different societies i n different time and space have influenced and, i n turn been influenced by, mathematics' evolution. Understandably, w h i l e its theoretical components may be the same across societies, its application and usage are culturally biased. In this section, I have examined the concepts of receptiveness and its conceptual components and delineated it from other closely related concepts. I have also explored the philosophical and theoretical perspectives of ethnomathematics i n order to set the stage for a review of research studies viewed as ethnomathematical. I have offered statements describing ethnomathematics which illustrate the approaches I w i l l use throughout this work. In the ensuing chapter, I w i l l review some related literature i n the area of the evolution of formal education i n Cameroon, mathematics education and ethnomathematics. 40 Chapter 3 REVIEW OF RELATED LITERATURE 3.1 Introduction This chapter begins w i t h a historical overview of the evolution of formal education i n Cameroon. This is followed by a brief review of studies on mathematics education from a social and cultural perspective. Finally, there is also a review of current literature on ethnomathematics studies. 3.2 Evolution of Formal Education Almost all literate societies ascribe to a central consensual orthodoxy that education, particularly formal schooling is the requisite and the basis for any type of progress including economic development, social mobility, cultural preservation and transmission, and military sophistication (Seidman & Anang, 1992; Tambo, 1992; Azevedo, 1989). This consensus on progress through education concurs w i t h a declaration at the W o r l d Conference on Education for A l l sponsored by U N D P , U N E S C O , U N I C E F and the W o r l d Bank, that "(E)ducation is a fundamental right for all people, w o m e n and men, of all ages throughout our w o r l d ... an indispensable key to, though not a sufficient condition for, personal and social improvement .... (A) sound basic education is fundamental to - self-reliant development...." (Seidman & Anang, 1992, p. 103) If we are to recognize that schools exist i n contemporary societies as agencies for the handing on of the common cultural heritage of a society (Lawton, 1973,1980), and how each society is connected to the rest of the w o r l d , so that at least i n part 41 their purposes must be seen i n terms of socialization or acculturation, then the k i n d of education being referred to is just as important as the above declaration itself. The k i n d of education i n question immediately conjures questions of the curriculum and its content. C u r r i c u l u m according to L a w t o n (1980) is "essentially a selection from the culture of a society" (p. vii). Lawton's (1980) notion of curriculum as a selection from the culture implies that curriculum has the function of promoting the acquisition and mastery of the core values of the society for which it is structured (Omulando & Shiundu, 1992). Lawton's (1980) point of view resonates w i t h that of Julius Nyerere, former president of Tanzania, when he stated that the purpose of education is "to transmit from one generation to the next the accumulated w i s d o m and knowledge of the society, and to prepare the young people for their future membership of the society and their active participation i n its maintenance or development". (Nyerere, cited i n Bishop, 1985, p. 55) The cultural heritage, the knowledge and w i s d o m accumulated by any society over the years, provides an important source for decisions such as what is to be included i n the curriculum. Thus, curriculum, as an instrument of society for educating the young, naturally reflects the ideals and values, the knowledge and skills, the attitudes deemed significant by that society w h i c h w i l l consequently develop a strong cultural identity and is transmitted through the process of enculturation (Bishop, 1988b). This basic belief on the use of local cultural resources i n the curriculum has long been upheld by Cameroon whose government has remained committed to providing the most basic educational opportunities to its people. The history of its school curriculum is inextricably linked to its political and social history. The Cameroonization of the University of L o n d o n General Certificate of Education (G.C.E.) examination by the Cameroon Ministry of National Education since 1961, 42 was motivated by the ardent need the Cameroon government felt to have an examination based on a system that reflects to a great extent, the socio-cultural and economic nature of Cameroon, and yet ensuring the same rigour and scope in an education and examination system geared towards maintaining a world outlook and standards (Ndongko & Nyamnjoh, 2000). This arrangement with the British Department of Examinations, the British Senate House, the University of London, the British Government and the then Federal Republic of Cameroon was therefore an assurance to some Cameroonians who viewed the British-run educational system as a source of acculturation at best, or cultural alienation at worst, to return into the fold at the time of unification and support the school in an attempt to foster 1 national unity (Ndongko & Nyamnjoh, 2000). There is no doubt that, in Cameroon, dramatic progress has been achieved in this respect since the advent of the Cameroon General Certificate of Education examinations in 1977. However, what this agreement between the British Government and the then Federal Republic of Cameroon did not envisage at this point was identify the philosophical and ideological dimensions of the cultures of Cameroonian peoples and interpret the same for curriculum development and implementation. Rather, they focused primarily on the expressive aspects of the traditional cultures such as songs, dances, art works which were treated as an after school activity (DeLancey, 1989). Most of these aspects were not even tested in a Cameroon educational system that was rooted in national examinations. Some school subjects were not even seen as emphasizing non-Cameroonian cultural value systems. Such was the case with school subjects like mathematics, physics, chemistry, and biology, to name but a few. The case of mathematics was strongly contested based on arguments that mathematics, especially school mathematics was culture-free. The strengths of such 1 In 1961 the British-ruled Southern Cameroons in a plebiscite organized by the United Nations, voted to re-unite with their French-colonised East Cameroon to form a Federal Republic of Cameroon. 43 arguments have waned thanks to ongoing discourse on culture and mathematics. Hence, what I think is required at this point is a deep understanding of the philosophical and ideological dimensions of the cultures of Cameroonian peoples and viewing those philosophies and ideologies to be significant to the process of development and implementation of curriculum for cultural identity i n not only mathematics and the sciences but the arts as well. The focus of this section is twofold. The first part w i l l be a chronological analysis of the historical evolution of the curriculum i n Cameroon from the precolonial, the colonial and then to the post-colonial era. This analysis is necessary for two reasons: (1) it provides an understanding of the school curriculum as a continuously negotiated terrain by highlighting the conflicts, compromises, and processes of negotiations, through time, of the present curriculum i n Cameroon; and (2) it places into context the significance and the rationale for this study as outlined i n Chapter 1, and makes the case for a m u c h needed culturally relevant curriculum for Cameroonian schools. It w i l l be argued that the school curriculum i n Cameroon hasn't changed by much since the d a w n of independence, notwithstanding considerable expansion i n educational facilities (DeLancey, 1989; Tambo, 2000) and an increase i n the proportion of school-age children i n school i n the country. In charting the evolution of the curriculum i n Cameroon, I w i l l discuss the plethora of efforts aimed at harmonizing the two systems of education (French and British), i n an attempt to create a k i n d of national curriculum. 3.2.1 Pre-colonial Education and the Birth of Formal Education This section introduces pre-colonial education i n Cameroon and lays the foundations for the analysis of the evolution of the school curriculum to follow i n the next section. Prior to European arrival, the Cameroonians identified themselves as members of a particular ethnic group. Most of the power among the ethnic groups was vested i n the traditional rulers (chiefs or kings) (Ardener, 1996). The ruler i n 44 traditional politics was the accumulator of interests and values, and responsible for the distribution of privileges and economic favours. Politically, the area comprised essentially two types of traditional systems - state and stateless (Dillon, 1990). Understanding the evolution of curriculum and curriculum innovation i n Cameroon is not possible without a history of context. A s W a r i n g (cited i n Goodson, 1988) maintained, "If we are to understand events, whether of thought or of action, knowledge of the background is essential. Knowledge of events is merely the raw material of history: to be an intelligible reconstruction of the past, events must be related to other events, and to the assumptions and practices of the milieu (italics i n original). Hence they must be made the subject of inquiry, their origins as products of particular social and historical circumstance ..." (p. 95) The region today called Cameroon derived its name from a SpanishPortuguese w o r d "camaroes." 2 A s a political unit, Cameroon was created i n the 1880s. Prior to that time there were numerous political entities i n this area, each w i t h its culture, history, government, and economy (Eyongetah & Brain, 1974; DeLancey, 1989) . The pre-colonial period was dynamic, characterized by the migration of peoples, the rise and fall of governments, and economic relationships that tied together large numbers of the entities. However, conflict was a part of life, and war and conquest between African groups were common (Eyongetah & Brain, 1974). L o n g before the annexation of Cameroon by Germany i n 1884 or even prior to the arrival of the L o n d o n Baptist Missionary Society i n 1844, indigenous education i n Cameroon focused on equipping the youth w i t h essential skills as a form of 2 In 1472 the Portuguese sailor Fernua do Pao and his crew exploring the coast of West Africa found a variety of prawns swarming in the Cameroon river, and called the river Rio dos Camaroes - meaning River of Prawns i n Portuguese. In 1494, at the dawn of the N e w World, this region was later referred to as Camerones. This version was later anglicized to Cameroons. The Germans called it Kamerun while the French took this over as Cameroun. For more on the history and evolution of the name Cameroon, see Eyongetah & Brain (1974) or Njeuma (1989). 45 preparation for adulthood w i t h i n the Cameroonian society (Eyongetah & Brain, 1974). Before European arrival during the 15 century, Cameroonians were divided th into several viable kingdoms and chieftaincies that were busy p r o v i d i n g for the needs of their populations (Azevedo, 1989). Pre-colonial (indigenous) education was essentially education for living, w i t h its m a i n goal being that of training the youths for adulthood w i t h i n the society (Ngwang-Gumne, 2000; DeLancey, 1989). This form of education was less formal and its success depended on the unwritten rules and regulations that governed the choice of subject matter, the teaching strategies and the evaluation procedures that were laid d o w n by the society (LeVine, 1964). These unwritten rules and regulations concur w i t h current discussions on curriculum and curriculum policy by educators. O m u l a n d o (1992) maintains, "curriculum consists of the continuous chain of activities necessary for translating educational goals into concrete activities, materials and observable behavioral change" (p. 41). This implies that there may have been a prior debate about what passes as curriculum (and as curriculum theory), albeit non-formally, i n pre-colonial Cameroon w h i c h was resolved i n one w a y b y the elders and then presented as a fait accompli, as a once and for all given since the values were part of what glued the community together. There were no structures set u p i n the name of a formal learning environment, hence no formal schooling. However, most learning often took place i n organized non-formal and informal settings such as evening family gatherings at the fire site for story telling. D u r i n g this period, each of Cameroon's numerous ethnic g r o u p s had evolved its o w n form of education. Adaptation to local climatic, topo3 graphic and ecological conditions, well reflected i n the indigenous economies, guided this evolution. Emphasis was placed on normative goals, w h i c h were 3 There are over 200 ethnic groups speaking over 230 languages in Cameroon. This heterogeneous ethnic mosaic - demographic, geographic and physical features - has led historians to describe Cameroon as a microcosm of Africa - an "Africa in miniature" wherein one can find all the major cultural types of the continent (See Dillon, 1990, for an extensive treatment of Cameroon polity.). 46 concerned w i t h accepted standards and beliefs governing correct behaviour and expressive goals that were concerned w i t h unity and consensus (Shu, 1985). The elders enjoyed the monopoly of instructing the youths based on their life experiences. This monopoly was rooted on the single fundamental principle of the supremacy of history as the source of legitimacy and power (political and mystical) (Dillon, 1990). This stemmed from a rather literal interpretation of the commonsense notion that older people or their representatives had a more precise knowledge because they had existed before those currently living and were seeing things before they were born. It was only common knowledge then that since the elders had lived longer than the youths, the success of the youths i n life was thought to lie i n learning from the elders and approximating, inasmuch as possible, the knowledge of the elders themselves (Dillon, 1990). U s i n g the concept of historical primacy as a source of legitimacy and power, each of Cameroon's numerous societies had established a reservoir of stories w i t h legendary heroes through w h i c h moral lessons were transmitted (Eyongetah & Brain, 1974). For example, 'Wandong', 'Wafirmwa'ah', 'Tautsagha-Bahtum', 'Kebegha' (fox, leopard, sense-pass-king, and the tiger respectively). These were stock characters i n many stories from the Bantu and grassfield settlers of the savannah 4 region of Cameroon, through w h i c h children learned traditional values such as humility, honesty, courage, kindness, and respect (Shu, 1985). Each story had a moral lesson to be passed to the audience. A l s o common to most communities i n the savannah region was the burial of the last piece of the umbilical cord as a form of connecting the child w i t h their ancestors. This connection was to ensure protection 4 The grassfields, a term that derives from the German colonial period refers to the well-watered highland savannah region in western Cameroon. It is bounded by forest to the east, south and west and drier country to the north (See Eyongetah & Brain, 1974, for an extensive description of the geography of Cameroon). 47 against evil spirits, guard against immorality and make the child spiritually whole (Eyongetah & Brain, 1974). A s the children got older, they were trained i n skills that would ensure courage, endurance, self-control, patience and dignity, and skills that would introduce them to the direct service of the community. For example, the bridge from childhood to adulthood in the savannah communities of northwest Cameroon was marked by such traditional rites of passage as circumcision. Though a painful operation, it was often used to test for endurance and courage and as a formal occasion to inculcate the values of the society. The progressive introduction of subject matter as governed by the age of the children resonates with Piaget's stages of human development (Tambo, 1992). While Piaget acknowledged that chronological age is not the obvious determinant i n the introduction of subject matter, mental readiness and sequential learning was certainly a crucial determinant to the gradual introduction of subject matter and guiding the child through to adulthood. The general approach to the education of the child i n the indigenous educational setting was on a communal basis w i t h the parents of the child acting as the first teachers. Gradually, these responsibilities shifted to other family members and eventually to the entire community (Shu, 1985). The method of teaching took many forms again depending on the age of the children. A t an early age, children were encouraged to learn through play and also by imitating the adult w o r l d i n a creative and symbolic way. Learning through play, work, observation and imitation was premised on children's minds being like tabula rasa, the adult being the fount of knowledge, of definition and convention. This was based on certain assumptions derived from experience that the child had an inherent capability to structure sense impressions and a preferment for morally good acts rather than bad (Adelman, 1984). The children then often imitated their parents and other grown-ups activities w h i c h they w o u l d pursue at a later stage (DeLancey, 1989). 48 A child who d i d not participate actively i n play was normally suspected of being i l l or even abnormal. Children were left to their o w n initiative to make toys. They made toys from local materials of their o w n choice and interests. They moulded from m u d and clay and made use of articles w h i c h were of little use to adults. (Azevedo, 1989, p. 5) Besides learning through play, other teaching methods such as story telling were often used. This was even more so because the indigenous education was totally oral. W i t h these stories, the children were gradually equipped w i t h the mental capability of understanding the abstract and applying that understanding to their o w n life. T h r o u g h story telling, children also developed listening skills. This k i n d of learning was later reinforced by the language games the children often played such as riddles, proverbs, tongue twisters, transposition of syllables, or through nonsensically but musically arranged words (Eyongetah & Brain, 1974, p. 26). In their enactive theory of cognition, Varela, Thompson and Rosch (1991) characterize this k i n d of learning as "embodied". According to this theory of cognition, "embodied" refers to the sense of dependency by cognition u p o n sensorimotor capacities and these in turn u p o n a larger context of biological, psychological, and cultural influences. Thus using cognition as embodied action (Varela, Thompson & Rosch, 1991, p. 205), a youth w o u l d be thought as functioning adequately when they become part of the ongoing existing [external] adult w o r l d i.e. culturally, socially, spiritually and emotionally. Another approach to teaching i n the indigenous education setting was through apprenticeship (Dillon, 1990). Some children w i t h i n the community were expected to assume special responsibilities such as being herbalists, mid-wives, weather forecaster, cleansing of the land, communicating w i t h the ancestors. For example, a Bantu girl w h o eventually became a midwife most likely had a mother or grandmother w h o was a midwife, since midwifery was handed d o w n i n family lines within the Bantu culture (Dillon, 1990). Some of these responsibilities were 49 designated to particular children at birth to inherit the knowledge from a close relative who had been bestowed by the creator w i t h that particular knowledge (Dillon, 1990). The apprentices learned by observation. Learning through apprenticeship was a life long process. Though the learner performed some duties i n the presence of the instructors when they were thought to be knowledgeable and proficient (Dillon, 1990), they only performed independently once the instructor retired due to old age, got incapacitated or died. This k i n d of indigenous education through apprenticeship and association to a particular community concurs w i t h Lave and Wenger's (1994) discussion of situated learning. In their discussion, Lave and Wenger (1994), view learning i n relational terms - as a form of legitimate peripheral participation. They maintain that, learning viewed as legitimate peripheral participation, "is not merely a condition for membership, but is itself an evolving form of membership" (Lave & Wenger, 1994, p. 53) to any given community of practice. Lave and Wenger (1994) believe that "the historical significance of apprenticeship as a form for producing knowledgeably skilled persons" (p. 62) i n a given community of practice "has been overlooked, only because it does not conform to either functionalist or Marxist views of educational 'progress.'" (Lave & Wenger, p. 62). They write, "Apprenticeship has been treated as a historically significant object more often than most educational phenomena - but only to emphasize its anachronistic irrelevance. It connotes both outmoded production and obsolete education. W h e n its history is the pretext for dismissing an issue as an object of study, there is good reason to reexamine its existing historical and cultural diversity." (p. 62) It is clear from this description of indigenous education that it was firmly based on some philosophical foundation (DeLancey, 1989; Dillon, 1990), w i t h functionalism as the main guiding principle. This k i n d of education was strictly utilitarian as an immediate induction into society and a preparation for adulthood. 50 There was a high level of communalism as another guiding principle. C h i l d r e n were brought up to value group cohesion rather than i n d i v i d u a l achievement because "freedom of the i n d i v i d u a l was completely subordinated to the interests of the clan... and cooperation was preferred to competition" (Dillon, 1990, pp. 43-44). Through education, members of the society made sure that values and skills necessary for the survival of the cultural heritage were learnt. In summary, indigenous education was education geared towards producing individuals w i t h a strong and well-informed cultural identity. A n d just because indigenous education d i d not conform to the ways of the Westernized system, it is not the case for some less informed writers to have considered it primitive or inferior. Such contentions, Reagan (1996) maintains, should be seen as the product of ignorance and due to a total misunderstanding of the inherent value of informal education. 3.2.2 Colonial and Post-Colonial Education The evolution of formal education i n Cameroon may be d i v i d e d into five cultural epochs that bear eloquent testimony to a very rich diversity i n the influences at work on Cameroon history as a whole and on educational development i n particular (Shu, 2000). First, there was the period of English Education from 1844 to 1886 under the Baptist Missionary Society. Next came the epoch of German Education from 1884 to 1914 when Imperial Germany explored and conquered the hinterland and thus established their authority of the entire region. The third and fourth eras w h i c h occurred concurrently were that of French Education juxtaposed w i t h that of English Education from 1922 to 1961 (Shu, 2000). The fifth and last period is the post-colonial (post-independent) and reunification period from 1961 to present day. W i t h these five eras, it is obvious that the philosophy of education varied w i t h the change i n the European power i n control i n Cameroon. The arrival of the L o n d o n Baptist Missionary Society (B.M.S.) i n the 1840's signaled dramatic changes i n ideology and philosophy of indigenous educational 51 setting. Formal institutions of learning were being initiated and established. The B.M.S. and other American and European missionary organizations "envisioned themselves as bringing a high view of life to benighted savages" (Beidelman, cited i n Goodson & Ball, 1984, p. 118). Their main objectives i n the provision of schools was not to educate Cameroonians but as an incentive for the Cameroonians to allow their children to be subjected to the missionary influence and hence convert them into ' G o o d Christians' (Ball, 1984). Hence, formal education became an agent of assimilation, exploitation and subordination. In 1844 the B.M.S. led by Rev. Joseph Merrick established the first formal education setting i n Cameroon. This was not an accidental occurrence but an outgrowth of a pattern of Christian missionary activities throughout West Africa at the time (Azevedo, St. Lawrence, 1989). Because the B.M.S.'s main objective was to spread Christianity, the k i n d of curriculum that was developed and implemented i n this first primary school and subsequent ones was evangelical i n nature, emphasizing a religious content. The m e d i u m of instruction was English w i t h p i d g i n English as the lingua franca. Literacy skills were limited to the reading of the scriptures. The end result was to be the conversion of a few Cameroonians who w o u l d then assist i n spreading the gospel as "interpreters, and priests" (Shu, 2000). The B.M.S. began achieving their objectives as more and more Cameroonians were attracted to school with the hope that they w o u l d stand a better chance of getting a white-collar job that went to the educated. The success of the B.M.S. led to an influx of missionary organizations from Europe and America to compete for converts. Prior to this influx, the B.M.S. had limited the content of the curriculum to catechistical religious k n o w l edge. A s Beidelman (cited i n Goodson & Ball, 1984) maintains, "The missionaries were reluctant to teach secular skills to Africans because some feared to teach such skills w o u l d be to teach the w r o n g aspects of the world..."(p. 120) 52 Each of the missionary societies w o r k e d hard on attracting Cameroonians to their schools by modifying their approach to the k i n d of formal education offered. The C h u r c h Missionary Society (C.M.S.), one of the new missionary societies to establish i n Cameroon after the B.M.S., played d o w n the role of English as its m e d i u m of instruction, preferring to teach i n the vernacular, but yet maintained a firmly catechistic curriculum (Ball, 1984). It d i d not take long for Cameroonians to become suspicious of the narrowness of this k i n d of curriculum as they turned to the Catholic H o l y Ghost Fathers who taught other subjects beside religious instructions. C u r r i c u l u m changes such as the use of vernacular languages for instruction by some of the missionary societies came as desperate efforts to attract and retain Cameroonians i n their schools rather than as suggestions from the Cameroonians. Not surprising, there was no impact on this 'adapted' curriculum. One of the missionary societies, the Basel Mission, found that it was only possible to establish themselves by making fundamental changes to the sort of schooling they were offering rather than to the curriculum (Eyongetah & Brain, 1974). For example, "they concentrated their efforts into boarding schools" (Ball, 1984, p. 119) - a deliberate policy of detachment of individuals from the traditional milieu - and "teaching exclusively i n vernacular languages" (Ball, 1984, p. 119). The Baptist Boys School i n K u m b a , Joseph Merrick School i n Bimbia, and the Girls Baptist School at Bonaku were such schools. These schools continue to exist i n Cameroon till today. A s the cost of running the schools increased the metropolitan government (colonial government and the missionary organizations, settlers, estate owners and traders), was forced to make drastic changes to their policy on education i n the colonies. In 1847, the Education Committee of the Privy Council backed by the British government, issued its first systematic report to the Colonial Office in Cameroon. It stated, " A short and simple account of the mode w h i c h the Committee of the Council on Education considers that industrial schools for the colored 53 races may be conducted i n the colonies and to render the labour of the children available towards meeting some part of the expenses of their education" (Cited i n Goodson & Ball, 1984, p. 123). While each denomination taught aspects of its beliefs i n the n e w l y opened schools, increasingly strong control was applied by the colonial government i n matters of curriculum. This control was enhanced by the British administration's 'grants-in-aids' under the system of 'payment-by-results' as recommended by the Education Committee of the P r i v y Council (DeLancey, 1989, p. 30). Such control by the colonial government meant that i n general, the schools followed a program much the same as those throughout Nigeria and other British colonies, a program modeled after that used i n Britain and based on the British philosophy of education (Ball, 1984). The colonial government, it w o u l d seem, cared little about the culture and well being of indigenous Cameroonians. They viewed education as having a utilitarian and job-training role and hence supported an education system that produced a class of individuals who performed clerical and administrative duties required i n various government and commercial sectors (Azevedo, 1989). The colonial government had created a need for formal education and then later established taxing conditions on its provisions. Meantime, the colonialists were busy collecting raw materials from Cameroon and exporting these to Europe to be converted into finished goods. These same finished goods such as guns were later on exchanged for tons of raw material produced by Cameroonians. This form of intellectual exploitation resonates w i t h Willinsky's characterization of "intellectual mecantilism" as a form of exploitation wherein "the raw materials for grand theories are imported from the colonies for refinement and redistribution" (Willinsky, 1999, p. 59). W i t h the grants-in-aid programme as an incentive, the interests of the colonial government, the European settlers, and the missions appeared to coincide 54 and this gave rise to a relatively well-coordinated schooling system. The only group in this instance, w h i c h d i d not find its needs and interests being served, was the Cameroonians themselves (Ball, 1984). The Germans exacerbated this situation w i t h the annexation of Cameroon i n 1884. The parents who had always played a major role i n determining what k i n d of education their children needed were not only to be sidelined and become mere observers to the changes made i n the curriculum, but forced to send their children to be indoctrinated and alienated. Cameroon's annexation by Germany i n 1884 meant that Cameroon was n o w a part of Germany and this precipitated educational change i n conformity w i t h Germany's view of the purposes of the colonies. W i t h complete control of the territory by Germany, there was a withdrawal of the predominantly Englishspeaking missionaries and a sudden infusion of Germanic people. Missionaries w h o had spoken the local languages and English were replaced w i t h German-speaking arrivals w h o imposed German i n the schools as a m e d i u m of instruction, without allowing a transitional period. A t the time the Germans arrived, formal education was i n its infancy w i t h about 300 Cameroonian children enrolled i n the five mission schools that had been established (Azevedo & St. Lawrence, 1989). This was a period of confusion as all the schools were invariably conducted i n German w i t h limited use of vernacular languages i n the early years of German rule. Cameroon was n o w under colonial rule proper. To reinforce German influence, the use of vernacular languages was reduced and German instruction intensified. To maintain this, financial grants from the colonial government to the mission schools were based upon the German language competency of the pupils. In addition, government scholarships were available to students w h o demonstrated competence i n German for more advanced study i n Germany (Azevedo & St. Lawrence, 1989). German policy on the nature of formal education i n Cameroon was predominantly technical and vocational. Thus the 55 curriculum that evolved emphasized Germanic studies, and the immediate educational objective i n view was to introduce indigenes to Teutonic Culture (Shu, 2000). A s Nyerere i n "Education for self-reliance" argues, the purpose of education provided by the colonial government was not to prepare the youth for the service of their o w n country. Instead, it was motivated by a desire to inculcate the values and ideals of the colonial society and to train individuals for the service of the colonial state (Bishop, 1985, p. 237). Nyerere's argument resonates w i t h Rodney's characterization of colonial education: "It was not an educational system that grew out of the African environment, or one that was designed to promote the rational use of material and social resource. It was not an educational system designed to give y o u n g people confidence and pride as members of African society, but one w h i c h sought to instill a sense of deference towards all that was European and capitalist." (Rodney, cited i n Azevedo, 1989, p. 140) External influences on Cameroon d i d not, however, begin w i t h the establishment of colonial rule. That was directly preceded by periods of product and slave trade between the Cameroon coast, Europe, and N o r t h America, as well as the beginnings of a wide spread Christian missionary movement i n the area (DeLancey, 1989). However, w i t h the annexation of Cameroon by Germany o n July 14, 1884, there followed a brief period of competition among European countries, mainly to make treaties w i t h as many local chiefs as possible to gain control of the land and trade, as per the agreements of the Conference of Berlin—an international meeting on the European colonization of Africa. Germany's control over Cameroon was, however, short-lived. A few years after the first formal educational conference i n Cameroon i n 1907 i n w h i c h Germany re-emphasized its commitment to Germanspeaking instructions i n all the schools, they were ousted by British and French at the beginning of W o r l d War I (Azevedo, 1989, p. 140). Quite abruptly, German 56 ceased to be taught i n the schools, and German printed materials disappeared. Overnight, English and French became the official languages and the instructional languages i n the respective geographic areas occupied by each country. D u r i n g the linguistic confusion w h i c h followed this transition, p i d g i n was used to span the gap as people resorted to whatever languages w o u l d enable communication (Azevedo, 1989). In the French occupied areas, schools were supervised b y the French army chaplains and education was based on French values and ideas about education and the education process (Eyongetah & Brain, 1974). The French viewed education as part of the process of converting Cameroonians into francophones. The mission civilisatrice, the view that France's role was to bring its culture to a waiting w o r l d eager to adopt that culture, was a significant current in French colonial thought. Thus, the French saw no role i n the education process for Cameroonian languages or culture (DeLancey, 1989, p. 141). While vernacular languages were tolerated, they were not encouraged and were sometimes punishable w h e n expressed i n the classroom (Eyongetah & Brain, 1974; Njeuma, 1989). Denying the children the use of their language had one aim: to make them despise their language, hence the values carried b y that language, and b y implication despise themselves and the people w h o spoke a language which n o w was the cause of their daily humiliation and corporal punishment. Colonial education appears i n this context as a process of denying the national character, alienating the Cameroonian from her/his country and origin and, i n exacerbating her/his dependence on abroad, forcing her/him to be ashamed of her/his people and culture. From the time the French arrived, the curriculum and the syllabi became French. The French saw little need for local variations i n curriculum; strong central control was exerted from the colonial office i n Yaounde and from Paris. Educational support from the government was dependent u p o n the mission school's adherence to the French educational system. To ensure that the graduates from these schools 57 were assimilated into the French culture, the French limited the provision of education facilities u p to the secondary while granting scholarships for Cameroonian students (mostly to the sons and other relatives of the chiefs and kings) to continue their education i n France. It was thought that by limiting access to post-primary education and the literary/academic curriculum, they w o u l d be able to reproduce and control a neo-traditional native elite and avoid the creation of an employed educated group. W h e n questioned by the United Nations i n the early 1950s, on w h y no university had been established i n Cameroon, the French responded that Cameroonian higher education d i d exist - i n France (Azevedo, 1989, p. 141). In contrast to the French-occupied areas, the British sectors adjoining Nigeria reflected pre-colonial policy and encouraged instructional use of vernacular languages i n the early years followed by English instruction at the higher levels. Under the British colonial rule, education i n the British controlled territory was largely delegated to the Nigerian government. This meant that i n order for Cameroonians i n this region to receive training at the secondary or technical levels, they had to travel to Nigeria, a trend that continued well after independence (Shu, 2000). The British believed that they were educating Cameroonians to be Cameroonians - useful i n cash crop and other raw material production - and not attracted away from the land into the modern sector of the colonial economy. This was clearly spelt out i n the report of the Committee of the C o u n c i l on Education w h i c h stated among other recommendations, that the principal objectives of the education of the natives i n the colonies should be: (1) to diffuse grammatical k n o w l edge of the English language as the most important agent of civilization; (2) improved agriculture is required to replace the system of exhausting the virgin soils and then leaving the natural influences alone the work of reparation. The education of the colored races would, therefore, not be complete for the children of small farmers, unless it included this object; and (3) lesson books should teach the natural 58 interests of the mother country, and her dependencies, the natural basis of this connection and the domestic and social duties of the colored races (Ball, 1984, p. 124). In this connection, they tolerated the use of Cameroonian languages i n the early years of primary school followed by increasing instruction i n English i n the higher levels (Azevedo, 1989). Unlike i n the French occupied territories where the French exerted strong central control i n the organization of the school, the British left more discretion i n the hands of local school officials. The students i n schools prepared for similar examinations written by students i n L o n d o n and other British colonies (DeLancey, 1989). The report of the Education Committee was the first i n a continuing series of failed attempts to 'adapt' the school curriculum i n British controlled Cameroon and other colonies. Foster points out that the failure of the recommendations rested on two assumptions: first, that 'the creation of schools and curriculums' based on the economic development of agriculture ' w o u l d generate demand for such education among the "colored races'" (Foster, cited i n Goodson & Ball, 1984, p. 125); and secondly, that 'African expectations regarding the potential functions of educational institutions were congruent w i t h those of Europeans, or could be made so' (Foster, cited i n Goodson & Ball, 1984, p. 125). The failure of these recommendations is undoubtedly evident, as the colonial government had overlooked the contribution of the indigenous Cameroonians to the establishment of the schools. The failure of the Privy Council report precipitated another series of evaluation reports. One of such reports was that produced by the British government appointed Phelps-Stokes Commission, an American funded investigation into the African education w h i c h visited West and South Africa i n 1921 and East and Central Africa i n 1924. According to Ball (1984), "the reports, Education in Africa: A Study of West, South and Equatorial Africa by the African Education Commission (1922) and Education in East Africa (1925) written by Jesse Jones, were very much influenced by the work of black American educators at the H a m p t o n and Tuskegee Colleges. 59 Parallels were d r a w n between the educational problems of the American Negro and the African. W i t h this transfer of concepts, the emphasis of the reports was on the role of rural education" (Goodson & Ball, 1984, p. 132). This report was to become the guiding document on overcoming Cameroonian resistance to the 'adapted', narrow, and academically oriented curriculum. According to the report (cited i n Goodson & Ball, 1984, p. 133), 1. the schools were to be adapted to native life; 2. emphasis was to be technical vocational and agricultural training at the expense of more 'traditional' subjects w i t h i n the curriculum; 3. there was to be an increase awareness of the need to expand educational facilities for women and girls. These recommendations meant that major changes were to ensue i n the contents of certificate and matriculation examinations organized by the British government for all its colonies. A t the instigation of the Colonial Office A d v i s o r y Committee, the University of London passed that, "the English universities involved i n overseas examining expressed a willingness, in the best tradition of the Phelps-Stokes, to modify their syllabuses ' i n order that the external examination system might be adapted to local needs' and to ensure that colonial subjects were 'not being forced into an educational m o u l d that might deform their particular attitudes' and unfit them for a life i n their o w n country" (University of London, Minutes, 1935-36). Graduates from the French and British colonial schools who had hoped for better jobs and hence better lives soon realized that there were no openings for them. They had seen education as the one source of the material superiority of the white colonialists, and a route for i n d i v i d u a l social mobility. Courade and Courade (1975) quote two extracts from essays written i n 1930 by Alliance H i g h School pupils under the title ' W h y I G o to School'; 60 If I go to school and get much knowledge I w o u l d not always work for others but I may have much money, I w i l l look for many workmen. I myself want to go to school. But every boy w h o does not want to go to school is stupid. Y o u can't be a great man without going to school. If y o u want to be a great man then go to school. (Courade & Courade, 1975, p. 10) Even w i t h this education, Cameroonian graduates realized that their aspirations were some what far fetched. The position and jobs i n the colonial government was restricted mostly to the whites w i t h a very few lucky Cameroonians appointed to posts of responsibility. Frustrated by competition i n the important roles i n the administration, kept d o w n by whites i n the missionary groups, these graduates of the primary and secondary schools came to see that the continuation of foreign domination meant their continuing relegation to the lower posts, the lower pay, the lower profits (DeLancey, 1989). O n l y through independence was there a chance to open those opportunities for Cameroonians. Events leading up to independence were set i n motion i n 1955. In that year, Southern Cameroons (formerly West Cameroon) withdrew from the Cambridge University Board to take the West African Certificate Examination which had been created i n 1951 (Ndongko & Nyamnjoh, 2000). A t the d a w n of independence, when colonial regimes were being ousted from the African continent, Cameroon, like most African countries (e.g. Nigeria, Ghana, Chad, Sierra Leon, etc) had a literacy rate of 10 percent (DeLancey, 1989). The rate of educational expansion had not been even as some schools were more equipped than others based on the 'grants-in-aid' program and on the performance of their graduates i n the French language. A s a Federal Republic, made u p of the British West Cameroon and the French East Cameroon, it had inherited two systems of education - the French and the British. W i t h these two systems of education founded on two different and competing philosophies, integration and consequently harmoniz61 ation was to become one of the major tasks i n Cameroon's drive towards literacy for all Cameroonians (Tambo, 2000). Independence for Cameroon as for most of Africa, d i d not come easily. There were negotiations, agreements, treaties and concessions made by both the colonial government and the government of independent Cameroon. There were agreements signed between the British government and West Cameroon. In the first agreement i n 1963, Cameroon agreed to withdraw from the West African Certificate Examination and to follow the University of L o n d o n General Certificate of Education (G.C.E.) Examination. Similar agreements were signed between the French government and East Cameroon. One of such was the 1959 agreement between Ahidjo and the French government i n w h i c h Ahidjo, as the father of independence, was to maintain close cooperation i n economic, cultural, political, and military affairs (Beuth, 1975). Through this agreement, a system had been established whereby the two European powers got their raw materials and sold their finished goods. In East Cameroon, the French had imported printing presses i n order to produce instruction materials i n French. The educational system was well-organized i n this part of the country. In the West, the educational system had virtually collapsed. Primary education had developed slowly and higher education was available only i n Nigeria, which maintained strict quotas for Cameroonians (Azevedo, 1989). Besides these differences i n education provisions i n the two regions, Cameroonians began the w o r k at harmonizing and standardizing the structure and organization of the two systems. In 1976, the Cameroon government organized a national education seminar w i t h representatives of the British government. Important decisions were made i n this seminar regarding the British system of education: "The main features of the examination such as name, organization, objectives of the ordinary level and advanced level subjects, number of sessions, per year, types of questions, marking, grading, security measures, etc. were defined...It was agreed that special link arrange- 62 merits be established w i t h the Department of Examinations, Senate House, University of L o n d o n and the British Government. This involved assistance i n drafting of the syllabuses, moderation, the supply of consultants and external examiners and the training of the Cameroonian personnel. Finally, it was agreed that subject to the maintenance of appropriate standards, the University of L o n d o n w o u l d be asked to do all that it could to ensure the wide scale recognition of the Cameroon G.C.E. examination." (Ministry of National Education, 1976, pp. 34-35) Since the Cameroonization of the G.C.E. there have been a series of reforms by the M i n i s t r y of National Education to harmonize the G . C . E . w i t h the French Baccalaureate. Most of these reforms were not successful, as concession was demanded only of the English-speaking Cameroonians. The French-speaking section strictly followed the baccalaureate as organized i n France, and students w h o pursued their studies i n French wrote the baccalaureate and/or the G . C . E . examination w i t h the hope of studying abroad i n a French university. Even w i t h Cameroonians n o w at the helm of all the institutions i n the country, the school curriculum remained unchanged. The school curriculum, totally foreign to the people of the Cameroons, emphasized disciplines and training that d i d not address themselves to the needs of the indigenous population (Eyongetah & Brain, 1974). Neither was the curriculum based on the culture of the Cameroonian society, through w h i c h the young could be socialized, nor was it framed i n terms of what was regarded as being best or most valuable among the intellectual and artistic achievements of the Cameroonian society. The curriculum employed, though controlled by Cameroonians, was essentially European i n nature, grounded w i t h i n theories of education that gave rise to a view of knowledge as timeless, objective, o w i n g nothing to the particular circumstances of individual eras, societies, cultures or human beings (Shu, 2000). Concentration was on the history, geography and the language of the mother country (English and French) and acquisition was accom- 63 plished through constant memorization. Rodney (cited i n Azevedo, 1989) poignantly noted that: "... the colonial system educated too many fools and clowns, fascinated by the ideas and ways of life of the European capitalist class. Some reached a point of total estrangement from African conditions and the African w a y of life ... Colonial schooling was education for subordination, exploitation, the creation of mental confusion, and the development of underdevelopment." (p. 140) This is reminiscent of m y elementary school history text i n the late 1970s and books on the history of Cameroon that often opened w i t h the following: "The Carthaginian Adventurers were the earliest to discover M o u n t Cameroon..." (Eyongetah & Brain, 1974, p. 9). A s if Cameroonians had not seen M t . Cameroon, the highest peak i n West Africa, w h i c h was i n active eruption at the time. M t . Cameroon was the worshiping place for the Bakweri people of southwest Cameroon. To say that the Carthaginians led by the north African Hanno (Eyongetah & Brain, 1974), were the first to see this conspicuous physical feature is to i m p l y that the communities w h i c h had lived i n these area for hundreds, if not thousand, of years were b l i n d or not "men". Such knowledge was used and continues to be used to show the superiority of a race that had the ability to name the w o r l d for the Other. It is worth noting that M t . Cameroon is locally k n o w n as M t . Bwea, named after the tribe that had settled there hundreds or thousands of years ago (Ardener, 1996, p. 52). This sense of superiority for one race and the consequent inferiority of the other race is presented as "the" objective fact i n order to perpetuate the status quo. H a v i n g little or nothing to identify with, the learner passively listens to the teacher talk, ready to recite and reproduce the teacher's notes i n the exam. The learners memorize mechanically the content w h i c h turns them into "containers" or "receptacles" to be "filled" by the teacher (Freire, 1994, p. 53). This k i n d of teaching is what Freire (1994) refers to as "banking concept of education". 64 "Education thus becomes an act of depositing, i n w h i c h the students are the depositories and the teacher is the depositor. Instead of communicating, the teacher issues communiques and makes deposits w h i c h the student patiently receives, memorizes and repeats." (p. 53) A s a high school mathematics teacher, I remember questions i n the mathematics texts d r a w n from cultures very alien to Cameroonians requiring students to problem-solve. For example, "If a cricketer scores altogether r runs i n x innings, n times not out, his average is r/(x-n) runs. F i n d his average if he scores 204 runs i n 15 innings, 3 times not out." (Bishop, 1990, p. 55) While some of the students were able to solve the problem and others similar to these, w i t h their good knowledge of algebra, most never understood what the problem was all about. What is an inning?, a run? What is the game of cricket all about? were questions teachers could not provide answers to. But since the teachers are viewed as the most knowledgeable persons i n the classroom, it meant that the students could not pursue such inquiry further than the teacher was prepared to. This is a good problem to encourage students' application of concepts but they serve no purpose if the concept is entirely developed with examples like these. The end result is, students finding it difficult to construct any understanding w i t h no experiential knowledge. Consequently, the evaluation process becomes a continuous source of anxiety for the learners. W i t h a curriculum or more appropriately, a syllabus that concentrates on imparting knowledge that w i l l be tested i n an exam, it is little wonder that certificates become the top priority and almost the sole determinant i n the job market. A n d since Cameroon inherited i n full measure the colonial educational tradition, the pervasive hold of irrelevance and inappropriateness calls for reexamination and re-statement of educational aims, goals, and objectives (Tambo, 2000). 65 While there has been some progress i n educational provision, almost forty years after independence, the situation has not changed by much. A l t h o u g h there has been recorded dramatic improvement i n terms of expansion i n educational facilities and an increase i n the proportion of school-age children i n school (DeLancey, 1989, Tambo, 2000), yet illiteracy among adults remains common. Very little attention is paid on educating the adult populace. The accomplishment i n the expansion of education for the youths has not come without several problems, some common to other African states, some peculiar to Cameroon. For example, for all children i n Cameroon, education is either i n French or i n English, yet most of these students come from homes where these languages are not spoken. A s a result, there is a high dropout rate after each academic year, partly due to linguistic difficulties, and partly to lack of basic school needs. The traditional curriculum emphasizes physical education and musical education at the nursery schools, the three R's i n the primary level, and classical education at the secondary level to prepare students for the G.C.E. and Baccalaureate examinations (Azevedo, 1989). The curriculum is heavily influenced by the examinations w h i c h lead to certificates. Initially aimed at preparing students to continue their studies at universities i n France and the U K , the curriculum has continued to turn out these graduates even though the chances of studying abroad have become fewer and fewer. This therefore calls for Cameroon to re-think and reformulate its educational policy to make formal education relevant and useful to the Cameroonian society. 3.3 T h e Current State of Education In the current state of education, teaching and learning are heavily guided by examination syllabuses that provide a list of topics, theorems, and computations which 'candidates' need to k n o w and execute ( C G C E Board, 1994). N d o n g k o and Nyamnjoh (2000) describe the current school curriculum as based on the "academic6 6 elitist tradition of education inherited from the colonial educators" (p. 246). N g o h (2000) also characterizes the curriculum as examination-oriented and geared towards "university preparation and white-collar jobs w i t h undue emphasis on cognitive development at the expense of social skills, cultural and technological literacy" (p. 161). The very high dropout rates after each year of school and an equally unfortunate high failure rate i n the terminal examinations at each level of education suggest curricular problems than the quality of teaching i n Cameroon today. The curriculum basically aims at preparing children to go to university, rather than preparing them for the life that the majority of them, unable to gain university admission, w i l l live. To this end, emphasis is placed on the arts, humanities and social sciences at the expense of mathematics, science and technology. A clear indication to this current predicament is the large number of school-leavers who, after several years of hard work to meet the schooling demands still cannot find employment - or are not w i l l i n g to accept the types of employment available. It is easy to understand w h y school-leavers are unable to find employment having been educated i n English and/or French, about the history concerned w i t h h o w people lived i n societies very distant i n space and time, students complete their schooling w i t h disdain for their o w n society and few, if any, marketable skills (Samoff, Metzler, & Salie, 1992). This is the k i n d of curriculum that does not inspire learning as school knowledge does not b u i l d u p o n the tacit knowledge derived from cultural resources that the student already possesses (Aronowitz & Giroux, 1991). Thus, designing a curriculum for the needs of the country, a national curriculum, is a goal of most African states, w h i c h unfortunately, inherited from their colonial rulers a set of educational practices and philosophies based on the needs of that ruler, rather than on local priorities and culture. For Cameroon, this situation is more difficult because of its dual inheritance. "Harmonization," the joining together of French and British educational theory and practice (Ndongko & Tambo, 2000), has 67 made more difficult the task of Cameroonizing the curriculum. Yet this dual inheritance may also hold a unique possibility of bringing together i n the new Cameroon educational system a wider variety of experience than i n other African states. While education today i n Cameroon remains heavily influenced by its colonial cultural heritage (an experience that cannot be ignored nor dispose of easily) as w e l l as a multiplicity of vernacular languages, religious heterogeneity, and diverse ethnology (Shu, 2000), one of the distinct effects of the recent emergence of post-colonial discourse has been to force a radical re-thinking and re-formulation of forms of knowledge and social identities authored and authorized by colonialism and western domination (Mongia, 1996). I think Cameroon needs to re-think its educational policy and Cameroonians i n the focus on designing a drive towards curriculum that self-reliant development. will assist This k i n d of curriculum w i l l need to have [Cameroonian] culture as its central focus, be free of past colonial domination and yet be comparable w i t h the rest of the w o r l d . It w i l l be a curriculum derived from sources w i t h which learners can identify and realize their contribution i n the development of their country. One of the ways Cameroonians can break away from this cycle of inappropriate education is by identifying individual characteristics and behaviours that are valued i n the society, analysing them, and according to priority, developing a curriculum that w o u l d develop such individuals. To do this, the Cameroon government needs to get all those involved i n the education of the child participate in crucial decision making sessions regarding the curriculum. The role of the parents need to be emphasized, and most importantly input from the students should be considered since they are the ones w h o are going to be affected most. In this section, I have traced the evolution of the school curriculum i n Cameroon from the time of pre-colonial Cameroon to post-independence. The understanding of the series of conflicts, compromises, and the various processes of 68 negotiations, through time, of the present curriculum w i l l enable Cameroonians realize that failure to treat culture/society as a fundamental determinant to curriculum decisions is a major contributor to cultural alienation. Cameroon already seems to be making some great strides as evident i n the recommendations of the 1995 National Education Forum and most recently i n the 1999 National Education workshop. The 1995 Forum was the first of its k i n d since independence involving representatives from all the ten provinces as well as from private and lay secular education. The delegates at the forum stressed the need to harmonize the two systems, the need for the government to encourage the production of textbooks, and above all, the continual revision of the school curriculum to serve the needs of Cameroonians ( M I N E D U C , 1995, p. 67). What has the literature informed us about the cultural aspects of mathematics and mathematics learning and teaching? This is where I turn to n o w to review and discuss some studies on ethnomathematics. 3.4 Studies on Ethnomathematics M u c h has been written about relationships between culture and mathematics. The writings span a range of topics that include the cultural bases for mathematics, mathematics development i n different cultures, the historical culture of mathematics, the effects of culture on mathematics learning and dispositions toward mathematics, and the political effects of mathematics and mathematics education on societies. But research studies framed as ethnomathematical are few and varied, dating back to the early 1980s. A s such, research i n this area is i n its infancy relative to other areas of research i n mathematics or mathematics education. The conceptual, philosophical and theoretical explorations of ethnomathematics i n the previous chapter (Chapter 2) have set the stage for identifying ethnomathematical studies to be reviewed i n this section. Generally, research i n ethnomathematics is driven by three broad questions: (1) Where do mathematical ideas come from, h o w are they organized?; (2) H o w does mathematical knowledge advance?; and (3) D o these ideas 69 have anything to do w i t h the broad environment, be it socio-cultural or natural? (D'Ambrosio, 1994). Barton (1996), however, classifies research i n ethnomathematics i n three dimensions: time, culture, and mathematics. " O n the time dimension, ethno- mathematics may be concerned w i t h conceptions of an ancient or a contemporary cultural group" (p. 220). It also may be concerned w i t h historical or contemporary practices of a cultural group. Examples might include research on h o w early settlers i n Cameroon used mathematics or on modern-day Cameroonian entrepreneurs' use of mathematics. "The cultural dimension of the definition extends from a distinct ethnic group, to a purely social or vocational group" (Barton, 1996, p. 220). Research on this dimension may focus on African Canadians i n British Columbia, Children selling candy on the streets of Brazil, sports statisticians i n the National Hockey League ( N H L ) i n N o r t h America, or teachers i n adult literacy. "The mathematical dimension of ethnomathematics is determined by the relationship of the mathematical ideas to mathematics itself, i.e., ethnomathematics is a study w h i c h may be internal to mathematics, or conceptually removed from existing mathematical conventions" (Barton, 1996, p. 220). Examples of this dimension might include the varied formal conceptions of mathematics held by Cameroonian teachers or the mathematics used by craftsmen i n Cameroon. Below (Fig. 3.1) is a modified illustration (Barton, 1996) of these three dimensions and h o w they might interact. 5 The body of literature on ethnomathematics incorporates research o n the mathematics practice of identifiable cultures and research on the mathematics practice i n everyday situations within particular cultures. In the first case, researchers have tended to look at the mathematics practice of a whole culture (e.g., Bockaire, 1988; Carraher, Carraher & Schliemann, 1987; G a y & Cole, 1967; Gerdes, 1988; Lancy, 1983; 5 For more on the representation of the dimensions of ethnomathematics research see Barton, 1996. 70 Posner, 1982; Zaslavsky, 1973), whereas researchers investigating mathematics practice i n everyday situations (Ascher, 1991; Barta, 1995; Saxe, 1985; Silverman, 1994) have focused on one situation or work context (e.g., grocery shopping, carpentry, masonry, candy selling) within a culture. Time Mathematics. Internal X .''Past Culture Ethnic Group ^ Future Social Group External Fig. 3.1: Dimensions of Ethnomathematics Research Some of these researchers (e.g., Barta, 1995; Brenner, 1985; Carraher, Carraher & Schliemann, 1987; Ferreira, 1990; Masingila; 1994) have contrasted mathematics practice i n school w i t h mathematics practice i n everyday situations and noted the gap between the two. Lester (1989) suggested that knowledge gained i n out-ofschool situations often develops out of activities that occur i n a familiar setting, are dilemma driven, are goal directed, use the learner's o w n natural language, and often occur i n an apprenticeship situation. Knowledge acquired i n school all too often is formed out of a transmission paradigm of instruction that is largely devoid of meaning. In mathematics, this process of knowledge acquisition has been referred to as acculturation - the process by w h i c h a social group, and ultimately each of its members, actively constructs mathematical knowledge on the basis of experience i n a sociocultural environment that is not their o w n - and ethnomathematical studies have shown that this process often leads to intellectual impasses i n the learner. A s Knijnik (1997) reaffirms, 71 When a specific subordinate economic, social, and political group becomes disadvantages conscious of which its the scarce knowledge brings about, and tries to learn erudite knowledge, this type of consciousness may contribute to the process of social change (p. 409). I n o w review ethnomathematics studies that aim at p r o v i d i n g some insight into the above three questions that researchers i n this area concern themselves with. 3.4.1 Where do mathematical ideas come from, how are they organized? The anthropological literature (Ascher, 1991; Gerdes, 1988b & 1994; Masingila, 1994; Saxe, 1988; Zaslavsky, 1973) demonstrates that the mathematics, w h i c h most people study i n contemporary schools, is not the only mathematics that exists (Lumpkin, 1983). Paulus Gerdes, a Mozambican mathematician and mathematics educator, is perhaps one of the leading researchers i n uncovering the mathematical ideas embedded i n African cultural practices and artifacts and i n presenting these findings to the mathematical community. In looking for ways i n w h i c h one can understand students' cultural ways of producing and expressing their mathematics, Gerdes and his colleagues i n Southern Africa have "developed a methodology that enables one to uncover i n traditional, material culture some hidden moments of geometrical thinking" (Gerdes, 1988b, p. 140). In their methodology, they "looked to the geometrical forms and patterns of traditional objects like baskets, mats, pots, houses, fish-traps, and so forth and posed the question: why do these material products possess the form they have?" (Gerdes, 1988b, p. 140) Observing the production techniques, Gerdes and his colleagues found that "the form of these objects is almost never arbitrary, but generally represents many practical advantages and is, quite a lot of times, the only possible or optimal solution of a production problem" (Gerdes, 1988b, p. 140). According to Gerdes, "the traditional form reflects accumulated experience and w i s d o m . It constitutes not only biological and physical knowledge about the materials that are used, but also mathematical knowledge, 72 knowledge about the properties and relations of circles, angles, rectangles, squares, regular pentagons and hexagons, cones, pyramids, cylinders, and so forth" (Gerdes, 1988b, p. 140). " A p p l y i n g this method," (Gerdes, 1988b, p. 140) Gerdes and his colleagues have unveiled "a lot of 'hidden' or 'frozen' mathematics." (Gerdes, 1988b, p. 140) They see "the artisan, who imitates a k n o w n production technique, (Gerdes, 1988b, p. 140) as, "generally, not doing some mathematics. But the artisan(s) w h o discovered the technique, did mathematics, was/were thinking mathematically. W h e n pupils are stimulated to reinvent such a production technique, they are doing and learning mathematics" (Gerdes, 1988b, p. 141). This is however, dependent on the abilities and convictions of the teacher. A s Gerdes and his colleagues state: "Hereto they can be stimulated only if the teachers themselves are conscious of hidden mathematics, are convinced of the cultural, educational and scientific value of rediscovering and exploring hidden mathematics, are aware of the potential of 'unfreezing' this 'frozen mathematics.'" (Gerdes, 1988b, p. 141). According to Gerdes, the incorporation of this "frozen mathematics" like the A n g o l a n sand drawings into the school mathematics curriculum provides the potential for the achievement of three important societal goals: (1) it may contribute to the revival, reinforcement and valuing of this practice and towards a more productive and creative mathematics education, as it avoids socio-cultural and psychological alienation; (2) the integration of this practice, the knowledge it reveals and its mathematical potential, w i l l become less monopolized, less regional and less tribal and w i l l contribute to the development of a truly national culture of Angola; and (3) its incorporation w i l l lead to the consolidation of the idea that mathematics does not come from outside African cultures and this w i l l contribute to mathematical confidence that has been long lacking i n most of the non-European countries (Gerdes, 1988, pp. 19-20). 73 The argument that children make sense of the w o r l d according to their cultural background has been repeatedly supported, i n many contexts, among students i n Liberia (Gay & Cole, 1967), Sierra Leone (Bockaire, 1988), and Brazil (Saxe, 1988; Silverman, 1994). In Sierra Leone, for example, "Bockaire (1988) developed a project to 'investigate the mathematics that existed i n Mende culture, the strengths and limitations of such mathematics, the mathematics that probably existed long before formal schooling was introduced i n the M e n d e land' (p. 1). Bockaire was aware that, as a consequence of engaging i n and reflecting on traditional activities, 'at the village levels, on the farms, at the rivers and on the foot paths or roads to their work place,' people develop mathematical understandings, albeit not i n the same codified form as 'Western' mathematics. Cognizant of severe limitations i n the colonialist-inspired school system, the aim of the ethno- mathematical investigation of Bockaire and his team was to create a culturally appropriate curriculum to educate those i n the illiterate and innumerate sectors since 'mathematics courses i n [Sierra Leone's] formal school system are structured without recognition and exploitation of the wealthy mathematical activities w i t h i n the culture' (p. 1). In addition to contributing to the education of illiterate farmers, investigators used the Mende's approach to adding two sets of objects to teach, at Njala University College i n Sierra Leon, the concept of homomorphism to a sophomore-level course i n abstract algebra and noticed marked improvement i n understanding and interest (p. 63)" (Cited i n P o w e l l & Frankenstein, 1997b, p. 256257). A number of research studies have demonstrated some encouraging results w i t h the introduction of ethnomathematics into the mathematics classroom. Gerdes (1988) studied the sand drawings of the Tchokwe people of A n g o l a and suggested some possible ways for using these i n the mathematics classroom. From the sand drawings, Gerdes was able to demonstrate the presence of diverse arithmetical relationships such as arithmetic progression, series summation, and the Pythagorean 74 triplet. Other geometrical ideas observed and identified i n these traditional A n g o l a n sand drawings included geometrical symmetry (bilateral symmetry, rotational symmetry, double bilateral and point symmetry) and similarity. The existence of these properties suggests that the sand drawings could serve as a starting point for the teaching of geometrical symmetries and similarities. Contrary to arguments made against ethnomathematics as 'not real' mathematics, Gerdes was able to use one of the sand drawings, the 'akwa kuta sona', to show how its physical properties (characteristics of the curves) could be used to determine the greatest common divisor (GCD) of two natural numbers. A n awareness of these characteristics i n the determination of the greatest common divisor of two natural numbers establishes a solid foundation for the learning of other mathematical concepts and relationships such as the Euclidean algorithm i n the determination of the greatest common divisor of two natural numbers. The extrapolation of the concept of G C D achieves two goals: (1) it demonstrates i n the learner the presence of mathematical concepts i n their culture, (2) it demonstrates that mathematics is a mosaic of cultural contribution, and (3) allows the learner to better appreciate the subsequent refinements made of the mathematics concepts. Gerdes maintains that the incorporation of these traditional Angolan sand drawings, both educational and artistic-mathematical, into the curriculum may contribute (1) to the revival, reinforcement and valuing of the practice; (2) towards a more productive and creative mathematics education, and (3) to the development of a truly national culture by including other popular practices from all regions of the country. Gerdes' analysis can be applied to different cultural contexts w o r l d w i d e . 3.4.2 How does mathematical knowledge advance? Henderson (1981) argued that many people viewed and learned mathematics i n a rigid and rote way that has hindered their creativity. This condition is further "systematically reinforced by our culture, w h i c h views mathematics 75 as only accessible to a talented few. These views and attitudes, besides affecting individuals, have become part of what separates and holds d o w n many oppressed groups, including women, w o r k i n g class and racial minorities" (p. 12). A study conducted by Saxe (1985) i n an effort to gain insight into the influence of Western schooling on the development of arithmetical understandings in children from a non-technological culture, the O k s a p m i n of Papua N e w Guinea, showed that O k s a p m i n children not only spontaneously use the indigenous system i n the context of school arithmetic, but also have created new forms of numerical symbolization and calculation based on that system to deal w i t h the arithmetical problems i n formal instructional setting. This phenomenon w i t h the O k s a p m i n children is quite general and occurs i n other ethnic groups as they apply knowledge they have generated i n their everyday social activities to the novel problems they identify as they participate i n Western school classrooms. In their study of the mathematics learning difficulties among the Kpelle of Liberia, G a y and Cole (1967) concluded that cultural differences are not necessarily associated w i t h deep cognitive differences and that there do not exist any inherent difficulties: what happened i n the classroom, was that the contents d i d not make any sense from the view point of the Kpelle-culture; moreover the methods used were primarily based on rote memory. Instead of taking school knowledge as the reference and elaborating tasks from this perspective to see whether individuals transfer it to other settings, the researchers used ethnographic observations outside school contexts. They organized research tasks based on activities w i t h i n the community studied. The results showed that differences i n favour of schooled western people i n tests (generally showing a 'decalage' of several years' advantage over natives from non-western cultures) disappeared. In some cases, these differences appeared i n the opposite direction. For example, Kpelle illiterate adults performed better than their N o r t h American counterparts, when solving problems such as the estimation of number of cups of rice in a given container, that belong to 76 their indigenous mathematics. W i t h these results, G a y and Cole became convinced that it is necessary to investigate first the "indigenous mathematics," i n order to be able to b u i l d effective bridges from this "indigenous mathematics" to the new mathematics to be introduced i n the school. According to G a y and Cole, this ethnomathematical approach calls for the teacher to begin w i t h the materials of the indigenous culture, leading the child to use them i n a creative way, and from there advance to the new school mathematics. 3.4.3 Do these ideas have anything to do with the broad environment, be it socio-cultural or natural? Vygotskian perspective contends that culture influences individual development. Vygotsky believed that "creative imagination" grows out of the play of y o u n g children (Williams, p. 117) and that learning occurs at two levels: the social level i n collaboration w i t h others, and the individual level, where learning is internalized (Vygotsky [1934], 1978). Therefore a rich learning environment provides a stimulating place for meaningful interactions to occur and concepts to form at the everyday or spontaneous level. From this viewpoint, the meanings of many mathematical concepts are socially or culturally determined and at times the meaning of a w o r d may vary according to the sociocultural context i n question. FitzSimons et al. (1996) noted that the term 'numeracy' has been used to carry different meanings and its meaning varies according to the social and cultural background of its users. In general, the term 'numeracy' is used more often in the Anglo-American countries while the term 'mathematics' is used i n the German speaking countries. Generally, "being numerate means having developed certain basic mathematical skills applicable to various situations i n everyday life" (FitzSimons et al., 1996, p. 756). O n the other hand, to be 'mathematically educated' encompasses a broader perspective. It means to "have a sound mathematical knowledge", to have "acquired knowledge about concepts and 77 methods typically utilized i n mathematics - about their power, and about their limitations" (p. 756) as w e l l as to have a critical stance about mathematics as a whole. In sum, numeracy is regarded as only an important part of mathematics that is required by any i n d i v i d u a l to function relatively effectively i n every day life situation. Boaler (1993) conducted a study i n the U K that considered the influence of context upon students' choice of procedure as well as a comparison of the relative effectiveness of two environments (one characterized by the complete integration of process and content w i t h one that concentrated mainly u p o n content w i t h some separate attention to process) w i t h relation to a student's capability i n situations requiring transfer to the 'real w o r l d ' . In contrasting the effectiveness of the two learning environments, Boaler presented students w i t h questions set i n different contexts and investigated whether their capability i n transferring their mathematics was related to the way they had learned mathematics. She observed that while the methods students used i n the questions d i d not reveal any particular reason for differential response, it d i d confirm their variance in methods w i t h contexts. It was also observed that procedure and performance were, to some extent, determined by the task contexts. This research highlights the importance of context i n learning, as this provides bridges and helps break d o w n barriers to growth i n mathematical understanding. Boaler's (1993) study corroborates findings of an earlier study conducted by Carraher et al. (1987), which investigated the effect of the situation on the choice of procedures and efficiency i n problem solving w i t h 16 third graders aged 8-13, i n Recife, Brazil. D u r i n g the interview sessions, each child was asked to solve 10 problems i n each of three different situations: (a) simulated store situation i n w h i c h the child played the role of the store owner and the experimenter played the role of customer, (b) embedded i n w o r d (story) problems, and (c) computation exercises. Carraher et al. found significant differences i n the oral versus written computation 78 and that the oral procedure was significantly superior to written procedure at the .002 level. However, there was no indication that the situation i n w h i c h arithmetic problems are embedded has an effect on the number of correct responses obtained. In South America, Carraher et al. (1987), w o r k i n g w i t h street children i n Brazil observed that children w h o knew before they went to school, h o w to solve creatively arithmetical problems w h i c h they encountered i n daily life, for example, at the marketplace, could later i n the school, not solve the same problem, that is, not solve them w i t h the methods taught i n the arithmetic class. This indicates that the children's spontaneous abilities are downgraded, repressed and forgotten, while the learned ones are not being assimilated. It has also been suggested that the spatial environment influences the development of spatial relationships. Pallascio et al. (1990) conducted a study to examine the influence of children's spatial environments on their development of spatial skills i n a situation focusing on the manipulation of small objects and their two dimensional representations. Their observations suggested that Inuit children and children from an urban environments, differed from each other i n their perception and representation of their geometric properties of different objects and i n their spatial skills. Another study conducted by Pallascio et al. (2002) on Inuit students had as its objectives, among others, to (1) understand more fully the number of determinants i n the metacognitive processes of Inuit students when they are faced w i t h solving problems related to spatial ideas; and (2) to identify the sociocultural elements that teachers and curriculum developers should consider along w i t h students' interest and motivation w h e n designing curricula and when planning instruction. After analyzing the data collected using learning activities and questionnaires on the students' interests, cognitive and metacognitive aspects of the tasks, Pallascio et al. observed that the students manifested the weakest interest i n the most decontextualized activities—that is the activities having the fewest cultural connotations, 79 whereas the more contextualized activities, w h i c h had greater cultural or everyday connotations, drew a more positive response. N o t only were the students more interested i n the learning activities, their confidence and self-esteem were greatly improved as they could easily relate to the learning activities since they were based on their everyday experience. The above observations by Pallascio et al. (2002) are at the heart of ethnomathematics pedagogy. W i t h i n this pedagogy, cultural relevance is the determining factor for both the learning activities and the classroom instructions. For instructions to be culturally sensitive, content and pedagogy must be culturally congruent. Ethnomathematics pedagogy also subscribes to a constructivist theory that emphasizes situated and contextualized learning. In classrooms governed by constructivist theory, students carry out tasks and solve problems that resemble those i n the real w o r l d (Glatthorn, 1994). Instead of doing exercises out of context, the student becomes engaged w i t h contextualized problems that allow the learner to connect prior knowledge to new knowledge and transfer new knowledge and understanding to real situations. The recurrence of research findings w h i c h demonstrate the relationship between an individual's cultural background and their ability to construct mathematical concepts suggest that assumptions regarding understanding and application of mathematical knowledge as a result of learning i n context may be over simplistic. The above research studies only illuminate h o w much success students w i l l experience i n mathematics education not only i n Cameroon but everywhere if their prior learning or "primitive k n o w i n g " (Pirie & Kieren, 1991) is used as the starting point for learning school mathematics. By building u p o n the mathematical k n o w l edge students' bring to school from their everyday experiences i n their o w n culture, teachers can encourage students to: (a) make connections between the mathematics embedded i n their culture and school mathematics i n a manner that w i l l help formalize the students' informal mathematical knowledge, and (b) learn school mathe80 matics i n a more meaningful, relevant way. School "mathematics teaching can be more effective and w i l l yield more equal opportunities, provided it starts from and feeds on the cultural knowledge or cognitive background" of the students (Pinxten, van Dooren, & Soberon, 1994, p. 28). 3.5 Conclusion Teachers, researchers, mathematicians, and policymakers have all argued about what curricula should be used i n classrooms. A l t h o u g h proponents of relevant curricula agree about the importance of contextualized mathematics instructions, they have rarely considered the ways i n w h i c h foreign and transplanted curricula and mathematics content can or do impede students liking, learning of mathematics and claim of ownership i n the development of mathematical knowledge. Yet, to be successful, the constant changes to school curricula call for different kinds of learning approaches and role changes i n students and teachers. While it is generally agreed that mathematics knowledge is needed by all m a n k i n d (Bishop, 1988a; D'Ambrosio, 1994; Gerdes, 1997), educating Cameroonians mathematically consists of m u c h more than just teaching them some mathematics. It requires fundamental awareness of the values w h i c h underlie mathematics and a recognition of the complexity of educating Cameroonian children about those values. It is not enough to merely teach them mathematics; we need also to educate them about mathematics, to educate them through mathematics, and to educate them with mathematics (Bishop, 1988b). Hence we need to move conceptually from the idea of 'teaching mathematics to a l l ' towards 'a mathematical education of all and for all'. Mathematics education i n Cameroon should be aimed at furthering understanding of mathematics as a social and cultural construct of all. To accomplish this, I belief it is necessary to broaden the status and functions of mathematics i n our society to go beyond being seen as a barrier or filter to educational and/or career 81 aspiration. This w i l l require looking for culture elements, that survived colonialism and that reveal mathematical and other scientific thinking. A n ethnomathematics conception to the mathematics education w o u l d assist i n this task. I see an ethnomathematics approach to mathematics education i n Cameroon as responding to the question, " H o w can the mathematics curriculum and teaching methods be adapted to the needs, backgrounds, interests, and abilities of students?" A n ethno- mathematics approach w i l l shift the focus away from students' misunderstanding of formal mathematics concepts by assuming that students possess mathematical knowledge acquired from the daily cultural activities they participate i n . The focus w i l l be on h o w to learn about and b u i l d u p o n that indigenous mathematics k n o w l edge. Such an ethnomathematical approach, connecting mathematics and culture, is precisely at the heart of culturally relevant teaching pedagogies. The general empowerment through ethnomathematical knowledge is, I feel, a very important part of the struggle to overcome a colonized mentality. In this Chapter, I have reviewed the state of formal mathematics education i n Cameroon and made the case for the need of a relevant cultural approach. I have argued that an ethnomathematics foundation to the school mathematics curriculum w i l l assist i n empowering Cameroonians to emerge from the grip of colonialism that has continued to plague development efforts. Success i n such a curriculum w i l l depend on h o w broad based the curriculum development and implementation process is. I have also argued that such curricular changes have to be systematic and the views of all the curriculum stakeholders seriously considered. In the next Chapter, I outline the methodology employed i n assessing the receptiveness to a proposed ethnomathematics curriculum foundation. 82 stakeholders' Chapter 4 RESEARCH METHODOLOGY 4.1 Introduction This chapter begins w i t h a brief statement of overview of the study followed by a presentation of the methodology adopted. This is then followed by a description of the design and the precise methods used i n the collection and analysis of the data. 4.2 Statement of Research Overview Formal education i n Cameroon has continuously been criticized for failing to prepare the k i n d of individuals Cameroon dearly needs at the d a w n of the 21 st century. Critics of the system of education have blamed the school curricula for being inappropriate and unresponsive to the needs of the Cameroonian. Mathematics education has received its o w n share of the blame for alienating Cameroonians from the very values the country has been trying to inculcate. The poor student performance i n school mathematics and the declining level of enrolment of students i n mathematics and mathematics related carriers at the secondary school and institutions of higher education are a major concern to the educational authorities i n Cameroon. While there has been a general consensus that this problem needs urgent attention, there has been little agreement as to what steps to take exactly to begin to address the problem. A sound understanding of the issues plaguing mathematics education i n Cameroon is a necessary first step towards providing some possible solutions. The impetus to this project is framed by the argument that "increasing the relevance of school mathematics to the lives of child- 83 ren involves more than merely p r o v i d i n g 'real-world' contexts for mathematics problems; real-world solutions for these problems must also be considered" (Silver et al., 1995, p. 41). Providing real-world solutions would require developing alternative curriculum conceptions and pedagogic orientations to mathematics education. A n d one of the ways Cameroonians can break away from a cycle of inappropriate education is by identifying individual characteristics, practices and behaviours that are valued i n the society, analysing them, and according to priority, developing a curriculum that w o u l d develop the k i n d of individuals needed to b u i l d a self reliant nation. To do this requires getting all those involved i n the education of the child to participate i n crucial decision m a k i n g sessions regarding the curriculum. The role of the teachers and students needs to be emphasized, and most importantly input from the students should be seriously considered since they are the ones who are going to be affected most. One of the methods that certainly lends itself w e l l to such an approach and to this project is that of ethnomathematics. Before developing an ethnomathematics informed curriculum, it w i l l be necessary to understand what effects ethnomathematics w i l l have on the secondary school curriculum. For reasons of both significance and strategy, the purpose of this study was to make a systematic inquiry into the stakeholders' interests and response to an ethnomathematics curriculum foundation. The precise question that this project is answering is: H o w receptive are the curriculum stakeholders i n Cameroon to an ethnomathematics foundation to the school mathematics curriculum? Knowledge of the stakeholders' interests and response to an ethnomathematics curriculum foundation is crucial if Cameroon is to develop a more appropriate alternative mathematics curriculum for its secondary schools. A cultural foundation of the curriculum is significant since every h u m a n society is often identified by its cultural practices and mathematics is a form of cultural knowledge (Bishop, 1988b). It is therefore important to k n o w h o w a relevant cultural approach to the curriculum can 84 be developed and implemented, what w o u l d need to change and what w o u l d be retained i n the current school mathematics curriculum. Hence, focusing on the stakeholders' interests and response is a deliberate strategy of w o r k i n g w i t h those w h o play a major role i n curriculum development and implementation process i n Cameroon. 4.3 Methodology Used in this Study The thrust of social science is gathering data that can help us answer questions about various aspects of society and thus enable us to understand society (Bailey, 1982). Hence, knowledge produced by social science is a powerful and effective means to influence decisions regarding people's everyday lives. Whether this knowledge is used for the advantage or disadvantage of the group of people under study depends on who controls the research process (Guba & Lincoln, 1989). Conducting research i n a country like Cameroon w i t h an inherited bicultural system of education and a multiplicity of cultural polity requires formulating a research methodology that is not only ethical but politically legitimate - sensitive and responsive to the political, social, and cultural community - and yet can be encompassing so as to allow one to tap into a much deeper understanding of the Cameroonian society and answer questions about various aspects of the (mathematics) society. Most research i n mathematics education until recently have been grounded i n cognitive theories that tend to privilege Western epistemologies and methodologies. Hence, when one decides to conduct research on other ways of mathematizing, one is immediately confronted w i t h the problem of choice regarding theoretical, conceptual, epistemological and methodological frameworks. Either one applies an already established framework on Western thought or one labours to assemble a framework that not only legitimizes non-Western epistemologies but empowers those whose voices and ways of k n o w i n g have been relegated to the 85 background. Before presenting the methodology employed i n this study, I w i l l briefly review some methodologies used i n related studies of ethnomathematics. There is a growing body of literature on studies related to ethnomathematics. Some of these studies have focused on demonstrating the cultural connections of mathematics knowledge (see for example, Ascher, 1981; Bishop, 1988a; Gerdes, 1988a; Restivo et al., 1993). Others have focused on understanding the social connections (see for example, Carraher et al., 1985, 1987; Frankenstein, 1983; Masingila et al., 1996; Presmeg, 1988) of school mathematics. Still, others have focused on understanding the mathematics present i n some given contexts (see for example, Boaler, 1993; D o w l i n g , 1991; Masingila, 1993; Nunes, 1992; Saxe, 1991). It has been suggested (see for example, D ' A m b r o s i o & D ' A m b r o s i o , 1994) that answers to these questions seen as germane to our understanding of the cultural connections of mathematics are better sought through the use of the so-called qualitative research paradigm. Extensive reviews of literature on mathematics and culture (see for example, Lave, 1988; de Abreu, 1998; Gerdes, 1986, 1991, 1993; Lerman, 1998, 2000, 2001; Nunes et al., 1993; Saxe, 1991; Stigler & Baranes, 1988) show that most research i n this area generally follow a more socio-cultural approach using qualitative methods such as ethnography, interview, observation, participatory-inquiry, researcher introspection (Masingila et al., 1996), journal entries (see for example, P o w e l l & Lopez, 1989; Powell & Ramnauth, 1992), and 'talking aloud' i n w h i c h the researched are encouraged to verbalize whatever ideas come to their head (see for example, Jurdak & Shahin, 1999; N e w e l l & Simon, 1972; Ginsburg, Kossan, Schwartz, & Swanson, 1983). O n the other hand, previous studies have tended to use an anthropologicaldidactic approach to understand the evolution of mathematical ideas (Gay & Cole, 1967) and even to uncover hidden moments of mathematics thought i n other cultures or social groups (see for example, Gerdes, 1988b). 86 Still others have maintained the more traditional paradigm of quantitative methods such as various types of questionnaires, translated intelligence tests, to investigate the relationship between a learner's culture and their ability to learn mathematics (see for example, Saxe, 1991). The positivist research paradigm is losing ground i n favor of an interpretative approach (Walker, 1992). This is because positivist research presupposes a structured design i n w h i c h the hypotheses and techniques are determined a priori. Large survey studies and studies that focus on general trends are no longer i n vogue, and the new trust is i n interpretative case studies. In most of these studies, the aim has often been to make sense of what is going o n i n the classroom. In recent years, more and more researchers have advocated the use of a w i d e variety of research techniques, including both quantitative and qualitative methods, for the best attainment of their research aims. M c L e o d (1992) proposes that "research on cognitive issues i n mathematics education should develop a wide variety of methods" (p. 591). H o w e (1988) also points out that we are yet to find any convincing evidence that qualitative and quantitative methodologies are not compatible. Boaler (1999) i n her analysis of the data from a 3-year case study of two schools suggests that our understanding of mathematics learning i n the classroom "cannot be validly portrayed without the long-term focus characterized by ethnographic accounts or the multiplicity of perspectives captured by a range of qualitative and quantitative research methods" (p. 279). Evans (1994) used both a qualitative and quantitative methodology i n his study on adults' mathematics anxiety and he too, suggests that we should make use of the strength of both paradigms as long as they provide the most effective ways to investigate the questions under study. A number of researchers (Closs, 1986; Gerdes, 1993) have recommended qualitative research methods as promising and appropriate for the study of mathematics and culture. Closs (1986) suggested that studies on the history of mathematics must take the "form i n w h i c h an almost total reliance on the historical approach is 87 supplemented or replaced by drawing on the resources and methodologies of other disciplines such as anthropology, archaeology and linguistics" (p. 2). Likewise, Gerdes (1993) suggested that a qualitative approach such as 'cultural conscientialization' which involves the replication of the production techniques of some cultural objects and then studying the mathematics embedded i n them, is another potential method for studies involving mathematics and culture. Gerdes and his colleagues i n Mozambique used this approach to uncover hidden mathematics i n traditional geometric objects like baskets, mats, fishtraps and even houses. Similarly, after extensive reviews of research on mathematics and culture, Boaler (1993) proposes that besides the traditional method of using questionnaires and the studying of cultural artifacts and practices, "additional measures such as open-ended and semi-structured interviews and observation of behaviour must be included if richer and more accurate references must be made" (p. 348). In order to develop a picture of the classroom situation, w h i c h made explicit the patterns, and events which comprised the learning environment, she used a combination of observations, interviews and material analyses. In addition, she also suggests that, current research i n everyday mathematics, mathematics learning i n context and socio-mathematics are some other promising research approaches. To summarize, there is no single encompassing method that purports to be the 'best' i n research involving mathematics and culture. The best combination of methods is that w h i c h can best help us effectively investigate issues at hand. The choice of the methodology is entirely dependent on the nature of the research design, namely the size of the sample, and the aims of the research. The foregoing review suggests that small sample sizes and in-depth studies call for qualitative methods such as interview, questionnaire, observation, document analysis, and researcher introspection. W h e n it comes to analysing the data collected, many studies have used a variety of techniques. Smaller in-depth studies or studies w i t h smaller sample-size 88 have favoured the use of categorization (see for example, Manouchehri & Goodman, 2000) while others have tended to identify facets as a means of understanding the phenomenon (see for example, Matthews, 2003). These categories are sometimes predetermined while others emerge i n the course of the research or pre-analysis of the data. Still, others have used an inductive approach to analyze the data by looking for concepts and processes involved i n the mathematics practice i n given contexts (see for example, Masingila et al., 1996). Still yet, others have looked at the link to affective or cognitive theory and i n some cases philosophical underpinnings to guide their analysis. For example, i n order to understand h o w teachers attempted to incorporate a 3-component model of culturally relevant teaching into their mathematics instructions, Matthews (2003) identified four facets or complexities i n his analysis collected through observations, interviews and group meetings. These are (i) b u i l d i n g empowering relationships, (ii) building on culture and fostering critical thinking formally, (iii) b u i l d i n g on cultural knowledge and fostering critical thinking informally, and (iv) building to cultural knowledge. It is interesting to note that the first facet or complexity focuses on empowerment while the other three focus on culture and knowledge acquisition, thus suggesting an emphasis on learners' cultural background as germane to success i n culturally relevant teaching. Similarly, i n analyzing the data collected from an initial interview conducted to profile the elements that seemed to have greatly influenced the teachers' use of a newly prescribed mathematics textbook, Manouchehri and G o o d m a n (2000) used a thematic approach by categorizing their data using key concepts and occurring themes. These themes then guided their classroom observations and their reconstruction of categories of the teachers' actions during final analysis. These themes or "relationships are often depicted i n diagrams, such as grids or other structured boxes, outline- or tree-shaped taxonomies, ... or anything else the researcher can invent" (Tesch, 1990, p. 82). 89 Other studies that have used categorization i n analysing the data collected have adopted already established philosophical orientations or conceptual categories. This may partly be because most of these studies have been anthropological i n nature, making use of "pre-existing categories schemes to organize and analyze data" (Merriam, 2001, p. 156). Examples are studies conducted on attempting to unveil cultural moments of mathematical thoughts (see for example, Ascher, 1981; Gerdes, 1986, 1988a, 1988b) or mathematical concepts embedded i n cultural practices (Boaler, 1993, Masingila, 1995; Masingila et al. 1996). Most of these studies have i n one way or the other used categories established from six fundamental activities by Bishop (1988a) as mathematical and common to every w o r l d culture. These categories are: counting, locating, measuring, designing, playing and explaining. Discernible from the above studies is the idea that constructing a suitable set of categories for the data might be a good way to analyse the variety of data obtained. However, the categories need to be grounded or made to emerge as one goes through the research and data. This is because cultures vary and w i t h cultural studies such as these involving mathematics, what is expressed verbally may be different from what is actually observed i n a given cultural context. This study was driven by the belief that successful curriculum change requires an understanding of the stakeholders' interest i n reform and with the assumption that stakeholders w o u l d hold to different ideologies regarding education i n general and mathematics education i n particular (curriculum content, purpose and subject areas for curriculum development). From the above review I opted for a qualitative (smaller in-depth research) methodology as the most appropriate approach to inquire into the stakeholders' receptiveness to an ethnomathematics curriculum foundation. The decision to use a qualitative methodology was influenced by the composition of the stakeholders university which faculty. comprised students, This decision was 90 teachers, further pedagogic personnel, and informed by Evans' (1994) pronouncements that "the qualitative case study approach is useful w h e n we w i s h to explore the richness, coherence ... and process of development of a limited number of cases" (p. 326). Hence, considering the limitation of time and available resources, the methodology used i n this study is similar to that employed by Masingila (1993) but sanctioned by the Kantian framework w h i c h employs categorical ethical principles by treating participants as ends in themselves rather than as means (Howe & Moses, 1999). The qualitative methods used i n this study derived from the above review of methods used i n related studies comprised questionnaire, semi-structured interviews, participant observation, and researcher introspection. Based on the thesis problem, the research was conducted i n three steps. Step one was the administration of the questionnaire. Step two involved the development and observing the teaching of an ethnomathematics unit. Step three was a combination of semi-structured interviews and questionnaire. Semi-structured interviews were conducted w i t h 4 secondary mathematics teachers (2 from public secondary school and 2 from private secondary school), 1 pedagogic adviser, and 1 university faculty (teacher educator). The interviews were intended to supplement information collected using the questionnaire. 4.4 The Research Site The study was conducted i n Cameroon, a country roughly the size of California and located i n central Africa. The study involved two schools located i n the N o r t h West province, one of the predominantly English speaking parts of Cameroon (the other being the South West Province), having been colonised by the British after the W o r l d War I. Hence the system of education i n these provinces is patterned after the British. Over the years, government's attempt to bring education and administration closer to the people has led to the creation of local governments at the provincial level such as the Delegation of Education, the Pedagogic Inspectorates for primary and nursery education, and for secondary education. 91 These provincial government offices oversee the smooth functioning of the various schools (both public and private) and educational programmes w i t h i n the province and report to the minister of education. The decision to limit the research to two schools was a pragmatic one. The two schools that participated i n the study were determined only after permission to conduct the research had been granted by the respective educational authorities i n Cameroon. Being able to function i n social situations such as educational contexts requires some level of intimacy w i t h complex social and cultural perspectives as w e l l as a range of personal language resources. Being an insider (a native of Cameroon) provided me w i t h the level of intimacy and cultural perspective required in this k i n d of research. I was also aware of Miller's (1995) cautions about the subjectivity and objectivity of the researcher i n interviews. M i l l e r (1995) warns that, Familiarity w i t h a situation may lead to some loss of objectivity on the part of a researcher who may be unable to stand back from the situation and to see it from a perspective of distance. Too many aspects of a situation may be taken for granted and it may be difficult to ask naive questions about situations w h i c h are already familiar. O n the other hand, Hockey (1993) suggests that an insider may have a rapport w i t h informants w h i c h may be lacking i n a stranger. There may be certain linguistic and cultural familiarity w h i c h enables the researcher to understand the situation and interact more readily, (p. 30) Being a native of Cameroon facilitated entry into the research site and establishing an immediate rapport w i t h the participants. A s a former teacher, familiarity w i t h the educational system and the culture of teaching was an advantage. Taking on the role of semi-insider/researcher helped maximize objectivity and minimize subjectivity. Gans (1982) captures this role as one "who participates i n a social situation but is personally only partially involved, so that he can function as a researcher" (p. 54). I was very aware that to conduct research i n a country like Cameroon where hierarchy is well-regarded, the very success of the 92 whole project depended on respecting this hierarchical structure of the entire system w h e n seeking for permission and even later w h e n conducting the study. After arriving at the research site, I requested a meeting w i t h the provincial delegate for education. The provincial delegate for education represents the minister of education at the provincial level. It is through his/her office that permission is requested to conduct research i n any school in the province. D u r i n g the meeting w i t h the provincial delegate, I presented h i m w i t h a written letter of information describing the purpose of the study and the role the schools play. Written permission was then requested and granted (see A p p e n d i x A for sample request for permission forms). Shortly after that, I met separately with the provincial pedagogic adviser for mathematics and the university mathematics (teacher educator) to seek their respective written consents to participate i n the study. A s w i t h the provincial delegate, consent from the pedagogic adviser and university mathematics (teacher educator) were secured through oral and written solicitations. The study was initially intended to include two university instructors (teacher educators) versed i n the secondary school mathematics curriculum. However, only one teacher educator was available to participate. Hence only one teacher educator participated i n the study. A l t h o u g h only one teacher educator participated i n the study, the information collected from h i m was significant thanks to his years of experience teaching secondary school mathematics, undergraduate mathematics and secondary mathematics teacher education courses. 4.5 The Schools H a v i n g secured written permission to conduct the study and obtained written consent from the pedagogic adviser and teacher educator, I decided w h i c h two secondary schools w o u l d be part of the study. Poor communication due to poor conditions of the roads and telephone network system were major determinants i n deciding on w h i c h schools to include i n the study. The two schools chosen for the 93 study were Cameroon College of Arts, Science and Technology (CCAST) Bambili and C i t y College of Commerce ( C C C ) M a n k o n . What made these two schools interesting for this study was the contrast i n their proportion of formally trained teachers and the variance i n their instructional approaches to school improvement, school organization, and professional cultures. These two schools apart from being interesting for this study, were also chosen because of: (1) their proximity to each other (25km apart). This was a pragmatic decision imposed by the poor road infrastructure i n the country and because the researcher d i d not have a car and had to rely on public transportation to commute between the schools; (2) one was public (CCAST) and the other private (CCC) - the two main providers of formal education; and (3) C C A S T represented a typical public school i n the country w i t h all the teachers being formally trained while C C C represented a typical private school w i t h a majority of the teaching corps not formally trained. I visited the schools and met separately w i t h the principals i n their respective schools. Written consent and permission to conduct the study i n these schools was sought from the principal of each school independently. Once the respective principals had granted permission (see A p p e n d i x A for sample request for permission forms), I scheduled a meeting w i t h all the mathematics teachers of each school w i t h the respective principals facilitating this process. The purpose of the research study and the requirements for voluntarily participating i n the study were outlined i n both written and oral form. In the public school ( C C A S T Bambili) 5 teachers teaching F o r m 2 mathematics to 13 6 and 14 year-olds volunteered to participate i n the study. Of the five teachers, three of 6 Junior secondary school in Cameroon spans 5 years and this is preceded by 7 years of elementary education. Form 2 is the second year of junior secondary school for students aged 13-14 years old. A t the end of the fifth year (Form 5) students following the Anglo-Saxon system of education write the General Certificate of Education (Ordinary Level) examination. Success in at least 4 papers is required for admission to the senior secondary (high school) where after 2 years, students can sit for the Advanced Level of the General Certificate of Education ( G C E A / L ) examination. 94 them taught mathematics to separate streams of Form 2 at the same time (07:0007:50) each day while the other t w o teachers had their sessions around m i d d a y (11:40-12:30 & 12:30-13:20). Since the study needed just two teachers, the two teachers w i t h classes around m i d d a y were selected while the two teachers selected i n the private school ( C C C Mankon), where there were only five mathematics teachers and three of them volunteered, were those with classes i n the morning (07:00-08:40). This purposeful selection allowed the researcher the ability to make observations i n both the private school (07:00-08:40) and public school (11:40-13:20) the same day. Written consent to participate i n the study was then granted b y the four teachers separately (see A p p e n d i x A for sample copy). CCAST Bambili C C A S T Bambili is a government high school located i n Bambili, a small rural town of under 25,000 inhabitants. It is just 20km from Bamenda, the main city and provincial capital of the N o r t h West province. C C A S T is a mixed school w i t h a student population of over 3200 of w h i c h 54% are males and 46% are females. It is a day/boarding school and the student hostels are coed. However, the majority of the students are day students living i n privately owned rented apartments outside the school. This school carries the perception of being one of the best schools i n West Cameroon and this is not difficult to see. This is because (1) it is the oldest public educational institution i n West Cameroon, (2) it graduates relatively large numbers of students each year, (3) many graduates from this school have moved on to occupy positions of responsibility i n the government, (4) its junior secondary section is often used for student teacher practicum, and (5) its proximity to the school of education (Ecole Normale Superieure Bambili) places it at a vantage point of luring the best teacher graduates from the school of education. Over 90% of the teachers i n C C A S T Bambili are formally trained as teachers while less than 10% have a university degree i n a specific subject like mathematics, language arts, social studies, etc. 95 CCC Mankon C C C M a n k o n is a private high school located i n the heart of Bamenda, an urban city of more than 800,000 inhabitants. It is the provincial capital and the m a i n commercial city hosting more than 80% of the banking sector. C C C is a mixed, day school w i t h an average student population of 2800 of w h i c h 49% are males and 51% are females. C C C M a n k o n , like many private secondary schools i n the country is reputed for admitting students w h o have been denied admission into other government secondary schools because of l o w academic achievement or w h o otherwise w o u l d not pursue secondary education for one reason or the other. Hence students i n this school are usually considered as l o w underachievers compared w i t h their counterparts of public educational institutions. Unlike C C A S T Bambili, C C C receives a very limited number of student teachers for practicum purposes. O n l y 30% of teachers i n C C C are formally trained, 36% have a university education and 34% are high school graduates. 4.6 T h e Stakeholders Participation i n this study was voluntary. The participants i n the study comprised 4 secondary mathematics teachers, one university faculty, one pedagogic adviser and 40 students from each of the two schools. Hence a total of 80 students participated i n the research. Parents were not included i n the research because they are not considered as stakeholders within the Cameroon education community and so do not play any role i n curriculum change. This is a mindset inherited from the colonial days when the colonial educators took all major decisions regarding the curriculum. A n d since this study was to investigate those w h o play a role i n effecting curriculum change, the parents were thus excluded. A l l the participants were chosen only after permission to conduct the study had been granted by the provincial delegate for education and the principals of the two schools. Students i n private schools follow the same syllabus and write the same examinations as those 96 i n government or mission schools. There is tremendous pressure on private school teachers to make sure that their students i n examination classes are adequately prepared for national examinations. Most teachers (95%) i n these private schools do not have any pedagogic training. A majority of the teachers are high school graduates ( G C E ' A ' Level holders) while less than half have a university degree. Most of these teachers turn to teaching as a last resort. 4.6.1 The Pedagogic Adviser Cameroon is administratively divided into ten provinces. In each province, there is one pedagogic adviser per school subject or subject area. This personnel, usually appointed by the Minister of national education, is often someone w h o had been a secondary school teacher for a reasonable length of time - one w h o is deemed as having garnered extensive relevant pedagogic skills to advise other teachers, especially novice teachers. These individuals work out of the pedagogic department of the provincial delegation of education. The pedagogic adviser is i n charge of coordinating the teaching of mathematics i n the province i n general. S/he is also responsible for looking into the problems teachers face i n the classroom and providing supervisory support to the teaching of mathematics i n the province. S/he ensures that teachers' pedagogic skills are up-to-date and this is usually done through regular visitations to the schools and by organizing pedagogic workshops and seminars. The pedagogic adviser who participated i n this study was one w h o also happened to be the coordinator of the provincial Teachers' Resource Centre (TRC). Hence he had the dual role of adviser and coordinator of the T R C . H i s duties as the coordinator of the centre, which at times overlap w i t h those of the pedagogic adviser, include coordinating the work of all the other advisers (primary and nursery, secondary, technical and commercial education) w i t h i n the T R C . The T R C as aptly named is charged w i t h producing resources for teachers within the entire province and enabling teachers to come i n and produce resources for themselves. A s 97 the coordinator of the centre, it is his duty to coordinate the production of teachers' resources, organize workshops and seminars and this means going to the field to see what the teachers need, returning to the centre to encourage the other subject area advisers to respond to those needs i n the field. It is his duty to facilitate the movement of advisers to the field, reporting to hierarchy what is needed i n terms of resources, and being able to follow-up to see that those resources are provided to the advisers. 4.6.2 The University Faculty Since gaining independence i n 1960, there have been two secondary school teacher-training institutions i n Cameroon - one for training teachers for technical and commercial secondary education (Ecole Normale Superieure d'Enseignement Technique - ENSET) and the other for training teachers for secondary grammar or general education (Ecole Normale Superieure - ENS). The one responsible for training secondary grammar teachers is being run under the auspice of the University of Yaounde I with campuses i n Yaounde and Bambili. The Bambili campus is solely responsible for training teachers to teach i n Anglo-Saxon schools while that i n Yaounde is principally for Francophone teachers. The university faculty w h o participated i n this study teaches at the Bambili campus. H e too, a former graduate of E N S Bambili had been teaching mathematics i n the secondary school for close to 30 years. N o w as a full-time lecturer, he teaches mathematics education courses for secondary school student-teachers. D u r i n g his tenure as a secondary school mathematics teacher, he participated i n numerous curriculum workshops and seminars, worked w i t h curriculum experts from abroad on the school mathematics curriculum for Anglo-Saxon schools i n Cameroon and also taught part-time at E N S Bambili. 98 4.6.3 The Teachers Teachers are important agents of change. In the context of curriculum change, the teachers' understanding of the connection between mathematics and culture and of teaching i n context can be quite pivotal i n the development and implementation of culturally relevant mathematics curriculum. The 4 teachers w h o participated i n this study all taught ' F o r m 2' mathematics to 13 and 14 year-olds. A s stated above, the teachers were selected based on their daily schedules so that it was possible for the researcher to observe lessons in both schools the same day. The four teachers varied i n their educational background and years of teaching experience. Of the four teachers w h o participated in the study, two were teaching i n C C A S T Bambili, a public high school while two were teaching i n C C C M a n k o n , a private high school. Both teachers i n C C A S T were graduates of the teacher training college (ENS Bambili) while the two teachers i n C C C were holders of a bachelor's degree i n mathematics from neighbouring Nigeria. One of the teachers i n the private school had been teaching for 22 years while the other had been teaching for 10 years. In C C A S T Bambili, one of the teachers had been teaching for 15 years while the other had been i n the teaching corps for 12 years. Both teachers i n C C C were males while one teacher i n C C A S T was female. In C C A S T , 5 out of 16 teachers are females (31.25%) while all the teachers i n C C C (5) were males. 4.6.4 The Students Students in the public school system are selected from the top 30% of those who pass the Cameroon elementary school C o m m o n Entrance examination. Most of the remaining 70% of students enroll i n private and mission schools. A s a consequence, enrolments i n these private schools are sometimes enormous. A typical grade level i n these schools has between 200 and 300 students compared to an average of only 180 per grade level i n most public schools (Johnson, 2000). Because generalizability was not an aim of this study, the decision to include 40 students 99 from each school (total of 80 students) i n the project was a logistic and pragmatic one. I w i l l however argue that this number of students was reasonable and the data collected from this size of the student participants was rich and considerable. For the students w h o participated i n this study, consent was obtained through written solicitation. Students w h o could not grant consent had to get the consent form signed by a parent or guardian (see A p p e n d i x for copy of Letter of Information and Consent forms). A l l the 80 students w h o participated i n the study were i n Form 2 and were day (non-boarding) students l i v i n g i n rented apartments outside the school environment. Each of the Forms (Grades) is divided into several streams. The grouping of students into streams is not based on ability but dictated by the number of students w h o are admitted to that particular Form. In C C A S T Bambili, Forms 1 to 3 each is divided into fives streams labeled A , B, C, D , E and Forms 4 & 5 are each divided into four streams labeled A , B, C, and D . This is because the student population decreases as we move higher, a decrease attributable to poor student performance and student dropout. Hence students w h o participated i n the study i n C C A S T Bambili were i n two of the streams i n Form 2. Each stream i n Form 2 had an average student population of 65 thus totaling more than 320 students i n Form 2. The student questionnaire was administered to 40 students (20 from Form 2 A and 20 from Form 2D) randomly selected from all the students i n Forms 2 A and 2D w h o had consented to participate i n the study. The group of 20 students from F o r m 2 A was designated the ethnomathematics group and was taught the ethnomathematics unit while the group of 20 students from Form 2D was designated the non-ethnomathematics group and was taught the regular mathematics unit. Forms 2 A and 2D were so selected because these streams were being taught by the two mathematics teachers w h o consented to participate i n the study. In C C C M a n k o n , all the Forms were also streamed w i t h Forms 1 to 3 each divided into four streams A , B, C, and D , and Forms 4 and 5 each divided into 3 streams A , B, and C . Like i n C C A S T Bambili, the student population also decreased 100 as we moved higher with Form 1 having the largest number of students and Form 5 the lowest. In Form 2, there were a total of 250 students divided into four streams of 63, 63, 62, and 62. Just like in CCAST Bambili, the student questionnaire was administered to 40 students (20 from Form 2B and 20 from Form 2C) randomly selected from all the students in Forms 2B and 2C who had consented to participate in the study. The group of 20 students from Form 2B was designated the ethnomathematics group and was taught the ethnomathematics unit while the group of 20 students from Form 2C was designated the non-ethnomathematics group and was taught the regular mathematics unit. Forms 2B and 2C were so selected because these streams were being taught by the two mathematics teachers who consented to participate in the study. 4.7 Data Collection Having secured the necessary permission and statements of consent from all the participants, it was now time for data collection. Before the study actually began, a pilot study was conducted on the questionnaires and semi-structured interview. The aim of the pilot study was to clarify any ambiguities that may exist in the wording or sources of misinterpretation of the questions. The comments and suggestions from the pilot group of mathematics students, student-teachers, and teachers were used to revise the questionnaires and the interview protocol which became what is found in the Appendix A. The semi-structured interview protocol was piloted with 2 mathematics student-teachers and 2 secondary mathematics teachers (one from a public and one from a private secondary school that did not participate in the study). This pilot interview was used to evaluate the questions and to refine the interview and analysis techniques. Data collection for this study occurred in three stages and this began after the permission and consent had been secured from all the authorities and participants. 101 4.7.1 Step One: Questionnaires A l l the questionnaires in step one were administered during the first week of the study. In step one, my aim was to establish an information base of the stakeholders on key issues to be further explored in the research. I recognized that this initial questionnaire might not provide in-depth information but at the very least, provide a sense of the stakeholders' response regarding an ethnomathematics curriculum foundation. Kerlinger (1986) maintains that information gleaned from questionnaires of this kind do not go deep enough. The scope of the information is "usually emphasized at the expense of depth" (p. 387). In view of this limitation, I structured the questionnaire with a variety of question types, including open-ended questions, structured questions with 'yes-no', five-point scale, 'ranking in order of importance', and 'choosing as many options described from a large selection'. Because the stakeholders comprised four groups of participants (students, teachers, pedagogic personnel, teacher educator), the questionnaires varied slightly in content and purpose for each stakeholder. For the students, the questionnaire was to establish a knowledge base of the students on their views on mathematics. For the teachers, the purpose of the questionnaire was to determine where the teacher stood regarding an ethnomathematics approach to the secondary school mathematics curriculum. The pedagogic inspector's questionnaire, while almost similar in focus to that of the teachers, had the added focus of role determination in curriculum reform. A n d lastly, the university faculty's questionnaire was also similar to the teachers' with the added focus of a teacher educator and curriculum reform participant. Details on the administration of the individual questionnaires are described below. Student Questionnaire: At the start of the data collection process, the researcher administered a questionnaire to the 40 students in the selected stream of Form 2 in each school. Hence a total of 80 students responded to the questionnaire. Each student who completed the questionnaire was given a randomly generated alpha102 numeric code, which could be used later on for verbatim quotation. This questionnaire was administered in the students' regular classroom during one of the mathematics period. The questionnaire contained 15 items of w h i c h 8 were open-ended questions. Of the remaining 7 questions, 4 were of the Likert-scale type w i t h 5-point scales, 2 were of the rank type, and 1 of the 'Yes-No' type. The students were given one hour to complete the questionnaire. I went through the questionnaire w i t h the students reading out each question and explaining what was required of each (see A p p e n d i x A for a sample). In some cases I had to re-read the question and explain what the question was seeking. Some questions were not responded to d u r i n g this time. This was expected since there were questions which the students could answer effectively only at Step three of the data collection. The researcher collected the questionnaires once the students were done. Teacher Questionnaire: The researcher administered the teacher questionnaire at the start of the data collection process, w h i c h was after the teacher participants had been determined and their written consent granted. This took place at the teachers' respective schools and before the ethnomathematics unit had been prepared with the teachers. The decision to administer this questionnaire at this time was to ensure that the teachers' response to the questions was not influenced by the ideas and notes found i n the ethnomathematics unit. Each teacher was given a randomly generated alphanumeric code, w h i c h could be used later on for verbatim quotation. The teacher questionnaire contained 26 items of w h i c h 14 were openended questions and the remaining 12 questions were multiple-choice of the Likertscale type w i t h 5-point scales (see A p p e n d i x A for a sample). The teachers were given sufficient time to go through the questionnaire. Teachers i n each school responded to the questionnaire the same day. This took place when both teachers had more than one hour of free time i n school. This ensured that the teachers d i d not influence each other's responses to the questionnaire items. The questionnaires were then collected once the teachers were through w i t h them. The questionnaires from 103 the two teachers i n each school were all analysed for this study. Hence a total of 4 teacher questionnaires were analysed. Teacher Educator and Pedagogic Adviser Questionnaire: The questionnaires for the pedagogic adviser and the university mathematics (teacher educator) were very similar to that of the teachers. The pedagogic adviser questionnaire contained 28 questions of which 15 were open-ended questions and 13 were multiple-choice questions of the Likert-scale type with 5-point scales (see A p p e n d i x A for a sample). The university faculty questionnaire contained 27 questions of w h i c h 14 were openended questions and 13 were multiple-choice of the Likert-scale type w i t h 5-point scales (see A p p e n d i x A for a sample). The teacher educator completed the questionnaire during office hours at E N S Bambili while the pedagogic adviser completed his during office hours at the Pedagogic Centre. The researcher administered the questionnaire on separate days because of logistic reasons and also because it was not necessary that both be administered the same day since the two participants didn't work together. Each respondent took about VA hours to complete the questionnaire. The researcher collected the questionnaires once they were done. 4.7.2 Step Two: The Ethnomathematics Unit The second stage of the research involved developing an ethnomathematics unit (see A p p e n d i x B for sample Cameroonian games and the mathematics i n them w h i c h were used i n the unit) for Form 2. This occurred after the questionnaires had been administered and I took about a month to develop the unit plan. The researcher developed this unit w i t h some input from the teachers, pedagogic adviser and university faculty. The unit was intended to last 4 weeks but lasted 6 weeks. This was because classes were disrupted for more than a week during w h i c h time schools 104 participated i n Youth Week activities. T w o teachers (one i n each school) taught the 7 ethnomathematics unit to one stream of Form 2 (2A i n C C A S T and 2B i n C C C ) . The other two teachers (one in each school) taught the regular mathematics unit (covering the same concepts as the ethnomathematics unit) to one stream of Form 2 (2B i n C C A S T and 2C i n C C C ) . In developing the ethnomathematics unit I made use of local resources by including cultural games (game of pebbles, game of sevens, game of whodunit, and game of the diamonds) and the mathematics embedded i n them (see A p p e n d i x B). Criteria for selecting illustrations of mathematical concepts to be incorporated and analyzed included that the illustrations: (a) contain some identifiable mathematical activities according to Bishop's (1988) six fundamental activities: "counting, locating, measuring, designing, playing and explaining" (pp. 182-183), (b) be rich i n cultural knowledge such as indigenous mathematics knowledge, (c) contain mathematical objects such as geometric shapes, mathematical implements, and (d) mathematical objects of inquiry. Since I was looking for the reaction to the notion of ethnomathematics, the aim of the unit was to raise the awareness i n the students. The unit was not assessing the teaching of ethnomathematics. The unit was appropriate because I knew about the activities and could identify the mathematical processes embedded i n them. The unit was also appropriate because I played the activities as a youth. The activities listed i n appendix B are intended for illustrative purposes only. These activities were part of the ethnomathematics unit that was developed to cover the concepts the students were supposed to learn during that period. The teachers were given ample flexibility on h o w to incorporate the material into their teaching but making sure 7 Youth Week is a public holiday in Cameroon. Festivities for this holiday range from presentations to intramural activities such as singing, dancing and playing competitions to educational programmes, and usually last a week culminating with the Youth Day (February 11) celebration involving all educational institutions in the country. 105 that the mathematics embedded i n the activities was explored and discussed w i t h the students. The teachers sometimes used group investigations to get the students develop an explanation for how some of the games were played. Students were then asked to present their findings from w h i c h the lesson proceeded. I realized that probing the teachers' interest and response to an ethnomathematics curriculum foundation w o u l d be incomplete without an observation of how the teachers were incorporating relevant cultural aspects into their mathematics instructions. This is because most of what a teacher knows or believes may be tacit, not easily accessible to the researcher and may be revealed only i n the teacher's actions (Pajares, 1992; Ross, Cornett & McCutcheon, 1992). A s M e r r i a m contends, "observations are also conducted to triangulate emerging findings; that is, they are used i n conjunction w i t h interviewing and document analysis to substantiate the findings" (Merriam, 2001, p. 96). Hence an observation of how this is carried out i n the classroom and the interest and response from the students was therefore necessary. Observations were guided by broad questions that looked at: > H o w teachers b u i l d on students' cultural knowledge and informal experiences i n teaching mathematics, > H o w teachers fostered critical mathematical thinking and used a historical approach i n teaching mathematics, > How teachers incorporate empowerment orientations toward culture and experience in the teaching of mathematics, and > H o w often teachers used a historical approach or a historical dimension in the teaching of mathematics. The teachers were observed three times each week until the ethnomathe- matics unit had been completed w h i c h lasted 6 weeks. A total of 12 observations w i t h each teacher were recorded. This is because there were no observations during the 5 and 6 week as classes were interrupted as students rehearsed for the national th th 106 Youth Week festivities. Each mathematics lesson lasted 50 minutes. These observations were tape-recorded using a m i n i pocket digital recorder and a laptop computer. In addition to the observations, I also had numerous informal discussions before and after class w i t h the teachers, and these discussions provided information that I followed up i n the semi-structured interview at the end of the ethnomathematics unit. I carried my m i n i pocket digital recorder w i t h me at all times and on several occasions when informal discussions were yielding particular useful insights, I asked whether the conversations could be recorded. Permission was always given readily w h e n sought. Data were collected inside and outside classrooms d u r i n g the teaching of the ethnomathematics unit. D u r i n g the observations, I regularly moved around the classroom, observing students' work more closely and occasionally talked to students and the teacher. I had no involvement i n the actual teaching of the unit beside developing the unit w i t h the teachers and preparing some teaching aids. Questionnaires, interviews and observations were m y primary data sources while various mathematical, curriculum and archival documents provided additional useful data. Data on the students' performances was also collected before and after the ethnomathematics unit. 4.7.3 Step Three: Questionnaires and Interviews This last stage of the research, which lasted two weeks, occurred shortly after the ethnomathematics unit had been taught. A questionnaire was administered to the students while semi-structured interviews were conducted with the rest of the participants. Students: The questionnaire administered to the students at the beginning was administered again to the same students i n an attempt to understand what effects, if any, the ethnomathematics unit might have had on the students learning of mathematics. It was hoped that students w o u l d be able to respond to all the questions this time after having attended 107 the ethnomathematics lessons. The questionnaire was administered exactly as the first time by the researcher and the completed questionnaires were collected once the students were done. Teachers, Pedagogic Adviser and Teacher Educator. I recognized that eliciting the stakeholders' interest and response is not an easy task. Some aspects of the stakeholders' response to the ethnomathematics curriculum foundation could not be readily represented by overt propositions made on the questionnaire by each stakeholder. These aspects included experiences, values, emotions, and concerns, all of w h i c h were both personal and professional (Carter, 1993; Johnston, 1990). Analysis of the data collected i n step one revealed some general information about where each participant stood regarding an ethnomathematics curriculum foundation. To further understand the stakeholders' interests i n this proposed curriculum foundation, a semi-structured interview was conducted w i t h each teacher, pedagogic adviser, and university faculty after the completion of the ethnomathematics unit. The semi-structured interview format was chosen because it allowed the interviewer (researcher) to lead the participants to focused and systematic inquiry on proposed topics while allowing the participants some freedom and flexibility i n expressing their views and feelings i n issues they feel are important to mathematics education in Cameroon. The interview questions were designed to collect data on the stakeholders (excluding the students this time) i n six key areas explored i n the questionnaire: (1) nature of mathematics, (2) uses of mathematics, (3) mathematics and culture, (4) curriculum reform, (5) mathematics curriculum, and (6) importance of formal education. This was to ensure triangulation i n the data as questions posed in the questionnaire were reposed or rephrased i n the course of the interviews and the respondents given another opportunity to respond extensively. The teacher interviews took place at their respective schools and interviews in each school were conducted the same day w i t h each lasting between one and two hours. Each teacher interview usually started off w i t h me seeking the teacher's impressions on the just completed ethnomathematics unit. Then I posed questions on the above six areas 108 and used probing to elicit their interests and response on the possibility of an ethnomathematics curriculum. Other questions posed included: (a) What they valued most about an ethnomathematics curriculum foundation, (b) What they valued least, and (c) What concerns they had for such a curriculum foundation to be adopted i n Cameroon. Probing was used to elicit additional information regarding each participant's interests and response to an ethnomathematics curriculum foundation. In the interviews, other teacher characteristics such as years of teaching experience, type of certification, the number of university mathematics courses completed, membership i n professional organizations and attendance at mathematics conferences, seminars and workshops were also collected. The interviews w i t h the pedagogic adviser and university faculty took place during office hours at their respective w o r k i n g locations. Similar, to the teachers, the interviews w i t h the pedagogic adviser and university faculty explored the above named six key areas. Each interview session was audiotaped and later analysed. The interviews were transcribed by the researcher for analysis. A l t h o u g h interviews and questionnaire were the heart of the data collection, documents and archival records were collected to provide an alternative perspective on the research question. Marshall and Rossman (1989) argue that the unobtrusive nature of document and archival record collection provides a rich data source without disrupting the research site. The documents collected and examined for this study were the mathematics text and scheme of work for Form 2. 4.8 Data Analysis This study was driven by the idea that successful curriculum change requires a deep understanding of the stakeholders' interest i n systemic reform and w i t h the assumption that stakeholders w o u l d have different ideologies regarding education in general and mathematics education i n particular (curriculum content, purpose and subject areas for curriculum development). A l l the research data was analysed 109 through a process of inductive data analysis by allowing for common themes, patterns, frequently repeated expressions to emerge. Padilla's (1991) concept modelling methodology was used as a strategy for organizing and displaying the data. According to Padilla, one w a y to explain a situation is to identify various assumptions contained i n the data and organize them into a coherent whole. In the concept modelling method, assertions contained i n the data become fundamental elements for analysis. First, I created a matrix (see Table 6.1) i n w h i c h to display responses to the six main areas explored i n the interviews and questionnaires. Next, I reduced long statements from interview transcripts, questionnaires and excerpts from documents to short paraphrases, and entered these data into appropriate cells of the matrix. I then observed how the data arrayed across the various stakeholders, and then highlighted areas of convergence and divergence among six main areas. The areas of convergence and divergence were used to assess the stakeholders' receptiveness to an ethnomathematics curriculum foundation. This receptiveness was displayed in terms of the stakeholders' levels of interests and concerns i n an ethnomathematics curriculum. 4.9 Validity and Reliability of Data Collection and Analysis Since the data collection i n this study was qualitative, Lincoln and Guba (1985) suggest using the term 'trustworthiness' as a more suitable and equivalent term of validity and reliability for the quantitative data. According to them, "'trustworthiness' includes four main criteria: 'truth value', applicability, consistency and neutrality. In other words, it is imperative to establish the 'creditability', 'transferability', 'dependability' and 'confirmability' of the study, w h i c h is 'the naturalist's equivalent to the conventional terms "internal validity", "external validity", "reliability" and "objectivity" (Lincoln & Guba, 1985, p. 300). Thus, I attempted to establish the trustworthiness of m y data by adopting some of the techniques suggested by Lincoln and Guba. 110 Firstly, by using the interview to pose similar questions featured i n the questionnaire, but this time rephrased, I was able to clarify w i t h the respondent as w e l l as validate the data collected i n the questionnaire. This cross-validation indirectly served to enhance the trustworthiness of the data collected by questionnaire. Secondly, since the majority of the data was collected outside the classroom setting, it is possible that some of what was espoused i n the questionnaires and interview differed w i t h what I observed during the teaching of the ethnomathematics unit. In such instances, I replayed segments of the tape-recording done i n the classroom and then used follow-up to gain clarity of the respondent's (teacher) responses. This provided some level of trustworthiness i n the data collected and also i n the analysis that followed. Thirdly, I was fully aware that as the stakeholders communicated their responses to the ethnomathematics unit to me, I then had to interpret them i n terms of statements indicating receptiveness to an ethnomathematics curriculum foundation. I acknowledge the possibility of using researcher introspection to interpret some responses. To ensure that this interpretation d i d not influence the conclusions d r a w n from the data, I grounded my interpretations i n the data by developing two subcategories to classify the stakeholders' statements of response to the ethnomathematics curriculum foundation i n terms of: > level of interests towards an ethnomathematics curriculum, and > level of concern towards an ethnomathematics curriculum. In addition, I tried to interpret the responses holistically and always looked for confirmation of meaning by taking into account their responses to similar questions on the questionnaire and the interview. This provided another level of triangulation i n the data. Ill 4.10 Conclusion To conclude, even though I have justified m y choice of the methods used i n this study, I acknowledged that there may still be problems and limitations that I have to bear i n m i n d when I analyze m y data and report the findings i n the next chapters. 112 Chapter 5 PRESENTATION OF FINDINGS This chapter presents the data collected on the participants' receptiveness (interest and response) to an ethnomathematical foundation to the secondary school mathematics curriculum. 5.1 Introduction The purpose of this project was to examine the stakeholders' (students, teachers, pedagogic advisers, and university faculty) interests and response towards a proposed ethnomathematical foundation to the school mathematics curriculum i n Cameroon. The m a i n research question as stated i n Chapter one is: How receptive are the curriculum stakeholders in Cameroon to a proposed ethnomathematics curriculum foundation? T o measure and understand this receptiveness, various methods of data collection were employed. These were: participant observation, interviewing, and questionnaires. A sample mathematics unit for secondary schools (Form 2) i n Cameroon was developed incorporating mathematics concepts present i n everyday Cameroonian cultural practices. This unit was then tried out w i t h secondary mathematics teachers for their suitability and efficacy i n arousing the students' interest towards mathematics and their ultimate successful acquisition of mathematics knowledge. The study was conducted over one full school term (January - A p r i l 2003). 5.2 Findings This study was driven by the idea that successful curriculum change requires a deep understanding of the stakeholders' interest i n systemic reform and w i t h the 113 assumption that stakeholders w o u l d hold varying ideologies regarding education i n general and mathematics education i n particular (curriculum content, purpose and subject areas for curriculum development). Because a reasonable proportion of schools are privately owned (more than 45% as oppose to 55% owned by government), it was decided that the research w o u l d be conducted i n both government and private school. A s is often the case w i t h qualitative research designs, it is virtually impossible to present all the raw data collected from the questionnaires interview sessions (Miller, Malone, & K a n d l , 1992). and Responses to the research questionnaires and interview questions are reported using both particular and general descriptions w i t h i n the framework of interpretive commentary (Erickson, 1986). Rather than present the findings as verbatim responses to each research question or sub-question, the research questions have been grouped into six subsections: I. The nature of mathematics II. Mathematics and culture III. School mathematics curriculum IV. C u r r i c u l u m reform process V. Importance of formal education VI. Possibilities of an ethnomathematics curriculum Each subsection w i l l present responses to questions derived from questionnaires and interviews administered to the group, as w e l l as from teacher observation fieldnotes. 5.2.1 The Nature of Mathematics U s i n g questionnaires and interviews, the stakeholders were asked questions to elicit their understanding and views regarding the nature and philosophy of mathematics. The questions posed were: What is your definition of mathematics?, 114 What comes to m i n d w h e n y o u hear the w o r d mathematics?, H o w w o u l d y o u design the cover of a mathematics textbook?, Is mathematics pre-given or logically derived from axioms? LI The whole group The overall response from the stakeholders i n this study shows that a majority of the stakeholders responded that mathematics is a science geared towards the development of logical and critical thinking and more than four fifths of them said it involved numbers. O n l y one quarter of the respondents stated that it was a scientific approach to better the living conditions of mankind. Their overall response was based on the argument that though mathematics, especially school mathematics involve a lot of abstract concepts w h i c h may not seem directly applicable to the basic needs of mankind, the overall purpose of mathematics knowledge was important i n man'sfsic] understanding and mastery of his environment. 1.2 The students There were two questions that explored students' images, views, feelings and notions of mathematics w h i c h to some extent suggest their developing understanding of the nature and philosophy of mathematics. These questions were: What comes to m i n d w h e n y o u hear the w o r d mathematics? H o w w o u l d y o u design the cover of a mathematics textbook?. In addition, the following questions were also administered to further solicit the students' notions of mathematics: A r e history and mathematics related i n any way?; D o students k n o w mathematics w h i c h they were not taught?. (a) What comes to mind when you hear the word mathematics? There were very few differences i n the students' developing understanding of the nature of mathematics between students of the public (government) secondary schools and those of the private secondary schools. There was also very little notice- 115 able difference between students w h o were taught the ethnomathematics unit and those i n the regular mathematics class w h e n the questionnaire was first administered. Most of the students i n the four groups characterized mathematics as that involving 'numbers and calculations'. However, w h e n the questionnaire was administered a second time, there was a noticeable difference i n the students' attitudes and images of mathematics. Students w h o took the ethnomathematics unit i n both schools showed an improved positive attitude toward mathematics and a broader image of mathematics while students w h o continued w i t h the regular mathematics unit showed no difference i n both attitude and image of mathematics. Since the student questionnaire was administered twice, one prior to the ethnomathematics unit and another at the end of the unit, the findings are reported here w i t h comments grouped into before (before the ethnomathematics unit) and after (after the ethnomathematics unit). CCAST Bambili: In the students' response to the question: what comes to m i n d w h e n y o u hear the w o r d mathematics?, the findings reveal that most students i n the ethnomathematics and non-ethnomathematics group who considered themselves as successful i n mathematics (doing well i n mathematics such as attaining satisfactory performance i n mathematics tasks) stated that the w o r d mathematics conjures such processes as 'calculating, solving, counting, comparing, and sharing,' while students w h o had been less successful expressed feelings of 'hatred, boredom, fear, and getting the lowest score i n a test.' Of all the students w h o responded to this question less than a third from both the ethnomathematics and non-ethnomathematics group stated that they hated mathematics and were always uneasy during mathematics lessons. There was no observed difference i n the students responses w h e n the questionnaire before and after the ethnomathematics unit i n both groups of students as the following excepts indicate: 116 Before: "In short, as for me, when I hear the w o r d 'mathematics' what immediately comes into m y head is the most important word: calculation. It makes me feel good." (Student, C C A S T ) " W h e n I hear the w o r d mathematics what comes i n m y m i n d is I feel like wasting m y time, the way mathematics is very difficult, the way I always fail i n maths, the way I hate it. I w i l l think of the w a y I don't understand it." (Student, C C A S T ) "Calculation, that boring subject, the way mathematics is very difficult to solve. I feel very bad because math disturbs me m u c h i n school and I always fail i n math. The w a y the teacher is teaching is boring too m u c h than the w a y he or she is unhappy." (Student, C C A S T ) After: "Calculations, addition, subtraction, multiplication, division. I feel bad when I hear the w o r d mathematics because I never pass i n math." (Student, C C A S T ) "I immediately think of calculation and 'x'. I remember h o w I don't do well i n it. I remember the difficulties of its problems. I think of drawing parms, rectangles, squares, etc." (Student, C C A S T ) " W h e n I hear the w o r d mathematics the w o r d that comes to m y m i n d is that the person w h o says it is calculating." (Student, C C A S T ) C C C Mankon: Among the two group of students (ethnomathematics and non- ethnomathematics) i n C C C M a n k o n , most respondents echoed similar verbs as i n C C A S T Bambili b y stating that mathematics conjured such actions as 'calculate, solve, count, compare, share' and the four basic operations (+, - , 117 x ) . About a quarter of the respondents expressed uneasy feelings w h e n they hear the w o r d mathematics although all of them said they w o u l d be happy to learn and excel i n it. Most female respondents characterised what immediately comes to their m i n d i n affective terms using such words as uneasy, unhappy, discourage, difficult time, and bored. Most male respondents characterised their state i n terms of mathematical processes such as wanting to calculate and to solve. Most female respondents including those w h o said they were not good at mathematics also expressed the desire to become a mathematics teacher or a mathematician i n the future if only they could do very well i n it (mathematics). It is worth noting that a female teacher taught this second group of students while a male taught the first group. This is important, as it seems to suggest a correlation between female students and a female mathematics teacher i n h o w students think of mathematics. Before: "Those things that come into m y m i n d are calculation, solving and most subjects like physics, chemistry, geography because these subjects use maths also." (Student, C C C ) "Calculating, solving, comparing different things, measuring, drawing." (Student, C C C ) "Since I'm not a maths student, w h e n I hear the w o r d mathematics, I become angry especially when I'm unable to solve the problem." (Student, C C C ) After: " W h e n I hear the w o r d mathematics, what immediately comes to m y m i n d is solving and coming out w i t h solutions." (Student, C C C ) "When I hear the w o r d mathematics I don't longer think of it as complex subject even though the w o r d mathematics sounds like a big 118 w o r d . I feel that if y o u concentrate, it is easy as it is for me now." (Student, C C C ) " W h e n I hear the w o r d mathematics, I think of calculating and using the computer. I then begin to feel b a d because I like maths but m y problem is I don't understand it well." (Student, C C C ) (b) How would you design the cover of a mathematics textbook? There was no noticeable difference in the description of the textbook cover design between students i n the ethnomathematics group and those i n the non-ethnomathematics group i n both schools w h e n the questionnaire was first administered. W h e n the questionnaire was administered again after the ethnomathematics unit, there was more illustrating and less describing by students of the ethnomathematics group than those in the non-ethnomathematics group i n both schools. What remains unclear is whether this difference i n response could be attributed to the ethnomathematics unit since there were still some similarities between what students i n the ethnomathematics group illustrated and what those i n the nonethnomathematics group described. CCAST Bambili: A s stated above, there was no noticeable difference between the two groups (ethnomathematics and non-ethnomathematics) describing h o w they (the students) w o u l d design the cover a mathematics text w h e n the questionnaire was first administered. W h e n asked about h o w they w o u l d design the cover of a mathematics text, about half the students either drew geometric shapes both i n 2 and 3 dimensions or listed arithmetic operations (+, - , +, x). Color seemed to be an important requirement i n the design of a mathematics text as about a third of the students listed colors they w o u l d use to beautify the cover and their explanation was 119 that if the text were attractively designed, then it might not scare away or intimidate less able students i n mathematics. Before: "I w i l l put mathematics signs on it and I w i l l write the name of the book. I w i l l write the name of the author." (Student, C C A S T ) "I w i l l use a colour pencil, a pencil and draw a flower o n it. I w i l l write the author's name, the title of the textbook and I w i l l colour it to look attractive so that many people can like to read it and learn mathematics." (Female Student, C C A S T ) "... I w i l l put myself i n it when I am learning, I w i l l put what is inside and I w i l l also put mathematical signs on it like x, =, +, - , \ , X, °°> I- I w i l l also put graphs." (Student, C C A S T ) W h e n the questionnaire was administered again, there was a slight difference between the two groups (ethnomathematics and non-ethnomathematics) with students i n the ethnomathematics group highlighting the importance of using w a r m and friendly colors, inviting (less intimidating) titles and captions, and to some extent, a more regionalized or localized title. The increased emphasis i n color was especially prevalent among female students while the male students focused more on the general contents of the cover. The following quotes are illustrative: After: "Mathematics for African Schools for Improving the Level of Mathematics" (Student, C C A S T ) "Cover should contain drawings of people conducting an experiment i n mathematics like trying to solve a mathematics problem." (Student, CCAST) 120 "I w i l l use green and purple colors because I think the colors are very friendly and inviting. I w i l l use a bold marker to highlight the title of the book. I w i l l call it Passport to Mathematics 2." (Female Student, CCAST) "I w i l l draw a rectangle, a square, a triangle and paint it red yellow and green and a person calculating mathematics." (Female Student, CCAST) "I w i l l put some problems that I have solved to encourage people to do this good subject called mathematics that we use i n other subjects like physics, geography." (Student, C C A S T ) C C C Mankon: Generally, there was a noticeable shift from describing h o w the cover w o u l d look like to making illustrations of it. A majority of the respondents suggested including mathematics symbols such as geometric shapes and operation signs. M o r e female respondents emphasized the particular color they w o u l d use suggesting the importance of color for attracting learners to mathematics. Some of the descriptions were suggestive of the students' sentiments about school mathematics i n Cameroon and the desire to make school mathematics easy to learn or at the very least, less intimidating as the following statements from three students indicate: Before: "It w o u l d look like it is showing a teacher i n front of the board w i t h children sitting and trying to solve a mathematical problem or children that have found a solution to a problem and are glad or shouting and jumping i n class. The title w i l l be Mathematics Made Easy for Cameroon School." (Student, C C C ) "I w i l l design it w i t h an attractive colour like yellow and put some maths equations and some people solving maths. I w i l l like to encour- 121 age the student to do mathematics. A n d as the producer, I w i l l put m y name and address for people to contact me to k n o w more about mathematics." (Student, C C C ) "I w i l l put some things that w i l l make it look as if there are questions on it and put some signs of subtraction, multiplication, addition, and so on so that when somebody sees it without opening it they will know that it is a mathematics textbook." (Student, C C C ) After: "I w i l l first start by saying African Exercise Book and I w i l l draw a map of Africa to show that we do mathematics i n Africa. After that I w i l l write the name, class, subject, date and the colour w i l l be green and I w i l l draw square lines on it to show that it is a mathematics book." (Student, C C C ) "I put students studying mathematics i n class and the teacher standing i n front of the class. I w i l l also draw people buying and selling things at a market and calculating h o w m u c h money they have to give as change and how much money they have sold for the day." (Student, C C C ) "I w i l l design the cover by first writing at the top of the cover, Mathematics. Next, I will indicate by drawing some pictures directing how it is being taught in class and sometimes advertising a bit of its importance to the society." (Student, C C C ) (c) Are history and mathematics CCAST related in any way? Bambili: A s k e d whether history and mathematics were related i n any way, there was a noticeable shift i n thinking between the first and second time the questionnaire was administered. D u r i n g the first time, a third of the respondents i n both the ethnomathematics and non-ethnomathematics group stated that there was no relation 122 between history and mathematics because mathematics deals w i t h numbers and calculations while history deals with past events. Before: "History and mathematics are related because i n Roman numerals, we learn about the history of the Romans and h o w they developed their numerals." (Student, C C A S T ) "History and mathematics are related i n the sense that there are some important events i n history w h i c h are being remembered due to dates, and dates are made up of figures, and figures are gotten from mathematics." (Student, C C A S T ) "History and mathematics are not related because i n mathematics we deal w i t h addition, subtraction, multiplication, division etc. but i n history we deal w i t h stories of the past and biographies." (Student, CCAST) D u r i n g the second time the questionnaire was administered, only 2 i n 20 i n the ethnomathematics group and 7 i n 20 i n the non-ethnomathematics group still said history had nothing to do with mathematics. A l l those w h o had initially responded that history and mathematics were related maintained their opinion and further substantiated w h y by stating some of the following: After: "... history and mathematics are related because i n history y o u have to calculate years of people and they're people w h o invented mathematical signs i n history." (Student, C C A S T ) "There is a relationship between history and mathematics because history w h i c h talks about the past told us about the discovery of Arabic and R o m a n numerals w h i c h is taught i n mathematics." (Student, C C A S T ) 123 Those w h o maintained that there was no relationship between history and mathematics provided the following logical arguments for their stance: "History and mathematics are not related because history talks about what people d i d i n the past while mathematics deals w i t h the calculations i n our daily lives." (Student, C C A S T ) "Mathematics and history are not related i n any way. History deals w i t h the past and it does not need any mathematics. Mathematics has to do w i t h a lot of calculation and maths characteristics are quiet different from that of history. So history requires just a retentive memory, while mathematics requires a lot of thinking." (Student, CCAST) For all those w h o felt that history and mathematics were related saw the relationship i n terms of content and process. Content wise, the students cited the presence of Roman and H i n d u - A r a b i c numerals i n both subjects; process wise, the cited relationship was the calculation of time between two events as students saw it as the mathematical process of difference or subtraction. C C C Mankon: In the initial administration of the questionnaire, three quarters of the students (almost evenly distributed between the ethnomathematics and non- ethnomathematics groups) believed that history and mathematics had nothing i n common. The reasons they supplied for this line of thinking are found i n the following statements: Before: "History and mathematics are not related i n any w a y because in history we are talking about stories and past events while i n mathematics we are talking about addition, multiplication, subtraction, . division and we are calculating." (Student, C C C ) 124 " N o , history is an art subject and mathematics is a science subject and in mathematics there are some calculations. In history, there is no solving. In mathematics we study what is around us daily while i n history we study what has past some years ago." (Student, C C C ) " N o because there is no solving of equations i n history. History deals w i t h dates and past events while mathematics deals w i t h figures and calculations." (Student, C C C ) "They are not related i n any way because history deals w i t h the past and the present events while mathematics deals w i t h calculations. A n d also, they are not related because i n any exams y o u cannot write history but w i t h maths we must use mathematics i n every exams." (Student, C C C ) The second time the questionnaire was administered, more than two thirds of the students (with majority from the ethnomathematics group) n o w believed history and mathematics were definitely related. This major shift i n thinking may signal the students' realization of the presence of mathematics i n history and the presence of history i n mathematics, as can be seen by the following except: After: "Yes, because if someone says the 1 W o r l d W a r began i n 1996 and st ended i n 2003, if y o u want to get for h o w long it lasted y o u have to calculate, w h i c h w i l l then be 2003 - 1996 = 7. Therefore the 1 W o r l d st War w i l l have lasted for 7 years and this is mathematics." (Student, CCC) "Yes. This is because i n history numbers are also used for example, reasons w h y the British annexed Cameroon. W h e n y o u start by giving the reasons y o u w i l l number them and by doing so y o u are doing mathematics." (Student, C C C ) "Mathematics is related w i t h history i n the sense that there are some important dates i n history w h i c h are represented by numbers, and 125 numbers are gotten from mathematics. We are also taught of some important things i n mathematics which were discovered years ago and that is history." (Student, C C C ) "Mathematics and history are related because i n mathematics we learn things w h i c h were discovered so many years ago." (Student, C C C ) (d) Do students know mathematics which they were not explicitly taught? W h e n the questionnaire was first administered, a majority of students i n the ethnomathematics and non-ethnomathematics groups from the public school and about half of both groups i n the private school said this was not possible. Those i n the public school d i d not change their opinion by much w h e n the questionnaire was administered again whereas those i n the private school showed a slight increase i n both groups, of those w h o felt that it was possible to k n o w mathematics without being taught explicitly. CCAST Bambili: A s k e d if it was possible to k n o w some mathematics that one had not been taught, most respondents i n both the ethnomathematics and non-ethnomathematics groups said this was not possible unless one had been taught. O n l y 8 students said this was possible if one had the mathematics textbook and studied on their o w n . To those students w h o thought it was not possible to k n o w something unless one had been taught, their reasoning went as follows: Before: "... y o u cannot k n o w something unless y o u have been taught." (Student, C C A S T ) "... one is not born k n o w i n g math." (Student, C C A S T ) 126 "... a student cannot k n o w mathematics w h i c h he or she has not been taught because y o u cannot go to fetch water without a bucket just as y o u cannot eat 'achu' without soup." (Student, C C A S T ) Students w h o believed that it was possible to k n o w mathematics that one had not been taught based their response partly on the fact that mathematics is present i n our daily lives. These students had a more broad view of mathematics and counting was one of the activities i n mathematics: "Yes, for example, to count money is mathematics, there are people who can count money w h e n they were not taught like me." (Student, CCAST) "Yes ... because anything y o u do mathematics must be involved." (Student, C C A S T ) W h e n the questionnaire was administered a second time, half of the respondents i n the non-ethnomathematics group felt that it was possible to k n o w mathematics w h i c h one has not been taught by reading a mathematics text, by using mere common sense or simply by the fact that everyone could count money a process according to them not taught. The other half of the respondents felt that it was not possible to k n o w mathematics w h e n one had not been taught. Some stated that the very fact that mathematics is being taught i n school is because one cannot k n o w it until they are taught. Other statements supporting the ' N O ' response were: After: " N o , because y o u cannot just sit like that and k n o w something w h e n you have not been taught or w h e n y o u have not heard from somebody or somewhere." (Student, C C A S T ) " N o , because mathematics first of all is difficult to understand and if you are not taught y o u can never k n o w something about it. So the 127 only thing is that they have to teach us mathematics before we can understand." (Student, C C A S T ) C C C Mankon: When the questionnaire was first administered, about half (11 out of 20) of the students i n the ethnomathematics group said it was possible to k n o w mathematics that one had not been explicitly taught. Of this proportion of students, a majority saw this possibility i n terms of the ability of some students to self-teach themselves wherein a student could study mathematics from a mathematics text and come to k n o w some mathematics without being taught by a teacher or someone else. A m o n g the non-ethnomathematics group, 8 out of 20 respondents said it was possible to k n o w mathematics that one had not been explicitly taught. Of these 8 respondents, 3 referred to intuition as the example of k n o w i n g mathematics without being explicitly taught. The following excerpts represent the arguments made by the respondents from both the ethnomathematics and non-ethnomathematics groups: Before: " N o , because the students are not G o d w h o are born k n o w i n g everything. Yes it is true some people are naturally talented but no one knows without being taught." (Student, C C C ) " N o , this is because y o u cannot k n o w what y o u have not been taught. Mathematics is not a simple subject because we do not guess answers." (Student, C C C ) "Yes, I k n o w mathematics that I have not been taught i n class because when I go home, I learn from the textbook what I have not been taught in class." (Student, C C C ) "Yes, this is because if y o u are having an elder that has just passed that class, the person may teach you. Y o u may also read from a text128 book ahead of the teacher so that w h e n the teacher is teaching y o u should understand h i m or her better." (Student, C C C ) "Yes, because mathematics is an interesting subjects that needs a lot of reasoning. So one can learn it without being taught by another person simply by reasoning." (Student, C C C ) After the completion of the unit containing cultural aspects of mathematics, responses to this same question showed a marked increase i n the proportion of those who thought it was possible for students to k n o w mathematics w h i c h they were not explicitly taught. This increase was noticed i n both groups although the proportion of students i n the ethnomathematics group was higher than i n the non- ethnomathematics group. The following statements are verbatim responses of the reasons the students provided as to w h y it was possible to k n o w mathematics w h i c h they had not been explicitly taught: After: "It is no because all mathematics that I k n o w I have been taught by m y teachers i n class and if I don't k n o w or understand I always go and see m y elders to show me better." (Student, C C C ) "Yes, because if y o u are not taught, y o u can still k n o w it because if y o u are selling and just calculate it shows that it is mathematics not only calculating money but many other things." (Student, C C C ) "Yes, m y illiterate grandparents knew h o w to calculate money very w e l l even though they were not literate. I say this because mathematics is calculation. A n d so was I w h e n I was about 5 i n nursery school." (Student, C C C ) "Yes, because mathematics is being carried out i n everyday life for instance if y o u are sent to buy bread i n the morning and y o u are given 500 Frs. that y o u should b u y it for 100 Frs., y o u immediately start calculating the change." (Student, C C C ) 129 "Yes, i n the sense that as the student was growing, he or she knows that '1 + 1 = 2' without telling or asking from a friend or he/she knows that 'a book + a book = 2 books'. G o d made man and put a little sense in the man's m i n d . It is just left for the teacher to add a little to what y o u k n o w . " (Student, C C C ) M a n y of the students' responses seem to suggest that they understood mathematics can only be taught i n a formal environment, say a classroom while the teaching of mathematics i n informal and non-formal situations was not considered as teaching or learning taking place. This explains w h y they w o u l d consider themselves as k n o w i n g that '1 + 1 = 2', the amount of money given to them and even the difference they w i l l receive after purchasing something that requires a balance. 1.3 The teachers W i t h i n the realm of the nature of mathematics, five questions were posed i n the questionnaire that sought to reveal the teachers' stances or positions regarding the nature/philosophy of mathematics. These questions were: (a) What is your definition of mathematics?; (b) If y o u were asked to design the cover of a mathematics text, what w o u l d it look like? Y o u can either draw or list the things y o u w o u l d put on the cover; (c) Mathematics has been described by some as a theoretical subject divorced from its human origins. H o w w o u l d y o u respond to such a description?; (d) H o w w o u l d y o u respond to the following statements: Different cultures and the course of history have contributed to mathematics, Mathematics exist as pre-given knowledge i.e. mathematics knowledge cannot be created but exist independent of mankind, Mathematics is logically derived from axioms?; and (e) D o teachers expect students to k n o w mathematics which they were not explicitly taught? 130 (a) What is your definition of mathematics? A l l the four teachers saw mathematics as a science and these views were very consistent throughout the research. One of the teachers (Teacher B) i n C C C went as far as identifying different types or uses of mathematics. The fact that only two teachers taught the ethnomathematics unit d i d not seem to influence the teachers already established views about the nature of mathematics. Here is how the teachers defined mathematics: " M a t h is a language of the sciences and plays the role of w i d e n i n g your m i n d to think and act reasonably and logically." (Teacher A , CCAST) " M a t h is the science that builds up logical thinking i n pupils. It is the science of numbers employed i n solving day-to-day problems. Thus it is geared towards imparting proper reasoning skills i n the learners at the beginning of their studies i n math. Furthermore, mathematics involves the building up of theories w h i c h serve as framework i n various fields e.g. binary theory i n computer programs etc." (Teacher N , CCAST) "Mathematics is the science of numbers, w h i c h has evolved throughout the years and is used i n our daily activities e.g. mathematics i n the kitchen, mathematics on the farm, mathematics i n economics, geography etc or simply mathematics i n school." (Teacher B, C C C ) "Science w h i c h involves logical study of shapes, arrangement, quantity, and many related concepts and may fall into the following branches: arithmetic, algebra, analysis, geometry, trigonometry etc." (Teacher K , C C C ) The above quotations suggest each teacher's philosophical position regarding mathematics and it is possible that these positions w o u l d have a bearing on h o w each teacher approaches the teaching of mathematics. 131 (b) If you were asked to design the cover of a mathematics text, what would it look like? Of all the four teachers, only one teacher (Teacher B, C C C ) suggested making the cover colourful and attractive and only one teacher (Teacher K, C C C ) suggested making the cover responsive and closer to the student's environment by including a local cultural background on the cover. Both teachers i n C C A S T focused on the mathematics content that should be presented. Considering both teachers in C C A S T were formally trained, it is surprising to note that both opted for a design that is more abstract and less welcoming i n its presentation. Here is h o w they each described their design: "It should carry the operation signs (+, -s-, *, -) and a small market situation w i t h people buying and selling fowls, other food crops including fruits. There should also be a bricklayer at work." (Teacher A, C C A S T ) "Book covers always have to be made so that a reader makes some sense out of it, or so that they give an impression of what the book is about. Depending on the level for w h i c h the book is geared, I w o u l d include the title at the top, some maths symbols in the middle showing a bit of what is inside the book and finally the name of the author at the base of the cover page." (Teacher N, C C A S T ) "Some common shapes and symbols, and numbers e.g. triangles, circles; alpha, lambda; the plus, multiplication signs; the set of numbers N,M,Z etc. Some interwoven colours attractive to the user." (Teacher B, C C C ) "The cover w i l l contain as many mathematical symbols, giving a background view of the environment, or area where the book has been written. Trying to insert the cultural background where the book has been written." (Teacher K, C C C ) 132 (c) and (d) How would you respond to the following statements: Different cultures and the course of history have contributed to mathematics. A l l four teachers agreed that different cultures and the course of history have contributed to mathematics. This fallibilist view is i n line w i t h Ernest's writing on social constructivism i n which he presents mathematics as cultural knowledge, more than a collection of subjective beliefs, but less than a body of absolute objective knowledge. The teachers' responses went thus: "Quite true. The Hindus, the Arabs etc developed our present counting numbers and numerals." (Teacher A , C C A S T ) "This is true as maths curricula keep changing to catch up w i t h changing world; from traditional math ways of counting w i t h pebbles to the development of numeral and number words i n the different cultures." (Teacher N , C C A S T ) "True because mathematics has evolved to this extent because of interaction of different cultures giving rise to many developments in mathematics skills and reasoning." (Teacher K , C C C ) "True. Early m a n had his sheep and cattle. In order to k n o w the size he h a d to count them. Hence mathematics originated. The w o r d algebra, alpha, sigma, Pythagoras theorem etc are from different cultures and history." (Teacher B, C C C ) (e) Do teachers expect students to know mathematics which they were not explicitly taught? In response to the question whether teachers expected students to k n o w mathematics w h i c h they (students) were not explicitly taught, both teachers w o r k i n g w i t h the ethnomathematics group said yes while the other two teachers i n the nonethnomathematics group said no. Whether this is related to the teachers w o r k i n g on the ethnomathematics unit or not was not clear. What is however evident i n the 133 teachers' responses to this question is that the teachers w h o held a view of mathematics as an organized body of knowledge tended to disagree while those w i t h a view of mathematics as arising from h u m a n daily activities felt it was possible for a student to k n o w mathematics without being taught. Their views were espoused i n the following statements: "Yes. F r o m observing carpenters or bricklayers students can have very good ideas on geometry etc." (Teacher A , C C A S T ) " N o . Because a lot of knowledge is built u p w h e n a topic is taught i n class or even read by a learner (in the case of out of class study by learner). Though this might involve from the students some remote knowledge of that particular maths it is usually better understood only w h e n there is an instructor to guide them." (Teacher N , C C A S T ) "Yes. If students explore with a lot of interest and determination they can easily discover the basic and related concepts which are required." (Teacher K, C C C ) " N o . Because math requires applications i n our day-to-day activities, meaning that ideas should be w e l l taught and explicitly taught." (Teacher B, C C C ) The various positions taken by the teachers i n responding to the above five questions suggest that teachers ascribe to more than one view about the nature of mathematics. If this is true, then it is possible that teachers are able to hold varying positions regarding the nature/philosophy of mathematics and access the position w h i c h is most appropriate for a given situation. The teachers various positions vis-avis the nature of mathematics concurs w i t h Lakatos' argument that no definitions i n mathematics are ever absolutely final and beyond revision. A s such the history of mathematics should always be given pride of place i n any philosophical account. 134 1.4 The pedagogic adviser In response to the question 'What is mathematics?', the pedagogic adviser provided the following response: "Mathematics is a w a y of expression ... a form of language." He supported this statement by arguing that mathematics cannot be pregiven knowledge since it's a form of language and language cannot exist independent of h u m a n beings. To h i m , mathematics is knowledge created by h u m a n beings as they come i n contact w i t h specific difficulties or specific situations. This knowledge, he maintains is created i n the form of (mathematical) models w i t h rules for manipulating them. These rules and knowledge change as man's knowledge of the situation improves. Hence mathematics is not static but changes w i t h man's increased experience of the w o r l d . This view of the nature of mathematics espoused by the pedagogic adviser leads h i m to conclude that since it (mathematics) is a way of expression it is therefore naturally linked to man's culture. Hence mathematics is a cultural product or a product of cultural expression. This view of mathematics as a cultural product suggests that different cultures approach mathematics differently depending on their culture and their experience of the w o r l d . Such a view w o u l d seem to encourage a need for a cultural approach to the teaching and learning of mathematics. 1.5 The university faculty The university faculty conceded to the fact that the definition of mathematics depends on w h o is offering that definition and under certain given conditions. To this end, the w o r d mathematics conjures several things to h i m for example, it could be seen as " . . . a way of reasoning. It is thought provoking, and enables critical thinking and this can be achieved through certain specific principles." In offering this view of mathematics, he seems to be suggesting that mathematical knowledge is 135 acquired only if one followed "certain specific" path of reasoning or line of argument. This is a view of mathematics w i t h an epistemological bent as it seems to suggest that there are specific ways one must go about to acquire mathematics knowledge or to engage i n mathematics. Another statement expressing what comes to his m i n d w h e n he hears the w o r d mathematics has more to do w i t h the uses of mathematics (knowledge and its principles of acquisition) by emphasizing its usefulness to other fields of study such as the sciences. H e states, " . . . mathematics allows other sciences, specifically, the physical sciences to express relations between certain variables." This utilitarian view of mathematics is often used to justify the importance of mathematics and the reasons for studying it thus suggesting that mathematics devoid of a utilitarian bent is ignoble, lesser mathematics or less important. Another view, w h i c h the w o r d mathematics connotes to the university faculty, was that of being a language like any of our many w o r l d languages: " . . . mathematics is a language on its o w n . Because as every other living language, it has its o w n syntax, its o w n symbols, its o w n vocabulary, but as a living language, borrows words from other languages but uses it i n its o w n context." (University Faculty) These different meanings or images held by the university faculty regarding the meaning of the w o r d mathematics seems to suggest that it is possible to compartmentalize different and sometimes even contradictory views about the subject of mathematics. This is possible since according to fallibilism, mathematical knowledge is always understood relative to the context, and is evaluated or justified w i t h i n principled or rule governed systems. According to this view there is an underlying basis for knowledge and rational choice, but that basis is context-relative and not absolute. Hence there is no such thing as absolute truth as mathematics transcends any particular individuals This is partly w h y the university faculty, like 136 the teachers i n the two schools, seem to hold to varying and competing views on what the w o r d mathematics means to h i m . 5.2.2 Mathematics and Culture The stakeholders were asked whether history, culture, cultural background, students everyday experiences were important i n the teaching and learning of mathematics. If they were important, h o w often they incorporated these into their mathematics lesson instructions. The stakeholders were then asked to state, on a five-point scale (from most important to least important, or from more relevant to least relevant, or from incorporating these all the time to never incorporating these at all) how relevant or h o w important these were. Other questions such as: Ts mathematics culture-free, culture-neutral, or value-free?'; ' A r e there any connections between school mathematics and the students' cultural background?'; ' H o w important are these connections?' were also posed. These questions were used both in the questionnaire and during the interviews to elicit the participants' views vis-avis school mathematics and culture. 71.2 The whole group Most of the questions i n this area were posed to the teachers, university faculty and pedagogic adviser. The questions focused on accessing teachers' k n o w l edge of mathematics and culture and h o w this connection affects their instructional planning and organization of learning experience. Most of the students' questions i n this area were of their knowledge of mathematics embedded i n their culture or cultural connections of school mathematics. 137 11.2 The students Do you find mathematics in your culture in any way? (Give examples to support your answer) CCAST Bambili: W h e n the questionnaire was first administered, 29 out of 40 students (16 i n the ethnomathematics group and 13 i n the non-ethnomathematics group) said mathematics was present i n their culture. Clearly a majority of the students i n both groups felt that mathematics was present i n their culture. The following excerpts illustrate their responses: Before: " W h e n there is a dead celebration, people celebrate and try to share food i n the way that everybody w i l l have and w h e n they want to share meat, they count the people so that everybody can have a piece." (Student, C C A S T ) "Yes, I can finds mathematics i n m y culture because w h e n a boy wants to marry a girl, the father of the girl w i l l ask the boy to give a certain amount of money. I think that is mathematics." (Student, C C A S T ) "Yes, mathematics is present i n m y culture because i n our culture they pay people dancing and they must count the money and they use mathematics i n counting the money and sharing it between the people dancing." (Student, C C A S T ) " N o , because I don't find mathematics i n m y culture, I find only my dialect."(Student, C C A S T ) Beside the above statements, most students stated that mathematics was present i n their culture because i n the local market place, people buy and sell w h i c h according to many of them is mathematics. O n l y one respondent maintained i n both questionnaires that mathematics was not present i n her culture because i n her 138 culture there is no w o r d for mathematics and besides, she had never heard of it. She explained: After: "There is no mathematics i n m y culture because we do not have a w o r d for mathematics. Otherwise I w o u l d have heard somebody use it in the dialect. Also, we cannot write i n m y dialect so h o w can we do mathematics?" (Student, C C A S T ) What the above statements suggest is that this student holds a view of mathematics as that which resides i n the textbook and does not think of the mathematical processes that she performs everyday without pen and paper. C C C Mankon: This last item on the questionnaire, w h i c h explored the presence of mathematics i n students' culture, revealed that when the questionnaire was first administered, 26 out of 40 respondents (11 i n the ethnomathematics group and 15 i n the nonethnomathematics group) felt that mathematics was present i n their culture i n some form. Hence a majority of the students i n both the ethnomathematics and nonethnomathematics groups felt that mathematics was present i n their culture i n one form or the other. Below are some of their responses: Before: "Yes, because i n m y culture the amount of plantains or jugs of p a l m wine to be brought if asked by the chief to any celebration is calculated using mathematical symbols and the total to be brought is then given i n terms of mid-size calabashes." (Student, C C C ) "Mathematics is not present i n our culture because our people have not yet got i n touch w i t h what we call mathematics. They lack teachers 139 to teach them mathematics and that is one of the reasons that I want to improve on mathematics very well." (Student, CCC) "Yes, in my culture there is the tendency of doing mathematics because when you are married you must teach it to your wife and children so that those who are jobless can start a new life either by selling in a store or counting money through trading and other businesses like banking." (Student, CCC) "Yes, I find mathematics in my culture when a man wants to get married to a woman from my tribe, certain things are demanded from him. But in situations where they cannot be easily got, the people calculate the price of all those things and ask him to pay in cash." (Student, CCC) "Most people in my village do not like mathematics because it is too difficult so we don't find it in our culture" (Student, CCC) However, when the questionnaire was next administered, all the students in the ethnomathematics group said mathematics was present in their culture while 3 students in the non-ethnomathematics group still thought that mathematics was not present in their culture. Those who felt that mathematics was present in their culture provided examples of occurrence of mathematics similar to those stated above as well as the following statements: After: "Yes, in my culture I find mathematics a bit like when calculating the 'country Sunday' (traditional Sunday), market day, and the day of a chief. Mathematics is also taught in my language." (Student, CCC) "Mathematics is present in my culture when: tailors use mensuration in sewing, bricklayers use mensuration in building, carpenters use mensuration in roofing and it is also compulsory in certificate and public exams." (Student, CCC) 140 Those students w h o still maintained that mathematics was not present i n their culture substantiated this by statements such as: "We don't have mathematics i n m y culture because i n m y village, we have only primary schools and i n primary school y o u learn only arithmetic. Mathematics is done only i n the secondary school w h i c h is far away from m y village." (Student, C C C ) "In m y culture we don't have mathematics because it is very difficult. People use common sense which is different from mathematics." (Student, C C C ) "Mathematics is not present i n m y culture because we don't not calculate the area or find the square root of numbers. What we do i n m y culture is mostly arithmetic w h i c h is easier than the mathematics we study i n school." (Student, C C C ) The above statements suggest that i n both the public and private schools most students felt that mathematics was present i n their culture although the w o r d mathematics was never uttered. The students seem to be identifying the presence of mathematical processes i n their culture or daily lives or the use of mathematics knowledge i n certain cultural situations. 11.3 The teachers O n the question of h o w often they presented a historical or cultural back- ground to the mathematics they teach, all four teachers (2 w h o w o r k e d w i t h and taught the ethno-mathematics unit and 2 who taught their regular unit not influenced by ethnomathematics ideas) responded that they sometimes but not regularly presented a historical or cultural background to the mathematics they are teaching. But w h e n asked h o w important and relevant students everyday experiences and cultural background were when they p l a n their lesson instructions, both teachers responded that this was relevant or very important. This suggests that 141 while both teachers agreed on the importance of students' background knowledge to acquiring new knowledge and the importance of relating what students were learning to what they experienced everyday, the teachers d i d not incorporate these ideas regularly. In response to whether teachers can use students' in-school and out-ofschool knowledge to connect the mathematics practices i n these contexts both teachers (2) of the ethnomathematics group responded i n the affirmative giving the following statements to support their opinions: "Yes. W h e n one recognizes the fact that maths exists in the various cultures, students' experiences are very important as they bring the students into real practical situations. Formal maths can be imparted w i t h examples d r a w n from the homes. This instils confidence i n students and makes learning easy." (Teacher A , C C A S T ) "Yes. B y planning our lessons to include practical experiences, this helps i n b u i l d i n g up examples w h i c h bring the math knowledge closer and easier to understand. A visit to an industrial firm w o u l d facilitate students' understanding of functions as this w o u l d be used to describe h o w the plan i n the firm operates." (Teacher K, C C C ) II. 4 The pedagogic adviser T w o m a i n questions were posed on the nature of the relationship between school mathematics and culture and h o w the latter affects the teaching and learning of the former. These questions were: (a) 7s mathematics and culture related in any way? (b) How is the teaching and learning of school mathematics affected by this relationship? To the question on whether mathematics and culture were related i n any way, the pedagogic adviser began by first looking at what mathematics is and what mathematics knowledge is. H e argued that if mathematics is a way of expression, then it is a form of language and since language cannot exist independent of h u m a n beings, it must be logical to assume the existence of a cultural connection. F o l l o w i n g 142 his line of argument, the pedagogic adviser is clearly suggesting that mathematics and mathematical knowledge are not pre-given (i.e. do not exist independent of mankind) but created by h u m a n beings to solve specific problems or situations. "Mathematics is a way of expression. In that light, it is a form of language. The question to ask then is: C a n language exist independent of human beings? It can't. In that case, we cannot consider mathematics as pre-given knowledge. It [mathematics] is knowledge created by h u m a n beings as they come i n contact or i n face w i t h specific difficulties or specific situations. This knowledge is created i n the form of mathematical models and rules for manipulating them. These rules and the knowledge change as man's knowledge of the situation improves. So we are saying i n effect that mathematics is not static. It changes with man's experience of the world, man's increased experience of the w o r l d . A s a w a y of expression, it is naturally linked to man's culture. If we say that the content of mathematics is almost universal, that does not mean that every person or every culture approaches it w i t h the same frame of mind, or w i t h the same facility. Because people approach mathematics differently depending on their culture, and their experience of the w o r l d . " (Pedagogic Adviser) W h e n pressed on what he meant by "...people approach mathematics differently depending on their culture, and their experience of the w o r l d " , he used an example to illustrate how mathematics is embedded i n culture and h o w this relationship therefore influences mathematical thought processes, representations and symbolizations. This is captured i n the following statements: " W h e n y o u look at certain things i n mathematics, i n statistics for example, y o u name a certain chart a pie chart why? I d i d not k n o w the meaning of this until I learned of a thing called 'apple pie', w h i c h is some sort of a cake w h i c h is round i n shape. That explains the origin of the pie chart. So if y o u approach the concept pie chart without k n o w i n g what a pie is, your interpretation w i l l be quite different. But if y o u have at the back of your m i n d the apple pie as a circular cake, then y o u k n o w w h y it is called a pie chart and y o u interpret, i n fact y o u approach it w i t h more insight. So i n the light of that, y o u see h o w 143 culture gets into mathematics. Every mathematical element can be traceable to one cultural element i n one form or the other. O u r present number system originated from the Arabs. It is true that it has been modified quite a lot by the Europeans and the Americans as it is today but it still has a lot of cultural background attached to it. So I think that history is important. It is very important for us to k n o w h o w some of these elements originated. We w i l l appreciate them better and we w i l l easily tie them to their cultural background, w h e n we are explaining them especially to learners." (Pedagogic Adviser) " W e l l , from the definition of mathematics, we say it's a w a y of expression. A n y way of expression is culturally bound. The way one expresses themselves depends on their culture. A n d the way one approximates, one reasons, one calculates, is culturally bound. A n d if mathematics develops from this, then y o u cannot divorce it from culture." (Pedagogic Adviser) 11.5 The university faculty O n whether mathematics existed as pre-given knowledge, the university faculty responded that mathematics was dynamic, culture free and some aspects of it [mathematics] were based on axioms. In response to the questions: (a) How often do you present a historical background to the mathematics you teach?; (b) How often to do you present a cultural background to the mathematics you teach? The university faculty responded "sometimes but not regularly" but w h e n asked how important or h o w relevant were the students' everyday experiences i n lesson planning and mathematics learning, the university faculty's response ranged from 'relevant' to 'crucial'. These responses suggest that while the university faculty ascribes to the fallibilist view of mathematics i n which history and culture are germane, these views are not explored or encouraged among student teachers. This is partly due to the fact that these teachers are being prepared to teach students preparing to write the G C E or Bac, exams w h i c h do not test for that k i n d of knowledge. A s result, there is no need wasting time on developing or encouraging these views. 144 Another reason is more a logistic one based on the fact there are only two faculty members i n mathematics education w h i c h makes it difficult to explore these views given the workload for each faculty member. 5.2.3 School Mathematics Curriculum Five general questions focusing on the school mathematics curriculum guided the interviews with the teachers, pedagogic adviser and the university faculty while one question was used i n the student questionnaire to collect information regarding the school mathematics curriculum. The five general questions were: (1) What purpose was the present mathematics curriculum designed to serve? (2) To what extent has this purpose been served? (3) What flaws i n the curriculum itself could have undermined its effectiveness? (4) H o w could these flaws be eliminated and/or remedied? (5) What mathematics should be taught to Cameroonian children? W h y ? There were two questions on the student questionnaire focusing on the school mathematics curriculum: (i) What can be done to make mathematics more 'interesting' to you?, (ii) Should mathematics remain a compulsory subject at the secondary school? III.l The whole group The general feeling expressed was that the present mathematics curriculum was designed to graduate students w h o w i l l move on to further studies abroad w i t h less regard on returning home to put those skills to use i n the community. The complete reliance on an examination intended for those to proceed to higher education has left many graduates the inability to be self-reliant and the total reliance on the government to absorb them into the public service w h i c h has not been possible. For the whole educational system to centre around exams meant for the top 20% of students has led to teaching for exams and not for understanding. The general 145 consensus was that a w a y forward w o u l d require major curriculum reforms w i t h a focus on self-reliance and self-sustainability. There was a general feeling that the k i n d of mathematics that should be taught to Cameroonian children should be one that besides developing numeracy skills, should equip them w i t h skills to become useful to the community and also be able to compete w i t h children i n other parts of the w o r l d . 111.2 The students (i) What can be done to make mathematics more 'interesting' to you? This question was intended to solicit students' suggestions regarding the current state of the school mathematics curriculum and ways at addressing the issues plaguing the mathematics curriculum. In response to the above question, a majority of the students i n each of the four groups i n both schools suggested that the government should send well-trained teachers w h o are less boring and are enthusiastic about the subject. A third of the students (31) i n all the groups also suggested the need for good textbooks and the need for mathematics knowledge that can be easily applied to daily life. Of all the student responses, 16 i n 80 students suggested the need to introduce a multi-method approach to problem solving i n school mathematics, as they believe that this w o u l d allow them to choose a simpler method to solve a given problem rather than memorizing particular algorithms. 8 i n 80 students suggested eliminating the variable 'x' i n mathematics as that w o u l d make mathematics more interesting and less difficult since they w o u l d not have to look for 'x' any more. To them, the use of the letter 'x' as an u n k n o w n or as a variable carries w i t h it the connotation of difficulty or complexity. A d d i t i o n a l suggestions included the need for government assistance i n mathematics texts and learning resources as well as additional mathematics periods per week to allow them better understand the processes during problem solving. 146 (ii) Should mathematics remain a compulsory subject at the secondary school? This question was posed to the students prior to and after the ethnomathematics unit. The responses are therefore recorded i n the two instances - before the ethnomathematics unit and after the ethnomathematics unit. CCAST Bambili: O n the question of whether mathematics should remain compulsory i n the secondary school, when the questionnaire was first administered, most students i n both the ethnomathematics and non-ethnomathematics group felt that mathematics should continue to be compulsory because it is important i n life and i n other school subjects. W h e n the questionnaire was administered again, only 2 students i n the ethnomathematics group and 1 i n the non-ethnomathematics group felt that students should still be able to opt out of doing mathematics since some are not very good at it. Most students w h o felt that mathematics should continue to be compulsory at secondary school saw the importance of its utilitarian aspects while the few w h o thought it should not remain compulsory argued that their performance i n school mathematics was affecting their overall performance i n school. Some of the frequently stated reasons were as follows: Before: "I think so because it is used i n our daily lives e.g. if y o u want to do a job such as banking, y o u w i l l have to k n o w h o w to calculate." (Student, C C A S T ) "It should remain compulsory because we need mathematics i n every thing we do e.g. calculating money." (Student, C C A S T ) "Yes, because without mathematics we cannot do anything because mathematics is connected w i t h our daily lives." (Student, C C A S T ) 147 "...It should not be compulsory because some people do not want to do it i n the future and some people hate it because it can dislocate the brain." (Student, C C A S T ) "It should continue to be compulsory because i n a situation like the G C E , you cannot do without mathematics." (Student, C C A S T ) After: "It should be compulsory because what y o u k n o w n o w i n mathematics cannot help y o u so y o u need to learn more." (Student, C C A S T ) "... because mathematics is needed i n every situation i n life e.g. counting money, doing business." (Student, C C A S T ) "It should remain compulsory because y o u cannot be a scientist or engineer without k n o w i n g maths." (Student, C C A S T ) "It should not be compulsory because some people don't like it and others like it. It should also not be compulsory because it is boring when doing it." (Student, C C A S T ) A third of the respondents ethnomathematics i n both the ethnomathematics and non- groups saw the importance of mathematics remaining a compulsory subject because it was one of the subjects written i n the G . C . E . and most other public examinations. It should be noted that these students are i n the last grade level before embarking on the rigorous preparation for the G . C . E . Ordinary Level w h i c h w i l l be written three years from now. A n d i n the G C E examination all students are automatically registered to write the exam i n mathematics although there is no penalty apart from simply receiving a failed grade i n the subject for those who elect to sit out. Critics of the compulsory policy of the secondary school mathematics have spotted this as the weakness of the policy and w o u l d like the policy to 148 be revised to ensure that students do not only register by actually attend the mathematics classes throughout the entire programme. CCC Mankon: O n the question of whether mathematics should remain a compulsory subject at the secondary school, 14 out of 20 students i n the ethnomathematics group and 16 out of 20 students i n the non-ethnomathematics group said it should remain compulsory w h e n the questionnaire was first administered. O n their reasons w h y it should remain a compulsory subject, 11 students i n the ethnomathematics group and 10 students i n the non-ethnomathematics group said because it was considered a prerequisite for admission into high school. Those w h o felt that it should not remain a compulsory subject argued that not everybody was good at mathematics and so people should have a choice of whether to do it [math] or not. Some of these respondents d i d admit however that they were aware of the usefulness of mathematics i n their daily lives. The following excerpts were fairly common among the respondents w h o said it should remain a compulsory subject: Before: "It should remain compulsory because without mathematics y o u cannot reason well, think well. Y o u w i l l always be saying things that are off. In Cameroon there is no concours without mathematics and if y o u don't k n o w mathematics y o u write a concours and pass i n other subjects, y o u may be forced to repeat because of mathematics." (Student, C C C ) "In secondary school mathematics is a compulsory subject because y o u must have it at the Ordinary Level examinations before y o u can be admitted to high school." (Student, C C C ) "This is because, for instance m y elder sister found difficulties i n the second cycle w h e n she was i n A 3 (which is economics, history, litera- 149 ture) and she had difficulties i n economics because she disliked and put no interest in mathematics and had no basic idea of it." (Student, CCC) "Yes. This is because mathematics is important i n everyday life and without the knowledge of mathematics one cannot be able to run a business." (Student, C C C ) Those w h o felt that mathematics should not remain compulsory argued their case as follows: " N o because not everyone understands it or even like it. For example I myself do not understand it." (Student, C C C ) " N o it should remain compulsory because in the course of calculations some students' brains become offset. A l s o not all students like maths because they may not involve themselves i n any occupation that w i l l make them to be calculating things. When some students solve mathematical problems over and over, they develop a mental problem later." (Student, C C C ) W h e n the questionnaire was administered a second time, all 20 respondents i n the ethnomathematics group and only 14 instead of the previous 16 respondents said it should remain a compulsory subject at the secondary school. O n their reasons w h y it should remain a compulsory subject, more than half of the respondents said because it was needed i n their daily lives and also because it developed their reasoning faculties. A good proportion of the respondents said because it was present in all public examinations. Those w h o said it should not remain compulsory cited difficulties in learning mathematics and the preference to chose what they liked to study. 150 After: "It should remain a compulsory subject at the secondary school because mathematics is interesting if only y o u k n o w it. M o r e to that though it is difficult so we should try to study it better but the minister [of education] should find any method that students should not be running away from it." (Student, C C C ) "Yes, because we are i n the computer age and y o u cannot k n o w h o w to work on a computer w h e n y o u don't k n o w mathematics. So it should remain a compulsory subject i n school." (Student, C C C ) "Yes, because a student cannot leave secondary school and go to any office or bank without mathematics. So it should remain a compulsory subject so as to encourage the student to k n o w mathematics very w e l l and w h e n they leave and go to any place there w i l l be no problem i n solving mathematics." (Student, C C C ) " N o , because I don't like it. It is too difficult to learn it and it is difficult to understand it." (Student, C C C ) "Yes, because it deals w i t h our daily life activities and because we cannot live without mathematics." (Student, C C C ) " N o , it should not remain a compulsory subject since i n the high school y o u chose what series [subject combination] y o u want to study, some of w h i c h do not have mathematics." (Student, C C C ) "Yes, this is because it helps people to k n o w h o w to calculate even money. Mathematics is almost compulsory i n all professional exams. There is no concours i n the country that y o u can write without mathematics. Also, maths makes students to be good i n calculations. H a r d l y w i l l y o u find any highly or richly regarded person i n the society and the government without mathematics." (Student, C C C ) W o r t h noting i n the students' responses is that even students w h o were not succeeding i n mathematics still felt that it was necessary to maintain the compulsory 151 nature of the subject at secondary school because mathematics knowledge was important i n life. This is a point the government may need to consider and decide on h o w to go about ensuring that students are made to stay i n during mathematics lessons. III.3 The teachers (1) What purpose was the present mathematics curriculum designed to serve? In response to this question, three of the teachers while lamenting the inappropriateness of the school mathematics curriculum agreed that the mathematics curriculum being used i n Cameroonian secondary schools was inherited from the former colonial government and was intended for the General Certificate of Education examination w i t h the hope that successful students w o u l d be able to pursue studies abroad especially i n the United K i n g d o m . The following statements below attest to their dissatisfaction w i t h the curriculum, sentiments commonly espoused by secondary school teachers i n Cameroon. "...the curriculum was adapted from the British and French and as such, one could say it was intended to enable Cameroonians carryout studies i n the U K . " (Teacher A , C C A S T ) "...there is actually no national curriculum for mathematics for the country. A n d what we are using, not that we are not using any form of curriculum but it is what we inherited from the colonial masters w h i c h have not been adapted to our o w n situation. So practically it's a syllabus, an examination syllabus ... broken d o w n into schemes of work for the various Forms (grade levels), ... to prepare students for the General Certificate of Education (GCE) examinations." (Teacher N , CCAST) "It's unfortunate that the mathematics curriculum was developed for the purpose of an examination, that the mathematics student's success is measured only i n terms of that examination. A n d that's w h y the curriculum was developed. The mathematics curriculum has never 152 been measured as a knowledgeable subject that y o u can just test everybody but the curriculum has been developed as a measure of the degree of examination and not just k n o w i n g mathematics." (Teacher K, C C C ) O n l y one teacher (Teacher B, C C C ) believed the curriculum was intended to develop i n the students, skills that w i l l make them useful citizens. "...it was intended to equip students w i t h the necessary skills to be useful not only to themselves but to the entire nation tomorrow." (Teacher B, C C C ) This view of citizenship education concurs w i t h the stated educational goals at the d a w n of independence but had never been revised to reflect the changing realities of today. Post-independence educational policies i n Cameroon, as far as contextualization was concern, showed very little regard for the various local cultural practices as a source of knowledge. Little wonder w h y the school mathematics curriculum has not changed by much since the 1960's and this has contributed to very little (educational) development. (2) To what extent has this purpose been served? Assessing the success to the curriculum according to the teachers really depends on the intended aims of the curriculum or of formal education. A l l the teachers expressed disappointment with the performance of the students especially i n national examinations like the G C E or the Baccalaureate. They maintain that if the purpose of the curriculum was to enable students pass the national examination, then it could be concluded, and rightly too, that the purpose of the curriculum is not being served. They all fault the curriculum and not the students on the basis that the curriculum was intended for the top 20% of students but is being administered to all 153 the students. The following statements capture their feelings regarding the level of success with the curriculum. "This has to do w i t h the educational policies i n the country. Cameroon's educational policies are still that of mass education trying to make the majority of the people read and write. That is the philosophy of education i n Cameroon. A n d so if that is the aim of education for Cameroon, then one w o u l d say that the goals at least are being achieved because at the beginning, the colonialist d i d not aim at producing producers at the end of the course. They were just geared at making people read and write to be able to find their way around." (Teacher A , C C A S T ) "If the Baccalaureate and G C E examinations were the reasons for developing the mathematics curriculum for secondary schools i n Cameroon, then one could say that the purpose is not being served well since the failure rate i n the examinations is very high." (Teacher N , CCAST) "We are not succeeding very well. Because of 1) our foundation, 2) even those w h o manage to come out without mathematics and the government says if y o u don't have mathematics no jobs for you, no further education, people are still going ahead, the government is not keeping to those rules. So what makes a student feel like it is important I must do it (mathematics)? ... If y o u say mathematics is compulsory, then establish measures to ensure that students follow that compulsory nature of it. So I think that we are failing. M a k i n g it only for a few." (Teacher B, C C C ) "We are not. We are failing woefully. Because the tendency is not to instil mathematics i n the students but to get the students write the mathematics for that examination. A n d as a result some students are finding it very difficult to study that mathematics of that examination." (Teacher K, C C C ) 154 (3) What flaws in the curriculum itself could have undermined its effectiveness? A l l the teachers agreed that the m a i n flaw i n the curriculum had to do w i t h the fact that it [the curriculum] is determined by the national examination w h i c h the students w i l l be writing instead of the curriculum determining what should be tested i n the examination such as the G C E and the Baccalaureate. A s a result, the teachers tend to focus more on preparing students for the exam instead of teaching for understanding. This view is captured i n the statements below: "...we produce consumers and not producers. ... most of what is done is solely for the purpose of the exam (GCE). A s such the teacher is focused on preparing students to be able answer questions at the end of the course. W h i c h is not good. That is like rushing to the answer without understanding the processes. They need to pass the G C E quite all right. But they have to understand what they are doing." (Teacher A , C C A S T ) Another flaw to the curriculum, which was also echoed by all the teachers, is captured i n the excerpts below w h i c h focus on the fact that the curriculum is meant for the exceptionally talented students but is administered to all the students. H e states: "We also realize that the curriculum we are using i n mathematics now is what the colonial masters left way back at independence. A n d this was mathematics for the top 20% of students. N o w we are teaching it to all the students. W h i c h means that we are assuming that all of them are supposed to be very good, w h i c h is not true. A n d that brought us to thinking of coming up w i t h a project to re-examine the curriculum." (Teacher N , C C A S T ) One of the teachers believes that the continued failure i n educating children mathematically is also attributable to the educational policies and quality of teachers vis-a-vis the pedagogy and mastery of the subject matter. 155 "We are not explicit because we memorized the processes and went and passed our exams. ... The student fears mathematics for the good reason that some of us were not w e l l educated mathematically. The teacher training is a government arrangement. A n d the government curriculum, w h e n the government says this subject is compulsory but fails to set the rules for ensuring that compulsory nature, hence the students are really not motivated about it." (Teacher B, C C C ) ".. .if we the people had actually realized that mathematics is practiced consciously and unconsciously, the curriculum w o u l d have just been a guide, it w o u l d have not been something implanted that this is what you have to cover at this level. It w o u l d have just been a guide that w o u l d help a child or anybody go through and learn mathematics." (Teacher K , C C C ) While the teachers partly blame the curriculum for the failure of students i n mathematics, it can be said that simply revising the curriculum cannot eliminate the flaws and improve performance. A major review of the educational policies is what may be required. (4) How could these flaws be eliminated and/or remedied? A starting point for eliminating the flaws i n the curriculum according to two of the teachers (both teaching i n C C A S T Bambili), is by clearly stating the learning objectives/outcomes of the subject i n terms of skills acquisition. The teachers also believe that designing other ways of assessing student learning such as continuous assessment should be considered. While the two teachers i n C C A S T Bambili were focused more on the learning outcomes and alternative assessment methods, those i n the private school felt the need to create a balanced curriculum by emphasizing real life applications of mathematics. To them, this is essential for self-reliant development. 156 "...I w o u l d like the curriculum to specify exactly what the student w o u l d be able to achieve studying that topic. ... There should be a balance between passing the G C E and k n o w i n g what y o u are studying. Sometimes, certificates don't speak for themselves." (Teacher A , CCAST) "School mathematics is not supposed to just be examination oriented, not to talk of examination oriented towards a particular examination board. Because what w o u l d happen w h e n we shall have more than one examination board? Each w i l l come up w i t h its o w n syllabus. A n end of course examination should be included to test part of a curriculum." (Teacher N , C C A S T ) "We don't find agriculture aspects i n the mathematics curriculum and yet we are an agricultural country. W i t h the presence of computers i n our society, I w o u l d have loved it to be incorporated into the curriculum as the excitement of their presence may motivate some students.... In all, we need to include practical aspects of mathematics e.g. mathematics i n weaving, mathematics i n farming, etc." (Teacher B, C C C ) "...there should be a level where we can think of bringing students to realize the importance of mathematics and doing mathematics for its usefulness i n life rather than just to pass an exam." (Teacher K , C C C ) (5) What mathematics should be taught to Cameroonian children? Why? M o s t of the teachers (3 out of 4) suggest the need for a curriculum w i t h the practical application of mathematics knowledge as it central focus. They argue that since not every student w i l l be able to proceed to university, there is no point i n having one curriculum whose m a i n focus is i n preparing students for advance studies i n mathematics. It is necessary to create alternative programs for those students w h o w o u l d not proceed to university. These alternative programs, they argue, w i l l allow those students w h o do not proceed to university to be able to look 157 for employment or create their o w n employment by applying their mathematics skills to their daily lives. " W h e n I look at the present situation, we turn out students from our schools and then we turn around and cry that they don't have any jobs. N o w , y o u cannot send a child to school solely to be employed by someone else always. Y o u should bring up a human being to be selfreliant, to be self-sufficient. This means that somebody coming out of school should be able to produce. So we should bring up producers and not consumers. So we should emphasize those areas that w o u l d produce skills as i n carpentry, building construction, etc. We should therefore develop a curriculum that is responsive to our local needs." (Teacher A , C C A S T ) "I w o u l d like to see the curriculum w i t h topics that w o u l d develop i n the students, problem-solving skills especially problems encountered i n their daily lives. W h i c h means it should include topics w h i c h even if the student were not to continue w i t h mathematics i n the high school, they should be useful, that student should be useful; it should also be topics that w o u l d develop computation skills i n the students; it should also be topics that w o u l d develop electronic skills if they have to get into the computer w o r l d they should have them even if the uses are not readily available. I w o u l d also like to include topics that w o u l d help them trace some historical and cultural roots for some of the maths concepts such as modular arithmetic." (Teacher N , C C A S T ) "We need a mathematics curriculum that can produce useful Cameroonians, mathematics that can get people to create jobs for themselves, mathematics that can develop the minds of the students to be logical thinkers." (Teacher B, C C C ) "Developing a curriculum that can help demystify the school mathematics is necessary if we want to encourage students i n mathematics. This w i l l require developing mathematical concepts by using local examples or situations. It is important for students to realize that mathematics exist w i t h i n their surrounding and that everybody can do mathematics and succeed." (Teacher K, C C C ) 158 (a) How important are the following qualities when deciding on a mathematics text? O n a scale of 0 - 4 (0 = not important, 1 = slightly important, 2 = important, 3 = very important, 4 = crucial) responses to all the five items ranged from important to crucial w i t h both 'correctness of content' and 'relevance to the national examination' being seen as crucial. Similar to the university faculty and pedagogic adviser, the teachers all stated that 'relevance to the national examination' and 'correct of content' was a determining factor i n deciding on a mathematics text. But w i t h a limited number of texts and a majority of them printed abroad, these preferences often do not play a major role especially as the texts are prescribed b y the ministry and must be followed. (b) To what extent would you agree or disagree with the following statements? O n a 5-point scale (0 = strongly disagree, 1 = disagree, 2 = neutral, 3 = agree, 4 = strongly agree), all the teachers i n both C C C M a n k o n and C C A S T Bambili either agreed or strongly agreed w i t h statements (a) the mathematics curriculum should be seen by all pupils as relevant to their future lives, (b) the mathematics curriculum should incorporate elements of the cultural histories of all the people of the region, and (d) the mathematics curriculum should resonate, as far as possible, w i t h diverse home cultures. One teacher disagreed with statement (c) the mathematics curriculum should be experienced as "real" by all children. III.4 The pedagogic adviser (1) What purpose was the present mathematics curriculum designed to serve? In response to the above question, the pedagogic adviser doesn't feel that the mathematics curriculum was even 'designed'. To h i m , it is not appropriate to use the w o r d 'designed' since the government holds the prerogative to curriculum reforms 159 and there has been little or no attempt since independence at reforming school curricula not only i n mathematics but i n all other school subjects. According to h i m , "Actually this curriculum was adopted, not designed by the Cameroon government. It's a colonial heritage, y o u see, we had the L o n d o n G C E at independence w i t h school i n this part of the country. Then i n 1966/67, the Cameroon G C E was introduced and patterned after the L o n d o n G C E . So, it's actually a colonial heritage as far as I am concerned. It was meant for the top 20% of the school population but is being taught to everybody. The remaining 80% is simply not responding because the program wasn't meant for them." The above statements suggest one of the main problems plaguing (mathematics) education i n Cameroon and other formerly colonised regions of the w o r l d the degree of irrelevance and the non-responsive nature of school curricula experienced by children i n these countries. It is understandable w h y the level of employment remains high because the k i n d of education being offered is besides being outdated, neither self-reliant nor self-sustaining. Such a system of education continues to rely on external forces for curriculum reform and innovation. (2) To what extent has this purpose been served? The pedagogic adviser believes that it is very difficult to k n o w if success is being made or not because the aims and goals of mathematics education are not clearly stated by the government. G i v e n the absence of statements of goals by the government, one could argue that we are succeeding if the intention is to provide mathematics education just for the top 20%. But if the goals of mathematics education were broadly stated to include the entire school going populace, then one can say that the objectives are not being met as the failure rate of students i n national examinations like the Cameroon G C E and the Baccalaureate continue to be very l o w (below 50%). This situation is captured i n the following statements: 160 "... our program is still patterned after the L o n d o n G C E and ... was meant for the top 20 percent of the school population. N o w , what happens to the bottom 80%? They are not just responding. N o w , when y o u go and take your G C E certificate and y o u have a grade of ' U ' i n mathematics, does it mean that your mathematics knowledge is zero? I mean, what does a ' U ' i n the Cameroonized L o n d o n G C E represent i n terms of mathematical ability? Certainly it does not represent n i l mathematical ability. That's the problem. The 80% of the populations is not simply responding because the maths program is not prepared for them." (3) What flaws in the curriculum itself could have undermined its effectiveness? One of the flaws i n the school mathematics curriculum according to the pedagogic adviser is the very nature of the curriculum. The other has to do w i t h government's decision i n making the learning of mathematics compulsory for all students i n secondary school. H e suggests that if the curriculum were reformed to make mathematics more meaningful and responsive to students, and different syllabuses created for the different ability groups of students then it won't need to be made compulsory for all students to learn. H e states, "The first problem emanates from the fact that we have only one program for all students i n the secondary schools (first 5 years of secondary education) writing the G C E Ordinary Levels examinations. That's not right. W e should have alternative syllabuses for students of different abilities. That's the first thing we should do. If y o u look at G C S E , there are at least four syllabuses for Ordinary Levels, so that y o u give every student the opportunity to do some form of mathematics. That's the first thing I w o u l d like to see done i n our school system if I were given the opportunity. The next thing w i l l be to be able to create situations where we develop support materials. There are a lot of materials available, w h i c h could be developed, but they are not easily accepted. It is necessary to develop a situation where teachers can create materials and these materials can then get into the school system." 161 One often cited impediment to the successful implementation of the secondary school mathematics curriculum is the lack of school textbooks as most of the texts used in Cameroon are either produced abroad or donated by foreign governments. As the pedagogic adviser decries, "...most of our maths programs have been adopted from abroad. We've tried to modify them but the modifications are still limited by the fact that we rely a lot on textbooks from abroad, on examination materials from abroad, on teachers and experts from abroad. ...In fact more than 80% of our textbooks come from abroad, or are produced by authors from abroad." (4) Hoiv could these flaws be eliminated and/or remedied? The pedagogic adviser believes that the main flaw to the current curriculum lies in the fact that it was intended for the top 20% of students but is being taught to everyone. He believes that the first step towards reforming the situation is the creation of another mathematics program for the bottom 80% of students. He argues, "If we were to allow the present math program to be taught just to the top 20%, we will not face the problem we are facing. We will not need to make it [mathematics] compulsory. So if we designed another curriculum to meet the needs of the remaining 80%, then they would respond more." Another cited flaw in the curriculum lies at the very heart of the nature of mathematics and mathematics education being emphasized in the two inherited systems of education (English and French). Because Cameroon practices two systems of education that it inherited from its colonial masters, the argument being made is that school mathematics in the English system is viewed as a service subject and is approached as such whereas in the French system, emphasis is on the philosophical orientation. The following excerpts by the pedagogic adviser illustrate this point: 162 "The Anglo-Saxon culture regards mathematics 80% of the time as a service subject meant to enable y o u solve practical problems i n engineering, science and so on. That's not the case w i t h the Francophones. They do mathematics from a theoretical point of view, abstract point of view, and philosophical point of view, defining the rules and then following them logically and arriving at what they want to do. I'll give y o u an example: a Francophone can define a Riemann integral meticulously and deal w i t h it to the end, but if y o u ask h i m to calculate a definite integral from say, 0 to ;r,of a certain function, he w i l l get stuck. That's the basic difference." Developing a national curriculum may be what is needed here. But the pedagogic adviser seems to believe that for that to happen, a change i n "thinking" (regarding the purpose of mathematics education) w i l l need to occur first. W i t h a national curriculum, it w i l l be possible to develop an appreciation for the various ways i n w h i c h people come to k n o w without looking at any particular approach as inferior or superior. "[T]he problem is that the underlying philosophy of mathematics education differs between the two systems of education. N o w , the A n g l o phone regards mathematics as a service subject and approaches it from a very practical point of view. The Francophone regards mathematics as a philosophy and approaches it from a very abstract philosophical point of view, and does not stress so m u c h on the practical implications of mathematics. N o w , unless these two philosophies are made to merge, w h i c h is quite difficult, we w i l l always have differences i n mathematics no matter what y o u do. ... A n d therefore, the difficulty here is not the mathematics program, but the thinking." (5) What mathematics should be taught to Cameroonian children? Why? It is the pedagogic adviser's belief that the k i n d of mathematics taught to Cameroonian children should be one that (1) enables them to be able to solve everyday problems easily, (2) develops critical thinking i n children, (3) provides the child163 ren the skills needed to lead a better life, and (4) helps children pursue further studies especially i n the sciences requiring a good knowledge of mathematics. (a) How important are the following qualities when deciding on a mathematics text? Five items and a 5-point scale were used to access the faculty's response to the above question. The five items were: a) correctness of content, b) adaptation to students' abilities, c) preparation of the students for the text by what they have already learned, d) preparation for what they w o u l d have to learn i n the future, and e) relevance of the text to national examinations. O n a scale of 0 - 4 (0 = not important, 1 = slightly important, 2 = important, 3 = very important, 4 = crucial) all the five items were rated as either very important or crucial, w i t h both 'correctness of content', 'adaptation to student abilities' and 'relevance of the text to national examinations' being seen as crucial. This particular question captured the attention of the pedagogic adviser because textbook production is a major problem and most of the textbooks i n use i n secondary schools i n Cameroon are produced abroad by authors w h o often not familiar the differences i n culture. Statements a, b, c, and d were seen as crucial while statement d was seen as important. (b) To what extent would you agree or disagree with the following statements? Four items and a 5-point scale were used to access the faculty's response to the above question. The four items were a) The mathematics curriculum should be seen b y pupils as relevant to their future lives, b) The mathematics curriculum should incorporate elements of the cultural histories of all the people of the region, c) The mathematics curriculum should be experienced as "real" by all children, and d) The mathematics curriculum should resonate, as far as possible, w i t h diverse home cultures. O n a 5-point scale (0 = strongly disagree, 1 = disagree, 2 = neutral, 3 = agree, 4 = strongly agree), the pedagogic adviser agreed w i t h statements (b) and (c) while strongly agreeing w i t h statements (a) and (d). 164 III.5 The university faculty (1) What purpose was the present mathematics curriculum designed to serve? There is no clearly stated government policy o n mathematics education i n Cameroon. However, statements suggesting the purpose of mathematics education i n Cameroon could be gleaned from the introduction to the secondary school schemes of work, a publication of the Cameroon General Certificate of Education (GCE) Examination Board which state: "(i) to train students to become mathematically literate; (ii) to prepare students for competition to enter the market of 'small jobs' i n Cameroon; and (iii) to prepare them for competitive national examinations such as the General Certificate of Education (G.C.E) and the Baccalaureate." ( C G C E Board, 1994, p. iv). According to the university faculty, this is a big shift from the initial purpose of mathematics literacy on the eve of independence. H e states: "The initial purpose, ...of the system of mathematics that persisted i n this country was that brought by the British system w h i c h was meant to train people to be able to assist the administrators for taxation. So it was meant to assist government to be able to count and carry out numeric aspects of the administration. A n d that was emphasized i n the curriculum i.e. numeracy was the main objective. This is not the case currently as the focus is on preparing students for national certificate examinations such as the G.C.E and the Baccalaureate w i t h the hope that they w i l l be able to study i n foreign universities. The importance attached to these exams has impacted not only what is taught but h o w it is taught." (2) To what extent has this purpose been served? Following the three statements gleaned from the schemes of work as outlined above, it could be argued that there is some success w i t h regard to statements (i) and (iii). A s far as statement (ii) is concern, there is very little evidence of success as there are many Cameroonian graduates unable to gain employment w i t h their mathematics education. A d d i t i o n a l statements gleaned from yearly presidential addresses 165 to the nation have maintained that school mathematics should remain a compulsory subject in primary and secondary schools. What these proclamations fail to do is elaborate or provide some guidelines on what is meant by "compulsory" and exactly how the teaching (and learning) of mathematics will be carried out. The university faculty believes that government's failure in providing any guideline is only exacerbating the situation as far as mathematics education for development is concerned. He maintains, "Yes, government policy says mathematics is compulsory at all levels, in fact primary and first level of secondary education. And to make this compulsory nature meaningful to the students, it is one of the three subjects (the others being English and French languages) taken at all competitive exams. But outside that, you find students going to high school who never did mathematics. You find children successful who dropped mathematics in Form 2 (the second year of secondary school). So what really are we talking about compulsory? The notion of compulsory is pen and paper activity. It doesn't make sense.... However, the usefulness and the importance in real life is what have caused government to think that everybody should do mathematics. .... The question comes back: does school mathematics actually give the people the mathematics knowledge they need to survive? .... If we want to make school mathematics compulsory, it is necessary to completely overhaul the mathematics curriculum in Cameroon so that school mathematics should tie more to real life situations and people will see the usefulness." (3) What flaws in the curriculum itself could have undermined its effectiveness? The university faculty sees the adoption of alien curricula by Cameroonian educational authorities as one of the main reasons why the current school mathematics curriculum is suitable for producing individuals with the much-needed skills in the Cameroonian society. He points to the generation gap that has emerged as a result of the disconnection between the mathematics being taught in Cameroonian 166 schools today w i t h the mathematics that was taught at the d a w n of independence. He states, ".. .the b i g drift from past mathematical processes, what was taught, to the recent so called modern mathematics, creates a big gap between old generation and the new generation. A n d we need to bridge this gap by making mathematics that is interwoven. A n o l d man w h o went through school i n the 1960s should be able to pick u p his grandson's arithmetic exercise book and say ok, o yeah, I looked at this. Where is your difficulty? But that linkage is not there." (4) How could theseflawsbe eliminated and/or remedied? The university faculty strongly believes i n the need for educational reforms and not just curricula reforms. H e is worried about the continued use of outdated curricula and the heavy reliance of the populace for employment from the government. H e strongly believes a curriculum that caters to the need of Cameroonians by training them to become more self-reliant w i l l bring about development. To this end, he sees the importance of developing a more responsive curriculum without relying on examinations to dictate what should be taught. "The West has m o v e d away from this (modern mathematics) because if I look at the G C S E program i n London, it caters more for the people there. But that's not what we are implementing here. We are still w i t h the o l d curriculum of the 1960's from Britain, w h i c h is a shame. Every person w h o goes to school here is looking forward to having a whitecollar job and this has proven to be very difficult. So this is a clear indication that we urgently need to reform, we need a reform; we need to change our entire educational system, the entire curriculum, to make it functional. A s long as we are still thinking of competition, making our students go and study abroad, make sure that they have certificates that are recognized outside Cameroon, we w i l l continue to teach this system that does not really cater for the development of Cameroon. This is one of the causes of unemployment, ... many people loitering around w i t h degrees because the education they've acquired is not functional." 167 (5) What mathematics should be taught to Cameroonian children? Wliy? The b u z z w o r d i n his response to this question was "functionality". The need to develop and streamline the school mathematics curriculum that emphasizes critical and creative thinking, and raising the awareness of the presence of mathematical objects and practices i n the daily lives of the students and w i t h i n the immediate surroundings. H e summed this up w i t h the following statements: "But that is where we are faced w i t h the need to streamline our school mathematics curriculum to the extent that people should be functional. Functional mathematics is what I w o u l d emphasize. A n d using mathematics to promote critical and creative thinking, and getting children to see mathematics as a language, a tool w i t h its o w n aesthetic values that should be emphasized is crucial...To this end, a Cameroonian developed curriculum w i t h a touch of local notions and applications ... w i l l cause learners to not only see the relevance of the subject but to develop a liking for it. More meaning w i l l also be given to real life mathematical concepts." (a) How important are the following qualities when deciding on a mathematics text? O n a scale of 0 - 4 (0 = not important, 1 = slightly important, 2 = important, 3 = very important, 4 = crucial) all the five items were rated as either very important or crucial, w i t h both 'correctness of content' and 'preparation for what the students w o u l d have to learn' being seen as crucial. (b) To what extent would you agree or disagree with the following statements? O n a 5-point scale (0 = strongly disagree, 1 = disagree, 2 = neutral, 3 = agree, 4 = strongly agree), the university faculty strongly agreed that it was important for the mathematics curriculum to be seen by students as relevant to their future lives and therefore it was important for the mathematics curriculum to resonate as far as possible w i t h the diverse home culture of the students. H e equally agreed w i t h the incorporation of cultural elements into the mathematics curriculum and making 168 mathematics real although this was not as highly rated as m a k i n g it [mathematics] relevant to students' future lives. 5.2.4 Curriculum Reform Process The process through w h i c h the curriculum of schools is designed and implemented is an important strategic concern i n curriculum development theory because it has to do w i t h the choice of the major arenas or locales where key decisions are made, the involvement of different categories of persons, the orchestration of material and other resources, and so forth. It can be said that the curriculum development process that is formulated or institutionalized i n a school system at a specific time determines, to a large extent, the quality of the curriculum products and their dissemination or implementation. The curriculum development process i n Cameroon utilizes top-down procedures and engages mainly persons w h o are i n line or authority positions. Hence exploring the curriculum reform process from the point of view of the stakeholders is essential i n understanding the requirements i n bringing about successful curriculum change. Questions i n this area therefore focused on the stakeholders' knowledge of curriculum reform, and curriculum development process i n Cameroon w i t h particular attention to mathematics. The three m a i n questions were: (1) W h o are those involved i n the curriculum development and implementation process taking the case of school mathematics?, (2) W h o are those involved i n selecting official mathematics texts to use i n secondary schools i n Cameroon?, and (3) If y o u were to be charged w i t h the responsibility of designing the mathematics curriculum for secondary schools i n Cameroon, what are some of those things y o u w o u l d consider or pay attention to? 169 IV.l The (limited) group Questions i n this area were intended to elicit the teachers', pedagogic adviser's and university faculty's knowledge of the curriculum reform process. The questions focused on the actual process and it was felt that students w o u l d lack the requisite knowledge and experience never having participated i n the reform process. Knowledge of the curriculum reform process varied w i t h each stakeholder (teacher, pedagogic adviser, and university faculty) and was indicative of his or her level of involvement i n the reform process. The teachers professed limited knowledge while the university faculty and the pedagogic adviser expressed extensive knowledge of the whole reform process. This variation i n the knowledge level relating to the curriculum reform process is not surprising. This is because curriculum reform i n Cameroon is a top-down, centre-to-periphery model wherein the ministry personnel such as the national and provincial inspectors, pedagogic adviser, university expertise such as teacher educators, subject specialists and a very limited number of school teachers are invited to participate. Overall, only one i n four teachers had some form of experience i n curriculum reform while the pedagogic adviser and teacher educators had extensive experience i n this process. This result is shown i n Table 5.1. Table 5.1: Experience in Curriculum Reform by Stakeholders No Experience 1-5 years Experience 6-10 years > 10 years Teaching Experience Experience Experience Teacher A X 15 Teacher B X 12 Teacher K X X 22 Teacher N Pedagogic Adviser 12 X 20 X Teacher Educator 170 30 Table 5.1 does not include the students as they are usually excluded from participating i n curriculum reform. The results seem to suggest that experience i n curriculum reform is related to teaching experience. In other words, the longer a stakeholder has been teaching, the higher the chances of him/her being invited to participate i n curriculum reform by the ministry of education. IV.2 The teachers Overall, the majority of teachers have little or no experience i n curriculum reform matters. A s a result, teacher's knowledge of curriculum development and innovation is very poor. Teachers' responses to the following questions seem to support this assertion: (1) Who are those involved in curriculum reform process in Cameroon? The system of government is very centralized and w i t h the curriculum being considered a political document, the government through its ministry of national education strictly controls any changes to it. One reason for this control is because curriculum development has been limited to the writing of syllabuses i n the traditional school subjects and to the requirements of examination systems such as the General Certificate of Education or the Baccalaureate. By controlling the reform process, the government hopes this w i l l ensure quality control i n the examinations. "The government, because there is an examination to be taken and then the government thinks that the w a y to measure the level of mathematics is by taking that examination. A n d so it is the government or the ministry of national education w h o does the planning of what goes into the curriculum document. When teachers are to be invited, the number is usually very insignificant." (Teacher K, C C C ) 171 (2) Who are those involved secondary schools in in selecting official mathematics texts for use in Cameroon? A crucial problem related to textbooks i n Cameroon, is their selection. National inspectors and other higher educational authorities are required to recommend a list of textbooks for each school level on a yearly basis. Since the book market i n Cameroon focuses essentially on students, both local and multi-national publishers are engaged i n a cutthroat competition at the Central Office of the ministry of education for their books to be included i n the annual national book list. Some of the teachers i n this study expressed dismay i n h o w this process of textbook selection is conducted: "In principle, it is the ministry of national education that selects the textbooks.... I think the teachers w i t h the approval of the inspectors should be the ones to come out w i t h the list of recommended textbooks. ... But I k n o w that what happens at times is that somebody writes a book and k i n d of finds out a w a y and before y o u k n o w the book is on the list whereas the book may not be good." (Teacher A , CCAST) "It is the government or the ministry of national education. This is unfortunate that i n Cameroon there are so many textbooks but they way the textbooks are recommended, it depends on who y o u are as the author or what I can say i n quote 'pushing your way through to the person w h o w o u l d recommend them by giving h i m what he wants i n order that the text should be there on the list. N o t necessarily that the text is the required text or it is very, very important.'" (Teacher K , CCC) "The ministry does the selection of the official texts. But what many schools do is ask their teachers to suggest books w h i c h they think are good and then these books are added to the official list from the ministry." (Teacher N , C C A S T ) 172 (3) If you were to be charged with the responsibility curriculum for secondary schools in Cameroon, you would consider or pay attention of designing the mathematics what are some of those things to? A l l the teachers were critical of the examination-focused nature of the curriculum and expressed the need for a responsive curriculum where examination is only one form of assessing success. The teachers believe that there is a need for the curriculum to include topics that facilitate the development of other non-academic life skills. Such skills are highly needed, they maintain, i n a country like Cameroon whose economy is agricultural. T w o of the teachers echoed the need for the curriculum to be reflective of current technological advancements. The following statements suggest their vision of the k i n d of curriculum needed i n Cameroon: "I w i l l try to include topics that w o u l d develop i n the students, problem-solving skills especially problems encountered i n their daily lives. W h i c h means it should include topics which even if the student were not to continue w i t h mathematics i n the high school, they should be useful, that student should be useful; it should also be topics that w o u l d develop computation skills i n the students; it should also be topics that w o u l d develop electronic skills if they have to get into the computer w o r l d they should have them even if the uses are not readily available. I w o u l d also like to include topics that w o u l d help them trace some cultural roots for some of the maths concepts such as modular arithmetic." (Teacher N , C C A S T ) "Just as the slide rule died off, I w o u l d w i s h that this four figure tables be thrown very far. Those are obsolete, but y o u still find teachers trying to bring out the idea oifour figure tables. I w o u l d love to see that taken off because we are i n the computer age, the computer has come and we don't need those things again. I think that is a major issue." (Teacher B, C C C ) "We need to create a curriculum for people who are not going to continue to university. ... By extending topics like modular arithmetic to include various local counting systems, traditional calendars, and other local kinds of measurement w o u l d not only make mathematics 173 meaningful, but w o u l
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Stakeholders’ receptiveness to an ethnomathematics curriculum foundation : the case of Cameroon Kang, Henry 2004
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Title | Stakeholders’ receptiveness to an ethnomathematics curriculum foundation : the case of Cameroon |
Creator |
Kang, Henry |
Date Issued | 2004 |
Description | The purpose of this study is to assess the curriculum stakeholders' receptiveness to a curriculum built on an ethnomathematics foundation. The stakeholders who participated in this study were secondary school students, secondary mathematics teachers, pedagogic personnel and teacher educator. The students and the teachers were from two secondary schools in Cameroon, one of which was public and the other private. Data collected for analysis include audiotaped interview transcripts, questionnaires and field notes from classroom observations of the teaching of an ethnomathematics unit. From an analysis and interpretation of the data, a picture emerged of the stakeholders' level of interests and concern about adopting an ethnomathematics curriculum foundation. Findings from this study indicate that the stakeholders are generally receptive to an ethnomathematics curriculum but are also concerned about the demands such a curriculum would have on the cultural knowledge background of those in the mathematics classroom. The study also indicates that the stakeholders' encounter with an ethnomathematics approach can help them develop a broader view of mathematics and raise awareness of the presence of mathematical processes in cultural practices. The study notes that the stakeholders demonstrated both situational and actualized interests that were complex and not fixed. When a particular cultural activity facilitated mathematics teaching and learning, the stakeholders exhibited actualized interest to an ethnomathematics curriculum. When the lesson activities demanded much from the stakeholders in terms of cultural background knowledge and the teaching and learning implements, the stakeholders showed situational interest. The study also suggests that stakeholders' interests in an ethnomathematics curriculum are complex and interrelated, and are influenced more by external factors than by a given phenomenon. The findings also suggest that stakeholders had some concerns regarding an ethnomathematics curriculum and that these concerns were more complex and varied with each stakeholder according to how each viewed her/his role in the education process. The study's analysis of the stakeholders' receptiveness provides useful and important implications for relevant mathematics education, teacher education and above all, curriculum reform. It also highlights the importance of involving all those concerned with the education process to play major roles in the curriculum development process. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2009-12-23 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0055159 |
URI | http://hdl.handle.net/2429/17325 |
Degree |
Doctor of Philosophy - PhD |
Program |
Curriculum Studies |
Affiliation |
Education, Faculty of Curriculum and Pedagogy (EDCP), Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2005-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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