Mathematics Connection

Volume 8

August 2009

Mathematics Connection Published by The Mathematical Association of Ghana (MAG) ISSN 08554706

Volume 8, 2009             

MATHEMATICS CONNECTION aims at providing a forum to promote the development of Mathematics Education in Ghana. Articles that seek to enhance the teaching and/or learning of mathematics at all levels of the educational system are welcome.

Executive Editor Prof. D. K. Mereku, Department of Mathematics Education (UEW), Winneba Managing Editor Dr. S. K. Aseidu-Addo, Department of Mathematics Education (UEW), Winneba Editorial Board Prof. N. K. Kofinti, Faculty of Social Sciences, Methodist University, Accra Prof. B. K. Gordor, Department of Mathematics (UCC), Cape Coast Dr. P. O. Cofie, Department of Mathematics Education (UEW), Winneba Dr. C. O. Okpoti, Department of Mathematics Education (UEW), Winneba Dr. I Yidana, Department of Mathematics Education (UEW), Winneba Dr.M. Nabie, Department of Mathematics Education (UEW), Winneba Mrs. B. Osafo-Affum, GES District Directorate, Asuogyaman, Akosombo

Enquiries can be forwarded to

Managing Editor, Mathematical Association of Ghana, C/o Department of Mathematics Education University of Education, Winneba P. O. Box 25, Winneba Tel. +033 2320986 E-mail: [email protected]

ISSN: 0855-4706 Typeset: Kofi Mereku, UEW Published by the Mathematical Association of Ghana

 Mathematical Association of Ghana (MAG) 2009 The points of view, selection of facts, and opinions expressed in the MATHEMATICS CONNECTION are those of the authors and do not necessarily coincide with the official positions of MAG.

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Mathematics Connection Volume 8, 2009

Published by the Mathematical Association of Ghana

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Contents Mathematics Connection Vol. 8 2009 Gaps between inside school and out-of-school mathematics in Ghana1 Davis, E. K., Bishop, A. J. & Seah, W. T. ...................................................................................... 1 Comparative analysis of performance of eighth graders from six African countries Asabere-Ameyaw, A. & Mereku, D. K. ........................................................................................ 17 Pre-service teachers’ content knowledge for teaching basic school mathematics Ntow, F. D., Tackie, N. A. & Sokpe, B. Y. ................................................................................... 28 Socio-economic background and the relative efficacy of self drills on factual recall and students’ achievement in mathematics in senior secondary schools in Nigeria Olagunju, O. P. & Owoyele, J. W. ................................................................................................ 33 Beliefs, attitudes and self-confidence in learning mathematics among basic school students in the Central Region of Ghana Ampadu, E. ................................................................................................................................... 45 Impact of in-service education and training programmes on teachers’ capacity to teach for the development of literacy and study skills in social studies Amedzake, G. A. & Amuah, P. ..................................................................................................... 57 An Investigation into factors influencing the choice of business education in two tertiary institutions in Nigeria Obijole, E.F. .................................................................................................................................. 69 Report on the baseline survey of untrained teachers in Ghana Osei-Anto, S., Mereku, D. K., Aboagye, J. K., Akwesi, C., and Kutor, N. K............................... 75

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Mathematics Connection, Vol. 8, 2009

Gaps between inside school and outofschool mathematics in Ghana

Davis, E. K., Bishop, A. J. & Seah, W.T1 Abstract Researchers are now pointing to the fact that mathematics that was once regarded as culturei and value free is no longer regarded as such. Views about the nature of mathematical facts being absolute and unquestionable have changed in recent times. Mathematics education researchers have therefore highlighted the need for the recognition of children’s out-of-school mathematical practicesii in scaffolding their learning in mathematics classroom. This paper reviews theories and research on cultural nature of mathematical knowledge. It also discusses the possible reason why Ghanaian primary school mathematics curriculumiii does not reflect much of the out-of-school mathematical practices and the way forward. Keywords

nature of mathematics, mathematics and culture, ethnomathematics, mathematics curriculum

Introduction: Mathematics as a cultural object Mathematics is a subject that is taught in schools all over the world. Like all other countries, Ghana, a country located on the West coast of Africa with a long history of formal education (since 1529) also includes mathematics in her school curriculum. However, what most mathematics educators in Ghana have perhaps not thought about is “why the local language is accepted in the formal school system, at least for the first three years of formal education as a medium of instruction and even beyond in some cases (as a school subject) but not local mathematics?” Some mathematics education researchers have highlighted the cultural nature of mathematics. Presmeg (1998) for instance has said that in the last few decades, many writers in the ethnomathematicsiv movement have argued that, "mathematics is a cultural product which needs to be acknowledged as such in classroom both for the purpose of meaningful learning of the subject in developing countries …” (p.320). Presmeg suggests that culture of both pupils and the teacher could be a useful tool in mathematics teaching and learning. Other researchers are also of the view that the successful study of mathematics must take into account the many and varied experiences with which children come to school (see, for example, Charborneau & JohnSteiner, 1988; Fleer & Robbins, 2005). For most countries that were colonized some of the things that came along with colonization such as religion have been contextualized but not mathematics education, in most cases. A look at the nature of church services in Christian churches (especially 1 Alan J. Bishop is Emeritus Professor and Wee Tiong Seah is a senior lecture both at the Faculty of Education, Monash University, Australia. Mr. Ernest Kofi Davis is a PhD student in mathematical education at the same university.

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Gaps between inside school and out‐of‐school mathematics in Ghana Davis, E. K., Bishop, A. J. & Seah, W.T .

African Initiated Christian Churches) in most African countries in general and Ghana specifically today, for instance, show that the nature of service has been shaped by the local culture. The literature in this area shows that the Bible is read with an African cultural background rather than western cultural background (Lettinga, 2000). Today one can see local drums, hear local gospel music other than the orthodox ones being sung during worship services. Christianity in Africa today has witnessed a very fast growth rate as a result of its being contextualized. The same however cannot be said about mathematics education after independence, even though both western mathematics and Christianity came together through the introduction of formal education during the colonial era. In situations where policy even emphasize the need to include cultural dimensions into mathematics education the will to execute it through school curriculum is absent (Kaleva, 2004). We suggest that this is probably because some developing countries still hold the notion that mathematical truth is absolute. Bishop (1988) asserted that "mathematics must be understood as a kind of cultural knowledge, which all cultures generate but which need not necessarily 'look' the same from one cultural group to another" (p180). He further postulates six fundamental activities which are both universal and appear to be carried out by all cultural group ever studied and also necessary and sufficient for the development of mathematics. These activities include counting, locating, measuring, designing, playing and explaining (Bishop, 1988). To the naturalist like Alan Bishop "Mathematics, as cultural knowledge, derives from humans engaging in these six universal activities in a sustained, and conscious manner" (Bishop, 1988; pp. 182 &183). Bishop (1988) further distinguishes two types of Mathematics as Mathematics with big “M” and mathematics with small “m”. He argues that that Mathematics with big “M” is the western/international Mathematics that students encounter in schools whilst mathematics with small “m” is the one that pertains to the local cultures, which is otherwise called out-of-school mathematics in this paper. It is clear from the standpoint of the naturalist that all cultures create their own mathematics by engaging in activities such as counting, measuring and designing. However some individuals and even societies undervalue some of the mathematics within cultures or assign more value to some of the practices within some cultures than others (Abreu, 1993; Abreu, 1995). Presmeg (2007) refers to the former situation as “historiography” and the latter as “valorization”. According to Presmeg (2007), historiography looks at some of the world's mathematical systems that have been ignored or undervalued in mathematics classroom whereas valorization is the social process of assigning more value to certain practices than others. Literature has shown that most children denied the existence of, or devalued, mathematics as used in practices which they encountered within out-of-school settings (Abreu, Bishop, & Pompeu, 1997; Abreu & Cline, 1998). This issue is critical in any future attempt to implement meaningful mathematics education in a developing country like Ghana, especially in the implementation of mathematics education programs that ensure bridging of gap between in-schoolv and out-of-school mathematics. Stakeholders’ notion of the value of out-ofschool mathematics within the Ghanaian culture will determine their willingness to use it to support the development of in-school mathematics in the classroom setting. Premeg (2007) asserts that until Abreu's (1993, 1995) research brought up this topic (valorization), "the value of formal mathematics as an academic subject was for so long 2


taken for granted that it became a given notion that was not culturally questioned"(p.141). Presmeg further asserts that "if ethnomathematics as a research program is to have a legitimate place in broadening notion of what counts as mathematics and of which people have originated these forms of knowledge, then issues of valorization assume paramount importance" (p.141). Related to the issue about the cultural nature of mathematics is the cultural nature of the mathematics curriculum. Literature points to the cultural nature of the mathematics curriculum (Bishop, 1988). After analysing five approaches to curriculum by Howson, Keitel and Kilpatrick (1981), namely the Behavourist Approach, New-Math Approach, Structuralist Approach, Formative Approach and Integrated-Teaching Approach, Bishop (1988) had proposed a sixth approach – the Cultural Approach – to mathematics curriculum. He postulates five characteristics of an enculturation curriculum or the cultural approach to mathematics curriculum as: 

Representing the Mathematical culture, in terms of both symbolic technology and values. Here the need to attend explicitly and formally to all values (objectism, control, mystery, rationalism, openness and progress) of the mathematical cultures is emphasized. Most especially the need to pay attention to values such as rationalism, openness and progress in mathematics.



Objectifying the formal level of culture of Mathematics. Here the emphasis is on the connection between the formal level of mathematics and the informal level of mathematics and their link to the technical level of Mathematics (as in pure Mathematics). The curriculum must reflect connection between Mathematics and present society as well as mathematics as a cultural phenomenon. Preparation for technical level of Mathematics is not the main aim of this curriculum. The structure used in the development of Mathematical ideas and concepts are to be based on six universal activities discussed above. According to Bishop, as Mathematics is part of learners’ culture it will be important to reflect that cultural basis in the structure of their Mathematics curriculum.



Being accessible to all children. Here the emphasis is on the need for Mathematics curriculum to be designed to meet the learning needs of all learners but not just the few who want to pursue mathematics at higher levels. This is reflected in Bishop’s assertion that “enculturation must be for all” (p. 96). Thus the ‘top-down’ approach (as described by Bishop) does not help students who either do not wish, or who are unable to go on to study further mathematics in school. This curriculum must however take cognizance of individual difference in learners and therefore provide opportunities for them to pursue certain ideas further than other children based on their interest and background. Related to this issue of accessibility is the fact that the curriculum content must not be beyond the intellectual capabilities of the children, nor must the material examples, situations, and phenomena to be explained, be exclusive to any one group of the society. Thus the choice of what counts as mathematics to be included in school curricula and also the need to create curriculum structures to cater for individual needs of children become important here.




Emphasizing Mathematics as explanation. Here the emphasis is not mainly on doing with very little emphasis on explaining but on explaining as well. Bishop asserts that Mathematics as a cultural phenomenon derives its power from being a rich source of explanations and that feature must shape significant understanding to emerge from the enculturation curriculum. According to Bishop, the power of explanation will only be conveyed if the phenomena-to-be explained are accessible to all children, and are ‘known’ by them but remain unexplained. He asserts further that both physical (natural and man-made) environments, and social environments, constitute the source of such phenomena. The need for the mathematics curriculum to be based in the child’s environment and the child’s society is the message here.



Being relatively broad and elementary rather than narrow and demanding in its conception. Here Bishop proposed for the need for a variety of contexts to be offered since the power of explanation, which is derived from Mathematics’ ability to connect unlikely group of phenomena, needs to be fully revealed. He argues that the constraint of a finite time for schooling means that, if breadth of explanation and context is to be an important goal, then mathematics content must be relatively elementary. Here he does not propose merely simple arithmetic content or Fun Maths or only childish games. The basis of his argument is that if ‘enculturation’ is the goal, and if ‘explanation’ is the power of the symbolic technology of the culture, then undue complexities in that technology will fail to explain, fail to convince and therefore ultimately, fail to enculturate. The message here is that a good enculturating curriculum must take cognisance of explanation and context in its structure and this has implications for the nature of mathematics content to be taught.

According to Bishop these attributes of good curricula are very important in the education of all manners of children including future mathematicians. Related to the issue of cultural nature of mathematics curriculum is the cultural nature of mathematical concepts. Cultural differences in mathematical concepts: Fractions in the Ghanaian context The literature also points to the cultural nature of learners’ formation of mathematical concepts (Draisma, 2006; Pixten and Francois, 2007). For the purpose of this paper we shall use the concept of fractions as our main example. This is because many writers refer to the fact that the teaching and learning of the concept of fractions continues to be a major problem (Duedu, Atakpa, Dzinyela, Sokpe, & Davis 2005; Driscoll, 1984; Theunissen, 2005) and therefore continues to attract the attention of mathematics teachers and education researchers worldwide (see Meagher, 2002). According to Freudenthal (1983), for instance, the concreteness of fractions does not end with breaking a given whole into parts but also in comparing objects which are separated from each other or experience. This he termed “fraction as comparer” and gives examples such as “in a room there are half as many women as men”, “the bench is half of the height of the table” among others. Related to this theme of fraction as a comparer is the issue of fractions in everyday language such as “half as many”, “half as much” which by

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analogy implies “equally as”, “twice as …”, etc. in comparing quantities and values of magnitude. This shows the cultural nature of the concept of fractions. It also raises a question about the universality of this concept (as most people might think). It is possible that different cultures may interpret this western view of halves in everyday language differently in different context. In the Ghanaian context, for example, it is common for people to describe three-fifths auditorium full of audience as half (“fã”) and two-fifths also as “fã”. Thus in the Ghanaian culture ‘a half’ does not necessary mean equal halves. Hence ‘a half of’ may not necessarily be interpreted as “twice as”. Also, the issue of fair share in fraction as fracturer (i.e. as part whole relationship) in the school context may not be the same as in the home context in the Ghanaian context and we suppose in many cultures as well. It is possible for a junior colleague for instance to accept less than a half of a parcel shared between him/her and a senior colleague as a fair share. Thus fair share may not always imply equal share as in the classroom situation. In some context it may be interpreted as equal share and in another context it may be interpreted as what is shared satisfactorily amongst those who are sharing. Gaps between inside school and out-of-school mathematics in Ghanaian school system Clearly, thus, gaps appear to exist between the mathematical practices that Ghanaian school children encounter at the inside school (ISM) and those they encounter in the outof-school (OSM) settings. In the home context it is very common for adults to describe three-quarters bucket full of water as “insu sin” (in Fante dialect), and three-fifths bucket full also as “insu sin”. "insu" means water whilst "sin" means less than a whole. Thus implying that in the home context there is no differentiation of fractions. That implies that half a bucket full of water, three-quarters bucket full of water etc are all described as “sin” which means less than a whole but there is a contrast with the school situation where children have to differentiate fractions and even compare them. In the home context children make use of empty tins in measuring (i.e. “cups”, “olonka”, etc). The metric system of measurement is not too often used in the local markets both in urban and rural settings. Financial news on national radio stations usually quote prices of commodities in these local units (olanka etc) but unfortunately these local units have no place in the classroom mathematics curriculum. A look at the mathematics curriculum for fractions in Ghana shows that fractions is introduced at grade 2 (Ministry of Education, 2001, p.28; Ministry of Education, 2007, p.42). It begins with the concept of half and the teaching strategies emphasize the use of cultural artefacts such as a loaf of bread, orange, a piece of string etc. However this trend does not go beyond grade two. The curriculum then suggests the introduction of onefourth and the teaching approach emphasize the use of paper folding (Ministry of Education, 2001, p 29). The approach becomes gradually abstract as one moves up the grade levels. The approach suggested in the development of the concept of fractions in this curriculum is very similar to what Freudenthal (1983) describes as “fraction as fracturer”. It begins with part(s) whole relationship using concrete objects and then becomes abstract. This approach limits children’s notion of what fractions are since according to Freudenthal (1983) it leads to the development of only proper fractions. The


use of the out-of-school knowledge acquired by children through their culture in sharing is not mentioned at all. The teaching strategy suggests the use of equal sharing of things between people. This in the opinion of the authors may create conflict in the minds of children since in the home context sharing between two people does not necessary mean sharing into equal parts (as already noted). The fact that lessons in fractions are deliberately designed to remain silent on children’s prior knowledge on sharing does not necessary mean children will automatically unlearn what they already know. They may accept the new rules for sharing as another way of sharing which is peculiar to the school and therefore exacerbate their view of school mathematical practices as different from home or reject it. In the latter case these pupils may not pay attention to equal divisions in partitioning of a whole for instance and the school may end up branding such pupils as failing pupils. One is therefore not sure whether some of these pupils who are branded as failing are actually failing or it is the school Mathematics curriculum which is failing. In the next section we would attempt to suggest a possible reason for gaps between ISM and OSM in Ghanaian primary schools. A Possible Reason for Gaps between Inside School and Out-of-School Mathematics Mathematics education researchers have addressed the issue of mathematics concept formation from different perspectives. In this paper we will look at two of those that have relevance to this paper namely Skemp’s (1987) and Burn’s (1992) explanation of concept formation in Mathematics. Skemp (1987) emphasizes the need to provide children with known experiences in the process of mathematical concept formation. Skemp explains further that it is possible, for example, to teach Ghanaian students the history of Europe before the history of Ghana without the students having problems understanding these two areas but in mathematics the situation is very different. Students will find a concept of algebra very difficult if their knowledge in arithmetic is very weak. Burns (1992) stresses the importance of children’s experience in the real world in the learning of mathematical concepts. He explains further that children attain equilibrium when their understanding is based on reality rather than perception and that there is a continuous interaction between mental conceptual structures and environment at the state of equilibrium. The implication here therefore is that children’s previous experiences play a vital role in the successful formation of mathematical concepts. Burns (1992) further suggests three other factors that influence students’ mathematics learning, these being maturity, physical experience and social interaction, and the “process of equilibrium coordinates these three factors” (p.28). A look at the literature by Skemp and Burns show that even though both of them mentioned the need for previous knowledge in the process of concept formation, the former’s explanation of prior knowledge did not make the issue of culture explicit. This has been characteristic of the western view of mathematics concept formation (see Sutherland, 1992). Culture seems to have no place in the process of concept formation (Barnett & Dickson, 2003; Bright, 2003). Concept formation therefore seems to concentrate on factors that are internal to the learner (learner’s cognition, maturation etc) with no emphasis on the culture (Piaget, 1953, 1954; Inhelder & Piaget, 1958). This approach to concept formation seems to influence the Ghanaian system very much. The tendency is that in the process of mathematics concept formation the communicator of

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the concept may emphasize the innate ability of the learner and neglect social and cultural factors, which equally play vital roles in the process of concept formation (Vygotsky, 1987). A look at a lesson observation on the teaching of perimeter as reported by Mereku (2004) succinctly describes a typical situation in curriculum delivery in the Ghanaian classroom. In this lesson Mereku reports of a grade six teacher who introduced the concept of perimeter by reviewing concepts of polygons using cut-out shapes such as squares and hexagons. The teacher showed each shape and asked pupils to describe the properties, such as the number of sides, square corners or right angles. The teacher then drew two rectangles on the chalk board and wrote down the formula for perimeter, ‘p = 2(l + w). Through questioning she guided pupils to identify lengths and widths of the rectangles and then asked them to substitute these into the formula to calculate the perimeter. The teacher worked one or two examples and asked for some volunteers to try some more on the chalkboard. The teacher then wrote five similar exercises from the mathematics textbook on the chalkboard and asked pupils to copy in their notes books to answer them as homework. A look at the lesson presented shows clearly that concept formation in measurement pays very little attention to culture. Unlike Primary One where a lot of outof-school context are used in the development of the concept of measurement the reverse is the case in this lesson (Ministry of Education, 2001; Ministry of Education, 2007). Thus neglect or probably ignorance of the role of culture of pupils and teachers on pupils’ mathematics learning may have contributed to the gaps between OSM and ISM in Ghana. In the next section we shall attempt to discuss some of the OSM in the Ghanaian society. Mathematics in the Ghanaian society Like other countries in sub-Sahara Africa (see Zaslavsky, 1978), there are numerous mathematical practices in the Ghanaian society, which may support mathematics learning in schools if they are employed in mathematics teaching in the classroom. Our argument on mathematics in the Ghanaian society will be based on the six fundamental mathematical activities proposed by Bishop (1988) which are universal in that they appear to be carried out by all cultural groups ever studied and also necessary and sufficient for the development of mathematics to highlight some of the mathematics within the Ghanaian society. We wish to emphasize that in most cases a number of mathematical activities are embedded in any one activity. It is possible for instance for a learner to be taught both measurement and shapes in the process of designing. In other words, it is rare to find somebody just learning to measure for the sake of it. Learning (in out- of- school context) in the Ghanaian society is mainly contextual in nature. A dressmaker for instance will teach an apprentice how to measure during the process of designing the dress. The dressmaker will not teach the apprentice how to measure at one time and how to apply the measurement skill in designing at another time. Our explanation will include counting, locating, measuring, designing and playing games. Explanation of all the activities listed above and other phenomenon are usually done by means of verbal and gesture explanations, instead of being implicit or hidden. To start with, the counting system in twi language which is shared by the Akans in Southern Ghana and some of the languages spoken in the Northern parts of Ghana make clear their base ten structure. This applies to the number words representing quantities


between ten and twenty. Ten in the Fante dialect (which is one of the dialects spoken by the Akans), for instance, is “du” eleven is “dubiako” which means “ten and one”, twelve is “duebien” which means “ten and two” etc. A lot of arithmetic also goes on in the Ghanaian society. Examples include the system of counting by some market women which is based on multiples of two and the use of arithmetic by children who sell candies, newspapers, cold drinking water, oranges etc in both rural and urban areas. These children (both schooled and unschooled) engage in the process of doing and explaining the arithmetic they go through in order to arrive at the total cost of items a customer purchases. Locating is also a common mathematical practice that is carried out in the Ghanaian society. Through the use of gestures and verbal explanations people explain the location of different objects and places. It is possible for an ‘unschooled’ person for instance to describe the location of a car park for example using vocabularies like turns, right, left, straight, north, south etc. Measuring is yet another mathematical practice in the Ghanaian society. Ghanaian market women make use of empty tins as a unit of measure, especially in the sale of grains, chilli among others. It may be possible for most Ghanaian rice sellers for instance to tell the number of empty margarine tins full of rice which make up one bag of rice. Liquids such as oil for example are usually measured using empty bottles. Thus some of these empty bottles are used as units of measure all over the country. It is very common for market women, for instance, to tell the number of beer bottles of oil in one 4.5- litre gallon. The use of the stretch of arms in the measurement of length, the measurement of time using the sun’s position in the day, and the crow of the rooster is also common. Hence measurement is done very much in the Ghanaian society. Designs especially the local designs used in the Ghanaian fabrics such as the “adinkra” (see Table 1) are very good examples of some of the regular and irregular shapes that students experience in school. "Adinkrahene" for instance could be used in the introduction of the concept of concentric circles. By so doing the learner will have the opportunity to learn the concept of concentric circles and also get the opportunity to learn about the Ghanaian culture. The patterns in “Kente”, a traditional Ghanaian clothing usually worn by traditional rulers is similar to the patterns in the twil weaving which Charinder (2002) used in her research on culture activity in mathematics. This is an indication that there may be a possibility of mathematizing (i.e. creating mathematics) from the process of making Kente. Apart from the Kente, twil weaving is also carried out in the Ghanaian society. The architectural design used in building houses in northern Ghana for example shows a lot of geometric shapes. The roof has the shape of a cone while the house is cylindrical in shape. In farms, the mounds in which farmers cultivate yams have the shape of a cone. Also the cultivation of palm trees for example usually follows a pattern. Farmers usually keep constant intervals between the trees. This can be used in the process of teaching concept of series for example. The collection of Adinkra symbols in Table 1 can be used for teaching concepts in geometry such shapes, congruence and symmetry (i.e. both line and rotational symmetry).

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Table 1 Symbol

Collection of Adinkra symbols and their meanings in Ghanaian society Adinkra label or Name

Interpretation

Meaning

Mframadan

Wind resistant house

Symbol of fortitude and readiness to face life's vicissitudes

Adinkrahene

Chief of the adinkra symbols

Symbol of greatness, charisma and leadership

Fawohodie

Independence

Symbol of independence, freedom, emancipation

Mpatapo

Knot of pacification/recon ciliation

Symbol of reconciliation, peacemaking and pacification

Source: Arthur Kojo, 2001

Games are yet another source of mathematics in the Ghanaian society. Some of the games in the Ghanaian society such as “Oware”, “Draught” and “Tumatu” are good examples of games that may be useful in developing a number of mathematical concepts and also prepare children for problem solving in mathematics. “Tumatu” for example could be used to develop the concept of addition (adding on). Figure 1 shows the diagram of the "Tumatu" game. This diagram is drawn on the floor. The game usually involves two or more people with players usually stepping in each of the regions starting from A on one leg to pick an object the player drops from O into the regions in a systematic manner. The winner of the game is the one who wins more of the regions numbered A to I. From this game it may be possible draw a child's attention to the fact that to find the total number of regions to be covered in order to get to region G for instance whilst the child is already in region C, the child could get the total number of regions between A and G by counting on from where the child is (i.e. C) instead of counting all from A. In this case the child will add the remaining four regions (D, E, F and G) on the three to get seven regions as the answer. This may help them to develop the concept of adding on in the learning of the concept of addition.


H

E O A

B

C

Figure 1

D

F

G

I

“Tumatu” game

Even though some researchers have highlighted the need for educators to take caution when introducing indigenous mathematical practices in the mathematics classroom, they concur with the need to use children’s OSM knowledge as an asset rather than a liability in the mathematics classroom (Seah, Atweh, Clarkson & Ellerton, 2008). Thus from the discussion so far on the mathematics in the Ghanaian society it is clear that there are numerous culturally relevant potential approaches to mathematics within the Ghanaian society – all of which could be usefully employed by teachers in the classroom. Even though Martin et al (1992) propose activities involving the use of draught in teaching arithmetic, a look at the teaching strategies proposed in the primary school mathematics syllabus in Ghana, however, generally gives very little attention to the use of these outof-school mathematics in mathematics learning in school (Ministry of Education, 2001; Ministry of Education, 2007). In most cases concrete materials from the environment (including cultural artifacts) are used in only primary one and two, as in the case of measurement for example (Ministry of Education, 2001, pp. 13-15, 30). This may be due to the influence of constructivist theories that play down on the role of culture in concept formation and rather create the notion that children need these materials only when they are operating at certain levels (concrete abstraction) of development (see Sutherland, 1992). If the aim of mathematics education is to help people to use mathematics efficiently within the society then the issue of what counts as mathematics to be learnt in schools should be looked at carefully in Ghana. The question that might come into the mind of our readers may be so what is the way forward for Ghanaians? In the next section we shall attempt to suggest what could be done to bridge the gap between OSM and ISM. The way forward Based on the characteristics of ‘cultural’ curriculum proposed by Bishop (1988), most especially with his emphasis on the connection between the formal level of mathematics and the informal level of mathematics, and their link to the technical level of Mathematics, Mathematics curriculum especially at the primary and the junior high school levels in Ghana must cater for the informal mathematics as well. It must also reflect the values prescribed in the ‘culture’ curriculum (i.e. rationalism, openness and progress), as much as possible. This is necessary because if mathematics educators believe that knowledge is a product of culture (see Bishop, 1991), then the culture of the people must be taken into consideration in the process of knowledge acquisition. Once

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this is done mathematics education in Ghana will not look more like inducting people into mathematics that is foreign to their own culture but also inducting them into their own culture as well. This will have implication on what goes into the Ghanaian mathematics textbooks. Developments of concepts have to start with what pertains in the Ghanaian culture and then eventually developed into school concept. In treating a topic like shapes for instance it has to begin with the “adinkra” symbols and then eventually take children through the shapes they usually encounter in the classroom. By so doing the element of horizontal mathematization, which is often missing during mathematics concept formation as it was seen in the example of the lesson, observed by Mereku (2002) will be present. This will go a long way towards improving students’ understanding of mathematics. It will also help students to value the mathematics they encounter in the out-of-school setting. They will be able to make the connection between the two sets of mathematics. Consideration of the values in ‘cultural’ curriculum (i.e. rationalism, openness and progress) will therefore require alternative means of assessment where different criteria would be used to assess the abilities of different groups of students based on their background and their learning needs. Thus in evaluating students’ learning the emphasis will be on the individual students but not looking at the performance of the student with respect to other students. Also teachers must be knowledgeable about children’s culture so that they can use children’s culture as an asset rather than a liability during mathematics lessons. Literature suggests that teachers’ knowledge about children’s culture does enhance children’s learning outcomes in mathematics (Presmeg, 1998; Draisma, 2006; Pinxten and Francois, 2007). Once the teacher is knowledgeable about the child’s culture the teacher will be able to find different pathways of presenting mathematics. Draisma (2006) for instance reports about his study in Mozambique where teachers made use of finger counting, which formed part of children’s culture to help them to add beyond the sum of their two fingers (ten). The study found out those students did better in addition using finger counting than using unstructured materials such as stones. This therefore has implication for mathematics teacher preparation. Mathematics teacher preparation must provide opportunity for the prospective teachers to learn about the mathematics that pertains in the Ghanaian culture, values (religious, ethical etc) in mathematics among others. Prospective teachers must therefore be introduced to Ethnomathematics, which has been described as mathematics of cultural practices (Presmeg, 1998) but which currently has no place in the mathematics teacher education curriculum at all levels in Ghana. Based on Bishop’s assertion that mathematics must be accessible to all and not just the future mathematicians, and considering the large number of school children who are not able to continue their education after Junior High School level (Ministry of Education, Youth and Sport, 2007), the content of mathematics should not be the same for all children from upper primary level onwards. Children who find school mathematics extremely difficult should not be bothered with topics such as indices and logarithms at the JHS level but should be taken through another mathematics curriculum, which has more of the OSM content. There should however be opportunities for students to be able to transfer from one stream to the other. For instance a student who is on the later mathematics curriculum (predominantly OSM) should be able to transfer to the other if that student is judged to be well prepared for it. In the same way a students who is on the


former (predominantly ISM curriculum with more difficult concepts) should be able to transfer to the later mathematics curriculum if the student is not able to cope with the curriculum. This will prepare them to use the mathematics they learn at school more efficiently after school. Conclusion It is evident from the literature reviewed that mathematics is a cultural object. As such the culture of both students and teachers must be respected in the teaching and learning of mathematics in schools. This point is buttressed by Scholnick’s (1988) assertion that: it is taken for granted that mathematics learning is embedded in a cultural context. Yet there are many cultural contexts within a society so every one does not approach adding and subtracting in the same way … it is equally important to specify the necessary bridging structures between home and school and between one concept and another that enable the child to learn mathematics. (p. 87)

There may be the need for stakeholders in education in Ghana to explore the possibility of helping teachers to use students’ out-of-school mathematical experiences in scaffolding students’ mathematics learning. We have shown that there are numerous culturally relevant potential approaches to mathematics within the Ghanaian society – all of which could be usefully employed by teachers in the classroom (as already noted). Teacher education in Ghana must therefore equip both prospective and in-service teachers with the requisite skills and knowledge to be able to do this, while the curriculum developers should also ensure that mathematics within the Ghanaian culture is fairly represented in the mathematics curriculum. Thus there is the need for curriculum developers to reconsider what counts as mathematics to be learnt in school. This may go a long way towards helping to improve Ghanaian school children’s understanding of mathematics and hence help them to improve upon their performance in mathematics both at the local and international levels. References Abreu, G. de. (1993). The Relationship between Home and School Mathematics in a farming Community in Rural Brazil. Unpublished PhD, University of Cambridge, UK. Abreu, G. de. (1995). Teachers' practices and beliefs in a community where home mathematics diverges from school mathematics. Paper presented at the PME 19. Abreu, G. de. & Cline, T. (1998). Studying social representations of mathematics learning in multiethnic primary schools: Work in progress papers on social representations, 7(1-2), 1-20. Abreu, G. de., Bishop, A. J., & Pompeu, G. (1997). What children and teachers count as mathematics? In T. Nunes & P. Bryant (Eds.), Learning and Teaching Mathematics: An international perspective. UK: Erlbaum.

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Arthur Kojo, G. F. (2001). West African wisdom: Adinkra symbols and meanings [Electronic Version]. Retrieved 20/9/2007 from http://www.welttemered.net/adinkra/htmls. Barrett, J., & Dickson, S. (2003). Broken rulers: teaching notes. In G. W. Bright & D. H. Clement (Eds), Classroom activities for learning and teaching measurement 2003 year book NCTM (pp. 11-14). Reston, V,: {Something’s missing here} Inc Bishop, A. J. (1988). Mathematical enculturation a cultural perspective on mathematics education. Dordrecht: Kluwer Academic Publishers. Bishop, A. J. (1991). Mathematical enculturation: a cultural perspective on mathematics education. Dordrecht: Kluwer Academic Publishers. Bright, G. W. (2003). Angle measurement. In G. W. Bright & D. H. Clement (Eds.), Classroom activities for learning and teaching measurement 2003 year book NCTM (pp. 25-26). VA, USA Inc. Reston. (check last few worss, refer to the entry above) Burns, M. (1992). How children learn mathematics. NY: Maths Solutions Publications. Charbonneau, M. P., & John-Steiner, V. (1988). Patterns of experience and the language of mathematics In R. R. Cocking & J. P. Mestre (Eds.), Linguistic and Cultural influences on learning mathematics (pp. 91-100). NJ: Lawrence Erlbaum Associate Publishers. Cherinda, M. (2002). The use of a cultural activity in the teaching and learning of mathematics: Exploring twill weaving with a weaving board in Mozambican classrooms. Unpublished PhD, University of The Witwatersrand, Johannesburg. Draisma, J. (2006). Teaching Gesture and Oral Computatio in Mozambique: Four case studies. Unpublished PhD, Monash University, Melbourne. Driscoll, M. (1984). What research says. Arithemetic Teacher 31(6), 34-35. Duedu, C. B., Atakpa, S. K., Dzinyela, J. M., Sokpe, B. Y., & Davis, E. K. (2005). Baseline study of Catholic Relief Services Assisted Primary Schools in the Three Northern Regions of Ghana Cape Coast: CRIQPEG, University of Cape coast. Fleer, M., & Robbins, J. (2005). 'There is much more to this literacy and numeracy than you realise...': Family enactments of literacy and numeracy versus educators' construction of learning in home Contexts. Journal of Australian Research in Early Childhood Education, 12(1), 23-41. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Holland: Reidel Publishing Company. Howson, G., Keitel, C., & Kilpatrick, J. (1981). Curriculum development in mathematics. Cambridge: Cambridge University Press. Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence London: Routledge & Kegan Paul.


Kaleva, W. (2004). The cultural dimension of mathematics curriculum in Papua New Guinea: Teacher beliefs and practices. Unpublished PhD, Monash University, Melbourne. Lattinga, M. (2000). African Christianity: A history of the Christian churches in Africa [Electronic Version]. Retrieved 24/7/2007 from www.bethel.edu/letnie/African chritianity/Author.html. Martin, J. L., Afful, E., Appronti, D. O., Apsemah, P., Asare, J. K., Atitsogbi, E. K., et al. (1993). Mathematics for teacher training in Ghana-student activities. Accra: Playpen Ltd. Meagher, M. (2002). Teaching fractions: New methods, new resources [Electronic Version]. ERIC Digest. Retrieved 23/1/08 from http://www.ericse.org/digest/dse02-01html. Mereku, K. (2004). Mathematics curriculum implementation in Ghana. Accra: Danjoe production. Ministry of Education. (2001). Mathematics syllabus for primary schools. Ghana Education Service. Ministry of Education. (2007). Mathematics syllabus for primary schools. Ghana Education Service. Ministry of Education Youth and Sports. (2004a). Ghana's performance in TIMSS 2003: Ghana Education Service. Ministry of Education Youth and Sports. (2004b). White Paper on the report of the Educational Reform Review Committee. Accra: Ministry of Education Youth and Sports. Ministry of Education Science and Sports. (2007). Annual Year Book Data (Publication. Retrieved 9/7/07, from Ministry of Education Sport and Science: http://www.edughana.net/emis%20data/html Piaget, J. (1953). The origins of intelligence in children. London: Routledge & Kegan Paul. Piaget, J. (1954). The child's construction of reality. NY: Basic Books. Pinxten, R., & Francois, K. (2007). Ethnomathematics in practices. In K. Francois & J. P. V. Bendegem (Eds.), Philosophical dimensions in mathematics education (pp. 214-227). NY: Springer. Presmeg, N. (2007). The world role of culture in mathematics education. In H. Iwasaki (Ed.), Empirical study on the evaluation method for international cooperation in mathematics education in developing countries - focusing on pupils' learning achievement- (pp. 131-159). Hiroshima, Japan: Hiroshima University. Presmeg, N. C. (1998). Ethnomathematics in teacher education. Journal of Mathematics Teacher Education, 1(3), 317-339. Scholnick, E. K. (1988). Why Should Developmental Psychologist be Interested in Studying Acquisition of Arithmetic? In R. R. Cooking & J. S. Mestre (Eds.),

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Linguistic and Cultural Influences on Learning Mathematics (pp. 73-90). NJ: Lawrence Erlbuam Associates, Publishers. Seah, W. T. (2004). Negotiation of perceived value differences by immigrant teachers of mathematics in Australia. Unpublished PhD, Monash University, Melbourne. Seah, W. T., Atweh, B., Clarkson, P., & Ellerton, N. (2008). Sociocultural perspectives in mathematics teaching and learning. In H. Forgasz, A. Barkatsas, A. Bishop, B. Clarke, S. Keast, W. T. Seah & P. Sullivan (Eds.), Research in Mathematics Education in Australia 2004-2007 (pp.223-253). Rotterdam, The Netherlands: Sense Publishers Skemp, R. R. (1987). The psychology of learning mathematics. Hillsdale, NJ Lawrence Erlbaum Associate Inc. Publishers. Sutherland, P. (1992). Cognitive development today, Piaget and his critics. London: Paul Chapman Publishing Ltd. Theunissen, E. (2005). Revisiting fractions. Mathematics Teaching, 192, 45-47. Vygotsky, L. S. (1934/1987). Thinking and speech (N. Minick, Trans). In R. W. Rieber & A. S. Carton (Eds.), The collected works of L. S. Vygotsky (Vol. 1, pp. 37-288). NY: Plenum Press. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge: Harvard University Press. Zaslavsky, C. (1978). African count number and pattern in African cultures. Chicago: Lawrence Hill Books ENDNOTES i

Culture as used in this paper refers to the complex of shared understanding which serves as a medium through which individual human minds interact in communication with one another (Bishop, 1988). ii

Out of school/everyday mathematical practices as used in this paper refers to mathematics practices experienced in everyday life which may not be the same as school/western mathematics. This knowledge constitutes knowledge acquired through cultures as the child grows (Authors’ interpretation for this paper) iii

Curriculum as used in this paper encompasses the aims, content, methods and assessment procedures but not only syllabus (Howson, Keitel and Kilpatrick, 1981). iv v

Ethnomathematics as used in this paper refers Mathematics of cultural practices (Presmeg, 1998)

In‐school mathematics refers to western/international mathematics which the child usually acquires through formal schooling (Authors’ interpretation for this paper)


Comparative analysis of performance of eighth graders from six African countries AsabereAmeyaw, A. & Mereku, D. K.2 Abstract School science and mathematics play a critical role in the development of scientists, engineers and technologists required for development. Although many African countries recognize this, the teaching and learning of science and mathematics have not received the necessary support and attention as foundational subjects in the school curriculum. In 2003, six African countries - Ghana Egypt, Tunisia, Morocco, Botswana and South Africa - participated in an international assessment programme in science and mathematics, called the Trends in International Mathematics and Science Study (TIMSS-2003). The study examined the performance of eighth graders in mathematics and science as well as the contextual factors that could have influenced the performance. This paper draws on the results of the TIMSS-2003 which indicated that the African countries performed poorly. The paper presents a comparative analysis of the performance of the eighth graders from the six African countries and discusses the contexts for learning science and mathematics that might have influenced the performance. Recommendations are made for African governments to pay attention to the teaching of science and mathematics in the primary and secondary schools. Keywords

international assessment, TIMSS, contexts for learning, school science and mathematics

Background Though national level data on student achievement is available in each country, in Africa, little information is currently available to assist policymakers, researchers, educators and the public obtain a comprehensive picture of how students perform in key subject areas like mathematics and science across the continent. The participation and collaboration of six African states in TIMSS-2003 generated data that have raised great concerns among educators on the continent (Anamuah-Mensah and Mereku, 2005; van de Linde, 2005). Though some researchers in Europe and North America have argued that international studies such as the TIMSS provide distracting and misleading information for researchers and policy makers (Wang, 2001; Holliday, 2003), the authors of this paper believe that, for many developing low income countries, such global comparative assessments could serve a great deal of purpose. TIMSS-2003 provided the participating countries, particularly those in Africa, with the opportunity to examine students’ achievement in mathematics and science using an international yardstick and to compare this to that of other countries both within and beyond the continent of Africa. It also provided rich information on the context for the teaching and learning of mathematics and science in African schools which could be used to identify strengths and weaknesses in teaching and learning of these subjects. 2

Prof. Asabere‐Ameyaw, A. and Prof. Mereku, D. K. are both lecturers in the Faculty of Science Education, University of Education,Winneba Ghana

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