CULTURAL OBJECTS AND MATHEMATICS TEACHING AND LEARNING Janet Kaahwa
In this paper I present my views and beliefs about cultural objects and their use in mathematics learning. I explain the potential in the cultural objects to enable students construct and understand mathematical concepts leading to enjoyment and love for the subject. I touch on how they may be used in teaching. Poor and developing countries stand a lot to gain from teachers' use of cultural objects.
Currently, I am working on ways of effectively training teachers to effectively use cultural objects in their classroom teaching so as to enhance mathematics learning in all children.(The terms objects and artefacts are used interchangeably).What is culture? Culture is "an historically developed, patterned way of life which includes beliefs and ideologies; formally and informally established interrelationships between persons and groups and material goods and technologies, all of which are systematically related so as to form an integrated whole" (Dobbert, (1982:10) quoted by Lancy (1993:35). It is shared meanings accepted by a society and the content of socialisation(Nickson,1994), p.8). Cultural artefacts are man-made objects traditionally found and used in homes in particular and communities in general. Among these (using examples from Uganda) are:
utensils used for household chores like cooking, storage, and decoration. Examples of these are pots, calabashes, gourds, winnowers, mats and baskets.
Objects that are used outside the house; in the homestead, courtyard, garden and farm. They include hoes, pangas, granaries, ladders, canoes, and many others.
Objects used for other purposes like leisure and entertainment:
a) 'ebinyege', 'ekiguuli', 'entogoro', 'ebisegenya' used in cultural dances;
b) drums of different shapes, the 'endingidi' 'amadinda', 'amakondere' used as musical instruments and
c) 'Umweso' used as game.. These objects vary from culture to culture. Cultural objects and mathematics teaching and learning Importance of use of cultural objects in mathematics teaching and learning can be explained as follows:(i) Mathematics is ingrained in cultural objects. A lot of cultural objects in Uganda and other parts of the world are made from the natural environment. Looking closely at the environment we can see mathematics ingrained in nature. Take for example trees; these have branches inclined at angles and alternating to create balance. The egg plant if sliced longitudinally will result in mathematical shape, a hyperbola. Now since cultural objects are made from this environment, they definitely have mathematics in them. Unless, of course, the inherent mathematics is lost during their making which is unlikely. So we can therefore say 'mathematics is ingrained in culture'. Examples of cultural objects from the environment are gourds, calabashes, mortar and pestle. Apart from cultural artefacts being made from the environment, for a long time in the past, they were made by people, (mainly women), that had never been to school. Surprisingly, these illiterates made objects that are very mathematical as seen in the examples given by Gerdes (1995) and Zaslavsky(1994). Examples from Uganda exhibiting mathematics have been cited above. This to me, is an indication that mathematics is ingrained in the human mind. Other writers that appear to imply this are: Victor and Kartz (1994)who wrote that mathematics originated as an answer to human practical daily life needs. Eves (1983) specifically points out that mathematics originated as a practical science to assist in agricultural and engineering pursuits. In case you find these assertions difficult to believe, you may be interested to know that Gerdes (1982, 1985) referred to the mathematics in cultural objects as hidden or frozen mathematics, thus explaining this difficulty. He points at the need to unfreeze / reconstruct this mathematics (Gerdes, 1994). Indeed African cultures have a lot of mathematics in them. Zaslavsky (1991) also subscribes to this when he quotes Barry's (1989) comments about the African American patchwork quilts; that possibly African slave women introduced patchwork in England I find this intriguing. Imagine illiterate slave women doing this! Dodson (1990) however confirms this in his writing that: NOTHING SEEMS MORE INNOCENT, perhaps, than the sight of a handmade quilt hanging on a line. Look again. What you may be seeing is the legacy of along history of secret codes, a hidden language that is believed to have guided enslaved Africans to freedom, a language used to help them travel during the days of the underground railroad and even before. ( P.146)These quilts had messages to and from slaves and they only could decipher them. According to Dodson (1999: 146), Tobin and Dobard provide the first documentation of the inner workings and meanings of quilt codes and link them to symbols Africans brought with them to the Americas in their publication Hidden in plain View: The secret story of Quilts and the underground Railroad (Doubleday, $27.50). Now coding of messages is done using words or symbols. The symbols used could be mathematical ones. Wilder's (1950)summary is quoted by Gerdes (1996) and says: In man's various cultures are certain elements called mathematical. In the earlier days of civilisation, they varied greatly from one culture to another so much that what was called 'mathematics in one culture would hardly be recognised as such in certain others. With the increase in diffusion due, first, to exploration and invention, and, secondly to increase in the use of suitable symbols and their subsequent standardisation and dissemination in journals, the mathematical elements of the most advanced cultures gradually merged until... there has resulted essentially one element, common to all civilised cultures, known as mathematics. This is not fixed entity, however, but is subject to constant change. Not all of the change represents accretion of new material; some of it is a shedding of material no longer, due to influential cultural variations, considered mathematics pp.269-270). All the above are a testimony to the fact that mathematics is ingrained in culture and cultural objects.
(ii) Cultural objects are familiar to students and teachers from a given culture Cultural objects are man-made and are used daily in homes and communal activities. They therefore are familiar to students. If they are brought to class, the students will recognise them as those things they use at home. This should create a lot of interest. In the two classes that I observed being taught by the two teachers that I worked with in my yet uncompleted research excitement and interest were visible among students as they viewed the cultural objects brought in by their teachers. Also, whenever I bring cultural objects into my sessions of training teachers(pre-service or in-service) I see great interest aroused. I must add that I also see teachers surprised because they usually have not thought about or looked at cultural objects in this light. Use of cultural objects in the introduction of mathematics content is likely to be of advantage to the learning process in that, the students, both boys and girls, are likely to feel more at ease with the content. I can foresee students imagining that what is to be learnt:
(a) will not be strange or threatening and should therefore not be feared. Consider as an example, using a table cloth that is home embroidered; the message in it is that girls also can do mathematics since they manage to do such complicated embroidery that is mathematical. As a matter of fact, one teacher in my research team used some girls in her class to make the tablecloths that she later used in the same class to introduce the idea of regions.
(b) Will not be unmanageable since what they do using these objects at home is most times manageable. Indeed the girls referred to above found the work on tablecloths manageable; this should have put the mat ease with the inherent mathematics.
(c) Be enjoyable depending on whether what they do at home with these objects is usually enjoyable. I can imagine (since I did not try to find this out from the girls themselves) the pleasure the girls referred to above felt as they made these tablecloths for the teacher and the even greater pleasure at discovering that what they made was mathematical. They should have enjoyed the process of learning this mathematics.
(d) Furthermore such enjoyment is likely to culminate in students' interest in the subject.
(e) Evidence of mathematics in cultural objects that they use at home is likely to result in the realisation that after all mathematics is constantly with them and not only in textbooks; they live and deal with it daily. This will also serve to demystify mathematics as a school subject (something urgently needed in schools).(iii) Cultural artefacts can enhance mathematics learning by construction
a) Constuctivism is a theory about the limits of human knowledge, a belief that all knowledge is necessarily a product of our own cognitive acts (Davis,1990); a view that knowledge is something that each individual must construct for oneself. (Lockhead quoted by Orton,(1992)).We construct our understanding through our experiences, and the character of our experience is influenced profoundly by our cognitive lenses. Constructivism is the philosophy my underlying my argument for use of cultural objects. My view like that of Piaget and others quoted above is that students learn through construction of their own reality. That by participating in various mathematical activities they can come face-to-face with and experience mathematical knowledge as they gradually abstract it making it their own. I argue like Gerdes (1982, 1985 ) that mathematics is frozen in artefacts and add that as students abstract this knowledge they 'unfreeze' it out of them making it part of their knowledge structure.
b) Cultural objects and constructive mathematics learning I therefore argue that cultural objects when brought to class can be used as instructional / learning materials. Teachers can use activities developed from them to involve learners in their mathematics learning by construction. The activities provide students with experiences. They enable them to work purposively as they organise their knowledge. Cultural objects being in plenty they allow a resourceful teacher to organise varied activities that facilitate knowledge construction and abstraction. Cultural objects being concrete materials allow learners to actively participate in learning by manipulating them. This is in accordance with Piagetian theory that children learn actively from varied experiences. In fact I wish to propose that cultural objects could contribute to issues discussed by Schliemann (1995). They can be the referents that he talks about. They can be used to design opportunities for children to develop mathematical knowledge that is wider than what they would develop outside school, but that preserves the focus on meaning found in every day situations. They can be used to start from what children already know and to proceed to take them to new strategies and new understandings. I wish to add that cultural objects could be used to develop activities that could serve as starting point on which the learner could later build mathematical knowledge that is not part of everyday situation. Such activities could serve as the basis for students' continuous reflection especially at home. This point is explored further in the next section.(iv) Cultural artefacts offer students more time for learning Because cultural artefacts are in homes, students are likely to follow-up their schoolwork from home. This claim I base on the fact that children are naturally curious. I have witnessed this often with my children. On many occasions when they come home and the teacher has done something at school that is connected with some materials available at home, they spend quite an amount of time trying to go over it. One of the teachers that I worked with on my current project in an interview mentioned that in his school students while at home do go over what the teacher has done in class usually with the aim of disproving the teacher. Cultural objects therefore have an advantage over other educational materials which cannot be taken home and therefore restrict students' active learning to school contexts only. Cultural objects are all the time available at home. Not only that but, these objects are being used by students in their home chores. This is reflected in a research conducted among first and second year university students at Makerere University. A number of them reported that doing home chores offers them time to practice some mathematical skills. (Kaahwa, 1999). This conserves time in various ways:
a) If a student can work on some piece of mathematics at school and at home he/she is availed more learning time.
b) When doing home chores there is opportunity for students to extend their learning time through exploration of mathematical concepts at the same time. This is likely to facilitate students' own discovery of the relationships of mathematics to other activities (home chores) which could increase the value of and students' love for mathematics. There is also a possibility of a student discovering mathematics in these home activities.v) Students' own knowledge construction and more time on task can produce interest and love for mathematics I have touched on this in an earlier section. If students have more time on learning tasks and construct their own knowledge, they are likely to develop love for mathematics. More time on learning task facilitates knowledge construction since it offers more opportunities for learning experiences. Knowledge construction results in understanding that demystifies the content. Understanding generates love for the subject since the student creates a warmer relationship between him/her and the content. Szendrei (1996: 427) puts it generally that: "Manipulatives (sic)help pupils develop and understand the concepts, procedures and other aspects of mathematics." Understanding creates confidence in the students, which in turn removes fear, and creates willingness to learn.
vi) Cultural artefacts and Gender in mathematics learning Cultural objects can have the potential to encourage both boys and girls to study mathematics. In most cultures (of Uganda) there is a substantive amount of cultural objects that are made by women. Thus if such cultural objects are brought to class and used in mathematics teaching, girls are likely to realise the fact that mathematics is not only for male but al so for females. This they will see through the realisation that these object sin which there is a claim that mathematics exists, are things that they themselves do. Which should be the evidence that they can do mathematics. Take of example, in the Tochkwe 'Sona' culture that Gerdes (1990) talks about, the baskets are made by women. Which to me means that women do mathematics without realising they are doing so. I am aware that some people might and indeed will argue that these women are just doing or replicating that which some individual at one to me thought out and did. And that, that individual is the one that did mathematics and not these other women that do a similar thing. But then proponents of such an argument should remember that even in doing mathematics many learners simply repeat or reproduce what other mathematicians did. Yet we evaluate their work and judge them as knowing and doing some form of mathematics. Thus I still would argue that the women that do these cultural objects do carry out some form of mathematics. The mathematics that gets frozen (as Gerdes (1994) would put it) in the resulting objects. But of course the extent to which girls might get encouraged by use of cultural objects will depend on the quality and style of teacher's usage of these. If the teacher is careful to point out this fact to the class, I can envisage two outcomes:
i) the girls getting the encouragement that they can do mathematics and
ii) the boys realising that girls too can and do mathematics. Use of Cultural Artefacts Can Enhance Ethnomathematics. According to the ethnomathematical movement quoted by Gerdes (1994), ethnomathematicians draw attention to the fact that mathematics and its techniques and truths are a cultural product. That looking for possibilities for improving the teaching of mathematics by imbedding it into the cultural context of students and teachers is one of the ethnomathematical research (Gerdes 1995 p.7). My interests are there for ethnomathematical. Gerdes ibid. provides an example of this when he used sipatsi, a cultural object in one of the South African cultures. He says there are many possibilities of using sipatsi to improve mathematics teaching (and I suppose learning) by imbedding it into the cultural context of students and teachers. Kartz (1994) is of a view that ethnomathematics if included in school curriculum
(i) can enable students to recognise that mathematical practices and ideas arose out of the real needs and interests of people. That a lot of school mathematics originated in Africa and Asia.
(ii) Can enable students to take pride in their peoples' achievements.
(iii) Can help students to see the usefulness of mathematics in other subjects and its relevance to their own lives and communities.
(vi) Provides a way of incorporating students' out of school experiences in their mathematics learning making it less strange and accessible to all children .
(v) Can enable students to identify and do something about societal factors that hinder their having fulfilled lives. I see in all this an avenue of making mathematics interesting to learners since the above-mentioned are among the things that motivate students to learn (Derville, 1966).In summary, we have seen that cultural objects have a lot of untapped potential in the areas of mathematics learning and teaching. There is need for mathematics teachers to explore ways of maximally utilising them for their own benefit and of the learners.REFERENCES
Davis, B.R.; Maher, A C; Noddings, N. (1990) Constructivist Views on the teaching and Learning of Mathematics. In Journal for Research in Mathematics Education Monograph Number 4 Constructivist views on the teaching and Learning Mathematics 1990 NCTM
Dodson, A. (1999) SECRETS OF THE QUILTS An elderly woman's recitation of the code passed down through her family sheds new light on how quilt shelped slaves escape to freedom, in ESSENCE FEBRUARY 1999 p.146, 148Comag (a magazine).
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