Teaching Gesture and Oral Computation in Mozambique: Four Case Studies

Transcription

1 Teaching Gesture and Oral Computation in Mozambique: Four Case Studies J. Draisma 5 Introduction This article reports on the implementation of an Experimental Mathematics Program (EMP) in which finger gesture computation with its verbalisation in Portuguese and the main local languages was explored for early numeracy learning. The purpose of this program was to find a more efficient model for early arithmetic than the sticks, pebbles and strokes in the exercise books which are used in Mozambique since colonial times (Draisma, 1999), and which are used, in general, for rudimentary counting strategies, according to the classroom observations of Kilborn and his Mozambican team (Kilborn, 1990). With Verschaffel and De Corte (1996) I share the believe that "a crucial role is played by carefully chosen mathematical tools and models. Manipulative, visual models, schemes, and diagrams can be used as scaffolds fulfilling this bridging function between children's intuitive notions and informal strategies, on the one hand, and the concepts and procedures of formal mathematics, on the other." (p. 103) Finger gestures were chosen, because they correspond to local traditions, and they constitute the historical basis for the verbal numeration systems of the Mozambican indigenous languages (Gerdes and Cherinda, lgg3). Finger gestures allow for computations with ones, fives and tens, stimulating the use of computation strategies instead of elementary counting strategies. Computation with ones, fives and tens is one of the characteristics of the educational traditions of Japan and The Netherlands. In these countries, specific didactic devices are used by the pupils in school, corresponding to a structure: tiles and the traditional abacus (soroban) in Japan (Hatano, 1982), and the arithmetic rack with 2 rows of 10 counters, painted in groups of 5, in The Netherlands (Treffers and De Moor, 1990). With finger gestures, children have access to visual and manipulative support for early arithmetic with a structure similar to that of the tiles or arithmetic rack. This structure is also present in the majority of the Mozambican indigenous languages. The importance of verbalised computations The attention for the verbalisation of computations corresponds to the fundamental importance of speech on the development of thought, including mathematical thought, as shown by Vygotsky (1934/1986). An interesting example for this thesis is given by the Brazilian psychologists T.N. Carraher, D' Carraher and A. Schliemann (1988), in their study on oral mathematics as practised "in the street" and written mathematics as leamed in school. They show that children's oral mathematics as practised "in the street" is well understood, very flexible, and leading to few effors. On the other hand, the algorithms of written arithmetic, as learned in school, are learned without real life context and lead to many mistakes, including absurd results, without the children being aware of the absurdities. But children's informal, oral strategies which they use for survival in the streets are not recognised by the school. Interviews with unschooled adults in Mozambique showed well developed and varied skills in verbal computation (Draisma, 1998). These interviews were conducted in ten different Mozambican languages, by students of the Pedagogical University. For the interviewers, all experienced adult educators or primary school teachers, it was an important discovery to see that unschooled adults were able to calculate verbally, without knowing how to write numbers and without resorting to counting. In addition to the general importance of verbal mathematics, the EMP wanted to explore some of the 5 This article is based on the author's doctoral dissertation, completed at Monash University, Victoria (Australia) in 2006, under the supervision of Alan Bishop and Barbara Clarke. The author is grateful to the supervisors for their help throughouthe study, and to the four case study teachers in Beira: Maria InCs Baet4 Ant6nio Francisco, Luis Uache Gulube and Luisa Jrilio Duce, for their dedication and interest with which they carried out the Experimental Mathematics Program. t97

2 characteristics of the Mozambican indigenous languages, which are: a) Regular numeration systems based on ten: eleven is said as ten-and-one, twelve as ten-and-two, twenly as two-tens, etc. b) The majority of the languages also use the auxiliary base five, i.e., srx is calledy've-and-one, seven is five-and-two, and, for instance, seventy-four is calledfive-and-two-tens, and four. Table I: Examples of numerals in Portuguese, Ndau, Chuwabo and Tshwa tr Portuguese Ndau (coast) Chuwabo Tshwa I um clmwe Modha xlnwe 2 dois zvlwlrl Bili zimbiri J tr6s zvitat:.t Tharu zinharu 4 quaffo zvrrongomuna Nai mune 5 clnco zvishanu Tanu ntlhanu 6 SEIS zvitandhatu tanu na modha, or tanamodha ntlhanu ni xinwe 7 sete zviiomwe tanu na bili, or tanabili ntlhanu ni zimbiri 8 oito zvlsere tanu na tharu, or tanatharu ntlhanu ni zinharu 9 nove zvipfumbamwe tanunanai, ortananai ntlhanu ni mune l0 dez gumr Kumi khume 20 vinte makurni mawiri, or makumawiri makumi meli, or makumeli makume mambiri, or makumambiri t) setenta e tr6s makumanomwe zvitat't:10x7+3 na makumatanameli na :l0x(5+2)+3 tharu nthlanu ni mambiri wa makhume ni ztnharu :(5+2)x10+3 Regular and explicit number names are considered an advantage for early numeracy learning. Examples are the numeration systems in Chinese, Japanese and Korean. (Fuson and Kwon, 1992; Hatano, 1982). Fuson et al. (1992) mention the irregular number names in English for the numbers 11-19, as one of the reasons why "children in the United States lack the cultural support for ten-structured methods identified above for Korean children", because the ten is not available in a sum in order to suggest a ten-structured computation method (p. 159). Portuguese and the indigenous Mozambican languages in education During the EMP ( ), Mozambican indigenous languages were not being used in general education, just as in colonial times. During the armed struggle for Independence ( ), the Mozambique Liberation Front (FRELIMO) had adopted Portuguese as the language of communication. The curriculum for primary education, adopted with Independence (1975), established tacitly Porfuguese as the only language of instruction. The curriculum calls Portuguese the 'National Language" and emphasises that Portuguese should be transformed into a true "element of the consolidation of national unity" (MEC, 1975, p. 25). 198

3 Only during the 1980's the first initiatives were taken to "attribute more relevant social functions to the indigenous languages" (Firmino 2002, p.277). Therefore, the revised Constitution of 1990 defines for the first time Portuguese as the official language of the country, and declares that "The State values the national languages and promotes their development and increasing use as vehicular languages and in education" (Boletim da Repriblica 1990, Article 5). Therefore, in the 1990s, literacy programs in local languages are launched by INDE, for which I had the opportunity to collaborate for the Mathematics component. Simultaneously, INDE launched an experimental project of bilingual education, in three rural schools in Tete province, where Nyanja is spoken, and in two village schools in Gaza province, where Changana is spoken (Benson, 1998). The positive results of these experiments led to the introduction of the indigenous Mozambican languages into the new curriculum of basic education, according to three alternatives: a) Bilingual education, with a local language as language of instruction and as a subject during Grades 1 and 2, and with Portuguese as a subject. During Grades 3 and 4 a transition takes place, in which Portuguese becomes the language of instruction for some subjects, and the local language for other subjects. From Grade 5 onwards, Portuguese will be the language of instruction, and the local language one ofthe subjects to be studied. b) Monolingual education in Portuguese, but with the resource to local languages whenever it is useful or necessary. c) Monolingual education in Portuguese, with a local language being taught as a subject (INDE, 2003, p.30-32). The new curriculum stresses that culture is best expressed through the mother tongue. Children should have the opportunity of studying in the language they know best. The preservation and development of peoples' languages are considered a human right, not only of the individuals but also of the linguistic communities. However, in the curriculum plan there is no mention of the implications of the use of a local language for the teaching ofthe different subjects. At present, the variant of bilingual education is being introduced in a limited number of schools for each language. These schools cover 16 languages and for each language at least one school per province is involved. For these l6 languages pupils' books are being produced, including books for Mathematics. The main challenge for teachers of mathematics and teacher educators will be to explore the particular opportunities which the local languages ofler for mathematics leaming. Hopefully the results of the Experimental Mathematics Program (EI\4P) will give ideas to teachers and teacher educators in this field. The Experimental Mathematics Program (Beira, ) The EMP program was carried out by four teachers in Grades 1-3 in Beira, Mozambique's second largest city. Two of them worked in an urban school, and two in a semi-rural school. In the EMP pupils were encouraged to use finger gestures for addition and subtraction within the limit of 100, and explain their computations orally, not only in Portuguese, but also in the main local languages: Ndau (mother tongue for 50% of the pupils), Chuwabo (mother tongue of 25%o of the pupils in one of the schools) and Tshwa (mother tongue of 20Yo of the pupils, in the second school). All pupils learned to verbalise their computations in at least three languages. Ndau speakers in Beira use a decimal system of number names, whereas Chuwabo and Tshwa use also the intermediate unit of five, as shown in Table 1. In 1999, a first experiment was conducted during 6 weeks, at the end of Grade One. The Mathematics lessons were given by Ant6nio Francisco to Maria InCs' pupils. In 2000 the four teachers started with the program in the beginning of Grade One and accompanied their pupils to Grade Two (2001) and Grade Three (2002).In 2001 they received also a second group of pupils in Grade One, which they accompanied to Grades Two and Three (in 2002 and 2003, respectively). In Mozambique, most urban and suburban schools function in three daytime shifts, and one evening shift. So the four teachers worked during two of t99

4 the three daytime shifts, during the school years 2001 and2002. Results of the Experimental Mathematics Program The four teachers succeeded having their pupils calculating, at the end of Grade Two, sums and differences within the limit of 100, using gestures and explaining all steps of the computations. The gestures allowed for powerful visualisations for recomposition methods arotnd five and ten. The teachers were unanimous in their conclusion that the results of the EMP are much better than the results they used to have in the past, when they and their pupils were working with unstructured manipulatives like sticks, pebbles and strokes, and using counting strategies. As the chosen mathematical model, finger gestures, plays a crucial role in children's learning, the main visualisations are presented in detail. They constitute an important part of the findings of the study. An essential ingredient of all computations is the role of the intermediate lunite of fivq corresponding to one hand, and component of the numbers 6-9. Powerful visualisations by gestures a) Complements infive (3 + 2 : 5 and 5-2 : 3) If three fingers are shown (unfolded), then two fingers are folded, showing also that 3 +2 : 5 or 5-2 : 3. With the Japanese tiles there is no easy way to show these facts. That's probably why Kobayashi (1988, p. 9, 10) suggests this gesture in order to visualise complements in 5. Figure 3 : The visualisation of a complement in five by a gesture b) Complements in ten ( : l0 and 10-3 : 7) If seven fingers are shown as one full hand and two fingers on the other, then three fingers are folded on the second hand, showing also that : 10 and 10-3 : 7. Again, this gesture is suggested by Kobayashi (1998, p. 32), because there is no easy way to show these complements with tiles. Figure 4: Thevisualisation of a complement in ten by a gestute c) The passage offive: : : 7 The EMP pupils invented three different visualisations for 4 + 3: (i) Show four on one hand. Four plus one, five (tnfolding the thumb). Five plus tlvo, seven (showing two onthe other hand. (recomposition up-over-five) (ii) 1. Show four on one hand and three on the other: 2. Separate the little finger from the three, totching the little finger of thefour as a sign of transformingfour and three into five and two. seven. Figure 5: Gestureshowing : 5 +2 (iii) A second method of transforming 4+3 into 5+2 was done by taking away one finger of the three, folding the middle finger, and adding a finger to thefour through the unfolding of the thumb. 200

5 In all three methods the pupils recognise that five-and-nuo is the same as seyen, "by definition". In Chuwabo and Tshwa, the name for seven is just five-and-two. These are important visualisations, which are not available if the visualisation of fives is not explored. If fivesarenotexplored,the"standardcomputation"of4+3consistsinusingthefactthat4+4:8,and concluding that4 + 3:8-1 (Padberg,7996, p. 83,84; Steinberg, 1985, p.3a0). But how is 4+4:8 visualised? Padberg and Steinberg only mention that the "doubles" are easily remembered, but not how they are calculated. d) Subtractingfrom l0 one of the numbers 6, 7, 8 or 9 The typical gesture found for 10-7 : 3 is to use one hand with which 2 fingers of the other hand are bent, leaving 3 fingers visible, which is the result. This is the gesture of taking away 5 * 2 from the two open hands. Figure 6: Typical gesturefor 10-7:3 This typical gesture of subtracting from 10 a number greater than 5 was used by nearly all pupils for the cases of subtraction with "borrowing", as they usually subtracted the units from a ten. e) The passage of ten: : (5 + l) + (5 + 2) : : 13 The visualisation of : 13 is similar with the arithmetic rack and tiles, if the decompositions into and are used, as shown by the following figures: T T Figure 7: : 13 on the arithmetic rack (Treffers et al., 1990, p. 45) Figure 8: Visualisation of : I 3 with tiles The visualisation with gestures is similar, the main difference being the fact that people have only ten fingers, whereas the arithmetic rack has twenty counters, and with tiles there is no limit to the numbers which may be represented. The EMP pupils discovered that the easiest way is to represent each number on its own hand, showing first 5 and then the remainder, as shown by the following images: l. Represent the number six, raising a full open hand (five). Then show one, on the same hand. Only one of the six fingers remains visible. 2. Represent the number seven) raising the other, full hand (five). Then show fwo, on the same hand. Only two of the seven ftngers remain visible. 3. Adding srx plus seven means joining the two full hands, which are no longer visible. They make a ten. Then add the visible fingers: one plus two, three; plus ten, thirteen. Figure 9: : (5 + I) + (5 + 2) : : 13, by gestures During the observed lessons, many pupils carried out this kind of computation without difficulty. They didn't have the difficulties "to reuse fingers to show numbers larger than ten", as imagined by Fuson er a/. (1992, p. 159). However, a few cases were observed of pupils showing difficulties with the "passage of 201 If*-

6 ten". fl Switching from fingers representing tens to fingers representing ones In the case of subtraction with "borrowing" with two-digit numbers, many pupils used a gesture computation, in which first the tens are subtracted (visualised by individual fingers); then from the remaining tens one ten is taken (one finger) and substituted by ten ones (ten fingers). From these ten fingers the required units are subtracted. In the end, the remaining tens and the remaining ones are added, together with the ones of the minuend, which were not touched during the computation. An example is Paulo's computation of :36: Szxry minus twenty equals forty. His gestures are faster than his words: he shows six frngers (five (LH) and one (Rlt)). With his right hand he bends one frnger (LFI), leaving four fingers visible. Keep thirty apart; ten are left (bends three of the four frngerc; the thumb is left, and is substituted by ten frngers - two full hands). Ten, take away nine, equals one (with his right hand he bends four fingers (LH), leaving the thumb visible). Phtsfive equals thirty-six. Gesture computation and the role of languages with the auxiliary base five As mentioned, the use of computation with fives is suggested by the use of gestures. The role of verbalisation is to support the computation, to allow for communication (explaining), and, finally, to allow for verbal and mental computation for which gestures are no longer necessary. The experience with the Beira EMP program confirms what is known from the Dutch and the Japanese experience, that computation with fives (and tens and ones) may be verbalised in a language in which the numbers for 6,7, 8, and t have their own name. The EMP pupils calculated using ones, fives and tens, because of the use of gestures, and they verbalised their computations in Portuguese and Ndau, both of which have a decimal system of numeration, but also in Chuwabo and Tshwa, which use the intermediate lunit of five. During the observed lessons, no pupils appeared to have a clear preference for computation in the mother tongue. It seems that in the present language situation in Beira Portuguese is not an obstacle which could stimulate or force pupils and teachers to use their mother tongue. The situation would probably be different in the countryside, far from the main urban centres. It is possible that the four teachers avoided using the local languages for computations, as the choice of language in a particular lesson was their own decision, and in a sense the EMP required teachers and pupils to use an artificial combination of languages: Portuguese as language of communication, together with a local language, from which only the number names were used for computation purposes. After having participated in the EMP, Luis asked to be transferred to the Gorongosa district, where he is now working in a village school. His present experience is that nearly all his pupils depend on explanations in the local language, Sena, as very few people in the village speak Portuguese. He is now in a better position to implement the EMP using Sena as the language of instruction and language for computation. In Luisa's opinion: "Any language is OK for use in school, as long as the teacher gives a good explanation. Some children know Portuguese but they also know how to count in local languages and even do sums in local languages. Therefore, even if the local languages are not used in the official schools, they do not impede the learning of mathematics. Some children who speak Portuguese well, when they enter school, they have difficulties in understanding mathematics, whereas some children who only know the local language have no difficulties in understanding mathematics. Therefore, any language will do in school." Luisa's comment seems to be important: teaching in the pupils' mother tongue is no guarantee of their mathematics leaming. Her comment is a warning against the unfounded expectation that, with the introduction of bilingual education, children's difficulties with mathematics will suddenly cease to exist. Teachers must know how to give "good explanations", i.e., they need specific knowledge of how to teach mathematics to young children. 202

7 Summary The four teachers and their pupils explored, in general, the opportunities to calculate with fives, as suggested by the gestures they were using. They showed also that the corresponding verbalisation is as easily done in languages like Portuguese and Ndau, which have single number names for 1-10, as it is done in Chuwabo and Tshwa, with number names based on five for 6-9. These are the main similarities with the use of the Dutch arithmetic rack and the Japanese tiles. Possible advantages of the use of gestures as compared to the use of the arithmetic rack and the tiles seem to be: o o the kinaesthetic experience the greater stimulus for imagination and memory use, which may be helpful for reaching the goal of mental computation. In addition to these aspects it must be mentioned that all children have access to finger gestures, therefore the schools do not have to spend their limited resources on factory-made materials. Although in the lessons of the four teachers the pupils' explanations were similar - based on gestures with a verbal explanation - there were some interesting differences between their computations. Three types of computation were observed: mental, oral and through gestures. In Luis' lessons, the pupils calculated menially, without showing any visible sign of gesture use. In Maria InOs' lessons, the pupils calculated orally, without the use of gestures. However, in both cases the pupils were able to explain their computations through gestures. In Ant6nio's lessons, all computations were carried out through Sestures, accompanied by the corresponding spoken computation. In Luisa's lessons examples of the three types of computation were observed. In Maria In0s' and Luisa's lessons there were some pupils who calculateo well, but who were unable to explain with gestures how they had obtained certain partial results. It seems that they had used the finger gestures for learning certain strategies, but their verbalisation had made the gestures superfluous. What can teachers do with these results? The results of the EMP of gesture computation accompanied by the corresponding verbalisation functioned well, mainly in Portuguese, but on a modest level also with verbalisation of the computations in local languages. As Ant6nio suggested in his first interview, full lessons should be given in a local language, "in order to give the pupils more confidence in their language." With the introduction of bilingual education, all teachers now have the opporfunity to use the local languages in support ofthe teaching-learning process, which is no longer forbidden. In the schools where a local language is used as medium of teaching and learning, teachers have the opportunity to implement fully Ant6nio's suggestion, as the curriculum requires that pupils should acquire skills of mental computation (INDEA4INED 2001a,2001b). But even in schools where Portuguese is the language of teaching and learning, the local languages should be used for explanations, whenever the teacher or the children think it is helpful. In this situation, a teacher can veriff whether some children calculate more easily in their mother tongue than in Portuguese. Switching from local language to Portuguese and back may help the pupils to understand the meaning of the irregular Portuguese numerals. For computations to be meaningful it is necessary that the number names have a clear meaning. 203

8 What are the implications for bilingual education? The modest results of the EMP seem to indicate that early arithmetic, visualised by gestures, can be verbalised in indigenous Mozambican languages. Therefore, teachers who are involved in bilingual education, either according to the l't or the 2"d variant, should explore the possibilities of the languages which they and their pupils are using. One open question is, whether the first variant of bilingual education will allow pupils to become fully bilingual, i.e., equally fluent in a local language as in Portuguese. This question relates directly to mathematical skills: every mathematics curriculum must guarantee that children build a repertoire of basic facts, which must be instantaneously available for more complex mathematical reasoning and computation. This repertoire of basic facts is stored in our brains in a verbal form (Dehaene, 1997) - although after several years of schooling some people will also acquire basic mathematical knowledge in a written form, e.g., in the cases of extended practice with the written algorithms or symbolic algebraic manipulation. For bilingual mathematics learning the open question is, whether pupils can acquire full bilingual basic facts, or whether the basic, verbalised facts learned in a first language, can be transformed into basic facts in a second language. The new curriculum for basic education projects the transition from local language to Portuguese as language of instruction during Grades 3 and 4. What will happen, during and after the transition, with the accumulated arithmetical knowledge in the local language? The pupils' knowledge of the local language may continue to develop, if the local language continues to be studied as a school subject. But the question remains: What about the pupils' mathematical knowledge, acquired through the local language? Will it be transformed into knowledge in the Porfuguese language, and further developec in the second language? No one can predict how the use of local languages in education will develop in the future. But the introduction of the use of the Mozambican indigenous languages in basic education is an important first step, which may have far-reaching consequences. References Benson, C. (1998). Alguns resultados da avaliaqdo externa da expericncia de escolarizagdo bilingue em Mogambique. In C. Stroud & A. Tuzine (Eds.), (Jso de l{nguas africanas no ensino: problemas e perspectivas ( ). Maputo: INDE (Instituto Nacional do Desenvolvimento da EducagSo). Boletim da Repriblica (1990). Constituiqdo da Repilblica de Mogambique. Boletim da Repfblica, lu Sdrie, No.44, Maputo. Carraher, T.N., Carraher, D., & Schliemann, A. (1988). Matem6tica escrita versus matem6tica oral. In T. Nunes Carraher, D. Carraher, & A. Schliemann: Navida dez, na escola zero (45-47). S5o Paulo: Cortez. Dehaene, S. (1997). The number sense. How the mind creates mathematrcs. New York: Oxford University Press. Draisma, J. (1998). On verbal addition and subtraction in Mozambican bantu languages. In A. Olivier & K. Newstead: Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, ). Stellenbosch: Universidade de Stellenbosch. Draisma, J. (i999). NumeragSo falada e gestual como recursos naturais na aprendizagem inicial da Matemftica. In Comissdo Organizadora do ProfMat 99, Actas do Pro/Mat 99 ( ). Lisboa: Associagdo de Professores de Matem5tica. Firmino, G. (2002). A "questdo lingutstica" no Afri"o pos-colonial: o caso do Portugu s e das l{nguas aut 6c t one s em Mog amb ique. Maputo : Promddia' Fuson, K.C., & Kwon, Y. (1992).oKorean children's single-digit addition and subtraction: numbers structured by ten', in Journal for Research in Mathematics Education, Y ol. 23, No 2, Gerdes, P., & Cherinda, M. (1993). Sistemas africanos de numeragio. Em P. Gerdes (Ed'.), A numeragdo em Mogambique. Contribuiqdo para uma reflexdo sobre cultura, l{ngua e educagdo matemdtica (8-204

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