«BATIKS: HOW TO LEARN MATHEMATICS A DIFFERENT WAY AND IN A PARTICULAR SCENARIO Lucília Teles & Margarida César Universidade de Lisboa, Centro de ...»
BATIKS: HOW TO LEARN MATHEMATICS A DIFFERENT WAY
AND IN A PARTICULAR SCENARIO
Lucília Teles & Margarida César
Universidade de Lisboa, Centro de Investigação em Educação da Faculdade de
The Dance School community is a minority but it is valued by society. This is a
vocational school but students also attend subjects from the mainstream educational
system. This research took place in a 9th grade class already that was used to work collaboratively. It assumed an interpretative/qualitative approach, based in an action-research project, inspired in ethnographic methods. This study was based on 2 research projects: Interaction and Knowledge (IK) and IDMAMIM. A microproject was implemented in order to elaborate batiks, using the school’s name. Results illuminated students’ (re)actions and accounts related to this microproject, to Mathematics’ classes, namely mathematical performances. They also illuminate how important this microproject was to students’ engagement and to their achievement.
In the Dance School everything is organized around dance and the artistic subjects:
timetables, classrooms, and so on. Mathematics plays a secondary role. The selective subjects are classic and modern dance techniques. Although students know that they cannot ignore academic subjects because they need to succeed on them too, in order to continue their dream, they also state that their favourite subjects and the ones they spend most of their time and effort are the ones directly related to dance. Thus, to learn and to teach mathematics in this scenario is different than teaching it in a mainstream school and it is a new and stimulating challenge to the teacher.
As we live on a multicultural society and we claim that intercultural practices are needed, we promoted an intercultural (and interdisciplinary) microproject based on Cape Verde culture. Cape Verde community is the biggest ethnic minority in Portugal and their uses, ways of being, reasoning or approaching mathematical tasks are not socially recognised and valued. Thus, gathering mathematical learning and valuing this culture seemed a promising enterprise.
Thus, the problem we were studying was the lack of significance that mathematics used to have for the Dance School students. The research questions we were addressing were: (1) What are the contributions of an intercultural microproject, associated to collaborative work, in order to facilitatestudents’ mathematical J.F. Matos, P. Valero & K. Yasukawa (Eds.) (2008). Proceedings of the Fifth International Mathematics Education and Society Conference. Lisbon: Centro de Investigação em Educação, Universidade de Lisboa – Department of Education, Learning and Philosophy, Aalborg University.
meaning construction and mathematical knowledge appropriation?; (2) What are the contributions of an intercultural microproject in order to develop students’ citizenship, namely their respect and valorisation of usually less valued minority cultures, like the one from Cape Verde?
THEORETICAL BACKGROUNDThere is an enormous diversity of definitions of culture. Many authors tried to define
culture and presented different claims. Nieto (2002) defined culture as:
“(…) the ever-changing values, traditions, social and political relationships, and worldview created and shared by a group of people bound together by a combination of factors (which can include a common history, geographic location, language, social class, and/or religion), and how these are transformed by those who share them.” (p. 53) Thus, at school we can find an enormous diversity of cultures. Not only origin cultures but also many others, including the school’s culture. In many cases this culture is so far away from students’ cultures that they focus their energies on others directions (Säljö, 2004). Thus, it is important to find a way of promote interactions among the different cultures which are part of a particular school. This illuminates the need of an intercultural education, namely in mathematics (D’Ambrósio, 2002;
Favilli, César, & Oliveras, 2004; Peres, 2000; Powell & Frankenstein, 1997). In 1991, Ouellet already stressed that the intercultural education was not only for the minorities but also for majority groups, based on the comprehension of each other, on the communication among them, and on the promotion of interactions. Intercultural education also include citizenship education, namely through mathematics (Skovsmose, 1998, 2005).
In the Dance School, we tried to contribute to students’ citizenship education during the classes and also through the microproject. We tried to show some elements of a minority culture that is highly represented in the Portuguese society, relating this culture with school mathematics. We aimed at facilitating students’ recognition of the value of Cape Verde culture and its mediation role in order to learn mathematics (Teles, 2005; Teles & César, 2005, 2006a, 2006b, 2007). Some authors argued that intercultural microprojects related to handicraft activities support an intercultural approach, giving a cultural dimension to the learning process, contributing to academic achievement (César & Azeiteiro, 2002; César, Mendes, & Azeiteiro, 2003;
Favilli, 2000; Favilli, César, & Oliveras, 2003; Favilli, Oliveras, & César, 2003).
They illuminated the potential of intercultural and interdisciplinary microprojects in order to promote mathematical knowledge appropriation, but also to mobilise/develop students’ competencies, including social and emotional ones.
During the whole school year these 9th grade students worked collaboratively in mathematics classes. They worked in dyads, discussing their reasoning and solving strategies, helping each other, and co-constructing their knowledge (Teles, 2005;
Teles & César, 2005, 2006a, 2006b). Thus, the microproject was part of a coherent didactic contract implemented during the whole school year and negotiated with students on the beginning of the year. Collaborative work was studied and promoted in other studies and stated to be a facilitator for students’ knowledge appropriation when it was part of a negotiated and coherent didactic contract (César, 1998, 2007;
César & Santos, 2006; Schubauer-Leoni & Perret-Clermont, 1997).
A community of learning emerged from the practices that took place in mathematics classes as there was a mutual engagement, a joint enterprise and a shared repertoire (César, 2007; Wenger, 1998). The nature of the mathematical tasks assumed a relevant role on that process (César, Oliveira, & Teles, 2004). Their social marking was essential to students’ engagement (Doise & Mugny, 1981), promoting their participation in the solving strategies and during the general class discussion.
Hummel (1979) stated that “culture and education are intensively linked as verse and reverse of the same reality. It is impossible determinate where the educational ends and the cultural starts and it would be nonsense separate them” (p. 234). Thus, teachers’ role also includes contributing to create bridges among cultures and education, promoting diverse and intercultural learning experiences.
METHOD This research is an interpretative/qualitative study, inspired in ethnographic methods and based in two research projects: IK and IDMAMIM. The first one was developed during 12 years and its main goal was to study and implement social interactions in formal educational scenarios. IDMAMIM project was developed in some towns of Spain (Granada), Italy (Pisa) and Portugal (Lisbon). The two main goals of this project were to identify didactic needs to develop intercultural Mathematics Education, and to elaborate intercultural didactic materials.
This study was an intercultural and interdisciplinary microproject also engaging the mathematics, drawing, Portuguese, and history teachers. The mathematics teacher was also the researcher. Students elaborated batiks based in the school’s name (EDCN). Later on the batiks’ elaboration process was used to explore some mathematical contents such as direct and inverse proportionality. These students worked collaboratively during the whole school year and they developed this microproject in 4-students groups after being used to work in dyads in mathematics classes. To explore direct proportionality we used tasks based on the first day of the batiks’ elaboration process. In that day students needed to make a paste with flour, water and lime. They had a recipe referring to the ingredients needs for 500g of dry cotton. But they only had pieces of dry cotton that weighed 80g/90g. So they needed to calculate some proportions in order to know the quantities they needed to make the paste. Through those calculations we explored direct proportionality notion and its properties. Inverse proportionality was explored through what students did on the third day: the tainting process. Each student chose two colours to his/her batiks and only two students chose the black colour. In another school, where batiks were elaborated during the previous year, only one group of students chose the black colour and all ink was used on it. Inverse proportionality was explored through this situation, supposing that the number of black batiks could be changed but the quantity of ink could not. In this paper we focus our analysis on the batiks’ elaboration process and on the first day after that, when we began exploring the mathematical contents based in the batiks’ elaboration.
This research was developed with sixteen 9th grade students from the Dance School.
As the Dance School is a small vocational and artistic school, these were all the students attending the 9th grade.
Data were collected through participant observation (audio and/or video taped), questionnaires (students and teachers), interviews (six students chose as main informants and the drawing teacher), several documents, and students’ protocols.
The participant observation took place all over the school year and was registered in the teacher/researcher’s diary. The audio and video tapes used in this paper were taped in May. Students answered to questionnaires in the beginning of the school year (September), in the beginning of the second term (January) and in the end of the school year (June). The teachers only answered in the end of the school year. All the interviews also took place after the end of the school year (July). The documents were mainly collected in the beginning and at the end of the school year, and students’ protocols were collected during the whole school year.
We created six inductive categories based on an in-depth and successive content analysis: school’s culture, interdisciplinarity, didactic contract, leadership, argumentation and mathematical knowledge appropriation. In this paper we focus in the microproject, students’ work, and how students used their own experience to appropriate mathematical knowledge.
RESULTS Promoting this kind of project in the Dance School was a great challenge to the teacher/researcher and it was also a great pleasure. Students were deeply engaged in the tasks related to the intercultural and interdisciplinary microproject. For instance, they have a very hectic life, as they have classes all mornings, afternoons and part of the evening and, sometimes, when they have shows, they also have the rehearsals.
But they asked the dance teachers for permission to enter a bit later in their class – something unusual and usually forbidden – in order to participate on the 2nd day of batiks’ elaboration, as this part had to be in an extra class time. Thus, the batiks’ elaboration illuminated the need of a great organization and depended on students’ motivation, responsibility and engagement. At the end of the school year these were
some students’ voices accounting for what they learned through the batiks:
I think that... I think that it is easier and more interesting to learn. Because many people don’t like Mathematics (Madalena, I., p. 4).
Because it is nice and funny. And we can know some cultures from other countries (Carlota, I., p. 5).
In order to elaborate the batiks students needed to make templates with the school name (EDCN). The school’s name is deeply connected to this community’s identity and the Dance School culture is a very powerful one. Thus, producing the templates and knowing that there would be an exhibition at the end of the school year contributed for students’ engagement. These templates were made in mathematics and drawing classes, and the two teachers worked together.
During the batiks’ elaboration, students had a paper with a detailed description of the steps they needed to do. On the first day, students had to do a paste with flour, water and lime. But they needed to adapt the receipt to their own conditions using direct proportionality. This is illustrated in Figure 1 which shows the computations of one of the groups, using direct proportionality.
Weigh of dry cotton: 70g Weigh of small glass: 29g Weigh of great glass: 46g Figure 1. Students’ computations (direct proportionality) In mathematics classes after the batiks’ elaboration the teacher/researcher proposed some tasks based on the microproject and what students did. The first task began with a question where students should explain they needed to keep the proportionality among all the ingredients of the first paste (1st day). And they explained it as we can
see through the following example:
Dyad’s answer: We think that all ingredients should be divided proportionally in order to the paste to be consistent. If it doesn’t happen, the paste doesn’t be well done and thus we never could to construct batik.