CONNECTING MATHEMATICS TO OTHER DISCIPLINES AS A MEETING POINT FOR PRE-SERVICE TEACHERS

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1 CONNECTING MATHEMATICS TO OTHER DISCIPLINES AS A MEETING POINT FOR PRE-SERVICE TEACHERS Javier Diez-Palomar, Joaquin Gimenez, Yuly Marsela Vanegas, Vicenç Font University of Barcelona Research contribution (type B) to sub-theme 1. In this paper we introduce the case of a group of children being confronted to their previous conceptions about floatability. Mathematics emerges from their discussion with the teacher and the facilitators. Analysing this episode leads us to discuss with prospective teachers the challenges when connecting mathematics with other disciplines, as a modelling perspective (Garcia, Gascón, Ruiz & Bosch, 2006). We are also interested to see the possible changes on math teacher positions from Primary to Secondary School. THEORETICAL FRAMEWORK Steiner & Vermandel (1988) more than twenty years ago already claimed that mathematics education appears to be connected to other disciplines of the human knowledge. In this paper, we assume that Mathematics is a human activity focused on solving problematic situations of the real world, and focusing on emerging objects within mathematical practices. Thus, in our perspective, we assume that pre-service teachers need to develop a critical thinking based upon didactic analysis of teaching processes that considers five levels or types of analysis (Godino, Batanero & Font, 2007): 1) identifying mathematical practices; 2) developing configurations of objects and mathematical processes; 3) analysing didactic trajectories and interactions; 4) identifying the system of rules and meta-rules; and 5) assessing the educational competences of the teaching process. In this presentation, we concentrate on modelling perspective by analysing possible Research Study Trajectories (Barquero, Bosch & Gascon, 2006). METHODOLOGY In this paper we want to discuss a case study to illustrate how to encourage future teachers to identify (and design) practices potentially mathematical, by analysing configurations of objects and mathematical processes in order to be more critical on their task design or implementation, drawing on Godino et al (2007) theoretical approach. In the case study we are discussing here, Courtney (a middle-age teacher) use examples of the real life to challenge children s [mis] conceptions about floatability. She uses mathematical objects (size, weight, volume, etc.) to challenge 11 years old children to think about their previous ideas. Data comes from a lesson videotaped. We presented this video to a group of future teachers in the Faculty for discussion. We draw on children dialogues and interactions to discuss how Courtney questions lead children to dealt with cognitive conflicts, and how they solve them at the end of the lesson. Using Godino s et al (2007) analysis, we provide future teachers with tools for investigating classroom practices, in which they are able to conduct a detailed analysis (both descriptive and explanatory) of the teaching and learning process. To this end it is necessary, firstly, to develop and apply descriptive and explanatory tools that enable teachers to understand and respond to the question: what has happened here, and why? Secondly, according to Godino et al (2007) there is a need to develop and apply criteria of suitability, which somehow connects with the questions mentioned above. Drawing on Godino s approach, we say that an analytical tool is adequate when it is suitable to allow teachers to understand the practice that

they are observing / discussing. Thus, using the suitability criteria enable teachers to evaluate (analyse) a classroom practice, which may serve as a guide for improving that practice.

Courtney said: Today we are going to discover what is floatability. Children felt excited and curious. The teacher continued: look, many years ago there was a man named Archimedes.

2 they are observing / discussing. Thus, using the suitability criteria enable teachers to evaluate (analyse) a classroom practice, which may serve as a guide for improving that practice. FINDINGS Let s explain the experience that had been analysed. The journey begun with the teacher and the facilitators (two teenagers from the community) introducing the topic of the lesson. Courtney said: Today we are going to discover what is floatability. Children felt excited and curious. The teacher continued: look, many years ago there was a man named Archimedes. Anybody knows who Archimedes was? No answers. Courtney explained that Archimedes was a man living in the Ancient Greece, who was well known because his discoveries. She narrated the story of Archimedes and the King crown, and how Archimedes was able to figure out that the crown was in fact not made in gold, but mixed with other metals. Starting from this point, Courtney introduced the activity of the day: why some objects float while others sink to the bottom of a bucket of water? Figure 2a (left) The 3 possibilities regarding objects floatability and Figure 2b (right) Children objects classification according their floatability. During interaction and intentional analysis, future teachers explain that the teacher objective seemed to be to introduce Archimedes principle to the children. Thus, it appears a nice discussion about such interdisciplinary historical approach. In fact many future teachers don t know the relation between floatability forces as a physical problem and the emergence of equivalence volume problem as a mathematical issue. Such parenthesis, gives opportunities for epistemic and philosophical reflection. It is also discussed some ideas coming from Archimedes' pursuit concerning geometrical analysis, legacy and influence in Engineering and Mechanisms Design (Paipetis & Ceccarelli, 2010). Next, when analysing media tools and their influence, it s observed that Courtney and the two other facilitators provided recipients with water to the children, as well as a collection of objects: cloves, a drill, an empty can, an empty plastic bottle, an orange, a cork, a boat made of aluminium foil, a big ball of aluminium foil, a small ball of aluminium foil, a volcanic stone, a pumice, a sponge, a fork, and a melon. The question Courtney raised to the children was: which of these objects will float and which ones will sink? There were 3 different possibilities: (1) float, (2) totally sink, or (3) sink halfway (see figure 2a). Children first had to decide what would happen with all the objects. They had pictures to paste them on the case they though it would occur. For example, if they thought that orange will sink, then they would take the picture of the orange and paste it on the picture representing an object in the bottom of the bucket. Figure 2b show their final decisions. During epistemic and cognitive analysis, future teachers found that the teacher started to ask questions to the children, inquiring them about different situations. For instance, if we have a bottle of water, what do

3 you think it would happen to it? It will sink down the bucket or it will go halfway? Some children replied: It will float, it will float! Then the children pasted the picture of the bottle in the window, on the picture representing an objected floating. The lesson continued with the rest of the objects. At some point, Courtney asked what would happen with the can. Nayara said: The Trina 1... it would go halfway under the water. So, Courtney replied: Look... the Trina why you have pasted it in the picture representing objects that go halfway under the water? Susanne answered: I do not know... ; and Courtney continued: Why you think? At this point, Nayara claimed: Because it is made of metal, so it s heavier? OK..., said Courtney, and the melon? Why you pasted it on the picture representing objects that would sink to the bottom? Isabel said: Because it is heavy, I would say. The discussion continued for a while. During the interaction analysis, future teachers recognized that the teacher was making questions and children were answering them, sometimes guessing sometimes knowing for sure their answers. After that, Courtney proposed to take the actual real objects and experience whether or not they really sink or float in a bucket full of water. She was raising questions to challenge children. Courtney: Just one question... why did you paste the big drill in the picture of the objects sinking to the bottom, and the cloves in the one representing objects that sink halfway under the water? Isabel: Because the size! Children, teacher and facilitators move to the bucket full of water. Courtney: Would you try with a clove? What would happen to the big clove (the drill) and to the small clove? There [pointing to the window with the classification] you have classified them according to their size, right? You said that the big clove (the drill) will sink, while the small one will go halfway under the water, isn t it? Can we check it? [Children introduce both cloves in the bucket] Nayara: Uh! The two of them sink down. When analysing the case, future teachers observe that the children realized that the size of the object does not matter in terms of considering if it would float or not. The same happened to other characteristics. A table 1 is presented to the future teachers organized to see the changes of ideas before and after the experiment. Table 1. Children prior thinking and answers after the experiment Answer a priori Answer after the experiment Epistemological dimension Objects sink down because its size. Bigger objects are more likely to sink that small ones. Porous objects sink because they have holes and the water fills the holes. Gravity provokes objects to sink. Objects sink because their weight. Two objects (cloves and drill) with different sizes sink down under the water. The size is not related to floatability. They observed a sponge, a pumice and a volcanic rock. The pumice, despite its holes, floated. Having holes is not enough to explain why some objects sink while others do not sink. Children observed the both the volcanic stone and the pumice fell down out of the water, when someone drops them. However, if we drop them on the water, the volcanic rock sinks, while the pumice floats. Gravity does not explain floatability. After dropping the melon in the water, children observed that it floated, despite its weight. Thus floatability does not depend on weight. Size Gravity Weight / Mass 1 Trina is the brand of the orange juice within the can.

4 Some objects made with the same material float, while others does not. Children observed that the fork, and the ball made of aluminium sink, while the boat made of aluminium floated. Hence the material is not connected to floatability. Material This table helps the group of future teachers to reflect about cognitive and epistemic changes observed as a learning trajectory for children. Finally, another discussion is to observe how mathematics objects as volume emerge in absolute connection to physical object as forces. CONCLUSION We have evidences (not described here because the lack of space) that future teachers assumed that children learn mathematics when they are able to recognize and verbalize the rules and meta-rules characterizing mathematical objects, processes and systems (at least in such examples). This is a difficult work that has been largely studied in the past. In our case we can also notice the central role played by interactions in the whole picture. Children are being challenges by Courtney. The future teachers assume that teacher confronts them to unexpected situations. Learning then becomes situated in particular situations. Situated learning is also a contribution emerging from previous research (Lave & Wenger, 1991). Children need to go to the real world and experiment themselves in order to discover situations that may led them to modify their previous conceptions (about floatability in our case). In the discussion mathematics emerge as connected to other disciplines (as physics, etc.), while teaching mathematics also needs from the knowledge coming out from psychology, pedagogy, sociology, etc., as stated by Steiner. What can we learn from this case study, in terms of teacher training? According to Godino and colleagues (Godino, Batanero & Font, 2007) mathematics education should provide preservice and in-service teachers (and other professionals working in the field of education) with tools to identify mathematical practices, and to develop configurations of objects and mathematical processes, by analysing didactic trajectories and interactions. The analysis driven gives the future teachers with tools for redesigning mathematical tasks as research study trajectories. ACKNOWLEDGMENT The work presented was realized in the framework of the Project EDU Development of a program by competencies in a initial training for Secondary School. REFERENCES García, F. J., Gascón, J., Ruiz Higueras, L. & Bosch, M. (2006). Mathematical modelling as a tool for the connection of school mathematics, ZDM International Journal on Mathematics Education, 38(3), Barquero, B., M. Bosch y J. Gascón (2006). La modelación matemática como instrumento de articulación de las matemáticas del primer ciclo universitario de Ciencias: estudio de la dinámica de poblaciones, en L. Ruiz Higueras, A. Estepa y F. J. García (eds.), Matemáticas, escuela y sociedad. Aportaciones de la Teoría Antropológica de lo Didáctico, Jaén, Publicaciones de la Diputación de Jaén, Servicio de Publicaciones Universidad de Jaén, pp Godino, J. D., Batanero, C., & Font, V. (2007). The Onto-Semiotic Approach to Research in Mathematics Education. ZDM. The International Journal on Mathematics Education, 39(1-2), Lave, J. and Wenger, E. (1991). Situated Learning. Legitimate peripheral participation. University of Cambridge Press. Cambridge. Paipeitis, S., Ceccarelli, M.(Eds.) (2010). The Genius of Archimedes - 23 centuries of influence on mathematics, science and engineering. Kluwer. Dordrecht.

5 Steiner, H.G & Vermandel, A. (1988): Foundations and methodology of the discipline of mathematics education. Proceedings of the TME Conference. Antwerp, Belgium.

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