24 SES 02 B, Representations in Mathematics Education
Teachers’ practice strongly influences students’ learning and a very important aspect of such practice is the way teachers use mathematical representations. In this communication we seek to understand how teachers promote their students’ learning and understanding of mathematical representations. To achieve this, we analyse the actions of two third grade teachers, as they work on a task involving the construction and interpretation of a statistical graph.
The NCTM (2000) defines “representation” as both the process of representing and the resulting product. It also states that students need to know a great variety of representations to understand concepts, to be able to choose the appropriate representation to deal with a specific situation and to make transformations between representations. Stylianou (2010) indicates that we need to use representations to interpret, organize, and understand the information given in a problem statement and also to find the right answer and to monitor and evaluate the solving process. However, the connection between a representation and its meaning is often difficult to understand and the transformations between representations are sometimes hard to make. Bishop and Goffree (1986) indicate that the interpretation and transformation of representations may be demanding for students, because they have to learn and understand the specific vocabulary and features of each representation.
Stylianou (2010) states that teachers should offer their students the opportunity of learning and understanding different representations. In her view, the representations that teachers use during a lesson may promote the discussion of ideas and lead students to generate new representations. She also indicates that, despite using various representations in their teaching, teachers sometimes do not do it consciously and with a well-defined intention. In addition, she claims that teachers tend to focus on representations as products (drawings, graphs, diagrams) and not so much on representations as processes (of representing mathematical objects and ideas).
In order to understand teachers’ practices, it is necessary to observe and analyse their classes, their actions and their discourse (Ponte, Quaresma & Branco, 2012). Jaworski and Potari (2009) also indicate that we may look at teachers’ practice paying attention to their actions. Teachers’ practice is guided by teachers’ motives and this practice only makes sense within the context of their activity. Ponte (2005) identifies three main moments that often occur as students work on a task and that tend to frame the nature of teachers’ practice: (i) presentation and interpretation of the task, usually in whole class discussion, (ii) students’ autonomous work (individually, in pairs or in small groups), and (iii) presentation and whole class discussion of the students’ solutions and final synthesis. Ponte, Mata-Pereira and Quaresma (2013) indicate four main teachers’ actions during a mathematical discussion: (i) inviting students to begin a discussion, (ii) supporting and guiding students’ participation and leading them through questioning, (iii) informing/suggesting by giving some information or validating students’ arguments, and (iv) challenging students to make inferences, justify statements or evaluate strategies. Of course, the tasks that teachers chose to propose to their students are also a very important element of their practice. For example, Swan (2007) states that the success of a task depends on what teachers do, the role that they assume, the way how they introduce the task, the kind of questions that they make to students, and the way they manage the whole class discussion.
Bishop, A., & Goffree, F. (1986). Classroom organization and dynamics. In B. Christiansen, A. G. Howson & M. Otte (Eds.). Perspectives on mathematics education (pp. 309-365). Dordrecht: D. Reidel. Jaworski, B., & Potari, D. (2009). Bridging the macro- and micro-divide: Using an activity theory model to capture sociocultural complexity in mathematics teaching and its development. Educational Studies in Mathematics, 72, 219–236. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author. Ponte, J. P. (2005). Gestão curricular em Matemática. In GTI (Ed.), O professor e o desenvolvimento curricular (pp. 11-34). Lisboa: APM. Ponte, J. P., Mata-Pereira, J., & Quaresma, M. (2013). Ações do professor na condução de discussões matemáticas. Quadrante, 22(2), 55-81. Ponte, J. P., Quaresma, M., & Branco, N. (2012). Práticas profissionais dos professores de Matemática. Avances en Investigación en Educación Matemática, 1, 65-86. Stylianou, D. A. (2010). Teachers’ conceptions of representation in middle school mathematics. Journal of Mathematics Teacher Education, 13, 325-343. Swan, M. (2007). The impact of task based professional development on teachers’ practices and beliefs: A design research study. Journal of Mathematics Teacher Education, 10, 217-237.
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