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The role of visual representations for structuring classroom mathematical activity

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Abstract

It is our presupposition that there is still a need for more research about how classroom practices can exploit the use and power of visualization in mathematics education. The aim of this article is to contribute in this direction, investigating how visual representations can structure geometry activity in the classroom and discussing teaching practices that can facilitate students’ visualization of mathematical objects. We present one illustrative episode that shows how drawings of geometrical figures have a powerful role in structuring and modifying the mathematical activity in the classroom. It was selected from a database that we have been building to investigate the learning of mathematics in public elementary schools in Brazil. The framework of Activity Theory helped in the characterization of the episode as a system of interconnected activities. We discuss the changes and transformations perceived in those activities; and we explore the idea of miniature cycles of learning actions to focus on the mathematical learning that is taking place. We describe the dynamics and the complexity of the ongoing activity in the calculation of areas; and, how drawings form a part, and show their influence, in it. We argue that part of this influence was associated with the contradiction between abstract mathematical ideas and their empirical representations, revealed by the tensions perceived in the activities analysed; and, simultaneously, that we could see as an impelling force for the learning of the rules and norms which regulate the use of visual representations in school mathematics.

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Notes

  1. According to Bishop mathematical knowledge in culture develops from the following six key activities: counting, locating, measuring, designing, playing, explaining (Bishop, 1991).

  2. One of these evaluations was the nationwide Prova Brasil (Brasil, 2010) (www.inep.gov.br), and the other one was the international evaluation Pisa (OECD, 2009).

  3. According to OECD (2009), the key aspects of space and shape are: recognising shapes and patterns in shapes; describing, encoding and decoding visual information; understanding dynamic changes to shapes; identifying similarities and differences; identifying relative positions; interpreting two-dimensional and three-dimensional representations and the relations between them; navigation through space.

  4. All names are pseudonyms

  5. Interestingly enough, he does not take the same amount of care when dealing with the interpretation of the outside rectangle as he did with the internal triangle. For the outside figure (from which we also do not know the measures of the angles and of all sides) he just states: “this here is a rectangle…isn’t it? with these measures here…” . In doing so, he was probably aware of the fact that this was not the most appropriate drawing to raise the tension, as it looked pretty much like the figure it was supposed to represent, as in the previous situations.

  6. \( \left( {8 \times 5} \right) \div 2 = 20 \)

  7. \( \left( {8 \times 5} \right) - \left[ {\left( {5 \times 5} \right) \div 2 + \left( {4 \times 3} \right) \div 2 + \left( {8 \times 1} \right) \div 2} \right] = 40 - 22.5 = 17.5 \)

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Acknowledgments

The authors want to declare, first of all, their gratitude to the teacher and students involved in this study, for all we have learned with them. We also wish to thank the participants of the Grupo de Pesquisa e Estudos Histórico-Culturais em Educação Matemática e em Ciências (CNPq) for their valuable comments on earlier versions of this article and the financial support received from the Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq and from Fundação de Amparo à Pesquisa de Minas Gerais—FAPEMIG. We express as well our gratitude to the anonymous reviewers and to the editor for their insightful and critical comments on the manuscript of this paper.

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  1. Universidade Federal de Minas Gerais, Minas Gerais, Brazil

    Maria Manuela David & Vanessa Sena Tomaz

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  1. Maria Manuela David
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  2. Vanessa Sena Tomaz
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Correspondence to Maria Manuela David.

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David, M.M., Tomaz, V.S. The role of visual representations for structuring classroom mathematical activity. Educ Stud Math 80, 413–431 (2012). https://doi.org/10.1007/s10649-011-9358-6

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