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Humans-with-media and continuing education for mathematics teachers in online environments

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This paper begins by situating online mathematics education in Brazil within the context of research on digital technology over the past 25 years. I argue that Brazilian research on technology in mathematics education can be divided into four phases, and then present an example that “blends” aspects of the second and third phases. Phase two can be characterized by research with software designed to address traditional mathematics topics, such as functions, while the third phase is characterized by online courses. The data presented show creative solutions for a problem designed for collectives of humans-with-function-software. The paper is analyzed from a perspective that emphasizes the role of different technologies as teachers and professors collaborate to produce knowledge about the use of mathematical software in regular face-to-face classrooms. A model of online education is presented. Finally, the paper discusses how technology may change collaboration and teaching approaches in continuing education, as it allows for greater integration of online learning with teachers’ classroom activities in schools. In this case, the online platform plays an active role in the learning collective composed of humans-with-media.

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  1. Technology, other Media and Mathematics Education Research Group (

  2. Details about this network of schools and the online environment will be provided in Sect. 4.

  3. The Bradesco Foundation is funded by the Bradesco Bank and has a social mission, as their schools are generally located in poor neighborhoods. Although it is a private foundation, their schools are free of charge and they develop intense continuing education activities with their teachers. It has at least one school in each of the 26 Brazilian states and in the capital, Brasília.

  4. This project was led by two members of GPIMEM, Marcelo C. Borba and Rubia B.A. Zulatto. Since then the second participant has changed her name to Rubia B. Amaral.

  5. For a more thorough discussion of use of artifacts in mathematics education, the reader should refer to Villarreal and Borba (2010).

  6. Literature in mathematics education also provides ample evidence that teachers can learn to use ICT in their classrooms through participation in face-to-face sessions that mirror those in which they are to teach with ICT (see, for instance, Stols and Kriek 2011).

  7. Original in Portuguese: “[…] A utilização do recurso—do software winplot—em sala de aula antes mesmo de finalizarmos o curso. […] O compartilhar de experiências cotidianas de sala aula, dos professores da Fundação, durante as videoconferências do curso”.

  8. Original in Portuguese: “As apresentações de colegas de outras escolas nos mostrou também que como pode ser diversificada a utilização e como vários pontos de vista podem ser desenvolvidos em cima de um só problema”.

  9. I would like to thank all the reviewers for their insightful comments, but in particular the one who brought to my attention the “mirror metaphor”, which I decided to incorporate into the paper.


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I would like to thank Ricardo Scucuglia Rodrigues da Silva, member of GPIMEM, doctorate student at the University of Western Ontario for his comments as I was preparing this paper. I would also like to similarly thank Anne Kepple for comments. Finally, I would like to thank Rubia Amaral and Aparecida Chiari for helping me select the important excerpts from the teacher evaluation of the online courses discussed in this paper. I would like to thank participants of the conference for comments and questions that led to improvements in this version.

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Correspondence to Marcelo C. Borba.

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An earlier version of this paper was published in 2010, in the proceedings of The Canadian Mathematics Education Study Group Annual Meeting held in 2009, in York University, Toronto.

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Borba, M.C. Humans-with-media and continuing education for mathematics teachers in online environments. ZDM Mathematics Education 44, 801–814 (2012).

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