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Building Mathematical Knowledge in Teaching by Means of Theorised Tools

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Mathematical Knowledge in Teaching

Part of the book series: Mathematics Education Library ((MELI,volume 50))

Abstract

The literature on mathematical knowledge in/for teaching demonstrates a concern in the mathematics education community to deepen understanding of what knowledge a teacher has/should have for the teaching of mathematics, and how this knowledge is constructed. In order to make progress from the present state of affairs, this concern must be translated into efforts to provide the necessary tools to further that understanding.

One can differentiate the tools or the studies presented in the foregoing chapters in terms of their scope (the contexts and the limitations involved), whether these tools are either generic or specific to the study of mathematical knowledge in teaching, whether the teacher’s learning environment or the context of the study is individual or collective. To this extent, it is important to analyse the role given to teacher reflection in these chapters, and think closely about the potential of a tool to promote professional knowledge and development.

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Notes

  1. 1.

    Research forum1: Teacher knowledge and teaching: Considering a complex relationship through three different perspectives. Several authors in Tzekaki, Klardrimidou, & Sakonidis (2009).

  2. 2.

    The introduction by Sullivan is titled “Knowledge for Teaching Mathematics: An Introduction”. Section 1 is headed “Mathematics Discipline Knowledge for Teaching”.

  3. 3.

    The application of these tools is typically realised through activities or tasks promoting learning on the part of the teacher. These tasks are not the object of analysis in this section. A repertoire of tasks oriented towards teacher training can be found in Clarke, Grevholm, & Millman (2008), including Section A, “Tasks as a Tool for Exploring the Cyclical Nature of Learning and Developing Reflection in the Teaching of Mathematics”, which considers the importance of reflection in mathematics teaching (to which I shall return), Section B, “Tasks as a Tool for Developing Mathematical Knowledge for Teaching”, and Section C, “Tasks as a Tool for Developing Knowledge through and for Practice”, tackling the importance of starting from the practice of teaching in order to construct teachers’ knowledge, including a chapter relating tasks in groups of practising teachers with tasks in initial teacher training (Carrillo & Climent, 2008).

  4. 4.

    Note that the KQ was originally developed in the context of intial primary teacher training, although it has subsequently been applied to in-service contexts (Turner, 2008).

  5. 5.

    Other theoretical tools can be employed according to the focus of the analysis of mathematical knowledge, such as: van Hiele levels for geometrical concept formation (Gutiérrez & Jaime, 1998; Van Hiele, 1986), the APOS theory (Dubinsky, 1994), and Sfard’s stages in the process of acquisition of mathematical notions (Sfard, 1991). Although not taking a combinatory approach to theory, as in the chapter by Tirosh et al., see the essay by Carrillo, Climent, Contreras, and Muñoz (2007) and Muñoz-Catalán, Carrillo, and Climent (2009) on applying Sfard’s work to the professional development of mathematics teachers.

  6. 6.

    Concern for student learning and for promoting the teacher’s learning from this is also a feature of Learning Study, a combination of Lesson Study and design experiment: “In a learning study teachers get the opportunity to observe colleagues teach the same thing. This is one of the features of a learning study that makes it appropriate to mathematics teacher education” (Runesson, 2008, p. 170).

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Authors and Affiliations

  1. Faculty of Educational Sciences, University of Huelva, Huelva, E21007, Spain

    José Carrillo

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  1. José Carrillo
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Correspondence to José Carrillo .

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Editors and Affiliations

  1. Fac. Education, University of Cambridge, Hills Road 184, Cambridge, CB2 2PQ, United Kingdom

    Tim Rowland

  2. Fac. Education, University of Cambridge, Hills Road 184, Cambridge, CB2 2PQ, United Kingdom

    Kenneth Ruthven

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Carrillo, J. (2011). Building Mathematical Knowledge in Teaching by Means of Theorised Tools. In: Rowland, T., Ruthven, K. (eds) Mathematical Knowledge in Teaching. Mathematics Education Library, vol 50. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9766-8_16

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