Developing Research in Mathematics Education

Twenty Years of Communication, Cooperation and Collaboration in Europe

Tommy Dreyfus, Michèle Artigue, Despina Potari, Susanne Prediger, Kenneth Ruthven, Tommy Dreyfus, Michèle Artigue, Despina Potari, Susanne Prediger, Kenneth Ruthven

• Publisher: Routledge
• Year: 2018
• ISBN:
  9781351625401
• Edition: 1
• Topic: Bildung
• Subtopic: Bildung Allgemein

Developing Research in Mathematics Education is the first book in the series New Perspectives on Research in Mathematics Education, to be produced in association with the prestigious European Society for Research in Mathematics Education. This inaugural volume sets out broad advances in research in mathematics education which have accumulated over the last 20 years through the sustained exchange of ideas and collaboration between researchers in the field.

An impressive range of contributors provide specifically European and complementary global perspectives on major areas of research in the field on topics that include:

• 
  
  

• the content domains of arithmetic, geometry, algebra, statistics, and probability;
• the mathematical processes of proving and modeling;
• teaching and learning at specific age levels from early years to university;
• teacher education, teaching and classroom practices;
• special aspects of teaching and learning mathematics such as creativity, affect, diversity, technology and history;
• theoretical perspectives and comparative approaches in mathematics education research.

This book is a fascinating compendium of state-of-the-art knowledge for all mathematics education researchers, graduate students, teacher educators and curriculum developers worldwide.

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1

FROM GEOMETRICAL THINKING TO GEOMETRICAL WORKING COMPETENCIES

Alain Kuzniak, Philippe R. Richard and Paraskevi Michael-Chrysanthou

1 Introduction

The chapter provides a comprehensive view of the research in geometry education that has been presented during CERME conferences. To write this chapter, all the CERME conferences proceedings since the beginning have been read again in order to track the main developments in trends, theories and methodologies and identify progress in research on the teaching and learning of geometry. Undeniably, this research field in the CERME conferences benefited from the creation of the group in 2003, which allowed the development of common theoretical frameworks and research trends concerning the teaching and learning of geometry in different educational levels and systems. This facilitated discussions and collaboration between researchers coming from different countries with various experiences in the field of geometry education, with different views on what research in didactics of geometry should be.

In the first section, the evolution of topics the group dealt with is traced. The emphasis is on the close link between theoretical and empirical aspects that has always guided the researchers of the group. Then, in the second section, some methodological and theoretical tools developed within the group are introduced and we show how they constituted a common support for studies in the field. In the third and fourth sections, the main findings from the group are given by emphasising the spirit of communication and collaboration built up through the different meetings. In a concluding section, a common research agenda is considered that could be organised to achieve a deeper understanding of geometrical thinking and competencies through the whole education.

2 Linking theoretical and empirical aspects: a brief history of the thematic evolution of the group

Through this short history of the group, we will show how the researchers involved in the group displayed a constant and growing interest in using frameworks that allow the connection between theoretical and empirical aspects. Beside references to general psychological (Piaget’s or Vygotski’s works) or didactical theories (Brousseau’s or Chevallard’s theories), specific theoretical and methodological developments were provided during the meetings. The van Hiele levels, Fischbein’s figural concept and Duval’s registers were among the most relevant cognitive and semiotic approaches that were adopted. Regarding the epistemological and didactical approaches, a decisive importance was granted to the geometrical paradigms and the Geometrical Working Spaces, and more recently to a lesser extent, on the notion of geometrical competencies. A great number of researches focused on different aspects of geometrical understanding, such as figural apprehension, visualisation and the effort of conceptualisation in geometry using different methodologies and frameworks. Among the numerous papers presented in the Working Group on Geometry, we can distinguish some recurrent topics that we present below in parallel with the history of the group.

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The first working group specifically dedicated to geometry was created in CERME 3. Since this meeting various names were given to this group including some nuances, from thinking of researching through teaching and learning, ending at the last edition (CERME 10) with the sole name Geometry. On Figure 1.1, which is drawn from the synthesis of CERME 7, themes and issues generated by the contributions are organised in a conceptual tree.

On Figure 1.2, some unifying characteristics related to geometric competencies that have been developed during CERME 8 are presented.

Further on, we will illustrate the evolution of the group with a short overview of the papers presented during these conferences, intending to show the richness of the themes developed throughout these meetings.

FIGURE 1.1 Main themes or issues of the group

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FIGURE 1.2 Geometrical competencies

2.1 Before the official creation of the group

In the first two sessions, even if the group did not officially exist, some communications were already centred on geometry with a diversity of viewpoints. Some reflexive papers (Kilpatrick, 1999) intended to provide references concerning curriculum issues and the reform of mathematics examination at a time when social and professional mobility was a major issue. Other contributions insisted on the need of metaphors and images (Parzysz, 2002) or focused on epistemological and cultural approaches about geometrical knowledge (e.g. Arzarello, Dorier, Hefendehl-Hebeke, & Turnau, 1999; Burton, 1999).

Mostly descriptive, the studies addressed the geometric content and often mentioned some general mathematical skills or cognitive and instrumental dimensions in geometric activity. They also indicated some “frictions” with other mathematical domains like algebra, and some typical interactions in the use of technologies with digital geometric software (Dreyfus, Hillel & Sierpinska, 1999).

In addition, the natural relationship of geometry to proof and proving was regularly examined, both in theoretical and philosophical essays and in reports of studies involving empirical research. They link proving, arguing, modelling and discovering processes. They also deal with the students’ way for proving and their use of a logical discourse, figural signs and technology when they are faced with proof exercises in geometry. Attention was also paid to the links between the discovery process, visualisation and instrumentation processes with software (Jones, Lagrange & Lemut, 2002).

There were also studies concerned with the teacher education and professional development in geometry and the organisation of courses (Houdement & Kuzniak, 2002).

Despite the fact that several works deal with mathematical proofs, visualisation and dynamic geometry, surprisingly, other natural links with geometry are virtually missing during the first congresses. In fact, few researches focus on modelling of physical phenomena using geometrical tools.

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2.2 Thematic from CERME 3 to CERME 5

During these conferences, the new geometry group starts the study of geometrical thinking using and developing the key concepts of paradigms, developmental stages and generalisation in space (Houdement & Kuzniak, 2004; Gueudet-Chartier, 2004; Houdement, 2007). At the core of many researches, visualisation is often directly connected to Duval’s registers of representation (Perrin-Glorian, 2004; Kurina, 2004; Pittalis, Mousoulides & Christou, 2007). In an original study, the visualisation issue is raised for sighted and blind students (Kohanová, 2007). Other papers focus on the notion of instrumentation, both with classical drawing tools (Vighi, 2006; Bulf, 2007; Kospentaris & Spyrou, 2007) or digital tools (Rolet, 2004). Education for future teachers (Kuzniak & Rauscher, 2006) and reasoning (Ding, Fujita & Jones, 2006; Markopoulos & Potari, 2006) appear as topics of continuity, while concepts and conceptions remain key themes in many studies (e.g. Modestou, Elia, Gagatsis & Spanoudes, 2007; Marchini & Rinaldi, 2006).

2.3 Thematic from CERME 6 to CERME 7

The theoretical and methodological dimensions of research in geometry remain prominent topics, especially in further development of the notions of geometrical work and Geometrical Working Spaces. Many points were considered as a common background, as they were developed during former sessions. These points were related to the use of geometrical figures and diagrams (Deliyianni et al., 2011) and to the understanding and use of concepts and proof in geometry (Gagatsis, Michael, Deliyianni, Monoyiou & Kuzniak, 2011; Fujita, Jones, Kunimune, Kumakura & Matsumoto, 2011). For an epistemological and didactical approach, researchers used the geometrical paradigms and geometrical work spaces. Attention is also paid to 3D geometry forms of representation through the possible use of digital tools (Mithalal, 2010; Hattermann, 2010; Steinwandel & Ludwig, 2011). In addition to the usual geometrical topics, special attention is paid to general or cross-cutting aspects, such as educational goals and curriculum in geometry (Girnat, 2011; Kuzniak, 2011), communication and language (Bulf, Mathé & Mithalal, 2011), the teaching, the thinking and the learning processes in geometry. Moreover, Mackrell (2011) questions the interrelations between numbers, algebra and geometry, especially in digital environments.

2.4 Restructuring in CERME 8 to CERME 10

More recently, the working group sought to revisit and extend the issue of geometrical thinking and geometrical work by reformulating it in terms of four geometrical competencies (reasoning, figural, operational and visual) organised in a tetrahedron (see Figure 1.2). Each geometric competence constitutes a pole of geometrical thought and it is the study of the link between these competences that makes it possible to better understand the global functioning of geometrical thought (Maschietto, Mithalal, Richard & Swoboda, 2013). Further on, at CERME 9, there were more specific contributions about the way geometry is, or should be, taught and the four competencies were used as a general way of describing the geometrical activity and for creating links between different points of view (theoretical and empirical). In this group the discussions were related to geometry teaching and learning (Douaire & Emprin, 2015; Kuzniak & Nechache, 2015) and issues such as teaching practices and task design (Mithalal, 2015; Pytlak, 2015). Furthermore, cultural and educational contexts modifying the geometry curricula were also discussed, introducing a new issue about the role of language and social interactions in the teaching and learning of geometry. In CERME 10, the four competencies were used to describe geometrical thinking: reasoning, figural, operational and figural. The group took these dimensions as a background that was very helpful to understand each other and to compare our approaches to the issue of what is at stake in the teaching and learning of geometry.

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Three main issues were addressed during the working group: The role of material activity in the construction of mathematical concepts, including using instruments, manipulation, investigation, modelling . . .; Visualisation and spatial skills; Language, proof and argumentation. In comparison to the previous CERME, this time psychological points of view, among others, were represented. This raised new questions, often with very different theoretical and methodological backgrounds sometimes far from mathematics.

3 Studying the teaching and learning of geometry through a common lens

Since the creation of the group, the development of shared theoretical frameworks was central to ground collaboration between participants. This point is particularly evident in CERME 3, where the main theoretical concerns of the group were summarised by Dorier, Gutiérrez and Strässer (2004). Their classification is used to highlight some theoretical approaches which have been reinforced by their presentation and discussion during CERME conferences.

3.1 Geometrical paradigms

The history of geometry shows two contradictory trends. First, geometry is used as a tool to deal with situations in real life but, on the other hand, geometry for more than two thousand years was considered the prototype of logical, mathematical thinking and writing after the publication of Euclid’s “Elements”. These contradictory perspectives are taken into account in geometry education by Houdement and Kuzniak (2004) with geometrical paradigms. Three paradigms are distinguished. “Geometry I: natural geometry” is intimately related to reality; experiments and deduction grounded on material objects. “Geometry II: natural/axiomatic geometry” is based on hypothetical deductive laws as the source of validation with a set of axioms as close as possible to intuition and may be incomplete. “Geometry III: formal axiomatic geometry” is formal, with axioms that are no more based on the sensory reality. These various paradigms are not organised in a hierarchy making one preferable to another, but their work horizons are different and the choice of a path toward the solution is determined by the purpose of the problem and the researcher’s viewpoint.

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Useful to provide a method to classify geometrical thinking, geometrical paradigms have also been helpful to interpret tasks eventually given to students and future teachers and can be used to classify the students’ productions, offering an orientation for the teacher of geometry. In contrast to van Hiele’s theory, this approach is not dependent on the general thinking and reasoning development, but relates more to an epistemological viewpoint.

3.2 Development stages

The so-called “van Hiele levels” of geometrical reasoning are among the most used theoretical frameworks for organising the teaching and learning of geometry according to development stages. The description of the “van Hiele levels” already gives hints to fundamental links between these levels and the model suggested by Houdement and Kuzniak (2004). There was some discussion whether geometrical knowledge progresses through sequences of stages. Some papers presented (Braconne-Michoux, 20...