Gestión del conocimiento

between teacher, pupil and the subject of mathematics. This interplay is known as the didactic triangle. In a special edition of the ZDM Mathematics Education Journal (Volume 44, 2012) several authors, including Jaworski, combined to share their research on the didactic triangle. As a model the didactic triangle linking mathematics, teachers and students has been used by researchers wishing to consider teaching-learning interactions in mathematics classrooms. The triangle originated in the work of Chevallard (1985) and later with Brosseau (1997). While acknowledging the seminal importance of the triangle as a way of describing the teaching and learning situation in the subject of mathematics, more recent research has focused on the inadequacies of the triangle as a means of capturing the complexity of what happens in mathematics classrooms. For instance, Chevallard expands on the original triangle by placing it within a circle called the ‘noosphère’. The noosphère is defined as the bureaucratic universe that shapes schooling, which influences what happens in classrooms. The noosphère in turn is placed within a rectangle called the ‘environnement’ which reflects the cultural and contextual factors that result in the

transformation of mathematics as practised to mathematics as taught (Schoenfeld, 2012). An example from Ireland could be the ‘high stakes’ Leaving Certificate

examination which determines what is taught in classrooms, how students view mathematics and, depending on their grades, who goes to university.

Brousseau labels the formal school environment the système didactique. Within this school environment he wishes to create and study Didactical Situations that assist student engagement with rich mathematics. These Situations have a number of properties. “They are intended to be mathematically and pedagogically rich, so that

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by engaging in them students will develop deep understandings of the mathematics” (Schoenfeld, 2012, p.589). The emphasis is not on the teacher ‘telling’ the pupils the

required information but on the students being asked to do a task in which their engagement with the mathematics to be learned is central. The creation of such situations would also be an aim of my own research. Therefore, what is envisaged is a task which requires some element of investigation by the pupils and not just the performance of routine algorithms.

The ‘didactical contract’ is another core component of ‘Didactique’. This is the classroom version of the ‘social contract’ which is the set of largely covert rules that

govern the interactions of students and teacher. This is akin to the term ‘sociomathematical norms’, as promulgated by Cobb and Yackel (1996) and

discussed elsewhere in this chapter. Schoenfeld (2012) gives the example of a classroom being either focused on ‘answer getting’ or alternatively on mathematical ‘sense-making’. If the former is emphasised students will obtain answers to given

tasks and see no need to explore the mathematics further. If the latter is emphasised students will seek out underlying mathematical reasons as to why things operate the way they do, and they will feel compelled to explain their understandings. In this way a sociomathematical norm or contract is established in the classroom as to what constitutes an appropriate mathematical explanation. If the emphasis is on obtaining correct answers to set written tasks then the authority of the textbook may suffice. If the emphasis is on explanation, the classroom community will require students to give clear and unified mathematical justifications. Through their classroom experiences students not only come to understand the rules of the game but also to shape such rules. In turn, the rules of the game shape not only students’ actions but also their beliefs about the nature of mathematics. Schoenfeld (2012) comments that

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a central part of Didactique is a concern for the nature of the didactical contract in any classroom, and the creation of Situations that are favourable to the higher levels of mathematical engagement and learning. Schoenfeld (2012, p. 592) cites Brosseau (1997) in stating that any mathematical knowledge to be attained should be in its full richness –“not merely a statement of a mathematical concept, but its meaning, its uses, its connections to prior knowledge, the context in which it is likely to be encountered, the language commonly used to express it.” Schoenfeld contends that

French authors like Chevallard and Brosseau have more of a cultural construal of the didactical triangle than has typically been the case in the English-speaking world. However, several authors have sought to remedy this situation.

For instance, Rezat and Sträβer (2012) argue for the addition of a fourth vertex to the didactic triangle to turn it into a tetrahedron. The fourth vertex would concern the use of tools or artefacts in the teaching of mathematics. The authors contend that mathematics content is dependent on artefacts (embodiments) to assist the teaching/learning process. Such tools could be physical like mathematical textbooks, rulers, compasses, log tables and, of course, digital technologies. However, tools could also be non-physical such as, for instance, language, gestures and diagrams. Vygotsky is quoted by Rieber and Wollock (1997, p.85) as he introduces another element of sophistication when he distinguishes between psychological and technical tools:

The most essential feature distinguishing the psychological tool from the technical one is that it is meant to act between mind and behaviour, whereas the technical tool, which is also inserted as a middle term between the activity of man and the external object, is meant to cause changes in the object itself. The psychological tool changes nothing in the object.

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Rezat and Sträβer (2012) comment that the central aim of tool use in didactical situations is to change the students’ cognition of mathematics and not the

mathematics itself and that, therefore, all tools used in the teaching and learning of mathematics, be they physical or not, can be considered psychological tools. According to Solomon et al. (2006) students use the tools and artefacts of culture to assist their conceptual development and express themselves more meaningfully. Since the notion of tools is easily tainted with the idea of something material Rezat and Sträβer (2012) prefer the broader notion of artefacts. Bringing in the fourth dimension of ‘artefact’ (artifact in American usage) means that the didactic triangle

becomes a tetrahedron as follows in Figure 2 below:

Figure 2: Tetrahedron model of the didactical situation

Straβer (2009) points out that it may be worthwhile to think of what I will term ‘spheres of influence’ surrounding the tetrahedron. An example would be the sphere

containing the personnel and institutions interested in the teaching and learning of mathematics; the ‘noosphere’ to quote Chevallard’s (1985) term. More recently, Geiger (2014) speaks of ‘spheres of social context’ (SSC) which are inspired by Chevallard’s ‘noosphere’ but they differ in that they are peculiar to the types of

interaction that take place in individual, small group and whole group settings. Geiger (2014) contends that social interactions, in harmony with available secondary artefacts, influence the transformation of students’ understanding of mathematical

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knowledge, as well as their ways of reasoning and sense making, in different ways according to the particular social setting in which learning is located. Therefore, his SSCs are not concentric and independent entities but rather that SSCs interact.

Rezat and Sträβer (2012) comment that Geiger’s extension of the tetrahedron model

draws definite attention to social settings in the classroom and their effect on instrumented learning but that it does not include societal and institutional influences. To overcome this shortcoming of the model they draw on Engestrom’s (1998) model of the activity system from the perspective of cultural-historical- activity theory. Engestrom sees activity as a collective systemic formation that has a complex meditational structure. The attraction for Rezat and Sträβer is that less visible social mediators of activity- rules, community and division of labour- are depicted at the bottom of the model. They contend that artefacts play a crucial role in the system because they serve to focalise the other aspects of the entire system. In turn, they derive a sophisticated ‘socio-didactical tetrahedron’ which is depicted in

Figure 3 which follows on the next page. It can be seen that the ‘socio-didactical tetrahedron’ is a sophisticated three-dimensional representation and expansion of the

original two-dimensional model (teacher-student-mathematics) as outlined by Jaworski (2012) at the beginning of this section.

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Figure 3: A socio-didactical tetrahedron (Source and StraBer 2012)

Rezat and Sträβer are quick to point out the limitations of their model of the socio-

didactical tetrahedron. For instance, they state that a direct connection between the student and the public image of mathematics/relevance of mathematics in society is missing. However, I have to comment that these can be linked indirectly via the vertex ‘mathematics’ of the original didactic tetrahedron. In other words, the public

image of mathematics is relevant in any discussion of the interplay between student and mathematics. Speaking of discussion, they also make the interesting point that they would place research by Yackel and Cobb (1996) on sociomathematical norms on the triangle linking artefacts with the conventions and norms about being a student and those about being a teacher. This is because they consider the role of ‘discussion’ in the negotiation of sociomathematical norms to be an artefact.

In this section I have considered various authors’ contributions to the research on the

didactic triangle and its expansions. I now wish to consider Jaworski’s contribution as her work is a central component of this thesis.

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