La integración horizontal

In my opinion Jaworski has made two significant contributions to research on the didactic triangle. The first contribution concerns the development and analysis of teachers’ work in classrooms. During her PhD research Jaworski came up with the ‘Teaching Triad’ as an analytical tool to look at the attempts of four secondary

school teachers to make their mathematics lessons more investigative in line with a constructivist view of children’s learning. As defined earlier the Teaching Triad

consists of three categories: management of learning, sensitivity to students and mathematical challenge. Goodchild and Sriraman (2012) believe the triad helps in answering a question which drives developmental research in mathematics education: How might teachers be empowered to become aware of and work on

relationships among themselves, their students and the mathematics? They further

state that research and development activity (like Jaworski’s) that has focused on problem solving, inquiry and investigation, and teachers’ engagement with students

in classrooms is basically concerned with students’ engagement with mathematics, and the mathematical challenge they experience. They believe that researchers taking these issues as the focus for their enquiries address the fundamental relationships represented within the didactic triangle. I believe this is a fine tribute to Jaworski’s work.

A corollary of Jaworski’s Teaching Triad is that it can be used as a developmental

tool in teacher education for those teachers seeking to follow a constructivist- compatible approach to their work. For instance, the notion of mathematical challenge reminds us of the Vygotskian zone of proximal development which purports that children should be challenged at the frontiers of their current knowledge. This connects with the idea of sensitivity to students as teachers have to

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be aware of pupils’ current knowledge levels and pitch their lessons accordingly. It also reminds us that pupils’ interests should be borne in mind by teachers when they

are planning lesson topics. Potari and Jaworski (2002) define harmony as the extent to which the degree of challenge in a lesson is appropriate to the particular cohort of students involved. Harmony involves achieving a balance between sensitivity and challenge. The third category of management of learning informs us that constructivist-compatible classrooms have to be set up in a certain way, usually involving groupwork, so that pupils can pursue lines of inquiry and develop their critical thinking skills. Leonard (2003) comments that constructivist lessons are often described as student-led learning where the teacher debriefs before and after but sets up a learning situation where the pupils discover the solution themselves. Therefore, the heart of constructivism in education is critical thinking. It can be seen that I am an enthusiast of Jaworski’s Teaching Triad and I intend to use it in my own research

both as a developmental and analytical tool. I will return to this issue in Chapter 4.

Another Jaworskian contribution is that she has expanded on the didactic triangle in her recent research. She suggests adding the role of researchers in the classroom, or didacticians, to use her term, as an additional node or adjunct to the didactic triangle. Her rationale is that teachers and didacticians share a reflexive relationship. She comments that although teachers’ knowledge in practice goes far beyond didacticians’ knowledge, the complementary knowledge of research and theory

brought by didacticians provides stimulus and inspiration to which cohorts of teachers are able to respond. The relationship is reflexive in that teachers develop new approaches to working with their students such as using inquiry modes of learning. In tandem with this didacticians learn about how theories and research findings can and do “influence the practice of real teachers in real schools and

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classrooms acting under all the constraints of institutional and political pressure”

(Jaworski, 2012). Jaworski states that as a didactician herself, she is aware of the power of this collaborative knowledge and associated developmental practice in addressing approaches to educating students in mathematics. She offers the diagram (Figure 4) below as a way of representing the reciprocal influence of didacticians and teachers on the didactic triangle.

Figure 4: The didactic triangle for several teachers, their students and didacticians (Source: Jaworski 2012)

However, she is quick to point out that the above diagram might be seen to capture relationships at a particular point in time but that it does not recognize teaching development in any clear way and that it does not recognise elements of situation and context. To reflect elements of learning and development for both didacticians and teachers over time she offers the schematic representation as in Figure 5 which follows on the next page.

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Figure 5: The expanded didactic triangle (Source: Jaworski 2012)

Jaworski states that the lower circle represents the traditional didactic triangle, connecting teacher, student and mathematics and attempting to characterise elements of the relationships involved within a community of teachers, their students and mathematics. It includes both the didactical and pedagogical thinking of the teacher in converting mathematics into classroom action, the interactions between teacher and students, the ways in which both teacher and students interact with mathematics and ways in which teachers themselves interact within the school context. It also encompasses the teaching and learning philosophies present and the sociocultural contexts in which the mathematics classrooms are located.

The upper circle is different in nature to the lower one. Whereas the lower circle strives to characterise situations, activity, events and relationships (what Jaworski calls the situational), the upper circle purports to represent the developmental processes which occur when teachers and didacticians inquire into all that is characterised in the lower circle. Therefore, the upper circle can be labelled as being developmental in nature. Jaworski states that the upper circle represents co- development between teachers and didacticians, a meta-dimension on the lower. It focuses on the learning of both groups as they participate in insider and outsider

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research with clear learning outcomes to be achieved as both groups build new identities and increase their agency. In my own research I hope to comment further on developing Jaworski’s work on the Teaching Triad in chapter 6. For now, I move the focus back to constructivist-compatible pedagogies.