Categorising constructivist teaching: paradigm and assumptions

acknowledging that in ontological terms there will be multiple, socially constructed realities. In terms of epistemology I can foresee an interactive link between myself as researcher and the participant teachers, especially when this interplay involves possible changes of practice for the teachers involved. As regards methodology the

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research will be primarily qualitative, dialectical in approach with ‘thick description’ of the contextual factors influencing the teachers’ work. These assumptions should

convey that I wish to work within the constructivist paradigm. From a sociocultural perspective it should be interesting to track how the relationship between the teacher participants and me changes over time and how it impacts on our evolving views of constructivism.

As I explained in chapter two different authors have chosen different ways to categorise constructivist teaching for research purposes. Jaworski (1994) is the author whose framework I draw upon in conceptualising teachers’ mathematical practice. She used the term ‘Teaching Triad’ to refer to three domains, which in her view, captured the important elements of such teaching. The domains were named as ‘management of learning’, ‘sensitivity to students’ and ‘mathematical challenge’.

Jaworski (1994) elaborates as follows:

Management of learning is manifested in a set of teaching strategies and beliefs about teaching which influence the prevailing classroom atmosphere and the way in which lessons are conducted. Sensitivity to students is inherent in the student- teacher relationship and the teacher’s knowledge of individual students and influences the way in which the teacher interacts with, and challenges, students. Mathematical challenge arises from the teacher’s own epistemological standpoint and the way in which she offers mathematics to her students depending on their individual needs and levels of progress (pps. 107-108).

The three categories are individual in identity, but are closely interrelated and Jaworski (1994) believes they have the potential to describe a complex classroom environment; provided the teacher involved is working to a constructivist agenda. Before I could analyse the data emerging on a constructivist-compatible approach to teaching I decided to take advice. In order to further develop my understanding of the origins of her work I contacted Barbara Jaworski by email, indicating my admiration for, and my use of, her work on constructivism. In 1991 she completed a

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PhD thesis with the Open University entitled Interpretations of a Constructivist

Philosophy in Mathematics Teaching. This study became the basis of her 1994 book

called Investigating Mathematics Teaching: A Constructivist Enquiry. She generously agreed to speak to me by telephone and I rang her on 24th January, 2012 at 11.20am. We spoke for about thirty minutes. She gave the background to her own research on constructivism in the late 1980s in British classrooms. She stated that contextual factors are crucial. By this, she meant that it was easier in the late 1980s for teachers to adopt a constructivist approach to their work, as the curriculum was less prescribed than it is now. She raised a concept for me called intersubjectivity; a concept upon which she had elaborated in work she co-authored with Potari (2009). Jaworski and Potari (2009) state that in some studies of classroom interaction, the social dimension of learning has been seen in terms of intersubjectivity between participants (Cobb, Yackel & Wood, 1992; Jaworski, 1994; Steinbring, 1998; Voigt, 1996), a position which has been criticised by Daniels (2001) as limiting analysis. Daniels (p. 86) quotes Wertsch and Lee (1984) who comment that many of the psychological accounts, which attempt to discuss factors beyond the individual level “tend to equate the social with the intersubjective”. The resultant criticism is that “the research focus stays within the interaction itself and does not address wider sociological factors with respect to which the interaction is meaningful” (Jaworski

and Potari, 2009). However, intersubjectivity should be perceived as deeply sociocultural in its demonstrations- “a function of the setting, the activity, the actors, the texts, and so on” (Lerman, 1996, p. 137). Lerman argues for an integrated account, which brings the macro and micro together to enable us examine “how social forces, such as a liberal-progressive position, affect the development of particular forms of mathematical thinking” (Lerman, 2001, p. 89). He cites Wertsch, del Rio, and Alvarez when he states that “the goal of a sociocultural approach is to

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explicate the relationships between human action, on the one hand, and the cultural, institutional, and historical situation in which this action occurs, on the other”

(Wertsch, del Rio, & Alvarez, 1995, p.11, cited in Lerman, 2001, p. 89). Moreover, Jaworski (2009) suggests a unit of analysis comprising structures and systems on one level and daily classroom practices on the other. She quotes Engestrom (1998) who highlights “the middle level between the formal structure of school systems and the content and methods of teaching” (Engestrom, 1998, p.76 cited in Jaworski and

Potari 2009, p. 221). Engestrom refers to this middle level of analysis as the ‘hidden curriculum’ which includes:

grading and testing practices, patterning and punctuation of time, uses (not contents) of textbooks, bounding and use of the physical space, grouping of students, patterns of discipline and control, connections to the world outside school, and interactions among teachers as well as between teachers and parents.

(Engestrom, 1998, p. 76, original brackets)

It can be surmised that embracing such a definition of the ‘hidden curriculum’ leads to a questioning of school and educational systems, as well as the place of family and friends in national political and economic systems. It is no surprise that Jaworski (2012) has broadened her research interests into activity theory. In my conversation with Jaworski, I asked if her Teaching Triad could be regarded as a tool for analysing classroom practice in constructivist situations. I was attempting to narrow the categories under which I might analyse such practice. Jaworski confirmed that her Teaching Triad could be used as an analytical tool. Jaworski (1994) found one counter-example in her research. When she tried to categorise the teaching of a teacher named Simon she failed to do so in terms of the Teaching Triad. She argued that this was because Simon was bound up in a transmission view of teaching, which did not allow for considerations of individual learners beyond their responses to what he offered. Furthermore, Simon’s “planning and presentation of lessons

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seemed to indicate an absolutist view involving the existence of invariant concepts, which it was his task to deliver, rather than of personal concepts which individuals could be encouraged to develop, share and negotiate” (pps. 183-184). The teacher’s view seemed to be that knowledge was a fixed, immutable product, which could be communicated easily. It is here that I wish to refer back to ‘teaching as telling’ as

outlined in the previous chapter. In a constructivist approach ‘teaching as telling’ can be somewhat justified as long as the teacher has the long term aim of engaging the pupils in an investigative approach to mathematics, which incorporates respect for their views on what mathematics to pursue and how it is to be pursued. Therefore, the issue is one of epistemology. The constructivist teacher allows for individualistic ways of knowledge construction, whereas the teacher as transmitter views knowledge acquisition as a fixed process requiring little choice on the learner’s behalf. However, the caveat is that even the constructivist teacher will need

to tell pupils information on occasions, because she knows that this will lead to further knowledge generation on pupils’ behalf. The need to tell pupils information

rather than let them discover it for themselves is often driven by time and curricular constraints placed on teachers. Such constraints may take the form of external assessments, such as standardised testing or entrance assessments to second-level schooling.

Another author who analysed teaching in Ireland from a constructivist perspective for his PhD thesis is O’ Shea (2009). His PhD thesis was entitled Endeavouring to

teach mathematical problem solving from a constructivist perspective: The experiences of primary teachers. O’ Shea (2009) analysed the teaching of five

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is interesting and informative for my work is O’ Shea’s (2009) analysis of teachers’ mathematical practice in terms of two main categories:

1. Teachers’ didactic teaching style

This included teachers’ emphasis on rote memorisation of number facts, and their

direct transmission approach to teaching problem solving, allowing pupils little freedom to construct their own methods.

2. Teachers’ constructivist approach to teaching and learning

This was the category where teachers showed evidence that they encouraged pupils to choose a problem solving strategy and to justify it to the teacher and other pupils. O’Shea (2009) found that themes emerged in his five individual case studies and across the case studies as a whole. These were “a focus on rote memorisation,

mathematical problem solving from a constructivist perspective, as enrichment activity, and teaching students with different learning abilities from a constructivist perspective” (p. 91). He states that although the mathematics curriculum espouses

constructivist principles, and he delivered in-service to the teacher participants, which reflected those principles; “the traditional understanding of and approach to

mathematics teaching that was characteristic of the teachers inhibited the acceptance of a constructivist approach” (p. 244). O’ Shea’s (2009) research shows that it is

extremely difficult to categorise teaching in absolute terms; as being purely constructivist, with the teacher acting in a facilitative role, or as being solely didactic with the teacher adopting a delivery mode. He found that all five teachers exhibited characteristics of both approaches. Therefore, in this research, I wish to remain open-minded to the possibility that teachers exhibit characteristics of both a constructivist and a didactic approach. In engaging with teachers’ professional development Borko (2004) offers a model which I propose to use in this research as it encapsulates the approach I wish to adopt.

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