Descripción de los resultados de la variable: Integración

In looking at constructivist theory a practical problem now arises: how does constructivism actually manifest itself in classrooms? What is needed is an eclectic form of constructivism, which allows for several forms of knowledge and knowledge acquisition. Herscovics (1989) describes such an approach as ‘rational constructivism’. For instance, the radical perspective is a psychological constructivist’s view of an individual pupil’s activity, as she engages with and

contributes to the development of communal processes. However, the social perspective is an interactionist view of collective classroom activity. As a researcher,

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do I focus on individual or communal activity to gain evidence of constructivism in action? Fortunately, there is a view called the emergent perspective, which seeks to combine both theories and in so doing justifies the need to look for both individual and socially mediated learning in classrooms. O’ Shea (2009) gave a comprehensive

elaboration of the emergent perspective and it is to this work I now turn. He cites Cobb and Bauersfeld (1995, p. 176) who state that the coordination of interactionism and psychological constructivism is the defining characteristic of the version of social constructivism that is referred to as the emergent perspective. This perspective highlights social processes and views knowledge acquisition as comprising both individual and social constituents and contends that these cannot be viewed as distinct in any meaningful way. Wilson (1996) comments that the difference between the rational and social perspectives is that radical constructivists emphasise how individuals create more sophisticated mental representations using information, manipulatives and other resources whereas social constructivists perceive learning as augmenting one’s ability to participate with others in meaningful activity. To put it

simply, the social constructivist’s view is similar to the old adage that ‘two (or more) heads are better than one’. Yet, from my own classroom experience, I have witnessed occasions when a pupil does not want to be helped by another pupil or the teacher; preferring to work a problem out by himself. Obviously, in this scenario, the pupil perceives the solution to be close at hand and the personal motivation required to solve the problem outweighs the need for social bonding. Tobin and Tippins (1993) surmise that the emergent perspective in a synthesis of radical and social perspectives, which claims that knowledge is both personally constructed and socially mediated. Windschitl (1999 p. 34) puts it well in declaring that learning is both an act of individual interpretation and negotiation with others. Therefore, in researching mathematics classrooms one has to ascertain if pupils are showing

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evidence of being cognitively challenged at an individual level and/or engaged productively in discussion at group level.

O’ Shea (2009) states that according to the social constructivist, the constructive

processes are subjective and developed in the context of social interaction. Pupils gain mathematical knowledge through participation in the social practice of the classroom, rather than through the discovery of external structures, which exist independently of them.

Cobb and Yackel (1996) and Stephen and Cobb (2003) declare the emergent perspective to be a version of social constructivism. It draws on constructivist theories, which see learning as a series of cognitive reorganisations of the individual (von Glaserfeld, 1995) and interactionist theories, which perceive learning as a social accomplishment (Bauersfeld 1992). Therefore, the emergent perspective tries to reconcile radical and social constructivism. Cobb and Yackel (1996, p. 177) elucidate when they state that pupils reorganise their learning “as they both participate in and contribute to, the social and mathematical context of which they are part.” They argue that mathematical knowledge is both an individual and a social construction and that both dimensions of learning complement one another. O’ Shea

(2009, p. 31) justifiably quotes Ernest in this regard:

The two key features of the account are as follows. First of all, there is the active construction of knowledge, typically concepts and hypotheses, on the basis of experience and previous knowledge. These provide a basis for understanding and serve the purpose of guiding future actions. Secondly, there is the essential role played by experience and interaction with the physical and social worlds in both the physical action and speech modes. This experience constitutes the intended use of the

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knowledge, but it provides the conflicts between intended and perceived outcomes which lead to the restructuring of knowledge, to improve its fit with experience.

(Ernest, 1991, p. 72)

Voigt (1992) elaborates on the relevance of the emergent perspective for mathematics education research, when he states that both cultural and social processes are integral to mathematical activity. Mathematical learning opportunities occur when pupils compare other solutions to their own and thereby try to make sense of their own solution in the broader perspective. To reiterate, the business of doing maths is both a social and an individual activity. The teacher has a pivotal role in initiating and guiding the formation of norms during mathematical activity, but the individual pupil also has an active role. Cobb and Yackel (1996) have done major research on the development of social norms and sociomathematical norms in the classroom and I now give a summary of their research.