Social constructivism and a socio-constructivist approach

Cowan states that through social constructivism “students can better construct their knowledge when it is embedded in a social process” (Cowan 2004, p. 4). He

elaborates on a socio-constructivist approach by describing it as being a type of social constructivism, which is developed only in mathematics education. The tenets of this type are broadly similar to the characteristics of RME in that it is suggested that mathematics should be taught through problem solving, whereby students

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interact with one another and the teacher. The theory draws on the Marxist idea of collective activity, wherein those who have more knowledge or are more skilled share that knowledge and skill with those who are less knowledgeable or less able in order to accomplish a task. Gravemeijer (1994) highlights two similarities between a socio-constructivist approach and RME. Firstly, both approaches have developed independently of radical constructivism. Secondly, in both approaches pupils are given opportunities to share their experiences with other pupils. Furthermore, mathematics is seen as a creative humanistic endeavour in which learning occurs as the pupils develop their own strategies and concepts in order to solve problems. De Lange (2006) states that the main difference between RME and constructivism is that RME is only applied to mathematics education, whereas constructivism is used in other subjects. However, Cowan (2004) contends, and I agree, that in RME the

integration of mathematical strands is essential as RME is a holistic approach to

mathematics, which incorporates several learning strategies within a mathematical problem solving foundation. I have to point out that in Ireland the word integration is normally reserved for the connection of mathematics with other subjects, whereas the word linkage is used for the connection of mathematical strands with one another. As far back as 1990, the Primary Education Review body recommended more frequent utilisation of integration in mathematics. This was to become one of the suggested methodologies in the revised curriculum of 1999. Linkage was also suggested as a methodology, with footnotes for it appearing on some of the curricular content pages. Returning to RME, Gravemeijer (1994) asserts that it offers heuristics for developing instructional activities for students, whereas a socio- constructivist approach does not offer heuristics, but seeks to find solutions through guided reinvention. Cowan (2004, p. 5) defines heuristics as “a method of solving problems by learning from past experiences and investigating practical ways of

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finding a solution”. In other words the socio-constructivist approach will not offer a

ready-made formulaic solution but instead seek to derive one. This is similar to Vygotsky’s notion of scaffolding, whereby pupils are assisted in working at the

boundaries of their current reasoning, so that they can construct new knowledge. However, in RME it may not be the teacher who provides the scaffolding, but rather the other pupils with their ideas and strategies. In the co-construction of the zone of proximal development (zo-ped) the children are permitted to explain their strategies to the class and for the class to test their hypotheses, without the teacher pushing to seek a collective solution. Ernest (1995) and Wood et al. (1993) state that an awareness of the social construction of knowledge suggests more of a pedagogical emphasis on discussion, collaboration, negotiation and shared meanings. This is a practical manifestation of the theory of the social construction of knowledge. In Elwood’s (2008) framework, as outlined in section 2.2, Ernest’s (1995) and Wood et al’s. (1993) ideas would appear to the centre-right of the continuum as they

emphasise that learning is both a social and cultural activity. As Gergen (1995, p. 30) remarks, “Knowledge is in continuous production as dialogue ensues. To be

knowledgeable is to occupy a given position at a given time within an ongoing relationship”. The teacher is not meant to be the oracle constructing the knowledge

for the pupils. Therefore, the teacher and pupils co-construct their evolving mathematical identities based on their mathematical experiences in the classroom. I hoped to witness and support this type of teacher interaction with pupils in this classroom research. Scaffolding evokes a construction metaphor in that Cowan (2004) states it has five major functions:

1. Provide support for the learner;

2. Function as a tool or methodology (to use a phrase more familiar to Irish teachers);

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3. Extend the range of the learner;

4. Allow the attainment of tasks not otherwise thought achievable; 5. Works best when used judiciously.

Cowan (2005) describes the type of classroom scenario in which he believes scaffolding evolves. Initially, a teacher tends to do most of the work, but gradually the teacher and the learners share responsibility. As the learners become more competent, the teacher steadily withdraws the scaffolding so that the learners can perform independently. In construction parlance it evokes for me an image of a ladder being removed from a structure once it has been completed. The key to this structure is that the scaffolding keeps the children at the zo-ped, which is reframed as the children enhance their competency and improve their understandings. Cowan says he observed such practices during his own observations over a two week placement in RME classrooms. At the beginning of lessons the teacher seemed to be speaking for lengthy periods at the blackboard, slowly proceeding through algorithmical concepts step by step. As pupils’ understandings developed the

teacher-talk ebbed and became less evident. Cowan graphically describes the difficulties he experienced as he endeavoured to stop himself putting words and explanations into the children’s mouths. He elaborates further:

I also had to make myself play devil’s advocate, which is a role I have trouble playing at times when I know that a correct answer has been arrived at; I tend to accept the answer without thinking about the process in which it resulted.

(Cowan 2004, p. 5)

Little did I know that I too would later experience such difficulties in this classroom research. Theoretically speaking, Cowan sees RME as a brilliant concept with the potential to revolutionise the way we teach mathematics. He cautions, however, that

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in practice RME can be very difficult to adopt, because it tests the faith the teacher has in her students to operate without the need for a tight framework or regime. He points out that during the first two days of his placement he found it difficult to deliver a lesson using RME, as he experienced anxiety when it came to leaving the children alone “long enough to free radical” (Cowan 2004, p. 6). It reminds me of Cobb’s comment that ‘trust’ is one of the most important elements of a constructivist

classroom. However, over time Cowan found it easier to let the pupils talk for most of the lesson in place of the teacher. This allowed the children to hypothesise and experiment with their strategies; thereby taking the focus off the ‘right’ and ‘wrong’ normally associated with mathematics. Furthermore, Cowan had the benefit of RME lesson guides, which allowed the teachers involved more time to concentrate on methodology as opposed to content. Cowan contends that RME aids the process of more of the class understanding a concept as opposed to less of the class. However, he adds the proviso that teachers have to be mathematically secure in their own understanding of concepts as they have to be able to interpret children’s strategies without necessarily imposing their own viewpoint too soon.