In the final session the students were required to plan a lesson based on the van Hiele Levels of Geometric Thought. The focus of the lessons was on one of the four content goals for geometry, namely shapes and properties, location, transformation and visualization of the lesson at a van Hiele level 1 activity. In addition the students were required to encompass all five strands of mathematical proficiency in their lesson. [Appendix Eight]
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4.3.4.1 Themes of enactivism that emerged
Experience
The final reflective task required the students to identify three facets of the practical tutorial sessions that the students attended in the first and second phase that had helped to develop their skills in teaching for mathematical proficiency. They were also asked to reflect on the link between lectures, tutorials and their teaching practice. From the perspective of conceptual understanding the team pointed to the use of resources and real-life objects to make the mathematics concepts clearer and to enhance their practice. In addition it was noted that “I have realised that teaching for proficiency requires planning and preparation on the part of the teacher, in order to ensure successful learning” (Research Participant 6, September 2011).
With regard to procedural fluency the team indicated they were more flexible in their teaching and acknowledged the need to be prepared in order to deal with any problems the learners might pose. Strategic competence was recognised by the team, who were of the opinion that the learners should be given the opportunity to explore problems for themselves and come up with their own methods (invented strategies) since “the exploration of the problems is very important as it encourages understanding and independence” (Research Participant 4, September 2011). “I have found that letting the learners come up with their own methods to be one of the strongest connections between lectures and the tutorial sessions. I battled with this concept in the beginning as I was always taught to follow the methods the teacher used, especially in maths. I find coming up with your own methods fun, rewarding and it makes complicated things seem manageable and easier to remember. I will definitely use this in my teaching practice as I think the learners will thoroughly enjoy the chance at coming up with their own methods and showing the class their way” (Research Participant 5, September 2011), indicating the importance of adaptive reasoning.
In terms of productive disposition both from a personal and teaching for proficiency perspective the team revealed that there had been an improvement in their practice and that in one case it was acknowledged that despite being able to do the Mathematics it was more challenging trying to teach the concepts, than was originally anticipated. The need for teachers not to “assume that they know everything
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about a subject; just because they have done the subject themselves as a learner at school”, but the importance of doing further research for “math related lessons” (Research Participant 6, September 2011) was also identified. There was also recognition of the value in linking mathematics to the real world and the need for additional research in order to develop proficiency. A third member expressed the connection succinctly in the following comment “during the lectures, we discuss themes and concepts, as well as teaching strategies, learning styles, and the importance of age appropriate teaching [conceptual understanding]. During the tuts, we take what we have learnt during the lectures, and apply it into practice [procedural fluency]. I can apply this to my teaching practice by making sure I do research and planning before I teach my lessons [conceptual understanding/productive disposition]” (Research Participant 6, September 2011).
Emergence
In each session the theme of emergence was developed since the teaching groups were required to complete questions in order to reflect on and create awareness with regard to their proficiency.
Autonomy
Autonomy featured more prominently in the final session’s reflection with the team recognising an improvement in their confidence, to varying degrees, and capability to teach Mathematics, with one member indicating that “the math tuts have helped me to transcend my own boundaries, and this will be beneficial for me as I have accepted an Intermediate Phase post for next year” (Research Participant 6, September 2011). The characteristics of flexibility and patience were mentioned as the team had either recognised these qualities in themselves or as qualities that they needed to develop. This is consistent with McGann (2008) and Thompson’s (2007) view that as autonomous beings we have the ability to generate ourselves in order to determine and bring forth our cognitive ability. Similarly the pre-service teachers were able to bring forth their cognitive ability with regard to teaching for proficiency and what they had gained from the module.
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Embodiment
Finally embodiment was identified through the value of the groups and the discussion and active participation that these generated. Since students had been placed in groups, not of their choosing, they had indicated that they felt more accountability toward the group members, with some taking on new roles like leadership, which they would not have done if they had been a group with their friends. The group discussions helped to maintain focus and provided clarity on the task. In addition, groups were required to think about the hermeneutic questions that they could ask the learners. This requirement had been included in an attempt to encourage strategic competence and adaptive reasoning.
What the students brought forth
The research participants acknowledged the benefit of using resources and real-life objects to make the mathematics concepts clearer and to enhance their teaching practice. They also recognised that teaching for proficiency and successful learning required planning and preparation on the part of the teacher. They were of the opinion that flexibility and patience is an important characteristic of teaching.
The participants indicated that learners should be given the opportunity to explore problems for themselves and come up with their own methods (invented strategies). This was based on their experience of lectures and the tutorial sessions. “I battled with this concept in the beginning as I was always taught to follow the methods the teacher used, especially in maths. I find coming up with your own methods fun, rewarding and it makes complicated things seem manageable and easier to remember” (Research Participant 5, September 2011). They all felt that there had been an improvement in their teaching practice and that there had been an improvement in their confidence and ability to teach Mathematics. They also remarked that teaching the concepts was more challenging than was originally anticipated and recognised the value in linking Mathematics to the real world. Finally the participants recognised the need for additional research in order to develop proficiency.
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Problems the participants posed in terms of teaching for proficiency
No problems were identified
Following session three it was evident that the inclusion of the five themes of enactivism in pre-service teacher education in a mathematics module develops flexibility in solving mathematical tasks and teaching for proficiency. This increases the ability to respond appropriately to different methods of approaching a problem or task put forward by pupils. Creating an awareness of mathematical identity through the theme of autonomy and reflective tasks (emergence theme) illustrates an increase in mathematics flexibility which in turn affects mathematical confidence positively and encourages an appropriate response to further perturbations and triggers.
4.3.5 Analytical observation through the lens of the five strands of teaching