In session two the pre-service teachers continued to focus on fractions, this time with the emphasis on fraction operations. The strands of proficiency that the students were asked to incorporate were strategic competence and adaptive reasoning. [Appendix Six]. For the reflective tasks, the students needed to discuss the role that they had played in helping the group come to a common understanding of the concept/task they were planning to teach. Furthermore, they were required to identify an incident (or discussion) during the planning phase that had contributed to their
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understanding of the topic. They also had to unpack how the incident contributed to their understanding for teaching for proficiency and discuss how old ideas that they may have had regarding the concept were changed and/or resolved. A third task was to discuss how, working as part of a group, both in the mathematics lecture and in preparing for the tutorial sessions, had changed their mathematics identity with regard to their personal mathematical proficiency and teaching for mathematical proficiency. This was in response to Leatham and Hill’s (2010) definition of mathematical identity as being one’s relationship with Mathematics with regard to the way one learns, does, thinks about, retains or chooses to associate with the subject. The final task was to explain what assumptions they had about the concept/task and how to teach it and how these assumptions changed through working with their group.
4.3.2.1 Analysis of Videos Phase Two Session 2
Tape 1: (Grade 5: Multiplication of fractions)
The first lesson was a very confident presentation, the participant was able to engage the students with her initial activity as it was something that they all could relate to, namely making French toast and she encouraged participation through various mathematics questions. Students were asked to explain how they arrived at an answer, thereby developing their adaptive reasoning and strategic competence skills. Her activities were very learner centred and she demonstrated good classroom management, for example, if a student’s attention wandered she brought them back to task by asking an appropriate mathematical question. The activities generated discussion and participation and the explanations often included visual representations. Again there was a good balance of incorporating the five strands of proficiency into the development of the lesson. There was evidence in part of the lesson that her conceptual understanding was not as strong as it could have been.
Tape 2: (Grade 6: Decimals – problem solving)
The second lesson was confidently presented by the research participant who was able to use different methods to explain and at the same time to demonstrate the link between the alternative methods. The participant encouraged learner participation through encouraging their strategic competence skill in determining a strategy to
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arrive at the solution and their adaptive reasoning by way of an explanation. Her conceptual understanding and procedural fluency was evident in the way she was able to explain the concepts using a variety of strategies and demonstrate the link between the alternative methods used. There was also effective use of board work which was linked with a table that the students had to complete. The participant was also comfortable working with the mathematical ideas that the students raised; an indication of a positive disposition and conceptual understanding. The activities that she used scaffolded one another.
4.3.2.2 Themes of enactivism that emerged
Emergence
There was an indication of emergence as students started to reflect on their own practice and that of others in addition to acknowledging that working in a mathematics community “takes the pressure off” (Research Participant 7, August 2011). An awareness of what an individual needed to do from a personal perspective to develop their own proficiency and improve their practice and approach to teaching was evident. For some students conceptual understanding of the topic was required before they could begin trying to teach the topic. However having understood the concept they were then happy to put forward ideas with an increase in the focus on linking mathematical concepts to real life. An increase in conceptual understanding emerged with students extending their pedagogical practice to include linking concepts to everyday ideas that the students were familiar with, for example making French toast as part of their introductory activity to build a mathematical concept. This extended to their procedural fluency as indicated by a member of the team “I feel that I have always taught in my own way and now can use other strategies” (Research Participant 2, August 2011). The fact that each session in phase two has been allocated specific strands on which to focus meant that the students continued to develop their repertoire of strategies, encompassing all five strands of proficiency. Furthermore students were recognising that learners think in different ways and that they needed to know how to approach this. As a team member pointed out, “I have always done math in one way and it was how I was taught – with our lessons I have how to
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teach all ways or to see how pupils would try to do the sums and therefore also let them participate” (Research Participant 2, August 2011).
Embodiment
There is evidence of embodiment as students indicated that being part of a group contributed to their understanding, and having understood the concept they were then able to contribute to the perturbation, the task, and to teach it. It has also resulted in developing confidence in the students. Initially students were reluctant to teach in front of their peers but by the end of the semester we had two or three students teaching together because they all wanted to try their hand at teaching their peers and we had run out of tutorial sessions. Furthermore participating in a group and watching other groups teach similar concepts exposed students to different approaches to teaching concepts, again building up their repertoire of strategies. In many cases the discussion during meetings to prepare for the forthcoming tutorial session contributed to the conceptual understanding and thus sense-making as well as the instructional approach of the students. There is more evidence of creativity in the planning of lessons since students were required to focus on strategic competence and adaptive reasoning for example having an introductory activity that captured the learners’ attention and that they could relate to. An increase in the use of invented strategies as a means of developing strategic competence and adaptive reasoning in the learners became apparent. Van De Walle, et al. (2010, p. 215) describe invented strategies as “personal and flexible strategies” used for computation that differ from the traditional algorithm.
Autonomy
Productive disposition started to emerge by this session in that students indicated that they were more confident to teach in the Intermediate Phase at the same time acknowledging the importance of attending group meetings and taking part in the discussions to avoid feeling insecure about what needed to be taught. Furthermore, in session two there was a noticeable increase in autonomy as the team indicated an increase in confidence to teach the content. This was brought about by “teaching and planning in a group” (Research Participant, August 2011) as all members had come to
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a common understanding. Some students felt more proficient at teaching Mathematics to the Intermediate Phase as the lectures and tutorials had created an awareness “of the different learners in the class and their needs and the different way each concept can be taught” (Research Participant 5, August 2011). This resulted in some students feeling less anxious about teaching in the Intermediate Phase.
Sense making
Students noted the benefit of playing an active and participatory role in emerging conversations within the group in the sense-making process as it assists in finding a solution to the problem and determining how to teach the concept. Davis, Sumara & Luce-Kapler (2008, p. 69) describe conversations as coupling that which occurs between individuals at the level of brain activity, resulting in a single cognitive system with enhanced capabilities. This is due to the various experiences and interpretations available.
What the students brought forth.
There was growth in conceptual understanding and procedural fluency in addition to an increase in their repertoire of teaching strategies encompassing the strands of proficiency. Furthermore there was an observable creativity in the planning of lessons as well as an increase in the use of invented strategies as a means of developing strategic competence and adaptive reasoning in the learners. Students indicated that they were feeling more proficient and less anxious about teaching Mathematics to the Intermediate Phase.
Problems the participants posed in terms of teaching for proficiency
No problems were identified.
The themes of embodiment and autonomy continued to be significant in increasing confidence, conceptual understanding and in developing an increasing repertoire of teaching strategies. This research study demonstrates that including the themes of enactivism, and in particular the experience theme, assists in the growth and
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development of teaching for mathematical proficiency in pre-service teachers by creating an awareness of identity and a sense of accountability when responding to perturbations.