4.2.3.1 Conceptual Understanding
An important aspect of this strand according to Kilpatrick, et al. (2001, p. 381) is that “teachers need to make connections within and among their knowledge of
102
Mathematics, students, and pedagogy” as well as a range of pedagogical practices. From this perspective the analysis revealed that the students’ repertoire of teaching strategies was limited and often their choice of method was the approach demonstrated during the lecture. This in itself is not to be discouraged since in some cases it helped to consolidate the students’ personal conceptual understanding and “it would benefit us to try it out in a lesson and get more understanding of it” (Research Participant, May 2011). One needs to factor in the realisation that these were pre- service teachers and so a limited repertoire of strategies is to be expected. In addition, I also took into consideration the possibility that some students did not have the confidence or courage to try new ideas in case they were unsuccessful in front of their peers. In some cases students were not too sure of the level at which to pitch the lessons.
4.2.3.2 Procedural Fluency
In some instances the instructions were not always clear and more explanation was needed for clarity. I noticed that the teaching strategy tended to be very procedural in nature and teacher centred, As a result of their previous experience they did not always have an alternative strategy to employ. However they did know that although this was an artificial situation, the expectation was to teach a lesson aimed at Intermediate Phase learners.
4.2.3.3 Strategic Competence
According to Kilpatrick, et al. (2001, p. 383) strategic competence requires that the teacher needs to determine what the learner knows, then decide firstly, how to respond to learners’ ideas and, secondly, whether or not to follow these ideas. In essence they see teaching as a form of problem solving in itself. This was a more difficult strand to implement, however, one group decided to anticipate the difficulties some learners might experience and created a story that would overcome these difficulties. “We thought it would help the learners to understand the concept more easily and remember it” (Participant, May 2011). This was not a strand that was very well embodied. However, it was noted that if the student ran into difficulties during a
103
lesson, the balance of her group readily supported her with ideas or answers, in this way bringing the enactivism theme of embodiment to the fore.
4.2.3.4 Adaptive Reasoning
There was far more evidence of adaptive reasoning firstly, in the conversations generated within the groups, to explain and justify the concept and teaching approach that should be adopted and secondly, in the reflective tasks that the teaching groups had to complete. Nevertheless, during the discussion at the completion of each session many students, including the teaching groups, contributed thoughtful and constructive comments and suggestions. Furthermore in some cases students engaged in discussion justifying the intention of the lesson, particularly if this had not been clearly demonstrated. With regard to the lessons taught, students seldom required their ‘learners’ to explain how they got to their answers. As one member of the observation team group pointed out, she “never required learners to explain their process just accepted their answers” (Student, May 2011). This could have been a confidence issue if the student herself did not have the necessary conceptual understanding. Alternatively, because they were teaching their peers they may have taken it for granted that they all understood.
4.2.3.5 Productive Disposition
Kilpatrick, et al. (2001, p. 384) indicate that to develop a productive disposition a teacher needs to ensure that the Mathematics to be taught, the teaching strategy to be used and their understanding of the learners thinking, needs to be synchronised. This is a process that can be broadened by listening to the learners and how they engage with the concepts and the strategies they employ.
Again this strand was not well addressed since most of the lessons were teacher centred making it difficult to ascertain the learners’ mathematical thinking. In addition, the tutorial sessions were contrived so that the ‘learners’ were their peers and not 10 and 11 year old pupils.
104
What the students brought forth
Enactivism views knowing as knowledge in action and so from phase one the following points were brought forth by the research participants in terms of what they know to be teaching for proficiency. It is, as Proulx and Simmt (2013, p.72) point out, “what they know, how they are, and who they are”.
Firstly, mathematical histories and experiences had had an impact on the team’s mathematical identity. Through the lens of Kilpatrick’s strands of proficiency it became evident that past experiences had affected productive
disposition which had resulted in procedural fluency becoming the dominant
strand when teaching for proficiency. As the observer I reflected on what past histories had brought to the lecture room, and to what extent lecturers need to deal with this influence. Is it enough to merely draw the students’ attention to the impact of their past experiences in an attempt to get the students to respond to the perturbation offered during lectures? The students need to take steps to address their mathematical identity, consolidate areas of strength and develop and remediate areas of weakness.
Their previous experience was generally teacher centred not student centred.
Students had to become accountable for their personal conceptual understanding in the event they would be required to teach during the tutorial session.
Co-emergence ensued between the different students as a result of having to engage in conversation in order to teach a mathematical concept.
Reflective tasks regarding their personal proficiency and teaching gave students insight into what was required to improve and develop their proficiency.
Embodiment had an influence on productive disposition. As they were working in a smaller mathematics community students were able to support those who were anxious and less confident by helping them to understand the problem and share ideas on teaching strategies and approaches.
Tutorial sessions provided students with an opportunity to link the theory of mathematical concepts and pedagogy when it came to teaching for mathematical proficiency.
105
Experience in the form of teaching tutorials resulted in students being introduced to different teaching strategies and approaches.
Students were most comfortable and confident with the conceptual understanding and procedural fluency strands of proficiency with regard to their teaching practice.
An unthreatening environment that did not include assessment, encouraged creativity and experimentation in teaching approaches.
The practical tutorial experience allowed the students’ to make mistakes in their choice of teaching approach and strategies as there were no repercussions.
The pre-service teachers received more feedback on their teaching than they would ordinarily from teaching practice. This was deemed to be helpful and constructive.
They had more confidence to teach certain concepts having listened to feedback and through observation of the lessons taught.
They knew from past experience what qualities they believed a teacher should have for proficiency.
Problems the participants posed in terms of teaching for proficiency
Firstly, in many instances conceptual understanding needed to be strengthened. Regarding the practical tutorial experience, students had difficulty in addressing the productive disposition, adaptive reasoning and strategic competence strands when teaching for proficiency. They also had difficulty in translating their ideas into teaching practice and often the practical sessions tended to be more teacher centred and procedural in nature. Lastly the students found teaching their peers created an element of anxiety as it seemed as though they were being judged.
On completion of the analysis of phase one the following points emerged with regard to the influence that the different themes of enactivism had on particular strands of teaching for mathematical proficiency (Table 4.3). Three themes of enactivism were revealed to be more significant in informing the students practice in proficient teaching. The theme of autonomy revealed that the students past experience affected their level of confidence to teach mathematics proficiently and tended to
106
result in a more teacher centred and procedural approach to teaching. Embodiment provided support in terms of understanding and the growth of confidence. Finally, the experience theme provided a safe environment in which to put theory into practice in addition to introducing students to various approaches to teaching the same concept. It is important that lecturers involved in the training of pre-service teachers take into consideration the impact that past experiences have on students and introduce reflective tasks and triggers to generate self-awareness of their mathematical identity in the students thus giving them the opportunity to problem pose and react to these perturbations.
Table 4.3
Matrix indicating the points raised during analysis of the influence of the themes of enactivism on the five strands of mathematical proficiency in Phase One