Re-observing the emergence of mathematics learning through conversations in a classroom from an enactivist perspective

: A methodological study

Abstract

From my observer perspective, how can I learn from interactions in a mathematics classroom? In this thesis, I am focusing on how my own learning began to evolve through the observation of mathematics knowing emerging from the details of the interactions during conversations in a mathematics classroom both between the teacher and students and amongst the students themselves. The contribution to knowledge is therefore methodological. I bring an enactivist approach to the study, which took place in a Chilean grade- eight classroom (ages 13–14 years old students), when they were doing mathematics in their usual way and also working with a mathematical modelling task for the first time. My methodological stance to this study is also enactivist, allowing me to carry out different observations as the mathematics knowledge emerges through the interaction of the participants. As a result, I conducted my work during recursive cycles of observation and re-observation of data. The first observation implies an explicit and descriptive categorisation of the mathematical action of the participants delivered by methods such as observations, video- and audio-recorded lessons and interviews. The second observation focuses on mathematical episodes, detailed and selected mathematics conversations recorded by re-observation of the participants. I discuss the role of the ‘second’ observer in the use of video-recordings when the observations of the classroom were carried out by the same observer, drawing the observations under two types of categorisation: basic and subordinate. I carried out my analysis of the data using episodes of mathematics, based on my observations of the conversations noting how my own learning emerges through the interactions that I have with this study. From these observations, as a theoretical framework, I bring forth emotions, the process of becoming aware and actions of distinction. I also considered the levels of categorisation of my observations. This analysis uncovered the following characterisation of learning when students and teacher are involved in solving a mathematics problem. They perform actions to become coherently aware mathematically in their classroom, noting moments of: suspension, redirection and letting go (Depraz, Varela, & Vermersch 2000, 2003; Varela, 2000). There is an empathy between student and teacher and amongst the students linked to an interpretation of what they are doing mathematically. The distinctions made in the micro-historicity of each teacher or student can trigger similar interactions in the mathematics classroom. As observer, I argue for a characterisation of the learning process to include the micro-historicity of each participant. Such an approach encourages me to observe the details and the shifts in action, which lead to considering the process of actions through which a sense of empathy can lead to an understanding or misunderstanding of a mathematical situation. By considering the distinctions of each participant, I am able to see the similarities and mathematical influences in the interactions between the teacher and their students, and the students themselves. The desired mathematical awareness begins for the student in the process of interactions with others. To observe this, it is necessary to follow the particular interactions of each participant noting the underlying awareness.

Date of Award	28 Nov 2019
Original language	English
Awarding Institution	The University of Bristol
Supervisor	Laurinda C Brown (Supervisor) & Rosamund J Sutherland (Supervisor)