| Summary: | The aim of the research is to rethink how students of mathematics learn, by employing the ‘late’ Husserl’s theory of knowledge. To do this the mathematical investigations of three prospective teachers of mathematics are explored, focusing on ‘objectification’ and ‘intuition’, thinking about how students acquire new knowledge in order to resolve mathematical tasks; in particular how students come to grasp mathematical objects in order to conceptualise the tasks. For this purpose I employed Husserl’s phenomenological attitude and his methods of reduction and bracketing, informed by Merleau-Ponty’s extension of the Husserlian frame. The course from where the data were collected included thirteen students, and the teacher facilitated my phenomenological perspective, by deliberately suspending guidance and any other intervention apart from introducing the operational aspect of the tasks. By using the Husserlian theory and methodology and by following the teacher’s non-intervention strategy I managed to track the ‘moments’ of objectification and the critical role of intuitions—in the Husserlian sense—in the process of objectification. The embodied, founding powers of the living body (the body-subject) and the pre-reflective and reflective intentional forces manifested their significance for the students’ objectification processes. Most importantly, intuitions appeared as the critical acts that enabled each objectification to take place. In summary, the main findings of the research and the related contributions of the study to knowledge are the following: Intuitions are critical objectifying acts, preparing as well as constituting mathematical objects. Three genetic features of intuitions are identified, thus allowing their tracing as such. Empirical and abstract intuitions were shown as essentially interrelated, and the description of the transition from empirical to abstract knowledge through according objects was exemplified in a number of cases. The general structure of the (Husserlian) abstract intuitions was clarified, leading to suggestions for teachers to introduce abstract objects in accordance to the aforementioned structure. A novel phenomenological gaze on the mathematical learning experience is introduced, one that transforms ready-made mathematical objects to objectified lived experience. The contribution to knowledge suggested by this gaze is that it takes into account the complexity and richness of the learners’ lived worlds, that it has the potential to reorient teaching practice into a student-oriented inclusive praxis, and finally, to enable the researcher to cash in Husserl’s theoretical and methodological reflections as a “working philosophy”.
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