Abstract
The goal of this text was to understand how the constitution of knowledge by being in contact with Mathematics, computers, and co-subjects takes place. We adopt the Husserlian phenomenology, in order to go to the thing-itself and seek the thing, as it appears to those who experience it. We held a distance learning course in geometry, looking for a philosophical view of the ideas that underlie that subject. The target audience of the course, which constitutes the co-subjects of our study, is comprised of teachers who teach geometry. The constitution of knowledge in geometry was shown in the thinking movement itself, triggered by significant questions addressed, by researchers, to the co-subjects while performing the requested activities. We understand that this movement does not occur in a straight line with chronologically marked sequences, but through paths evidenced by working with the question, therefore bringing memories of teaching and learning situations experienced by the co-subjects, intentionally aware of what is being said by those with whom they share the situation of thinking/knowing/producing. The production of knowledge was unveiled while the activities envolved in that production were experienced, perceived, analyzed. It was articulated through language. It was evaluated being presented to the community of mathematical educators.
Taís Alves Moreira Barbariz deceased at the time of publication.
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Notes
- 1.
Informational screen (Bicudo & Rosa, 2010).
- 2.
In Crisis of European Science (1970), Husserl makes a very relevant and enlightening phenomenology of Lebenswelt. However, it is a work written in the 1930s, when cyberspace was not yet a reality.
- 3.
Spoken language refers to the expression of understandings made by the subject expressed in sound, structured according to the spoken language present in the socio-cultural context in which they live.
- 4.
The phenomenological reduction is characterized as a thinking movement, exposing, through the articulations of ideas, the complexity of senses, and meanings that intertwine in broader ideas. The movement of exposing the understandings of the phenomenon encompasses the process of highlighting the phenomenon of the background from which it arises, expressing understandings open to the researcher about its senses and meanings. Through Husserl we understand that the research phenomenon is put in suspension or evidence or epoché. This means that the researcher’s preconceptions about the object investigated are put in suspension so that the researcher is on alert and their previous experiences do not deterministically lead the research. The reduction movement, still considering what is said by the afore mentioned author, is taken as intentionality, determined by the acts of consciousness, in motion. As intentional subjects, we are open to the life-world that presents itself to our perception, so, the reduction movement can help us understand essential aspects of this world. The reduction movement presents itself as important for the investigation, because, through reduction, acts of conscience are exposed, that is, it is possible that one becomes aware of them so that, by reflection, the cognitive roots of the affirmations themselves are explicit (Bicudo, 2011). It should be clarified that reduction, within the scope of phenomenology, does not refer to a simplification or a summary of what is presented in the text, but to an articulating thought movement in which the meanings are intertwined with more meanings and by the assigned meanings, put into language, they are shaped into ideas that encompass them in a comprehensive whole.
- 5.
Actualization as Baumann (2013, authors’ translation) says of a creative action that, when performed, makes present possibilities present [...]. Such, updating is understood here as a movement to make current, to make the ideas present [in the project] through actions that may or may not innovate. By dismembering the word, we can reaffirm that updating is the action of making current.
- 6.
In the case of the research mentioned here, despite recognizing that the expression “distance learning” can refer to any kind of teaching and learning process that considers the teacher and the learner in this condition, such as correspondence teaching or by using media such as radio and television, we are focusing on those processes that take place through computer intermediation and with the Internet.
- 7.
This investigation, which led Taís Barbariz to the title of “Doctor” was supervised by me, Maria Aparecida Viggiani Bicudo, very closely. Taís defended her doctorate thesis on 3/16/2017 and passed away on 04/03/2018.
- 8.
This tells us about the professor–researcher, Taís Barbariz, who conducted the course and me, Maria Bicudo, who was with her throughout all stages of the research.
- 9.
Nuclear Ideas—NI—in the research cited in this article, refers to the broader convergences articulated through successive movements of phenomenological reduction.
- 10.
Hans Freudenthal, (Luckenwalde, Germany, September 11, 1905– October 13, 1990), the creator of Realistic Mathematics, was a Dutch-born mathematician. He made substantial contributions to algebraic topology and was also interested in literature, philosophy, history, and Mathematical Education.
- 11.
The course uses the Moodle platform; its conduction for the constitution of research data was made possible by CECEMCA—Center for Continuing Education in Mathematical, Scientific and Environmental Education; made available by UNESP Rio Claro campus, and is certified by the Dean of Institute of GeoSciences and Exact Sciences of UNESP. It is described in its entirety in Barbariz (2017).
- 12.
The living-experience (Erlebnis) tells, at first glance, about the life that flows, as it is lived. We live experienced acts being in motion for the duration of their temporality. At each moment we live the present moment of the act taking place. Psychical acts, such as perceiving, imagining, fantasizing, remembering, reflecting, which are inherent to human beings, even if they occur uniquely in each individual. Living-experiences flow, slide from now to what has been, making room for other living-experiences. We know we are living, but only by an act of consciousness do we realize what we are experiencing. This act is to perceive the experience as being lived and Husserl calls it “Erlebnisse.”
- 13.
I speak in the first person singular, for Taís is now present in the flow of my memory. In the text there is a variation between me and us. When I speak in the first person singular, it is me, Maria, revisiting the memory of what was experienced with Taís and also the understandings that were opened to me; when I speak in the first person plural, I bring the thought and what was accomplished by the both of us.
- 14.
Detailed data is found in Barbariz (2017).
- 15.
T.N.: all abbreviations, initialisms, and acronyms correspond to the original words in Portuguese.
- 16.
Perception of the self while questioning the execution of what was proposed. Opening horizons of understanding of geometry; Perception of the institutional means for the actualization of the course; Perception of the complexity of course planning; Perception of experiences lived as a student, a teacher, and a researcher; Willingness to understanding pedagogical practice; Self-perception of the researcher when conducting her own research; Psychical ways of being a teacher/researcher; Perception of the possibility of course actually happening; Self-perception as a teacher–researcher; Perception of glitches in the technological apparatus; Perception of the actual beginning course; Focus on the technological apparatus supporting the course; Perception of the discrepancy between the activity developed as a teacher and that expected as a researcher; Perception of tensions and compatibilities between the course which was planned and what was effected; Perception of change in the way of being a teacher; Perception of concern regarding the research. Perception of oneself producing knowledge; Ways to be with each other at a distance; Perception of mathematics teaching practice according to the available materiality; Perception of dynamics of the course; Perception that others and the computer are present in the very act of designing the course; How to make search records; Perception of the assessment of the way of being, of others and themselves; Concern about student attendance and participation (Barbariz, 2017, p. 97, authors’ translation).
- 17.
Self-perception designing the course; Self-perception producing knowledge; Self-perception as a teacher/researcher; Perception of the movement of actualizing the course; Perception of the psychical ways of being a teacher/researcher; Perception of ways of being with others at a distance; Perception of others articulating ideas; Perception of others constituting knowledge; Perception of the importance of language in actualizing the course; Perception difficulties of articulation by the subjects (Barbariz, 2017, p. 98, authors’ translation).
- 18.
The living-body makes us appear and touch the world, while making us perceive ourselves by touching what is in front of us and being touched by what is there. It is a totality consisting of “flesh and bone,” that is, of physical/psychic and spiritual aspects. Thus, the human body is not reduced to a structure composed by the pair psyche and body, looked at in its materiality, which, in the works of Husserl (2002) and Merleau-Ponty (1994) is explicit by the word Körper. Husserl exposes his understanding of the living-body as an interweaving of the physical, psychic, and spiritual aspects. Thus, it shows actions nurtured by the intention and willingness to do something in a situation, that is, as always moving toward something to do. The conception of the living-body—a body that lives and feels that it is living is expressed by the word Leib.
- 19.
The natural attitude is to take the world as existing in its objectivity.
- 20.
Intropathy is knowledge of the other that occurs directly in the experiences in which the other is given (brought, exposed) to the self in its corporeality. It is a constituent perception of intersubjectivity. It is not, therefore, a theoretical concept or a predicatively constructed statement.
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Bicudo, M.A.V. (2020). Constituting Mathematical Knowledge Being-with-Media in Cyberspace. In: Viggiani Bicudo, M. (eds) Constitution and Production of Mathematics in the Cyberspace. Springer, Cham. https://doi.org/10.1007/978-3-030-42242-4_5
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