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Phenomenological Mathematics Teaching Päivi Portaankorva- Koivisto The University of Tampere, Finland Námsstefna Flatar 29.-30.9.2006

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Something about Tampere The city was founded by Gustav III in 1. Oct.1779, on the bank of the Tammerkoski rapids. Population 202 932 Tampere

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The University of Tampere As the University of Tampere since 1966 About 15 400 students Faculties: - Economics and Administration - Education - Humanities - Information sciences - Medicine - Social Sciences

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About Finnish Schoolsystem

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The Teacher Education at the University of Tampere Early Childhood Education Department of Teacher Education, Hämeenlinna for primary school teachers Department of Teacher Education, Tampere Tampere Hämeenlinna

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About Mathematics Teacher Education

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”The mountains of mathematics”

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Phenomenology (Lehtovaara, M., Rauhala, Husserl) Listening, Emotions Senses, experiences, uniqueness Openness Aestetic, Individuality Intuition, genuinity Meanings Phenomenological Mathematics Teaching Interactive Experiential Cooperative, collaborative Mathematics as a language Using illustrations Exploratory

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What kind of challenges does the development of phenomenological mathematics teaching pose for prospective mathematics teachers? They should take the pupils as individuals They should encourage the pupils to talk and use all of their senses They should help the pupils to identify relevant mathematics and to make sense of the mathematical solution and its limitations

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What kind of challenges does the development of phenomenological mathematics teaching pose for teacher education? more opportunities to reflect and work together encourage the practice of dialogical and cooperative methods of learning as part of student teaching more opportunities to understand the pupils’ learning processes

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The six components of phenomenological mathematics teaching - working in the classroom manipulativesauthentic situations Experiential drawings Using illustrations Cooperative Interactive Exploratory Mathematics as a language element individually investigations a pupil, orally mindmaps tables, graphs demonstrations structurelessonplancurricular Kagan & Kagan, 2002 in pairs Vuorinen, 2001 in groupsdemonstrations classroom discussion lecture open tasks projects shared exploratory process a teacher, literally a pupil, literally a teacher, orally meanings As a tool for the pupil As a tool for the teacher

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Stages 1/3 Experiential pupil cutting, glueing, folding manipulatives, using computers authentic situations concept enlargening Using illustrations teacher alone teacher and pupils together pupils together concept enlargening

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Stages 2/3 Cooperative a single element a tool for the pupils integrated in all classroom work using cooperative learning regularly Interactive teacher-pupils, pupil-pupil pupil-teacher, teacher-pupil, pupil-pupils pupils-pupils, pupils-pupil, pupils-teacher various interactions

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Stages 3/3 Exploratory investigations projects working inductively exploratory ways of teaching Mathematics as a language teacher orally and literally the differences between the teacher’s language and the pupils’s language meanings, deeper understanding mathematics becomes a language

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Stages in the development of the student teachers 1/2:

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Stages in the development of the student teachers 2/2:

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The six components of phenomenological mathematics teaching (I’m introducing today) manipulativesauthentic situations Experiential drawings Using illustrations Cooperative Interactive Exploratory Mathematics as a language individually investigations a pupil, orally mindmaps tables, graphs demonstrations elementlessonplancurricular Kagan & Kagan, 2002 in pairs Vuorinen, 2001 in groupsdemonstrations classroom discussion lecture open tasks projects shared exploratory process a teacher, literally a pupil, literally a teacher, orally meanings structure

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Authentic situations Something familiar (paradigm, prototype) Something unfamiliar (contrast) something really unfamiliar (boundary)

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Mindmaps (Clarke,1990) Identify the major concepts Place the concepts on paper from most abstract to most concrete Link the concepts and label each link Include definitions and illustrations use cross-links to analyze additional relationships

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Learning together and alone (Vuorinen, 2001) Interaction The size of the group VerbalVisualActiveMusicalDramatic As a one group demon- stration, discussion transpa- rencies, movies games, excursion singing and listening together sociodrama Small groups experi- ences, group- discussion posters, collages investi- gations, exhibition choir, improvi- sation pantomimes Individuals reading, exercises artlearning skills, activities composing, lyrics improvi- sations

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Individualistic Learning (Johnson & Johnson,1987) adequate space for each student each student can work at own pace each student takes responsibility to complete the task each student evaluates own progress and quality of learning simple skill or knowledge acquisition assignment clear, no need for help or confusion goal is important task is relevant materials for each student

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Competitive Learning (Johnson & Johnson, 1987) skill practice, knowledge recall assignment is clear with rules for competing goal is not so important each student can win or loose teacher referees disputes, judges correctness and rewards the winners activity is captivating set of materials for each triad any group can win possible to monitor the progress of competitors possible to compare abilities, skills or knowledge with peers

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Cooperative Learning (Johnson & Johnson, 1987) 1.positive interdependence 2.face-to-face interaction 3.individual accountability 4.interpersonal and small group skills 5.conceptual and complex tasks with problem solving or decision making or creativity 6.goal is perceived to be important

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Mathematics as a language (Freudenthal, 1983) What is Length? ”Length” has more than one meaning. ”At length”, going to the utmost length”... If length is something long, what about width, height, thickness, distance, latitude, depth,... 170 cm

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Thank You!

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