Phenomenological Mathematics Teaching

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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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Senses, experiences, uniqueness
Experiential
Listening, Emotions
Openness
Interactive
Cooperative, collaborative
Phenomenology (Lehtovaara, M., Rauhala, Husserl)
Phenomenological Mathematics Teaching
Meanings
Intuition, genuinity
Aestetic, Individuality
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
As a tool for the pupil
As a tool for the teacher
manipulatives
authentic situations
Experiential
drawings
mindmaps
tables, graphs
demonstrations
Using illustrations
element
structure
lessonplan
curricular
Cooperative
Kagan Kagan, 2002
individually
in pairs
in groups
demonstrations
classroom discussion
lecture
Interactive
Vuorinen, 2001
investigations
open tasks
projects
shared exploratory process
Exploratory
a pupil, orally
a teacher, literally
a pupil, literally
a teacher, orally
meanings
meanings
Mathematics as a language
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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 teachers language
  and the pupilss 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 (Im introducing today)
manipulatives
authentic situations
Experiential
drawings
mindmaps
tables, graphs
demonstrations
Using illustrations
element
lessonplan
curricular
structure
Cooperative
Kagan Kagan, 2002
individually
in pairs
in groups
demonstrations
classroom discussion
lecture
Interactive
Vuorinen, 2001
investigations
open tasks
projects
shared exploratory process
Exploratory
a pupil, orally
a teacher, literally
a pupil, literally
a teacher, orally
meanings
meanings
Mathematics as a language
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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 Interaction Interaction Interaction Interaction
The size of the group Verbal Visual Active Musical Dramatic
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 art learning 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!