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Phenomenological Mathematics Teaching
Päivi Portaankorva-Koivisto
The University of Tampere,
Finland
Námsstefna Flatar 29.-30.9.2006
Something about Tampere
• The city was founded by Gustav III in 1. Oct.1779, on the bank of the Tammerkoski rapids.
• Population 202 932Tampere
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
About Finnish Schoolsystem PRIMARY SCHOOL LOWER UPPER SECONDARY SECONDARY High school / vocational PRIMARY SCHOOL SUBJECT TEACHER TEACHER CLASS TEACHER (almost every subject, (maths, physics, chemistry, IT) pure mathematics about 3/160)
6 year s Age 7 grades 1 -6
3 years grades 7 -9
3 – 4 years
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
About Mathematics Teacher Education
MASTER OF SCIENCES MATHS TEACHER OPTIONAL AFTER BOLOGNA BACHELOR MASTER OF SCIENCES OF SCIENCES 25 points 35 points
MATHEMATICS DEPARTMENT
TE, 1 year
MATHEMATICS DEPARTMENT
TE, 1 – 2 years 60 points ( o f that 15 -20 points training)
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 illustrationsExploratory
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
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
The six components of phenomenological mathematics teaching - working in the classroom
manipulatives authentic situationsExperiential
drawingsUsing illustrations
Cooperative
Interactive
Exploratory
Mathematics as a language
element
individually
investigations
a pupil, orally
mindmaps tables, graphs demonstrations
structure lessonplan curricularKagan & Kagan, 2002
in pairsVuorinen, 2001
in groups demonstrationsclassroomdiscussion
lecture
open tasks projectsshared
exploratory process
a teacher,literally
a pupil,literally
a teacher,orally
meanings meanings
As a tool for the pupil As a tool for the teacher
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
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
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
Stages in the development of the student teachers 1/2:
Interactive
teacher -pupils pupil-pupil
1
pupil-teacher pupil-pupils teacher -pupil
3
pupils -pupils pupils -pupil pupils -teacher
2 Experie ntial
pupil cutt ing, g lue ing, fo lding
2
manipulatives comp uters
1
authe ntic sit uations
1 Illustra tive
teacher using grap hs , tables, mindmaps e tc.
4
teacher a nd pupils using graphs e tc. tog ether
0
pupil us ing grap hs etc. as for lear ning tool
2
Stages in the development of the student teachers 2/2:
Exploratory
pupil inves tig ating mathematical pro blems alo ne
4
pupils carrying o ut mathematical inves tig atio ns in g ro up
0
teaching and learning as a s hared explo rato ry pro ces s
2
Co o perative
as a s ing le e lement
4
as a reg ular part o f le s s o ns
2
integ rated in all teaching and learning pro ces s
0 Mathematic s as a lang uag e
fo cus o n the teacher’s us e o f mathematical lang uag e o rally and literally
5
awarenes s o f t he differences be tween teacher’ and pupils ’ mathematical lang uag e
0
integ rated meanin g s , deep unders tanding o f the us e o f mathematical lang uag e
1
The six components of phenomenological mathematics teaching (I’m introducing today)
manipulatives authentic situationsExperiential
drawingsUsing illustrations
Cooperative
Interactive
Exploratory
Mathematics as a language
individually
investigations
a pupil, orally
mindmaps tables, graphs demonstrations
element lessonplan curricularKagan & Kagan, 2002
in pairsVuorinen, 2001
in groups demonstrationsclassroomdiscussion
lecture
open tasks projectsshared
exploratory process
a teacher,literally
a pupil,literally
a teacher,orally
meanings meanings
structure
Authentic situations
• Something familiar (paradigm, prototype)
• Something unfamiliar (contrast)
• something really unfamiliar (boundary)
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
TRIANGLE three sides three angles
Right triangle
Equilateral triangle
Isosceles triangle
Two sides are equal in lenght
The three angles of any triangle add up to 180°.
The t woangl es at t hebase a reequ .al
Learning together and alone(Vuorinen, 2001)
InteractionThe 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
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
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
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
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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