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)

”The mountains of mathematics”

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

Thank You!