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MATHEMATICS AS CULTURAL PRAXIS

EECERA conference 3-6.9.2008

Jyrki Reunamo

Jari-Matti Vuorio

Department of Applied Sciences of Education,

UNIVERSITY OF HELSINKI 2008

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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008

Mathematics is considered as a content orientation, in

which children start to acquire tools and capabilities by

means of which they are able to gradually increase their

ability to examine, understand and experience a wide

range of phenomena in the world around them.

Mathematical orientation is based on making

comparisons, conclusions and calculations in a closed

conceptual system. In ECEC, this takes place in a playful

manner in daily situations by using concrete materials,

objects and equipment that children know and that they

find interesting.

Finnish national curriculum guidelines on ECEC (2005, 24-25)

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Research question

What does mathematics look like through Vygotskian

lenses? What kind of educational questions Vygotskian

mathematics provoke? How to apply Vygotskian mathematics?

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Culturally existing math (Proximal development)

Mathematics is out there. The problem is how to find it. People can get access to the existing mathematics by

reaching out for the physical or social content of

mathematics. There is a lot of existing mathematics. The problem is to

find the important or relevant mathematics. There may be a mathematical truth. Math is still

incomplete and open for new organizational principles or

a more profound foundation.

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Closed doctrine (Actual development)

Mathematics is a doctrine, philosophy or science defined

by mathematicians. Mathematics represents itself in human understanding,

operations and schemas. Mathematics is what one sees it being or defines it being. There is a lot of mathematical beliefs. The problem is their

preference and their questionable relation to reality. There are many mathematical models with respective

axioms and theorems. New axioms may be added to a

closed model. It is not possible to always tell if the

statement is true or false.

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Math application (Instrumental tools)

The power of mathematics can be seen in the application

of it in real life situations. Pure mathematic thinking can

have an unexpected relation to reality. Math explains reality and has an effect on reality. Math is

a tool to get things done or understood. Mathematics is a powerful instrument for constructing and

analyzing reality. The problem is in the practical

enforcement of mathematics. The environment can be seen as organizing along

mathematical principles. Math is the origin, foundation or

explanation of environmental change.

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Math production (Producing tools)

Mathematics is a cultural product without predefined

content or axioms. The problem is to use culturally

relevant mathematics. Culture and mathematics have an effect on each other. Mathematics is reflected e.g. in ICT, science and

information society. The problem is that when pure

mathematics is used in cultural contexts it has ethical and

esthetic connections. Math and historical context are related and reflect each

other, e.g. stone age, agriculture, modern, postmodern.

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Math education: Proximal development

The child’s open and involved contact to the math content

in the environment, more advanced math helps the child

in producing more advanced interaction. The child learns the uses and contents of math to better

correspond to the socially shared society. It can be

appreciated and benefited by others too. Learning is reaching for even more advanced math used

by more skilful partners.

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Math education: Actual development

The math skills the child has learned and can use without

help from others. The developmental phase of the child. The internalized math tools and restrictions for processing

things. The child’s use of math tells about child’s mental

operations and schemas, imagination and orientation. Learning is adding elements and inventing new ones,

ability to use new elements without external help.

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Math education: Instrumental tools

Math is the connection between the child’s motives and

reality. Child tests the different outcomes of different

mathematics. Math is a tool to get things done. The child’s personal application of math in the

environment. The impact is not wholly restricted by

deficiencies in math. Math is a tool for influencing environmental changes.

Learning is to find ways to control and organize the

environment using math.

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Math education: Producing tools

A child’s contribution to the math content. A child tests,

stretches and remolds the limits of math. For example 2

pieces of clay + 2 pieces of clay = 3 apples. Dialogue produces a common workspace. Creative

expression with play. The child redefines and tests the

structure of clay. Participative math learning is producing dynamic versions

of mathematical time and space. Math is a cultural

product without predefined axioms.

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Solid shapes: Proximal development

Blocks are discussed, feeled, smelled and guessed by their sound. The teacher presents and uses the concepts of ball, cube etc. The mobility of the objects are studied, same shapes are looked

after in the environment. The properties, differencies and similarities are discussed. Playing with the shadows of the shapes. Covering the blocks under

a cloth. The teacher helps children to perceive aspects of the blocks. Children’s involvement is important.

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Solid shapes: Actual development

Children do exercises with the blocks. Children solve math problems. The blocks are counted,

identified, remembered, classified and compared. The objects are measured and their properties

investigated. Memory games are played, the properties of the shapes

are learned and repeated again and again. The teacher teaches the proper use of mathematical

concepts. Children’s independent mastery of the concepts related to

the blocks is important.

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Solid shapes: Instrumental tools

The blocks are relocated from the teaching tool cabinet to readily

available playing material. The use of blocks is encouraged. The blocks are of good quality

and there is enough of them. The teacher participates in children’s play when opportunity arises

enriching and offering new ideas to play with the blocks. Children’s play is appreciated and given time. Children’s products

are left for others to see and they are discussed together. The use of the blocks in children’s personal play is appreciated.

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Solid shapes: Producing tools

The teacher makes a puppet theater in which the puppet uses the

blocks to build a house, but the puppet does everything wrong.

Luckily the children help him. The finished house is awesome! In small groups children plan and build their own houses of the

blocks. In the end the finished houses are evaluated by all. A village of the houses is created. New shapes are discussed and

introduced. The blocks are material for a social and cultural

development. Children adventure in a village filled with mathematical content.

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The cycle of math development

The four points of view produce a cycle: first the math

content of the blocks is perceived and interactively

contacted (PD). Then the mathematical content is practiced, repeated,

remembered and learned (AD). After possessing the mathematical tools the blocks can

be used as personal instruments for personal production

(IT). In the end the products and tools become part of cultural

development, which in turn is a new platform for proximal

development (PT).