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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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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Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 2008
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).