MATHEMATICS AS CULTURAL PRAXIS EECERA conference Jyrki Reunamo Jari-Matti Vuorio Department of Applied Sciences of Education, UNIVERSITY OF HELSINKI 2008

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 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)

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki Research question What does mathematics look like through Vygotskian lenses? What kind of educational questions Vygotskian mathematics provoke? How to apply Vygotskian mathematics?

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 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.

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 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.

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 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.

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 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.

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 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.

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 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.

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 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.

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 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.

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 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.

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 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.

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 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.

Jyrki Reunamo & Jari-Matti Vuorio University of Helsinki 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.

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