MATHEMATICS AS CULTURAL PRAXIS

1. 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

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

3. Research question • What does mathematics look like through Vygotskian lenses? • What kind of educational questions Vygotskian mathematics provoke? • How to apply Vygotskian mathematics?

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

19. 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.

20. 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.

21. 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.

22. 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.

23. 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).