1. Mathematics by images Piotr Zarzycki University of Gdańsk, Poland http://math.univ.gda.pl/~Piotr.Zarzycki/ matpz@math.univ.gda.pl

2. DES-TIME-2006 Example 1: A ball This is a ball. seeing touching measuring actions

3. DES-TIME-2006 A ball This picture represents a ball. icon

4. DES-TIME-2006 A ball A ball is a subset of three-dimensional Euclidean space R 3 which consists of points (x 1, x 2, x 3 ) such that x 1 2 + x 2 2 + x 3 2  1. symbolic description

5. DES-TIME-2006 Three stages of intellectual development (by Jerome Bruner)  Enactive where a person learns about the world through actions on objects.  Iconic where we learn through using models and pictures.  Symbolic which describes the capacity to think in abstract terms. iconicsymbolicenactive

6. DES-TIME-2006 enactive + iconic = building images (models)

7. DES-TIME-2006 Example 2 Imagine a skeleton model of a cube n  n  n which consists of n 3 unit cubes. We consider (n+1) 3 knots of the model. Suppose the cube is plunging in water in such a way that its main diagonal is perpendicular to the surface of water. What are the numbers of consecutively plunged knots?

8. DES-TIME-2006 tonący_szescian.cg3

9. DES-TIME-2006 Example 3 is linear, then drezno1.fig

10. DES-TIME-2006 Comments on Example 3  For many mathematical concepts the order of three stages may be different.  building = enactive + iconic + symbolic mathematical objects by programs, using technology enactiveiconicsymbolic

11. DES-TIME-2006 Example 4 - proving by clicking dresden1.dfw

12. DES-TIME-2006 Example 5 – images Take any four points in the plane, A, B, C and X. Going from X to A take the mid-point X 1, from X 1 to B take the mid-point X 2, from X 2 to C take the mid-point X 3. Continue with this process, where will you finally come to after this „middlings”? drezno2.fig

13. DES-TIME-2006 Example 5 – images + symbols Mathematics by images does not mean we neglect working with symbols. dresden2.dfw

14. DES-TIME-2006 Example 6 – from images to research problem What is the set ?

15. DES-TIME-2006 Example 6 – What is your guess? dresden3.dfw

16. DES-TIME-2006 Some final remarks  For many mathematical concepts the best way to work on them, to play with them is building images using programs like CABRI or DERIVE.  To be competent in mathematics should also mean ability to make images using mathematical software.  Task for educators: criteria for such a competence.  Task for educators: methods, examples to teach this competence.  Task for educators: how to assess this competence.

17. DES-TIME-2006 Thank you. Good bye.