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

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

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

A ball A ball is a subset of three-dimensional Euclidean space R3 which consists of points (x1, x2, x3) such that x12 + x22 + x32  1. symbolic description 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. enactive iconic symbolic DES-TIME-2006

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

Example 2 Imagine a skeleton model of a cube nnn which consists of n3 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? DES-TIME-2006

tonący_szescian.cg3 DES-TIME-2006

Example 3 is linear, then drezno1.fig 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 enactive iconic symbolic DES-TIME-2006

Example 4 - proving by clicking dresden1.dfw 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 X1, from X1 to B take the mid-point X2, from X2 to C take the mid-point X3. Continue with this process, where will you finally come to after this „middlings”? drezno2.fig DES-TIME-2006

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

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

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

Thank you. Good bye. DES-TIME-2006