1. University of Gdańsk, Poland
Mathematics by images
Piotr Zarzycki
University of Gdańsk, Poland

2. Example 1: A ball This is a ball. actions seeing touching measuring
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3. A ball
This picture represents a ball.
icon
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4. 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
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5. 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
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6. building images (models) enactive + iconic =
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7. 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?
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8. tonący_szescian.cg3
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9. Example 3
is linear, then
drezno1.fig
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10. 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
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11. Example 4 - proving by clicking
dresden1.dfw
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12. 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
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13. Example 5 – images + symbols
Mathematics by images does not mean we neglect
working with symbols.
dresden2.dfw
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14. Example 6 – from images to research problem?
What is the set
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15. Example 6 – What is your guess?
dresden3.dfw
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16. 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.
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17. Thank you.
Good bye.
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