The geometric image of the world

To be able to:

• build a sequence from a starting point to a foreseen result
• compare  the concept of angle as change in direction with the Euclidean concept of angle as space between two lines starting from a common point
• compare the trace left by the turtle with the Euclidean concept of edge
• recognize the regular polygons as modulus in translations, rotations, symmetries  (and  recall modular arithmetic aspects)
• compare the idea of circle as  “the limit” of regular polygons with the Euclidean definition of circle as set of all points in the plane at a given distance from a fixed point
• describe the position of an object by using different reference points ( right or left related with….) and use properly  the Cartesian reference system

• break down a problem  in simpler steps, and then to rebuild the whole problem, by using tree-graphs.
• distinguish variables from constants in a graphic project. This is a very important training that prepare pupils towards the analysis of  algebraic problems.
• use  the structure: “if…then…otherwise” in elementary situations. This logical structure permit the control of recursive steps in geometrical problems (e.g. .spirals) or in numerical ones (e.g. series)

An effective teaching/learning of Geometry should take account of the following dimensions (2):

• the pedagogical one, raising questions like: ‘How to motivate pupils?’’, ‘How to help them better understand, draw generalizations and construct meanings and competencies?’, ‘What kind of tools are more effective?,’ What role can ICT play?’….
• the historical one,that is a source of suggestions for teachers, is motivating for pupils and show them the problematic nature of the process that led to our geometric knowledge.
• the epistemological one,because a reflection about the structure of the discipline and on its  way of thinking allows to identify key concepts and procedures (e.g. the importance of the heuristic thought, the internal consistency, the study of patterns and relationships, the connection with the real world ..)

• Thales (642 B.C. – 548 B.C. ?): an angle inscribed in a semicircle is a right angle.
• Pythagoras (580 – 500 ?) and its theorem, the golden section of a segment, the figured numbers…
• The incommensurability: the diagonal of a square…
• The regular solids (cube, tetrahedron, octahedron, dodecahedron, icosahedron) and Plato
• Euclid
• Archimedes
• Apollonius of Perga (262 – 190 ?): p (radian), ellipsis, parabola and hyperbola.

• The perspective in art: Leon Battista Alberti, Piero della Francesca…(around 1400)
• Cavalieri and its method for calculating volumes
• Descartes and its reference system (1596 – 1650)
• Klein and the Erlangen Program (1872) about the geometric transformations.
• Papert and Logo (1980)

Starting from this last step, we come back  to  the Ancient Greek period, planning activities where pupils are guided to explore figures and relationships among them.
They can  study, for example,  how to build the procedure for drawing  regular polygons. Stimulated to identify relationships between side number and angle measurement to solve this problem, they often discover by themselves the “Turtle theorem” (3).  Pupils  can also be  encouraged to draw regular polygons in the Euclidean way,  using pencil, rule and compass or dynamic geometry software like Cabri.  
Combining Cabri with Logo is particularly interesting not only because the use of different  tools usually contributes in raising interest, but also and mainly  because, if the children have the opportunity to observe shapes that gradually change from one into another, it become easier for them making conjectures on their geometric properties (see Erlangen Program).

Programming with Logo is a good way to approach the heuristic-mathematical process, because pupils  can develop their own projects,  build procedures by themselves and correct them, guided by their own mistakes (‘bricolage’ method).
Making a procedure is a construction with an internal consistency, i.e. pupils cannot break the rules of the Logo commands or syntax.  Pupils can see in real time and step by step  if their procedure works/doesn’t work and are stimulated to find corrections by reasoning both on the procedure and  on the draw generated by the procedure. This is an excellent way of training children in  inductive/deductive thinking and then of getting  them used  to the algorithmic method. Later, pupils will recognize and use the same way of thinking in the demonstrations of classical theorems (1; 3).

An other strong point of Logo is the opportunity to break down a problem  in simpler steps, and then to rebuild the whole problem ( top-down/bottom-up method). Sometimes one of the simpler steps is the main problem itself (4): this is a recursive situation, like this Italian nursery-rhyme:
‘Once upon a time there was a king, sitting on a sofa, who was telling to his servant: -Tell me a story!-, and the servant  started telling him - Once upon….’.
Geometrical problems developed with Logo, like a spiral construction with the Fibonacci series, can help the pupils to use this form of thought.

Geometry gives a complex representation of reality, because it ‘perceive’ and represent the reality in different ways and with different theories, sometimes contradictory between them, and because often moves far away from the reality.
The ICT help children grasp this complexity, because they give them the opportunity to analyze geometric problems with different approaches, from the more intuitive to the more deductive. We found useful  to integrate the use of Logo with other software, i.e. Cabri II Plus, a basic CAD software and  a graphics package. While the CAD software has a deductive and abstract method of analyzing a geometric problem, like the 3D representation of a solid, Cabri offers a good environment to stimulate intuitive reasoning and modelling,  for example  when children use Cabri to find out what shape better fits the image of a real  object (5; 8; 10).
Graphics packages can be used to  make more appealing Logo drawing and to animate them, in order  to capture the interest of pupils, who like a lot this kind of activities.

For an exemplification of  contents and approaches for 11-14 year old pupils - Table 1. Examples of contents in different learning environments

1. Scotto di Clemente, G. (1989) Tre anni di Logo. In Compuscuola. Edited by Jackson Editore. Milan, February 1989. 36-43.
2. Scotto di Clemente, G. (1991) Perchè la matematica è difficile da studiare. La scuola SE, 77, 22-25. English version in: http://www.infinito.it/utenti/martini12/italy1/article.htm
3. Abelson, H. Disessa, A. (1986) La geometria della tartaruga (Turtle geometry). Muzzio Editore, Padua, 1986.
4. Roberts, E.S. (1987) Il pensiero ricorsivo (Thinking Recursively). Franco Angeli, Milan, 1987.
5. Adrian Oldknow, Cabri geometry and digital images in bringing geometry to life, and life to mathematics, 10/09/2004, Third Cabri Geometry International Conference, Rome.
6. J.M. Laborde cit. in Cabri2004 – Rome, September  2004.
7. Scotto di Clemente, G. (2003-2006), Logo geometry: some examples, http://www.infinito.it/utenti/martini12/italy1/ logo_flash_en_03.htm
   http://www.infinito.it/utenti/martini12/italy1/ /logo_flash_en_06.htm
8. Scotto di Clemente, G. (2005), Cabri and modelling, http://martini.xoom.it/cabri2005/index.htm
9. Turra, S. (2003-2006), Turtle Geometry, http://www.infinito.it/utenti/martini12/italy2/logo.htm
10. Turra, S. (2003-2006), Studying quadrilaterals with Cabri, http://www.infinito.it/utenti/martini12/italy2/cabri.htm