Scienze e Viaggi

Geometry with Logo

1. Team members: Graziano Scotto di Clemente and Sandra Turra

2. Academic level of the *students*: pupils 11 – 14 years old; first class, second class and third class of Italian lower secondary school.

3. Reflection on the *mathematical content* from a didactical point of view which might focus in epistemology, history, social issues, etc:

History - the evolution of geometry, in some great steps

• During Greek – Roman antiquity

• Talete (642 b.C. – 548 b.C. ?): an angle, inscribed in a semicircle, is a right angle.
• Pythagoras (580 – 500 ?): the Pythagoras’ 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…
• Apollonio by Pergamo (262 – 190 ?): ? (radian), ellipsis, parabola and hyperbola..

• During and after the Renaissance:

• The prospective in the arts: Leon Battista Alberti, Piero della Francesca…(around 1400)
• Kepler, Galilei and Cavalieri: Archimedes Method in the calculation of volume solids
• Descartes (1596 – 1650)
• Klein and the Erlangen’s Program (1872) about the geometrical transformations.
• Papert and Logo (1980)

From this last step we can come back until the ancient Greek with some simple uses of Logo, sometimes it will be possible to study the same problem with a different point of view: for example the constructions of the regular polygons with Logo or with Euclidean way.
So it is very important to study the structures of geometrical figures in a dynamic approach ( see the Erlangen’s Program), with transformations of modulus etc..

4. Epistemology:

• Programming with Logo it is a strong exercise of heuristic, mathematical process: a pupil can build his procedure helped by his self mistakes (“bricolage” method).
  How is it important in geometry an heuristic thought?
  The construction of a procedure is an algorithmic construction, pupils cannot break the rules of the instructions or the rules of Logo syntax.
  In this way every heuristic process becomes an algorithmic process.
  This is the process of a good demonstration of classical theorems of geometry, also.
  For the young students, a Logo Lab is a good training of inductive – deductive demonstrations.
  
• The importance of a recursive thought:
  Many problems can be decomposed in more simple down problems (bottom-up design), sometimes one of these is the same main problem. This is a recursive situation. There are many situations in that the recursive thought is shown, in an Italian nursey-rhyme:
  “ Once upon a time there was a king, sitting up a divan, who was telling to his servant: “ Tell me a tale!”, and the servant begin with once upon….”
  We have tried that some geometrical problems, with Logo, can help the pupils to use this form of thought: for examples the constructions of spirals (like a spiral with the Fibonacci’s series).

5. A list of *goals/objectives* to be reached:

• To build a sequence to arrive to the foreseen result. This is a typical goal for a mathematical curriculum.
  
• To know the concept of angle like a rotation (left and right).
  
• To know the concept of egde when you want to draw a geometrical figure, from a starting point.
  These geometrical points of view can be compared with the euclidean ones.
  
• To know the regular polygons like modulus in translations, rotations…. (if you want it is possible to connect these situations with some simple aspects of the modular arithmetic.
  
• To learn how the circle is “the limit” of the regular polygons.
  
• To be able to use a reference system, in different conditions ( right or left respect for….)
  
• And then to know the Descartes’ plane. In this geometrical environment the pupils can draw envelopes.
  
• To be able to decompose a problem in more simple down problems and then to be able to recompose their to reach the solution (bottom-up design):
  o the pupils will use tree-graphs;
  
• To be able to find variable parts and constant parts in a graphic project. It is very important a long training in the analysis of variables and constants before to study the algebric problems in the lower secondary school. Also the use of mathematical structured aids, like Dienes’ logical blocks, could be useful.
  
• To know the structure: “if…then…otherwise”in elementary situations. This logical structure is very important to control the use of recursive thought in same geometrical problems (like spirals) or numerical problems like series:
  In these cases the pupils will be able to use flow charts.

6. Description of the *used materials*, availability, and classification according to the process followed (v. gr.: initial assessment, classroom presentation, homework, group work, final assessment, etc.):

At the beginning the class works without computer only with pencil and paper and a pupil behaves like a robot to understand the difference between to teach a procedure and to execute a procedure.
Then the teacher shows the program (we use Berkeley’s University MSWLogo, with a little translator from English instructions to Italian instructions).
The first times the pupils work in pairs, so they can help themselves.
This work is completely operative: the used materials allow both an help and an autonomous resolution of the problem.
The realization of the project is often the final assessment, some things are important:

• conformity beetwen the final object and the starting project;
  
• choice of the more economic way;
  
• presence of original observations and reflections…

Papert

Programmazione scolastica e Logo

Carrellata geometrica con il Logo

Perchè il Logo