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The Theoretical Dimension of Mathematics: a Challenge for Didacticians

The Theoretical Dimension of Mathematics: a Challenge for Didacticians. Mariolina Bartolini Bussi Dipartimento di Matematica Modena - Italia. Plenary speech given at the 24th Annual Meeting of the Canadian Mathematics Education Study Group Université du Québec à Montréal - May 28th 2000.

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The Theoretical Dimension of Mathematics: a Challenge for Didacticians

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  1. The Theoretical Dimension of Mathematics:a Challenge for Didacticians Mariolina Bartolini Bussi Dipartimento di Matematica Modena - Italia Plenary speech given at the 24th Annual Meeting of the Canadian Mathematics Education Study Group Université du Québec à Montréal - May 28th 2000 bartolini@unimo.it Montreal 2000

  2. Theoretical knowledge All grades Epistemological Complementary 2nd order approaches 4 teams Genoa (Boero) Modena (Bartolini) Pisa (Mariotti) Turin (Arzarello) Didactical PME reports forum 97 plen. 2000 Cognitive Montreal 2000

  3. Theoretical knowledge All grades Epistemological Complementary 2nd order approaches 4 teams Didactical Book Kluwer to appear Cognitive Montreal 2000

  4. Theoretical knowledge From ‘empirical’ to ‘theoretical’ compass 5th grade Overcoming conceptual mistakes 7th grade Semiotic Mediation Montreal 2000

  5. From ‘empirical’ to ‘theoretical’ compass(5th grade)Field of experience The functioning of gears in everyday objects: predictive hypotheses interpretative hypotheses Montreal 2000

  6. From ‘empirical’ to ‘theoretical’ compass(5th grade)Field of experience The functioning of gears in everyday objects: predictive hypotheses interpretative hypotheses Montreal 2000

  7. Algebraic and geometrical modelling TERC MA T T Montreal 2000

  8. The Task (5th grade) Draw a circle, with radius 4 cm, tangent to both circles. Explain carefully your method and justify it. Montreal 2000

  9. Veronica’s solution The first thing I have done was to find the centre of the wheel C; I have made by trial and error, in fact I have immediately found the distance between the wheel B and C. Then I have found the distance between A and C and I have given the right 'inclination' to the two segments, so that the radius of C measured 4cm in all the cases. Then I have traced the circle. Montreal 2000

  10. Veronica’s solution JUSTIFICATION I am sure that my method works because it agrees with the three theories we have found : The points of tangency H and G are aligned with ST and TR ; II) The segments ST and TR meet the points of tangency H and G ; III) the segments ST and TR are equal to the sum of the radii SG and GT, TH and HR. Montreal 2000

  11. The classroom discussionof Veronica’s protocol Teacher : Veronica has tried to give ‘the right inclination’. Which segments is she speaking of ? Many of you open the compass 4 cm. Does Veronica use the segment of 4 cm? What does she say she is using ? [Veronica's text is read again.It becomes clear that she is using segments of 6 and 7 cms] Montreal 2000

  12. The classroom discussion Jessica : She uses the two segments ... Maddalena : .. given by the sum of radii [Some pupils point with thumb-index at the ‘sum’ segments on Veronica's drawing and try to 'move' them like sticks. They continue to rotate them till the end of the discussion] Montreal 2000

  13. The classroom discussion Teacher : How did she make ? Giuseppe : She has rotated a segment. Veronica : Had I used one segment only, I could have used the compass […]. I planned to make RT perpendicular and then I moved ST and RT until they touched each other and the radius of C was 4 cm. Montreal 2000

  14. The classroom discussion Alessio : I had planned to take two compasses, to open them 7 and 6 and to look whether they found the centre. But I could not use two compasses. Stefania P. : Like me ; I too had two compasses in the mind. Montreal 2000

  15. The classroom discussion Elisabetta [excited] : She has taken the two segments of 6 and 7, has kept the centre still and has rotated : ah I have understood ! Stefania P. : ... to find the centre of the wheel ... Elisabetta : ... after having found the two segments ... Stefania P. : ... she has moved the two segments. Montreal 2000

  16. The classroom discussion Moved? Teacher : Moved ? Is moved a right word ? Voices : Rotated .. as if she had the compass. Alessio : Had she translated them, she had moved the centre. Andrea : I have understood, teacher, I have understood really, look at me … Voices : Yes, the centre comes out there, it's true. Rotated! Translated? Montreal 2000

  17. The classroom discussion Alessio : It's true but you cannot use two compasses Veronica : You can use a compass first on one side and then on the other. Teacher: Good pupils. Now draw the two circles on your sheet. [All the pupils draw the two circles on their sheet and identify the two solutions]. Montreal 2000

  18. The two solutions Montreal 2000

  19. Veronica’s first solution Dynamic / Procedural A circle is the figure described when a straight line, always remaining in one plane, moves about one extremity as a fixed point until it returns to its first position (Hero) Montreal 2000

  20. The final shared solution Static / Relational A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another (Euclid) Montreal 2000

  21. An old (yet topical) problem …. Montreal 2000

  22. … for students Excerpt from an interview 11th grade I have already proved that that segment [KM] is always constant. … No,I haven't proved it because I haven't proved that this one [KM] rotates ...or something like that.… Now I must also say why the locus is a circle, shouldn’t I? Shall I prove it? After having proved that while C moves on C1, the segment KM (M is the midpoint of EF) does not change Montreal 2000

  23. INT. Haven't you done it? you said that this one [KM] always remains constant. It remains constant.... INT. How do you define a circle? I define it as locus.. you are right... locus of points equidistant from the centre... it crossed my mind that I had to prove also... no... maybe it is stupid ... that I had to prove that it was rotating around the centre … from Mariotti, Mogetta & Maracci, 2000 Montreal 2000

  24. Do the two circles surely meet? WHY? … and for mathematiciansThe compass and the continuum Montreal 2000

  25. EUCLID: Look at the lines in the drawing HERO: Rotate the two lines (sticks, fingers, arms …) until they clash DEDEKIND: If in a given plane a circle C has one point X inside and one point Y outside another circle C’, the two circles intersect in two points (continuity). Different answersThe compass and the continuum Montreal 2000

  26. In the experiment the compass is used V. Kandinsky To draw round shapes But also ... Montreal 2000

  27. … the compass is evoked in the mind and simulated by means of gestures To draw circles and to find points at a given distance Montreal 2000

  28. Semiotic Mediation The simple stimulus-response process is replaced by a complex, mediated act, which we picture as S --------------- R X [This auxiliary stimulus] transfers the psychological operation to higher and qualitatively new forms and permits humans, by the aid of extrinsic stimuli, to control their behaviour from the outside. The use of signs […] creates new forms of a culturally based psychological process (Vygotskij). Montreal 2000

  29. The ‘enriched’ compassmay be a tool of semiotic mediation drawing devices were used for centuries to construct and ‘prove’ the existence of the solutions • of geometrical problems • of algebraic equations Cavalieri’s instrument for parabola Montreal 2000

  30. Our intuition about the continuum is built from invariants which emerge from a plurality of acts of experience: Time, Movement, The Pencil on a sheet Trajectories ……. ‘L’acte de prévoir, anticiper une trajectoire constitue le fondement antique, l’embryon pré-humain de l’abstraction géométrique humaine’ Giuseppe Longo, 1997 http://www.dmi.ens.fr/users/longo/geocogni.html Montreal 2000

  31. First Example The compass (and other drawing instruments) and the problem of continuum Montreal 2000

  32. Second Example The Abacus and the polynomial representation of numbers Montreal 2000

  33. Third Example The Perspectographs and the roots of projective geometry Montreal 2000

  34. Semiotic mediation Concrete artefacts only? Concrete artefacts Embodied cognition Montreal 2000

  35. Further examples: microwolds Geometry as a theory (Mariotti - Handbook - LEA - to appear) Algebra as a theory (Cerulli - to be presented in ITS 2000 Montreal - June) Montreal 2000

  36. from Plato’s Meno: Overcoming conceptual mistakes (5/7th grade) “doubling the square” Montreal 2000

  37. The teaching experimentthe students • Solve individually the problem posed by Socrates to the slave. • Read Plato’s dialogue and detect, with the teacher’s guide, the three phases. • Discuss the content and the different roles played by Socrates and by the slave, with the teacher’s guide. Montreal 2000

  38. Scheme of Plato’s dialogueThe problem: doubling a square1 The slave is self confident Socrates asks questions The mistake is detected by visual evidence Montreal 2000

  39. Scheme of Plato’s dialogueTowards the awareness that …2 The slave is insecure Socrates asks questions and makes comments A new attempt Montreal 2000

  40. Scheme of Plato’s dialogueTowards a general solution3 Socrates guides the slave with suitable questions The slave follows Socrates with suitable answers Montreal 2000

  41. The teaching experimentthe students • Choose another conceptual mistake in a different area, well known by the students • Discuss collectively about the chosen mistake, with the teacher’s guide. • Construct individually a ‘Socratic’ dialogue about the chosen mistake • Compare in collective discussion some ‘dialogues’ produced by the students Montreal 2000

  42. The Task(7th grade) Write a Socratic dialogue about the following conceptual mistake By dividing an integer number by another number, one always gets a number smaller than the dividend Montreal 2000

  43. A closer lookat the two examples Compass Dialogue Aim To realise productive classroom activities about the theoretical the overcoming nature of shared of a physical conceptual instrument mistakes Montreal 2000

  44. A closer lookat the two examples Compass Dialogue Task To produce a method a dialogue of construction according and its to Plato’s justification model Montreal 2000

  45. A closer lookat the two examples Compass Dialogue Instrumental use To use the compass Plato’s dialogue to learn how to find points a square at a given with a double distance area Montreal 2000

  46. The instrumental use of the compass Montreal 2000

  47. A closer lookat the two examples Compass Dialogue Mediational use To internalize the activity the model with the physical of Socratic compass dialogue to control one’s own behaviour Montreal 2000

  48. A closer lookat the two examples Compass Dialogue Mediation takes place when? In the collective discussion AFTER BEFORE the individual task with the teacher’s guide Montreal 2000

  49. A closer lookat the two examples Compass Dialogue Mediation takes place how? With an essential role played by IMITATION of gestures of genre of words of structure started, encouraged and explicitly required by the teacher Montreal 2000

  50. Nec manus nuda nec intellectus sibi permissus multum valet: instruments et auxiliis res perficitur (Bacon: The New Organon …, 1690 quoted by Vygotskij and Lurija, 1930) Neither the naked hand nor the understanding left to itself can effect much: it is by instruments and aids that the work is done Rembrandt Montreal 2000

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