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

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

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Montreal Theoretical knowledge Semiotic Mediation From empirical to theoretical compass 5th grade Overcoming conceptual mistakes 7th grade

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

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

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Montreal Algebraic and geometrical modelling T T TERC MA

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

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Montreal Veronicas 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.

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Montreal Veronicas 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.

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Montreal The classroom discussion of Veronicas 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]

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Montreal 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]

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Montreal 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.

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Montreal 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.

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Montreal 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.

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Montreal The classroom discussion 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. Moved? Rotated! Translated?

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Montreal 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].

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Montreal The two solutions

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Montreal Veronicas 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)

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Montreal 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)

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Montreal An old (yet topical) problem ….

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Montreal … 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, shouldnt 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

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Montreal 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

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Montreal … and for mathematicians The compass and the continuum Do the two circles surely meet? WHY?

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Montreal Different answers The compass and the continuum 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).

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

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

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Montreal 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).

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Montreal The enriched compass may 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 Cavalieris instrument for parabola

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Montreal 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 ……. Lacte de prévoir, anticiper une trajectoire constitue le fondement antique, lembryon pré- humain de labstraction géométrique humaine Giuseppe Longo, 1997

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

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Montreal Second Example The Abacus and the polynomial representation of numbers

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Montreal Third Example The Perspectographs and the roots of projective geometry

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Montreal Semiotic mediation Concrete artefacts Embodied cognition Concrete artefacts only?

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Montreal 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)

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Montreal Overcoming conceptual mistakes (5/7th grade) from Platos Meno: doubling the square

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Montreal The teaching experiment the students Solve individually the problem posed by Socrates to the slave. Read Platos dialogue and detect, with the teachers guide, the three phases. Discuss the content and the different roles played by Socrates and by the slave, with the teachers guide.

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Montreal Scheme of Platos dialogue The problem: doubling a square 1 The slave is self confident Socrates asks questions The mistake is detected by visual evidence

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Montreal Scheme of Platos dialogue Towards the awareness that … 2 The slave is insecure Socrates asks questions and makes comments A new attempt

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Montreal Scheme of Platos dialogue Towards a general solution 3 Socrates guides the slave with suitable questions The slave follows Socrates with suitable answers

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Montreal The teaching experiment the students Choose another conceptual mistake in a different area, well known by the students Discuss collectively about the chosen mistake, with the teachers guide. Construct individually a Socratic dialogue about the chosen mistake Compare in collective discussion some dialogues produced by the students

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Montreal 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

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Montreal A closer look at the two examples CompassDialogue Aim To realise productive classroom activities about the theoreticalthe overcoming natureof shared of a physical conceptual instrumentmistakes

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Montreal A closer look at the two examples CompassDialogue Task To produce a methoda dialogue of constructionaccording and itsto Platos justificationmodel

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Montreal A closer look at the two examples CompassDialogue Instrumental use To use the compassPlatos dialogue to learn how to find points a square at a given with a double distancearea

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Montreal The instrumental use of the compass

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Montreal A closer look at the two examples CompassDialogue Mediational use To internalize the activitythe model with the physicalof Socratic compassdialogue to control ones own behaviour

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Montreal A closer look at the two examples CompassDialogue Mediation takes place when? In the collective discussion AFTERBEFORE the individual task with the teachers guide

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Montreal A closer look at the two examples CompassDialogue Mediation takes place how? With an essential role played by IMITATION of gesturesof genre of wordsof structure started, encouraged and explicitly required by the teacher

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Montreal 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

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Montreal A question for the Working Group D Dynamic geometry Pointwise generation of loci Continuous generation of curves ANIMATION Which epistemological analysis?

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Montreal A question for the Working Group A Which kind of mathematics preparation for primary school teachers if the aim is the approach to theoretical knowledge?

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Montreal A question for the Working Group C Is the task of producing the Socratic dialogue a problem that may (must) be solved by division?

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Montreal A question for everybody Have you ever tried to (re)construct a Socratic dialogue about a conceptual mistake of yours? Try and become teachers of yourselves!

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Montreal A remind of some references on this collective project PME reports from 1993 first author: Arzarello, Bartolini, Boero, Garuti, Mariotti (et al.) Mariotti et al., forum, PME 1997 (Lahti). Arzarello, plenary, PME 2000 (Hiroshima). Some other conferences Bartolini, plenary, ICM98, Berlin, 1998 Mariotti, Mogetta & Maracci, NCTM presession, Chicago, 2000 Cerulli & Mariotti, Montreal, ITS 2000 (June) International Journals and volumes Bartolini, ESM 96 Bartolini et al. ESM 99 Bartolini & Mariotti FLM 99 Mariotti, in English et al. (ed.), Handbook …, LEA (to appear) Book Boero (ed.), Theorems in school, …, Kluwer, to appear

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