Montreal 20001 The Theoretical Dimension of Mathematics: a Challenge for Didacticians Mariolina Bartolini Bussi Dipartimento di Matematica Modena - Italia.

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The Theoretical Dimension of Mathematics:

a Challenge for Didacticians

Mariolina Bartolini Bussi

Dipartimento di Matematica

Modena - Italia

[email protected]

Plenary speech given at the 24th Annual Meetingof the Canadian Mathematics Education Study

GroupUniversité du Québec à Montréal - May 28th 2000

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Theoreticalknowledge

PME reportsforum 97

plen. 2000

4 teamsGenoa (Boero)

Modena (Bartolini)Pisa (Mariotti)

Turin (Arzarello)

All grades

Complementary2nd order

approaches

Epistemological

Didactical

Cognitive

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BookKluwer

to appear

4 teams

All grades

Complementary2nd order

approaches

Epistemological

Didactical

Cognitive

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SemioticMediation

From ‘empirical’ to ‘theoretical’ compass

5th grade

Overcoming conceptual mistakes

7th grade

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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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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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Algebraic and geometrical modelling

T T

TERC MA

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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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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.

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Veronica’s solution

JUSTIFICATIONI 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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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]

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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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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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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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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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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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Alessio : It's true but you cannot use two compassesVeronica : 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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The two solutions

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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)

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The final shared solutionStatic / 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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An old (yet topical) problem ….

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… for studentsExcerpt from an interview

11th gradeI 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

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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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… and for mathematiciansThe compass and the continuum

Do the two circles surely meet?

WHY?

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Different answersThe 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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In the experiment the compass is used

To draw round shapes

But also ...V. Kandinsky

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… the compass is evoked in the mind and simulated by

means of gestures

To draw circles andto find points at a given distance

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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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The ‘enriched’ compassmay be

a tool of semiotic mediationdrawing 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

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Our intuition about the continuum is built from invariants which emerge from a plurality of acts of experience:Time, Movement, The Pencil on a sheetTrajectories …….‘L’acte de prévoir, anticiper une trajectoire constitue le fondement antique, l’embryon pré-humain de l’abstraction géométrique humaine’

Giuseppe Longo, 1997http://www.dmi.ens.fr/users/longo/geocogni.html

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First Example

The compass

(and other drawing instruments)

and the problem of continuum

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Second Example

The Abacus

and the polynomial representation of numbers

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Third Example

The Perspectographs

and the roots of projective geometry

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Semiotic mediation

Concreteartefacts

Embodied cognition

Concrete artefactsonly?

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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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Overcoming conceptual mistakes (5/7th grade)

from Plato’s Meno:

“doubling the square”

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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.

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Scheme of Plato’s dialogueThe problem: doubling a square

1The slave

is self confident

Socrates

asks questions

The mistake is detected by visual evidence

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Scheme of Plato’s dialogueTowards the awareness that …

2The slave

is insecure

Socrates

asks questions

and makes comments

A new attempt

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Scheme of Plato’s dialogueTowards a general solution

3Socrates

guides the slave

with suitable questions

The slave

follows Socrates

with suitable answers

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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

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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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A closer lookat the two examples

Compass DialogueAim

To realise productive classroom activitiesabout

the theoretical the overcomingnature of sharedof a physical conceptualinstrument mistakes

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Compass DialogueTask

To produce

a method a dialogueof construction according and its to Plato’sjustification model

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Compass DialogueInstrumental use

To use the compass Plato’s dialogue

to learn how to findpoints a squareat a given with a double distance area

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The instrumental use of the compass

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Compass DialogueMediational useTo internalize

the activity the model with the physical of Socratic compass dialogue

to control one’s own behaviour

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Compass DialogueMediation takes place

when?In the collective discussion

AFTER BEFOREthe individual task

with the teacher’s guide

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Compass DialogueMediation takes place

how?With an essential role played by

IMITATIONof gestures of genreof words of structure

started, encouraged and explicitly required by the teacher

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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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A question for the Working Group DDynamic geometry

Pointwise generation of loci

Continuous generation of curves

ANIMATION

Which epistemological

analysis?

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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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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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A question for everybody

Have you ever tried

to (re)construct a Socratic dialogue

about a conceptual mistake of yours?

Try and becometeachers of yourselves!

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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

Montreal 20001 The Theoretical Dimension of Mathematics: a Challenge for Didacticians Mariolina Bartolini Bussi Dipartimento di Matematica Modena - Italia.

Documents

discussion slide

grade slide

euclid slide

cms slide

position hero slide

theoretical compass

segments st

classroom discussion