B

A

C

P

P1

P2

P3I

A problem (1)

Construct a triangle ABC. Construct a point P and its symmetrical point P1 about A. Construct the symmetrical point P2 of P about B, construct the symmetrical point P3 of P about C.Construct the point I, the midpoint of [PP3].What can be said about the point I when P is moved?

From Capponi (1995) Cabri-classe, sheet 4-10.

B

A

C

P

P1

P2

P3I

“... when, for example, we put P to the left, then P3 compensate to the right. If it goes up, then the other goes down...”

“... why I is invariant? Why I does not move?”

A problem (1)

“The others, they do not move. You see what I mean? Then how could you define the point I, finally, without using the points P, P1, P2, P3?” [prot.143.]

B

A

C

P

P1

P2

P3I

Students rather easily proved that ABCI is a parallelogram

The tutor efforts...

... can be summarized, by the desperate question: ‘don’t you see what I see?’

Seeing is knowing

B

A

C

P

P1

P2

P3I

invariance of I

phenomenon

facts

Internaluniverse

Interface

Externaluniverse immobility of I

geometry

knowledge

“… why I is invariant?Why I does not move?”

B

A

C

P

P1

P2

P3I

invariance of I

phenomenon

facts

Internaluniverse

Interface

Externaluniverse immobility of I

geometry

knowledge

“… why I is invariant?Why I does not move?”

Modeling

• Learners and teachers could…… have different “understanding”

… have different “reading”

… be actors of different “stories”

• How can we inform these differences in understanding, or reading, or stories

First hint: investigate representations?

Question: what/where is the problem?

The Mendelbrot set for z z+c

The picture shows the non connectivity of M

A crucial example, The case of fractals

A proof first...

then…the picture of the filaments

A crucial example, The case of fractals

A proof first...

then…the picture of the filaments

A crucial example, The case of fractals

Back to students

Sin(exp x)

x4-5x2+x+4

how to balance trust and doubt?

Back to students

Sin(exp x)

x4-5x2+x+4

how to balance trust and doubt?

résoudre: Ln(ex-1) = x

Back to students: to balance trust and doubt

résoudre: Ln(ex-1) = x ex-1=ex

Back to students: to balance trust and doubt

« ƒ is defined byf(x) = lnx + 10sinx

Is the limit + and in +? »The environment plays a role in the number of errors we observe: - with a graphic calculator 25% of errors- without the graphic calculator 5% of errors

D. Guin et L. Trouche

And more, if needed• π=3.14• a convergent series reach its limit• the Fibonacci series

U0=1, U1=(1+√5)/2, Un=Un-1+Un-2

is divergent

Back to students: the pragmatic origin of meaning

The need to bridgeknowing and proving

Nicolas.Balacheff @ imag.fr

• our knowledge (connaissance) is the result of our interaction with our environment

• learning is the outcome of a process of adaptation (ie ecological)

the learner environment could be physical, social, symbolic…but

only certain features of the environment are relevant from the learning point of view: the “milieu”

An agreed ecological perspective

Individual ways of knowing could be ...

• Contradictory depending on the nature of context (in and out of school, on the work place and at home, at the grocery and at the laboratory, …)• Even though potentially attached to the same specific concept

Contradiction, a familiar characteristic of human beings

Let’s look at knowing as holding a set of conceptions .

Conceptions are accessible to falsification

A conception is validation dependentThe claim for validity which is at the core of

knowing requires(i) the possibility to express a statement(ii) the possibility to engage in a validation process(iii) the hypothesis of transcendence

Conceptions, validity and proof

Problems as the fundamental criteria for the characterization of a conception

a state of the dynamic equilibrium of a loop of interaction, action/feedback, between a subject and a milieu under viability constraints.

“Problems are the source and the criterium of knowing” (Vergnaud 1981)

action

   S Mfeedback

constraints

A first characterization of a conception

P... a set of problems

R... a set of operators

L... a system of representation

.... a control structure

- describe the domain of validity of a conception (its sphere of practice) - the educational characterisation of P is an open question

- the system of representation could be linguistic or not - it allows the expression of the elements from P and R

- ensure the logical coherency of the conception, it contains at least under the form of an oracle the tools needed to take decisions, make choices, express judgement on the use of an operator or on the state of a problem (solved or not)

- the operators allow the transformation of problems- operators are elicited by behaviors and productions

A characterization of a conception

Construct a circle with AB as a diameter. Split AB in two equal parts, AC and CB. Then construct the two circles of diameter AC et CB… an so on.How does vary the total perimeter at each stage ?How vary the area ?

A problem (2)

9. Vincent : the perimeter is 2πr and the area is πr2

10. Ludovic : OK 11. Vincent : r is divided by 2 ?12. Ludovic : yes, the first

perimeter is 2πr and the second is 2πr over 2 plus 2πr over 2 hence …. It will be the same

[…]17. Vincent : the other is 2πr

over 4 but 4 times18. Ludovic : so it is always 2πr19. Vincent : it is always the

same perimeter….

20. Ludovic : yes, but for the area…

21. Vincent : let’s see …22. Ludovic : hum…. It will be

devided by 2 each time23. Vincent : yes, π(r/2)2 plus

π(r/2)2 is equal to…[…]31. Vincent : the area is always

divided by 2…so, at the limit? The limit is a line, the segment from which we started …

32. Ludovic : but the area is divided by two each time

33. Vincent : yes, and then it is 034. Ludovic : yes this is true if we

go on…

A problem (2)

37. Vincent : yes, but then the perimeter … ?

38. Ludovic: no, the perimeter is always the same

[…]41. Vincent : it falls in the segment…

the circle are so small42. Ludovic: hum… but it is always 2πr43. Vincent : yes, but when the area

tends to 0 it will be almost equal…44. Ludovic: no, I don’t think so45. Vincent : if the area tends to 0,

then the perimeter also… I don’t know

46. Ludovic: I finish to write the proof

A problem (2)

Algebraic frame

area /perimeter

formulaLudovicalgebraic conception

Vincentsymbolic-arithmetic

conception

Valid

atio

n

Valid

atio

n

Representation and control

Algebraic frame

area /perimeter

Ludovicalgebraic conception

Vincentsymbolic-arithmetic

conception

Valid

atio

n

Valid

atio

n

Representation and control

Algebraic frame

area /perimeter

Ludovicalgebraic conception

Vincentsymbolic-arithmetic

conception

Valid

atio

n

Valid

atio

n

Representation

Representation and control

Let z be the sum of the two given even numbers, z is even means z=2p. We can write p=n+m, thus z=2n+2m. But 2n and 2m are a manner to write the two numbers. So z is even.

An even number can only finish with 0, 2, 4, 6 and 8, so it is for the sum of two of them

OOOOOOO   OOOOOOOOOOOO   OOOOO

OOOOOOOOOOOOOOOOOOOOOOOO

+

=

Let x and y be two even numbers, and z=x+y. Then it exists two numbers n and m so that x=2n and y=2m. So : z=2n+2m=2(n+m) because of the associative law, hence z is an even number.

2, 2= 4  4, 4= 8  6, 8= 42, 4= 6  4, 6= 0  2, 6= 8  4, 8= 22, 8= 0  

(1)

(2)

(3)

(4)

(5)

Problem (3)If two numbers are even, so is their sum

What is a mathematical proof?

From a learning point of view, there is a need to give a status to something which may be different from what is a proof for mathematicians, but still has a meaning within a mathematical activity.

ExplanationProof

Mathematical proof

The search for certaintyThe search for understandingThe need for communication

A specific economy of practice

The rôle of mathematical proofin the practice of mathematicians

Internal needs

Social communication

mathematicalrationalism

non mathematicalrationalismVersus

Rigour Efficiency

On the opposition theory/practice

The opposition theory/practice is a reality of the learning of elementary mathematics...

In the case of geometry, it takes the form of the opposition between practical geometry (geometry of drawings and figures) and theoretical geometry(deductive or axiomatic geometry)

An other opposition is that of symbolic arithmetic and algebra, which I propose as a possible explanation of the complexity of the use of spreadsheets

Which genesis for mathematical proof from a learning perspective

the origin of knowing is in actionbut the achievement of

Mathematical proof is in languageknowing in action

knowing in discourse

         construction

formulationaction validation

representation

means for action control

formulationaction validation

representation

means for action

Proofand

controlunity

formulation

demonstration

language of afamiliar world

language asa tool

naïveformalism

validation

Pragmaticproofs

Intellectualproofs

mathematicalproof

action

practice(know how)

explicitknowing

knowingas a theory

validation

Pragmaticproofs

Intellectualproofs

mathematicalproof

certaintyunderstandingcommunication

A long way to mathematical proof

naïve empiricism

crucial experiment

validation

Pragmaticproofs

Intellectualproofs

mathematicalproof

certaintyunderstandingcommunication

A long way to mathematical proof

generic example

thought experiment

statement calculus

naïve empiricism

crucial experiment

validation

Pragmaticproofs

Intellectualproofs

mathematicalproof

certaintyunderstandingcommunication

A long way to mathematical proof

Possible support from computer-based microworlds and simulation

An educational problématique of proof cannot be separated from that of constructing mathematical knowing

Specific situations are necessary to allow an evolution toward a mathematical rationality

Look for the potential contribution of the theory of didactical situations

Mathematics call for a milieu which feedback could account for its specific character

The need to clarify the epistemological and cognitive rational of didactical choices

Setting the didactical scene

Possible support from computer-based microworlds and simulation

An educational problématique of proof cannot be separated from that of constructing mathematical knowing

Specific situations are necessary to allow an evolution toward a mathematical rationality

Look for the potential contribution of the theory of didactical situations

Mathematics call for a milieu which feedback could account for its specific character

The need to clarify the epistemological and cognitive rational of didactical choices

Setting the didactical scene

Possible support from computer-based microworlds and simulation

An educational problématique of proof cannot be separated from that of constructing mathematical knowing

Specific situations are necessary to allow an evolution toward a mathematical rationality

Look for the potential contribution of the theory of didactical situations

Mathematics call for a milieu which feedback could account for its specific character

The need to clarify the epistemological and cognitive rational of didactical choices

Setting the didactical scene

Possible support from computer-based microworlds and simulation

An educational problématique of proof cannot be separated from that of constructing mathematical knowing

Specific situations are necessary to allow an evolution toward a mathematical rationality

Look for the potential contribution of the theory of didactical situations

Mathematics call for a milieu which feedback could account for its specific character

The need to clarify the epistemological and cognitive rational of didactical choices

Setting the didactical scene

Thank you !