The complexity of the epistemological and didactical genesis of mathematical proof

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What would be your comments on these proofs?

An even number can only finish with 0, 2, 4, 6 and 8, so is it 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 distributive law, hence z is an even number.

2, 2= 4 4, 4= 8 6, 6= 2 8, 8=62, 4= 6 4, 6= 0 6, 8=42, 6= 8 4, 8= 22, 8= 0

If two numbers are even, so is their sum

+=

from Healey & Hoyles

Nicolas Balacheff, CINESTAV, September 2015

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What is a mathematical proof?

From a teacher point of view?

From a mathematician point of view?

From a learning point of view?

From your point of view?

Where does the difference between mathematical and non mathematical proof lie?

Our statements about learners’ conceptions of mathematical proof and argumentation is bound by our own conception (and/or research problématique).


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What is a mathematical proof?Fawcett “The concept of proof is one concerning which the pupil should

have a growing and increasing understanding. It is a concept which not only pervades his work in mathematics but is also involved in all situations where conclusions are to be reached and decision to be made. Mathematics has a unique contribution to make in the development of this concept…”

Harel & Sowder “One's proof scheme is idiosyncratic and may vary from field to field, and even within mathematics itself ”

Healey & Hoyles “Proof is the heart of mathematical thinking, and deductive reasoning, which underpins the process of proving, exemplifies the distinction between mathematics and the empirical sciences”

Hanna & Janke “The most significant potential contribution of proof to mathematics education is the communication of mathematical understanding” […] “in order to understand the meaning of a theorem and the value of its proof, students must have extensive and coherent experience in the appropriate application area.”


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Theorem = system of (statements, proof, theory)

“a theoretical fact, a theorem is acceptable only because it is systematized within a theory, with a complete autonomy from any verification or argumentation at an empirical level”

The existence of a reference as a system of shared principles and deduction rules is needed if we were to speak of proof in a mathematical sense

The presence of concrete and semantically pregnant referents for performing concrete actions that allow the internalisation of the visual field where dynamic mental experiments are carried out, presence of semiotic mediation tools, construction of an evolving student internal context […] a polyphony of articulated voices on a mathematical object

What is a mathematical proof? Boero, Mariotti & Bartolini


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Is mathematical proof…

(1) a universal and exemplary type of proof

(2) at the core of mathematics

(3) specific to mathematics as an autonomous field

(1’) of an idiosyncratic nature

(2’) a tool needed by

mathematics

(3’) getting its meaning from applications


COMPLEXITY OF THE EPISTEMOLOGICAL & DIDACTICAL GENESIS OF MATHEMATICAL PROOFnotes from a research journey on learning proof for the CINVESTAV doctoral colloquium 2015

Nicolas.Balacheff @ imag.fr

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A working positionProof is part of the development of rationality(the way one takes decisions, makes choices, performs judgements) Tension between practical reasons and theoretical

reasons (economy of practice) Tension between argumentation and mathematical proof Central role of systems of representation (e.g. linguistic,

graphical, symbolic)

Mathematical proof is tightly related to The specific nature of mathematical objects, ontological

and semiotic Three interrelated objectives: understanding,

communicating and validatingTheorem = system of (statements, proof, theory)


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Case 1: the sum of the angles of a triangle

This theorem/property is taught at the 7th grade.Mathematical proof has not been introduced as such, but experimenting with geometrical objects is common.

Is it possible:- to introduce the conjecture?- to raise the problem of proving it mathematically?

and to which extent?

Create conjecturing conditions Disqualify measurement as a means to prove Raise the problem of proving at a theoretical level


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Draw a triangle, measure the angles and calculate the sum… all results are displayed...

One same triangle is for all students… they first bet a result, then make measurements and computations shared on the blackboard…

Three triangles of very different in shapes are proposed , work in groups of three or four… bet, results, sharing on the blackboard...

A familiar and practical context: drawing, measuring, computing. No result is privileged

The same but… if all have the same triangle, all should have the same result…

Very different shapes stimulate a debate on what and why

The collective confrontation install the condition for the conjecture and the disqualification of measurement


A mathematical argument should be constructed using the repertoire of the known and accepted statements , following rules accepted by the class


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Draw a triangle, measure the angles and calculate the sum of the results… all results are displayed on the black board...

One same triangle is given, the same for all students… they have first to bet a result, then to make measurements and computations shared on the blackboard…

Three triangles very different in shapes are proposed to the students who, this time, work in groups of three or four… a bet, results, sharing on the blackboard...

Statement of a conjecture Searching a proof…



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Case 1: the sum of the angles of a triangleA look at the discussion

S144 - The protractor is not very accurate. We don’t necessarily find 180 […]S150 - A square [drawing a rectangle] , it has four angles, 4 times 90 makes 360, right? And a triangle 180 [he cuts the rectangle in two parts] then a triangle is 180S159 - We have found 190 because we have measuredT161 – and each time you found 190… for the first triangle and the one I gave youS162 – yes, we have measured, so…


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Case 1: the sum of the angles of a triangleA look at the discussion

S476-78 – one can draw whatever… it will be equal to 180… more or lessS480 – we can even draw very small triangle, I thinkS487 – it’s not the triangle which counts, it is the anglesS500 – if we find measures different from 180… it’s because angles are badly measured. The angle sectors extend indefinitely and we will always find 180 for the sum of the angles of a triangle


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Case 1 debriefing

Theoretical geometry

Practical geometry

The sum of the angles of a triangle

Pragmatic proofNo way to go beyond

approximationbut conformance to the rule book

Intellectual proofNeed to organize objects and

relationswithin a theoretical framework

argumentation

mathematical proof


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Case 1 debriefingProof and argumentationMathematical proof holds two characteristics, which oppose it to argumentation (see Duval).

First, it is based on the operational value of statements and not on beliefs which may be attached to them (epistemic value).

Second, the development of a deductive reasoning relies on the possibility of chaining the elementary deductive steps, whereas argumentation relies on the reinterpretation or the accumulation of arguments from different points of view

in this sense….argumentation is an epistemological obstacle to

the learning of mathematical proofbut…depending on the reasoning tools and representations available, argumentation remains a possible mathematical validation framework (quasi-empiricist approaches)


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Case 1 debriefingThe core didactical structure“Each item of knowledge can be characterized by a (or some) adidactical situation(s) which preserve(s) meaning ; we shall call this a fundamental situation” (p.30)

devolution institutionalization

adidactical situation

didactical situation

actual teaching situation

fundamental situationrestrictionadaptation

Knowledge analysis


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

Case 2: the fix point

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, midpoint of [PP3].

What can be said about the point I when P is moved?

“... 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?”


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The student easily proved that ABCI is a parallelogram but couldn’t conclude

The tutor efforts...“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?”

... can be summarized, by the desperate question:

‘don’t you see what I see?’

“... 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?”

B

A

C

P

P1

P2

P3I


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invariance of I

geometrical phenomenon

facts

Within the machine

Interface

immobility of I

e-geometry

knowledge

B

A

C

P

P1

P2

P3I


conocimiento


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Case 2 debriefingLet’s consider the student from an epistemic perspective: the subject and the environment as well: the milieu

Both interacts (act/react) based on representations and means for actions.

Interaction is driven by embedded controls: subject rationality, feedback rules of the milieu

action

S Mfeedback

constraints

B

A

C

P

P1

P2

P3I

and characteristics of a situationNicolas Balacheff, CINESTAV, September 2015

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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 criterion of knowing” (Vergnaud 1981)

action

S Mfeedback

constraints

Characterizing conceptions

Conceptions• may be contradictory even though

potentially attached to the same concept

• are accessible to falsification• are validation dependent


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action

S Mfeedback

constraints

Characterizing conceptionsA conception can be characterized by- the set of problem (sphere of practice) in which it is efficient under the current means of control available- the set of operators which allows to solve the problem- the representation systems (linguistic, symbolic or graphical) which allows to express and treat the problem- the control structure which allows to make choices, take decisions, assess operation, validate solutions


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Conceptions - interaction - situation

The relationships between a student and a milieu correspond to different forms of knowledge

[3] forms which allow the explicit “control” of the interactions in relation to the validity of a statement.

[2] formulations of the descriptions and models [1] models for action governing decisions

They correspond to three major categories of situations

[1] Action → actions and decisions that act directly [2] Formulation → exchange of info coded into a language [3] Validation → exchange of judgment

(Brousseau 1997 p.61)Nicolas Balacheff, CINESTAV, September 2015

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Back to case 1: the sum of the angles of a triangleThe bigger the triangle, the bigger its perimeter

Geometry of shapesDrawing and measurementRepertoire of geometrical factsArgumentation

Spatio-graphic objects

Situation 1 & 2- each has its own triangle- one triangle for all

Devolution of a problem

Situation 3social interaction, confrontation of conceptions

Birth of a conjecture

Problem of proof

The sum of the angles of a triangle is invariant

Geometrical objectsGeometryStatements, diagramsRepertoire of propertiesMathematical argumentation

C1

C2


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The design of didactical situationslearning dynamic

design teachingdynamic dynamic

engage in the situation / mobilize

conceptionsshare

communication means, language,

representationexplicit

validation pattern

action

formulation

validation

devolution

institutionalization

Knowledge should be the only legitimate reference for decision making


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The design of didactical situationsGiven a content to be taught-learned

identify which proof or argument is accessible and mathematically acceptable

the repertoire of the accepted statements the representations needed and available (or accessible) to

studentsthen specify situations to introduce it.

Guiding the design by pragmatic questions like“Why would the student do or say this rather than that?”“What must happen if she does it or doesn’t do it?”“What meaning would the answer have if the student had given it?”…/…

it is possible to elicit the conditions to be imposed on the milieu. does the milieu include a feedback function

adapted to the need for adjustment of the interaction to the targeted knowledge?


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Back to case 1: the sum of the angles of a triangle

S159 - We have found 190 because we have measuredT161 – and each time you found 190… for the first triangle and the one I gave youS162 – yes, we have measured, so…

How far can the teacher step back?

The core of the didactical complexity lays in… the didactical contract / interaction with the teacher,

devolution of the situation the regulation of social interactions among (rules of the

“game”)

The institutionalization must be explicit on the status of the proven statement (statement and proof) the means to prove it (proof and theoretical framework)


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Conclusionon the side of proof

The role of mathematical proofin the practice of mathematicians

Internal needsCommunication

mathematical

rationalism

non mathematicalrationalism

Versus

Rigour Efficiency

organizing and structuring the mathematical content

Specific economy of practice


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Conclusionon the side of the learner

formulationaction validation

representation

operators control

Proofand

controlunity

Nicolas Balacheff, CINESTAV, Septembre 2015

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Conclusionon the side of the situation

Situations of validation involve players who confront each other over an object of studies composed- of messages and

descriptions- of the didactical milieu

which serves as a reference

There is room for- pragmatic exchanges- theoretical exchanges

Nicolas Balacheff, CINESTAV, Septembre 2015

A's stake

player A

proposer, opposer

player B

opposer, proposer, executor

statements proofs refutations

statements, theories allowed by A

information

actions

actionsinformation

stake constraints of debate

B's

stak

e

statements, theories allowed by B

action milieu messages

mathematical proof

argumentation

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Which relation between argumentation and mathematical proof? dialectic of proofs and refutations argumentation as a ground and an epistemological

obstacle Is there anything like "mathematical argumentation" and

what are its characteristics? switch from epistemic to logical values of statements ground of agreed statements (repertoire, theoretical

framework) What are the characteristics of situations demonstrating the

benefit of mathematical proof? proof as an object Vs proof as a tool switch from a practical to a theoretical problématique

ConclusionSome questions and issues


Education

The complexity of the epistemological and didactical genesis of mathematical proof