learning mathematical proof, lessons learned and outlines of a learning environment

© N. Balacheff Oct. 2005

Learning mathematical prooflesson learned from research

&principles of design of a learning environment

[email protected]


A controversial question...

What is a mathematical proof ?


The rôle of mathematical proofin the practice of mathematicians

Internal needs

Social communication



Internal needs


mathematicalrationalism

non mathematicalrationalismVersus



Internal needs




VersusRigour Efficiency



Internal needs




VersusRigour Efficiency

The specific economyof the practice of mathematics


Argumentationvs

Mathematical proof

Argumentation

content count

epistemic value

Mathematical proof

operational value count

structural value


Mathematical proof can be considered as an answer to...

The search for certainty

The need for communication

The search for understanding


Mathematical proof can be considered as an answer to...

The search for understandingThe search for certainty

The need for communicationYes, and the three

dimensions cannot be separated....


Learning mathematical proof,which genesis ?

the origin of knowledge is in actionbut the achievement of

Mathematical proof is in language





knowledge in action

knowledge in discourse





knowledge in action


construction



the origin of knowledge is in action




Yes, but...why is that true ?


formulationnature of

knowledgevalidation



knowledgevalidation

demonstration

practice(know how)

Pragmaticproofs



knowledgevalidation

demonstration

language of afamiliar world

practice(know how)

Pragmaticproofs



knowledgevalidation

demonstration


language asa tool

practice(know how)

explicitknowledge

Pragmaticproofs



knowledgevalidation

demonstration


language asa tool

naïveformalism

practice(know how)

explicitknowledge

knowledgeas a theory

Pragmaticproofs

Intellectualproofs



knowledgevalidation

language

conceptualization

control


The origin of knowledge is in problems

in the case of proof, mathematical problem-solving is not enough

the need to stimulate a movementfrom proof as a tool to proof as an object

contradictions as means to give rise to the problem of proof

counter-examplesa revealers

socio-cognitifconflicts as catalysts


The origin of knowledge is in problems

in the case of proof, mathematical problem-solving is not enough

the need to stimulate a movementfrom proof as a tool to proof as an object

contradictions as means to give rise to the problem of proof

counter-examplesa revealers

socio-cognitifconflicts as catalysts

counter-examplesa

revealers


Counter-examples as revealers

Let ABCD be a parallelogram, A’ the symetric image of Awith respect to point B, B’ the symetric image of ...

A’B’C’D’ isa parallelogram





SAS





SAS

He!What about a square ? !



Let ABCD be a square...

A’B’C’D’ is alsoa square



Let ABCD be a square...

A’B’C’D’ is alsoa square

This is great!It holds!



So, let ABCD be a rectangle...

A’B’C’D’ is NOTa rectangle



So, let ABCD be a rectangle...

A’B’C’D’ is NOTa rectangle

Too bad!So what?


How to deal with a counter-example?

The mathematics classroom tends to be a manichean world:

to be or not to be true is the only question

Whereas an example proves mathematically nothing, a counter-example just destroyes every effort...

Could we revisit the old classical position?


ProofConjecture


ProofConjecture

Counter-example


ProofConjecture

Counter-example

The Lakatosiannightmare, again...


ProofConjecture

Rationality Knowledge

Counter-example


Where are we?

A constructivist approach is possible, but a problématiqueof proof must not be separated from knowledge construction

Regulation of cognitive processesrelated to proving in mathematics,treatment of refutations


Where are we?



Specific situations are needed in order to elicitthe meaning of mathematical proofs

The rôle of the teacher, negociationnot on the objects but on the means


Computer-based microworldscould offer a virtual reality tomathematical abstractions

Where are we?





Mathematics needs a specific milieu which feedbackcan reflect the specificities of its objects


Computer-based microworldscould offer a virtual reality tomathematical abstractions

Where are we?





Mathematics needs a specific milieu which feedbackcan reflect the specificities of its objects

Computer-basedmicroworldscould offer avirtual reality tomathematicalabstractions


Knowledge as the equilibrium state of a

Subject/Milieu System

S M

action

feedback

Which characteristics for M in the case of mathematics?


S M

action

feedback

The limits of the “real” world


S M

action

feedback


N°16 - The rainwater from a flat roof 15m by 20m drains into a tank 3m deep on a base 4 m square. What depth of rainfall will fill the tank...(O level, 1978)


A domain of phenomenology

A formal system

S M

action

feedback


The potential of computer-based environments



A formal system

S M

action

feedback


mathematical properties as perceptual phenomena

computational representation of objects and relations

The potential of computer-based environments



A formal system

S M

action

feedback

direct manipulation of graphical objects

formal objects and relationships, a cartesian model

The case of geometry



Cabri-géomètre, a dynamic geometry software




Construct the symmetrical point P1 of P about A,then the symmetrical point P2 of P1 about B, etc. Then, construct the point I, the midpoint of [PP3].

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




I move P and I does not move.When, for example, we put P to the left,then P3 compensate to the right.If it goes up, then the other goes down...





knowledge in action


construction


TheoreticalGeometry

PracticalGeometry


TheoreticalGeometry

PracticalGeometry

geometrical figure

satisfiabilitytheoretical existence


TheoreticalGeometry

PracticalGeometry

geometrical figure

geometrical drawing

satisfiability

constructibility

theoretical existence

effective construction


TheoreticalGeometry

PracticalGeometry

geometrical figure

geometrical drawing

satisfiability

constructibility



rules of the art

mathematical proof


TheoreticalGeometry

PracticalGeometry

geometricalobject

geometrical figure

geometrical drawing

satisfiability

constructibility



rules of the art

mathematical proof


The end?





in a mathematical microworld

for a rational agent, understanding the constraints of the mathematical discourse

V. Luengo 1997

From action...

... to formulation


DrawingDrawing


DrawingDrawingTextText


DrawingDrawingTextText

StructureStructure


This property appears to be true on your drawing,but it is not the case in general;

would you like a counter-example


The statement “[AC] is parallel to [KL]”cannot be obtained using this theorem.


The proof is correct but you have to provethe remaining conjectures


The (very) end!

learning mathematical proof, lessons learned and outlines of a learning environment

Education

rle of mathematical

case of proof

origin of knowledge

language knowledge

intellectual proofs

learning environment

symetric image

practice of mathematics