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

Learning mathematical proof
lesson learned from research
&
principles of design of a learning environment
Nicolas.Balacheff@imag.fr

© N. Balacheff Oct. 2005

A controversial question...

What is a mathematical proof ?


The rôle of mathematical proof
in the practice of mathematicians


Internal needs
Social communication


mathematical
rationalism


Versus

Internal needs
non mathematical
rationalism


mathematical
rationalism

Rigour


Versus

Versus

Internal needs
non mathematical
rationalism

Efficiency


mathematical
rationalism

Versus

Internal needs
non mathematical
rationalism

The specific economy
of the practice of mathematics
Rigour


Versus

Efficiency

Argumentation
vs
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 understanding
The search for certainty
The need for communication

Yes, and the three
dimensions cannot be
separated....


Learning mathematical proof,
which genesis ?

the origin of knowledge is in action
but the achievement of

Mathematical proof is in language


which genesis ?


knowledge in action

knowledge in discourse

which genesis ?


knowledge in action
con
str
uct
ion


which genesis ?

the origin of
knowledge is in
action


which genesis ?

the origin of
knowledge is in
action

Yes, but...
why is that true ?


formulation


nature of
knowledge

validation

formulation

nature of
knowledge

validation

demonstration
practice
(know how)


Pragmatic
proofs

formulation

nature of
knowledge

validation

demonstration
language of a
familiar world


practice
(know how)

Pragmatic
proofs

formulation

nature of
knowledge

validation

demonstration
language of a
familiar world

practice
(know how)
explicit
knowledge

language as
a tool


Pragmatic
proofs

formulation

nature of
knowledge

validation

demonstration
language of a
familiar world

practice
(know how)

Pragmatic
proofs

explicit
knowledge
language as
a tool

naïve
formalism

Intellectual
proofs
knowledge
as a theory

formulation

nature of
knowledge

validation

conceptualization

language


control

The origin of knowledge is in problems
in the case of proof, mathematical problem-solving is not enough

the need to stimulate a movement
from proof as a tool to proof as an object
contradictions as means
to give rise to the problem of proof

counter-examples
a revealers


socio-cognitif
conflicts 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 movement
from proof as a tool to proof as an object

counter-examples
a contradictions as means
to give rise to the problem of proof
revealers

counter-examples
a revealers


socio-cognitif
conflicts as catalysts

Counter-examples as revealers
Let ABCD be a parallelogram, A’ the symetric image of A
with respect to point B, B’ the symetric image of ...

A’B’C’D’ is
a parallelogram



A’B’C’D’ is
a parallelogram

SAS



A’B’C’D’ is
a parallelogram

He!
What about a square ? !
SAS


Let ABCD be a square...

A’B’C’D’ is also
a square


Let ABCD be a square...

A’B’C’D’ is also
a square

This is great!
It holds!


So, let ABCD be a rectangle...

A’B’C’D’ is NOT
a rectangle


So, let ABCD be a rectangle...

A’B’C’D’ is NOT
a 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?

Proof
Conjecture


Proof
Conjecture

Counter-example


Proof
Conjecture

The Lakatosian
nightmare, Counter-example
again...


Rationality

Knowledge

Proof
Conjecture

Counter-example


Where are we?
A constructivist approach is possible, but a problématique
of proof must not be separated from knowledge construction
Regulation of cognitive processes
related to proving in mathematics,
treatment of refutations


Where are we?

Specific situations are needed in order to elicit
the meaning of mathematical proofs
The rôle of the teacher, negociation
not on the objects but on the means


Where are we?

not on the objects but on the means

Mathematics needs a specific milieu which feedback
can reflect the specificities of its objects
Computer-based microworlds
could offer a virtual reality to
mathematical abstractions

Where are we?
Computer-based

microworlds
could offer a
virtual reality to
mathematical
abstractions on the means
not on the objects but

Mathematics needs a specific milieu which feedback
can reflect the specificities of its objects
Computer-based microworlds
could offer a virtual reality to
mathematical abstractions

Knowledge as the
equilibrium state of a
Subject/Milieu System

action

S

M
feedback

Which characteristics for M in the case of mathematics?

action

S

M
feedback

The limits of the “real” world


action

S

M
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)


action

S

M
feedback


A formal system

The potential of computer-based
environments

A domain of phenomenology

action

S

M
feedback


A formal system

The potential of computer-based
environments

computational representation of objects and relations
mathematical properties as perceptual phenomena


action

S

M
feedback

The case of geometry

A formal system
formal objects and relationships, a cartesian model
direct manipulation of graphical objects



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...

which genesis ?

knowledge in action
con
str
uct
ion


Theoretical
Geometry

Practical
Geometry


Theoretical
Geometry

satisfiability
theoretical existence
geometrical figure

Practical
Geometry


Theoretical
Geometry

satisfiability
geometrical figure

geometrical drawing

Practical
Geometry


effective construction
constructibility

mathematical proof

Theoretical
Geometry

satisfiability
geometrical figure

geometrical drawing

Practical
Geometry

constructibility
rules of the art


mathematical proof

Theoretical
Geometry

satisfiability
geometrical figure

geometrical
object

geometrical drawing

Practical
Geometry

constructibility
rules of the art


The end?


which genesis ?

From action...
in a mathematical microworld
... to formulation
for a rational agent, understanding the constraints
of the mathematical discourse

V. Luengo 1997

Drawing


Drawing


Text

Drawing

Structure


Text

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 prove
the remaining conjectures


The (very) end!


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

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