Bridging knowing and proving in mathematics

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
Internal
universe
Interface
External
universe 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
Internal
universe
Interface
External
universe 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 → z2
+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

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 by
f(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
• =3.14π
• a convergent series reach its limit
• the Fibonacci series
U0=1, U1=(1+ 5)/2, U√ n=Un-1+Un-2
is divergent
Back to students: the
pragmatic origin of meaning

The need to bridge
knowing 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 dependent
The 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 M
feedback
constraints
A first characterization of
a conception

P... a set of problems
R... a set of operators
L... a system of representation
Σ.... 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 2 rπ
and the area is r2π
10. Ludovic : OK
11. Vincent : r is divided by 2 ?
12. Ludovic : yes, the first perimeter
is 2 r and the second is 2 rπ π
over 2 plus 2 r over 2 hence ….π
It will be the same
[…]
17. Vincent : the other is 2 r over 4π
but 4 times
18. Ludovic : so it is always 2 rπ
19. 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 time
23. Vincent : yes, (r/2)π 2
plus (r/2)π 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 0
34. 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 small
42. Ludovic: hum… but it is always 2 rπ
43. Vincent : yes, but when the area tends to 0
it will be almost equal…
44. Ludovic: no, I don’t think so
45. 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
formulaLudovic
algebraic conception
Vincent
symbolic-arithmetic
conception
Validation
V
alidation
Representation
and control

Algebraic frame
area /perimeter
Ludovic
Vincent
symbolic-arithmetic
conception
Validation
V
alidation
Representation
and control

Algebraic frame
area /perimeter
Ludovic
Vincent
symbolic-arithmetic
conception
Validation
V
alidation
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 OOOOO
OOOOOOO OOOOO
OOOOOOOOOOOO
OOOOOOOOOOOO
+
=
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= 4
2, 4= 6 4, 6= 0
2, 6= 8 4, 8= 2
2, 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.
Explanation
Proof
Mathematical proof
The search for certainty
The search for understanding
The need for communication

A specific economy
of practice
The rôle of mathematical proof
in the practice of mathematicians
Internal needs
Social communication
mathematical
rationalism
non mathematical
rationalismVersus
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 action
but the achievement of
Mathematical proof is in language
knowing in action
knowing in discourse
construction

formulationaction validation
representation
means for action control

formulationaction validation
representation
means for action
Proof
and
control
unity

formulation
demonstration
language of a
familiar world
language as
a tool
naïve
formalism
validation
Pragmatic
proofs
Intellectual
proofs
mathematical
proof
action
practice
(know how)
explicit
knowing
knowing
as a theory

validation
Pragmatic
proofs
Intellectual
proofs
mathematical
proof
certainty
understanding
communication
A long way to
mathematical proof

naïve empiricism
crucial experiment
validation
Pragmatic
proofs
Intellectual
proofs
mathematical
proof
certainty
understanding
communication
A long way to
mathematical proof

generic example
thought experiment
statement calculus
naïve empiricism
crucial experiment
validation
Pragmatic
proofs
Intellectual
proofs
mathematical
proof
certainty
understanding
communication
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

Bridging knowing and proving in mathematics

More Related Content

What's hot (19)

Viewers also liked (20)

Similar to Bridging knowing and proving in mathematics (20)

More from Nicolas Balacheff (15)

Recently uploaded (20)

Bridging knowing and proving in mathematics