Students’ mathematical knowledge is first rooted in pragmatic evidences and in the effort to make sense of the content and procedures taught. They develop a true knowledge which works as a tool in problem situations, and is accessible to falsification and argumentation. They can validate what they claim to be true, but based on means which may not conform to current mathematical standards. The theory of didactical situations (TSD) is based on the recognition of the existence of this true knowledge and the analysis of the specific complexity of the teaching situations from an epistemological perspective. It is in this framework that I propose to address the problems raised by the teaching and learning of mathematical proof. The main issue which I will discuss is that the evolution of the students understanding of what count as proof in mathematics implies – and is constitutive of – an evolution of their knowing of mathematical concepts. This discussion will support the claim that the “situation of validation” conceptualized by the TSD must be the starting point of any didactical engineering.

1. 1
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 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 distributive law, hence z is an even number.
2, 2= 4 4, 4= 8 6, 6= 2 8, 8=6
2, 4= 6 4, 6= 0 6, 8=4
2, 6= 8 4, 8= 2
2, 8= 0
If two numbers are
even, so is their sum
+
=
from Healey & Hoyles
Nicolas Balacheff, CINESTAV, September 2015

2. 2
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).
Nicolas Balacheff, CINESTAV, September 2015

3. 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.”
3
Nicolas Balacheff, CINESTAV, September 2015

4. 4
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
Nicolas Balacheff, CINESTAV, September 2015

5. Is mathematicalproof…
5
(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
Nicolas Balacheff, CINESTAV, September 2015

6. COMPLEXITY OF THE
EPISTEMOLOGICAL & DIDACTICAL
GENESIS OF MATHEMATICAL PROOF
notes from a research journey on learning proof for the
CINVESTAV doctoral colloquium 2015
Nicolas.Balacheff @ imag.fr

7. 7
A working position
Proof 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 validating
Theorem = system of (statements, proof, theory)
Nicolas Balacheff, CINESTAV, September 2015

8. 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
Nicolas Balacheff, CINESTAV, September 2015
8

9.  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
Case 1: the sum of the angles of a
triangle
A mathematical argument should be
constructed using the repertoire of the
known and accepted statements ,
following rules accepted by the class
Nicolas Balacheff, CINESTAV, September 2015
9

10.  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…
Case 1: the sum of the angles of a
triangle
Nicolas Balacheff, CINESTAV, September 2015
10

11. Case 1: the sum of the angles of a
triangle
A 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 180
S159 - We have found 190 because we have measured
T161 – and each time you found 190… for the first
triangle and the one I gave you
S162 – yes, we have measured, so…
Nicolas Balacheff, CINESTAV, September 2015
11

12. Case 1: the sum of the angles of a
triangle
A look at the discussion
S476-78 – one can draw whatever… it will be equal to
180… more or less
S480 – we can even draw very small triangle, I think
S487 – it’s not the triangle which counts, it is the
angles
S500 – 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
Nicolas Balacheff, CINESTAV, September 2015
12

13. Case 1 debriefing
13
Theoretical geometry
Practical geometry
The sum of the
angles of a triangle
Pragmatic proof
No way to go beyond approximation
but conformance to the rule book
Intellectual proof
Need to organize objects and relations
within a theoretical framework
argumentation
mathematical
proof
Nicolas Balacheff, CINESTAV, September 2015

14. Case 1 debriefing
Proof and argumentation
Mathematical 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 proof
but…
depending on the reasoning tools and representations available,
argumentation remains a possible mathematical validation
framework (quasi-empiricist approaches)
14Nicolas Balacheff, CINESTAV, September 2015

15. Case 1 debriefing
The 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 situation
restriction
adaptation
Knowledge analysis
Nicolas Balacheff, CINESTAV, September 2015
15

16. 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?”
Nicolas Balacheff, CINESTAV, September 2015
16

17. Case 2: the fix point
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
Nicolas Balacheff, CINESTAV, September 2015
17

18. invariance of I
geometrical
phenomenon
facts
Within the
machine
Interface
immobility of I
e-geometry
knowledge
B
A
C
P
P1
P2
P3I
Case 2: the fix point
conocimiento
Nicolas Balacheff, CINESTAV, September 2015
18

19. Case 2 debriefing
Let’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
19
action
S M
feedback
constraints
B
A
C
P
P1
P2
P3I
and characteristics of a situation
Nicolas Balacheff, CINESTAV, September 2015

20. 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 M
feedback
constraints
Characterizing conceptions
Conceptions
• may be contradictory even though
potentially attached to the same
concept
• are accessible to falsification
• are validation dependent
Nicolas Balacheff, CINESTAV, September 2015
20

21. action
S M
feedback
constraints
Characterizing conceptions
A 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
Nicolas Balacheff, CINESTAV, September 2015
21

22. 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
22

23. Back to case 1: the sum of the angles
of a triangle
23
The bigger the
triangle, the bigger
its perimeter
Geometry of shapes
Drawing and measurement
Repertoire of geometrical facts
Argumentation
Spatio-graphic objects
Situation 1 & 2
- each has its own triangle
- one triangle for all
Devolution of a problem
Situation 3
social interaction,
confrontation of
conceptions
Birth of a conjecture
Problem of proof
The sum of the
angles of a triangle
is invariant
Geometrical objects
Geometry
Statements, diagrams
Repertoire of properties
Mathematical argumentation
C1
C2
Nicolas Balacheff, CINESTAV, September 2015

24. The design of didactical situations
24
learning dynamic
design teaching
dynamic dynamic
engage in the
situation / mobilize
conceptions
share communication
means, language,
representation
explicit
validation
pattern
action
formulation
validation
devolution
institutionalization
Knowledge should be the only legitimate reference for decision making
Nicolas Balacheff, CINESTAV, September 2015

25. The design of didactical situations
Given 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
students
then specify situations to introduce it.
25
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?
Nicolas Balacheff, CINESTAV, September 2015

26. Back to case 1: the sum of the angles
of a triangle
S159 - We have found 190 because we have measured
T161 – and each time you found 190… for the first triangle and
the one I gave you
S162 – 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)
26Nicolas Balacheff, CINESTAV, September 2015

27. Conclusion
on the side of proof
The role of mathematical proof
in the practice of mathematicians
Internal needs
Communication
mathematical
rationalism
non
mathematical
rationalism
Versus
Rigour Efficiency
organizing and
structuring the
mathematical content
Specific economy of practice
Nicolas Balacheff, CINESTAV, September 2015
27

28. Conclusion
on the side of the learner
28
formulationaction validation
representation
operators control
Proof
and
control
unity
Nicolas Balacheff, CINESTAV, Septembre 2015

29. Conclusion
on 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
29Nicolas 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'sstake
statements, theories
allowed by B
action
milieu messages
mathematical proof
argumentation

30. 30
 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
Conclusion
Some questions and issues
Nicolas Balacheff, CINESTAV, September 2015