Along history or across educational traditions, the space given to mathematical proof in compulsory school curricula varies from a quasi-absence to a formal obligation which for some has turned into an obstacle to mathematics learning. The contemporary evolution is to give to proof the space it deserves in the learning of mathematics. This is for example witnessed in different ways by The national curriculum in England (2014), the Common Core State Standards for Mathematics (2010) in the US or the recent Report on the teaching of mathematics (1918) commissioned by the French government; the latter asserts: The notion of proof is at the heart of mathematical activity, whatever the level (this assertion is valid from kindergarten to university). And, beyond mathematical theory, understanding what is a reasoned justification approach based on logic is an important aspect of citizen training. The seeds of this fundamentally mathematical approach are sown in the early grades. These are a few examples of the current worldwide consensus on the centrality proof should have in the compulsory school curricula. However, the institutional statements share difficulty to express this objective. The vocabulary includes words such as argument, justification and proof without clear reasons for such diversity: are these words mere synonymous or are there differences that we should pay attention to? What are the characteristics of the discourse these words may refer to in the mathematics classroom? Eventually, how can be addressed the problem of assessing the truth value of a mathematical statement at the different grades all along compulsory school? I shall explore these questions, starting from questioning the meaning of these words and its consequences. Then, I shall shape the relations between argumentation and proof from an epistemological and didactical perspective. In the end, the participants will be invited to a discussion on the benefit and relevance of shaping the notion of mathematical argumentation as a precursor of mathematical proof.

1. MATHEMATICALARGUMENTATION
as a
PRECURSOR OF MATHEMATICAL PROOF
Nicolas Balacheff – LIG – UGA & CNRS - Grenoble
1Nicolas Balacheff, Cambridge MERG talk, 191118 /30

2. 2
Early learning of mathematical proof
UK
“[To] reason mathematically by following a line of enquiry, conjecturing relationships and
generalisations, and developing an argument, justification or proof using mathematical
language” (Key Stage 3 and 4, Dep. of education, 2014, p.40)
USA
"All students, especially the college intending, should learn that deductive reasoning is the
method by which the validity of a mathematical assertion is finally established" (NCTM, 1989,
p.143).
“One hallmark of mathematical understanding is the ability to justify, in a way appropriate to
the student’s mathematical maturity, why a particular mathematical statement is true or where
a mathematical rule comes from.” (NGA Center & CCSSO, 2010, p. 4).
Chile
“Se apunta principalmente a que los y las estudiantes sepan diferenciar entre una
argumentación intuitiva y una argumentación matemática; a que sean capaces de interpretar
y comprender cadenas de implicaciones lógicas y puedan convencer a otros y otras de que
la propuesta es válida matemáticamente y aceptada por todos y todas. De esta manera,
serán capaces de efectuar demostraciones matemáticas de proposiciones, en un lenguaje
matemático, apoyadas por medio de representaciones pictóricas y con explicaciones en
lenguaje cotidiano.” (Min. De Educación, Mat. Primero Medio, 2016, p.42)
.
2Nicolas Balacheff, Cambridge MERG talk, 191118 /30

3. 3
Early learning of mathematical proof
France
The notion of proof is at the heart of mathematical activity, whatever the level
(this assertion is valid from kindergarten to university). And, beyond
mathematical theory, understanding what is a reasoned justification approach
based on logic is an important aspect of citizen education. The seeds of this
fundamentally mathematical approach are sown in the early grades. (Villani &
Torossian, 2018, pp. 25–26)
Targeted competence (official program)
To prove “mathematically” (Démontrer) : logical reasoning using well defined
rules (properties, theorems, formula).
To support claims based on established results and a good command of
“argumentation”.
However…
It is required not to prove mathematically all taught theorems or properties
In order to positively support students having difficulties, their spontaneous
production should be valued
.
3Nicolas Balacheff, Cambridge MERG talk, 191118 /30

4. Early learning of mathematical proof
Arguing and proving are competences targeted through all grades
From an educational perspective
 to distinguish belief from knowledge
From a didactical perspective
 to learn to respond to the question of truth
 to distinguish argumentation from proof
 to understand the role of proof in mathematics
Hence the challenges for both research and teaching
• Meaning of: argumentation, proof, mathematical proof
• Relation between proving and knowing (proving/explaining c.f. Hanna)
• The tensions between convincing and persuading (c.f. Harel proof schemes)
4Nicolas Balacheff, Cambridge MERG talk, 191118 /30

5. Vocabulary – argumentation … proof
Argumentation is a discourse
Oriented - it aims at the validity of a statement
Intentional - it seeks to modify a judgment
Critical - it analyzes, supports and defends
Argument is an act
that structures socialization
that instruments language
that changes the epistemic value of a statement
that modifies the relationship to knowledge
characterized by the pair
[statement, argument]
5Discourse analysis Nicolas Balacheff, Cambridge MERG talk, 191118 /30

6. Vocabulary – proof, proof and proof
Proof (preuve)
A proof is “a fact or piece of information that shows that something
exists or is true”. (online Cambridge dictionary)
“the process or an instance of establishing the validity of a statement
especially by derivation from other statements in accordance with
principles of reasoning” (online Merriam-Webster dictionary)
Mathematical proof (démonstration)
“A mathematical proof is an inferential argument for a mathematical
statement, showing that the stated assumptions logically guarantee the
conclusion.” (Wikipedia)
Formal proof
“A formal proof is a finite sequence of sentences (called well-formed
formulas in the case of a formal language), each of which is an axiom,
an assumption, or follows from the preceding sentences in the sequence
by a rule of inference.” (Wikipedia)
6Nicolas Balacheff, Cambridge MERG talk, 191118 /30

7. Vocabulary – reasoning … explanation
Reasoning
Organization of statements that is oriented towards a target statement to modify the
epistemic value that this target statement has in a given state of knowledge, or in a
given social environment, and that, consequently, modifies its value of truth when
certain particular organizational conditions are met.
 to modify the epistemic value of a target statement
The question of the epistemic value being answered, arises that of the construction of coherence or
belonging of the new production to the knowledge system.
Explication
The explanation is not intended to modify the epistemic value of a target statement:
it is not at all based on the epistemic values of the mobilized statements, but only
on their content.
 system of relationships within which the
statement to be explained finds its place
7Duval 1992 (my translation) Nicolas Balacheff, Cambridge MERG talk, 191118 /30

8. Vocabulary – argumentation … proof
8
Argumentation  likelihood and convicing (others/oneself)
Explanation  validity
Explanation (validity) + Speech (orientation, intention, criticism)
argumentation collectively accepted  Proof
particular form of proof in mathematics  Mathematical proof
argumentation
proofExplanation
private public
Mathematical proof
Transition space
Nicolas Balacheff, Cambridge MERG talk, 191118 /30

9. Vocabulary – argumentation & statement’s value
9
Argumentation  rhetoric vs heuristic
Explication  epistemic vs ontic
« […] while the content of any particular proof is the fruit of a person’s
epistemic work, it can be separated as an object independent of a
particular mind. Other people can read this proof and be convinced by
it. This leads us to the question whether showing why a theorem is true
is a feature of the proof itself or a feature of communicative acts, texts
or representations. »
(Delarivière, Frans, & Van Kerkhove, 2017. p.3)
Argumentation  likelihood and convicing (others/oneself)
Explanation  validity
Explanation (validity) + Speech (orientation, intention, criticism)
Duval 1992/ Hanna 2017 Nicolas Balacheff, Cambridge MERG talk, 191118 /30

10. Vocabulary – argumentation & statement’s value
Agent = oneself, an other, the others
Argumentation: rhetoric vs heuristic
rhetoric  to convince an agent
heuristic  to make progress in problem-solving
o in relation to a given state of knowledge
o within a given context
 modifies the epistemic value of a statement
epistemic vs ontic value
epistemic  depends on the assessment of an agent
ontic  independent from the assessment of any agent
10Duval 1992/ Hanna 2017 Nicolas Balacheff, Cambridge MERG talk, 191118 /30

11. Vocabulary – explanation … mathematical proof
11
argumentation
proofExplanation
private public
Mathematical proof
Space for debate
Argumentation
Orientation (towards others)
Intention (to obtain acceptance)
Critical (anticipation, rejection, agreement)
Explanation
making explicit, organization
within a knowledge system
Heuristic argumentation
 Target (epistemic value)
problemsolving
Proof
argumentation collectively valid
 when not accepted, it is kept in the
collective space for debate (conjecture)
Mathematical proof
proof which satisfies the
specific mathematical norm
That a proof
explains could be
- shared
- discussed
Nicolas Balacheff, Cambridge MERG talk, 191118 /30

12. Interlude 1 – tools and representation
Let “(I) u0, u1, u2 … un, un+1, etc…” be a series
…/…
If the series is convergent and its various
terms are continuous functions of x in
a neighborhood of some particular
value of this variable, then
sn, rn and s
are also three functions of the variable x,
the first of which is obviously continuous
with respect to x in a neighborhood of the
particular value in question.
Given this, let us consider the increments
in these three functions when we
increase x by an infinitely small
quantity α. For all possible values of n,
the increment in sn is an infinitely small
quantity.
The increment of rn, as well as rn itself,
becomes infinitely small for very large
values of n.
Consequently, the increment in the
function s must be infinitely small.
(trans. Bradley & Sandifer, 2009, pp. 89-90)
12Arsac 2013
Cauchy “Cours d’analyse” (calculus)
1821
 he called a remark this narrative
The variable x is implicit in the
formulation of the terms of the series,
whereas the writing f(x) is used
elsewhere in the same course.
un and x are both variables, with un
depending on x, the independent
variable.
Continuity
- In the neighborhood of a point
- Linked to the continuity of the a curve
- Evocation of a temporal dynamic
Nicolas Balacheff, Cambridge MERG talk, 191118 /30

13. Interlude 1 – tools and representation
Let n’ be a number larger than n. […]
Let’s imagine now that, giving to n a value
large enough, we could -- for all values of x
between the given limits – make the module
of sn’-sn (for all n’), and hence the module of
rn, inferior to a number ε as small as we want.
Since the increase of x could be close
enough to zero for the corresponding
increase of sn to be smaller to a number as
small as we want, it is clear that it is sufficient
to give a value infinitely large to n, and to
give a value infinitely small to the increase of
x, so as to prove (démontrer) – between the
given limits – the continuity of the function.
(free translation)
The variable x remains implicit in the
formulation of the terms of the series,
whereas the writing f(x) is used elsewhere in
the same course.
The order of the text is not congruent to the
logical order which underpins it
∀ ε ∃ N ∀ x ∀ n>N ∀ n’>n |sn-s n’ |< ε
the natural language “hides” that n depends
on ε, not on x
∀ ε ∃ N ∀ x …
13Arsac 2013
Cauchy proof revised 1853
 closer to a modern mathematical
proof but…
Nicolas Balacheff, Cambridge MERG talk, 191118 /30

14. Interlude 1 – controls and representation
Arsac's analyses Cauchy’s text taking into account the situation of
mathematics in the first half of the XIX° century.
• The notion of variable dominates that of function (dependent variable) with a
“dynamic” conception of convergence which influences that of limit and continuity.
• Continuity is defined on an interval and not at a point, it is tightly linked to the
perception of the graphical continuity of the curve.
• The formal notation “|…|” of the absolute value is missing
• Quantifiers are not available (not until the XX° century)
 difficulties to identify dependencies present in the discourse
 difficulty to express the negation of a statements incorporating them (e.g. discontinuity as
negation of continuity)
Cauchy’s proofs were bound by the available representations and, in the
case of the 1821 remark, by the epistemic controls (e.g. lex continuitatis)
not by his underlying rationality and his cognitive maturity.
14
14
Arsac 2013 Nicolas Balacheff, Cambridge MERG talk, 191118 /30

15. Controls and representation
From an educational perspective
aside cognitive, motivational and contextual determiners
 the level and nature of validation depends on the type of access
(representation) to mathematical objects involved
Problem solving in mathematics and proof are bound by the available:
• representation systems
• operators
• means for validation
Criteria and tools to judge, choose, decide, validate
15
Vergnaud model
of “concept”
Nicolas Balacheff, Cambridge MERG talk, 191118 /30

16. Illustration 2 – area and perimeter
Grade 7, groups of two students, the task statement is…
In a classroom, students work on the area
and perimeter of a rectangle. Here is what
some of them are saying:
• David: Two rectangles that have the same
perimeter have the same area.
• Suzanne: two rectangles that have the
same area have the same perimeter.
• Guy: If you increase the perimeter of a
rectangle its area also increases.
• Serge: If you increase the area of a
rectangle its perimeter also increases.
• Brigitte: all rectangles that have an area of
36cm have a perimeter that is no less than
24cm.
• Louise: For every rectangle there is
another one that has the same area but
with a larger perimeter.
What do you think of what each of these
students is saying: do you agree or disagree?
Explain why.
yes, because they have the same perimeter so
they are isometric, otherwise they would not
have the same area, because the area is delimited
by the perimeter so as the perimeter is [the]
same, the area is the same.
Theorem-in-action
if two shapes are different then so are the
measures associated with them.
16Balacheff 1988 chap. IV Nicolas Balacheff, Cambridge MERG talk, 191118 /30

17. Illustration 2 – area and perimeter
yes, because the delimitation of the surface will be increased so the
surface will also be increased there, the numbers that multiply...
that are added [... ] well yes, because when you increase the
perimeter, the length and width increase. So when we multiply
them both, it also increases.
17Balacheff 1988 chap. IV
Grade 7, groups of two students, the task statement is…
In a classroom, students work on the area
and perimeter of a rectangle. Here is what
some of them are saying:
• David: Two rectangles that have the same
perimeter have the same area.
• Suzanne: two rectangles that have the
same area have the same perimeter.
• Guy: If you increase the perimeter of a
rectangle its area also increases.
• Serge: If you increase the area of a
rectangle its perimeter also increases.
• Brigitte: all rectangles that have an area of
36cm have a perimeter that is no less than
24cm.
• Louise: For every rectangle there is
another one that has the same area but
with a larger perimeter.
What do you think of what each of these
students is saying: do you agree or disagree?
Explain why.
Nicolas Balacheff, Cambridge MERG talk, 191118 /30

18. Knowing, conception, argumentation
Conception surface-contour line
• spatio-graphic setting
• perception of the shape
Conception area-perimeter
• Arithmetical setting
• formulas
… they have the same perimeter so they are isometric, otherwise
they would not have the same area, because the area is delimited
by the perimeter so as the perimeter is [the] same, the area is the
same.
-----
… the delimitation of the surface will be increased so the surface
will also be increased there, the numbers that multiply... that are
added [... ] well yes, because when you increase the perimeter, the
length and width increase. So when we multiply them both, it also
increases.
18
One knowing  multiples conceptions
activated according to the problem to be solved.
distinguished by means of representation systems
and control structures (choice, judgments, decisions)
Nicolas Balacheff, Cambridge MERG talk, 191118 /30

19. Interlude 3: problems and conceptions
19
Case of L. and J (free translation from French).
Problem 1: construction of the mirror image (symmetrical)
of a segment
• 28. J : it’s ok there.
• 29. L : a right angle…, then, we take the compass like that…
you see?
• 30. J : yes.
• 31. L : oups, wait… if we fold it like that… yes it fits, it’s ok
• 32. J : hum.
Problem 3: to recognize that two figures are symmetrical
• hyp:
• ABCD parallelogram
• M milieu de [AD]
• N milieu de [BC]
• Proof:
• A is the symmetric point of D with respect to M, because A and D
are at the same distance to M and the 3 points are aligned.
• B is the symmetric point of C with respect to N, because B
and C are at the same distance to N and the 3 points are aligned.
• Conclusion: AB and DC are symmetrical with respect to the line MN
Miyakawa 2005 Nicolas Balacheff, Cambridge MERG talk, 191118 /30

20. The central role of the control structure
The control structure ensures the operators relevance, their
sequencing, the correctness of their use, and
the conception coherence
 conceptions are validation dependent.
Three main types of controls
Referent controls
They guide the search for a solution and its validation
Instrumentation
They guide the choice of the operators and their sequencing
Operational
They ensure the correct use of the operators.
20Nicolas Balacheff, Cambridge MERG talk, 191118 /30

21. Types d’argumentation / preuve
David Tall et al., 2011
- « Children or novices do not initially think deductively. »
- « It is only much later—usually at college level—that axiomatic formal proof arises in terms of
formal definitions and deductions »
to rethink the nature of mathematical proof and to consider the use of different
types of proof related to the cognitive development of the individual.
Piaget
The development of reflective thinking involves two main stages characterized by
the nature of the relationships to the objects it considers and by the nature of
the logical structures that underlie the conduct and notions of the subjects.
What types of proof are available before learning mathematical proof?
What place should they have in teaching?
21
Not only  epistemic perspective
Nicolas Balacheff, Cambridge MERG talk, 191118 /30

22. cK¢ and types of proofs
22Nicolas Balacheff, Singapore talk 180918
For the same students, along the same
problem solving process or within the same
period but with different activities, several
types of proofs could be observed.
 These type of proofs are not
necessarily steps in a student
development
The meaning of these types of proofs
cannot be separated from an
understanding of the conceptions
enacted by students and their
understanding of the situation in which
they are engaged.
 A type of proof reflects both a
conception and a principle of economy of
logic.
Balacheff 1987/1988 /30

23. Generic example
Coordination of several modalities
- discourse,
- visualization
- body expression (gesture)
explanation of the reasons for the
validity of a statement by performing
operations on an object which is a good
representative of a class and not
present for itself.
possible move towards a thought
experiment by a kind of seamless
replacement of the specific numbers by
a evocation of (whatever) “numbers”
23Teo Kwee Huang & Ng Swee Fong, 2018
Singapore, Grade 5, study of the properties
of odd and even numbers.
Nicolas Balacheff, Cambridge MERG talk, 191118 /30

24. generalization and
construction of an
argument
rethoric  heuristic
epistemic  ontique
Generic example
Challenge of the representation
 of the objects
 of the relations
Making reasons explicit
24Balacheff 1987/1988
Balacheff 1976 replication of
Bell 1976 “Add and Take” problem
It will always be 10+10
I chose 2 and it canceled each of them
out, so if I choose an other number
between 1 and 10 it will always
canceled each of them out and it will
always be equal to 20.
ADD AND TAKE (Bell 1976)
Choose any number between 1 and 10. Add it to 10 and
write down the answer. Take the first number away from
10 and write down the answer. Add your two answers.
Nicolas Balacheff, Cambridge MERG talk, 191118 /30

25. 25
enacting operators,
providing evidences
Informal language
to express actions
and relations
language as
a tool
naïve
formalism
Pragmatic
proofs
Intellectual
proofs
mathematical
proof
Implicit models
of action
explicit
operators
Structured
organisation
of operators
formulation validationaction
theory
as a tool
Balacheff 1987/1988 Nicolas Balacheff, Cambridge MERG talk, 191118 /30

26. Proof as a discursive object
From the analysis of a Ball sequence with
third graders.
- Challenge of representation
of objects and their
relations
- Negotiation of the generic
character of a particular
case
- Argumentation as part of
the negotiation of the
validity
argumentation as a tool,
not identified nor discussed
as such
didactical challenge of
taking the argumentation as
an object.
26Stylianides 2007-a Nicolas Balacheff, Cambridge MERG talk, 191118 /30

27. The complexity of the epistemological genesis
of mathematical proof
Control structure : means and procedures
to assess, verify, choose, validate  decide
27
Referent control
Instrumentation control
Operational control
Heuristic argumentation
Explanation
proof
Solving
process
Process of
validation
Private
Public
(in effect or potential)
cognitive & structural
unity
Nicolas Balacheff, Cambridge MERG talk, 191118 /30

28. The complexity of the epistemological genesis
of mathematical proof
Argumentation emerges as a (natural) tool
 rhetorical
 heuristic
Consider proof as an object
 norms of communication
mathematical argumentation
28Nicolas Balacheff, Singapore talk 180918
Which definition?
Which situation?
« mathematical proof »
a tool for validation
Potential obstacle ?
Nicolas Balacheff, Cambridge MERG talk, 191118 /30

29. “Proof is a mathematical argument”
(Stylianides 2007)
Proof is a mathematical argument,
a connected sequence of assertions
for or against a mathematical claim,
with the following characteristics:
• 1. set of accepted statements
It uses statements accepted by the classroom community that are true and available
without further justification;
• 2. modes of argumentation
It employs forms of reasoning that are valid and known to, or within the conceptual
reach of, the classroom community;
• 3. modes of argument representation
It is communicated with forms of expression that are appropriate and known to, or
within the conceptual reach of, the classroom community.
29Stylianides 2007-b Nicolas Balacheff, Cambridge MERG talk, 191118
How would it work for generic examples
and thought experiments?
/30

30. How to make a step further?
30Nicolas Balacheff, Cambridge MERG talk, 191118 /30
How would it work for generic examples
and thought experiments?