The Complexity of Mathematical Proof

THE COMPLEXITY OF THE
EPISTEMOLOGICAL GENESIS OF
MATHEMATICAL PROOF
Nicolas Balacheff – LIG – UGA & CNRS - Grenoble
1 / 32Nicolas Balacheff, Tokyo – Singapore talks

Early learning of mathematical proof
the French context (grades 1 to 9)
Targeted competence (learning – grades 7 to 9)
• 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
Following official recommendations, pedagogical sequences should include (1) time
for sharing and argumentation targeting the production of a proof, and (2) time for
shaping and writing the mathematical proof (démonstration)
2 / 32

Early learning of mathematical proof
the French context (grades 1 to 9)
Two difficulties are identified:
• Passing from problem solving to proving
• Shaping a proof so that it can be communicated and accepted as a
convincing argument
 Proving as a collective activity supported by the teacher
Competences appearing at earlier grades
Grade 4 to 6
• Introduce students to the ideas of proof and argumentation. Passing from
the manipulation and observation of geometrical objects, to reasoning based
on their properties and the use of instruments.
Grade 1 and 3
• Create situations to allow students to experiment, to explain, to exploit and
to communicate results. The discourse should be based on arguments and
observations, not beliefs.
3Nicolas Balacheff, Tokyo – Singapore talks / 32

Summary, the French context (grades 1 to 9)
…. Arguing and proving are competences targeted through all the
grades
From an educational perspective
 the aim is to distinguish beliefs from knowledge
From a didactical perspective
 the aim is to respond to the question of truth
to distinguish argumentation, proof and mathematical proof
 to understand mathematical proof (démonstration) as a type of proof
Hence challenges for both research and teaching
• The need for clarification of the terms used (argumentation, proof,
mathematical proof) and the relationships of the related concepts.
• The tension between proving and understanding (proving Vs explaining -
c.f. Hanna)
• The tensions between convincing and persuading (c.f. Harel proof schemes)

Explaining, proving and mathematics
Reasoning and explanation are key words often used in the research
literature about the learning of proof, I use them with the following
meanings:
Reasoning: organization of propositions that is oriented towards a targeted
statement to modify the epistemic value that this target statement has in a
given state of knowledge, or in a given social milieu, and which,
consequently, modifies its value truth when certain particular conditions of
organization are fulfilled
Explanation: system of relations in which the data to be explained finds its
place; the question of grounding the epistemic value arises from that of the
construction of the coherence or belonging of the new production to the
system of knowledge.

Explanation  ensures validity within one’s knowledge context
Argumentation  articulates the reasons backing the validity of a statement for oneself or somebody
Argumentation could be accepted or rejected based on relevance (semantic coherence) and strength (“positive”
epistemic value), as well as on mathematic (including logical considerations).
Raymond Duval distinguishes between “rhetorical argumentation” (including the aim to convince) and “heuristic
argumentation”(focused on statements and objects independently of the objective to convince).
Proof  argumentation which successfully goes through the review process of a community.
Mathematical proof  proof which technically meets standards of the mathematical community.
Exploring the explaining character of a proof, Gilla Hanna draws the attention to and discusses the philosophical
distinction between “epistemic proof ”, involving an agent (mathematician, teacher, learner), and “ontic proof ”,
considered independently of its explanatory character from an agent perspective.
Explaining, proving and mathematics
6
argumentation Mathematical proof
proofExplanation
private
public
Nicolas Balacheff, Tokyo – Singapore talks / 32

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.
7Nicolas Balacheff, Tokyo – Singapore talksArsac 2013
Cauchy “Cours d’analyse” (calculus) 1821
Theorem of uniform convergence
/32
When the various terms of series (1) are
functions of the same variable x,
continuous with respect to this variable in
the neighborhood of a particular value
for which the series converges, the sum s of
the series is also a continuous function of x in
the neighborhood of this particular value.
Cauchy called
remark this
narrative 
(trans. Bradley & Sandifer 2010 pp.89-90)
The notion of variable dominates that of function
(dependent variable) with a “dynamic” conception of
convergence which influences that of limit and
continuity. The reference to the curve underpins the
reasoning.

Let n’ be a number superior to 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.
(my free translation)
The order of the conditions in the text is not congruent to
their logical order as required by the Cauchy criterion
∀ ε ∃ N ∀ x ∀ n>N ∀ n’>n |sn-s n’ |< ε
8
The variable x remains implicit in the formulation
of the terms of the series, whereas Cauchy used the
writing f(x) elsewhere in his mathematical texts.
Nicolas Balacheff, Tokyo – Singapore talksArsac 2013 /32
Cauchy “Cours d’analyse” (calculus) 1853
Theorem of uniform convergence
This new proof is proposed after three decades of
work of the mathetical community which is analysed
by Gilbert Arsac in his book « Cauchy, Abel, Seidel,
Stokes et La Convergence uniforme » Paris:
Hermann (2013)
Now Cauchy used the French word « démontrer » (i.e. to
prove mathematically) as he did elsewhere in his
mathematical work.

Arsac's analyses Cauchy’s text takes into account the development and
the context of the mathematics of 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.
• The algebraic notation of the absolute value is missing
• 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.
• 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)
9
9
Nicolas Balacheff, Tokyo – Singapore talksArsac 2013 /32

Making sense of the mathematical understanding
 different levels of validation are present in the 1821 course,
depending on the available access to mathematical objects (i.e.
semiotic means and registers)
 problem solving and proof are bound by the nature and technicalities
of the available:
• representation systems
• Operators
• means for validation
Criteria and tools to judge, choose, decide, validate
10Nicolas Balacheff, Tokyo – Singapore talks
Vergnaud model of “concept”
/ 32

Making sense of the mathematical understanding
Caveat : the English word “knowledge” could refer to (1) the understanding someone has of something (e.g. a learner
knowledge of calculus), or (2) to the theoretical or practical understanding of a subject shared and recognized by a
community (mathematical knowledge). In order to avoid misunderstanding, the model I will present through the next
slides uses …
… “knowing” as a noun when referring to the knowledge of an individual or an
informal group (meaning 1), and
… “knowledge” is kept for “knowings” which have a social and institutional status
(meaning 2).
While not being a too rude modification of English usage, this allows to keep the French nuance between
“connaissance” and “savoir”, which can be found in most Romance language.
Back to students:
“many times a child’s response is labeled erroneous too quickly and […] if one were to imagine
how the child was making sense of the situation, then one would find the errors to be reasoned and
supportable” (Jere Confrey 1990).
Human beings have conceptions which are adapted and efficient in different
situations they are familiar with. They are not naïve or misconceived, nor mere
beliefs. They are situated and operational in the adequate circumstances.

subject<>milieu -- a fundamental system
Modeling learners knowing raises
a difficult challenge since it is desirable
(1) to account for its situatedness
while (2) being fair with the hypothesis of
rationality of the learners, and
(3) preserving its epistemological character.
A principle of the model presented here
is to make the following “simplifications”:
• The human being is “reduced” to the epistemic subject
• The environment is “reduced” to these features
which are relevant from an epistemic perspective: the milieu
This does not mean that other dimensions (esp. social, emotional) are ignored. They must be
reintroduced when reconsidering the results based on the model when back to the classroom situation.
Reminder: «Epistemic subject » (Piaget) refers to the structures of actions and ideas common to subjects at the same
level of development, as opposed to the «psychological subject» who uses these structures.
action
feedback
constraints
S M
/ 32

The French word “milieu” is difficult to translate in English. The spontaneous choice, for years, has been to burrow the word
“milieu” which pertains to the English lexicon. It may bot have been the best choice…
“Milieu” has an English meaning rather large, dictionaries refer to “surrounding”, with a social connotation in British English. One of
his synonymous is the word “environment”; as a matter of fact we have long stories of referees and editors suggesting to use
“environment” instead of “milieu”. It happens that both words exist in French, with for “environnement” the same meaning as for
“environment” in English.
So why “milieu”?
Learning occurs in an environment, partly organized by the teacher, which is complex. Not all features of this
environment are relevant from the perspective of the learning of a given content.
Let’s take a quick example. Using a dynamic geometry environment (DGE) asking students to make a
construction, the brand of the computer and its OS are most likely not relevant elements to understand what
happens, but knowing the available functionalities and the properties (action-feedback) of the direct
manipulation of objects on the screen are essential. The latter will possibly resist the student attempt to
achieve the task; it is this resistance which specifically contributes to the leaning.
Within the learning environment, given a task or a problem, the milieu is this part delineated by the features
which are key in orientating the student strategies and learning.
Here is the meaning of “The milieu is the system opposing the taught system” (Brousseau 1997 p.57).
In other words, “milieu” can be understood as the subset of the environment which has a key epistemic
relevance.
So, let’s try this:
Milieu: epistemological niche within the learner environment,
with respect to a given content to be learn, or task to be achieved.
Nicolas Balacheff, Tokyo –Singapore talks 13 / 32

• S and M exchange information through
actions and feedback
• Constraints may disrupt the system,
giving rise to a problem
The milieu is the subject’s
antagonistic system (Brousseau
sense)
• The stake is to preserve the system
equilibrium (solving the problem)
action
feedback
constraints
S M
A conception is the state of
dynamical equilibrium of an
action/feedback loop between
a learner and a milieu under
proscriptive constraints of
viability
/ 32

• S and M exchange information through
actions and feedback
• Constraints may disrupt the system,
giving rise to a problem
The milieu is the subject’s
antagonistic system (Brousseau
sense)
• The stake is to preserve the system
equilibrium (solving the problem)
P - a set of problems (sphere of
practice)
R - a set of operators.
L - a representation system (linguistic,
graphical, etc.)
Σ - a control structure
/ 32
This can be formalized by a quadruplet
action
feedback
constraints
S M
This is not developped in this talk, for more see
Balacheff (2017)

Interlude 2: representations and controls
Let’s consider the following
problem proposed to 11th/12th
grade students:
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 ?
This example will serve to
illustrate the importance of the
interaction between
representations and controls.
16Nicolas Balacheff, Tokyo –Singapore talks / 32Pedemonte 2002

17Nicolas Balacheff, Tokyo –Singapore talks
9. Vincent : the perimeter is 2πr
and the area is πr2
10. Ludovic : OK
11. Vincent : r is divided by 2 ?
12. Ludovic : yes, the first
perimeter is 2πr and the
second is 2πr over 2 plus 2πr
over 2 hence …. It will be the
same
[…]
17. Vincent : the other is 2πr over
4 but 4 times
18. Ludovic : so it is always 2πr
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, π(r/2)2 plus
π(r/2)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…
/ 32Pedemonte 2002
students were recorded, the two following
slides are excerpts from the dialogues.

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 2πr
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
/ 32Pedemonte 2002

Activity under the
control of the algebraic
syntax and theorems
area /perimeter
formulaalgebraic conception
symbolic-arithmetic
conception
Activity under the
control of the graphical
representations
(drawings)
Pedemonte 2002 /32

The central role of the control structure
Interlude 2
Shared by students
•P – problems about shapes, perimeter and area common to grade 9
•R – geometrical manipulation of drawings, transformation of arithmetical formulas
•L – syntax of arithmetical formulas, geometrical drawings
Semiotic mediation
Their key difference
•Σ - At a deep level of the control structure: the referent control (Gaudin) ensures the
coherence of the solution
The control structure
• ensures the conception coherence, the operators relevance and their
sequencing, and the correctness of their use
 a conception is validation dependent.
• is bound by the system of representation (linguistic, graphical, iconic, etc.).
Nicolas Balacheff, Tokyo – Singapore talks 20/ 32

The central role of the control structure
 a conception is validation dependent.
The control structure ensures the operators relevance and
their sequencing, the correctness of their use, and the
conception coherence
Three main types of controls
Referent controls
They guide the search for a solution
Instrumental controls
They guide the choice of the operators and their sequencing
Operational controls
They ensure the correct use of the operators.
Nicolas Balacheff, Tokyo – Singapore talks 21 / 32

The TSD (Theory of Didactical Situations) framework
“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
/ 32
In the context of the TSD, the subject<>milieu system is a subsystem of the global didactical system which can be
analyzed along different layers from an initial knowledge analysis to the design of a didactical situation. This slide
outlines the relations between these layers, for more refer to (Brousseau 1997)

• 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. It is composed of…
• a description or model expressed in a certain “language”
• a judgment on the adequacy of this description
• [2] formulations of the descriptions and models
• [1] models for action governing decisions
• They correspond to three types of situations
• [1] Action → actions and decisions that act directly
• [2] Formulation → exchange of information coded into a language
• [3] Validation → exchange of judgments
(Brousseau 1997 p.61)

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
theory
as a tool
formulation validationaction
representations controlsoperators

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)
cK¢ and types of proofs

Interlude 3: problems and conceptions
Let’s consider the following problem
proposed to 9th grade students:
Problem 1: construction of the mirror
image (symmetrical) of a segment
To construct the mirror image of [AB] with
respect to (d), students can use a ruler, a
square, a protractor, or just to draw it freely if
then do not know how to use these
instruments.
Problem 3: to recognize that two figures
are symmetrical
The quadrilateral ABCD is a parallelogram.
M, N are the middle points of the opposite
sides AD and BC. Are the segments [AB] and
[DC] symmetrical with respect to the line (MN)?
Your answer could be either “yes”, “no” or “not
always”; it should be proved.
/ 32Miyakawa 2005

Case of L. and J (pp. 225 sqq - free translation from French).
Problem 1
• 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
• 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
/ 32Miyakawa 2005
Conflict of conceptions which are
available; the taught procedure
does not disqualify a conception
based on perception, pragmatic
control and an informal sense of
“being the same” (that
mathematics may structure…)

Given a concept (c) a student may evidence different
conceptions depending on the problem s/he is considering;
this set of conceptions is what we refer to as is his or her
knowing (k) of c.
A conception (¢) is the instantiation of a
knowing by a situation
It is determined
- by the set of problems it efficiently contributes to solve
with respect to a specific control structure;
- and the operators and representation systems mobilized
a knowing may not be coherent, a conception is
Nicolas Balacheff, Tokyo – Singapore talks 28/ 32Miyakawa 2005

TSD schema of validation
Validation requires a milieu
including a linguistic dimension
which provides the instruments for
supporting/disputing validity
29
milieu for validation
the learning of mathematical proof can be conceived as
the learning of a discursive practice
 what does mean reducing mathematic to a language
beyond representations
discourse
Nicolas Balacheff, Tokyo – Singapore talks / 32Brousseau/MargolinasBrousseau1997p.70

The complexity of the epistemological genesis of
mathematical proof
Control structure : means and
procedures to assess, to verify,
to choose, to validate  to decide
30
Referent control
Instrumentation control
Operational control
Heuristic argumentation
Explanation
proof
Solving
process
Process of
validation
Private
Public
(in effect or potential)
epistemic & structural
unity issues
Nicolas Balacheff, Tokyo – Singapore talks / 32
Boero, Pedemonte, …
The challenging transitions, with or
without ruptures, from problem solving
to proving, from rhetoric to content
centered (ontic), from private to public

The complexity of the epistemological genesis
of mathematical proof
Argumentation emerges as a (natural) tool
 rhetorical
 heuristic
open the possibility
to consider proof as an object and
to search for  norms of communication
mathematical argumentation
which definition?
which situation?
« mathematical proof »
as a tool for validation
potential obstacle raised by the strength of
rhetoric encouraged by the social context
/ 32

References
• Arsac, G. (2013). Cauchy, Abel, Seidel, Stokes et la convergence uniforme. Paris : Hermann.
• Balacheff, N. (1988). A study of students’ proving processes at the junior high school level. In I. Wirszup & R. Streit (Eds.), Proceedings of
the Second UCSMP International Conference on Mathematics Education (pp. 284–297). Chicago: National Council of Teachers of
Mathematics, Reston, VA,.
• Balacheff, Nicolas. (2013). cK¢, a model to reason on learners’ conceptions. In M. V. Martinez & A. Castro Superfine (Eds.) (pp. 2–15).
Presented at the PME-NA 2013 - Psychology of Mathematics Education, North American Chapter, Chicago. Retrieved from
https://hal.archives-ouvertes.fr/hal-00853856
• Balacheff, Nicolas. (2017). cK¢, a model to understand learners’ understanding – Discussing the case of functions. El Calculo y Su
Ensenanza, IX (Jul-Dic), 1–23.
• Boero, P. (2017). Cognitive unity of theorems, theories and related rationalities, 8. https://keynote.conference-
services.net/resources/444/5118/pdf/CERME10_0612.pdf
• Bradley, R. E., & Sandifer, C. E. (2010). Cauchy’s Cours d’analyse: An Annotated Translation. Springer Science & Business Media.
• Brousseau, G. (1997). Theory of Didactical Situations in Mathematics: Didactique des Mathématiques, 1970–1990. Springer Netherlands.
Retrieved from //www.springer.com/fr/book/9780792345268
• Confrey, J. (1990). A Review of the Research on Student Conceptions in Mathematics, Science, and Programming. Review of Research in
Education, 16, 3–56. https://doi.org/10.2307/1167350
• EDUSCOL. (2008). Raisonnement et démonstration. Ministère de l’éducation nationale, de l’enseignement supérieur et de la recherche.
• EDUSCOL. (2016). Raisonner [institutionnel]. Retrieved 30 September 2018, from
http://cache.media.eduscol.education.fr/file/Competences_travaillees/83/6/RA16_C4_MATH_raisonner_547836.pdf
• Hanna, G. (2017). Connecting two different views of mathematical explanation [Conference]. Retrieved 25 September 2018, from
https://enablingmaths.wordpress.com/abstracts/
• Harel, G., & Sowder, L. (1998). Students’ Proof Schemes: Results from Exploratory Studies. CBMS Issues in Mathematics Education, 7,
234–283.
• L’épistémologie de Piaget. (n.d.). Retrieved 30 September 2018, from
http://www.fondationjeanpiaget.ch/fjp/site/ModuleFJP001/index_gen_page.php?IDPAGE=320&IDMODULE=72
• Miyakawa, T. (2005). Une étude du rapport entre connaissance et preuve : le cas de la notion de symétrie orthogonale. Université Joseph
Fourier, Grenoble. Retrieved from http://www.juen.ac.jp/math/miyakawa/article/these_main.pdf
• Miyakawa, T. (2016). Comparative analysis on the nature of proof to be taught in geometry: the cases of French and Japanese lower
secondary schools. Educational Studies in Mathematics, 92(2), 20 pages. https://doi.org/DOI 10.1007/s10649-016-9711-x
• Pedemonte, Betina. (2002). Etude didactique et cognitive des rapports de l’argumentation et de la démonstration dans l’apprentissage des
mathématiques. Université Joseph Fourrier, Grenoble. Retrieved from https://tel.archives-ouvertes.fr/tel-00004579/file/tel-00004579.pdf
• Pedemonte, Bettina. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics,
66(1), 23–41. https://doi.org/10.1007/s10649-006-9057-x
• Yackel, E., & Cobb, P. (1996). Sociomathematical Norms, Argumentation, and Autonomy in Mathematics. Journal for Research in
Mathematics Education, 27(4), 458–477. https://doi.org/10.2307/749877

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The Complexity of Mathematical Proof