Content uploaded by Philippe R. Richard
Author content
All content in this area was uploaded by Philippe R. Richard on Dec 27, 2018
Content may be subject to copyright.
1
Issues and challenges about instrumental proof
Philippe R. Richard
Université de Montréal
Phone: (+1) 514-343-2064 • Fax: (+1) 514-343-7286
philippe.r.richard@umontreal.ca
Fabienne Venant
Université du Québec à Montréal
Canada
Michel Gagnon
École Polytechnique de Montréal
Canada
Abstract
Our article aims to define the notion of instrumental proof based on didactic,
epistemological and cognitive considerations. We raise issues and challenges related to
the use of this type of proof in mathematical work and mathematical thinking. The theory
of mathematical working spaces serves as a construct on which we address questions
about proof, reasoning and epistemic necessity, taking advantage of the possibilities
offered by the development geneses and fibrations in an instrumented perspective. The
coordination of the semiotic, discursive and instrumental geneses of the working space
founds discursive-graphic proofs, mechanical proofs and algorithmic proofs that are
activated at school in the subject-milieu interactions. We end with a discussion on some
consequences of the computer-assisted modelling of the learning conditions of
mathematics, and we conclude on a necessary reconciliation of heuristics and validation.
Notre article vise à définir la notion de preuve instrumentale en partant de considérations
didactiques, épistémologiques et cognitives. Nous soulevons des enjeux et des défis liés à
ce type de preuve au regard du travail mathématique et de la pensée mathématique. La
théorie des espaces de travail mathématique sert de charpente sur laquelle nous abordons
des questions sur la preuve, le raisonnement et la nécessité épistémique, profitant des
possibilités qu’offrent le développement des genèses et des fibrations dans une
perspective instrumentée. La coordination des genèses sémiotique, discursive et
instrumentale de l’espace de travail fondent des preuves discursivo-graphiques, des
preuves mécaniques et des preuves algorithmiques qui s’activent à l’école dans
l’interaction sujet-milieu. Nous terminons par une discussion de quelques conséquences
de la modélisation des conditions d’apprentissage des mathématiques assisté par des
dispositifs informatiques, et nous concluons sur un rapprochement nécessaire entre
heuristique et validation.
2
Keywords
Didactics of mathematics• Mathematical working space • Discursive-graphic proof •
Mechanical proof • Algorithmic proof • Instrumented reasoning • Inference and
connection of epistemic necessity • Subject-milieu interactions • Mathematical work and
mathematical thinking • Genetic developments and fibrations
1 Introduction
Optics appears to be a mathematical art based on instrumental proof.
Sven Du pré (2006)
The notion of instrumental proof is relatively new; but if the term is as yet of little use in
didactic literature, its association with technologies, old and new, seems self-evident.
On the epistemological side, the discovery of Archimedes’ palimpsest recently gave us a
better understanding of how the weighing method was considered a kind of mechanical
proof, and this suggests that the association between proof and artifacts/tools is rather
old. Similarly, in computer proofs such as the proof of the four-colour theorem — first
shown in 1976 by Kenneth Appel and Wolfgang Haken, then formally addressed in 2005
using Coq software by Georges Gonthier and Benjamin Werner — the decision or the
verification of all cases rely on programs, and thus reflect an unavoidable reality of
contemporary mathematical work. Whether the tools are physical or logical, their use in a
validation certainly renews the common ideas we have about the concepts of proof,
modelling and representation of knowledge.
On the didactic side, it seems there is a constant struggle with paradoxes. Nowadays,
when a student is asked to prove propositions, she has an automated reasoning
informatics device at her disposal; but at the same time she is required to work with
meaningful knowledge of her own, and to transform this knowledge by working more
and more at the interface of computer tools that deal with some part of the representation
and of the treatment, sometimes experimenting on mathematical objects (e.g. dynamic
figures) as a physicist does with objects of his own domain. And all of this happens while
the teacher cannot refer to mathematics that could be described as «technological», since
he was educated in a deductive science traditionally developed in writing.
It is then by developing ideas already expounded in our previous work, including the
recent paper The Concept of Proof in the Light of Mathematical Work (Richard, Oller, &
Meavilla, 2016), and adopting the conclusions of our current research projects on the
design of the tutorial system QED-Tutrix in high school geometry (Richard, Gagnon, &
Fortuny, 2018; Font, Richard, & Gagnon, 2018), and the use of automated reasoning tools
in teacher training (Kovács, Richard, Recio, & Vélez, 2017), that we address the question
of instrumental proofs, keeping in mind that the interaction between subject and milieu
1
is a unit of epistemic necessity. With this consideration, the subject can be either a reader,
considering traditional proofs, or the user of software or of a mathematical machine. The
notions of reasoning in action or through algorithms, and of reasoning that unfolds by
other means than discourse, will be addressed, as will the Theory of Mathematical
1
From Brousseau’s theory of didactical situations in mathematics (1998).
3
Working Spaces, where the question of the coordination of discursive, semiotic and
instrumental geneses arises between epistemological and cognitive planes (Kuzniak,
Richard, & Michael-Chrysanthou, 2018).
2 Towards an instrumental proof
This assumed inference carries with it its own demonstration.
Alexis Claude Clairaut (1741)
2.1 Reasoning, proof and demonstration
The notions of proof and reasoning have always been closely linked; but if we can hardly
imagine proving a proposition without some use of reasoning, there is no causal link or
anteriority of one to the other. The classical definitions of these notions speak rather of
operations in a generic sense, which allow the logical sequence of ideas or propositions
(for the reasoning), or by which the accuracy of a result is controlled (for the proof). One
could believe that the discourse constitutes the essential unifying foundation of these
operations. Duval (1995) considers that inference is a particular form of discursive
expansion, like calculations for processing, and that reasoning is a form of discursive
expansion like demonstration, when certain particular conditions of discursive
organization are met. But beyond any mechanism of discursive expansion, it must be
remembered that proofs or reasoning can also be expressed in action or with tools, and
that they may invoke any of several registers of semiotic representation (natural
language, symbolic languages, graphs, geometrical figures, etc.). Moreover, this
deployment raises questions about the mobilization of registers, their articulation and
their coordination. But whether we prove or we reason, we must remember that in
addition to natural language and its means of expression there is a whole range of
instruments for performing these operations.
We must remember that mathematical proof has its particular conditions of discursive
organization. If the demonstration is essentially a text (Barbin et al., 2001), it allows the
introduction of several registers of semiotic representation. In node theory, as an
example, Sullivan (2000) shows reasoning steps in which justification relies on figures or
graphs. Despite the formal style of the text, the deductive nature of the lemma-theorem-
corollary organization and the high epistemic level of the journal, the author adds several
graphs to his demonstrations -- even going so far as to say, in the demonstration of a
lemma (p. 309) : «The proof of the next lemma is given in Figure 22», and to formulate all
his reasoning visually in the style of a comic strip. So it is not just a diagrammatic
reasoning about the understanding of concepts and ideas, visualized with the use of
imagery instead of by linguistic or symbolic means; it is authentic proof, based on a well-
constituted semiotic representation register, and it fulfils the functions of a proof (de
Villiers, 1993), as understood in mathematics. This type of demonstrative initiative is in
line with Proof Without Words (Alsina & Nelsen, 2006) which succeeds in proving
mathematical properties by all sorts of semiotic means, scrupulously avoiding the use of
natural language. But in these proofs there is always some sort of treatment and control
between what we know and what derives from that. Our conclusion is that in
mathematical activity, the inferential capacity of the means for expressing the reasoning
leads us to a mode of validation, even when conveyed by a tool or a machine.
4
When, at the beginning of the section, we quote Clairaut, it was in order to go in the same
direction. That is, if «this assumed inference carries with it its own demonstration»
2
, it
was not the type of rationality that supported the shift from «presumed» to «established»,
but the inferential nature of operations that consists in acknowledging a result by virtue
of its connection with other results already acknowledged. It is therefore not the type of
rationality that carries itself from inductive to deductive reasoning, nor the same registers
that are at stake. We can even add that in its inductive argument, Clairaut tries to make
dynamic a figure by reasoning, playing on the fact that the text will be read and that the
model reader will be able to visualize the animation while, in his demonstration, a
classical figure like those in Euclid's Elements is proposed. In current terms, we can easily
express this idea of inferential connection with the following functional notation:
ƒ(antecedents) = consequent,
where the «antecedents» are the previously acknowledged results and the «consequent»
the newly acknowledged result, according to the connection of epistemic necessity ƒ. We
thus approach didactic definitions of reasoning, such as the one given by Balacheff (1987),
where the reasoning designates «the intellectual activity, mostly non-explicit, of
manipulating information to, from data, produce new information» (p. 148), while
specifying that the type of «inferential» connection is also «of epistemic necessity ». If the
quality of being necessary is shared, at a given moment, within a community, it is indeed
a proof, extending the meaning given by Balacheff (1987) where «signification is the
requirement to determine a validation system common to the interlocutors» (p. 148).
Thus, by seeking his reader's conviction, Clairaut wants him to first join the community
of those for whom his conjecture is necessary (inductive proof), to the point of showing
him why it is mathematically necessary (deductive proof).
From an instrumental-proof perspective, we will examine the nature of ƒ and the issues
and the challenges that it poses; this is the main purpose of our paper. But before
understanding more specifically what we mean by «instrument», we must clarify what is
mathematical work.
2.2 Mathematical work and mathematical thought: a temporal
invariance?
If writing is so important for the expression of reasoning, it is because mathematical
models began with the first written documents. We can even suppose that it is at that
time that mathematical science begins. If our distant ancestors had a form of
mathematical thought, which we can describe as protomathematics, they had to interact
in one way or another with the objects they could represent, beyond the act of
visualization itself or the implicit treatment then imposed. It is known that, long before
writing, Homo sapiens and their Neanderthal cousins could express a symbolic thought
with geometric shapes that seem to be evidence of premeditated creations (Hoffmann et
al., 2018). But in the absence of direct testimony, we do not know whether they reasoned
2
This translation was provided in 1881 by J. Kaines from the edition of the Éléments de géométrie published
in Paris in 1830. In the original in French («cette induction présumée porte avec elle sa demonstration»),
Clairaut (1741) attempts to convince the reader of the relevance of a conjecture through inductive reasoning,
before embarking on a deductive demonstration.
5
with or about these forms, or whether these forms had any instrumental function.
Conversely, the oldest potentially mathematical artefacts, such as Ishango's bone, seem to
tell us more. Although the problem of prehistoric archaeological (strictly speaking),
ethnographic and didactic sources calls for avoiding any over-interpretation (Keller,
2004), we can have in mind that if these artefacts had any mathematical function, it was
the production of new information, like a calculation or some reasoning. In other words,
even before the invention of writing, a form of mathematical thought had to exist to
preside over the interaction between reality and what would become models, the
expression of this interaction implying both signs and tools. In short, mathematics has
been an instrumented activity since its beginnings.
Even today, and to limit ourselves here to the world of education, it is not easy to
distinguish whether it is mathematics (as a science) or mathematical thought (that does
not necessarily proceed by writing) that is at stake (in a given situation/task/activity). In
Canada, for example, a non-profit organization devoted to promoting chess in schools
proudly displays its benefits:
When learning the movement of the knight, the bishop and the rest of the pieces, did you
know that you were doing geometry? Yes, I assure you.
3
If one considers mathematics a science, this assertion seems absurd: chess shares neither
rationality nor means of expression with geometry. Of course, one can do mathematics
when modelling some part of a game in graph theory, possibly with the help of a
computer to support the discovery of a winning strategy. Such activity does not reflect
the specifics of the game itself - but if the sentence above is replaced by: «when learning
the movement of the knight, the bishop and the rest of the pieces, did you know that you
were developing your geometrical thought?» one seems more inclined to answer «yes»,
just as one needs a sense of numbers to do arithmetic or a sense of structure to do algebra.
We can push the question a little further: does the pupil do mathematics when
programming with Scratch$
4
? We can easily propose in the classroom tasks defined in
mathematics (or projects, as in designer jargon like Boutin (2017)), knowing that the
means of expression of the graphic signs manipulated to accomplish the task (to realize a
project) remain rather close to mathematical writing. At the same time, the combination
of these elements is instrumented by the gesture at the interface of the computing device,
which somewhat distances us from the writing. The choice and ordering of the elements
evoke the development of some deductive reasoning or proof — such as a construction
protocol in geometry too. But we can change the order of elements (or some parameters)
dynamically in noting the effect of its algorithm on the interface, even when it is not
completely executed. In this situation, the student develops his mathematical thinking
with Scratch in terms of expression, reasoning and proof (when checking and executing
an algorithm), and is thus «doing mathematics» as an instrumented activity, similarly to
what was done at the time of Ishango's bone, 20,000 years ago.
3
This is the association Échecs et Maths, a pun in French that also means "checkmate" (retrieved April 17,
2018 from https://echecs.org/les-bienfaits-des-echecs).
4
Coding and algorithmic learning platform using a visual and dynamic programming language in which
programs are designed by assembling graphic elements (accessible April 17, 2018 at https://scratch.mit.edu/).
6
2.3 The Mathematical Working Space
In this mathematical science and activity, both sides of the same coin, which is studied
and practised, we consider that mathematical work is the visible part of mathematical
thought. For over ten years, the concept of mathematical work in mathematics didactics
has been the object of collaborative research among various researchers, mainly from
French and Spanish speaking countries (Kuzniak, Tanguay, & Elia, 2016). The
Mathematical Working Space theory (MWS) aims to provide a tool for the specific study
of mathematical work engaged during mathematical sessions. Mathematical work is
progressively constructed, as a process of bridging the epistemological and the cognitive
aspects in accordance with three yet intertwined genetic developments, identified in the
theory as the semiotic, instrumental and discursive geneses (Kuzniak & Richard, 2014).
MWS appears as a theoretical and methodological model that allows one to report on
mathematical activity, potential or real, during problem solving or mathematical tasks. In
particular it allows the description of dominant interactions, whether finalized or not,
depending on the nature or issue of significant moments (e.g. didactic interactions during
the devolution of a task, adidactic interactions while solving a proof problem, etc.). In the
next section, we interpret the types of proofs in the light of the MWS.
Figure 1. The components of the MWS from Kuzniak & Richard (2014) in which the concept of proof is
traditionally related to the epistemological plane by the discursive genesis.
The MWS model is presented in a basic form as a skeleton to which different frameworks
or theories ‘add flesh’, depending on the questions, problems or difficulties involved in a
research study. Thus, in Fig. 1, the vertical planes are related to different phases of the
mathematical work, as discovery, reasoning and communication in a broad sense of
mathematical competencies. The effective realization of these phases can define, for
example, some cognitive mathematical competencies based on the coordination of the
geneses, in order to think the integration of the phases of mathematical work. However,
in the base form of the diagram, these planes are presented only with the generic labels
sem-ins, ins-dis and sem-dis so that the coordination of the geneses can be adapted to the
Communication
(sem-dis)
i
cat
i
o
n
di
s
)
Com
m
(
s
e
Cognitive plane
Epistemological plane
Semiotic genesis
Instrumental genesis
Discursive genesis
Visualization
Construction
Proof
Representamen
Artefacts
Referential
7
task at hand. In Section 3, we will take advantage of the adaptability of the model to
describe our instrumental proofs.
Figure 2. Internal fibrations in an MWS from Lagrange, Recio, Richard, & Vivier (2017) that shows the roles of the
tool (operational means), of representation and control that have been retained within the model.
The concept of fibration has been suggested to label moves, transitions and specific
activities between the different elements of the MWS (Tanguay, Kuzniak, & Gagatsis,
2015; Recio, Richard, & Vivier, 2015). In Fig. 2, we see the internal fibrations that can
intervene in the process of conceptualization, during both the formation of a
mathematical conception and its implementation. In addition to internal fibration
(between planes, between poles, between registers of representation, etc.), the model
considers external fibrations in the same logic between some MWSs from various
mathematical domains such as: during intra-mathematical modelling activity between
analysis and statistics (Derouet, 2016), extra-mathematical modelling with Physics
(Moutet, 2016) or transversal modelling with algorithms (Laval, 2018). The MWS model
can then be articulated in a wide range of situations involving a mathematical task, at one
time or another.
2.4 Instrumented reasoning, instrumental proof
So far, we have used the notion of instrument to designate either a tool or a system of
signs (diagram, figure or other registers of semiotic representation) used by someone in
order to get something done. With such a broad definition, we could consider the many
figural inferences we can find in Alsina & Nelsen (2006) as some kind of instrumented
inferences. Thus, to prove the equality of areas a + b + c = d in the partition of the
following parallelogram (Richard, 2003):
8
the following sequence of figures is used as justification:
It is then a question of a structure inference: ƒ(a + b + c, ▱, d) = (a + b + c = d), where ƒ is
the inference that emerges from the sequence of figures and ▱, the signifying figural
propositions from the top figure. Can we really say that this is instrumentation? In terms
of the MWS model, the visualization of the properties represented by ƒ is typical of
semiotic genesis. One can also grant a role of semiotic tool to the sequence of figures
(fibration), because the material support and the sequence itself, as a means of action
animated by the reader, are not specific to the figural register, and allow him not only to
view a movement but also to compare areas going back and forth from one figure to
another. If, instead of a sequence of figures, the situation was set at the interface of a
dynamic geometry software (see Appendix A), it would be easy to interpret the situation
as one of instrumentation in the user-milieu interaction (IT milieu). However, we will see
in Section 3.1 that the activity of reading a figure and its interactive manipulation
generate different connections of epistemic necessity. In our example, we consider that
linking the rationale to the antecedents and to the consequent involves the discursive
genesis: one can recognize here a step of discursive-graphical reasoning (in the sense of
Richard, 2004a, b), highlighting the coordination of these two geneses (sem-dis plane).
In the French didactic tradition, the notion of instruments is inspired by the work of
Rabardel (1995). In his cognitive approach to contemporary instruments, it is no longer
«the artefact that is explicitly or implicitly considered as the instrument» (p. 4), but a new
entity that is both a subject and an artifact. According to Trouche (2005): «The word
‹instrument› will designate a mixed entity consisting of ‹the technical object and its
modes of use› constructed by a user» (p.93). Thus, since the instruments are not
immediately given to the user, he must develop their use through his mathematical
activity of instrumental genesis. In this perspective, we consider that a figural inference
5
as instrumented during the recognition of figural invariants (inductive) or instrumented
construction (deductive), whether by the intervention of the ruler and the compass, a
software of dynamic geometry or «mathematical machines» (see especially Bartolini-
Bussi & Maschietto, 2005, but also Bryant & Sangwin, 2008). In his work, Rabardel does
not address the question of proof, and he speaks little of reasoning. He mentions, with
regard to Gérard Vergnaud's theory of conceptual fields, the inferences (reasonings)
which allow the treatment and the anticipation from the schemas of a subject, while
5
See Richard (2004 a, b) for a definition of a figural inference, and Coutat, Laborde & Richard (2016), for
the instrumented figural inference.
ab
c
d
X
Y
9
specifying that «there is always a lot of implicitness in a schema, and therefore difficulties
in making it explicit for subjects.» (Rabardel, 1995, p. 88,). Rather, this limit encourages us
to exploit the subject’s interaction with a milieu to speak about reasoning.
Besides, the very idea of instrumental proof is not at all contemporary. Recently,
Cormack (2017) stresses the importance of so-called practical mathematics in early
modern Europe, in order to show how the mathematization (and modelling) of natural
philosophy became an investigation of the interplay between useful mathematics and its
practitioners, and natural philosophers. For example, the book explains that cartographer
Edward Wright first explained Mercator’s cartographic projection, providing an elegant
Euclidean proof of the geometry involved. It is a typical modelling approach between, on
the one hand, a cartographic representation system (the situation model, in the meaning
of Blum, & Leiß, 2007; see also Fig. 3 for details) and its mathematization in geometry on
the other, the proof remaining attached to a discursive genesis activity within the
mathematical model. But a diametrically opposed approach is also evoked in the
collective book, which we readily assimilate to a cycle of «antimodelling» that starts from
the mathematical model to be interpreted first in the reality of a situation model. We
particularly consider as such Archimedes' mechanical approaches, like the weighing
method to discover the area of a parabola segment or the volume of a ball (Netz, & Noel,
2008). In this approach the validity is assumed by the physical coherence under the
constraints of a proper use of the method, before going back to the mathematical model
to infer the areas or the volumes. Unlike Wright's problem, Archimedes's task is set in
mathematics, so in order to solve a mathematical problem Archimedes needs his method
to be anchored in physical reality.
Figure 3. Modelling cycle from Blum & Leiβ (2007). According to Borromeo-Ferri's (2006) cognitive point of view,
the situation model is a mental representation of reality.
Although the subject goes well beyond the purpose of our article, we can briefly raise the
question of the validity of this mechanical proof. If it is at least a heuristic means of great
pedagogical value —in this sense, Archimedes and Clairaut participate in the same
movement— it can be restricted to an empirical method which would need mathematical
discourse to give it a high epistemic value, understanding that Archimedes might have
kept hidden in his drawers a more formal proof equivalent. Moreover, as Keller (2017)
says:
For Plutarch, his biographer who lived long after Archimedes, the great mathematician
could not really have found intellectual satisfaction in his machines; he could only have
meant them to serve to impress the vulgar (…) who could not appreciate more abstract
ideas. Supposedly for Archimedes and Plato (Plutarch would have assumed),
mathe-
matical
model
mathematical
results
real
results
real
model
situation
model
real-
situation
Reality Mathematics
12
3
4
5
6
1 understanding
the task
2 simplifiying/
structuring
3 mathematizing
4 working
mathematically
5 interpretation
6 validation
7 presenting
7
10
mathematical theorems and proofs dealt with ideal situations and one should not think of
them as applicable to real life, which is by necessity so untidy. (p. 116).
Nevertheless, Vitrac (1992) shows that some scientific historians have long considered
that the mechanical method is mathematically admissible. That is, if he does not deny
that Archimedes «never assigns the status of proof to the mechanical method» (p. 76), it
testifies to an Archimedes «axiomatizing mechanics [which] includes at least a part of it
in geometry» (pp 75-76). From this we deduce that physical coherence is induced not
only by matter itself, but also by a scientific method that enables the interpretation of
what is happening. So that if this type of proof appeared as justification for an inference,
to the epistemic necessity connection ƒ, it must be associated with its instrumental
validity, as much in terms of use (of the method, of the artifact) as of the machine (its
constitution, its domain of validity).
From the mental stress from which theory and practice derive, Dupré (2017) gives the
example of the way in which Ettore Ausonio, mathematician and instruments creator
6
,
appropriated the reading of Witelo’s optical treaty Perspectiva:
His teaching was based above all on his reading of Witelo’s Perspectiva , but his lecture
notes reveal highly selective reading practices. These notes listed only Witelo’s
descriptions of the instruments to measure reflection and refraction and those propositions
in which the Polish perspectivist claimed the use of these instruments as proof of the
proposition. Ausonio left out all of Witelo’s propositions not established with the
instruments, and in the selected propositions, he discarded the geometrical
demonstrations and the geometrical diagrams. In sum, Ausonio appropriated Witelo’s
optics in such a way that optics appeared to be a mathematical art based on instrumental
proof. (p. 140)
Compared to Archimedes's method, Ausonio's approach seems quite radical. Because not
only does he avoid the geometrical demonstrations, he rejects geometrical diagrams. His
mathematical activity would be limited mainly to the instrumental genesis, or if we
prefer to reuse the spirit of the «mathematical art based on instrumental proof» by Dupré,
it reflects a competence and a practice of validation requiring planning and intelligence,
as an art. Unlike Archimedes, who deals «with ideal situations» in mathematics, the
initial questioning of Ausonio, and the purpose of his work, is in op. And unlike Wright,
who seeks to validate mathematically, Ausonio wants to perform a validation that
remains in the universe of his instruments (artifact sense). From these considerations, we
draw a first type of instrumental proof in mathematical work, the mechanical proof, which
proceeds essentially by coordination of semiotic and instrumental genesis. In choosing a
term relating to mechanics, it is not so much to highlight the fact that the proof depends
on the operation of a machine or a mechanism, but that the justification is based on some
sort of laws of motion or balance that objects exercise in relation to each other. This
definition allows us, first, to account for certain proofs that already exist in mathematical
education, such as the mathematics of, in and for the reality from Emma Castelnuovo or
those that frequently emerge from pedagogical initiatives (Fig. 4), and it is consistent,
second, with the use of mathematical machines, and especially with dynamic geometry
6
We also write "Ettor Eusobio". The Thesaurus of the Consortium of European Research Librairies (CERL)
considers him as an instrument maker (from German «instrumentenbauer», see
https://thesaurus.cerl.org/record/cnp02134922), and in the 1678 edition of Leonardo Fioravanti's Dello
specchio di scientia universale, it is said of him: «the great philosopher and mathematician, Mr Ettor Eusobio
da Venetia; inventor of the most beautiful mathematical material ever seen» (p. 94).
11
software. For the «laws of motion» and the «balance of forces» relate to the operation of
the software and the logic of the construction of the figure, with the particularity that by
acting on the figure or on its elements, the user also acts on the register of semiotic
representation, which is not seen in Archimedes's approach. From a didactic perspective,
our point of view is the idea of the physicist geometer described by Tanguay and
Geeraerts (2014).
Figure 4. Examples of mechanical proofs of Pythagoras's theorem in mathematical education. The first on the left is
justified by comparison of weights (Castelnuovo & Barra, 1976) and the second, on the right, by the transfer of
liquid volumes (YouTube, 2009)
7
.
2.5 Algorithmic and proof
Various algorithms were already known in antiquity, in arithmetic or in, geometry,
including, among the most familiar:
- rules for calculating the length of arcs and the area of surfaces, in Egypt and in
Greece;
- several methods for solving integer equations, following the work of Diophantus
of Alexandria in the 4th century AD;
- the Euclid algorithm (c. 300 BC) that calculates the greatest common divisor of
two natural numbers;
- the calculation schema of the number ! due to Archimedes.
If there were to be a tool-object dichotomy at play, like the theory-practice dichotomy in
Section 2.4, it would only be purely functional. Because the determination of geometric
measurements or the approximation of irrational numbers like ! refers to the essential
nature of real numbers and suggests, as Gray and Tall (1994) show in a learning context
with the notion of procept, that mathematical objects are formed by encapsulating
processes. An algorithm therefore solves not a single problem but a whole class of
problems differing only by the data and the specific course-of-values, but controlled by
7
Extract from the original video entitled Pythagorean theorem water demo (retrieved August 3, 2018 from
https://youtu.be/CAkMUdeB06o).
12
the same requirements -- that is, it must operate with certainty regardless of the given
problem.
Let's imagine Archimedes today trying to solve a problem of counterfeit coins with his
weighing method and a Roberval scale. The problem goes like this:
In a set of coins, indistinguishable by sight or touch, there are false coins. Real coins all
have the same weight, and so do counterfeits, but their weight is different from that of real
coins. With the help of a scale and without being able to have a reference weight, how can
Archimedes find the counterfeit coins? Which is the method that would allow him to find
them with as little weighting as possible?
Such is the problem posed by Modeste, Gravier, & Ouvrier-Buffet (2010), but without
Archimedes (!), to pre-service teachers, playing on the constraints of weight and the
number of counterfeit coins. This problem does indeed concern an algorithmic
problematic, because it is a matter of elaborating an effective method to search for
counterfeit coins, proving it and studying its optimality, e.g. by means of successive
weighing operations or trees. In this type of open problem, students need to think about
both the definition of the situation and the search for valid solutions, the design factors
associated with the problem, the plausible creative processes in some kind of iterative
problematic and an innovative line of reasoning behind the constraints, factors and the
design choices made. This activity can be seen as a problematic-modelling dynamic
between reality and the mathematical world (Fig. 3) which allows, under certain
conditions, to get some results and enrich the understanding of reality (Fig. 5). According
to the authors: «this problem and most of its variants, are not yet completely solved as far
as mathematical research is concerned» (p. 61), which presupposes from the outset that it
will be necessary to be inventive. The algorithmic approach would first be used for
programming a mathematical work by gradually approaching a reference situation
through successive problem solving or judiciously considered cases.
Figure 5. When we attack a complex real problem, the modelling cycle (Fig. 3) is characterized by some back and
forth processes that we wish to converge (infundibuliform path), at least until we obtain a stable problematic. At
this tipping point, the cycle can continue for results (cycle tightening), or the real situation can be rethought by
considering new constraints (cycle widening). The situation model is modified and the cycle starts again (vignette).
13
Traditionally, algorithmic is the study and production of rules and techniques that are
involved in the definition and design of algorithms, which are structured steps that
transform well-defined inputs into desirable outputs. We can then talk about
disconnected algorithmic. However, the algorithms can be encoded in a machine to
reproduce these steps very efficiently, especially when it comes to solving complex
problems that require a large computing capacity. Moreover, to function effectively, i.e.
to ensure that it is executed quickly and that it converges towards desirable outputs, the
algorithm must be stable and not modify itself. Otherwise it would be autonomous, even
intelligent (in the sense of adapting to a new situation), and we could no longer predict or
optimize how it would work towards the desired results. This algorithmic determinism
may seem the complete opposite of human intelligence, which must constantly adapt to
the unexpected. But it is this capacity for human adaptation that makes it possible to
govern the solution of a complex problem, in the sense that it is the human being who
breaks down the problem, chooses the constraints, builds the necessary algorithms,
possibly delegates the execution of the algorithm to a machine, and carries out the
necessary verification (validity of the algorithm, encoding, response, calculability in a
reasonable time, etc.). And it is also s/he who, if necessary, remodels the model situation
to enrich the understanding of reality, the dimension of the initial problem or the validity
domain of the resolution process that s/he has thus systematized. This human attitude to
solve a problem in such a dynamic and which invites to «think about the tasks to be
accomplished in the form of a series of steps» (cf. Venant, 2018, p. 58) is what we call the
algorithmic spirit.
To remain in the mathematical sphere, the idea of a line of action or a series of operations
proposed to achieve a result is highlighted by the algorithmic approach in solving the
areas problem developed by Trahan (2016)
8
:
Let any two polygons have the same area. It is possible to cut the first polygon into a finite
number of pieces and then reform into the second polygon.
This is the classic statement of Wallace-Bolyai-Gerwien's theorem. Exactly as in the
previous problem, the statement is simple and it is by playing on constraints that we
succeed in demonstrating this result, in verifying certain effects and wondering about
possible implications or generalizations, whether it be to curvilinear figures or higher
dimensions. Without entering into a particular formalization, we can show the structure
of its approach in four steps (Fig. 6). Although this does not appear in our paper, the
author carefully demonstrates each of his propositions: his mathematical work is
essentially activated through a discursive genesis, with several steps of discursive-
graphic reasoning when he considers particular examples. But his reasoning is broader
than that. Although it is easy to describe a posteriori his approach as systematic and to
see only a series of traditional demonstrations, possibly presentable in the lemma-
theorem-corollary mode, his program is an inventive mathematical work that shows
vividly the search for a validation of the original problem. Despite this connection
between the heuristics of mathematical discovery and validation by this type of proof
approach, algorithmic is much more than a mathematical way of working. Moreover, this
algorithmic spirit, already implicitly present in teaching, could be worked out and
brought to light thanks to algorithmic instruments.
8
The author explained that part of his resolution program is inspired by the website Choux romanesco,
Vache qui rit et intégrales curvilignes accessible from http://eljjdx.canalblog.com/.
14
Figure 6. Structure of the process of solving of the Area Puzzle Problem from the Trahan's algorithmic approach
(2016): setting the situation (part 1), first resolution program (part 2), second resolution program with effect
verification (step 3) and opening towards new problematizations with results anticipation (step 4). The derived
proposition in step 3 is an illustration of a modified situation model as suggested in the vignette in Fig. 5. For the
convenience of the reader, a transcript of the contents of the boxes is presented in the Appendix B.
Trahan's attitude is close to the way that mathematicians treat long proofs. By the 1980s,
according to Krieger (2004), a variety of rigorous proofs were provided of various
fundamental facts about our world, many of them lengthy and complex and involving
much calculation (Krieger, 2004):
Actually, many of the preliminary theorems motivate the proof and indicate what is
needed if a proof is to go through. And the lemmas might be seen as lemmas hanging
from a tree of theorems or troops lined up to do particular work. As in many such
calculations, the result almost miraculously appears at the end. (…) The achievement (of
lengthy and complex proofs) is again the ability to divide up the problem into tractable
parts, to orchestrate the parts so that they work together, and to be able to tell a story of
the proof. (p. 1227-1228)
Some of these "facts about our world" begin with questions initially asked in physics, but
the proofs in question do indeed stem from mathematical work. We consider that this
idea of orchestration of parts, which works with the same goal of proof, is typical of the
algorithmic spirit. Of course, a general algorithm makes it possible to produce a proof
and can possibly constitute other algorithms to verify parts of it, as it is also done with
traditional proofs of lemmas.
Besides, in the research of Modeste, Gravier, & Ouvrier-Buffet (2010), the natural link
between the algorithmic approach and the learning of proof is particularly emphasized:
The algorithmic approach uses arguments that are common to mathematics, but also to a
specific way of thinking. It seems essential that the study of the algorithm as an object
allows for a challenge to this way of thinking. A central concern of mathematics is to
Theorem (not surprising)
If you cut a polygon into a finite number of
pieces to reform another, then both
polygons will have the same area.
Theorem (more surprising) !
Should any two polygons have the same
area, it is possible to cut the first into a
finite number of pieces to then form the
second.
New situations (to pose and solve)!
•Generalization: curved surface?!
•Generalizations in 3 dimensions!
•(…)
Hilbert's 3rd problem: given two polyhedrons of equal volume, can we
cut the first polyhedron into polyhedra and bring them together to form
the second polyhedron?!
- No: Dehn found an invariant (Dehn invariant) that is preserved during a
cut; the cube and the tetrahedron do not have the same invariant.!
- Some possible cuttings...
Conclusion
Any polygon can be cut into a finite number of pieces to
then form a square.
Propositions (program)!
1.Any polygon can be cut into a finite number of triangles. !
2.Any triangle can be cut to form a rectangle. !
3.Any rectangle can be cut to form a square. !
4.Any pair of squares can be cut to form a square.
Resolution (breakdown by case) !
1.Demonstration in two cases: convex polygon and concave polygon. !
2.Check the four cor ners of the rectangle and check the
«joints» (alignment of points). !
3.In a rectangle L by ℓ: case ℓ < L ≤ 4 ℓ, checking of the angles, of the
«joints» then sides; case L > 4 ℓ, check also lengths and sides.!
4.Verification of angles, "joints" then sides.
Can we do better? Yes, the demonstration makes it
possible to obtain a division between two polygons, but
this is not optimal.
Verification with interesting cuttings !
(demonstrated in other research work)!
•Square in equilateral triangle.!
•Square in nonagon.!
•Some cutting of the pentagon.
Deriving proposition
If two polygons of the same surface can be cut into a finite number of
pieces to each form a square, then there is a split of the first polygon
to form the second.
Solution
Superposition of the two cuttings.
1
2
3
4
15
provide demonstrations of its results, so it is inevitable, if one looks at the algorithm as an
object of mathematics, to query the algorithm-proof link (p. 55).
Even if it is not a surprise for the authors, the tone is set: the algorithm is not only a
mathematical way of working, but also an object that must be implemented in school. At
the same time, since the introduction of algorithms as objects, new questions emerge,
such as the efficiency of the algorithm in terms of the number of operations or the
memory necessary for execution, all of which may divert thematical work. Anyway,
computer science has changed mathematics, in particular by allowing objects to be
studied from new perspectives, by bringing new questions, by creating emerging
mathematical fields that are now booming, and by transforming the mathematician's
activity thanks to new tools (ibidem, p. 57). In addition, computer programming allows
some formalization of reasoning to support the student in mathematical tasks like
derivation, discovery, and proving geometry statements (Kovács, Richard, Recio, &
Vélez, 2017) or in learning mathematical proof (Tessier-Baillargeon, Richard, Leduc, &
Gagnon, 2017).
2.6 Another instrumental proof
At an epistemological level, many situations ask for rethinking mathematical work: too
many cases to study, intrinsic inability to demonstrate by mathematical induction,
research by repeated tests or the conjecture invalidation iterations are not convergent, etc.
The proof of the four-colour theorem, which we cited in our Introduction, is an example of
a mathematical work that we cannot yet achieve without a computer. In a critical
commentary that goes back some forty years, Appel and Haken (1977) already said:
Our proof of the four-colour theorem suggests that there are limits to what can be
achieved in mathematics by theoretical methods alone. It also implies that in the past the
need for computational methods in mathematical proofs has been underestimated. It is of
great practical value to mathematicians to determine the powers and limitations of their
methods. We hope that our work will facilitate progress in this direction and that this
expansion of acceptable proof techniques justifies the immense effort devoted over the
past century to proving the four-colour theorem. (p. 121).
The fundamental problem for the acceptability of this type of proof is that, despite its
great interest, algorithmic thinking does not convey a mode of validation in itself
9
. On the
contrary, the theorem-demonstration problem moves towards a problem of validation of
the algorithm and its execution by the machine. In other words, you must be sure that the
algorithm is achieving the proper result, whatever the pending problem might be. In a
very pragmatic way, it often happens that computer experts validate an algorithm by
using verification methods. But in computability and complexity, one usually tries to
prove mathematically the effectiveness of the algorithm, even if it is necessary to
reformulate its domain of validity. The algorithmic way of thinking, as a characteristic of
mathematical work, as well as the execution of an algorithm by a machine, suggest the
existence of a type of instrumental proof different from mechanical proof. We call this
9
Although we know that solving algorithmically a problem of proof and using a program to verify a set of
particular cases is not the same thing, we can state as Clairaut said that an algorithm often carries with it its
own demonstration. This wink to our quotation of Clairaut in an algorithmic context comes from Simon
Modeste.
16
type of proof algorithmic proof: it usually proceeds by coordination of the discursive and
instrumental geneses.
In some cases, the learning of the proof is instrumented by a computing device and it can
go through an algorithmic process, constituting rightly an algorithmic proof according to
the definition given above. Thus, in secondary school, we can mention the resolution of a
problem of proof at the interface of the QED-Tutrix tutorial system (Leduc et al., 2017). In
this kind of tutorial system, we tend to support the student in elaborating a deductive
proof, mostly by forcing him to construct his proof by forward chaining (from the
hypotheses of the problem to the conclusion) or backward chaining (the other way
round). However, the QED-Tutrix system accepts that the student can enter the
propositions of his demonstration in whatever order he wants, whether they are
assumptions, justifications or results that he deduced or he supposes are valid (Fig. 7). It
even happens that the student tests the system itself by watching its reaction to one of his
propositions (instrumental genesis). But the system is programmed to support the
student in the logic of the problem, and to lead him gradually to write a deductive proof
(discursive genesis).
Figure 7. QED-Tutrix interface, an intelligent tutorial system that supports solving problems of proof in geometry.
From the problem statement (top), the student writes propositions (bottom), acts on the figure of the Geogebra
dynamic geometry module (left) and discuss with an artificial pedagogical agent (right).
The instrumental proof would thus not materialize only by the engagement of artefacts,
tools or methods to a process of mathematical proof. It would also open up to physical or
algorithmic modes of thinking, renewing several questions about the epistemic necessity,
the control of knowledge or the decision-making that arise from different types of
reasoning. The example of proof learning and its quest using QED-Tutrix testifies that
deductive reasoning is not necessarily an instruction tool for learning demonstration,
17
contrary to what is sometimes heard among supporters of the old school
10
. This is also
what Brousseau (2004) reminds us with his paradoxes about the study of the teaching of
reasoning and proof:
The "orthodox" presentation of mathematical texts gives the impression that formal logic
(modus ponens, with perhaps a few other tools of logic) is the fundamental and necessary
instrument of mathematics, and that the aim of mathematics is to demonstrate that its
author has not produced any contradictions (with himself or with known mathematics).
Many teachers tend to deduce from this that since mathematical reasoning is the sole
means of establishing publicly that a mathematical statement is true, this reasoning must
also necessarily describe (or serve as a model for describing) the thinking that correctly
constructs mathematics, hence that describes the thinking of mathematicians and of
students. As a result, they try to teach directly how to think, and then how to reason as
one does in making a proof. They thus mix up the activity and mathematical reasoning of
the students with their cultural product: the standard method of communication.
If, on the other hand, one assumes that the natural functioning of thought produces exact
knowledge by some processes (rhetorical, heuristic, psychological…) which cannot be
reduced to the presentations and notations that are most convenient (for mathematicians
and their research), then what are those processes, and how can they be realized? or how
can someone be led to achieve them?
Certainly, one might fear that by its versatility, instrumental proof inherently creates a
problem of validity for mathematical theory, or for mathematics that is constructed as a
science. Nevertheless, mathematical activity is much more than the slow establishment of
a non-contradictory reference system, and this is even truer when it comes to
mathematical work at school. But it is necessary from the outset to distinguish the
mathematical work of the student from that of the teacher: the latter must develop not
one but two kinds of geneses: one of them personal, for his or her mathematical work,
and the other one professional, for teaching (see Haspekian, 2011, about a double
instrumental genesis; Derouet, 2017, on the student-teacher contrast or Leduc and al.,
2017, for the genetic distinction between mathematical work and didactic work). As a
result, in addition to appearing legitimate from a didactic and epistemological point of
view, the instrumental proofs become particularly useful in answering Brousseau's
questions about «those processes».
3 Three types of proof in the mathematical work
I would also claim that, in a very specific sense, mathematical work is a form of philosophical analysis.
The mathematicians and mathematical physicists find out through their rigorous proofs just which features of the world are
necessary if we are to have the kind of world we do have.
Martin H. Krieger (2004)
10
It is still common to hear at conferences that automated reasoning, however used, is not appropriate for
learning mathematical proof in school. Even very recently, a colleague showed us an anonymous assessor's
comment for the evaluation of an important project: «automatic proofs are not necessarily suitable for
educational purposes, therefore most parts of action are somewhat out of interest in this context». This point
of view seems to us similar to the one where, in the 1970s, it was feared that students would no longer learn
to calculate, after the introduction of calculators in schools. Paradoxically, this artefact is today practically
tossed in the dustbin of history in favor of tools once unthinkable.
18
3.1 In the model of the mathematical working space
By introducing the mathematical working space theory, we were able to deal with the
notion of proof both in school and in the work of the mathematician. Even if, since
Chevallard's anthropological theory of the didactic (1992), we have a better
understanding of why school mathematics is not a reduction of scholarly mathematics,
Kuzniak, Tanguay and Elia (2016) remind us that the MWS «involved in mathematics
education, are related to different kinds of vigilance – epistemological, didactic, and
cognitive» (p. 729). Moreover, among students who are destined to become mathematics
teachers, the knowledge conveyed often moves from one mathematical paradigm to
another, to the point that even the meanings of the mathematical concepts at stake seem
to be agglutinated (Arzarello, 2006; Tanguay, & Venant, 2016). Facing the inescapable
complexity of mathematical work at school and in order to set our types of proof
according to MWS, we have to make assumptions about the mathematical task, the
subject-milieu interaction, the proof/reasoning nesting and the valence of the proof
activity in mathematics, as follows.
1. The proof results from a mathematical task and not from a problem set in physics or in
algorithmics. For example, some situations in special relativity at school involve
Minkowski diagrams at the interface of a dynamic geometry software, playing on
rationality frameworks in both physical sciences and mathematics (Moutet, 2016). In such
a case, we must consider two epistemological planes, modelling links between these
planes and external fibrations respectively between elements of the semiotic and
instrumental geneses of each plane. If a mechanical proof had to be recognized in the
resolution of the initial situation, then it would be necessary to start from a well-
identified mathematical task during the resolution process and to deal with external
coordination issues between geneses. Similarly, some algorithmic situations are already
close to mathematics, such as introducing the dichotomy algorithm to find a square root
bounding of prime numbers, or determining the zeros of a function by the sweeping-out
method (Laval, 2018). In each case, one can start working with some software like
Algobox
11
and ask the software to test the algorithm that has been composed, evoking the
algorithmical proof. Nevertheless, the transition between the worlds of Algorithmics (A)
and Mathematics (M) is not symmetrical: although the A à M direction is essentially
based on a mathematization, the M à A direction is closer to a conversion, or even to
intramathematical modelling. In the problems of bounding or determination of zeros, the
mathematical work is expressed conceptually in a dynamics of K(epsilon)-game (Bartle &
Sherbert, 2011) because of the very nature of real numbers (see Section 2.5). The
activation of an MWS and its proof types would start at the same time as the
mathematical task.
2. The proof is defined by the interaction between a subject, who can be a reader and/or a
user, and a milieu. This idea of interaction is inspired by the adidactic situations
described by Brousseau (1998) and from the model to reason on learners’ conceptions of
Balacheff, & Margolinas (2005). We have previously shown that the epistemic necessity is
characteristic of this interaction for the instrumented learning of properties in geometry
(Coutat, Laborde, & Richard, 2016) and that the type of proof of the same property may
vary depending on the nature of this interaction (Richard, Oller, & Meavilla, 2016). It
follows that even traditional proofs are not the prerogative of individuals, but also of the
11
Freeware by Pascal Brachet (2018) available at http://www.xm1math.net/algobox/.
19
milieu that supports them (paper-pencil, computers, etc.). When a proof involves an
automated reasoning tool, the interest for the process is not in the calculation by the
machine, as a tooled necessity, but in the questioning that triggers the calculation and in
the interpretation of the results. If a proof encapsulates a procedure by semiotic,
instrumental or discursive means, one must be able either to explain how the procedure
works, despite the black box effect that could result, or to vary it with an additional,
compensatory or confirmatory proof. The MWS can then be considered a system of
activities that evidences the types of proof.
3. A proof may consist of several steps of reasoning, and reasoning may consist of several
proofs. In addition to our considerations from Section 2.1, reasoning can consist of a
sequence of inferences ƒ1, ƒ2, …, ƒn, where the ƒis are the inferential justifications, both in
traditional or instrumental reasoning. But justifications may depend on the very structure
of the propositions at stake, such as in a syllogism, in a semantic inference, according to a
third-party statement (from the French «énoncé-tiers», by Duval, 1995), or in a deduction
or in a discursive inference. Besides the discursive register, these justifications can be
expressed through a discursive-graphic reasoning, the use of an artifact or within an
encapsulated procedure. In the same way, a proof can be structured by a sequence of
connections of epistemic necessity, by adopting deductive, inductive or abductive
validation modes. What must be emphasized here is not the structuring aspect of the
inferences, but rather the functional purpose of the connection of epistemic necessity in
the interaction between a subject and a milieu
12
. This hypothesis is that of a possible
structuring for reasoning and a functional validation for the one who advances a proof in
his mathematical work.
4. The ability to prove that results from subject-milieu interactions is affected if the
semiotic, instrumental and discursive means of mathematical work are opened or
constrained. In Richard, Oller, & Meavilla (2016), we define the set of the potential
subject-milieu interactions related to a type of epistemic necessity as the space of
epistemic necessity. This allows us to consider the ability to prove, when interacting with
different milieus, in terms of tolerance to possible variations in the coordination of
geneses. Like the concept of valence of mathematical work, there is a valence of proof
activity in mathematics in a given space of epistemic necessity. Originally, the concept of
tolerance is based on the engineering tolerance which focuses on the permissible limit(s)
of the potential interactions in a MWS. Thus, the tolerance analysis of a proof is the study
of the operating domain of these interactions related to a given space of epistemic
necessity. Because a specific milieu conveys mathematical knowledge and processes that
are revealed in the use of a tool or of a semiotic production, a variation that involves the
geneses into a MWS allows one to test the ability to produce a proof.
The first type of proof we retain is the discursive-graphic proof which is represented in
the back vertical plane in Fig. 8 (the sem-dis plane in the base form of the diagram, see
Fig. 1). This is the most common proof in the mathematical work, operating essentially by
the coordination of semiotic and discursive geneses. Traditional demonstrations or those
involving well-defined registers of semiotic representation in mathematics are examples
of this type of proof. In other words, proofs without words would be the proofs least
dependent on the discursive genesis in the mathematical activity, and formal proofs,
12
For more information, see the analysis of student texts and of the editorial organization from Duval
(1995), and the analyses of the strategic contexture of proofs in secondary school by Richard (2004a).
20
those least related to the semiotic genesis. As we already mentioned Section 2, the
question of medium for representation is important. In the absence of dynamism due to
the material device (i.e., any form of physical or electronic data carrier), the subject has to
animate and control the properties represented. To understand the articulation of
connections of epistemic necessity, especially for someone who is not the author of «what
to see», it may be very helpful to give some wording to the reasoning, such as a
discursive description or the use of plastic means (colours, textures, grain, lighting,
frame, arrows, etc.) to focus on significant elements.
Figure 8. In the model of MWS, the instrumental proofs appear on the salient vertical planes and the discursive-
graphic proof, in the background plane, in order to highlight the coordination of the dominant geneses in
mathematical work.
The second type of proof is the mechanical proof, which we have already introduced in
Section 2.4. These proofs, represented in Fig. 8 in the left vertical plane (the sem-ins
plane), proceed above all by the coordination of the semiotic and instrumental geneses. In
the same way that discursive-graphic proofs can be described as effective, thanks to non-
verbal semiotic representations that do not necessarily have to bear the weight of the
discursive processing, the mechanical proofs are only functional, and if they "carry with it
their own demonstrations", it's not according to a purely mathematical rationality. But
this advantage has its drawback, because it raises the question of the operative
transparency. The joint use of an artifact raises what Rabardel (1995) calls the
phenomenal material causality. It concerns the structure of the artifact, its functioning,
even its conduct (e. g. systems producing reasoning as the Automated Reasoning Tool
(ART) with dynamic geometry software), or at least what is relevant for the subject’s
action. To this causality is added the subject's instrumented action oriented towards the
finality of the task. Thus, during a mechanical proof, knowing that it is not the artifact,
the method or the implemented reference model that is at stake, but the way they are
used, is an example of causality of the instrumented action in proving. Some mechanical
proofs can then introduce some ART with a certain independence from what is done with
these tools, without compromising the very fact that it is proof.
21
Some mathematical properties are clearly revealed in an instrumented perspective, while
engaging reasonings are activated on different planes within the MWS. Thus, to show
that the tangent to a circle is perpendicular to the radius, we can construct a figure-
situation at the interface of a dynamic geometry software where the centre of the circle
and the end of the radius are defined on grid nodes, the tangent line being able to pivot
around the point of tangency (Fig. 9, top left). To see the property, it is necessary to
oscillate between satisfactory and unsatisfactory configurations: it is the invariance
during these back-and-forth draggings that allow the property to be induced. This is a
mechanical proof. Now, if we leave the figure in a position where the perpendicularity is
visible, we can certainly see the property according to the visual appearance, but also by
counting on the grid nodes (verification of the orthogonality criterion) to notice that we
are definitely in a situation of two perpendicular lines. This time it is a discursive-graphic
proof that could very well have been done without the computer tool, assuming that
constraints are considered when defining the object. The hypothesis of nesting between
proof and reasoning is clarified here: a connection of epistemic necessity that concludes
similarly (being perpendicular to the radius) can be both instrumental and cognitive,
depending on the type of interaction involved such as an action at the interface or a
reading.
Figure 9. Two types of epistemic necessity in the subject-milieu interaction (Richard, Oller, & Meavilla, 2016).
The third type of proof is the algorithmic proof. In the same way that it is easy to
consider the instrumental genesis when the execution of an algorithm by a machine is
governed by the user, the setting up or the development of an algorithm is naturally
associated with the discursive genesis. In an algorithmic proof process, in the meaning
we introduced in Section 2.6, it is not so much a matter of validating an algorithm in a
sort of disconnected computer activity, but rather of considering the algorithmic
connected to a mathematical task whose purpose is to prove. Indeed, if we present the
algorithmic proofs in the right vertical plane in Fig. 8 (the ins-dis plane), it is in order to
highlight the interplay of geneses during the design-in-use of proving, that is the
adaptive design of an algorithm while using a machine. However, the algorithm thinking
mode in mathematical work can appear independently of the execution of an algorithm
Cognitive necessity
of the link constraints-conclusion
Recognition of invariants by
induction on the operational
dynamic figure
Unsatisfactory
configurations
Satisfactory
configurations
Conception
(link constraints-conclusion)
Reading and accepting of the conclusion
by discursive-graphic reasoning
Instrumental necessity
of the link constraints-conclusion
Interactions
(model reader-user)
Procept
(object-process expressed using
the register of dynamic figures)
22
by a machine, as in Trahan's example (Fig. 6), or during an instrumented reasoning that
responds to the student's discourse, as does the QED-Turix system (Fig. 7). Like the
mechanical proofs, these two examples of algorithmic proofs show that it is possible from
time to time to engage in reasonings on separate planes in the MWS, which is the case
with the joint support of discursive-graphic steps (proofs or reasonings).
3.2 Didactic implications
If we were not to consider the types of proofs we have just defined, we would still have
the essentially discursive proofs, such as traditional demonstrations, the semiotic proofs,
such as proofs without words, and the machine-computable proofs, such as proofs tooled
by automated reasoning; all these types of proofs would constitute a very good core. But
to take into account the complexity of the proving activity, the «other proofs» become
necessary. According to the model of mathematical working spaces, there is no
executable representamen, hence the need to separate the semiotics from the instrumental
and the discursive. For instance, when a geometric figure is represented at the interface of
a dynamic geometry software, one who manipulates the figure is also manipulating a
semiotic representation system managed partly by a tool, and this evokes the mechanical
proof. Also, every time we use a semiotic method to draw conclusions, such as a Venn
diagram to determine the probability of a composed event or with the spatial
organization of the subtraction of equations to identify the general term of a geometric
series, we are exactly in a process of discursive-graphic proofs. In the past, without any
software, it was quite difficult to act on a dynamic representamen — there are very few
material signs that react by themselves to an action — and the conjecture had to be
suggested before the proof because the geneses involved in each process were
fundamentally distinct. Such a work could be laborious, even going so far as to blatantly
mask the process of discovery. Just remember how we could determine a locus of points
using ruler and compass, starting with the production of several drawings. As soon as we
had a good enough idea of the locus, we hid with shame our research activity to show
discursively the necessity for it, quickly closing our sketchbook, and struggling to prove
it properly. However, the significant heuristic moments could have been encapsulated in
an instrumental proof worthy of interest, a prelude to the elaboration of reasoning
considered as a routine for a class of problems. Whether one needs a discourse or a
computer tool to animate the representamen, the opening of the perspective of
instrumented proofs brings the heuristics of validation closer in the same crucible that the
mathematical experience. In what follows, we focus on the issues concerning automated
reasoning tools and intelligent tutorial systems.
Adopting this open attitude is particularly important when it comes to using ART in
schools or teacher training. Until now, we have explained that a proof can consist of a
series of connections of epistemic necessity, that these connections can be in deductive,
inductive or abductive reasoning modes, but that the questioning must be at the initiative
of the student so that the computer-milieu remains a partner. Thus, questioning an oracle
at the interface of a dynamic geometry software may seem worrisome if the objective of
the task is limited to an immediate production of some deductive reasoning. But this
same questioning appears to be very powerful when it comes to support the derivation,
discovery and proof of geometry statements (Kovács, Richard, Recio, & Vélez, 2017).
With ART tools, we can easily imagine the student's mathematical activity realized in a
situation (context, problem or task) where she proactively questions the milieu, in the
specific logic of the situation and in the more general logic of the didactic contract that
23
links her to the knowledge at stake. The student then seeks answers adapted to the
context, to solve the problem or to accomplish the task, without having to bear all the
weight of the logical artillery of the «orthodox presentation of mathematical texts» (see
Section 2.6). With a dynamic geometry software, for example, we can refer to the many
ways that promote investigating geometrical properties of a figure or generalizing some
observed/conjectured geometric properties (cf. the nine tools of Kovács, Recio & Vélez,
2018a), and to the combine use of LEGOs and the software to link proving, computation
and experimental views in modelling tasks (Kovács, 2018).
In terms of reasoning, ART helps the student in producing valid abductions, as in Pierce's
meaning, which brings his experience to the property to be discovered, fostering rigorous
creativity during the solving of surprising problems. In fact, researches in didactics of
mathematics do not sufficiently address the modelling of physical phenomena using
geometrical tools. The same applies to problem-solving: most of the time, we work on
problems of proof that rarely go beyond the simple discovery of well-defined properties
already known by the teacher or even by the student himself. Undoubtedly, the student's
adherence to geometric science requires the development of competence in modelling
form, shape and space; but, unfortunately, modelling activity is generally not widely
practised in compulsory education - the tasks can be solved using standard
representations and definitions, routine procedures, predetermined heuristics or well-
defined methods. While few mathematics teachers in compulsory education or training
initiatives seem to be concerned with solving open problems, we believe that the
functionalities of ART are particularly useful for designing situations that engage
mathematical work on the basis of well-founded judgments, relying on discovery and
proof during the quest for problematization-modelling as instrumental proofs do.
The computer-assisted mathematical proof and the interactive proof assistants in
schooling are two very different concepts. The QED-Tutrix system is based on this
difference. The computer-assisted mathematical proof in an automated proof perspective
allows users to check statements and to discover new ones. According to Font, Richard &
Gagnon (2018), one of the main goals of the research community in automated theorem-
proving is to operate efficiently (here meaning fast and focused). Since synthetic
approaches are typically slower, most solvers rely on algebraic resolutions. The problem
with these systems of automated proving is that the algebraic model makes it possible to
say if a geometric statement is true or not, or maybe true (or false) in parts (cf. Kovács,
Recio & Vélez, 2018b), but it does not provide any proof of that, let alone a proof that a
student can master. With the interactive proof assistants, the major problem is often the
rigidity of the system that forces the student to work in forward or backward chaining,
and this brings us back to Brousseau's paradoxes (2004). It is indeed the modelling of the
learning conditions of mathematics in an instrumented perspective by the IT tool that
must appear in the design of these tutorial systems. In other words, it is not towards the
acceptance of systems to be used as they are, but towards those designed to integrate
students, and this very early in the design process. The didactic advantage of an
approach like QED-Tutrix is that it allows the student to prove jointly, with verbal and
figural statements, as in discursive-graphic proofs, and that it accepts statements in the
order suggested by the user. This is an indispensable condition for producing any
algorithmic proofs. In fact, if it is no longer the user who manages the structure of the
proof, he or she would be condemned to proceed as in a deductive calculation (e.g. by
forward or backward chaining). Furthermore, he or she could not integrate the
recognition of invariants when dragging, which would unnecessarily hinder the
realization of instrumental proofs. If automated reasoning tools can be integrated into the
interactive proof assistants, it is mainly to operate with readable proofs.
24
If we were to pursue research beyond this paper, consideration could be given to the
development of a catalogue of instrumental proofs, both proofs that already exist in
educational practice and those that should be encouraged in the classroom. If we
consider that teachers, trainer and researchers may take different avenues in their proofs,
it is obvious that the notion of mathematical work will still be playing a unifying role. We
knew that our initial idea about merging thought and activity may be considered as
surprising but bright, because we know that they are not similar. But it appears to be
useful because it is difficult to distinguish, in algorithmical thinking, when an algorithm
«carries with it its own demonstration», independently of any execution by a machine,
just as it is difficult to distinguish in interaction with physical reality, which is related to
semiotics or to instrumented representation.
4 Conclusion
Two proofs are better than one. "It is safe riding at two anchors”.
George Pólya (1973)
The notion of proof has long been a subject of study in mathematics education. Thematic
working groups, such as the International Congress on Mathematical Education (ICME)
or the Congress of the European Society for Research in Mathematics Education
(CERME), and some websites, such as La lettre de la preuve
13
, offer a very good inventory
of published papers and monographs on the subject. As for the consequences of proof in
the teaching and learning of mathematics, synthetic works such as the book Developing
research in mathematics education - twenty years of communication, cooperation and
collaboration in Europe (Dreyfus, Artigue, Potari, Prediger, & Ruthven, 2018) show very
well its transversal character in mathematical work. It is precisely by taking advantage of
this idea of overlapping of the proof on several mathematical, scientific and thematic
fields that we have supported our discourse. The added value of the three types of proofs
we have defined in this paper is rooted from the outset in this transversality. But what we
wish to highlight above all is that the recognition of these proofs, particularly the
instrumental ones, implies an opening on the different ways that proof can get in
mathematical work in school. In a culture of this difference, Pólya (1973) said:
When the solution that we have finally obtained is long and involved, we naturally
suspect that there is some clearer and less roundabout solution: Can you derive the result
differently? Can you see it at a glance? Yet even if we have succeeded in finding a satisfactory
solution we may still be interested in finding another solution. We desire to convince
ourselves of the validity of a theoretical result by two different derivations as we desire to
perceive a material object through two different senses. Having found a proof, we wish to
find another proof as we wish to touch an object after having seen it. (❡) Two proofs are
better than one. "It is safe riding at two anchors”. (pp. 61-62)
If it is obvious that with his heuristics, Pólya was an apostle of the combined use of
semiotics and discursive in his reasonings. We believe he would have been very
comfortable with the concept of the discursive-graphic proofs. But he rarely addressed
the issue of working with instruments, except in a classical perspective as «shall we draw
the figures exactly or approximately, with instruments or free-hand?» (p. 105). Even so,
13
This international newsletter on the teaching and learning of mathematical proof, whose current name is
simply Preuve, is available at http://www.lettredelapreuve.org (ISSN 1292-8763).
25
knowing the influence that his book
14
has had among some physicists and computer
scientists, we can believe that if he had had a computer on hand easily as today, he would
no doubt have found a way to engage us wisely in the instrumental opportunity:
The expert has, perhaps, no more ideas than the inexperienced, but appreciates more what
he has and uses it better. A wise man will make more opportunities than he finds. A wise man
will make tools of what comes to hand. A wise man turns chance into good fortune. Or, possibly,
the advantage of the expert is that he is continually on the lookout for opportunities. Have
an eye to the main chance (p. 224).
Following Pólya's footsteps and willingly adding the discursive-graphic proofs, we
deduce from this that the integration of instrumental proofs in mathematical work in
school is a desirable enrichment of the means of validation, and in doing so an approach
to the expert's work. If these «other proofs», compared to traditional written
demonstrations in natural language, appear to involve more obviously heuristic
characteristics, then the cross-cutting nature of proving could be formulated through the
geneses of mathematical work, their coordination and the fibrations weaving the working
space. Furthermore, while the ability to prove at school depends on the students'
resourcefulness, it also depends on the learning opportunities they have been offered, the
quality of the milieu chosen by the teacher and the habits conveyed by the didactic
contract, particularly the tolerance to possible variations in the coordination of geneses.
For the notion of valence of proof activity to become relevant, it must be understood that
mathematical proofs intervene in many different tasks and that the epistemic necessity is
likely to vary greatly from one task to another. If mechanical proofs and algorithmic
proofs serve very well this idea of valence, more exploration has to be done about their
usefulness for teaching and research in mathematics didactics. If two proofs are better
than one, now imagine three proofs!
Acknowledgments
We wish sincerely to thank Prof. Annette Braconne-Michoux for her devoted and far-
sighted work of linguistic review.
Appendix A - Area Partition Activity
In an activity of dynamic geometry, de Villiers (2018)
15
proposes the study of situations
which allow us to show that «to the working mathematician, proof is not merely a means
of verifying an already discovered result, but often also a means of exploring, analyzing,
discovering and inventing new results». It begins by proposing an interactive version of
the proof without words presented in Section 2.4, automatically providing the measures
on both sides of equality:
14
The first English translation of How to solve it still dates from 1945.
15
From http://dynamicmathematicslearning.com/area-parallelogram-partition-richard-theorem.html.
26
This is a proof that is both mechanical and discursive-graphic, as is our considerations
around Fig. 9. At the interface of the situation, the author adds a help button that
participates in the devolution of the problem (hint) and a generalization button that
allows the user to move to the next situation (pentagon). In fact, the user can always
move to a more general situation (pentagon or to another (k+1)-gon, until an octagon)
without even having solved the previous problem (parallelogram or k-gon):
This dynamicity of the activity makes it possible to identify invariants in the equality of
areas of figures and to reinvest them in a set of proof problems:
Carefully reflect on your proof, and consider how this same proof can also apply to a
certain type of pentagon, hexagon, etc. Make generalizations and check your generalized
conjectures by clicking on the Link buttons on the right to go to pentagons, hexagons, etc.
with a similar area partition property. (extract from the activity instructions)
It would then be a particularly rich activity of proof which also makes it possible to
engage an algorithmic proof.
27
Appendix B - Transcription of text from dense figures
This is the sequential transcription of Trahan's approach structured in Fig. 6.
Step 1
Theorem (not surprising)
If you cut a polygon into a finite number of pieces to reform another, then both polygons will have
the same area.
Theorem (more surprising)
Should any two polygons have the same area, it is possible to cut the first into a finite number of
pieces to then form the second.
Step 2
Propositions (program)
1. Any polygon can be cut into a finite number of triangles.
2. Any triangle can be cut to form a rectangle.
3. Any rectangle can be cut to form a square.
4. Any pair of squares can be cut to form a square.
Resolution (breakdown by case)
1. Demonstration in two cases: convex polygon and concave polygon.
2. Check the four corners of the rectangle and check the «joints» (alignment of points).
3. In a rectangle L by ℓ: case ℓ < L ≤ 4 ℓ, checking of the angles, of the «joints» then sides; case
L > 4 ℓ, check also lengths and sides.
4. Verification of angles, "joints" then sides.
Conclusion
Any polygon can be cut into a finite number of pieces to then form a square.
Step 3
Deriving proposition
If two polygons of the same surface can be cut into a finite number of pieces to each form a square,
then there is a split of the first polygon to form the second.
Solution
Superposition of the two cuttings.
Can we do better?
Yes, the demonstration makes it possible to obtain a division between two polygons, but this is not
optimal.
Verification with interesting cuttings
(demonstrated in other research work)
• Square in equilateral triangle.
• Square in nonagon.
• Some cutting of the pentagon.
Step 4
New situations (to pose and solve)
• Generalization: curved surface?
• Generalizations in 3 dimensions
• (…)
Hilbert's 3rd problem: given two polyhedra of equal volume, can we cut the first polyhedron into
polyhedra and bring them together to form the second polyhedron?
28
- No: Dehn found an invariant (Dehn invariant) that is preserved during a cut; the cube and
the tetrahedron do not have the same invariant.
- Some possible cuttings...
References
Alsina, C., & Nelsen, R. (2006). Math Made Visual. Creating Images for Understanding Mathematics.
Washington: the Mathematical Association of America.
Appel, K, & Haken, W. (1977). The Solution of the Four-Color-Map Problem. Scientific American,
237(4), 108-121. doi:10.1038/scientificamerican1077-108
Arzarello, F. (2006). Semiosis as a multimodal process. Revista latinoamericana de investigación en
matemática educativa, 9(1), 267–299.
Balacheff, N. (1987). Processus de preuve et situations de validation. Educational Studies in
Mathematics, 18(2), 147-176.
Balacheff, N., & Margolinas, C. (2005). Ck¢, modèle de connaissances pour le calcul de situations
didactiques. In A. Mercier, & C. Margolinas (Eds.), Balises pour la didactique des mathématiques
(pp. 75-106). Grenoble: La Pensée Sauvage.
Barbin, É., Duval, R., Giorgiutti, I., Houdebine, J. & Laborde, C. (2001). Produire et lire des textes de
démonstration. Paris : Éditions Ellipses.
Bartle, R.G. & Sherbert, D.R. (2011). Introduction to real analysis, 4th ed. John Wiley & Sons.
Bartolini Bussi M.G., & Maschietto M. (2005). Macchine Matematiche: dalla storia alla scuola. Springer.
Blum, W., & Leiß, D. (2007). How do students and teachers deal with modelling problems? In C.
Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling (ICTMA12):
Education, Engineering and Economics (pp. 222-231). Chichester, UK: Horwood Publishing.
DOI 10.1533/9780857099419.5.221
Borromeo-Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process.
ZDM - The International Journal on Mathematics Education, 38(2), 86-95 (DOI:
10.1007/BF02655883).
Boutin, F. (2017). Le cahier Transmath d’algorithmique Cycle 4 (5e/4e/3e). Nathan
Brousseau, G. (1998). Théorie des situations didactiques. La Pensée Sauvage: Grenoble.
Brousseau, G. (2004). Introduction to the study of the teaching of reasoning and proof: paradoxes.
La lettre de la preuve – International newsletter on the teaching and learning of mathematical proof,
retrieved April 17, 2018 from
http://www.lettredelapreuve.org/OldPreuve/Newsletter/04Ete/04EteThemeUK.html.
Bryant J., & Sangwin C. (2008). How round is your circle ? Where engineering and mathematics meet.
Princeton University Press.
Castelnuovo, E., & Barra, M. (1976). Matematica nella realtà. Torino : Boringhieri.
Chevallard, Y. (1992). La transposition didactique. Grenoble : La Pensée Sauvage.
Clairaut, A.C. (1741). Élémens de Géométrie. Paris: David Fils.
Clairaut, A.C. (1881). Elements of geometry. Traduction du texte de l’édition de 1830 publié à Paris.
London : C. Kegan Paul & Co.
29
Cormack, L.B. (2017). Introduction: Practical Mathematics, Practical Mathematicians, and the Case
for Transforming the Study of Nature. In L.B. Cormack, S.A. Walton, & J.A. Schuster (Eds.)
Mathematical Practitioners and the Transformation of Natural Knowledge in Early Modern Europe
(pp. 1-8), Studies in History and Philosophy of Science 45, DOI 10.1007/978-3-319-49430-2_1
Coutat, S., Laborde, C., & Richard, P. R. (2016). L’apprentissage instrumenté de propriétés en
géométrie : propédeutique à l’acquisition d’une compétence de démonstration. Educational
Studies in Mathematics, 1-27. DOI 10.1007/s10649-016-9684-9.
Derouet, C. (2016). La fonction de densité au carrefour entre probabilités et analyse en terminale S : Étude
de la conception et de la mise en oeuvre de tâches d’introduction articulant lois à densité et calcul
intégral (Thèse de doctorat, Sorbonne Paris Cité). Tiré de https://tel.archives-
ouvertes.fr/tel-01431913/document.
Dreyfus, T., Artigue, M., Potari, D., Prediger, S. & Ruthven, K. (2018). Developing research in
mathematics education - twenty years of communication, cooperation and collaboration in
Europe. Oxon, UK: Routledge (ISBN: 978-1-138-08027-0).
Dupré, S. (2006). Visualization in Renaissance optics: the function of geometrical diagrams and
pictures in the transmission of practical knowledge. In S. Kusukawa & I. Maclean (Eds)
Transmitting knowledge : words, images, and instruments in early modern Europe (pp. 11–39).
Oxford University Press.
Dupré, S. (2017) The Making of Practical Optics: Mathematica Practitioners’ Appropriation of
Optical Knowledge Between Theory and Practice. In L.B. Cormack, S.A. Walton, & J.A.
Schuster (Eds.) Mathematical Practitioners and the Transformation of Natural Knowledge in Early
Modern Europe (pp. 131-148), Studies in History and Philosophy of Science 45, DOI
10.1007/978-3-319-49430-2_7
Duval, R. (1995). Sémiosis et pensée humaine: registre sémiotique et apprentissages intellectuels. Berne:
Peter Lang.
Font , L., Richard, P.R. & Gagnon, M. (2018). Improving QED-Tutrix by Automating the generation
of Proofs. In P. Quaresma and W. Neuper (Eds.): Proceedings 6th International Workshop on
Theorem proving components for Educational software, (ThEdu’17), Electronic Proceedings in
Theoretical Computer Science, 267, 38-58 (DOI:10.4204/EPTCS.267.3).
Gray, E., & Tall, D. (1994). Duality, Ambiguity and Flexibility: A Proceptual View of Simple
Arithmetic. The Journal for Research in Mathematics Education, 26(2), 115-141.
Haspekian, M. (2011). The co-construction of a mathematical and a didactical instrument. In M.
Pytlak, T. Rowland, & E. Swoboda (Eds), Proceedings of the Seventh Congress of the European
Society for Research in Mathematics Education (CERME 7) (pp. 2298–2307). Rzeszów, Poland.
Hoffmann, D.L., Standish, C.D., García-Diez, M., Pettitt, P.B., Milton, J.A., Zilhão, J., … Pike, W.G.
(2018). U-Th dating of carbonate crusts reveals Neandertal origin of Iberian cave art. Science,
23, Vol. 359, Issue 6378, pp. 912-915. (DOI: 10.1126/science.aap7778).
Keller, A.G. (2017) Machines as Mathematical Instruments. In L.B. Cormack, S.A. Walton, & J.A.
Schuster (Eds.) Mathematical Practitioners and the Transformation of Natural Knowledge in Early
Modern Europe (pp. 115-127), Studies in History and Philosophy of Science 45, DOI
0.1007/978-3-319-49430-2_6
Keller, O. (2004). Aux origines de la géométrie: le Paléolithique, le monde des chasseurs-cueilleurs. Paris :
Vuibert.
Kovács, Z. (2018). Motion with LEGOs and Dynamic Geometry. Virginia mathematic teacher, vol. 44,
no. 2, pp. 43-48.
Kovács, Z., Recio, T. & Vélez, M.P. (2018a). Using Automated Reasoning Tools in GeoGebra in the
Teaching and Learning of Proving in Geometry. International Journal of Technology in
Mathematics Education, vol. 25, no. 2, pp. 33-50 (DOI: 10.1564/tme_v25.2.03).
30
Kovács, Z., Recio, T., & Vélez, M.P. (2018b). Detecting truth, just on parts, in automated reasoning
in geometry. In F. Botana, F. Gago & M. Ladra (Eds.): Proceeding of the 24th Conference of
applications of computer algebra, 32-35.
Kovács, Z., Richard, P.R., Recio, T. & Vélez, M.P. (2017). GeoGebra Automated Reasoning Tools: A
tutorial with examples. In G. Aldon, and J. Trgalova (Eds.): Proceedings of the 13th
International Conference on Technology in Mathematics Teaching, 400-404. https://hal.archives-
ouvertes.fr/hal-01632970.
Krieger, M.H. (2004). Some of What Mathematicians Do. Notices of the AMS, 51(10), pp. 1 226-1 230.
Kuzniak, A. & Richard, P.R. (2014). Espaces de travail mathématique. Point de vues et perspectives.
Revista latinoamericana de investigación en matemática educativa 17.4(I), 5-40.
Kuzniak, A., Richard P.R. & Michael-Chrysanthou, P. (2018). Chapter 1. From geometrical thinking
to geometrical working competencies. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger & K.
Ruthven (Eds.), Developing research in mathematics education - twenty years of communication,
cooperation and collaboration in Europe. Oxon, UK: Routledge - New Perspectives on Research
in Mathematics Education series, Vol. 1., pp. 8-22. (ISBN: 978-1-138-08027-0)
Kuzniak, A., Tanguay, D. & Elia, I. (2016). Mathematical Working Spaces in schooling: an
introduction. ZDM - The International Journal on Mathematics Education, 48(6), 721–737 (DOI
10.1007/s11858-016-0812-x).
Lagrange, J.B., Recio, T., Richard, P.R., & Vivier, L. (2017). Synthesis of Topic 2—Specificities of
tools and signs in the mathematical work. In I. Gómez-Chacón, A. Kuzniak, K.
Nikolantonakis, P.R. Richard & L. Vivier (Eds.), Actes du 5e symposium Espaces de Travail
Mathématique (ETM 5) (pp. 207-239). Greece: University of Western Macedonia.
Laval, D. (2018). L’algorithmique au lycée entre développement de savoirs spécifiques et usage
dans différents domaines mathématiques. Thèse de doctorat. Sorbonne Paris Cité.
Leduc, N. (2016). QED-Tutrix : système tutoriel intelligent pour l'accompagnement des élèves en situation
de résolution de problèmes de démonstration en géométrie plane (Thèse de doctorat, École
Polytechnique de Montréal). Tiré de https://publications.polymtl.ca/2450/.
Leduc, N., Tessier-Baillargeon, M., Corbeil, J.P., Richard, P.R., & Gagnon, M. (2017). Étude
prospective d’un système tutoriel à l’aide du modèle des espaces de travail mathématique.
In I. Gómez-Chacón, A. Kuzniak, K. Nikolantonakis, P.R. Richard & L. Vivier (Eds.), Actes
du 5e symposium Espaces de Travail Mathématique (ETM 5) (pp. 281-295). Greece: University of
Western Macedonia.
Modeste, S. (2012). Enseigner l’algorithme pour quoi ? Quelles nouvelles questions pour les
mathématiques ? Quels apports pour l’apprentissage de la preuve ? (Thèse de doctorat, Université
de Grenoble). Tiré de https://tel.archives-ouvertes.fr/tel-00783294/document.
Modeste, S., Gravier, S., & Ouvrier-Buffet, C. (2010). Algorithmique et apprentissage de la preuve.
Repères IREM, 79, 51-72.
Moutet, L. (2016). Diagrammes et théorie de la relativité restreinte : Une ingénierie didactique (Thèse de
doctorat, Sorbonne Paris Cité). Tiré de https://hal.archives-ouvertes.fr/tel-
01611332/document.
Netz, W., & Noel, W. (2008). Le codex d’Archimède - Les secrets du manuscrit le plus célèbre de la science.
Paris : JC Lattès.
Polya, G. (1973). How to solve it. A new aspect of mathematical method. New Jersey : Princeton
University Press.
Rabardel, P. (1995). Les hommes et les technologies, une approche cognitive des instruments contemporains.
Paris: Armand Colin.
Recio, T., Richard, P.R. & Vivier, L. (2015). Synthesis of Topic 2—Specific features of tools and signs
in the mathematical work. In I. Gómez-Chacón, J. Escribano, A. Kuzniak, & P. R. Richard
31
(Eds.), Actes du 4e symposium Espaces de Travail Mathématique (ETM 4) (pp. 197-226). Spain:
Universidad Complutense de Madrid.
Richard, P.R. (2004a). Raisonnement et stratégies de preuve dans l’enseignement des mathématiques.
Berne: Peter Lang.
Richard, P.R. (2004b). L’inférence figurale: Un pas de raisonnement discursivo-graphique.
Educational Studies in Mathematics, 57(2), 229-263.
Richard, P.R., Gagnon, M. & Fortuny, J.M. (2018). Chapter 20. Connectedness of Problems and
Impasse Resolution in the Solving Process in Geometry: A Major Educational Challenge. In
P. Herbst, U.H. Cheah, P.R. Richard & K. Jones (Eds) International Perspectives on the Teaching
and Learning of Geometry in Secondary Schools, 19 pages. Cham, Switzerland: Springer (ISBN:
978-3-319-77475-6).
Richard, P.R., Oller, A.M. & Meavilla, V. (2016). The Concept of Proof in the Light of Mathematical
Work. ZDM - The International Journal on Mathematics Education, 48(6), 843–859 (DOI:
10.1007/s11858-016-0805-9).
Tanguay, D., & Geeraerts, L. (2014). Conjecture, postulats et vérifications expérimentales dans le
paradigme du géomètre physicien : comment intégrer le travail avec des logiciels de
géométrie dynamique ? Revista latinoamericana de investigación en matemática educativa 17.4(II),
287-302.
Tanguay, D., & Venant, F. (2016). The semiotic and conceptual genesis of angle. ZDM - The
International Journal on Mathematics Education, 48(6), 875–894 (DOI: 10.1007/s11858-016-0789-
5).
Tanguay, D., Kuzniak, A., & Gagatsis, A. (2015). Synthesis of Topic 1—The mathematical work and
mathematical working spaces. In I. Gómez-Chacón, J. Escribano, A. Kuzniak, & P. R.
Richard (Eds.), Actes du 4e symposium Espaces de Travail Mathématique (ETM 4) (pp. 20-38).
Spain: Universidad Complutense de Madrid.
Tessier-Baillargeon, M. (2015). GeoGebraTUTOR : Développement d’un système tutoriel autonome pour
l’accompagnement d’élèves en situation de résolution de problèmes de démonstration en géométrie
plane et genèse d’un espace de travail géométrique idoine (Thèse de doctorat, Université de
Montréal). Tiré de
https://papyrus.bib.umontreal.ca/xmlui/bitstream/handle/1866/15902/Tessier-
Baillargeon_Michele_2016_these.pdf.
Tessier-Baillargeon, M., Leduc, N., Richard, P.R. et Gagnon, M. (2017). Étude comparative de
systèmes tutoriels pour l’exercice de la démonstration en géométrie. Annales de didactique et
de sciences cognitives, 22, 91-117. IREM de Strasbourg: Université Louis Pasteur.
Trahan, A. (2016, October). Casse-tête de polygones. Workshop session presented at the 60th Congress
of the Association Mathématique du Québec under the theme Des Mathématiques
Surprenantes, Québec, Québec.
Trouche, L. (2005). Construction et conduite des instruments dans les apprentissages
mathématiques : nécessité des orchestrations. Recherche en Didactique des Mathématiques, 25,
91-138.
Venant, F. (2018). Programmer les mathématiques : la pensé informatique à l’école primaire.
Bulletin AMQ, vol. LVIII, n° 3, pp. 57-70.
Villiers, M. (1993). El Papel y la Función de la demostración en matemáticas. Epsilon, 26 , 15-30.
Vitrac, B. (1992). À propos de la chronologie des œuvres d’Archimède. In J.Y. Guillaumin
(Ed), Mathématiques dans l’Antiquité (pp. 59-93). Publications de l’Université de Saint-Etienne.
Weber, K. & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in
Mathematics, 56, 209-234.
