Julie Stalder

What is the place of examples in the teaching of abstract algebra?

Keywords: Abstract Algebra, Anthropological Theory of the Didactics, Ideal of a ring, Epistemology, Structuralist praxeologies Algèbre abstraite, Théorie Anthropologique du Didactique, Idéal d'un anneau, Épistémologie, Praxéologies structuralistes

Teaching abstract algebra, seen as the study of structures and properties of structures, at a university level appears to be a challenge for both students and faculty. Some professors describe this passage through abstraction for the student as a "killing game", an "impassable wall" or a "leap through abstraction". In this paper, based on a case study (Candy, 2020) we investigate the choices of professors teaching abstract algebra at university. These professors were chosen because they teach abstract algebra at the university in France, and they gave us access to their course corpus and agreed to be interviewed. In this article, we choose to study in particular the teaching of the concept of ideal.

We rely on an epistemological analysis to highlight its central role in the construction of abstract algebra. Then, using the Anthropological Theory of the Didactic (Chevallard, 1999), we will try to specify the place of examples and exercises in the students' praxis.

Introduction

Our study focuses on the teaching of abstract algebra, seen as the study of structures and properties of structures, at the university. To obtain results on the place of examples in the teaching of abstract algebra at a fine granularity level, we have chosen to focus our research on the teaching of the concept of ideal. The epistemological analysis which follows this introduction will aim at putting forward the relevance of this choice. Before that, it appears important in this introduction to highlight the previous works on the teaching of algebraic concepts at the university.

The notion of Formalising, Unifying and Generalising (FUG) concepts is introduced by [START_REF] Robinet | Ingénierie didactique de l'élémentaire au supérieur[END_REF]. Robert (1987, p. 3) points out that, in higher education, "the concepts to be acquired are generalising, unifying and formalising concepts: they have been developed by mathematicians after many particular problems have been solved (in a particular way) and often correspond to a new formalisation". The notion of Simplifier in the term FUGS appeared a little later with the work of Dorier on linear algebra [START_REF] Dorier | Analyse historique de l'émergence des concepts élémentaires d'algèbre linéaire[END_REF]. Following a historical analysis of the emergence of the concepts of linear algebra, Dorier concludes that The use of the axiomatic theory of vector spaces as a privileged (or even unique) framework for the study of concepts and results in linear algebra is a recent discovery and choice. This choice essentially obeys a concern for organisation, unification and simplification of the study of different fields with similar tools and methods. (Dorier, 1990, p. 89) [START_REF] Robert | Didactique dans l'enseignement supérieur : une démarche[END_REF] mentions the fact that not all concepts have the same status, and these differences certainly entail differences in acquisition. She thus specifies that "linear algebra, convergences, R, have a formalising, unifying and generalising status (c.f. Robinet); in fact, historically, these notions have been introduced, it seems to us, as an outcome, whereas the solutions of part of the problems concerned were already working well in each particular case" (Robert, 1986, p. 9, our translation). She then points to the difficulty of finding open-ended problems that would allow students to mobilise the concept as a tool before the course.

Robert and Robinet, in a study of linear algebra (Robert & Robinet, 1987, our translation), point out that "for a majority of students, finally, linear algebra is nothing but a catalogue of very abstruse notions that they are unable to imagine; moreover, they are submerged under an avalanche of new words, new symbols, new definitions and new theorems". Thus, according to these authors, linear algebra concepts that are FUGS concepts "drown" students. They find themselves trying to make sense of these concepts whose formalisation is already difficult for them. The above quote suggests that students only see the axiomatized side of these concepts and cannot immediately access the simplification, unification and generalization that these concepts provide. This observation by Robert and Robinet is, in our opinion, valid for abstract algebra, whose formalisation characteristics are the same as for linear algebra.

For [START_REF] Robert | De quelques spécificités de l'enseignement des mathématiques dans l'enseignement post-obligatoire[END_REF], the FUGS status of a concept is a characteristic of concepts in postcompulsory education, which implies that at the time of introduction of the concept, it is very difficult to find problems, accessible to students, where the concepts intervene in an optimal way. She, therefore, proposes that, in order to teach these concepts, one should give, after the course, problems that allow students to work on several frameworks in which their knowledge is unequal. We also retain the idea of the importance of reinvestment, including in problems apparently relating to another field, allowing the acquisition of tools that are available and not only mobilisable. (Robert, 1987, p. 12, our translation) In [START_REF] Jovignot | L'analyse épistémologique du concept d'idéal et ses apports à l'étude didactique[END_REF] we had shown that the concept of ideal belongs to this class of FUGS knowledge. This research on FUGS concepts and the teaching of linear algebra highlights the complexity of finding fundamental situations to introduce these concepts due to their epistemological nature.

Regarding abstract algebra, Simpson and Stehliková (2006) explain that the power of much abstract algebra comes from this shift -working from the generality of the definitions to discover further relationships which tell us about properties which may have been hidden in examples of the structures. (Simpson & Stehlíková, 2006, p. 351) In their article, they focus on the shift of attention from the familiar and specific of objects and operations to the interrelationships between objects expressed by the properties of operations (Simpson & Stehlíková, 2006, p. 352). This is what they call 'apprehending the structure'.

Durand-Guerrier, [START_REF] Durand-Guerrier | Définitions et exemples : prérequis pour l'apprentissage de l'algèbre moderne[END_REF] provide some insights into this structure. The article presents the analysis of a questionnaire proposed to third year undergraduate students at the University of Montpellier 2. After an epistemological analysis, they define the notion of paradigmatic example: "an example (or a class of examples) is paradigmatic when it allows us to identify a set of concepts and operative processes that have a general scope and can be implemented on other examples according to the same pattern" (Durand-Guerrier et al., 2015, p. 7, our translation).

To illustrate this notion, they cite or as paradigmatic examples of Euclidean rings or as paradigmatic for the extension of the arithmetic of and the construction of an abstract theory of ring divisibility. According to the authors: with regard to the learning of algebraic structures, the setting up of these paradigmatic examples is all the more fundamental as the conceptual mode of thinking that is mobilised by abstract algebra constitutes a paradigm shift relative to previous algebraic practices (historically, the algebra of equations and in the school context, those of high school) (Durand-Guerrier et al., 2015, p. 7, our translation) The results of their study include the following (Durand-Guerrier et al., 2015, pp. 42-43): the articulation between abstract formal definitions and concrete object domains is difficult for students. They are, for example, not able to produce complex examples if they have not worked on them recently. The authors conclude that "the epistemological analysis highlights the fundamental role of the articulation between axiomatic definitions and examples". (Durand-Guerrier et al., 2015, p. 44, our translation).

Fukawa-Connelly and Newton (2014) investigate the place of examples in the teaching of abstract algebra. Their paper begins with a review of the literature that justifies the importance of illustration through examples. Based on the study of a lecture-based introduction to the basic concepts of abstract algebra, they highlight, among others, the following results: first, the teacher uses several group examples by varying the different group classes (e.g. commutative or noncommutative, finite or infinite groups, or counterexamples of groups). Thus, according to Fukawa-Connelly and Newton (2014, p. 342): "the students had the opportunity to develop a well-defined example neighbourhood as they were exposed to a collection of examples in which many aspects of the examples were allowed to vary". They also show that in his teaching, the teacher under study uses examples in different functions: firstly to illustrate definitions and also to instantiate propositions after introducing them. He [the teacher] also […] introduced the idea via a concrete example by asking, "What properties of the set and operation are necessary in order to be able to solve this equation?" Finally, Dr. P used examples to motivate claims (he was not observed generalizing from examples, so we have chosen "motivate" rather than "create") before giving their formal statement, such as when he used solving equations in the integers to motivate the question of whether cancellation is possible in groups (Fukawa-Connelly & Newton, 2014, p. 344). This research thus illustrates the importance of examples in the teaching of abstract algebra. However, they are limited to examples given by the teacher. In our study, we aim, in a complementary way, to study the use of examples in the problems the students have to solve (type of tasks and praxis) and not only in the explanation given by lecturers (logos).

The research discussed in this section has pointed at the fact that finding problems accessible to students that make sense of the concept of ideal appears complex, due to its FUGS nature. However, the works of [START_REF] Durand-Guerrier | Définitions et exemples : prérequis pour l'apprentissage de l'algèbre moderne[END_REF] and Fukawa-Connelly and Newton (2014) highlight the role of examples in the learning of abstract algebra. This leads to our research question in this article: what is the place of examples, in the activities proposed to students for the teaching of abstract algebra in Bachelor's degrees in France?

To answer this question we will start with an epistemological analysis of the concept of ideal, which will allow us to situate the role of examples in the birth of this concept and more generally of abstract algebra. Then, we will present the theoretical framework, and in particular the elements specific to abstract algebra. Based on this framework, we will present, in the following part, the methodology of the study as well as the analyses of the corpus. Finally, in the conclusion, we will provide some answers to our research question.

Epistemological analysis

In this part, we are be able to retrace a complete epistemological analysis (that the reader can find in [START_REF] Candy | Étude de la transposition didactique du concept d'idéal : écologie des savoirs et problématique de l'entrée dans la pensée structuraliste en France et en Suisse romande[END_REF])) but we will cite the main milestones of the birth of the concept of ideal and more widely of the abstract algebra. We chose to use [START_REF] Corry | Modern Algebra and the Rise of Mathematical Structures[END_REF] as main resource because, in his book, he highlighted the role of the concept of ideal. However, the reader must be aware that this choice implies a particular point of view on the construction of the abstract algebra and thus does not aim at a faithful historical reconstruction but allows us to throw light on the construction of the concept of ideal in interrelation with other concepts of the structuralist algebra. In particular, we will link the formal character of the concepts and the place of examples in the construction of abstract algebra. This is why we will only use the study of the body of knowledge1 [START_REF] Corry | Modern Algebra and the Rise of Mathematical Structures[END_REF] and not the analyse of the image of knowledge [START_REF] Corry | Modern Algebra and the Rise of Mathematical Structures[END_REF]. This epistemological analysis intends to allow us, as mathematicians or mathematics education specialists, to get rid of the "illusion of transparency" [START_REF] Artigue | Épistémologie et didactique[END_REF] which surrounds the concepts of modern algebra and the concept of ideal in particular, which our contemporary use tends to naturalize. This illusion of transparency, in the case of ideals, leads teachers to think that the concept, as they currently use it, has always been used and formalised as such.

2.A. Formal character

We will see in the following sections that the teachers in this study sometimes go so far as to teach the formal character of concepts first, as enabling the capacity for abstraction. This part of the epistemological analysis wishes to highlight that having the formal definition alone does not appear to be sufficient to motivate the use of the structure in the mathematical uses. Corry (2004, p. 32) explains, "the fact that many abstract definitions of mathematical concepts had already been formulated by that time seems not to have in itself sufficed to re-orient research in the abstract direction". He adds that the use of axiomatic allows for the effective treatment of structuralist theories, but that the developments of these theories […] were not directly connected with the use of the modern axiomatic method or of abstract formulations in general, but rather they were dictated by the solutions proposed for the existing problems of the discipline. It was the fact that similar sorts of problems arose in the different disciplines now included under the heading of modern algebra, and that at the same time they were addressed from an increasingly common perspective, that led to the unified exposition of van der Waerden. The full adoption of the abstract axiomatic formulation was only a part of a wider development. (Corry, 2004, p. 52) In order to better apprehend what guides the structures, we will concentrate the epistemological analysis on the development of the concept of ideal. This will allow us, however, to obtain results that can be applied more widely to abstract algebra because, as Corry explains about the case of ideals From the beginning of its development, its interplay with other, newly introduced algebraic concepts, such as fields, modules, groups, lattices, and polynomial rings, was clearly manifest. In fact, this interplay was central to the main concerns of the theory. Thus the theory of ideals had a rich pre-history, characterized by strong interconnections with other incipient structures, even before turning into a useful tool of central importance for the theory of abstract rings. (Corry, 2004, p. 15)

2.B. The birth of the concept of ideal and abstract algebra

According to Corry (2004, p. 85), it was in 1844 that Kummer became interested in the set (now) called the cyclotomic field. He discovered that factoring the number where and is a 23 th root of the unity does not have a unique factorisation into prime factors. In order to overcome this uniqueness, he developed the theory of ideal numbers.

Inspired by Kummer's work, Dedekind, considering will extract number systems whose properties are the following: "-The sums and differences of any two numbers of system will always be numbers of the same system .

-Any product of a number of system and a number of system is a number of system . [...] We shall now call any system , composed of numbers of the domain and enjoying the two properties I and II, an ideal." (Dedekind, 1877, pp. 84-85, our translation) Thus, Dedekind gives a definition of the concept of ideal which is almost the modern definition we use except that it has been formalised by relying on the properties of . In the rest of the paper he will extend the definition of the concept to bodies of numbers. Thus, this extension allows, for example, the simplification of the proof of the quadratic reciprocity law.

Dedekind wrote four versions of his theory of ideals. According to [START_REF] Corry | Modern Algebra and the Rise of Mathematical Structures[END_REF], Dedekind's formulation becomes more and more structuralist. Indeed, his first editions have theorems stated in terms of elements of the ideals whereas from the second version the treatment is done in terms of ideals and no longer in terms of elements that compose them. Finally, in the 4 th version Dedekind will add a part on the Galois theory. Thus, in his research, Dedekind works more at the level of sets of numbers than of numbers themselves. However, he takes support on the properties of particular sets. Major modifications will still be necessary to arrive at the concept of ideal as a structure as we know it today.

Hilbert's axiomatic programme will be part of these changes. In 1893, Hilbert published Grundlagen der Geometrie (Hilbert, 1899, cited by [START_REF] Corry | Modern Algebra and the Rise of Mathematical Structures[END_REF] which had a huge impact on the image of mathematics and made his name associated with the modern axiomatic method. He presents his axioms for geometry formulated for three systems of indefinite objects (points, lines and planes) and the axioms establish the mutual relations that should be satisfied by these objects.

Moreover, the impact of Hilbert's axiomatic research coupled with the "formalism" associated with his name in the framework of the so-called "foundational crisis" of the 1930s have been occasionally seen as promoting the view of mathematics as an empty, formal game. But as a matter of fact, Hilbert's own axiomatic research was never guided by such a view, and he often opposed it explicitly. Hilbert's own conception of axiomatics did not convey or encourage the formulation of arbitrary, abstract axiomatic systems as starting points for mathematical research. Rather, his work was directly motivated by the need to better define and understand existing mathematical and scientific theories. In Hilbert's view, the definition of systems of abstract axioms and their analysis following the above described guidelines were meant to be conducted for established and elaborated mathematical entities. (Corry, 2004, p. 161) It was not Hilbert who changed the image of algebra, although he apparently had the cards in his hand to do so. Two subsequent pieces of research played a major role in the construction of the image of algebra as a study of structures. These were Steinitz's work on the theory of abstract fields and Fraenkel's work on the theory of abstract rings. Indeed, according to [START_REF] Corry | Modern Algebra and the Rise of Mathematical Structures[END_REF], Steinitz was the first to make an abstract investigation of the concept of field, guided by Hensel and Fraenkel's p-adic numbers theory. He was also the first to make systematic use of set theory in his research on algebra. Thus, we witness a change of point of view: it is no longer necessary to start from number systems to construct these fields as was the case with Dedekind. This time, on the contrary, number systems appear as a specification of these fields. As for Fraenkel, the axiomatic treatment of g-adic number systems (where g is not prime) led him to the definition of an abstract ring in 1914 and the study of its properties. However, Fraenkel did not make the link with Dedekind's work and in particular, the concepts of ideals or modules, which are, however, in modern algebra, related concepts.

It was in 1905 that Lasker was inspired by Dedekind's work on ideals to work in the factorisation of polynomials. He introduces the notion of a prime ideal but, again, without pointing out the connection with a notion introduced by Dedekind. This notion explained that, in some cases, these ideals might not be powers of prime ideals contrary to what happens in the algebraic numbers. [START_REF] Corry | Modern Algebra and the Rise of Mathematical Structures[END_REF] points out that this proves that at that time the concept of ideal, for example, did not yet play a unifying role. The theory of polynomials was still independent of the theory of algebraic numbers.

Macaulay, in 1913, published results in Mathematische Annalen which refined those of Lasker by proving the uniqueness of the decomposition of polynomial ideals into primary ideals. Thus, the decomposition theorem of polynomial ideals into prime ideals is known as the Lasker-Macaulay theorem. However, as we have just seen, those studies still contextualised the concept of ideal and not investigated it in a more general way.

It was not until Emmy Noether, whose work on ideals began according to [START_REF] Corry | Modern Algebra and the Rise of Mathematical Structures[END_REF] in 1917. She brought the concept of the ideal into the modern form we use. In 1920, Noether published his first research results with ideals. Indeed, Noether and Schmeidler proved that in differential operators the decomposition theorem can be generalised by formulating it in terms of the least common divisor of ideals. In this paper, they establish the existence and uniqueness conditions of the decomposition into ideals.

It was in 1921 that Noether gave birth to abstract algebra as we know it. It will unify and generalise theories such as those of algebraic numbers or polynomials, thanks to the abstract concept of the ring. [START_REF] Noether | Idealtheorie in Ringbereichen[END_REF] begins by writing: the aim of this article is to transfer the theorems of decomposition of the relative integers or that of decomposition into ideals in the algebraic numbers into ideals in the commutative rings and even in the general rings. (Noether, 1921, p. 25, our translation) Noether defines an ideal in an axiomatic way similar to the one we know today (Noether, 1921, p. 29). A footnote explains that she adopts Fraenkel's definition except that she deletes properties that she says are too restrictive. In this paper (i.e. [START_REF] Noether | Idealtheorie in Ringbereichen[END_REF], she will focus mainly on the study of integral domain whose ideals are all generated by a finite number of elements. She then proves the equivalence between the finiteness of ideals and the ascending chain condition. Based on this condition, Noether proves several theorems of decomposition of ideals (for more details, see [START_REF] Noether | Idealtheorie in Ringbereichen[END_REF] or [START_REF] Corry | Modern Algebra and the Rise of Mathematical Structures[END_REF]) and states a theorem (Noether, 1921, p. 50) which, according to Corry, is the major innovation of this paper:

Noether's result is therefore not only a generalized formulation of the known theorem on polynomials, but in fact a new result which could not have been attained in the particular case of polynomial rings. (Corry, 2004, p. 233).

Noether, in this paper, will then reformulate from an abstract point of view properties obtained in her paper with Schmeidler using the concept of module on a ring. She shows that an ideal is a special case of a module and then reformulates the concepts introduced on ideals in terms of modules.

In 1927, [START_REF] Noether | Abstrakter Aufbau der Idealtheorie in algebraischen Zahl-und Funktionenkörpern[END_REF] gave a general formulation of the first and second isomorphism theorems for modules over a general ring. Thus, rings are no longer only interesting to study for the factorization properties they allow to be stated. They become a sector that raises structuralist questions.

According to Mc Larty :

Noether entirely reconceived the scope of the theorems and the relation between them. Dedekind proved the homomorphism theorem for finite groups, and the isomorphism theorems for infinite additive subgroups of the complex numbers. Noether gave a uniform method of proving isomorphism theorems from homomorphism theorems for many categories of structuresall groups, commutative groups, groups or commutative groups with a given domain of operators, all rings, commutative rings, rings with operators, and more. (Mclarty, 2006, p. 226) It is for this work that Noether is considered the mother of structuralist algebra and the creator of the concept of ideal as it is used today.

2.C. Epistemological conclusions

This analysis already allows us to get rid of the illusion of transparency that surrounds the concept of ideal. As in [START_REF] Jovignot | L'analyse épistémologique du concept d'idéal et ses apports à l'étude didactique[END_REF], the concept of ideal appears to be a FUGS concept. Moreover, the role of the formal character is well highlighted by the example of the creation of the concept of ideal and of its transformation to arrive at its modern definition. Indeed, we notice that the structuralist treatment as we know it today was not immediately accepted by the mathematic community. In addition, it is noteworthy to underline that the changes in the body of knowledge and the image of mathematics concerning structures have been long in coming. Thus, even if today this work in terms of formal axiomatic and structure has become a familiar way of thinking for us, this was not always the case, even for expert algebraists. A shift of point of view was necessary to allow for these modifications, and this was done by using examples (number fields, polynomials, differential operators). Moreover, as Lipschitz suggests to Dedekind for his theory of ideals, the presentation of the problems that the theory of ideals allows to solve could facilitate the acceptance of ideals theory. Thus, this theory, even for expert mathematicians of the time, is costly and to be accepted must be able to prove its power by showing its usefulness in many problems.

Theoretical framework

Our main theoretical framework is the anthropological theory of the didactic (ATD, [START_REF] Chevallard | L'analyse des pratiques enseignantes en théorie anthropologique du didactique[END_REF].

3.A. Praxeology

ATD postulates that all human activities and thus mathematical activities in particular, can be analysed in terms of praxeologies. A praxeology is a quadruplet where is a type of tasks to be performed; the technique to do it; the technology, i.e. a discourse that justifies the technique and theory, i.e. a discourse that justifies the technology. The block is the bloc of the praxis and the block of the logos.

To illustrate the concept of praxeologies we will analyse the following exercise extracted from our corpus.

Exercise.

Let be a ring. Let be two ideals of .

Show that is an ideal of .

Show that is an ideal of .

In this exercise, the type of tasks is "show that a subset of a ring is an ideal". The technique is "show that satisfies the properties of the definition". The technology includes the"definition of an ideal" and the theory the generalities over the ring and the fields that supports the definition of an ideal at stake. For more information on the role of theories in abstract algebra, we refer the reader to Candy (2020).

3.B. Structuralist praxeology and transition in abstract algebra

The epistemological analysis of the construction of abstract algebra, but also didactic analyses lead [START_REF] Hausberger | Structuralist Praxeologies as a Research Program on the Teaching and Learning of Abstract Algebra[END_REF] to put forward transitions that would be at stake in the construction of abstract algebra. The author (Hausberger, 2016, p. 302) introduces the notion of structuralist praxeology. It is a praxeology used in abstract algebra. It aims at the achievement of a type of tasks by placing itself at a level of generality. In doing so, it provides simplification, based on concepts and structuralist technological tools (e.g. properties or theorems about structure). In [START_REF] Hausberger | Structuralist Praxeologies as a Research Program on the Teaching and Learning of Abstract Algebra[END_REF], the author gives two steps for the development of structuralist praxeologies. For example, the type of tasks : "show that a ring is a principal ideal domain", can first be handled by elementary techniques and the associated technology will be reduced to the definition of the structures. However, in examining the technique it will become apparent that Euclidean division is the key to the demonstration. Thus, we can replace the technique by a technique : "show the existence of a Euclid algorithm". Thus, the logos block incorporates a theorem that deals with the structure "any Euclidean ring is a principal ideal domain". According to the author, this is a first transition from a praxeology to a structuralist praxeology Furthermore, the author highlights a second type of transition that leads to praxeologies whose praxis and logos lie in the abstract. Following our example, students might encounter tasks like "show that a Noetherian integral domain such that any maximal ideal is principal is a PID". [START_REF] Hausberger | Structuralist Praxeologies as a Research Program on the Teaching and Learning of Abstract Algebra[END_REF] specifies the transitions of (Winsløw, 2008) in the framework of structuralist algebra: the praxis block of praxeology is built on the block of logos of earlier structuralist praxeologies . In addition, does not show any more concrete objects. These transitions will be illustrated in more detail in the rest of the article via praxeological analyses.

Figure 1 -The two types of transition in the Abstract Algebra curriculum at the university for a student who is learning structuralist praxeologies (Hausberger, 2018, p. 89) Hausberger ( 2018) concludes that structuralist praxeologies take their place during the type 1 transition. As we will see, this is this transition that is at stake in our study.

The use of examples and the place of formal aspects in our corpus

4.A. Methodology

In Candy (2020), we studied the data of 9 French-speaking professors who teach an abstract algebra course dealing with ideals at the Bachelor's level. Among these 9 professors, 2 teach in the "classe préparatoire aux grandes écoles" mathematics and physic in France (i.e. during the second year of a curriculum that aims at allowing the integration of students into French engineering schools). We will call this institution CPGE. We will call the teachers P1 and P2. Two others are professors in the second year of the Bachelor's degree in France, we will call them P3 and P5. Finally, three are professors in the third year of a Bachelor's degree in France, we will call them P4, P6 and P7. Finally, two are professors in the first or second year of a Bachelor's degree in Switzerland. In this article, we will only analyse the data from the French teachers with the idea of studying transitions.

We carried out detailed praxeological analyses of the course materials provided by the teachers. For the most part, we had access to the lectures, tutorials and their answer keys. To carry out our praxeological analysis, we studied each exercise that deals with the concept of ideal, as well as the technological elements that accompany them. For this, we used the ATD and, more specifically, the praxeological analysis. In addition, in order to strengthen this praxeological analysis, we conducted interviews with the teachers participating in the study. These interviews were analysed using a mixed method [START_REF] Mayring | Qualitative Content Analysis: theoretical foundation, basic procedures and software solution[END_REF]. We previously fixed some coding thanks to the theoretical framework as the interviews were analysed. This analysis aimed to obtain various elements to specify the teachers' relationship to the teaching of structures or the use of examples in their lessons.

In the following, we present the results of the cross-analysis of the praxeological data and the interviews from the point of view of the management of transitions, the place of the formal and the role of examples in the teaching of the concept of ideal and more widely of abstract algebra.

4.B. Classes Préparatoires aux Grandes Écoles (CPGE)

The analyses of the official program of the mathematics-physics preparatory class show that structures are a learning objective. Indeed, we find a sector "usual algebraic structures" in which the first mention of the concept of ideal is made: "ideal of a commutative ring" in which we find the subject "divisibility relation in an integral domain". Later on, we find mention of the ideals of and and in the sector "decomposition of an endomorphism". Thus, the program suggests, a priori, a teaching strategy that starts with the general definition of the structure for its application in and as a paradigmatic example of Euclidean ring or principal ideal domain.

In the corpus of teacher P1, out of 18 tasks that deal with ideals, seven are abstract, i.e. they work in a general structure with properties. The introduction of structures is first formal and abstract and is, only later, worked on in paradigmatic examples (here and decimal numbers as Euclidean rings², and as principal ideal domain). Professor P2, on the other hand, chooses to introduce ideals first in the area of reduction of endomorphisms in order to give them motivation. Out of 24 tasks that deal with the concept of ideal, only 5 are abstract. Furthermore, the teacher uses examples as support for the construction of structures. For example, the type of task "demonstrate relations between inclusions of principal ideals and divisibility relations between generating elements of these ideals" is first presented contextualized in and then decontextualized and demonstrated in a general ring . P2 gives importance to contextualization with paradigmatic examples before a generalization or institutionalization. Indeed, this phenomenon reappears several times in the course, notably before the introduction of the Great Common Divisor (GCD). Here, the game with the "nature of the ring" leads the students, based on the demonstration carried out in the paradigmatic example of integral ring , to generalize the result to integral rings and thus demonstrate a structuralist theorem. This is a different didactic paradigm from that of P1, which, on the contrary, uses examples to illustrate technological notions stated in a general way beforehand.

In order to favour transition elements of type 1, P1 uses examples to show that a Euclidean ring is principal. Even if he does not go as far as the formal statement of this property, he will put forward, in an answer key, the spring of the proof of this property. Indeed, he will write, "the same structure is found in all Euclidean rings". He wants to accompany the type 1 transition in his students. He explains, "I try to do as much as possible demonstrations that show that nothing is specific to integers or polynomials". However, this support on paradigmatic examples remains punctual and comes only after a formal introduction of the concepts. Thus, although the official curriculum is common to both teachers, they make different didactic choices regarding the use of examples. P1 starts with an abstract and formal chapter and then contextualises later. P2, on the other hand, first deals with concepts in a contextualised way with paradigmatic examples and then allows their formal and general introduction. We could explain these differences between these teachers by their different relationships (following the sense of [START_REF] Chevallard | Approche anthropologique du rapport au savoir et didactique des mathématiques[END_REF] to algebra. Indeed, in his interview, P1 explains, "it is the discipline that makes people find maths abstract that they never understood etc. and not at all analysis or probabilities or geometry". P2 explains that "if I were to write an algebra book I would start with the problems and I wouldn't start with the structures [... that's] why we do algebra". P1 thus emphasizes the abstract character while P2 tends to emphasize the tool character of the concept of ideal. P2 will explain that he has taken a freedom, when I discussed it with certain colleagues, it didn't seem justified to them to introduce the ideal independently of a chapter of algebra [...] for the people of my generation there was still for some the notion of structures was very important. They had a spirit, for some, we'll say caricatured: mathematics had to be organised, sometimes the discussions were rather folkloric.

4.C. Second year of the Bachelor's degree

The analyses of the official programmes of the two B2 teachers show that the programme is mainly formulated in terms of properties and theorems. For example, there is mention of "GCD and LCM" or "Bezout and Gauss theorem". P3 program did not explicitly mention the concept of ideal. P4 program mentioned it but contextualised to polynomials: "ideals and GCD of polynomials". In both programmes, structures do not appear as a learning objective. The type 1 transition does not appear as a learning objective in this second year of the Bachelor's degree.

Moreover, the praxeological analysis shows that the concept of ideal intervenes in a few tasks and few are contextualized. One calls contextualized a task that works in a contextualized structure, such as as a Euclidean ring and not a general structure with properties. The first teacher (P3) has only 8 tasks that deal with ideals and only two are contextualised in (once as a paradigmatic example of a principal ideal domain and the other as a paradigmatic example of a Euclidean ring). These tasks consist mainly of a formal axiomatic game from the definitions, such as, for example, "showing that a set is an ideal". Thus, in the framework of the concept of ideal, the praxeological baggage constructed is weak. It is based on very few tasks, a majority of which are decontextualized to allow a formal axiomatic game. This could be explained in part by P3's relationship to structuralist algebra and its teaching. Indeed, the teacher explains that it is necessary for students "to know how to play with the definition". He "thinks that abstract algebra also starts with things like that where we just have an ideal, we don't really know, but it's something that has properties and we play with it [...] yes, here we don't say it's such and such polynomial times such and such polynomial, it's the set of all the things like that". The second teacher, P5, only deals with the concept of ideal in the parts of the course and there is no task in the students' praxis.

Thus, despite the absence in the syllabus of any mention of structure and the desire to bring properties in paradigmatic examples (such as ) to work, teachers have preferred to introduce a formal chapter on structures. On the one hand, we can explain it by the difficulty of using other paradigmatic examples. Indeed, P3 explained that he could not use as an example because is in the program of another module independent of his own. On the other hand, in their interviews, starting with a formal treatment of structures appears to be a way of facilitating the type 1 transition that takes place between the second and third year of the Bachelor's degree. They describe this transition as an "impassable wall" or "conceptual leap". To this end, they adopted the same didactic strategy: introduce structures in second year of bachelor, insisting on their formal character (they are "vocabulary" or "words"). Moreover, despite the programmes suggesting contextualised work, and in particular to the rings of polynomials, the few tasks present favoured decontextualized formal work. Thus, contrary to the CPGE teachers, in the second year of the Bachelor's degree, the work is essentially formal, almost detached from the examples, and this in a desire to encourage work in the abstraction to come in the third year of the Bachelor's degree.

4.D. Third year of the Bachelor's degree

P4 is the third year Bachelor teacher who teaches after P3. P4's syllabus includes both concepts and properties of concepts, but also specific examples (like or ). We could identify these examples with paradigmatic examples (of Euclidean ring or principal ideal domain) and it will be interesting to see if the teacher builds on these examples to generalise as P2 did. Secondly, this syllabus and contrary to those of the second year of the Bachelor, explicitly mentioned ideals many times. Structures are a learning objective of this course.

The praxeological analyses of P4 show that out of 45 tasks, 10 are abstract tasks. Within some of the praxeologies, we find occurrences of type of tasks first contextualised and then decontextualized. This is the case, for example, of the type of tasks "show that an ideal I of a ring A is prime". The first occurrences of this type of tasks are contextualised to particular ideals of (exercise 7, below). Then follows the type of tasks that aims at proving prime avoidance lemmas based on the definition of a prime ideal and thus extending the definition of a prime ideal in terms of inclusions of ideals (exercise 8, below). It seems to us that this last praxeology leads to a type 2 transition. Indeed, the praxeology in Exercise 8 below is abstract and the praxis block of this praxeology builds on earlier contextualised structuralist praxeologies constructed in Exercise 7.

Figure 2 -Illustration of tasks leading to type 2 transition Finally, we find the paradigmatic examples of classical principal rings of and , as well as of non principal ideals ( in , in in the exercise 7). These examples had been highlighted as paradigmatic by the teacher in the interview ("already, there are examples that are a bit 'pie in the sky', where you have to look at, the ideals of and . You have to look at or ] to show that there are rings that are not principal ideal domains"). Thus, P4 uses paradigmatic examples to illustrate formal properties or punctually to lead to a type 2 transition. P6 is the teacher who teaches after P5. P6's programme is formulated in terms of structures. There is no mention of examples except for "ring of polynomials and its universal property", where the domain of objects is considered with a view inherited from category theory (notion of "universal property"). Ideals, operations on ideals, principality properties and prime and maximal ideals come at the beginning of the course.

The interview of P6 shows that the teacher in charge of the seminars takes into account the ecology of the course to choose his exercises while being guided by the will to contextualise abstract notions "by putting concrete exercises and by making them work on familiar objects". Thus, in the praxeologies for which the student is responsible, only 9 out of 41 are abstract. One of the praxeologies, abstract by nature, is the one generated by the task type "demonstrate formal properties of operations on ideals". The task "determine the result of operations on principal ideals" is first worked on with a technique based on the dictionary, on particular cases. Then the abstract task which /will consist in the demonstration of the existence of the GCD, via ideals, in a commutative principal ring will allow the construction of a technology "in a principal ideal domain " which makes it possible to calculate the sum of two ideals by using the GCD One of the exercises of the seminars aims at proving that is a principal ideal domain by proving first that it is Euclidean. The work is done on a paradigmatic example, opening the door to a transition of type 1 when, in the course, the demonstration is carried out in the general case. The success of the transition, on this example, thus rests on a good articulation between the lectures and the seminars, because two different professors give those without necessarily a strong coordination. Moreover, in the following of the seminars, the teacher in charge do not used the structuralist theorems for the demonstrations. On the contrary, in the sectors "maximal ideals" and "prime ideals" the technology is mainly the definition of these concepts, leading to a handwork on the objects. Thus, the type 1 transition within the P6 course seems unfinished at the end of the module and this is partly due to the coordination between lectures and tutorials.

P7 is a professor who gives the first course in which ideals take place. The syllabus of the P7 course is very succinct and consists of a list of mathematical concepts related to or being structures, e.g. "homomorphisms, ideals, subrings, polynomial ring, Euclidean rings". In the part "rings and fields", the concept of ideal is explicitly mentioned. There is only one contextualisation mentioned by the programme in this part and that is "polynomial ring". The programme as it stands therefore leaves a lot of freedom to the teacher but emphasises concepts and structures. The structures are therefore a learning objective of this programme.

In the P7 teaching material, out of the 19 tasks, only 5 are abstract, the others are contextualised. The type of tasks "tasks which mobilise the ideals only via the quotient " (8 occurrences on the corpus) will be generalised then to "tasks which mobilise the ideals only via a quotient " by a play on the variables A and I. This is the only type of task for which we observe this phenomenon. For the others, the occurrences of the types of tasks are always 1 and even in the sector "first and maximal ideals", the tasks are always contextualized to . The study of the contextualisation of the tasks highlights the weak vitality of the back and forth between work on structures and work on objects. While the course introduces concepts at a high level of generality in a structuralist way, in the exercises the work is mainly done at the structuralist level 1. Moreover, the dissemination of tasks between the types of tasks will lead to a difficulty in systematising the techniques. We linked this dissemination to one of his learning objectives with regard to ideals: P7 expects students to reach what he calls the "creative" level, in the sense that "the student appropriates the reasoning and is able to modify it to make another".

Half of the structuralist praxeologies have a definition in their technology and require "handwritten" verifications. Moreover, even if the other praxeologies have structuralist theorems as technology, the unique occurrences of the tasks of these praxeologies make it difficult to understand the power of these theorems from a generalizing point of view. Thus, it seems to us that there are possible impediments to a type 1 transition. The interview with P7 suggests that the lack of time given the official programme could explain this phenomenon.

Thanks to the praxeological analysis and the interviews, we can conclude that in the second year of the Bachelor's degree, structures appear as a learning objective in their own right for some teachers. The type 1 transition takes place between the first and second year of the Bachelor's degree. Instead of being formulated in terms of objects, the programmes are this time treated in terms of structure. All third-year Bachelor teachers in this corpus start by formally introducing structures and then using them in contextualised praxeologies. The interviews show that this is strongly related to the teachers' institutional relationships to the teaching of structuralist algebra. Thus, the teachers only use paradigmatic examples to enable the creation of structures using the potential for generalisation.

Conclusion

Our analyses have highlighted that the type 1 transition appears to take place between the second year of the Bachelor's degree (or CPGE) and the third year of the Bachelor's degree. Occasionally, teachers use examples to facilitate transitions. For example, the lecturers work with the same praxeology on examples before generalising by putting forward what makes the spring of this praxeology as a generalising principle.

However, we note that due to many constraints, teachers can no longer rely on paradigmatic examples. Let us highlight some of these constraints as presented in [START_REF] Candy | Étude de la transposition didactique du concept d'idéal : écologie des savoirs et problématique de l'entrée dans la pensée structuraliste en France et en Suisse romande[END_REF]. On the one hand, the separation of modules can sometimes make it difficult to use paradigmatic examples. Indeed, as students do not all follow the same modules, they may have different praxeological baggage in terms of paradigmatic examples. This coupled with other constraints such as time or the number of ECTS 2 credits complicates the support on paradigmatic examples for teachers.

Finally, we have seen that the teachers in the second year of the Bachelor's degree wanted to favour the type 1 transition by preparing the students for abstraction. To do this, they opt, under the effect of their reports on the teaching of algebra, for a formal presentation of structures and a predominantly axiomatic work. They reduce the use of structures to a 'vocabulary', which, according to these lecturers, will facilitate their use in the third year of the Bachelor's degree. However, [START_REF] Candy | Étude de la transposition didactique du concept d'idéal : écologie des savoirs et problématique de l'entrée dans la pensée structuraliste en France et en Suisse romande[END_REF] highlights the fact that third-year Bachelor's professors still feel that students have difficulty moving towards abstraction despite an abstract introduction. One of the CPGE lecturers who first teaches structures in contexts and then decontextualizes them, pointed out a strong cultural connection to the teaching of abstract algebra. The formal character of structures appears culturally as the necessary premise for abstraction. Yet both our epistemological analysis, previous work on examples [START_REF] Durand-Guerrier | Définitions et exemples : prérequis pour l'apprentissage de l'algèbre moderne[END_REF] as well as theoretical framework [START_REF] Hausberger | Structuralist Praxeologies as a Research Program on the Teaching and Learning of Abstract Algebra[END_REF] highlight the place of examples in the construction of abstraction. In order to foster teaching strategies that use paradigmatic examples as a catalyst for abstraction, it appears that we should modify the institutional and personal relationships to the teaching of algebra.

According to Corry "the body of knowledge includes theories, 'facts', methods, open problems. The images of knowledge serve as guiding principles, or selectors. […] Thus the images of knowledge cover both cognitive and normative views of scientists concerning their own discipline"(Corry,

2004, pp. 3-4)

 Winsløw, C. (2008). Transformer la théorie en tâches : la transition du concret à l'abstrait en analyse réelle. In A. Rouchier & I. Bloch (Eds.) Actes de la XIIIème école d'été en didactique des mathématiques. La Pensée Sauvage.

Affiliation

Julie Stalder

Haute-Ecole Pédagogique du Valais e-mail: julie.stalder@edufr.ch


Samir Si-Mohammed
email: samir.si-mohammed@ens-lyon.fr

Zakaria Fraoui

Thomas Begin
email: thomas.begin@ens-lyon.fr

Isabelle Guérin Lassous
email: isabelle.guerin-lassous@ens-lyon.fr

Pascale Vicat-Blanc
email: pascale.vicatblanc@inria.fr

StackNet: IoT Network Simulation as a Service

Keywords: Internet of Things, Smart Industry, Connected Solutions, No-code, Software as a service, Network, Simulation

The Internet of Things (IoT) is transforming all economic sectors by connecting physical assets to the virtual world. The range of low-power connectivity options is continuously widening the range of possible IoT applications. However, too many possibilities often make it hard for industrial specialists to choose the right technology and configuration settings, yet these are crucial decisions. To deeply analyze and compare the performance and the scalability of various solution designs, one proven method is simulation. In this article, we show how IoT network simulation can help to future-proof an IoT connectivity design, without the burden of installing a lot of hardware and writing complicated scripts. Then, as the network simulation process is too complex for most IoT teams, we propose a no-code online IoT network simulation platform to make this powerful tool accessible to all. In particular, we explain how we hide the simulation workflow complexity via relevant abstractions and transform it into intuitive interactions. We illustrate the method and the usage of this promising approach, which can be integrated in a network digital twin. We show how it permits to easily evaluate what-if scenarios in order to answer a set of questions that may arise all along the life cycle of a smart connected solution.

I. INTRODUCTION

The Internet of Things, or IoT, defined as the convergence of the digital and physical worlds, is a fundamental pillar of the digitization of industrial enterprises and is becoming the beating heart of their operations. Amongst popular IoT solutions, we can list asset tracking for smarter transportation, smart building for more energy-efficient commercial areas, predictive maintenance for Industry 4.0, smart metering for efficient utilities, or video-surveillance for safer cities.

For more than ten years, remarkable progress has been made in network technologies and protocols to serve the increasing number of connected applications. The range of connectivity options offered to IoT architects keeps growing. From new low-power communication technologies like LoRaWAN, or Sigfox, Wi-Fi HaLow, 6LowPAN, NB-IoT to Wi-Fi, 5G or satellites, the choice is huge. These technologies differ from each other in terms of radio spectrum, hardware, protocols, medium access and sharing, environmental footprint, parameters, complexity, operations and capacities.

The variety of network technologies is continuously widening the range of possible IoT use-cases. However, too many possibilities often make it hard for industrial specialists to choose the right technology and configuration settings, yet these are crucial decisions. In particular, an under-sized network can be prejudicial to the quality of experience and consequently to the solution adoption. On the other hand, over-sizing can challenge the whole solution's profitability and environmental impact, thus the stakeholder's commitment. The current approach to test the adequation of a technology for a specific use-case is to deploy a Proof of Concept in labs and potentially a small pilot in real situation with a sample of instrumented assets. However, these experiments do not strongly secure a production deployment at large scale. Analytical models can be of a great use to overcome the deployment complexity of real experimentation. For instance, [START_REF] Stojanova | Conflict graph-based model for IEEE 802.11 networks: A Divide-and-Conquer approach[END_REF] proposes a graph-based model for estimating the attained throughput in a Wi-Fi network. This model is then used as a decision-support tool by a network architect for channel allocations and frame aggregation. A stochastic geometry framework for modeling the performance of a single gateway with LoRa is, for instance, proposed in [START_REF] Georgiou | Low power wide area network analysis: Can LoRa scale?[END_REF]. Nonetheless, we argue that analytical models sometimes lack precision and neglect important parameters for the network technologies analysis.

To address the need for in-depth evaluation, network simulation appears as a key enabler. Indeed, it can provide good insights about the performance of a technology and permits to test what-if scenarios at scale to trade-off cost, QoS and energy efficiency. Network simulation complements small scale PoC for large scale assessment and is cost efficient since there is no need to massively deploy real IoT equipment.

However, simulators like ns-3 [START_REF] Riley | The ns-3 network simulator[END_REF] require network expertise and knowledge in C++ programming to design experiments, to run them but also to analyze and exploit the results. Network simulators have been designed by network experts targeting network researchers and programmers, not product managers, industrial teams or even IoT architects, who do not have the time and skill to perform sophisticated simulations.

In this paper, we present the work we have done to make the network simulation power accessible and at the service of IoT teams. We propose an online no-code IoT network simulation platform to democratize the use of simulation and reach a community of non-network-experts. For this, we have developed a model-based approach and its associated intuitive interface for setting up, running and analyzing simulations without writing a single script. We believe that such a tool, potentially integrated within a network digital twin toolbox, can help accelerate the standardization of IoT practice to boost industry digitization and encourage contributions to open source IoT network simulators.

The remainder of this paper is organized as follows: Related works are analyzed in Section II. The problem statement is presented in Section III. An overview of our approach is given in Section IV. Section V illustrates how our proposal can be leveraged to answer typical IoT network questions around a smart building use-case. Conclusion and future works are given in Section VI.

II. RELATED WORKS

Numerous academic works on evaluating the performance and the scalability of IoT protocols or applications have been carried out and analyzed in [START_REF] Angelo | Simulation of the Internet of Things[END_REF] The authors stress that the design of complex IoT systems requires the implementation of testbeds on a very large scale, or the use of scalable simulation tools. A substantiated study of the challenges of IoT modeling and simulation is offered in [START_REF] Kecskemeti | Modelling and simulation challenges in internet of things[END_REF]. In [START_REF] Si-Mohammed | ADIperf: A Framework for Application-driven IoT Network Performance Evaluation[END_REF], the authors proposed a framework for the evaluation of IoT network technology using simulation, giving special attention to the energy consumption efficiency. [START_REF] Verhoeven | Performance Evaluation of Wi-Fi HaLow, NB-IoT and LoRa for Smart City Applications[END_REF] compares between Wi-Fi HaLow, LoRaWAN and NB-IoT for smart city applications, using simulation. In [START_REF] Pandey | Experimental Tools and Techniques for Wireless Sensor Networks[END_REF], the authors discuss the different types of IoT simulators used in the academic world. The full-stack simulators like iFogSim [START_REF] Gupta | iFogSim: A toolkit for modeling and simulation of resource management techniques in the Internet of Things, Edge and Fog computing environments[END_REF] provide support for simulating the end-to-end IoT chain, not specifically the network performance. Other simulators focus on the big data processing aspects of IoT applications like IOTSim [START_REF] Zeng | IoTSim: A simulator for analysing IoT applications[END_REF] and SimIoT [START_REF] Pandey | Experimental Tools and Techniques for Wireless Sensor Networks[END_REF]. Classical network simulators were not created specifically for the IoT, but have progressively evolved to include aspects necessary for IoT studies. The survey conducted in [START_REF] Pandey | Experimental Tools and Techniques for Wireless Sensor Networks[END_REF] includes more than 30 network simulators like OMNeT++ [START_REF] Varga | An overview of the omnet++ simulation environment[END_REF], ns-3 [START_REF] Riley | The ns-3 network simulator[END_REF], Cooja [START_REF] Osterlind | Cross-level sensor network simulation with cooja[END_REF]. It is worth noting that in all these studies, the accessibility of network simulation to non-network-experts like IoT architects has never been taken into account. Therefore these powerful tools are rarely used by IoT teams. On the other hand, the concept of no-code is becoming a very popular trend in IoT, as it permits manufacturers and operation managers to program their IoT applications while reducing the time and expertise needed. According to analysts, 70% of IoT applications are going to be coded via no-code in 2025 compared to 25% of them in 2020 [START_REF] Gartner | [END_REF]. Ultimately, IoT architects want to be able to program their IoT solution from building blocks without the need of having a deep expertise in networking, or facing the IT human resource bottleneck with its project backlog. In this perspective, [START_REF] Pantelimon | Towards a seamless integration of IoT devices with IoT platforms using a low-code approach[END_REF] proposes an end-to-end low-code mechanism for managing the relationship between heterogeneous hardware sensors and IoT platform. [START_REF] Si-Mohammed | SIFRAN: Evaluating IoT Networks with a No-Code Framework based on ns-3[END_REF] is a first attempt to simplify the programming of IoT network simulations. This tool proposes ns-3 templates but requires network expertise. In our work, we address IoT architects to let them seamlessly benefit from network simulation via an interactive tool, for what-if scenarios creation and comparison. This enables them to future-proof the infrastructure when developing IoT applications or to test a configuration adaption during operations. This opens a path towards the incremental realization and adoption of network digital twins [START_REF] Öhlén | Network digital twins-outlook and opportunities[END_REF], [START_REF] Almasan | Network digital twin: Context, enabling technologies, and opportunities[END_REF] in IoT solution design and management.

III. PROBLEM STATEMENT

A. Evaluating an IoT network technology at scale

Selecting and tuning the network, that best fits the needs of an envisioned smart application, is one important and complex problem the IoT architect faces. To examine this issue, let us use a smart building example. We consider the case of the instrumentation of a commercial building already equipped with a BMS (Building Management System) in charge of the remote control of HVAC (heating, ventilation, and air conditioning) systems as well as water and energy regulation. The facility manager wants to add a smart solution to finely monitor the building, to gain better visibility on its real usage and to better adapt the building services. For this, the facility manager would like to deploy a range of sensors: Entrance detectors, occupancy monitors, air quality sensors, temperature sensors, smart lighting and other end-devices. This customer is working with an IoT architect, who is proposing several sensors supporting various communication technologies. To demonstrate the feasibility of the project and to qualify the sensors for the solution, the solution architect has developed a proof of concept (PoC) of the smart solution that shows how sensors collect data and how the prototype application exploits and visualizes them.

1) Asking the right questions: After having convinced the customer, the architect has to select and define the detailed configuration of the network together with the sensors for the targeted deployment, and integrate the final solution. They have several options for the network, typically LoRaWAN, Wi-Fi, Wi-Fi HaLow and 6LowPAN and a range of questions that the small PoC has not truly answered:

• How would LoRaWAN and Wi-Fi capacities compare for this application? • How long will the battery last for this traffic workload?

• What are the best settings for maximizing the solution quality in this specific commercial building? • How many devices would one gateway support? 2) Analyzing the network behavior under application workload: To answer this type of questions, network experts would typically simulate the behavior of network candidates under application workload, considering the environmental conditions and the application traffic and topology (number of sensors, their location, etc.). They will evaluate if the traffic sent by all devices will be properly supported by the network, then estimate what percentage of packets will be successfully transmitted and how much energy will be consumed in different scenarios.

To analyze their smart building scenario in depth and at scale, the IoT architect would have to adopt the same approach: Define the application requirements as well as the network setup to be tested using a network simulator, then run a first simulation to establish the base-line and validate the parameter assumptions with one technology and a basic application scenario. Then, they would have to proceed with what-if scenarios comparison. This process requires specific skills as we detail below.

B. Coding a simulation

The network expert generally codes network experiments into the appropriated language for their simulator, for example, in C++ for the ns-3. This code globally works as follows: (i) It takes input parameters, (ii) creates and executes the corresponding network nodes and traffic, (iii) calculates the Key Performance Indicators (KPIs) obtained from the simulation.

1) Technology comparison: After the base-lining step, one has to conduct technology scenario comparisons. This step consists in setting up and running a set of simulations for: (i) Various network technologies to select the best of them and (ii) different network configurations to select the best network settings for the best network technology.

2) Analyzing the scalability: After the network choice has been established, the scalability analysis consists in analyzing how an increase in load in terms of number of devices or traffic intensity affects the application performance of the best network technology. This analysis is crucial because the applications and device deployments supported by the connectivity infrastructure are supposed to evolve and grow over time. For example in the smart building case, the number of sensors is expected to increase and the existing topology to change when various floors of the building are equipped. Estimating in advance the KPIs at the limits of the system capacities will ensure that the choice of the technology will survive the solution's scaling and that the cost or the footprint will not explode.

C. Dealing with simulation complexity

As we can see, this simulation workflow is far from the reach of IoT architects, who are not necessarily network experts. Consequently, they naturally tend to overlook the systematic analysis of the technology they deploy. This often leads to project failure at short, mid or long term [START_REF] Behrendt | Digital manufacturing-escaping pilot purgatory[END_REF].

To change this, we propose to abstract as much as possible the complexity of the simulation process and guide IoT architects in the in-depth performance evaluation of IoT network technologies for ensuring their IoT solutions design and evolution.

IV. PROPOSED SOLUTION

In this section we explain how we transform the complex simulation process detailed above and encapsulate the simulation and analysis code described in a no-code automated network simulation software named StackNet.

A. Principles

Guiding the IoT architect in the methodological evaluation via a high-level, easy-to-use interface, requires i)to simplify and structure the network performance evaluation process, ii) to propose pre-built components for application and network parameter settings, iii) to automatically code simulation scripts, iv) to deploy the simulator underneath and give online access to it, v) to gather and organize the simulation results and vi)to automatically create easy-to-understand, explainable and interactive dashboards.

B. Methodology

To analyze the performance and the scalability of various network options with StackNet framework, an IoT architect will perform the tasks listed below:

1) Identify key questions to answer for correctly supporting the targeted application. 2) Describe the application by initializing its important parameters. 3) Define a set of simulation scenarios to compare various networking alternatives. 4) Define a set of simulations to analyze the impact of the potential increase of fleet size or of workload in the future.

C. Modeling

The role of a no-code network simulator is to abstract what is an IoT scenario in order to be able to capture any IoT usecase and "code" it automatically in the simulator (for example in ns-3) language. To achieve this, and as illustrated in Figure 1, we segregate the description of an IoT scenario in two parts: The application part and the network part. Indeed, application parameters can be easily defined by the IoT architect or the application developer while the network settings are the difficult aspects of the simulation. Fig. 1: Scenario simulation building blocks. 1) Abstracting the application: An IoT solution often relates to sensors/actuators sending/receiving data to/from a remote application. The application is characterized by the number of connected sensors or actuators and the workload, which is the traffic they submit to the network. For example, in the smart building use-case described above, the application will be modeled by the number of sensors, the size and frequency of temperature samples sent to an IoT Hub, then to analytics pipeline and finally to the application's back-end, generally located in the cloud. This similar flow applies to other types of sensors. From a network simulation perspective, the number of sensors as well as the profile of data collected and transmitted over the communication channels have to be defined by the IoT architect. They are the best person to define these application characteristics. They have the knowledge of the sensors' details, the application and usage of the data. All this information should be formalized and initialized with a set of valued parameters, typically the traffic direction and workload. The external conditions affecting the radio signal also matter: We distinguish indoor and outdoor environments. The latter can be rural, urban or semi-urban presenting various behaviors in terms of radio propagation. In StackNet, a collection of pre-defined IoT application templates are proposed to accelerate the initialization process.

2) Abstracting the network: The specification of an IoT network simulation scenario needs a network model defined by a list of parameters representing the network technology and topology. These parameters can be divided as follows: (i) The IoT network technology as defined by its physical and MAC layers, (ii) the number of gateways and their location (topology) and (iii) the low-level parameters related to the network technology (configuration). In StackNet, pre-built network models are made available to the IoT architect, so they can integrate them in simulation scenario without network expertise.

D. Designing the no-code interface

A simulation scenario is defined by the application model, which is set once for several experiments, and the network model and settings on the other side. Various network settings can be selected to compare what-if scenarios. To automatically generate the scripts for running a simulation, our framework provides a dynamic interface to initialize the values of the application model's parameters and the selected network settings of a scenario, as illustrated in the left column of the StackNet's interface of Figure 2. Pre-defined network settings with default values are proposed for each supported technology. These predefined settings can be easily uploaded then adjusted manually. The interface enables to create and save multiple scenarios.

Once the application model and the network settings are initialized, the simulator is invoked. The simulation script is automatically generated, the simulation executed, and the resulting output metrics (KPIs) interactively visualized in scenario dashboards. To permit the user to analyze and explain one particular scenario in context, he can open the input parameters panels together with the results dashboard.

The results dashboard gathers and highlights the KPIs related to the IoT connectivity solution under study in the specified application context as illustrated in the Figure 2. The network performance evaluation focuses on five KPIs. These KPIs are: (i) Packet goodput, (ii) packet delivery, (iii) packet latency, (iv) energy consumption and (v) battery lifetime. We consider that, together, these metrics provide a fair representation of the performance of an IoT network technology for a given scenario.

If the application requires high quality data, like for precision air quality monitoring in the operating room of a hospital, then, estimating the packet delivery is of utmost importance.

For a use-case like real-time equipment location application, packet latency is what matters the most. If the deployment of sensors and the battery change are difficult, energy consumption and battery life time have critical importance.

As the IoT architect is looking for scenario comparison, a dedicated and intuitive dashboard (e.g., Figure 4) is proposed to display comparative results and charts, and to let the best technology and configuration immediately stand out.

The evaluation methodology has been seamlessly integrated within the interactive front-end while the programming and networking expertise have been encoded and hidden within the back-end. This makes the simulation experience smooth and quick, allowing the IoT architects to focus on the data and decision process rather than bothering them with programming and results collection complexity. Figure 3 illustrates the nocode workflow for creating and setting various scenarios for comparison, and the corresponding mapping to the simulation workflow activated underneath.

V. USE-CASE EXAMPLE

This section details how an IoT architect would use Stack-Net for the aforementioned smart building use-case. The application model is defined by the following: 100 enddevices are placed in a building. They are separated by a distance of around 200 meters. They send one packet of 100 bytes every 2 minutes. For the scalability analysis, the network density is scaled (up to 600 end-devices) and the traffic workload is increased (one packet of 110 bytes is sent every 90 seconds). The IoT architect introduces the application model inputs. Then, they create various scenarios for different network technologies, typically Wi-Fi, 6LoWPAN, 802.11ah and LoRaWAN. Let us consider how our IoT architect can leverage the network simulation service to easily answer the questions asked in Section III:

1) How would LoRaWAN and Wi-Fi capacities compare for this smart building application? Figure 4 displays the performance synthesis of the different technologies explored. We can easily observe that all the technologies perform the same in terms of goodput and packet delivery, i.e. all the packets are correctly received. However, LoRaWAN clearly outclasses the other technologies, including Wi-Fi, in terms of battery lifetime (up to 700 days). Therefore, we can say that LoRaWAN is more relevant than Wi-Fi in the basic version of this use-case.

2) How long will the battery last? The battery lifetime of Wi-Fi can last approximately 3 months (90 days) while 6LoWPAN can go up to 5 months (150 days). The difference is not that big, but it shows that 6LoWPAN is better designed and suited for this IoT applications than Wi-Fi. For LoRaWAN, it can go up to approximately 2 years.

3) What are the best settings with the LoRaWAN technology (spreading factor) to maximize the solution quality in this specific commercial building? Figure 5 shows how the IoT architect can further explore some configurations of LoRaWAN with different spreading factors (7, 8 and 9) to determine the impact of this parameter on the performance. Except covering longer distances, we observe that higher SF is less interesting in terms of all the considered KPIs. Packet latency is increased, which makes collisions more likely, and therefore lessens packet delivery. Additionally, end-devices spend more time to send a packet, and thus consume more energy, leading to lower battery lifetime. We see that a lower SF results in lower message latency, better message delivery and longer battery lifetime. In this case SF7 is the best setting.

4) How many devices would one gateway support? Figures 6a and 6b highlight the various network technologies' performance in terms of packet delivery and packet latency for the scaled settings. We see that Wi-Fi HaLow, 6LoWPAN and Wi-Fi behave well in terms of packet delivery. However, it drops faster for LoRaWAN (down to 60% for 600 enddevices). Setting and running these experiments only took a dozen of minutes. The scalability study shows that on one hand, one LoRaWAN gateway can handle up to 200 sensors before deteriorating the performances, while on the other hand one Wi-Fi HaLow gateway can handle 400 sensors without noticing any degradation in the performances. The node's placement and the radio environment in this use-case make it impossible to use only one gateway for Wi-Fi and 6LoWPAN.

VI. CONCLUSION AND FUTURE WORKS

In this work, we have presented StackNet, a no-code framework to ease and automate IoT network simulation and its integration in a SaaS (Software as a Service) platform. This approach enables IoT architect to test and compare IoT network technologies without learning and deploying any network simulator nor coding any script. We began by identifying the network evaluation problems that IoT architects face on their journey, and the difficulty of running network simulations to overcome them. Then, we described our approach by highlighting the salient aspects that need to be taken into consideration for hiding the complexity of IoT simulation while empowering architect with an intuitive solution to design and compare alternative scenarios.

An application of our methodology and tool on a smart building use-case has been presented. We put the emphasis on the ease of simulation initialization, on results visualization, on what-if scenarios comparison. We show how the nocode framework helps IoT architects ask and answer their own design questions. The tool currently supports simulations with Wi-Fi, LoRaWAN, Wi-Fi HaLow, 6LoWPAN and 5G mmWave, but is limited by the availability of validated sim-ulation modules in ns-3 for other new technologies. As the no-code and as a service approach can democratize the usage of cost-effective evaluation methods like simulation in IoT, we hope it will encourage the systematic development of ns-3 simulation modules for other IoT network technologies.

As the user interface is simulator-agnostic, we project to adapt the back-end to support other simulators like Omnet++ and even experimental test-beds to leverage the same userfriendly tool for technology and configuration comparison. In terms of future works, we will also extend StackNet with a decision support engine to generate recommendations for network selection and configuration. We will also combine network simulation with end-to-end solution simulation and monitoring for precise workload estimation, cost and environmental impact analysis. StackNet can be freely accessed via the Stackeo digital twin platform here https://app.stackeo.io/.

Fig. 2 :

 2 Fig. 2: Dynamic interface for setting and analyzing a scenario.

Fig. 3 :

 3 Fig.3: Illustration of the no-code workflow and its mapping to expert simulation for IoT scenarios analysis.

Fig. 4 :

 4 Fig. 4: Results for the network technologies comparison (Q1 & Q2).

Fig. 5 :

 5 Fig. 5: Results for the LoRaWAN configuration (Q3).

  (a) Packet delivery. (b) Packet latency.

Fig. 6 :

 6 Fig. 6: Results for the scalability study (Q4).

ACKNOWLEDGMENT This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the ANR-11-IDEX-0007 program and the support of Stackeo company (https://stackeo.io).