# The challenge of structuralist thinking in abstract algebra: a transition problem

Thomas Hausberger, Julie Jovignot

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## To cite this version:

Thomas Hausberger, Julie Jovignot. The challenge of structuralist thinking in abstract algebra: a transition problem. International Journal of Mathematical Education in Science and Technology, 2024, pp.1-23. 10.1080/0020739X.2024.2311284. hal-04471840

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# The challenge of structuralist thinking in abstract algebra: a transition problem

Thomas Hausbergerª and Julie Jovignotᵇ

ªUniversité de Montpellier, France; ᵇHaute Ecole pédagogique Fribourg, Switzerland

## ARTICLE HISTORY

Compiled 31st January 2024

### Abstract

This paper deals with the issue of transitions in the teaching and learning of abstract algebra throughout the Bachelor in France, focusing on the notion of ideal in ring theory which is seen as a gateway to structuralist thinking. In the framework of the Anthropological Theory of the Didactic, the notion of structuralist praxeologies, endowed with structuralist levels, is used to analyse a corpus of course documents from three upper-secondary teachers which cover the teaching of ideals through the Bachelor. Hausberger's three-phase model for structuralist transitions, that generalizes Winsløw's model of the concrete to abstract transition in analysis, is applied to discuss the issue of transitions. Our research results cover a description of structuralist praxeologies developed throughout the Bachelor around the notion of ideal, a discussion of discontinuities and ruptures that are observed, and the identification of strategies developed by teachers to handle these transitions as they carry out the internal didactic transposition.

### KEYWORDS

abstract algebra teaching and learning; transitions across university mathematics; Anthropological Theory of the Didactic; structuralist praxeologies; ideal concept in ring theory

## 1. Introduction

The issue of transitions has received increasing attention in research on mathematics education, resulting in a recent topical ICME survey (Gueudet, Bosch, diSessa, Kwon & Verschaffel, 2016). The state of the art of the literature reviewed in the survey underlines different facets (cognitive, epistemological, socio-cultural, institutional,...) of the transitions. The latter are investigated in terms of continuities, discontinuities, ruptures that occur at different transition points (e.g. from school to university) and research-based devices are proposed to accommodate them. Even more recently, Hochmuth, Broley & Nardi (2021) report on the works on this theme carried out within INDRUM.

This paper focuses on transitions across university mathematics courses. Such transitions have been mainly investigated in the context of the analysis path (Hochmuth et al., 2021, p. 203–204). In particular, several studies presented at INDRUM conferences adopt the institutional perspective offered by the Anthropological Theory of the Didactic (ATD; Chevallard & Bosch, 2020) and refer to a three-phase model of
the calculus to analysis transition introduced by Winsløw (2008). In their concluding section, Hochmuth et al. (2021) raise the question of the situation for the other fields, and, in reference to Winsløw, “what does the observed jump from the first to the third stage of the model mean in a longer-term perspective, e.g. taking into account what students learn in more advanced mathematics studies?” (p. 210).

Our goal is to investigate these questions in the context of the abstract algebra (Hausberger, 2020) path, with an epistemological and institutional lens. Indeed, the teaching and learning of abstract algebra at university is the occasion of an epistemological rupture that may be expressed in the following terms:

> Then in 1930 a textbook [Moderne Algebra; (van der Waerden, 1930)] was published that presented a totally new image of the discipline. [...] Van der Waerden’s new image of the discipline inverted the conceptual hierarchy of classical algebra. Groups, fields, rings, and other related concepts became the main focus, based on the implicit realization that all of these concepts were, in fact, instances of a more general, underlying idea: the idea of an algebraic structure. [...] The systems of real numbers, rational numbers, and polynomials were studied as particular instances of certain algebraic structures; the properties of these systems depended on what was known about the general structures of which they were instances, rather than the other way round. (Corry, 2016)

In our study, university teachers characterize the learning of abstract algebra as “an impassable wall”, “a threshold to pass”, or a “leap into abstraction”, which is confirmed by the literature (e.g. Durand-Guerrier, Hausberger & Spitalas, 2015; Simpson & Stehlíjová, 2006). In view of Corry’s quote, we call it the “challenge of structuralist thinking”. This raises the issue of understanding this challenge in terms of potential discontinuities of both epistemological and didactical nature.

To fulfill this endeavour, we draw on the perspective opened up by Hausberger (2018): the concrete to abstract transition identified by Winsløw in analysis is generalised in the form of the wider perspective of the teaching and learning of mathematical structuralism at large – in ATD terms, the development of structuralist praxeologies with their structuralist levels (see theoretical framework). A new three-phase model has been proposed by Hausberger and applied to abstract algebra in a small-scale pioneering study (Hausberger, 2018, 2016) with a view to a better understanding of transition issues related to abstract algebra.

A larger-scale study, centred on second and third year post-secondary teaching practices in France and Switzerland, has been carried out by Candy (2020a) in her PhD project with a focus on the concept of ideal in ring theory. Let us recall that a subset *I* of a commutative ring (*A*, +, ·) is called an ideal if it is an additive subgroup of (*A*, +) which is stable under multiplication by any element of *R*. This choice is motivated by the role of this concept as a gateway to mathematical structuralism:

> Most types of algebraic structures developed within particular contexts which stressed the specific role played by them within individual theories, rather than their interrelations with other kinds of structures. [...] The case of ideals is somewhat different in this respect. 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. (Corry, 2004, p. 15)

Partial results have been presented at INDRUM conferences in two directions. Firstly, Candy (2020b) offers a comparative study of structuralist praxeologies involving ideals and taught at the second year of upper-secondary studies in two different teaching institutions in France: the Bachelor at university (B2) and the Classes Préparatoires
aux Grandes Ecoles Mathématiques (CPGE; these are classes reserved for the best students who are destined to enter the French Grandes Ecoles). Secondly, the three-phase structuralist model is applied to study structuralist transitions throughout the Bachelor (Jovignot & Hausberger, 2023), thus adding the third year of Bachelor (B3) but with a coarser account of the structure of praxeologies (see our descriptive tools in the theoretical section).

This paper aims to combine both accounts in a coherent whole, also deepening both directions. It therefore addresses the following research questions: How are structuralist praxeologies that involve the concept of ideal structured in B2, CPGE and B3, and which are their structuralist levels? What continuities and ruptures can be observed in the development of these praxeologies from B2/CPGE to B3? How are transitions in the development of structuralist praxeologies handled by abstract algebra teachers?

We begin by presenting our theoretical framework and then describe the methodology for analyzing the data. We illustrate the methodology through its application to selected excerpts from the data and provide detailed praxeological analyses. Then, we discuss the results obtained in relation to the research questions, before concluding with the scope of the study and prospects for further development.

## 2. Theoretical framework

We first recall the key notions of ATD that are used in this research, and then present the additional tools, specific to the point of view of mathematical structuralism, that have been developed.

### 2.1. Didactic transposition

Knowledge as it is taught, even in higher education, does not coincide with scholarly knowledge as it has been developed in the scientific community. Chevallard (1989) explains that scholarly knowledge is transformed into *knowledge to be taught* in a process called external didactic transposition. This process is performed by the *noosphere* – at university, the members of the programme committee – and the product of the external didactic transposition is made of the curricula and syllabi (Bosch, Hausberger, Hochmuth, Kondratieva & Winslow, 2021). Then the knowledge to be taught itself undergoes a process of *internal didactic transposition* by which it is transformed into *taught knowledge* by university teachers. In this article, we will present the analysis of the product of the internal didactic transposition of the concept of ideal.

Didactic transposition processes are submitted to various institutional *conditions and constraints* that govern the *ecology* of the knowledge to be taught and taught (how this knowledge lives or fails to live in university programmes). These may be organized according to what is called in ATD the *scale of levels of didactic codeterminacy*:

$$ 
\begin{align*}
\text{Humanity} &\rightleftharpoons \text{Civilization} &\rightleftharpoons \text{Society} &\rightleftharpoons \text{School} &\rightleftharpoons \text{Pedagogy} &\rightleftharpoons \text{Discipline} \\
& &\rightleftharpoons \text{Domain} &\rightleftharpoons \text{Sector} &\rightleftharpoons \text{Theme} &\rightleftharpoons \text{Subject}
\end{align*}
$$

We will use the *specific levels* of the scale (below the discipline) to account for the structure of the courses (see section 4). *Generic levels* will also be considered, at the very end of the paper, to interpret the results of the analyses of the course documents, which cannot be explained on the basis of specific levels alone.
Finally, we will spin the ecological metaphor further and call, after Artaud (1997), *habitats* the different living places of the concept of ideal (described in terms of sectors, themes and subjects), and *niches* the function that the concept occupies within each of these habitats. In fact, the concept of ideal is embedded in a *trophic network* of other concepts, since it acts on other mathematical objects which also act on it, and this environment (or ecosystem) needs to be described. In what follows, we will therefore say that a given concept (or sector/theme where concepts are developed) feeds the *trophic needs* of another concept (or sector/theme) when it contributes to the “life” of the latter. For instance, the notion of ring is needed in the ecosystem of the ideal concept, which in turn feeds the trophic needs of the notion of quotient ring.

## 2.2. The theory of praxeologies

According to ATD, every human activity consists in the coordination of a praxis and a logos, hence the key notion of a *praxeology*, represented by a quadruple $[T/\tau/\theta/\Theta]$. Its practical-technical block (or know-how) consists of a type of tasks *T* together with a corresponding technique *τ* (useful to carry out the tasks *t* ∈ *T*). The technological-theoretical block (or know-why) comprises the technology *θ*, a discourse on the technique, and the theory *Θ*, the ultimate level of justification.

To be precise, a quadruple $[T/\tau/\theta/\Theta]$ is called a *punctual praxeology* (PP). Several PPs may be unified by a common technology $\theta$ to form a *local praxeology* (LP). In turn, LPs may be unified at the theoretical level $\Theta$ to make up a *regional praxeology* (RP). Our goal is to draw *institutional praxeological models* (IPM) – for B2, CPGE and B3 separately – that describe the *praxeological equipment* of students regarding the concept of ideal, that is the complexes of (punctual, local, regional) praxeologies in which the concept of ideal intervenes at one component (at least).

## 2.3. Structuralist praxeologies

The starting point is the consideration of mathematical structuralism as a methodology, which consists of reasoning in terms of classes of objects, relations between these classes and stability properties for operations on structures (Bourbaki, 1950). The general view of structures thus allows particular properties of objects to be demonstrated by making them appear as consequences of more general facts (theorems about structures). Dually, general statements are put to the test of objects, hence a *dialectical relationship between objects and structures* (Hausberger, 2018).

In praxeological terms, the structuralist method consists in the passage from a praxeology $P = [T/?/?/\Theta_{\text{particular}}]$ where it is unclear which technique to apply, to a structuralist praxeology $P_s = [T^g/\tau/\theta/\Theta_{\text{structure}}]$ where, modulo generalization of the type of tasks ($T^g$), the theory of a given type of structure guides the mathematician in solving the problem.

As a paradigmatic illustration in the context of abstract algebra, Hausberger (2018) analysed a thread from an online forum. In order to prove that the ring of decimal numbers is a principal ideal domain - PID<fnref n="1" /> (*T*: show that a given ring is a PID), a forum participant proposes to prove that any subring of $\mathbb{Q}$ is a PID (*T*⁹: show that any subring of a given PID is a PID), with the (erroneous) structuralist technique *τ*: show that any subring of a PID is again a PID. The underlying structuralist technology

<footnote n="1">¹A PID is an integral domain (without divisors of zero) such that every ideal is principal, i.e., can be generated by a single element.</footnote>
is the idea that striking features of rings are preserved by taking subrings, which unfortunately isn't true for being a PID ($\mathbb{Z}[X] \subset \mathbb{Q}[X]$ is a counter-example).

Furthermore, Hausberger (2018) distinguishes two main *levels*<fnref n="2" /> of structuralist praxeologies. At level 1, structures act as a vocabulary and appear mainly through definitions (e.g., *T* is solved by showing, by hand, that the definition is satisfied, i.e. that any ideal is monogeneous). At level 2, the technique used mobilizes general abstract results about structures (on our example, one may show the existence of an Euclidean algorithm, which invokes in the logos of the praxeology the structuralist theorem that any Euclidean<fnref n="3" /> ring is a PID).

In this example, the ring *A* plays the role of a *didactic variable* of the type of tasks: the structuralist praxeology is thus *contextualized* to domains of mathematical objects, whose variation is crucial to lead - in fine - to a *decontextualization* (the ring *A* is defined abstractly). We will be particularly attentive to the choices made by university teachers in relation to these didactic variables which are essential to operate the objects-structures dialectic.

## 2.4. The three-phase model of structuralist transitions

Following Winslow, Hausberger (2018, p. 89) proposes a new three-phase model (Figure 1) to account for the concrete-abstract movement that characterize upper-secondary studies, yet with a view to the development of structuralist praxeologies. While the first type of transition amounts to going from *P* (a praxeology centred on a non-structuralist praxis Π) to *P*<sub>s</sub> (a praxeology composed of a structuralist praxis block Π<sub>s</sub> and logos block Λ that refers to mathematical structures), the second type leads to praxeologies whose entirety of praxis Π' and logos Λ' lies in the abstract. To unfold our example, the student then encounters tasks like “show that a Noetherian<sup>4</sup> integral domain such that any maximal<sup>5</sup> ideal is principal is a PID”. In principle, the concepts mobilised in the praxis Π' emanate from the logos of structuralist praxeologies of the previous type, i.e. from Λ (in the notations of the figure).

<figure><img src="image_3.png" /><figcaption>**Figure 1.** The three-phase model of structuralist transitions</figcaption></figure>

<footnote n="2">²A third level is considered but this additional distinction will not be used in this article.</footnote>

<footnote n="3">³A Euclidean ring *A* is an integral domain endowed with a Euclidean algorithm which allows a suitable generalisation of the Euclidean division of integers: if *a* and *b* are in *A* and *b* is nonzero, then there exist *q* and *r* in *A* such that *a* = *bq* + *r* and either *r* = 0 or *f(r)* < *f(b)*, where the function *f*: *A* \ {0} → *N* is called the stathm.</footnote>

<footnote n="4">⁴A Noetherian ring is a ring that satisfies the ascending chain condition on ideals: every increasing sequence $I_1 \subseteq I_2 \subseteq I_3 \subseteq \dots$ of ideals is stationary, i.e., $I_n = I_{n+1} = \dots$ for some integer $n$.</footnote>

<footnote n="5">⁵*M* is maximal with respect to the inclusion of ideals: if *M* ⊆ *I* then *I* = *M*.</footnote>
## 2.5. Unification of structuralist praxeologies and mixed praxeologies

An LP (resp. RP) usually corresponds to a theme (resp. sector) where it is developed. In our context, the situation is more complex. The study of a textbook (Escofier, 2016), conducted as a preliminary study (Candy, 2020a, chap. 8), has pointed out different ways of aggregating praxeologies around themes and sectors. In some cases, an LP develops within a theme; in other cases, the praxeologies turn out to be more fragmented and isolated punctual praxeologies take place in different themes; or the construction of an LP by unification of praxeologies may take place transversally to different themes, possibly in different sectors.

The case of RPs is even more complex: the theoretical level of structuralist praxeologies needs to be clarified. Let us start with the origin of the notion of theory in ATD:

> In Greek, *theôria* took on the modern meaning of “abstract speculation” from Plato onwards. [...] In fact, theoretical statements frequently appear as *abstract*, far distant from the concerns of “simple” technologists and technicians. This effect of abstraction is correlated to what founds the great *generativity* of theoretical statements - their capacity to justify, explain, produce. (Chevallard, 1998, p. 95, our translation)

In abstract algebra, the scope of theoretical statements is undeniably linked to their abstract and general nature. However, there is another phenomenon that is not described in Chevallard’s quote, the idea of a *structure at the base of a theory*:

> Little by little a general idea emerged which would become more precise in the 20th century, that of a structure at the base of a mathematical theory; it is the consequence of the observation that what plays the primary role in a theory are the relations between the mathematical objects which appear in it, rather than the nature of these objects, and that in two very different theories, it may be that relations are expressed in the same way in both theories; the system of these relations and their consequences is the same structure “underlying” the two theories. (Dieudonné, 1987, p. 114, our translation)

Let us briefly describe the different theories involved and their interrelationships. First of all, there are theories which are at the foundation of mathematics: these are Set Theory and Logic. However, logical and set-theoretic praxeologies often remain implicit (Jovignot, Hausberger & Durand-Guerrier, 2017) so that we choose not to mention these theoretical levels. Then there are theories which relate to domains of objects: notably the *arithmetic of Z* and *K[X]*. Finally, there are structuralist theories (e.g. groups, rings and fields). The study of the arithmetic of integers, inherited from the work of Diophantus, Fermat, Gauss, and rewritten in the structuralist perspective of Noether, thus summons two theoretical levels which are “amalgamated”: the arithmetic of Z and Ring Theory. We call such a theoretical level a *mixed theory*.

Whenever a RP is constructed within a sector centred on a domain of objects (integers, polynomials) by the unification of praxelogies around a mixed theory, we call it a *mixed regional praxeology*. In our study, we will pay attention to the interplay between structuralist RPs (unification by a structuralist theory) and mixeds RPs, as a result of the objects-structures dialectic. A structuralist RP may be developed within a sector dedicated to it, or transversally to sectors where mixed RPs develop.

## 3. Methodology

In order to study the internal didactic transposition in relation to transitions in abstract algebra throughout the Bachelor, we conducted a case study of 9 university
teachers considered representative of second and third year (the concept of ideal is not encountered in the first year) teaching levels in post-secondary education, in France and Switzerland (Candy, 2020a). In this article, we rely on data from three teachers: MP1, EC2 and EC4. MP1 teaches mathematics in the second year of the CPGE, EC2 in the second year of the Bachelor (B2) a 7.5 ECTS course called “Linear Algebra” and EC4 in the third year (B3) a 5 ECTS course called “Elements of Ring and Field Theory”. EC4 teaches in the same university as EC2. Students of B3 either come from B2 or CPGE<fnref n="6" />. Thus, in this article, we can study the transitions through two possible curricula experienced by B3 students.

In our study, we began by conducting an ecological analysis of the official sylabi (the product of external didactic transposition) in order to bring to light the habitats and niches of the concept of ideal. Then, we focused on the internal didactic transposition and used course handouts to identify sectors and themes where ideals lived. Exercises from the tutorial sheets, which followed chronologically the lectures, were linked to those sectors and themes and subjected to a praxeological analysis when they mobilized the concept of ideal. When a correction was not available, we looked for other sources from a similar institution to obtain a solution of a task of the same type. Course handouts were also used to identify technological and theoretical elements that remained partially implicit in tutorials.

The next step consisted in unifying PPs to obtain LPs and RPs. Our IPMs are drawn from these informations and represented according to the following rules: i) mixed RPs are represented on a hatched background; ii) a mixed RP intersects a LP or PP if tasks that belong to the latter praxeologies are contextualized to the corresponding domain of objects; if this is the case for all tasks, the LP or PP is represented inside the mixed RP; iii) a structuralist RP intersects a LP or PP if solving the latter tasks uses the structuralist theoretical level of the RP; iv) dotted lines indicate that the RP remains partly implicit; v) the overlay of RPs and LPs reflects the chronology of the course; vi) light grey for LPs and PPs (in contrast to dark grey) indicates that the ideal concepts is only present in the theoretical level of praxeologies. These representation rules aim to make explicit the functioning of the dialectic between objects and structures (a didactic dialectic linked to the internal didactic transposition) within the IPMs.

Finally, the structuralist level was carefully noted as well as the value of the didactic variables of contextualization of the structuralist praxeologies. The aim was to analyze the continuities and ruptures that could be observed in relation to the two types of transitions described in the model, under the hypothesis that the type 1 transition would be situated at the institutional transition between B2/CPGE and B3 while the type 2 transition would be linked to that of the Bachelor's to the Master's degree. We consider the development of level 2 structuralist praxeologies as an indicator of the acquisition of the structuralist method, hence an achievement of the type 1 transition. By contrast, a predominant level 1 indicates a stage reduced to the identification of structures. The values of the didactic variables of contextualization make it possible to situate the dynamics at play in the relationship between the concrete and the abstract. An emphasis on decontextualized structuralist praxelogies is an indicator of the type 2 transition while level 1 decontextualized praxelogies should appear punctually in the context of the type 1 transition to prove the structuralist theorems that are needed to develop contextualized level 2 structuralist praxelogies.

<footnote n="6">⁶A number of CPGE students realise that they do not wish to go on to the Grandes Ecoles, which essentially lead to careers in engineering, and they continue their studies at university.</footnote>
# 4. Institutional praxeological models

In this section, we illustrate our methodology on salient excerpts from the corpus and report on the main findings of our analyses. The discussion of the results in relation to the research questions will be the subject of the next section.

## 4.1. The case of CPGE (as exemplified by MP1)

The analysis of the official syllabus of CPGE<fnref n="7" /> allows to identify three main habitats of the concept of ideal. The latter is introduced by its general definition in the section "common algebraic structures", at the level of the item "ideals of a commutative ring" where it is linked to the notion of ring homomorphism (as kernel) and to the notion of divisibility (a | b if and only is (a) ⊃ (b), where (x) denotes the ideal generated by an element x), and it is illustrated in the case of $\mathbb{Z}$. It is then taken up again in the item "rings of one-dimensional polynomials" of the same section, through the topic "ideals of $K[X]$". One can note that the syllabus does not mention the principality property (of an ideal, of a ring) and that it does not underline the analogy between the arithmetic of $\mathbb{Z}$ and that of $K[X]$. Nevertheless, the implicit organization of the content is based on this analogy. Finally, the concept of ideal is mentioned in the item "polynomials of an endomorphism, of a square matrix" of the section "reduction of endomorphisms and square matrices" where properties of ideals serve to justify the existence of the minimal polynomial.

Let us now consider the internal didactic transposition performed by MP1. The latter has provided 4 course handouts which we consider as 4 sectors (named according to the headings): "common algebraic structures", "polynomials and rational fractions", "algebra structure" and "linear algebra and reduction of endomorphisms". Ideals are present in each sector, confirming the potential of this notion to highlight structuralist praxeologies.

The concept of ideal is introduced in the first sector, in the theme "rings, ideals". The characterisation of divisibility in terms of ideals and that of the kernel of a ring morphism are given. The niches of the concept of ideal within this habitat are the principality properties of $\mathbb{Z}$ and $K[X]$. We consider that MP1 develops in this sector a structuralist RP that we call "common algebraic structures (rings)". In this sector, MP1 also points to the analogy between $\mathbb{Z}$ and $K[X]$, but the abstract notion of Euclidean domain is not formalized. This is why we indicate a structuralist RP "Euclidean rings" while representing it in dotted lines on figure 3. Dually, $\mathbb{Z}$ serves as a main contextualization (arithmetic is taught in the first year of CPGE), which points to a mixed RP "arithmetic of $\mathbb{Z}$".

We now detail the praxeologies that are part of these RPs, by analysing the tasks assigned to the students (exercise sheets and problems from the annals of the Grandes Ecoles). The first sector is the one that contains the most occurrences of tasks on ideals (8 in total) and all the praxeologies belong to the theme "rings, ideals". We note the type of tasks $T_1$ (prove that a subset $I$ of a ring $A$ is an ideal), which are solved by the technique $\tau_1$ (check that the axioms are fulfilled), thus giving rise to a level 1 structuralist praxeology. In fact, we find a single task of type $T_1$ (figure 2), with a choice of values of the didactic variables $I$ and $A$ that makes it an abstract task: $I$ is the radical of any ideal and $A$ is any ring. This problem is the place of development of a second praxeology generated by $T_2$ (prove properties of operations on ideals of

<footnote n="7">⁷https://cache.media.education.gouv.fr/file/special_1_MEN_ESR/42/4/MP-mathematiques_287424.pdf</footnote>
17. Radical d'un idéal.

Si $A$ est un anneau commutatif et $I$ un idéal de $A$, on note

$$ \sqrt{I} = \{x \in A, \exists n \in \mathbb{N}^{*}, x^n \in I\}. $$

Montrer que $\sqrt{I}$ est un idéal de $A$ et que si $I, J$ sont des idéaux de $A$, on a

$$ \sqrt{IJ} = \sqrt{I} \cap \sqrt{J} = \sqrt{I} \cap \sqrt{J}. $$

Montrer aussi que

$$ \sqrt{I+J} = \sqrt{\sqrt{I} + \sqrt{J}}. $$

Déterminer le radical d'un idéal $I$ de $\mathbb{Z}$.

<caption>**Figure 2.** A problem assigned by MP1</caption>

a ring $A$). This type, introduced during the lectures on the greatest common divisor (gcd) and least common multiple (lcm), is carried out here about a new operation whose behavior with respect to the two previous ones (intersection and addition) is studied. The key ingredient of the technique $\tau_2$ is to relate operations on ideals and relations between elements. Subsequently, the praxeology is contextualized to $A = \mathbb{Z}$ through the task of determining the radical of an ideal of $\mathbb{Z}$. The general properties of the radical allow a reduction to the case of prime integers so that $\sqrt{(n)}$ is generated by the product of the distinct prime factors of $n$. These tasks therefore show a dialectic between contextualization and decontextualization of praxeologies, in other words a local implementation of the dialectic between objects and structures within the mixed RP “arithmetic of $\mathbb{Z}$”. In this process, the praxeology generated by $T_2$ becomes a LP that we call “properties of operations in ideals”. Nevertheless, the notion of radical itself remains weakly motivated.

It is also in the sector “common algebraic structures (rings)” that the PP generated by $T_3$ (prove that a Euclidean domain is a PID) develops, whose technique is $\tau_3$: “take an ideal $I$ of $A$; choose an element $b \in I$ minimal for the Euclidean stathm; let $a \in I$, perform the Euclidean division of $a$ by $b$ then conclude that the remainder is zero by minimality of $b$, minimality of the remainder and absorbance of an ideal; deduce that $I = (b)$ and that any ideal of $A$ is principal, so that $A$ is a PID”. Remarkably, MP1 has chosen to introduce $\mathbb{Z}[i]$ (the Gaussian ring of integers, whose historical importance in the development of abstract algebra is well known) and the set $\mathbb{D}$ of decimal numbers as values of the didactic variable $A$ (figure 3, tasks 21 & 22). In spite of the genericity of the technique $\tau_3$, one proceeds by hand from the definition of principality. We relate this praxeology to the structuralist RP “Euclidean rings”, without going so far as to quote a structuralist theorem. Indeed, such a theorem is not on the syllabus; its status is that of a “cultural element” (bottom of figure 3: “culture: one finds the same structure in any Euclidean ring”).

Finally, the sector contains two isolated tasks (show that the ring $\mathbb{Z}[X]$ is not principal by considering the ideal $I = (2) + (X)$) and (show that if $(a) + (b)$ is principal then so is $(a) \cap (b)$), which have a theoretical scope but no application (since all the rings considered in the course are Euclidean hence principal). We do not relate them to a type of tasks but note that techniques to solve these tasks use the praxeology generated by $T_2$, which may be the didactic intent pursued by the teacher.

The second sector “polynomials and rational fractions” is dedicated to the arithmetic of $K[X]$, which we model as a mixed RP called “arithmetic of polynomial rings”. The notion of ideal is used in the theme “polynomials, gcd, lcm” as a tool to introduce new concepts (definition of gcd/lcm for $K[X]$) and to prove Bezout's theorem (the
21. Montrer que l'anneau Z[i] des entiers de Gauss est principal.

22. Montrer que l'ensemble des rationnels du type 10ⁿx avec n ∈ Z et x ∈ Z est un anneau principal.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** A structuralist theorem that remains implicit</figcaption></figure>

niche). The concept of ideal is then reused in the final theme of the sector, “rudiments on algebraic numbers” as a tool to define the annihilator and the minimal polynomial of an algebraic number (another niche). The task “prove that $A[X]$ is principal if and only if $A$ is a field, where $A$ is supposed to be an integral domain” is posed as an isolated task. On the other hand, we can group different tasks in the form of classes that we do not wish to describe further (in terms of types of tasks, nor to count) because the notion of ideal intervenes only at the theoretical level of the praxeologies (and does not appear in written proofs). We thus note the classes $\tilde{T}_4$ (tasks involving the gcd or lcm in $K[X]$; e.g. “to calculate the gcd of $X^a-1$ and $X^b-1$”), and $\tilde{T}_5$ (tasks involving the minimal polynomial of an algebraic element; e.g. “to find the minimal polynomial of $a = \sqrt[3]{2}$”). In our model, we consider that these 2 classes generate 2 LPs, represented in light grey in figure 5.

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Abstract from the annals of grandes écoles</figcaption></figure>

In the sector “algebra structure”, which gives rise to a structuralist RP “A-algebra”, the concept of ideal intervenes in the theme “algebra generated by an element”, at the level of the definition of the annihilator of an element $a$ of an algebra $A$, via the morphism of algebras $\phi_a : K[X] \to A$ defined by $\phi_a(P) = P(a)$. This construction is presented as a generalization of the case of algebraic numbers ($K = \mathbb{Q}$ and $A = \mathbb{C}$). We note that the progression from sector 2 to 3 is bottom-up (from the particular to the general), unlike the move from sector 1 to 2 which is top-down (general to particular). We count 3 tasks that relate to 2 different types: $T_1$, which is taken up with a new technique $\tau'_1$ (show that $I$ is the kernel of a ring homomorphism), figure 4, and a new type $T_6$ (prove that an ideal of $K[X]$ is generated by a unique monic element), also illustrated in the same figure (second part of the question). These tasks live in the context of the competitive exams to the Grandes Ecoles. In our model, we consider that $\tau'_1$ and the new technology $\theta'_1$ (the kernel of a ring homomorphism is an ideal), hence this time a level 2 structuralist praxeology, come to enrich the PP
issued from $T_1$ to form an LP named "belonging to the class ideal", located in the RP "usual algebraic structures (rings)". We also consider that $T_6$ generates an LP "form of ideals of $K[X]$" located in the mixed RP "arithmetic of polynomial rings". Finally, the sector "algebra structure" takes up the class $\tilde{T}_5$, in its wider context, so that the LP "minimal polynomial of an algebraic element" will finally be located in the RP "A-algebra".

<figure><img src="image_6.png" /><figcaption>Figure 5. The CPGE IPM</figcaption></figure>

In the last sector “linear algebra and reduction of endomorphisms”, the concept of ideal appears only in the theme “reduction: minimal polynomial” where it serves to define the minimal polynomial of an endomorphism (the niche). The general results obtained in the “algebra structure” sector are taken up on this particular domain of objects, which leads us to position a mixed RP “reduction of endomorphisms” above the structuralist RP “A-algebra”. It includes only the class $\tilde{T}_7$ (tasks of reduction of endomorphisms involving properties of the minimal polynomial), hence a last LP “minimal polynomial of an endomorphism” represented in light grey.

The structuring scheme of the praxeologies developed by MP1 is drawn in figure 5. To summarize the structuralist levels of praxeologies, we have two praxeologies of level 2 (in the two LPs in dark grey), with an implementation of the dialectic between objects and structures (the calculation of the radical in the case of $\mathbb{Z}$). The two PPs are of structuralist level 1 and the other LPs involve the notion of ideal only at the theoretical level of the praxeologies.

## 4.2. The case of $B_2$ (as exemplified by EC2)

The official syllabus of EC2 mentions notions (e.g. polynomials, linear systems), techniques (e.g. elementary operations on matrices) and theorems (e.g. D'Alembert-Gauss, Cayley-Hamilton). The course content is organized around objects (e.g. $K[X]$ for $K = \mathbb{R}$ or $\mathbb{C}$, vectors and matrices of dimension $n$ on $\mathbb{R}$ or $\mathbb{C}$), which is the occasion to approach structures (groups, rings and algebras), and it culminates with general results on endomorphisms and their reduction. This indicates a bottom-up organisa-
tion but the concept of ideal is not mentioned. To identify sectors and themes, we follow the course handout provided by EC2, divided into five parts (thus 5 sectors): “a short panorama of algebraic structure”, “the symmetric group $S(n)$”, “the algebra of polynomials $K[X]$”, “determinants” and “reduction of endomorphisms”.

EC2 chooses to introduce the ideal concept at the level of the theme “arithmetic of $K[X]$” (under the subject “the point of view of ideals”) of the sector “the algebra $K[X]$”, which constitutes its main habitat with the principality of $K[X]$ as niche. The gcd (defined from the divisibility relation) is reinterpreted in terms of ideals, hence the name of the subject. Not surprisingly, these results are subsequently applied to the theme “polynomials of endomorphisms” of the sector “reduction of endomorphisms”.

An exercise sheet accompanies the first sector. We therefore consider that EC2 constructs a structuralist RP named “algebraic structures (rings)” (a bottom-up approach). The concept of ideal is introduced in the mixed RP “algebra $K[X]$”. EC2 strictly limits the theoretical development to the trophic needs of the ecology suggested by the official program (principality of $K[X]$, existence of gcd/lcm). Precisely, we find 4 instances of the type of tasks $T_1$, which generate the only praxeology of the corpus (except application in linear algebra) and which are all decontextualized: it consists in proving that the sum and intersection of ideals is still an ideal, starting with the case of principal ideals. Hence, we call the developed PP “belonging to the class ideal (via definition)” and place it in the structuralist RP. The mixed RP contains only two isolated tasks, this time contextualized to $K[X]$ and which aim at proving theoretical elements stated in the course (principality of $K[X]$ and existence of the lcm).

The RP “reduction of endomorphisms” is the place of development of the LP “minimal polynomial of an endomorphism”. A last abstract task is assigned in this RP to define the minimal polynomial: prove that the kernel of a homomorphism of $K$-algebras is an ideal. It shall be noted that the definition of a ring homomorphism is not mentioned in the course. We therefore don’t connect the task to the PP based on $T_1$. Finally, the structuralist level of praxeologies is always 1 and most tasks are decontextualised, which shows a deficit of the dialectic between objects and structures. The structuring scheme of praxeologies is given in figure 6.

<figure><img src="image_7.png" /><figcaption>Figure 6. The B2 IPM</figcaption></figure>
### 4.3. The case of B3 (as exemplified by EC4)

The syllabus of EC4 is quite detailed: it includes both a large panel of concepts (ideal, ideal generated by a subset, quotient<fnref n="8" /> rings, prime<fnref n="9" />/maximal ideal, PID/Euclidean and unique factorization<fnref n="10" /> domains) and the study of their properties in a structuralist perspective (behavior of ideals under homomorphisms, isomorphism theorems), from which structuralist theorems result (e.g. characterization of prime and maximal ideals by quotient properties), but it also mentions specific examples that must be treated ($\mathbb{Z}$, $K[X]$, $\mathbb{Z}[X]$, $\mathbb{Z}[i]$). We thus find the paradigmatic examples (Durand-Guerrier et al., 2015) used by MP1 and the perspective of unification between the contexts of numbers and polynomials, which gives meaning to the abstract theorems. Unlike MP1, the learning goals are organized around the structures. The institutional perspective is top-down, by contrast with the bottom-up perspective of B2.

EC4 has provided 3 course handouts, each with an exercise sheet. In the first one, entitled “the language of rings and fields”, he defines and gives the properties of groups, rings, fields, algebras and corresponding morphisms. The concept of ideal does not appear there, but we consider in our model (figure 9) that it is the place of development of a structuralist RP called “generalities on rings and fields”, on which praxeologies on ideals will be based. The second handout, entitled “ideals and quotients”, includes 5 parts which we consider as sectors and of which we mention in brackets the sub-parts, considered as themes: Abelian group quotient, ideals and quotients (ideals, quotient by an ideal, factorization of morphisms, ideals of a quotient), polynomials and ideals (Euclidean division, ideals of $K[X]$, connection with annihilator and minimal polynomials), prime and maximal ideals (prime ideals, maximal ideals, maximal implies prime), operations on ideals (sum/intersection/product, Chinese remainder theorem). This second handout illustrates well the fact that the notion of ideal is at the heart of this course, with quotients as a main niche, and all the exercises dealing with ideals (12 in total) are located in the corresponding exercise sheet.

The notion of ideal still appears in the third handout, entitled “divisibility in integral domains” (PID, Euclidean and unique factorization domains), but no exercise is associated to it. In the generalisation of arithmetic to abstract rings, the teacher thus focuses attention on the divisibility between elements and not on the set-theoretic point of view of ideals. This choice of didactic transposition is classical at university: if the notion of ideal emerged from Dedekind’s work on Kummer’s ideal numbers, leading to the notion of factorization in product of prime ideals (Dedekind’s rings), and if the criterion of factoriality leads to the notion of Noetherian ring, these aspects are often postponed to the master’s level.

The analysis of the 12 exercises on ideals leads to the identification of 44 tasks which are divided into 4 sectors. Figure 7 shows the different themes involved in each sector, where the size of the block reflects the number of tasks. The arrows indicate that one sector/theme feeds the trophic needs of another sector/theme.

As an example, we find in the theme “quotient by an ideal” an exercise entitled “a few ideals of $\mathbb{Z}[i]$” (figure 8) which contains 4 tasks related to institutional types of

<footnote n="8">⁸The quotient $A/I$ of a ring $A$ by an ideal $I$, e.g. $\mathbb{Z}/n\mathbb{Z}$, is composed of equivalence classes for the congruence relation modulo $I$ and it inherits from $A$ its ring operations.</footnote>

<footnote n="9">⁹A prime ideal is a subset $P$ of a ring $A$ that shares important properties of a prime number in $\mathbb{Z}$: if $a$ and $b$ are two elements of $A$ such that $ab \in P$, then $a \in P$ or $b \in P$. It is required that $P \neq A$.</footnote>

<footnote n="10">¹⁰A unique factorization domain $A$, also called factorial ring, is a ring in which a statement analogous to the fundamental theorem of arithmetic holds: $A$ is integral and every non-zero non-unit element can be written as a product of irreducible elements, uniquely up to order and units. Irreducible elements generalize the notion of prime numbers in $\mathbb{Z}$.</footnote>
<figure><img src="image_8.png" /><figcaption>**Figure 7.** Ecology of the ideal concept: location of tasks assigned by EC4</figcaption></figure>

tasks. We thus identify $T_1$ (prove that a subset $I$ of a ring $A$ is an ideal) - question

6. **Quelques idéaux de $Z[i]$.** On travaille dans l'anneau des entiers de Gauss $Z[i]$ (voir l'Exercice 3 de la feuille de TD 1).

1) Soit $I$ l'ensemble des $a+bi \in Z[i]$ tels que $a$ et $b$ ont la même parité.

a) Montrer que $I$ est un idéal de $Z[i]$.

b) Montrer que $I = (1+i)$.

c) Montrer que le quotient $Z[i]/I$ est isomorphe à $Z/2Z$. (On pourra utiliser soit la première description, soit la deuxième description de l'idéal $I$).

2) Montrer que l'idéal $(3+i, 5+i)$ est principal.

<caption>**Figure 8.** EC4 - exercise 6</caption>

1a), $T_8$ (prove that an element $a$ generates the ideal $I$) - 1b, $T_9$ (prove that an ideal $I$ is principal) - question 2. The question 1c is a task that we rather consider as belonging to a class of types of tasks, noted $\tilde{T}_{10}$ (tasks which mobilize the ideal concept only via the elements of a quotient $A/I$) and whose technique consists in working with representatives of equivalence classes modulo $I$. The ideal notion intervenes only through the theoretical idea that it is the right notion to construct quotient rings. We model $\tilde{T}_{10}$ as an LP denoted "equivalence classes in a quotient ring" and represent it in light grey in figure 9. Removing the class $\tilde{T}_{10}$ brings us down to 41 tasks.

These 41 tasks may be linked to 17 institutional task types while 4 tasks remain isolated. The entire set of praxeologies is described in the thesis manuscript, by theme and sector, on 7 pages (Candy, 2020a, pp. 405-411). In the following, we limit ourselves to illustrating some of them, in order to highlight the most striking features of our IPM. In particular, we study the unification of praxeologies: for example, $T_8$ and $T_9$ generate two praxeologies of the LP "principality of ideals". In accordance to the theoretical framework, some LPs are constructed transversally to different sectors and themes: this is the case of principality, where $T_9$ is taken up in the theme "factorization of morphisms" (contextualized to $\mathbb{Z}^2$) and in the sector "polynomials and ideals" (contextualized to $K[X]$) with the same technique. Then a new PP based on $T_{11}$ (show that an ideal is not principal) is aggregated in the sector "prime and maximal ideals". The teacher takes advantage of the conceptual contributions of this sector to put into perspective the case of the ideal $(2, X)$ of $\mathbb{Z}[X]$ (the ideals $(2)$ and $(X)$ are prime but not maximal), although this is not involved in the technique used.
In the same way, the LP “quotient ring” is initiated in the theme “factorization of
morphisms” with $T_{12}$ (prove the existence of an isomorphism $B \simeq A/I$), around the
application of the “first isomorphism theorem”<fnref n="11" />. Then a new PP is aggregated in
the sector “polynomials and ideals”, generated by $T_{13}$ (simplify a quotient ring $A/I$),
contextualized to $K[X, Y]$, and taken up again in the sector “prime and maximal
ideals” (case of the quotient $\mathbb{Z}[X]/(X^2 + 1)$). The development of the LP ends in
the sector “operations on ideals” with $T'_{13}$ (simplify a quotient $K[X]/(P)$ where $P$ is
irreducible). The additional hypotheses allows to refine the technique to a context less
general but of great applicative scope (construction of $\mathbb{C}$ and theory of field extensions).
We also note $T''_{13}$ (simplify a quotient $A[X]/(P)$) where the technique consists in
decomposing $P$ into irreducible factors and applying the Chinese remainder theorem<fnref n="12" />.
This is a new technological element that enriches the available isomorphism theorems.
It also results in a new type of tasks $T_{14}$ (show that two products of quotient rings of
$\mathbb{Z}$ are isomorphic), as an application of the latter theorem.

If we now consider the unification by theories, we note that two mixed RPs are graf-
ted onto the structuralist RP “generalities on rings and fields”. One, which we denote
“arithmetic of $\mathbb{Z}$” (containing the definitions of divisibility and of gcd/lcm, reformu-
lated in terms of ideals), provides the theoretical framework of the first encounter with
the concept of ideal, within the theme “ideal”. The corresponding praxeological devel-
opment is reduced to a single PP “operations on ideals in $\mathbb{Z}$”. The second mixed RP,
called “polynomials over a field” (with emphasis on ideals of $K[X]$ but also $K[X, Y]$),
is built on the sector “polynomials and ideals”. Polynomials (to a lesser extent $\mathbb{Z}$)
constitute the main objects to contextualize structuralist praxeologies. The latter are
also developed in the form of abstract tasks, hence the representation of LPs and PPs
“straddling” the structuralist and mixed RPs. Finally, the LPs “belonging to the class
prime/maximal ideal”, whose types of tasks are reduced to showing that an ideal $I$ is,
or is not, prime/maximal, call for a theory that goes beyond the generalities on rings
and fields, hence their insertion within a second structuralist RP called “prime and
maximal ideals”. This brings us to the structuring scheme drawn in figure 9.

Let us now study structuralist levels. Of the 41 tasks previously mentioned, 18 use
a definition and 23 use a theorem about structures. This highlights EC4’s didactic
intention to reach a structuralist level 2. More precisely, the praxeologies mobilizing
prime or maximal ideals (for example, the praxeology generated by $T_{15}$ (show that
an ideal $I$ of a ring $A$ is prime), figure 10 exercise 7, mostly use as technology an
isomorphism theorem and thus work at a level 2. On the other hand, those concerning
the principality of ideals (e.g. $T_9$) have as technology the definition of a principal
ideal and are thus mostly done “by hand”. Indeed, they are mostly situated in the
sector “polynomials and ideals” and are contextualized to $K[X]$. We do not note
any subsequent development of the praxeology at the structuralist level 2, when the
principality of Euclidean domains is known.

Regarding the dialectic of contextualization and decontextualization of structuralist
praxeologies, 10 out of 41 tasks are abstract tasks. Within the corpus, we find occur-
rences of task types that are first contextualised and then decontextualised in order
to demonstrate a general result. This is the case, for example, of $T_{15}$, contextualized
twice in $\mathbb{Z}[X]$ in exercise 7 (figure 10), before being posed to the case of the inverse
image of any prime ideal by a ring homomorphism, and then to the case of an ideal

<footnote n="11">¹¹Let $\varphi: A \to B$ be a ring homomorphism. Then the image of $\varphi$ is isomorphic to the quotient ring $A/\ker \varphi$. See also Hausberger (2017).</footnote>

<footnote n="12">¹²Let $I_1, \dots, I_k$ be ideals of a ring $A$ and let $I$ be their intersection. If the ideals are pairwise coprime, we have an isomorphism $R/I \simeq R/I_1 \times \dots \times R/I_k$ defined by $x \bmod I \mapsto (x \bmod I_1, \dots, x \bmod I_k)$.</footnote>
<figure><img src="image_9.png" /><figcaption>Figure 9. The B3 IPM</figcaption></figure>

7. **Idéaux de Z[X].** On travaille dans l'anneau Z[X].

1) Montrer que l'idéal (2) est premier mais pas maximal.

2) Montrer que l'idéal (X) est premier mais pas maximal.

3) Montrer que l'idéal (2, X) n'est pas principal. Montrer que c'est un idéal maximal.

4) Montrer que l'idéal $(X^2+1)$ est premier mais pas maximal. Reconnaissez-vous le quotient $Z[X]/(X^2+1)$?

8. **Idéaux premiers.** Soit $A$ un anneau et $P$ un idéal de $A$ avec $P \neq A$. Montrer que $P$ est premier si et seulement si pour tous $I, J$ idéaux de $A$ on a : $IJ \subset P \Rightarrow I \subset P \text{ ou } J \subset P$.

<caption>Figure 10. Illustration of $T_{15}$ in EC4's corpus</caption>

$P$ defined abstractly by a condition that may be interpreted as a rewriting, in the set-theoretic terms of ideals, of the definition of a prime element (figure 10 exercise 8). The corresponding praxeology is of structuralist level 1, unlike the one applied to $Z[X]$. We can regret that the usefulness of these results is not highlighted by new contextualisations, in the spirit of the dialectic between objects and structures. The other abstract tasks are related to the discussion of the consequences of the presence of invertible elements in an ideal, to the determination of the ideals of a Cartesian product of rings and its consequences for a quotient of such a product. These are therefore theoretical results which are intended to feed contextualized structuralist praxeologies of level 2. For example, the question of the determination of the ideals of $Z^2$ is posed in application of the principality of $Z$ and the results on the Cartesian product. A praxeology generated by $T_9$ is thus obtained, at structuralist level 2.
# 5. Discussion

Our analyses are limited to the case of three professors chosen as representatives of CPGE, B2 and B3. However, Candy (2020a) provides results on a larger corpus that support these analyses. In the context of this article, we can provide some initial answers to our research questions. We begin by synthesizing our findings regarding the structure and structuralist levels of praxeologies focusing on ideals in relation to the issue of structuralist transitions (research question 1).

The analysis of the syllabi shows that the type 1 transition (access to structuralist thinking) is not an institutional learning goal of CPGE and B2. Accordingly, when tasks are contextualized to object domains, mainly $K[X]$ and $Z$, the technique consists in applying the definitions of concepts without relating the properties at stake to general results. Thus, the structuralist praxeologies involved are all level 1 and the objects-structures dialectic remains largely invisible to students in these institutions.

Nevertheless, MP1 has chosen to introduce a relative diversity of examples of PIDs which are all Euclidean, and he uses a meta discourse to allude to the underlying general principle (a structuralist theorem). This didactic gesture may be considered an intention to facilitate the type 1 transition. At university, EC2 introduces structures as a “vocabulary” and assigns abstract tasks on basic formal properties of ideals. The type 1 transition is therefore envisaged from a top-down perspective by the teacher: although the course content is organized around domains of objects (polynomials, matrices, endomorphisms), concepts are introduced beforehand and taken as a given. This strategy may be questioned since it may be argued that the resulting level 1 structuralist praxeologies will tend to be weakly motivated.

In praxeological terms, the top-down perspective translates into the initial development of a structuralist RP dedicated to generalities about structures, notably rings and fields. Mixed RPs dedicated to the arithmetic of $Z$ and $K[X]$ are then grafted on, so that the dialectic between objects and structures results in a pattern alternating between structuralist and mixed RPs. The praxeological development remains limited at B2 while it shows a much wider spectrum of LPs and RPs in the CPGE. The compartmentalisation of mathematics teaching into modules at university is probably a factor inhibiting the unifying perspective of structuralist algebra: the emphasis in the second year is on linear algebra and the teaching of ideals is weakly supported by the arithmetic of $Z$. Conversely, teaching by a single teacher throughout the school year and the prospect of entrance examinations to the Grandes Écoles seem to stimulate the structuralist perspective in CPGE. MP1 does not hesitate to go off the beaten track to mobilize notions such as algebraic integers or even Lie algebras in an extra-curriculum structuralist RP dedicated to A-algebras.

It is from the third year of the Bachelor's degree onward that the type 1 transition appears as a real objective. We have seen that the B3 syllabus at EC4's university is organized around structures. Moreover, EC4 deploys praxeologies, in numerous contexts, which mobilize the concept of ideal at a structuralist level 2, notably around prime/maximal ideals and isomorphism theorems. The objects-structures dialectic is at play and assigned abstract tasks punctually serve to subsequently develop contextualized level 2 structuralist praxeologies. Accordingly, the structuring scheme of praxeologies shows a pattern alternating between structuralist and mixed RPs as well as a much wider spectrum of structuralist LPs compared to CPGE/B2. Assigned tasks no longer focus on the question of whether or not given objects belong to a type of structure (ideal or principal ideal, PID); the diversification of types of tasks is impressive and due to a large extent to the new niche offered by the notion of quotient
of a ring by an ideal and its consequences in abstract algebra.

The type 2 transition does not appear as a learning goal in the institutions under study. Abstract tasks are rare in CPGE and restricted to the first basic properties of ideals in B2. At the third year, EC4 assigns tasks to explore structuralist properties of ideals (e.g. behavior under inverse image, Cartesian product). Nevertheless, the main application is the enrichment of contextualized level 2 structuralist praxeologies. A body of abstract knowledge in which ideals play a major role (e.g. Elimination Theory, Algebraic Geometry) is not in the horizon. However, the scope of exercise 8 (figure 10) may be connected to Algebraic Geometry. The teacher probably had this connection in mind, but it remains invisible to the students. Thus, the type 2 transition is left to the Master's degree.

To answer the question of continuities and ruptures which can be observed in the abstract algebra track around the concept of ideal through the Bachelor in France (research question 2), it seems important to us to recall that the students who take the course of EC4 could have first followed the course of EC2 or that of MP1. However, the treatment between EC2 and MP1 appears different. In both cases, the praxeologies involved are of structuralist level 1. But, if EC2 chooses to work on praxeologies in a decontextualized way, MP1 chooses to work on contextualized praxeologies and to accompany the type 1 transition by meta-discourse. Thus, students from the MP1 course could be better prepared for the upcoming transition since the structuralist perspective is pointed out as an horizon.

Structuralist praxeologies at the third year of Bachelor are either of level 1 or 2. The enrichment of structuralist technologies is considerable and simultaneously produces a rise in the structuralist level of praxeologies and a complexification of its structuring in LPs that is clearly visible in our IPMs. The resulting epistemological and cognitive rupture should not be underestimated.

Hausberger (2018, p. 89) makes the assumption that “a type 1-transition problem typically arises whenever the teacher adopts a top-down approach by setting out first the logos block Λ (figure 1) and then more or less artificially connecting that part to a praxis blocks Π<sub>s</sub> that is, without referring to the praxis block Π from which it historically originates, as a means to highlight the rationale of the new concepts and techniques”. In B3, we have observed that structuralist praxeologies are most of the time elaborated from theoretical elements provided a priori, in a top-down perspective. The analogy between the arithmetic of numbers and polynomials, carefully developed by MP1 in a bottom-up perspective, remains a missed opportunity to develop a structuralist praxeology in the EC4 course. On the whole, the problematisation of ideals by quotients remains very formal and not very much driven by the investigation of concrete problems anchored on earlier practices with mathematical objects.

Finally, the transition of type 2 is not worked out (as we expected). It would be necessary to analyze a corpus of teaching material at the Master's degree to measure the epistemological gap in praxeological terms.

As for the management of transitions by teachers (research question 3), our study emphasizes the central role played, for both transitions, by the objects-structures dialectic. Regarding the type 1 transition, continuities may be favoured by means of a meta-discourse that emphasizes the potential of a technical element of a structuralist level 1 praxeology to be generalized to a structuralist technology of a forthcoming level 2 praxeology (figure 3, MP1). The type 1 transition can also be managed through the choice of variables contextualizing the tasks which favour the use of level 2 praxeologies and make level 1 praxeologies costly. This is the case, for example, of EC4 in the framework of prime and maximal ideals (figure 10, exercise 7). Finally, we have seen that
the type 2 transition can be punctually favored by the dialectic of contextualization and decontextualization of praxeologies (figure 10).

## 6. General conclusion and outlook

The research presented in this article deals with the notion of ideal, which constitutes a privileged entry point on the transpositive phenomena linked to structuralist algebra. It allows in particular to pose the problem of the designation of the elements of knowledge to be taught, their delimitation and organization in domains, sectors and themes. The study of the knowledge taught at university makes it possible to show how the processes of didactic transposition also act on these levels which could be qualified as “mesodidactic” and which one sometimes tends to leave in the shade. To address these aspects, we have proposed an original methodology combining an ecological analysis (in terms of niches, habitats and trophic networks) of the notion of ideal within the corpus of course documents, followed by a praxeological analysis leading to representations called “structuring schemes of praxeologies”, whose codification highlights certain choices of didactic transposition. In particular, we introduced the notion of mixed RP and showed that the dialectic between objects and structures appears in this scheme through the way RPs are built on previously constructed RPs in an alternating pattern between structuralist and mixed RPs.

We have also, in this paper, made use of the model of structuralist transitions introduced by Hausberger (2018). With reference to this model, the predominance of the theoretical functions of the notion of ideal results from a discontinuity which we have analysed as a type 1 transition (with type 2 transition moments too). These transitions pose many learning problems. Of course, the chosen methodology allows us to investigate these issues only indirectly, through an institutional praxeological model accounting for the praxeological equipment of the students in an ideal way, i.e. under the assumption that the relationship of the students to the notion of ideal is in conformity with the one foreseen by the institution. We can thus highlight fragile links with previous practices and knowledge (such as the study of the arithmetic of $Z$ and $K[X]$) and didactic organisations which certainly put in place certain local strategies to accompany the development of structuralist praxeologies of level 2 (in the sense of our model), but which globally struggle to motivate the theoretical development. The notion of quotient of a ring by an ideal and all the structuralist technologies linked to it remain an end in itself (intra-theoretical) in a classical approach (top down, logos then praxis) of the algebraic structures, which unfold in a complex network of structuralist LPs in B3. These structures are already laid down upstream from CPGE/B2, taking advantage of a use, quite anecdotal all in all, of ideals in the definition of the minimal polynomial (confined to the theoretical level of praxeologies). The strong constraint of the official syllabus, located at the level of the discipline in the scale of the levels of didactic co-determinacy, that of the fragmentation into modules at university (level of pedagogy), of didactic time (level of the School) and of the competitive examinations at the Grandes Ecoles (level of Society) contribute to explain an overall inertia at the university already highlighted by Bosch et al. (2021) as well as a greater creativity in the capacity of the institution to leave the beaten track in CPGE compared to B2.

The analysis of conditions and constraints needs to be strengthened in order to deepen our understanding of the ecological dimensions of the didactic phenomena of transposition that have been identified. For this purpose, Candy (2020a) conducted complementary semi-structured interviews of the university teachers, which we can-
not report here for lack of space and which deserve to be the focus of an independent
article. The constraints outlined above were highlighted by the interviewees. Jovignot-
Candy also conducted a more systematic study than the three cases presented here,
with at least two different post-secondary institutions analysed by level of study, as
well as representing the case of French-speaking Switzerland. This reinforces the results
presented here while offering the possibility of contrasting the case of France (while
remaining within the Francophonie). The methodological problem of aggregating in-
stitutional praxeological models of a given level into a dominant praxeological model
of that level then arises. We have chosen not to address these aspects here in order to
focus on the explicitation and illustration of our methodology on representative case
studies, which already shows a thorough use of our theoretical framework. We there-
fore limit ourselves to mentioning these aspects as avenues for deepening the study
presented in this paper.

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