Renaud Chorlay
email: renaud.chorlay@inspe-paris.fr

Zoé Mesnil
email: zoe.mesnil@univ-paris-diderot.fr

From an analysis of content to an analysis of ordinary lecturing practices: A case-study in mathematical analysis

Keywords:

This paper bears on teaching practices in lecture courses on analysis in the first year of tertiary education. This case-study shows how the knowledge accumulated by the didactical community on the challenges in the teaching of a specific notionthe formal definition of limits of sequencesallows for a fine-grained analysis of teachers' practices. On the basis of this knowledge, three lecture courses were analysed and compared. Post-teaching interviews were used to test our hypotheses as to the didactical knowledge, choices and repertoire of the lecturers. This progress reports aims to sketch the research rationale and to discuss a small sample of results.

INTRODUCTION

This paper reports on a research project developed in the framework of activity theory [START_REF] Hache | Why and how to understand what is at stake in a mathematics class[END_REF]. It aims to contribute to its adaptation to the study of the teaching practices in lecture-courses [START_REF] Bridoux | Les moments d'exposition des connaissances. Analyses et exemples[END_REF], with a specific emphasis on the challenges of higher education (as in (Grenier-Boley, [START_REF] Bridoux | Les moments d'exposition des connaissances. Analyses et exemples[END_REF]). In this context, the focus is on "teaching", regarded as a professional activity. We hypothesize that teaching practices can be described, analysed, (to some extent) accounted for, and (possibly) altered.

Within this larger context, this paper is of a methodological nature. We aim to study to what extent the analysis of the relief of the mathematical content at stake in a given teaching contexti.e. an analysis combining mathematical, epistemological, didactical, and institutional aspects [START_REF] Bridoux | Les moments d'exposition des connaissances. Analyses et exemples[END_REF] allows the researcher to carry out an analysis of the empirical data based on the identification of observables. These observables will be denoted as "control points" in this paper. In particular, this approach provides means to overcome two common difficulties in the analysis of lecture-courses: The lack of information on actual student activity, on the one hand, and the difficulty to objectively identify what is not done, on the other hand. Moreover, the list of control points supports the construction of questionnaires which enable to researchers to test their hypotheses trough post-lectures interviews with the lecturer.

Beyond its contribution to this research program on teaching practices in higher education (with a focus on lecture-courses), this case studies suggests avenues for the professional development of teachers in higher education [START_REF] Lison | Reflective practice in higher education: From a theoretical approach to a professional development practice[END_REF], [START_REF] Rogalski | De l'analyse de l'activité de l'enseignant à la formation de formateurs. Le cas de l'enseignement des mathématiques dans le secondaire[END_REF]. This aspect can only be touched upon in this short progress report.

THE RELIEF OF THE FORMAL DEFINITION OF LIMITS IN THE FIRST YEAR OF TERTIARY EDUCATION

The challenges of the transition from an intuitive notion of limit and the associated techniques (algebra of limits, connections with inequalities) to a formal definition used to prove theoremsin the larger context of the proof system of analysishas been a continuous focus of didactical investigation since the 1970s [START_REF] Artigue | Mathematics Education Research at University Level: Achievements and Challenges[END_REF]. Numerous works have identified the epistemological, didactical and cognitive challenges students face in this transition. On this basis, several didactical interventions have sought occasionally with some successto identify and put to the trial teaching paths which can prove conducive to the formal definition (see [START_REF] Chorlay | A pathway to a student-worded definition of limits at the secondary-tertiary transition[END_REF] for a recent survey). Moreover, this vast literature forms a fairly coherent whole, since, in spite of the variety of theoretical frameworks, the same key-phenomena are generally identified.

Consequently, we drew on this solid body of didactical knowledge to devise a large list of control points on the basis of which actual teaching practices can be described. The fact that we chose to derive our grid of analysis from a part of the didactical literature that is subject-specifici.e. which bears on the formal definition of limits of sequencescomes at a price. Indeed, the results could be too subject-specific, thus making transfer to other contexts more difficult. Even so, we deemed the price worth paying, for two reasons. First, we wanted to see to what extent a body of knowledge bearing of mainly on students (their misconceptions, their documented behaviour in specific milieus whose didactic variables can be finely-tuned) could provide tools for the analysis of "ordinary" teaching practices, i.e. teaching practices which are not based on this research literature, by lecturers whose professional identity [START_REF] De Hosson | Du savoir savant au savoir enseigné, analyse de l'exposition des connaissances en cours magistral de physique: une etude de cas[END_REF] and trajectory are independent from the didactical community. Second, in terms of professional development (see [START_REF] Lison | Reflective practice in higher education: From a theoretical approach to a professional development practice[END_REF], [START_REF] Rogalski | De l'analyse de l'activité de l'enseignant à la formation de formateurs. Le cas de l'enseignement des mathématiques dans le secondaire[END_REF]), studies suggest that professionals are keener to engage in a reflective practice when the focus is on very specific issues rather than on fairly general challenges.

In the two tables below, we list a series of control points which we identified in the literature on the teaching of limits. This is only a samplealbeit a significant sampleand, for lack of space, each of the points is described only sketchily and without systematic references to the literature. A wide-ranging list of references can be found in [START_REF] Chorlay | A pathway to a student-worded definition of limits at the secondary-tertiary transition[END_REF]. The first table lists negative control points, bearing mainly on cognitive difficulties and misconceptions. The second table mentions more positive control points, since the literature allows for the identification of teaching strategies and moves which can prove conducive to the concept of limit, or, more generally, to advanced mathematical concepts.

The concept image of limits usually encompasses "primitive" models which are non-congruent with the formal definition. In particular, primitive models are usually "x-first"or "covariant", and resort to temporal, dynamic and causal imageries (the values of the variable n goes to infinity, and, as a consequence, the corresponding values of the sequence does this or that), whereas the definition is "y-first" or "contravariant" (the condition on the values of the function or sequence determines conditions on the values of the variable) and string of quantifiers does not reflect or capture any notion of temporal evolution or causality.

The concept image of limits usually encompasses misconceptions, erroneous beliefs, or in-act-theorems such as:

• Every convergent sequence is monotonic (at least as from a certain rank).

• Every sequence which tends to + is monotonic increasing (at least from a certain rank). • If a sequence converges, then the distance to the limit is monotonic decreasing. • If the distance between a sequence and a given number L is monotonic decreasing, then the sequence tends to L.

The nested quantifiers can be interpreted incorrectly in several ways:

• The order of quantifiers does matter, and a  sentence is not equivalent to the corresponding  sentence. • In a  sentence, the second variable depends on the first, but this dependence is not of a functional nature (more explicitly: in ">0  N  ℕ, N can be written N or N(), but N is not actually determined uniquely by the value of ). • The presence of the third quantifier is typically not regarded as necessary by students, who, for instance, tend to regard "A>0  N  ℕ u N >A" as an acceptable definition of the infinite limit. This incomplete definition echoes standard informal formulations such as "the values become arbitrarily large". • More generally, in the absence of the third quantifier, distinct mathematical concepts conflate: finite limit and that of subsequential limit; positive infinite limit and non-boundedness.

The transition between several formulations which are mathematically equivalent can prove challenging for students. For instance:

• For finite limits:

|𝑢 𝑛 -𝐿| < 𝜀  𝐿 -𝜀 < 𝑢 𝑛 < 𝐿 + 𝜀  𝑢 𝑛 ∈ ]𝐿 -𝜀, 𝐿 + 𝜀[ • For infinite limits: ∀ 𝑀 ∈ ℝ ∃ 𝑛 𝑀 ∈ ℕ ∀ 𝑛 ∈ ℕ 𝑛 ≥ 𝑛 𝑀 ⟹ 𝑢 𝑛 ≥ 𝑀  ∀ 𝑀 ∈ ℝ 𝑢 𝑛 ≥ 𝑀 except for (at most) a finite number of terms.
The definition of a convergent sequence can be used in two different contexts, and a non-trivial shift of viewpoints is necessary to use it in a relevant way.

• Context 1 (C1): In order to prove that a given sequence tends to a given number, any positive  has to be taken into account, and the existence of at least one corresponding N value has to be proved. • Context 2 (C2): If some property is to be derived for sequences which are known to converge, then one is at liberty to select and use one specific value for , and the existence of N is warranted. There are diagrams, such as the -strip diagram, which can be conducive from a change of perspective, from an x-first to a y-first perspective.

Discussing and comparing several correct definitions can have a positive effect in terms of conceptual understanding.

Some non-standard yet correct definitions can more likely be in the ZPD of students that the standard definition. This has been documented for the "nonimplicative" form of the definition (here for infinite limits):

∀ 𝑀 ∈ ℝ ∃ 𝑛 𝑀 ∈ ℕ ∀ 𝑛 ∈ ℕ 𝑢 𝑛 𝑀 +𝑛 ≥ 𝑀
Discussing and assessing incorrect definitions can have a positive effect in terms of conceptual understanding, in particular to help student distinguish between neighbouring concepts (e.g. + limit and non-boundedness).

Examples can come in several types, and can be used to serve a variety of purposes, all of which with a positive effect in terms of conceptual understanding:

•

TERRAIN AND METHOD

Three lectures were video-recorded in the first semester of the 2018-2019 academic year at Paris Diderot University. The three lectures were delivered by three experienced lecturers (C, F, and B), two research mathematicians, and a secondary-school teacher with a university teaching position. The lectures were part of the same, 1 rst year, introductory course to mathematical analysis, covering, in this order: functions, continuity (without proofs), properties of real numbers; a final period was dedicated to the formal notion of limits (finite and infinite) for sequences and to proving in analysis.

All the recorded sessions covered: formal definition of converging sequences, proof of the uniqueness of the limit for a converging sequence, proof that converging sequences are bounded. Lecturer B also proved that a subsequence of a converging sequence does converge to the same limit. Lecturer F also covered infinite limits and a few properties (such as: tends to +  not bounded above, while the converse is invalid).

We hypothesize that the lecture-courses under study are "ordinary", or "standard", for several reasons: They took place in standard institutional contexts; they were delivered by experienced lecturers with stable practices, with no involvement with didactical research, and with no claim to pedagogical innovation (as was checked in preliminary interviews); they were not designed specifically for this study, and we have no indication that they might have been affected by the fact that they would be recorded for research purposes. As mentioned above, the use of the control points allows us to identify things which did not happen, even in cases when the comparison among the three lectures would not point to them (lines 2 to 5 of table 3).

The purpose of this paper is not to analyse these data in detail, but to account for how they were used to further the analysis of lecturing practices. The long post-lecture interviews (about 50' each) were not carried out right after the teaching sessions. The videos were first analysed by the researchers. On this basis, the list of topics and the specific questions for the interviews were chosen to allow the researcher to study to what extent the observed practices reflected choices, and on what form of knowledge (lato sensu) these choices rested. The topics addressed in the three interviews were the same, but the wording of the questions differed. To promote reflective analysis, the questions were first based, for each lecturer, on her/his own practice, as in a explicitation interview [START_REF] Vermersch | L'entretien d'explicitation en formation initiale et en formation continue[END_REF]. Then, alternative elements of practicepreferably based on the other two lectureswere mentioned, for the lecturer to discuss and assess.

More precisely:

• When one of the control points was positive in table 3 ("X"), we attempted to investigate to what extent this reflected a choice (for instance, did the lecturer mentioned alternative moves?) and, if, so, how it was justified: What item of knowledge or what belief did the lecturer but forthif anyto account for it? • When one of the control points was negative in table 3 (" "), we also attempted to investigate to what extent this reflected a choice (i.e. a choice not to do something). o If so, how was it justified? What alternatives had been considered and rejected? What item of knowledge or what belief did the lecturer but forth if anyto account for this choice? o If not, we mentioned alternatives so as to study how lecturers made sense of them and assessed them. When their assessment was globally positive, we attempted to spot signs indicating that it could be a lever for professional development: Was the reaction very positive? Did the lecturer readily imagine ways to include this new element of practice in her/his own lecture (extension of the repertoire 1 )? don't have the problem I mentioned earlier" (she had mentioned the belief that converging sequences are monotonic). Other didactic variables went undetected.

Scaffolding the shift of viewpoints when switching between C1 and C2 contexts

As can be seen in the interview below, it took C a little time to identify the issue at stake, but she eventually formulated it in the clearest of ways:

Int.: Maybe it didn't happen in your lecture this time, but we prove the "convergent  bounded" property, it is not uncommon for students to ask questions. The reaction was roughly the same for the other two lecturers. It shows that not mentioning this difficulty in their lectures was not a choice. Also, it suggests that this question is a lever for professional development, since it triggers the realization of a difficulty. However, the reactions of the lecturers were not as enthusiastic as for the strip, and none of them spontaneously mentioned taking this issue into account in their future teaching.

DISCUSSION AND PERSPECTIVES

Our first analyses show that for two of the three teachers, most of the elements observed, then discussed in the interviews, can be justified in some way, but do not reflect choices insofar as no alternatives are clearly mentioned. This conclusion needs to be qualified for the case of B, who mentioned more alternatives and justified choices on the basis of his experience of students' behaviour. The interviews also show a potential for teacher training, as all three teachers adopt a reflective position that leads them to consider changes in their practices. Further analysis is needed to understand why discussing control points and alternatives elicits different reactions.

The point of this admittedly sketchy progress report was to describe the goal and the method of an on-going project. We could not present any results in detail, nor discuss the connections with several theoretical frameworks and issues. From a theoretical viewpoint, this research aims to contribute to collective work of adaptation to the

  The videos were analysed by the researchers. Both the comparison among lecturers and the comparison between what was done and what could have been done (as identified in the list of control points) provides a wealth of information. Table3below shows a sketchy sample of results ("X" stands for "Yes"):C F BUses informal formulations before the formal definition X X X Comments on the two contexts of use for the definition (C1, C2)Mentions alternative correct definitionsMentions boundary examplesUses an example space to assess a candidate-definition Use of x-first, dynamic,

Table 1 : Challenges in the teaching of the formal definition of limits of sequences

 1

Table 2 : Affordances for the teaching of the limit concept and other advanced concepts

 2

  The proof typically begins with "The sequence being convergent to L, I want to show it is bounded. So let's take a neighbourhood of size 1, let's say  = 1" [C nods, approvingly]. Students sometime say "I don't get it,  has to be indeterminate, that's the definition!" Did you ever get such questions? How would you respond to "You can't do that,  has to be indeterminate".

	C:	[Merely rephrases the definition]
	Int:	So, later on, if in exercises students study the convergence of sequences by
		saying: "I say  = 1" …
	C:	[laughs] Oh no! Of course not!
	Int.:	How come?

C:

No, but, there's a difference of … of … in one case we know the sequence converges (…) but when I want to prove that it does (…).

We use "repertoire" to denote the "stock of skills or types of behaviour that a person habitually uses" (Lexico dictionary, https://www.lexico.com/, accessed Jan.

th , 2020.)

Retrieved from http://gilles.dubois10.free.fr/analyse_reelle/suitesconvergentes.html (accessed oct.9, 2019).

specificities of tertiary education of all-purpose theoretical tools such as the MKT framework or the ergonomic and didactic approach to teaching practices (Hache and Robert, 2013). It should also prove instrumental in the adaptation of tools for the assessment of the quality of instruction (LMTP, 2011), in a context where the correctness of the mathematical content taught is not an issue.

SAMPLE OF RESULTS

Use of x-first and/or dynamical informal formulations

In the interviews, the lecturers were first asked to comment on their informal introduction to the definition. In the lecture courses, F was the only one to use contravariant, dynamical formulations; a fact with which he felt completely comfortable in the post-lecture interview, arguing that this formulation carries an intuitive representation of the concept of limit. He mentioned no possible effects in terms of misconceptions:

Int.:

All three of you made a similar choice, in so far as you introduced the notion of limit informally, in everyday language, before diving into the formal definition. Let me quote you: "a convergent sequence, a sequence which has a limit, is a sequence which, as it grows, has values which come closer to the limit value". What is the role of this sentence for you? How would you describe this sentence?

F: What matters is for students to retain the meaning [oral emphasis] of this definition. (…) It's a difficult notion, so it seems to me it's important to emphasize the intuitive side of the notion of limit.

By contrast, C et B used only contravariant rhetorical formulations such as "If (𝑢 𝑛 ) is a complex sequence and L a complex number. One says that sequence (𝑢 𝑛 ) converges to L when 𝑢 𝑛 is "as close to L as one might want provided n is large enough" " (lecturer C, sentence projected). The selection of this formulation reflects a choice for C and B. In C's case, little justification is provided:

Relevance of the -strip diagram to introduces or illustrate convergence

Neither C nor F used the -strip diagram, whereas B used an on-line applet which not only displayed the diagram but allowed the user to change the -value and visualize the effect on the horizontal strip and on the associated value of N() (see figure 1 below). B explained that he chose this applet after watching several of them, and accounted for his choice by mentioning students' concept image of the limit in a way which could reflect the x-first/y-first distinction:

Int. : This on-line applet […] did you sometimes use something else, is that something you chose from among others ? B :

Yes, I chose it, I watched several. What is not intuitive in the definition, compared with the students' representation … students tend to think: there, it's coming closer; while we say that it enters the interval, an interval which becomes ever smaller. It's not exactly the same idea. When commenting on the diagram, B mentioned another didactic variable in the milieu provided by the applet to the class: the fact that the distance between the sequence and the limit is not decreasing. But the fact that N is not necessarily the rank of the first value of u which enters the strip was not mentioned, neither to the class, neither during the interview It turns out that not using this type of diagrams was not a choice for C or F: They readily admitted they had never come across it, nor thought about it. By contrast, both reacted very positively, which suggests avenues for professional development. In spite of his attempts, the interviewer did not manage to elicit any specific analysis of the qualities (or drawbacks) of this diagram. Lecturer C was equally enthusiast, and, when pressed to comment on the choice of the displayed sequence, managed to spot one of the didactic variables in the milieu provided by the applet: The specific sequence on display is not monotonic, "what's not bad is that is goes above then below, so we


Stéphanie Bridoux
email: stephanie.bridoux@umons.ac.be

Martine De Vleeschouwer
email: martine.devleeschouwer@math.unamur.be

Nicolas Grenier-Boley
email: nicolas.grenier-boley@univ-rouen.fr

Rita Khanfour Armalé

Nathalie Lebrun
email: nathalie.lebrun@univ-lille.fr

Zoé Mesnil
email: zoe.mesnil@u-paris.fr

Céline Nihoul
email: celine.nihoul@umons.ac.be

Nicolas Genier-Boley

Rita Khanfour-Armalé
email: rita.khanfour-armale@u-cergy.fr

The professional identity of teacher-researchers in mathematics

Keywords: teachers' and students' practices at university level, preparation and training of university mathematics teachers, teacher-researchers, professional identity

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Many French universities have created professional development structures for higher-education teachers with a perspective of "pedagogical transformation" in order to respond, in particular, to the diversity of the student population [START_REF] Endrizzi | Learning how to teach in higher education: a matter of excellence[END_REF]. While research has focused primarily on the pedagogy of teaching practices [START_REF] Annoot | Pratiques pédagogiques dans l'enseignement supérieur: enseigner, apprendre, évaluer [Pedagogical practices in higher education: teach, learn, evaluate[END_REF], this issue has seldom been addressed through an approach based on the discipline of teacher-researchers [START_REF] Henkel | La relation enseignement-recherche [The relationship between teaching and research[END_REF][START_REF] Neumann | Disciplinary Differences and University Teaching[END_REF], even though several authors stress the need for it [START_REF] Becher | The significance of disciplinary differences[END_REF]. In this context, we have conducted a research based on three academic disciplines (Mathematics, Physics, Chemistry) [START_REF] Bridoux | L'identité professionnelle des enseignants-chercheurs en mathématiques, chimie et physique[END_REF]: in this contribution, we report on the results of this research for teacherresearchers (TRs) in mathematics. Our objective is to better know this community of teacher-researchers by highlighting several aspects of their professional identity, be it transversal aspects or aspects related to mathematics. This could be a first step for building teacher training courses that take into account the values and qualities they emphasize and specific elements of knowledge they enhance. By this approach, we hope to encourage the transformation of teaching practices and to understand teacher-researchers' expectations about students or the secondary-tertiary transition.

THEORICAL FRAMEWORK AND RESEARCH QUESTIONS

The concept of professional identity is complex and appears in the literature under different approaches. From the standpoint of pedagogical practices, the higher-education teacher is a researcher before being a teacher [START_REF] Fave-Bonnet | Les enseignants chercheurs et l'enseignement [Teacher-researchers and teaching[END_REF][START_REF] Musselin | Les universitaires[END_REF][START_REF] Van Lankveld | Developing a teacher identity in the university context: A systematic review of the literature[END_REF]. In mathematics education, the concept of identity of mathematics' teachers is defined in various ways and has been extensively studied [START_REF] Graven | Mathematics identity research: The state of the art and future directions[END_REF]. From the point of view of sociology of work, professional identity is defined as "a set of specific elements of professional representations, specifically activated according to the interaction situation and to respond to an aim of identification/differentiation with societal or professional groups" [START_REF] Blin | Les représentations professionnelles: un outil d'analyse du travail [Professional representations: a tool for analysing work[END_REF]. This relation with the community is found in [START_REF] Cattonar | Les identités professionnelles enseignantes. Ébauche d'un cadre d'analyse [Teachers' professional identities[END_REF] who defines the professional identity as "the characteristics that identify him as a teacher and that the teacher shares with other teachers, which he shares in common with other teachers because he belongs to the same professional group". In this context, [START_REF] De Hosson | Approcher l'identité professionnelle d'enseignants universitaires de physique: un levier pour initier des changements de pratiques pédagogiques [Approaching the professional identity of university physics teachers: a lever to initiate changes in pedagogical practices[END_REF] have retained several dimensions of professional identity understood as "the way in which an individual teacher is defined in relation to his or her professional teaching practice" and have highlighted tensions among TRs in physics. The dimensions retained in this work are related to the profession: norms, qualities and skills required, values [START_REF] Dubar | La socialisation: paradigmes, méthodes et implications théoriques [Socialisation: paradigms, methods and theorical implications[END_REF]. Other studies show that teachers identify strongly with their research disciplines [START_REF] Henkel | La relation enseignement-recherche [The relationship between teaching and research[END_REF]. In the latter case, tensions appear, with TRs valuing the research. [START_REF] Drucker-Godard | Le ressenti des enseignants-chercheurs: un conflit de valeurs [Teacherresearchers' feelings: a conflict of values[END_REF] have shown that French TRs feel "a conflict between values they initially approve (freedom, independence, autonomy, public service) and new values emerging from recent reforms of the university system (scientific productivity, effectiveness, efficiency, individualization of the career, fairness and unequal treatment and esteem" (p. 19). The perception of this conflict seems to be common to many TRs, regardless of their research and teaching disciplines. It is reinforced by the results of [START_REF] De Hosson | Approcher l'identité professionnelle d'enseignants universitaires de physique: un levier pour initier des changements de pratiques pédagogiques [Approaching the professional identity of university physics teachers: a lever to initiate changes in pedagogical practices[END_REF] who find that the professional identity of the physics TRs [...] interviewed appears to be strongly marked by tensions [...] that sometimes appear under the form I know that this should be done and yet I do the opposite. However, we may wonder if there is a discipline imprint on values, and thus on these tensions, whether at the level of public valuesthe epistemology of each discipline leads to potentially different beliefs and organizations -, or at the level of individual values that can influence each TR in the choice of his/her discipline. [START_REF] Van Lankveld | Developing a teacher identity in the university context: A systematic review of the literature[END_REF] also mention that teacher training is perceived as positive in terms of teaching capacity through peer-to-peer exchanges, making them more credible in their institution. This socialization effect is all the more pronounced among teachers who have not received initial teacher education [START_REF] Goodson | Exploring the teacher's professional knowledge: Constructiong identity and community[END_REF]. However, professional identity can be negatively affected when teacher education is perceived as a supervisory mechanism [START_REF] Van Lankveld | Developing a teacher identity in the university context: A systematic review of the literature[END_REF].

The aim of our research is to study the following research question for TRs in mathematics: what is the discipline's imprint on teaching practices at the university? We have two main motivations for conducting this study. On the one hand, we would like to understand the way in which the interviewed TRs perceive training courses in "university pedagogy". On the other hand, we would like to know their expectations towards students in terms of difficulties they mention. These motivations are thus taken into account in both the design of the interview protocol and the analysis of the interviews. Our methodology will also allow us to test the relevance of the following assumptions: a TR is a particular teacher because of its multiple missions, the research discipline is a marker of its professional identity, or even of the institution in which he works.

METHODOLOGY

Our data are taken from the anonymous transcripts of 12 semi-directive individual interviews, lasting between 30 and 90 minutes. To see if the institution or the length of service has any influence with respect to our question, the volunteer interviewees come from four universities2 and have teaching experiences of varying length (3 to 40 years). The interview protocol was initially constructed in such a way that we can identify the norms, qualities and values assigned by the TR to his teaching practices [START_REF] De Hosson | Approcher l'identité professionnelle d'enseignants universitaires de physique: un levier pour initier des changements de pratiques pédagogiques [Approaching the professional identity of university physics teachers: a lever to initiate changes in pedagogical practices[END_REF], which give a pertinent access to his teaching professional identity by hypothesis [START_REF] Cattonar | Les identités professionnelles enseignantes. Ébauche d'un cadre d'analyse [Teachers' professional identities[END_REF][START_REF] Dubar | La socialisation: paradigmes, méthodes et implications théoriques [Socialisation: paradigms, methods and theorical implications[END_REF]. We have amended the initial interview protocol to take into account the teaching discipline, in particular mathematics. In the table below, we give examples of questions that are related to each of these dimensions (norms, qualities, values, discipline). During the interviews, we discussed the following topics: organization of teaching, innovative practices, difficulties and evaluation of their students, TRs' training, the profession of TR, didactic questions (objective and content of a course). To analyze the interviews, we have conducted an empirical categorization by identifying verbatim excerpts that seem to be relevant for our research question. We consider a verbatim excerpt as a unit of meaning [START_REF] Bardin | L'analyse de contenu[END_REF], which may fall within several significant categories related to the discipline of the TR interviewed and to the attributes of its research activity. We have thus tried to spot regularities and variabilities that are intra or interdisciplinary. More specifically, we have looked for verbatim statements reflecting the influence of the research profession on the teaching profession, student assessment or teacher training. We have thereby tried to identify elements that refer to a researcher posture in teaching practices (creativity, freedom, collegiality, peer review and peer learning) or to the transposition of research methodological elements into the activities proposed to students (problem solving, reflexivity, student presentations, group work, etc.). We have also identified tension elements [START_REF] De Hosson | Approcher l'identité professionnelle d'enseignants universitaires de physique: un levier pour initier des changements de pratiques pédagogiques [Approaching the professional identity of university physics teachers: a lever to initiate changes in pedagogical practices[END_REF] often stemming from contextual factors (institution, relationship with students, teacher training).

RESULTS

Different aspects emerge that allow us to provide various types of answers to our research question.

Mathematical difficulties and transversal competences

During the interviews, TRs were asked to identify students' difficulties when entering university. Whereas the answers to this question are varied, we can spot regularities that concern either mathematical difficulties or mathematical conceptions and skills that are related to their researcher position.

Concerning mathematical difficulties, the majority of TRs point to gaps with specific concepts taught in secondary school: continuity, derivation, tangent to a curve, lack of logical knowledge.... Thus, students have "learned to perform calculations" and do not "know how to pass from an equation to a straight line or from a curve to an equation". The fact that students misunderstand, forget or have compartmentalized knowledge is mentioned by some TRs, with doubts concerning the origin of their difficulties.

M7:

And so we cannot know whether students' difficulties are due to the fact that they have not seen or assimilated these notions, or that they have assimilated and forgotten them.

Most TRs also highlight examples of high school mathematical practices that they see as blocking factors at the university level: favoring "recipes" rather than understanding the concepts, having a local understanding of courses rather than a global vision, having problems with formalizing an intuition. Many TRs also speak of a particular difficulty in mathematics related to the production and understanding of proofs. This is both an epistemological (understanding what a proof is) and mathematical issue (understanding the different statements inside a proof and the relations between them).

M5: I think that, at least unconsciously, we think that the students know what a proof is, when to say that something is true or false, etc. and they didn't actually learn that in secondary school.

In particular, TRs deplore a difficulty for students to enter into an abstraction and formalization process, and to conceive the relations between the different mathematical objects. The different excerpts show a tension among TRs between a will to present a rich and complex mathematical universe and a feeling that students are rather in a process of learning in a "fragmented way" without trying to "put the pieces back together".

M7:

More and more, a level course deletes skills from the level course.

Most TRs agree that the attitudes of many students in courses or in exercise sessions do not allow them to overcome all these difficulties: lack of personal work, difficulties with concentration or attention, lack of autonomy or selfconfidence, difficulties to concentrate, be attentive, or to master the language. Some TRs, however, find mitigating circumstances for some students: financial hardship, lack of time, transportation difficulties, having a part-time job.

M8:

There are some people who have a job, for example, so they have little time to work or etc. There are others who just don't have a job and do nothing.

However, what seems to be important for the TRs interviewed, is not that students work more, but rather that they really try to confront themselves to exercises or questions, so that they can "become theirs".

M3:

The work during exercise sessions cannot replace personal work, for example the course or even trying to do exercises by yourself.

For some TRs, the most important thing in a course is to insist on study processes in order to encourage students to continue the work personally, and to equip them methodologically for such a work.

M2:

The subject matter itself is important, but it is somewhat the mechanism that students need to learn in thinking that can be interesting.

For some TRs, this contradicts students' expectations: to have a well-structured course and a clear identification of the elements that will be evaluated.

All TRs point the difficulty for students to adapt to university mathematics. For a large majority of TRs, they "do not do math" in secondary school, in the sense that there is a change in mathematical practices between secondary school and university, "that they do not speak the same language". M8: I think we have a lot of students who don't know what it's like to do math and who are in this kind of attitude where they're going to try to learn a little bit of algorithmic things, some ready-made methods, that they will reproduce in a very similar context without any thought on the substance…

The following excerpt illustrates a mathematical activity that seems unsuitable for higher education.

M1:

In the first year, perhaps, I think that there is a first difficulty in adapting to our requirements. Which is not necessarily related to the matter itself but which is linked to when we define a concept, what does it mean to understand a concept?

This inadequacy is sometimes expressed as a difference between a research practice that requires almost permanent questioning, and students' expectation of a reassuring practice.

M4:

As a teacher-researcher, the researcher is not safe because he or she is looking for a way to disrupt everything he or she knows, as soon as there are things he or she doesn't understand, he or she is often faced with difficulties. That's what we'd like to do in our classes.

Assessment practices

The actual evaluation practices of TRs are quite diverse. The assessment may consist of a written and/or oral examination or of a continuous assessment followed by a written examination. It may concern questions about the course, exercises closely related to those presented in exercise session, new problems or a combination of these possibilities. Although the practices differ from one TR to another, we identify a similarity between them: they are generally not satisfied with the way they evaluate students.

M5:

Then I'd like to evaluate them on their ability to look a little bit, ask questions that are more open than those I'm actually asking in the exam, where they would look, think, etc. and we don't have time to do that. The exam is too short.

The ideas developed in this excerpt are shared by most of the TRs. In fact, for a large majority of interviewees, the main objective of the evaluation is to determine if students have a thorough understanding of the courses. For this reason, many people want to assess this understanding, particularly through problems in a different context from the exercise sessions. However, they do not do so because of several reasons: lack of time, lack of staff or because it would require too much investment on their part to the detriment of research. In addition, they want to mix various evaluation methods such as continuous monitoring, lectures, homework, mid-semester midterms, etc. The following excerpt clearly shows this trend.

M6:

The ideal is still to have a mix of continuous evaluations and evaluations, say at the end of semester or possibly mid-semester.

As a result, very few TRs assess what they really want to test and they seldom vary the forms of assessment. In our view, this reflects a tension between what they value as evaluation methods and what they actually practice. In addition, they are aware that students' success or failure at an examination is not significant in terms of their understanding of the course M3: It seems horrible to me, but at the same time I can't put the whole system back into play like this, this is the way it works, so I find that sometimes the grades aren't, apart from the very good and the very bad students, but sometimes the grades don't always correspond to the student's understanding of what he's doing.

In the analysis of the interviews, we then identify two tensions related to TRs' assessment practices: one between what is valued and what is evaluated in the exam and another between students' understanding and success.

Teacher-researchers' training

TRs were asked whether they thought it was desirable to have access to continuing education and if so, what form it could take. A first striking aspect is that all TRs agree that, in the case where there is such a training, it must necessarily be non-disciplinary. Indeed, mastery of discipline is taken for granted in the profession of TR.

Second, two major trends emerge from the interview analysis. The first concerns TRs who believe that the training of higher-education teachers is not useful. This concerns about half of the respondents. Some TRs also question what "learning to teach" might mean, as the following excerpt shows:

M9:

There is nothing that trains to teach in the strict sense. At the same time, I believe that in a university context, it's not really necessary... when someone knows his field well and... spends time preparing for his course... it's very rare that they actually make a bad course.

The other half of TRs are not opposed to continuing education, provided it has practical aspects that take many forms in their comments. For example, training could cover aspects that are not necessarily disciplinary, such as managing a group of students or how to get them to work more effectively. Finally, some TRs mention their lack of knowledge concerning didactics and believe that it would be positive to better understand this field of research through training.

DISCUSSION AND PERSPECTIVES

In this contribution, we have deliberately limited ourselves to presenting results in mathematics. We return here to our general issue concerning the impact of the discipline on teaching practices by pointing out some comparisons between our results and those of Physics.

We have identified that the vision of the discipline is clearly reflected in the statements made by TRs in mathematics, particularly when they point out students' difficulties (difficulties in entering into an abstraction and formalization process, in questioning themselves, etc.). In addition, their posture as researchers is highlighted when they evoke the elements they propose (or would like to propose) to students (research problems, for example). They also raise tensions in the way in which students are evaluated at university.

Thus, the norms, qualities and values mobilized by TRs are strongly marked by the image they have of their own research discipline. They are also marked by the institutional constraint represented by the evaluation standard, particularly at the beginning of university. Similar results have been obtained among TRs in physics that have been interviewed, such as a tension between what they value in terms of forms of assessment (oral, open-ended questions, etc.) and the evaluations they actually propose. Although mathematics are seen as an obstacle for learning physics, TRs in physics say that they still manage to "do physics" with their students: this contrasts with TRs in mathematics who do not manage to "do math" with theirs. For TRs in physics, it is possible to "do physics" because their discipline is at the crossroad of several disciplines and very much related to phenomena that can be observed in everyday life. We also find a strong imprint of their discipline in teaching objectives and practices (group work, reflexivity, creativity, etc.).

These first comparative results between mathematics and physics show some disciplinary disparities but also a strong anchoring of the research profession in teaching practices. In this respect, we find similarities with the theoretical framework presented above, which we need to deepen: the influence of the previous experiences of the TRs interviewed, identification with the research discipline (particularly in relation to learning), the socialization effect, particularly in teacher training. Through a more detailed comparison between the three disciplines, it would be necessary to analyze our results on the basis of the five psychological processes that influence teachers' professional identity [START_REF] Van Lankveld | Developing a teacher identity in the university context: A systematic review of the literature[END_REF]. At least, we observe certain similarities with these processes, thus reinforcing our research hypothesis.

In addition, this research should allow us to identify enough variables to design an online questionnaire (simple or multiple responses, or simple text) to confirm or refute our results and move towards statistical representativeness of the TR population. Another objective is to study in situ TRs' practices in order to refine our results [START_REF] Van Lankveld | Developing a teacher identity in the university context: A systematic review of the literature[END_REF].

Finally, the results presented in this contribution are also encouraging to provide some answers to the problem of university pedagogy mentioned in the motivation for this study. This would enable us to report on the results obtained on TRs in universities where interviews were held, or even during training courses dedicated to university pedagogy. The objective here would be to show TRs a representation of their own practices in order to initiate a discussion on different aspects that could better capture the complexity of practices. Here again, the disciplinary aspect is highlighted, in particular by the desire expressed by some TRs in the three disciplines to deepen their didactic knowledge.

Table 1 -Some examples of questions related to study dimensions

 1

	Dimension	Questions
	Norms	What is the objective of a course? How do you ensure that the objective is
		achieved? What is the purpose of an evaluation?
	Qualities	What do you find difficult in your job as a teacher? Do you feel the need to
		be trained as a teacher?

Values What do you enjoy most about being a TR? What would you be willing to delegate/not delegate? Discipline What are the sources of difficulty for students? What is a good course? What is a good teacher?

  I think, for example, of the possibilities to use the ENT 3 , computers, this kind of thing, it's a little bit up to each person to go and discover it for himself […], let's say, but nothing very pedagogically precise.

M10:

Laboratoire de Didactique André Revuz (LDAR, EA 4434), Université Artois, CY Cergy Paris Université, Université de Paris, Université Paris Est Créteil, Université Rouen Normandie.

Université Paris 13, Université Rouen Normandie (France) , UMONS, UNamur (Belgium).

ENT : digital work environment.


João Serôdio
email: jserodio@ua.pt

Johann Lavaud
email: johann.lavaud@bio.ulaval.ca

Diatoms and Their Ecological Importance

. Their photosynthetic activity accounts for 40% of the marine primary production, being comparable to the total amount fixed by all the terrestrial rain forest combined (Armbrust 2009). However, their critical role in the functioning and biodiversity of oceanic and coastal zones, while representing a major carbon sink and supporting important marine food webs, is threatened by ongoing climate change, namely by ocean acidification and eutrophication.

Introduction. Diatoms are unicellular or colonial photoautotrophic microalgae.

Diatom cells vary between a wide range of sizes, from 5 µm to above 1 mm in diameter or length [START_REF] Sabater | The diatom cell and its taxonomical entity[END_REF]. They are most commonly found as single cells, but can form colonies, living in suspension in the water column or attached to substrata. Diatoms are found in all marine and freshwater habitats and in moist terrestrial habitats, covering extreme environments like sea ice [START_REF] Arrigo | Sea Ice Ecosystems[END_REF] or deep marine sediments well below the photic zone of the ocean [START_REF] Cahoon | Viable diatoms and chlorophyll a in continental slope sediments off Cape Hatteras, North Carolina[END_REF]). Diatoms are a major constituent of the phytoplankton in oceanic and coastal waters, where they often dominate over other groups of microalgae or cyanobacteria [START_REF] Armbrust | The life of diatoms in the world's oceans[END_REF]. They are typically the dominant group in the microphytobenthos, the highly productive biofilm-forming communities of microalgae and cyanobacteria that colonize intertidal flats or shallow subtidal sediments [START_REF] Underwood | Primary production by phytoplankton and microphytobenthos in estuaries[END_REF]. Their ubiquity in virtually all marine habitats [START_REF] Mock | Genomics and Genetics of Diatoms[END_REF] and ability to strive in extreme habitats, such as polar systems [START_REF] Lyon | Polar Microalgae: New Approaches towards Understanding Adaptations to an Extreme and Changing Environment[END_REF] or under prolonged darkness [START_REF] Frankenbach | Functional resilience of PSII, vertical distribution and ecosystem-level estimates of subsurface microphytobenthos in estuarine tidal flats[END_REF][START_REF] Kennedy | Dark metabolism: a molecular insight into how the Antarctic sea-ice diatom Fragilariopsis cylindrus survives long-term darkness[END_REF], is believed to be based on their unique physiological and metabolic features [START_REF] Wilhelm | The regulation of carbon and nutrient assimilation in diatoms is significantly different from green algae[END_REF][START_REF] Gruber | Intracellular metabolic pathway distribution in diatoms and tools for genome-enabled experimental diatom research[END_REF].

The siliceous cell wall. The most distinctive feature of diatoms is the presence of a silicified cell wall ('frustule') made of two identical pieces ('valves'), which gave origin to the name of the group (from the greek word 'diatomos', meaning 'cut in half'). The two pieces overlap each other like the parts of a Petri dish, with a larger one ('epitheca') partially enclosing a smaller one ('hypotheca'). Because the valves do not increase in size after being formed, cell division implies the formation of new, smaller valves within the ones of the parental cell, causing a gradual reduction in cell size over several generations [START_REF] Tommasi | Diatom Frustule Morphogenesis and Function: a Multidisciplinary Survey[END_REF]. Original cell size is restored through sexual reproduction, by the production of diploid zygotes called auxospores [START_REF] Davidovich | Reproductive properties of diatoms significant for their cultivation and biotechnology[END_REF].

The siliceous cell wall has been thought to be a major factor explaining the adaptive success of diatom, as it confers multiple advantages, such as the following (Pickett-Heaps 2003): i) although rigid and impermeable, it is transparent and porous, allowing light penetration and the diffusion of carbon and nutrients, thus permitting photosynthesis; ii) silica is an abundant element, that is available from sand and suspended silt, and only in oligotrophic waters limits diatom growth; iii) the uptake and precipitation of silica is efficient, being energetically cheaper than and equivalent organic wall.

Diatom cell walls are rich in complex and intricated ornamentation that result from the deposition of siliceous material in regular patterns. Due to its enormous intricacy and diversity, the morphology of the frustule has been the basis for traditional diatom taxonomy and classification since the first microscopy observations [START_REF] Round | The Diatoms -Biology & Morphology of the genera[END_REF]). The advent of scanning electron microscopy, revealing with finer detail the ornamentation of the cell wall, caused an increase in the number of diatom taxa, as compared to the previously described on the basis of light microscopy observations [START_REF] Round | The Diatoms -Biology & Morphology of the genera[END_REF]). However, phenotypical changes in the frustule morphology may occur as responses to varying environmental conditions, affecting their taxonomical value. Recent approaches based on molecular sequence data are contributing to improved phylogenetic-based taxonomical classifications [START_REF] Sims | Evolution of the diatoms: insights from fossil, biological and molecular data[END_REF].

Diversity and evolution. Diatoms are an extremely diverse and evolutionarily successful group. They appeared between 190 and 250 million years ago, depending if the estimates are based on the fossil record or on molecular-clocks [START_REF] Benoiston | The evolution of diatoms and their biogeochemical functions[END_REF].

Diatoms are the most diverse group of algae, the number of existent species being estimated to reach above 100,000 [START_REF] Mann | An inordinate fondness? the number, distributions, and origins of diatom species[END_REF] and even up to 200,000 [START_REF] Armbrust | The life of diatoms in the world's oceans[END_REF]. Diatoms are eukaryotic organisms, classified as protists of the group of the Bacillariophyta. Despite long and intensive research, the phylogeny and classification of the group is still under debate [START_REF] Yu | Evolution of the Plastid Genomes in Diatoms, 1st edn[END_REF]. Diatoms have evolved through two successive endosymbiotic events from which originate both their secondary plastid [START_REF] Dorrell | Secondary Plastids of Stramenopiles[END_REF] and chimeric genome harboring a unique mix of bacterial, algal and animal-like features [START_REF] Tirichine | Recent progress in diatom genomics and epigenomics[END_REF]. The diatom plastid is thought to derive from red algae, originally suggested by the conservation of chlorophyll c and a plastid four membrane-bound ultrastructure, and later supported by plastid gene trees [START_REF] Dorrell | Secondary Plastids of Stramenopiles[END_REF].

There are two main groups of diatoms, the centrics and the pennates, which differ markedly regarding key cytological, biological and ecological aspects, that include cell wall symmetry, chloroplast number and morphology, sexual reproduction and motility [START_REF] Pickett-Heaps | The pan-Arctic biodiversity of marine pelagic and sea-ice unicellular eukaryotes: A first-attempt assessment[END_REF]. The most obvious difference between the two groups is cell wall symmetry.

Many centric cells show radially symmetrical valves, with radially organized patterns in valve ornamentation (Fig. 1A). Pennate species, while showing a large variety of forms and ornamentation complexity, typically show a well-defined main axis conferring a clearly nonradial symmetry (Fig. 1B). Another major morphological difference regards chloroplasts, with the centrics having a large number of small chloroplasts, and the pennates often possessing one or two large chloroplasts per cell (Fig. 2A,B). Centric diatoms are typically planktonic, often dominating the phytoplankton, especially in turbulent, nutrient-rich marine waters [START_REF] Malviya | Insights into global diatom distribution and diversity in the world's ocean[END_REF]. Pennate diatoms are predominantly benthic, dominating the microphytobenthos, the biofilm-forming communities that colonize intertidal and well-lit subtidal sediments [START_REF] Underwood | Primary production by phytoplankton and microphytobenthos in estuaries[END_REF], and sea-ice habitats (Poulin et al.

2011).

These differences in habitat preference between centrics and pennates seem to be associated with the evolution of directed motility in the presently larger group of pennate diatoms, the raphid pennates, enabled by the development of the 'raphe', a longitudinal thin and long slit through the surface of the valve [START_REF] Nakov | Accelerated diversification is related to life history and locomotion in a hyperdiverse lineage of microbial eukaryotes (Diatoms, Bacillariophyta)[END_REF]). However, not all pennates possess a raphe, for which reason the pennates are further separated between the araphids (without raphe) and the raphids (with raphe).

Motility in pennate diatoms. The raphe and associated cell motility are relatively recent traits in diatom evolution, having appeared during the Palaeocene, ca. 30 million years ago [START_REF] Armbrust | The life of diatoms in the world's oceans[END_REF]. Their appearance is hypothesized to have conferred an adaptive advantage to colonize new niches including the sedimentary microenvironment, due to an improved efficiency in fast responding to environmental gradients (light, nutrients) and exploiting habitat heterogeneity [START_REF] Cohn | Comparative analysis of light-stimulated motility responses in three diatom species[END_REF][START_REF] Nakov | Accelerated diversification is related to life history and locomotion in a hyperdiverse lineage of microbial eukaryotes (Diatoms, Bacillariophyta)[END_REF]. The evolution of motility is thought to be a primary driver of diatom diversification, explaining the rapid and large expansion of raphid pennate species, which are currently the most numerous group of diatoms [START_REF] Kooistra | The origin and evolution of the diatoms: their adaptation to a planktonic existence[END_REF].

The most common form of motility in pennate diatoms is known as 'gliding', consisting in directed cell movement, typically along the direction parallel to the longitudinal axis of the cell, and when the cell is in close contact with hard surfaces [START_REF] Edgar | Diatom locomotion[END_REF]. Directed motility, for example towards a stimulus, is achieved by varying the time between the reversal of direction, causing forward progression when the movement in the direction of the stimulus lasts longer than away from it [START_REF] Cohn | Characterisation of the diatom photophobic response to high irradiance[END_REF][START_REF] Apoya-Horton | Movement modalities and responses to environmental changes of the mudflat diatom Cylindrotheca closterium (Bacillariophyceae)[END_REF]). Diatom gliding is a complex process, involving the extrusion of adhesive, mucilaginous extracellular polymeric substances (EPS) through the raphe. According to the widely accepted model proposed by Edgar and Pickett-Heaps [START_REF] Consalvey | The ups and downs of life in a benthic biofilm: Migration of benthic diatoms[END_REF][START_REF] Molino | The biology of biofouling diatoms and their role in the development of microbial slimes[END_REF], diatom gliding is based on an actin-myosin mechanism, resulting in the transient attachment of the cell to the substratum while moving. Actin filaments run the length of the raphe in the cytoplasm immediately adjacent to the cell membrane and allows the cell to attach and to glide over of the substratum [START_REF] Poulsen | Biosilica from diatoms microalgae: Smart materials from bio-medicine to photonics[END_REF].

Ecology. Diatoms have an enormous ecological importance. Through their photosynthetic activity, the group contributes to a massive 20-25% of the Earth's global primary production, carbon fixation and oxygen release into the atmosphere [START_REF] Field | Primary production of the biosphere: Integrating terrestrial and oceanic components[END_REF][START_REF] Sarthou | Growth physiology and fate of diatoms in the ocean: A review[END_REF]. The amount of carbon fixed by diatoms' photosynthesis per year represents 40% of total marine primary production, being equivalent to the total amount fixed by all the terrestrial rain forest combined [START_REF] Armbrust | The life of diatoms in the world's oceans[END_REF]. Diatom productivity is thus a key source of carbon for marine, coastal and estuarine trophic webs, supporting a range of animal populations, from small crustacean to commercially-valuable fishes, seabirds and marine mammals [START_REF] Benoiston | The evolution of diatoms and their biogeochemical functions[END_REF]. Because of the heavy siliceous frustule, planktonic diatoms tend to sink through the water column, carrying organic carbon to the deep zones of the ocean. There, it is used as food for deep-sea organisms, remineralized back to CO2, fueling the ocean's carbon pump, or converted to carbonates and incorporated in deep sediments [START_REF] Tréguer | Influence of diatom diversity on the ocean biological carbon pump[END_REF].

The vertical migration of large centric diatoms along the water column represents a form of 'nutrient mining', through which substantial amounts of nutrients are transported upwards across the nutricline, replenishing the otherwise nutrient-depleted photic layer and contributing to new production [START_REF] Singler | Nitrogen inputs into the euphotic zone by vertically migrating Rhizosolenia mats[END_REF]. Vertical migration also causes that the photosynthetically-fixed carbon near the ocean surface is transported downwards and respired at sub-photic layers. These processes have a large impact on the vertical fluxes of nitrogen (and other nutrients like phosphorous), and have been estimated to contribute to more than one quarter of the ocean surface nitrate pool [START_REF] Singler | Nitrogen inputs into the euphotic zone by vertically migrating Rhizosolenia mats[END_REF]. The vertical migration of centric diatoms is widespread, and has major biogeochemical consequences, calling for the reassessment of the role of motility in marine phytoplankton and of the predicted implications of global warming on changes in phytoplankton diversity [START_REF] Villareal | Upward nitrate transport by phytoplankton in oceanic waters: balancing nutrient budgets in oligotrophic seas[END_REF].

Diatoms also strongly influence the carbon cycling in benthic sedimentary habitats.

Diatom gliding causes the excretion from the raphe of large amounts of carbon-rich mucilage formed by extracellular polymeric substances (EPS) (Pickett-Heaps 2003) which represent a major source of organic carbon fueling the growth of heterotrophic bacteria and their remineralization activity [START_REF] Bohórquez | Different types of diatom-derived extracellular polymeric substances drive changes in heterotrophic bacterial communities from intertidal sediments[END_REF]. A similar high excretion of EPS has been reported for sea-ice diatoms [START_REF] Underwood | Broad-scale predictability of carbohydrates and exopolymers in Antarctic and Arctic sea ice[END_REF]. The critical role of diatoms in the global carbon cycle makes them major players, as well as sentinels of environmental disturbances, in the context of global change scenarios [START_REF] Raven | The possible roles of algae in restricting the increase in atmospheric CO2 and global temperature[END_REF]).

Due to their abundance in planktonic and benthic habitats, and to the strict dependence of silicic acid for forming their cell walls, diatoms are main drivers of the silicate cycle at the global scale [START_REF] Tréguer | The World Ocean Silica Cycle[END_REF]. In the ocean, diatoms are a major sink of silica, ultimately causing the accumulation on the sea floor of massive deposits of cell walls from dead cells, called 'diatomaceous earth' or 'diatomite'. On the other hand, the availability of dissolved silicic acid is often a limiting factor of diatom growth, determining the productivity and species composition of phytoplankton communities [START_REF] Tréguer | The World Ocean Silica Cycle[END_REF]. On estuarine tidal flats and subtidal sediments, diatom-dominated microphytobenthic biofilms also strongly affect the fluxes of silica across the sediment-water interface, influencing the silica concentrations in the water column [START_REF] Longphuirt | Diurnal heterogeneity in silicic acid fluxes in shallow coastal sites: Causes and implications[END_REF]Bondoc et al. 2016a;[START_REF] Welsby | The role of benthic biofilm production in the mediation of silicon cycling in the Severn Estuary, UK[END_REF].

Reasons for success.

Several factors seem to contribute to the success of diatoms, including specific subcellular light energy distribution and allocation patterns of carbon into macromolecules [START_REF] Dorrell | Secondary Plastids of Stramenopiles[END_REF][START_REF] Wagner | Towards an understanding of the molecular regulation of carbon allocation in diatoms: the interaction of energy and carbon allocation[END_REF]). In comparison with other photoautotrophs, diatoms appear as particularly efficient in coping with high and/or fluctuating solar light intensities [START_REF] Ruban | The super-excess energy dissipation in diatom algae: Comparative analysis with higher plants[END_REF][START_REF] Wilhelm | The regulation of carbon and nutrient assimilation in diatoms is significantly different from green algae[END_REF]), which are a major cause of photoinhibition, the light-induced damage to the photosynthetic apparatus, in many situations a main limitation of microalgae productivity [START_REF] Raven | The cost of photoinhibition[END_REF]. This ability has been attributed to the efficient operation of energy-dissipation photoprotective processes of diatoms, allowing withstanding the high light levels and rapid fluctuations in light exposure occurring in the turbulent upper layers of the ocean's photic zone [START_REF] Lavaud | Non-Photochemical Quenching and Energy Dissipation in Plants, Algae and Cyanobacteria[END_REF].

Diatoms are known to be highly efficient for thermal dissipation of excess absorbed energy [START_REF] Buck | Lhcx proteins provide photoprotection via thermal dissipation of absorbed light in the diatom Phaeodactylum tricornutum[END_REF]. One of the main mechanisms underlying this capacity for thermal dissipation is the so-called xanthophyll cycle, a biochemical process consisting in the reversible conversion of photosynthetic pigment under high light conditions [START_REF] Goss | Biodiversity of NPQ[END_REF]. Unlike land plants, in diatoms and other microalgae group, the xanthophyll cycle involves the conversion of the pigment diadinoxanthin into the energy-dissipating form diatoxanthin [START_REF] Lavaud | Influence of the diadinoxanthin pool size on photoprotection in the marine planktonic diatom Phaeodactylum tricornutum[END_REF].

Cell motility also plays a role in enhancing productivity, by allowing cells to actively exploit spatial heterogeneities in the resource distribution. Planktonic centric species can regulate their buoyancy and move vertically in the water column, between the surficial nutrient-poor photic zone and the nutrient-rich deeper layers [START_REF] Villareal | Upward transport of oceanic nitrate by migrating diatom mats[END_REF].

This capacity to move vertically in the water column has been thought to represent an adaptation to living in stratified waters, typical of warm, oligotrophic regions [START_REF] Kemp | High diatom production and export in stratified waters -A potential negative feedback to global warming[END_REF]. Vertical migration confers these cells the capability for obtaining nutrients from sub-euphotic layers and for photosynthetizing in well-lit conditions, and for avoiding direct competition with much more abundant, non-migratory smaller phytoplankton, overcoming the disadvantages of their larger size [START_REF] Villareal | Upward nitrate transport by phytoplankton in oceanic waters: balancing nutrient budgets in oligotrophic seas[END_REF]).

In the benthos, directed vertical migration by pennate diatoms appears to have an even more important significance in terms of exploiting environmental heterogeneity.

Directed motility allows benthic diatoms to respond behaviorally to a wide array of abiotic and biotic factors and cues, such as visible light (intensity and spectrum) [START_REF] Cohn | High energy irradiation at the leading tip of moving diatoms causes a rapid change of cell direction[END_REF], ultra-violet radiation [START_REF] Waring | Responses of estuarine intertidal microphytobenthic algal assemblages to enhanced ultraviolet B radiation[END_REF], gravity [START_REF] Frankenbach | Evidence for gravitactic behaviour in benthic diatoms[END_REF], temperature [START_REF] Cohn | The effect of temperature and mixed species composition on diatom motility and adhesion[END_REF], salinity [START_REF] Sauer | Effects of salinity, light and time on the vertical migration of diatom assemblages[END_REF], desiccation [START_REF] Coelho | Effects of desiccation on the photosynthetic activity of intertidal microphytobenthos biofilms as studied by optical methods[END_REF], pH [START_REF] Cohn | Environmental factors influencing diatom cell motility[END_REF], chemical gradients (Bondoc et al. 2016a), or pheromones (Bondoc et al. 2016b). Responses to intensity and spectral composition of light are amongst the most important for diatoms, as they not only directly determine photosynthetic rates and growth, but also photodamage and possibly cell death [START_REF] Serôdio | Microphytobenthos vertical migratory photoresponse as characterised by light-response curves of surface biomass[END_REF]. The motility response of pennate diatoms to changes in light intensity is characterized by the avoidance of both darkness and low light and of very high light intensities, and the preference of intermediate irradiance levels [START_REF] Serôdio | Microphytobenthos vertical migratory photoresponse as characterised by light-response curves of surface biomass[END_REF]. Furthermore, due to the comparable scales of spatial variability of resources like light or nutrients, the diatom cell size and the distances covered through gliding, motile diatoms are able to rapidly move towards regions with more favorable light, carbon or nutrient conditions within the sedimentary microhabitat [START_REF] Cohn | Comparative analysis of light-stimulated motility responses in three diatom species[END_REF]Bondoc et al. 2016a) or to avoid unfavorable conditions such as photoinhibitory irradiances [START_REF] Serôdio | Microphytobenthos vertical migratory photoresponse as characterised by light-response curves of surface biomass[END_REF], or predators [START_REF] Kingston | Wave effects on the vertical migration of two benthic microalgae: Hantzschia virgata var. intermedia and Euglena proxima[END_REF]. Vertical movement by raphid diatoms is partially endogenously controlled, exhibiting a self-sustained rhythm synchronized with environmental day-night and tidal cycles [START_REF] Consalvey | The ups and downs of life in a benthic biofilm: Migration of benthic diatoms[END_REF]. This allows the cells to anticipate environmental periodicity, such as main periodic events including sunrise or sunset, and tidal ebb or flood [START_REF] Coelho | Endogenous versus environmental control of vertical migration by intertidal benthic microalgae[END_REF].

Diatoms also have the ability to survive for prolonged periods in continuous darkness, often while buried in anoxic sediments, and to regain photosynthetic activity and carbon fixation upon exposure to favorable surface conditions [START_REF] Wasmund | Micro-autoradiographic determination of the viability of algae inhabiting deep sediment layers[END_REF][START_REF] Frankenbach | Functional resilience of PSII, vertical distribution and ecosystem-level estimates of subsurface microphytobenthos in estuarine tidal flats[END_REF]. Survival in darkness or while buried has been documented for deep sea sediments [START_REF] Wasmund | Micro-autoradiographic determination of the viability of algae inhabiting deep sediment layers[END_REF][START_REF] Cahoon | Viable diatoms and chlorophyll a in continental slope sediments off Cape Hatteras, North Carolina[END_REF]) and Antarctic sediments [START_REF] Wulff | Exposure to sudden light burst after prolonged darkness -A case study on benthic diatoms from Antarctica[END_REF]. This is enabled by the capacity to live heterotrophically, based on organic energy sources [START_REF] Lewin | Heterotrophy in diatoms[END_REF][START_REF] Tuchman | Differential Heterotrophic Utilization of Organic Compounds by Diatoms and Bacteria under Light and Dark Conditions[END_REF],or by the formation of morphological unchanged resting cells [START_REF] Jewson | Co-existence and survival of diatoms on sand grains[END_REF] or spores [START_REF] Sugie | Resting spore formation in the marine diatom Thalassiosira nordenskioeldii under iron-and nitrogen-limited conditions[END_REF]. Facultative heterotrophy of diatoms seems more common among pennate, benthic forms (Lewin and Hellebust, 1970;Rivkin and Putt, 1987), it was shown to also occur in centric diatoms (Kamp et al., 2013;White, 1974). A similar ability to survive the winter polar night prolonged darkness and rapid reactivation of photosynthetic activity has been reported in polar diatoms [START_REF] Kvernvik | Fast reactivation of photosynthesis in arctic phytoplankton during the polar night1[END_REF][START_REF] Kennedy | Dark metabolism: a molecular insight into how the Antarctic sea-ice diatom Fragilariopsis cylindrus survives long-term darkness[END_REF].

Applications. Diatoms have been used for a variety of applications. The inert nature and long-lasting durability of the diatom silica wall makes it possible to use the deposits of frustules from dead cells (known as diatomite) for industrial and commercial purposes, or for geological and paleontological analysis of aquatic sediment [START_REF] Miettinen | Diatoms in Arctic regions: Potential tools to decipher environmental changes[END_REF]). More recently, diatoms have been proposed as valuable sources of bioactive compounds [START_REF] Lopez | Prospects in diatom research[END_REF] as well of lipids and biofuels, due to their high lipid content, as an alternative to plants or green algae [START_REF] Ramachandra | Milking diatoms for sustainable energy: Biochemical engineering versus gasoline-secreting diatom solar panels[END_REF][START_REF] Hess | Production of chemicals from microalgae lipids -status and perspectives[END_REF]). The ability to build complex three-dimensional frustules with light-interacting properties has raised considerable interest in the fields of nanotechnology and nanophotonics [START_REF] Ellegaard | The fascinating diatom frustule -can it play a role for attenuation of UV radiation?[END_REF][START_REF] Poulsen | Biosilica from diatoms microalgae: Smart materials from bio-medicine to photonics[END_REF].

Addressing SDG14 .FiguresFig. 1

 SDG141 Photoinhibition

  Fig. 1. A, C. Scanning electron micrographs of frustrules of a centric (Thalassiosira angulata; A) and a pennate diatom (Navicula phyllepta; C). B, D. Optical microscopy photographs of a centric (Coscinodiscus granii; B) and a pennate diatom (Nitzchia sigma; D), showing the differences in chloroplast number and size. A, courtesy of James M. Ehrman, Digital Microscopy Facility, Mount Allison University. B, courtesy of NCC -Benoit Tesson, Université de Nantes. D, courtesy of NCC-Pierre Gaudin, Université de Nantes.