M puq u s`1 du, which is the link between M pxq and the Riemann hypothesis, he easily deduces Riemann hypothesis. But when Mittag-Leffler asks him for explanations, they are vague. When Stieltjes died on december 31, 1894, no proof of Riemann's hypothesis or proof of his statement on M pxq was found.

The same misadventure happened to Jensen in 1899, in the memoir that demonstrates the formula that bears his name [START_REF] Jensen | Sur un nouvel et important théorème de la théorie des fonctions[END_REF]. Mertens, in 1897 [START_REF] Mertens | Über eine zahlentheoretische Funktion[END_REF], states an hypothesis of growth of the function M pxq from numerical findings. M pxq would be majorized by ?

x. This leads, as for Stieltjes, the Riemann hypothesis and the simplicity of the zeros. At the International Mathematical Congress of 1912, Von Sterneck [START_REF] Sterneck | Neue empirische Daten über die zahlentheoretischen funktion σpnq[END_REF] goes further and proposes M pxq ď 1 2 ?

x. This conjecture is numerically refuted by Neubauer [START_REF] Neubauer | Eine empirische Untersuchung zur Mertensschen Funktion[END_REF] [START_REF] Hurst | Computations of the Mertens function and improved bounds on the Mertens conjecture[END_REF]) while we conjecture that the true values are respectively `8 and ´8 as shown Ingham [START_REF] Ingham | On two conjectures in the theory of numbers[END_REF] in 1942, under the Riemann Hypothesis and a hypothesis of linear independence of zeros.

On the order, without hypothesis, of M pxq, we have succeeded in showing that M pxq " Opx expp´c 8` ? ln xq at the beginning of the twentieth century (Landau [START_REF] Landau | Handbuch der Lehre von der Verteilung der Primzahlen[END_REF] ) that was brought back to M pxq " Opx expp´c ? ln xq before Vinogradov-Korobov's zero free region is known and allows to have M pxq " x exp `´cpln xq 3{5 pln ln xq ´1{5

(Walfisz [START_REF] Walfisz | Weylsche Exponentialsummen in der neueren Zahlentheorie[END_REF], 1963) which remains, in 2018, the best known result. But we still do not know the exact order of M pxq which is related to the Riemann Hypothesis in a narrow way.

2' The purpose of this paper is to demonstrate that Riemann Hypothesis is false. For this, it is shown that M pxq can not be Opx β q, for β ă 1 without having a contradiction. The calculations will be done from the function 1{ζ according to the formula, valid for σ ą 1,

(2) 1 ζpσ `itq "

8 ÿ n"1 µpnq n σ`it " pσ `itq ż 8 1 M puq u σ`it`1 du,
where µpnq is the Möbius function defined by

(3) µpnq " $ ' & ' % 1 if n " 1.
p´1q r if n factors into r distinct prime numbers. 0 if n is divisible by an integer square other than 1.

The function ζpsq admits a single pole, which is simple, in s " 1. We therefore extend by continuity the function 1{ζpsq to s " 1 by setting 1{ζp1q " 0 (simple zero). The Schwarz symmetry principle makes it possible to look at only the case of t ě 0, which we will do.

We put M pxq " ř x n"1 µpnq for the summation function of the Möbius function. We put β ˚" sup ρ epρq. We know that β ˚P r1{2, 1s, 1{2 corresponding to Riemann's hypothesis. β ˚is also the infimum of β such that |M puq| " Opu β q.

The schema of the proof is as follows:

We start by admitting that M pxq ď mx β with β ă 1. The proof begins by generalizing theorem 8.7 of the Titschmarsh's treatise which expresses that

sup t ˇˇˇˇ8 ÿ n"1 µpnq n σ`it ˇˇˇˇ" ζpσq ζp2σq ,
in order to minorize for some t the integral ˇˇˇż 8

x M puq u σ`it du ˇˇˇ, using the Gegenbauer's theorem

N puq " ÿ nďu |µpnq| " 6 π 2 u `Op ? uq.
We majorize the integral and we obtain, for each x, an upper bound and a lower bound for the integral for any t satisfying proposition 2. The ratio of the lower bound on the upper bound must remain always smaller than 1. Now this ratio has its main term Opx 1´β q and therefore tends towards infinity, if β ă 1, when x Ñ 8, which is the contradiction sought. We deduce that β ˚" 1 and therefore that the Riemann hypothesis is false as well as the conjectures of Mertens.

3' We recall the Abel's summation formula ([4], theorem 1.6). Let a, b two positive numbers, a sequence pa n q of complex numbers and ϕpuq a continuous complex function that is derivable over the interval ra, bs with derivative ϕ 1 puq Riemann-integrable. Let Apuq " ř u n"1 a n . We have the identity By direct application, we have the formula (5)

x ÿ n"1 µpnq n σ`it " M pxq x σ`it `pσ `itq ż x 1 M puq u σ`it`1 du.
4' The Titchmarsh's treatise [START_REF] Titchmarsh | The theory of the Riemann zeta function[END_REF] on the Riemann zeta function contains the following proposition as theorem 8.7, where s " σ `it.

Proposition 1. If σ ą 1, then (6)
ˇˇˇ1 ζpsq

ˇˇˇď ζpσq ζp2σq

for all values of t, while

ˇˇˇ1 ζpsq ˇˇˇą p1 ´ q ζpσq ζp2σq (7)
for some indefinitely large values of t.

In the following we will use the following result, in the same vein as the proposition 1.

Proposition 2. Let x ą 1 and σ ą 1 two real numbers. We have for every t ą 0 (8)

ˇˇˇˇÿ něx µpnq n σ`it ˇˇˇˇď ÿ něx |µpnq| n σ ,
and there exists for all ą 0 at least one value of t ą 0 such that (9)

ˇˇˇˇÿ něx µpnq n σ`it ˇˇˇˇě p1 ´ q ÿ něx |µpnq| n σ .
Proof. Suppose instead that the second part of the proposition is not exact. Let σ ą 1, x ą 1 and ą 0 given such that for all t ą 0 we have ( 10)

ˇˇˇˇÿ něx µpnq n σ`it ˇˇˇˇă p1 ´ q ÿ něx |µpnq| n σ .
Let's cut the series into two parts (11)

8 ÿ n"1 µpnq n σ`it " ÿ năx µpnq n σ`it `ÿ něx µpnq n σ`it .
The sum is majorized in module by ( 12)

ˇˇˇˇ8 ÿ n"1 µpnq n σ`it ˇˇˇˇă ÿ năx |µpnq| n σ `p1 ´ q ÿ něx |µpnq| n σ . Let (13) θpxq " ÿ něx |µpnq| n σ M 8 ÿ n"1 |µpnq| n σ " ζp2σq ζpσq ÿ něx |µpnq| n σ ă 1.
The proposition 1 tells us that there are t ą 0 such as ( 14)

ˇˇˇˇ8 ÿ n"1 µpnq n σ`it ˇˇˇˇą p1 ´ θpxqq 8 ÿ n"1 |µpnq| n σ " ÿ năx |µpnq| n σ `p1 ´ q ÿ něx |µpnq| n σ ,
which contradicts the hypothesis formulated [START_REF] Hardy | An introduction to the theory of numbers[END_REF] through [START_REF] Ingham | On two conjectures in the theory of numbers[END_REF]. The hypothesis [START_REF] Hardy | An introduction to the theory of numbers[END_REF] is therefore false and the proposition 2 is demonstrated.

Corollary 3. For a given Ps0, 1r, the set T pxq of t ą 0 that satisfy to the inequality

(15) ˇˇˇˇÿ něx µpnq n σ`it ˇˇˇˇě p1 ´ q ÿ něx |µpnq| n σ .
is never empty, whatever x P r1, 8r.

Proof. Immediate.

Remark 4. T pxq has no reason, at first sight, to be the same for another value of x.

5' Our memoir is wholly based on the following hypothesis Hypothesis 5.

( where k " Op1q is a majorant of ? σ 2 `t2 λpx, σ, tq @t. We obtains

ÿ něx µpnq n σ`it `M pxq x σ`it " pσ `itq ż 8 x M puq u σ`it`1 du. (24)
We apply the proposition 2 and the hypothesis 5. We deduce that for every Ps0, 1r, whatever t P T pxq, we have (25)

ˇˇˇż 8 x M puq u σ`it`1 du ˇˇˇą p1 ´ q ? σ 2 `t2 ÿ něx |µpnq| n σ ´|Rpx, tq|, with |Rpx, tq| " |M pxq| x σ ? σ 2 `t2 ď mx β´σ ? σ 2 `t2 .
9' As Gegenbauer proved (see [START_REF] Gegenbauer | Asymptotiche gesetze der zahlentheorie[END_REF] or [START_REF] Hardy | An introduction to the theory of numbers[END_REF] §18.6)

N puq "

ÿ nďu |µpnq| " 6 π 2 u `Op ? uq,
we conclude from the Abel's summation formula (27)

ÿ něx |µpnq| n σ " σ ż 8 x N puq u σ`1 du ´N puq x σ " 6x 1´σ π 2 pσ ´1q `O `x1{2´σ ˘.
10' Let's summarize. For all x ě 1 and all t P R,

M puq u σ`it`1 du ˇˇˇď kmx β´σ pσ ´βq ? σ 2 `t2 .

But, after carrying ( 27) in ( 25), for the same x, whatever Ps0, 1r, whatever t P T pxq, we have (29)

ˇˇˇż 8 x M puq u σ`it`1 du ˇˇˇą p1 ´ q ? σ 2 `t2 " 6x 1´σ π 2 pσ ´1q `O `x1{2´σ ˘ ´mx β´σ ? σ 2 `t2 .
We will show that it is impossible. Recall that, whatever x ě 1, the set T pxq is never empty. So we always take t P T pxq from now on. We thus have the limits defined by ( 28) and ( 29), valid for all x ě 1:

kmx β´σ pσ ´βq ? σ 2 `t2 ě ˇˇˇż 8 x M puq u σ`it`1 du ˇˇˇą p1 ´ q ? σ 2 `t2 " 6x 1´σ π 2 pσ ´1q `O `x1{2´σ ˘´m x β´σ ? σ 2 `t2 .
Consider only the right hand sides of ( 28) and ( 29). The ratio of the right hand side of (29) on the right hand side of (28) is therefore smaller than 1, for all x ě 1. After simplification, this ratio is (30) p1 ´ q 6pσ ´βq kmπ 2 pσ ´1q

x 1´β `O `x1{2´β ˘´σ k `β k .

and we see that, when x goes to infinity, if β ă 1, this quantity tends to infinity and therefore exceeds 1 at a given moment, which is impossible. Then the hypothesis 5 is false. So β can not be chosen as β ˚ă β ă 1, and, as β ˚ď 1, this implies β ˚" 1. Theorem 6.

β ˚" supt eρ | ζpρq " 0u " inftβ | M puq " Opu β qu " 1.
The Riemann's hypothesis is false. More precisely • M pxq " Ω ˘px δ q pour tout δ ă 1;

• The simple convergence abscissa of the Dirichlet series ř Proof. We have previously shown that β ˚" 1, but the Riemann Hypothesis implies M pxq " Opx 1{2` q ( [20], theorem 14.25(C), page 370), whatever ą 0. So it implies β ˚" 1{2. It is therefore false as well as the generalized Mertens conjecture that M pxq " Op ? xq. The first point results from β ˚" 1 by application of the standard corollary of the Phragmen-Landau theorem ( [START_REF] Dress | Théorèmes d'oscillations et fonction de Möbius[END_REF], page 33-04).

The second point results from β ˚" 1 by the formulas which give the abscissae of simple and absolute convergence of a series of Dirichlet ([9], theorems 7 and 8 page 8).

For the third point, if there was a vertical line epsq " σ ă 1 such that 1{ζpsq had no pole to the right of this line, we could deduce M pxq " Opx σ q and β ˚ď σ. Since this does not happen and ζpsq has no zero on epsq " 1, the band r1 ´δ, 1r has at least one pole for all δ ą 0. There are therefore infinitely many poles in the immediate neighborhood of the line epsq " 1.

The hypotheses of Mertens', generalized Mertens' and weakened Mertens' involve the Riemann Hypothesis, so they are false. Remark 7. Charles Pisot, after Saffari [START_REF] Saffari | An Ω-theorem of the 'non-effective' type[END_REF], has conjectured that the set of real parts of the non-trivial zeros of Riemann's ζ function is dense in r0, 1s.

  (4) ÿ aănďb a n ϕpnq " Apuqϕpuq ˇˇu "b u"a ´ż b a Apuqϕ 1 puqdu.

σ 2

 2 `t2 . 8' Abel's summation formula gives, for all x ě 1, the equality

• 1 M 2

 12 with its abscissa of absolute convergence; • The ζ function has an infinite sequence of zeros tρ n u in the critical band as (31) lim nÑ8 epρ n q " 1; The generalized Mertens' hypothesis M puq " Op ? uq and weakened Mertens' hypothesis (32) ż x puq u 2 du " Oplnpxqq are false.

  in 1963 showing that M p7760000000q " 47465 ą 1

2

?

7760000000 " 44045. . . . whereas Jurkat

[START_REF] Jurkat | Eine Bemerkung zur Vermutung von Mertens[END_REF]

had shown, in 1961, that the conjecture of Von Sterneck had an infinity of counterexamples, but without exhibiting a single one. In 1985, Odlysko & Te Riele [17] refuted the Mertens conjecture by showing that lim sup xÑ8 M pxq{ a pxq ą 1.06 and lim inf xÑ8 M pxq{ a pxq ă ´1.009. Since then, these values have been improved to, respectively, 1.826054 and ´1.837625 (Hurst

  7' Let x ą 1, σ ą 1 et t ě 0 three real. From Abel's summation formula, we have, by setting s " σ `it,

	(19)
	ˇˇˇs	ż 8 x	M puq u s`1 du ˇˇˇ" ˇˇˇˇ1 ζpsq ´ÿ nďx	µpnq n s	`M pxq x s	ˇˇˇˇď ζpσq ζp2σq	`mx β´σ ď	ζpσq ζp2σq	`m
	because for the partial sum, as |µpnq| ď 1, we have the upper-bound
	(20)				ˇˇˇˇÿ nąx	µpnq n s	ˇˇˇˇď ζpσq ζp2σq
	in the region σ ą 1; finally x β´σ , which is decreasing, is majorized by 1. So there
	is a number K ą 0 such that
	(21)			? σ 2 `t2 ˇˇˇż x 8	M puq u σ`it`1 du ˇˇˇď K.
	By multiplying and dividing by	? σ 2 `t2 , which is no zero,
	(22)
	? σ 2 `t2 mx β´σ λpx, σ, tq pσ ´βq ? σ 2 `t2 "	? σ 2 `t2 λpx, σ, tq	mx β´σ pσ ´βq ? σ 2 `t2 ď	kmx β´σ pσ ´βq ? σ 2 `t2
	)							β ˚ă 1.
	An immediate corollary is the existence of a β such that
								ˇˇˇˇÿ
	(17)			|M puq| "
	6' By the hypothesis 5 on the integral
						ˇˇˇż x 8	M puq u s`1 du ˇˇw
	e have
			ˇˇˇż x 8	M puq u s`1 du ˇˇˇď	ż 8 x	|M puq| u σ`1 du "	ż 8 x	mu β´σ´1 " m	x β´σ σ ´β .
	So there exist a λpx, σ, tq of modulus less than 1 like
	(18)			ˇˇˇż x 8	M puq u s`1 du ˇˇˇ" m	x β´σ σ ´β λpx, σ, tq.

nďu µpnq ˇˇˇˇď mu β

where β ˚ă β ă 1, m " mpβq ą 0 being an appropriate real.

Acknowledgment 8. I thank Mr. Jacques Gélinas for ensuring the careful reading of this memoir throughout its realization..


T Duron

V Chiprianov

L Gallon

Co-construction of Computer Science knowledge-to-be-taught in a French context

Keywords: K-12, Computational Thinking, Didactics, curricula design and analysis

The French national curricula for elementary and secondary schools introduced teaching Computer Science (CS) concepts as mandatory, beginning the 2016-2017 school year. This also raised questions related to specifically what CS concepts should be taught and how. Several proposals of textbooks, pedagogical kits and other knowledgeto-be-taught have been made; some of them contain apparently surprising and even what seems, at a first glance, scientifically incorrect knowledge, which could prove to be obstacles in pupils' learning. In this paper we analyze such proposals, and advance explanations based on the Theory of the Didactic Transposition of Knowledge (TDTK). The TDTK considers that the knowledge-to-be-taught is the result of a complex process of various interactions and negotiations between the numerous actors of the educational system. We identify such interactions, which explain the existence of didactic obstacles. Being aware of such caveats may reduce the apparition of this type of obstacles in future construction of similar CS bodies of knowledge-to-be-taught.

Introduction

The mandatory introduction of Computer Science (CS) concepts in the French elementary (6-10 years old) and middle (11-14 years old) school national curricula, beginning the school year of 2016-2017 [START_REF] Gallon | Programmes d'enseignement du cycle des apprentissages fondamentaux (cycle 2), du cycle de consolidation (cycle 3) et du cycle des approfondissements (cycle 4)[END_REF], has had profound impacts on what and how CS is taught in France. However surprising may this have been for some, it was not totally without precedent: in the 1980s there was a movement of teaching CS concepts (and training teachers accordingly), which was replaced in the 1990s with teaching Digital Literacy; for details on the history of teaching CS in France cf. [START_REF] Baron | Computer science education in french secondary schools: Historical and didactical perspectives[END_REF].

Following the international trend on Computational Thinking [START_REF] Wing | Computational thinking[END_REF], and reports, such as that of the Academy of Sciences [START_REF] De Sciences | L'enseignement de l'informatique en france. il est urgent de ne plus attendre[END_REF], pinpointing France's lateness in adopting proper CS teaching in elementary and middle school, the political decision was taken to introduce CS concepts in the national curricula for these levels. It should be noted that, due to historical decisions [START_REF] Baron | Computer science education in french secondary schools: Historical and didactical perspectives[END_REF], at this point France did not have any primary or middle school teachers that had been formally trained in CS, only in Digital Literacy. Fig. 1: Excerpt from Mission 3, IniRobot [START_REF] Guitard | IniRobot. Activités robotiques avec Thymio II pour l'initiation a l'informatique et a la robotique[END_REF] As a consequence of this introduction, a number of resources, in the form of textbooks, (robot) pedagogical kits, recommendation documents, etc. have been proposed. Designing such proposals is important, however, it is only part of the continuous improvement education process. To evaluate the potential of such resources, questions need to be answered, such as: What is the knowledge such resources aim to teach? How is this knowledge actually taught by teachers and learned by pupils? Moreover, some of these resources contain a number of "features" that could, at first glance, be classified as errors, and which could become obstacles in pupils' learning. How were such "features" introduced and possible in the first place?

One such example concerns the IniRobot [START_REF] Guitard | IniRobot. Activités robotiques avec Thymio II pour l'initiation a l'informatique et a la robotique[END_REF][6] pedagogical kit, using the Thymio1 robot for introducing robotics and programming, especially in elementary 2 , but also middle school. IniRobot is distributed under a CC-BY license and has engendered several developments and versions. We focus here on the 2014 version [START_REF] Guitard | IniRobot. Activités robotiques avec Thymio II pour l'initiation a l'informatique et a la robotique[END_REF].

IniRobot proposes a sequence of 14 missions; of which mission 3 If ... then ... (fr. "Si ... alors ...") introduces, in its own terminology, the concepts of event and action, from event-driven programming. Thymio has 6 preprogrammed behaviors, of which, mission 3 proposes exercises, to discover 4. The connect-the-dots exercise for the behavior called The explorer (yellow) (fr. "L'explorateur (jaune)")is presented in Fig. 1. The "events" (or rather, as we will see below, the conditions on which the events are filtered) are introduced by the word IF (fr. "SI"), and the actions are introduced by the word THEN (fr. "ALORS"). The expected result is the discovery of the explorer behavior (using the IniRobot terminology): "IF Thymio detects an object in front Fig. 2: The didactic transposition process, from [START_REF] Bosch | Twenty-five years of the didactic transposition[END_REF] of it, THEN it goes back"; "IF Thymio detects an object at its right, THEN it turns left"; "IF Thymio doesn't detect anything, THEN it advances"; "IF Thymio detects an objects at its left, THEN it turns right". The use of the word IF may come as a surprise, and rightly so, as the concept it introduces is related to an event. The Aseba textual editor for programming Thymio even uses the keyword WHEN, but the Aseba Visual Programming Language (VPL) uses the keyword IF. One would expect the use of the same word in the graphical editor as in the textual one -WHEN (fr. "QUAND"). While this may seem insignificant, there are cases in which pupils are disturbed (c.f. Fig. 3).

In what follows we argue that such "features" are not simple "errors", but the result of a complex process of transformation of the scholarly knowledge into knowledgeto-be-taught. In this process, many actors are involved, with different concerns, numerous constraints, whose solutions are sometimes surprising ...

Didactic Transposition of Knowledge

Knowledge, as it is taught in the School, is not immutable. It is a human construct, so as to fulfill a particular goal. As goals evolve through time and to follow changes in society, the knowledge taught in the School evolves as well. As some of the main goals are related to teaching beginners, the knowledge taught in school differs from the knowledge as it was first created by researchers: it is "simplified", differently structured, overlaps and contradictions have been limitedthe scholarly, scientific knowledge has been subject to a series of transformations, for it to become knowledge to be taught in School. It has been argued that "[b]odies of knowledge are, with a few exceptions, not designed to be taught, but to be used. To teach a body of knowledge is thus a highly artificial enterprise." [START_REF] Chevallard | On didactic transposition theory: Some introductory notes[END_REF].

In this paper we investigate how such transformations are happening in the CS curricula that is being defined currently in France. The decisions of which subjects, from the broad CS scientific knowledge, to choose for teaching is influenced by various actors, from high political levels such as ministers and academicians to actors closer to the terrain such as teachers, researchers as collaborators of teachers and as analysts of the system. To analyze this system and how it influences the transformations on the CS knowledge to be taught, we adopt a theoretical background based on the Theory of the Didactic Transposition of Knowledge (TDTK).

Didactics is the science of the diffusion of knowledge in any institution (e.g. class of pupils, society at large). More particularly, it is the scientific study (and the knowledge resulting thereof) of the innumerable actions taken to cause (or impede) the diffusion of such and such a body of knowledge in such and such an institution [START_REF]Readjusting didactics to a changing epistemology[END_REF]. The TDTK has originally been proposed in French, in the 1980s, and has achieved a wide spread and acceptance in the Frenchspeaking communities, also in the Spanish-speaking, but much less so in the English-speaking communities, no doubt also because of relatively few and late translations (of which we selected a few in the bibliography of this paper, cf. infra). It is a theory initially proposed in the context of mathematics teaching, but has since evolved to encompass teaching of other science subjects, such as biology and geography.

The Theory of the Didactic Transposition of Knowledge (TDTK) [START_REF] Chevallard | On didactic transposition theory: Some introductory notes[END_REF], [START_REF]Readjusting didactics to a changing epistemology[END_REF], [START_REF]A theoretical approach to curricula[END_REF], studies the "transition from knowledge as a tool to be put to use, to knowledge as something to be taught and learned". Let us note that for us, in this paper, the concept of knowledge, in all of its further declinations, comprises both skills, know-how (fr. savoir faire), as well as "theoretical" knowledge (fr. savoir).

The TDTK distinguishes thus several types of knowledge. The scholarly (bodies of) knowledge (fr. savoir savant) denotes "an organized and more or less integrated whole" [START_REF] Chevallard | On didactic transposition theory: Some introductory notes[END_REF]. It is produced by researchers and scholars, usually integrated in theories, to be found in research articles and scientific books. Taking this type of knowledge as starting point, the TDTK studies how it is transformed into knowledge-to-be-taught (fr. savoir à enseigner), which is the "scholarly knowledge that exists only in contexts than cannot be faithfully replicated within school" [START_REF] Chevallard | On didactic transposition theory: Some introductory notes[END_REF], usually as it appears in curricula, textbooks and other similar resources. This is further transformed into taught knowledge (fr. savoir enseigné) -"the knowledge which becomes visible, so to speak, in the classroom" [START_REF]A theoretical approach to curricula[END_REF], as it is presented in the particular context of a class, by a particular teacher, to a particular group of pupils etc. Of course, there is a further difference between the taught knowledge and what is actually learned by pupils, the learned knowledge (fr. savoir appris, connaissances). A synthetic view of the TDTK is presented in Fig. 2.

These transformations are the result of interactions between actors of the educational system. The totality of these actors form what is called, in the TDTK, the didactic noosphere -the "sphere" of those who "think" about teaching [START_REF]A theoretical approach to curricula[END_REF], all those who share an interest in the teaching system. It consists of the agents -those actors who are in charge of knowledge (e.g. teachers), but also of the laity -those standing outside the teaching sphere. Obviously, different people have different positions in respect to education. Therefore, "it is the task of the noosphere to cope with the demands made by society on the teaching system, by transmuting them into conditions acceptable to both parties -society and its teaching system" [START_REF]A theoretical approach to curricula[END_REF]. Consequently, the central function of the noosphere is the negotiation with the society, taking into account different conditions, constraints, resulting in compromises.

We focus in this paper on analyzing the didactic transposition of CS knowledge related to the introduction of event programming based on IniRobot, as it happens in the context of a research and training project -PERSEVERONS 3 . We do emphasize that other actions, knowledge and research are happening in PERSEVERONS, which are out of the scope of this paper. In this context, we have identified as agents of the educational system: the Curricula Superior Council (fr. Conseil supérieur des programmes) -the body which defines the national French CS curricula, CS and Learning Sciences researchers, CS pedagogical counselors (fr. Enseignants référents pour les usages du numérique), primary and secondary school teachers. In the next sections, we analyze how these agents are interacting in a process of negotiation and co-construction -a process of defining together, through interactions more or less direct -of a CS curricula.

The co-construction process

In the context of our case study based on IniRobot, it seems as a fair assumption that the knowledge-to-be-taught should revolve around robots and their programming. We investigate in this section the transposition of such knowledge, from the definition of scholarly knowledge of what is called event-driven programming (one of the main paradigms used for programming robots), to its transformation into knowledge-to-be-taught, as found in the French national curricula and in the IniRobot pedagogical kit (and one of its declinations, the IniRobot for School), and further into taught and learned knowledge, as observed in regular French schools, while emphasizing the actions of different actors involved in each phase.

Defining the scholarly knowledge

Providing a complete historical and epistemological study on the CS concepts attached to event-driven programming is out of the scope of this paper. However, we do indicate that the concepts usually considered as in relation with the socalled event-driven paradigm have been defined in several places, in different manners, with different names, and although there are commonalities among these definitions, there are also numerous and important differences.

For example, [START_REF] Jusak | Language and runtime support for event programming in a distributed system[END_REF] defines a computational event as "anything that happens in the course of a computation [...] both occurrences in the program itself [...] as well as occurrences outside the computation proper". A similar and more precise 3 http://perseverons.espe-aquitaine.fr/ definition is that of [START_REF] Caspi | An approach to real-time systems modeling[END_REF], for which events are "the transitions between states that may appear in a system or in its environment". Please notice that these definitions consider both the external and the internal nature of events.

Other authors, like for example Turing-award winner L. Lamport [START_REF] Lamport | Time, clocks, and the ordering of events in a distributed system[END_REF], offer a more complete picture, speaking of several concepts: events (which are not formally defined), processes, messages (defined as the means of communication among spatially distinct processes -distributed systems -and whose sending or receiving are considered an event), partial ordering (of the sending and reception of messages), logical and physical clocks (for synchronization), etc.

[14] speaks about waiting for an event -in which case the program cannot complete an operation immediately and thus it registers a callback -a function that will be invoked when the event occurs. This waiting is typically done in a loop that polls for events and executes the appropriate callback when the event occurs. To differentiate between events, mechanisms of "filtering [...] by a condition" [START_REF] Caspi | An approach to real-time systems modeling[END_REF], which provide an event every time a condition is true, are needed. A concept similar to the callback is the event handler -a method "able to respond to one kind of external action (or event)" [START_REF] Chin | Responders: Language support for interactive applications[END_REF]. An event is defined as an external action.

Other researchers make a difference between what it is called threaded (or procedure-oriented) and event-driven (or message-oriented) programming models/systems [START_REF] Dabek | Event-driven programming for robust software[END_REF], trying either to show their duality e.g. [START_REF] Lauer | On the duality of operating system structures[END_REF], or the precedence of one over the other (at least in some contexts) e.g. [START_REF] Dabek | Event-driven programming for robust software[END_REF] [START_REF] Von Behren | Why events are a bad idea (for high-concurrency servers)[END_REF].

Defining the concepts related to the so called "eventdriven programming paradigm" has therefore been a long process, in which several CS researchers (the only type of actors -and more precisely laity in TDTK terminology -in this phase) have built on or against the work of previous researchers. The resulting scholarly knowledge may have attributes that enable construction of real-world applications (use of knowledge), but it is in itself a rather difficult and specialized body to understand, sort and classify.

Defining the knowledge-to-be-taught : the curricula

In France, a distinction is usually made between 2 types of knowledge-to-be-taught: the national curricula, resulted from the external transposition process of scholarly knowledge, and which serves as a reference document for the second type of knowledge-to-be-taught, which comprise textbooks, teacher training documents and other similar documents. The curricula is itself the result of a negotiation process between several organizations and councils. The Curricula Superior Council is the body which defines the national French curricula for elementary and middle schools. However, it does not work in isolation, and an analysis of its CS 2016 curricula [START_REF] Gallon | Programmes d'enseignement du cycle des apprentissages fondamentaux (cycle 2), du cycle de consolidation (cycle 3) et du cycle des approfondissements (cycle 4)[END_REF] shows a number of influences, from a report of the Academy of Sciences [START_REF] De Sciences | L'enseignement de l'informatique en france. il est urgent de ne plus attendre[END_REF], to other organizations such as EPI 4 and ATIEF 5 .

Regarding knowledge related to robots and event-driven programming, the French national curricula for middle school for example [1, pp. 365, 380], includes requirements such as: "write a program in which actions are triggered by exterior events", or "triggering of an action by an event". It is important to note that the CS concepts are mentioned in the curricula in 2 distinct parts: Mathematics and Technology. While the requirements evoked above are common to both parts, there are CS concepts specific to only one of them. As such, the Mathematics part also includes: "pupils are introduced to event programming (fr. programmation événementielle)", "programming actions in parallel", "control structures related to events", while the Technology part includes: "sensor, actuator, interface". One can see that the Mathematics part is more focused on the programming side, while the Technology part also includes more pronounced engineering and mechanical elements.

One can identify, in the curricula, CS concepts that have been selected from the scholarly knowledge, such as event (which in the curricula seem to concern mainly external events -so messages?), action (probably similar to the concepts of callback/handler), triggering and control structures related to events (probably related to waiting and loop polling), actions in parallel (related to processes? or another name for events?). Why and how have been these concepts selected? A possible answer may come from what seems to be a major source of inspiration for the CS part of the curricula, a report of the Academy of Science [4, pp. 23, 25], which indicates, in relation to event-driven programming, concepts such as: "notion of parallel algorithm", "sensors and actuators", "algorithms [...] control the system by acting on actuators depending on the sensed values", "the retroactive command in a closed loop".

However, one can notice that, while the Academy report presents together more mechanical (sensors, actuators) and more programmatic (algorithm, loop) concepts, the curricula segregates them into a Mathematics and a Technology part. How could this be explained? As mentioned in the introduction on the history of teaching CS in France, starting from the 1990s, teachers were trained mainly to Digital Literacy, but not to CS (programming). Therefore, in the 2010s, when the (political) decision to introduce CS teaching was taken, the Curricula Superior Council found itself confronted with the reality of having no teachers properly trained. It seems the retained solution was to separate the CS concepts into 2 subjects and assign their teaching to teachers which were the most likely to have connections to (and hopefully interest in) these concepts. The French national curricula is therefore a good example of how different actors, from political decision makers, the Academy of Science and the Curricula Superior 4 https://www.epi.asso.fr/ 5 http://atief.fr/ Council negotiated (indirectly), according to their purposes and with real-world constraints.

One can also notice that the curricula identifies CS concepts to be taught, but does not enter into details. For example, triggering is mentioned, but not filtering by a condition; the concept of event appears, but no discussion of the time implications, on questions related to the partial ordering of sending and/or reception of events. As we shall see, this may have had implications on the further transposition of the knowledge-to-be-taught.

Defining the knowledge-to-be-taught: the case of pedagogical kits

If a number of decisions regarding choices among scholarly knowledge, such as focusing on certain concepts of event-driven programming, as well as a separation, at middle school, into Mathematics and Technology, were taken, as a result of an external transposition process between laity of the educational system such as the Academy of Science and political decision makers, on the one hand, and agents such as the Curricula Superior Council, on the other hand, the identified concepts remain described rather vaguely and a number of choices still need to be made. One such more detailed specification is provided by the IniRobot [START_REF] Roy | IniRobot : a pedagogical kit to initiate children to concepts of robotics and computer science[END_REF] pedagogical kit. With a version made available in 2014 [START_REF] Guitard | IniRobot. Activités robotiques avec Thymio II pour l'initiation a l'informatique et a la robotique[END_REF], so somehow in parallel with the development of the curricula itself, mainly by actors which seem to be primarily CS researchers or school teachers, with interests in Education Sciences research, IniRobot has several declared objectives, among which "the acquisition and practical use of a number of fundamental concepts", such as, for example: "sensors", "actuators", "instruction", "algorithm" and "how to use basic concepts of event-based programming", by elementary and middle school pupils [START_REF] Roy | IniRobot : a pedagogical kit to initiate children to concepts of robotics and computer science[END_REF]. It seems therefore safe to assume the authors of IniRobot followed the principle of the lowfloor (easy to get started) [START_REF] Resnick | Scratch: Programming for all[END_REF], and tried to avoid as much as possible known difficulties.

"Programming in explicitly event-driven models is very difficult" [START_REF] Haller | Event-based programming without inversion of control[END_REF]. Several causes have been identified for this, among which we find particularly relevant for our discussion the fact that the interactive logic of a program is fragmented across multiple event handlers [START_REF] Dabek | Event-driven programming for robust software[END_REF] [START_REF] Haller | Event-based programming without inversion of control[END_REF]. While sequential programming is structured as a single flow of control, with control structures such as branching (i.e. IF) and loops, event-driven programming requires a series of small callbacks/handlers. The control flow among these handlers is not obvious, because of the inversion of the control [START_REF] Haller | Event-based programming without inversion of control[END_REF]: a program merely registers with the execution environment its interest to be resumed on certain events; it is the execution environment which dispatches the events to the event handlers; it is not the program which calls the handlers. Thus, the control flow among handlers is expressed only implicitly, through manipulation of shared state [START_REF] Chin | Responders: Language support for interactive applications[END_REF].

It is maybe to avoid such difficulties related to understanding the fragmented and inverted control flow that the authors of IniRobot proposed the use of the word "IF" to introduce what they call an "event", as we have seen in Fig. 1. This may help "lowering the floor" [START_REF] Resnick | Scratch: Programming for all[END_REF], the entry point for pupils, when first encountering event-driven programming, by approaching the fragmented event-driven control flow to a sequential one. This seems the more likely as, in special (simple) cases, the control flow of an event-driven program may actually be sequential. Therefore, as long as the situations/exercises only require sequential code, pupils having a sequential representation of the event-driven programming does not show to be problematic. However, are they, in this case, really taught event-driven programming? Moreover, the similarity of using logical conditions: from the filtering of an event by conditions when triggering/waiting/loop polling, and, respectively, of the branch condition, increases the similarity between the 2 concepts.

Nonetheless, the semantics of filtering of events by conditions (let's denote it by the keyword WHEN) is different from that of a branch (IF). A filter has to be unique in a program, while branching conditions may appear multiple times. This becomes visible in some cases, such as that presented by the situation introduced by IniRobot in Mission 10, analyzed in more detail in Fig. 3.

The 2014 version of IniRobot has evolved, engendering several branches and multiple versions. We focus here on a 2016 version, IniRobot for School [START_REF] Sagné | Séquence IniRobot scolaire[END_REF], the basis chosen for activities in PERSEVERONS. First change to notice is the dimension of the document: from 24 pages for IniRobot, to 80 pages for IniRobot for School. The authors of IniRobot for School are CS pedagogical counselors, whose declared objectives comprise, among others: "working numerous competencies related to mastering the language (oral and writing), mastering mathematical languages and mastering scientific languages" [20, p. 2]. While there still are important objectives related mainly to teaching CS concepts, IniRobot for School uses the activities with Thymio "to practice languages", "to practice scientific and technological procedures" and "to organize the personal work" [20, p.3] as well, in relation with other objectives of the French national curricula (defined for all school subjects) [START_REF] Gallon | Programmes d'enseignement du cycle des apprentissages fondamentaux (cycle 2), du cycle de consolidation (cycle 3) et du cycle des approfondissements (cycle 4)[END_REF]. While IniRobot consists of 6 sessions containing a total of 14 missions, with a total estimated completion time of about 6 hours (possibly longer, depending on audience), IniRobot for School consists of 12 sessions, during which contents from IniRobot missions 1, 2, 3, 4, 5, 6, 7, 8, 10 are partially reused, reordered, developed and enriched, for a total of an estimated completion time of about 12 hours.

The authors of IniRobot for School have thus selected, according to their objectives, from IniRobot, contents related to CS concepts found in the curricula, excluded others (e.g. missions 11, 12, 13 from IniRobot) and added other knowledge (e.g. mission 1 on drawing Thymio), not necessarily re-lated to CS, in order to augment the interdisciplinary content of their kit. Let's also note that, in IniRobot for School, while concepts such as event and action are mentioned several times and exemplified, no mention is made to conditions or filtering/waiting; which is, as we have seen, in accordance with their lack of mention in the curricula.

Defining the taught knowledge

At this phase in the transposition, it is the teachers' turn to intervene in the process. Based on the curricula, textbooks, pedagogical kits, other resources and the training they (are supposed to) have received, the French educational system expects teachers to define situations to be taught in the classroom, that introduce and argue the need of CS concepts. These are high expectations, difficult to meet, especially by teachers with no initial training in CS.

To support the teachers in this rather daunting task, the decision makers, most notably the Prime Minister and the Ministry of Education, have intervened through the e-FRAN 6Spaces for training, research and digital organization call for projects. It has selected 22 projects, for a total of 19.5 Million Euros. The successful projects demonstrated that they federate schools, county local authorities, companies, research laboratories, and other actors, around an innovative project with objectives regarding digital objects -whether to use them as a pedagogical resource, or in relation to new skills to be acquired, or as a research object -, taking place on diverse territories which enable a precise monitoring and evaluation (especially of pupils involved in experiments).

As part of one such project, PERSEVERONS, we have the opportunity of observing and analyzing interactions between the numerous actors involved. These interactions have driven the transposition of CS knowledge-to-be-taught, as defined by the curricula, textbooks and pedagogical kits. We have observed several teachers and classrooms, keeping records in the form of notes, videos and photos, following an observation-based, case study, document and artifact-driven content analysis, qualitative research method7 [START_REF] Ary | Introduction to research in education[END_REF].

It should come as no surprise that, even though the teachers used the same main pedagogical kit, IniRobot for School, there were differences in the way the class time was organized, the emphasis on some explanations or situations, the general and CS-specific teacher experience etc. However, some conditions were fairly common, such as pupils working in groups of 2 or 3, with 1 computer and 1 Thymio, in halfclassrooms of similar sizes, between around 10 and 15, for about 1 hour, in 1-2 sessions per week, for 4 -10 weeks. This allowed observing some recurring phenomena across Fig. 3: Expected solution to programming Thymio's explorer behavior, from [START_REF] Guitard | IniRobot. Activités robotiques avec Thymio II pour l'initiation a l'informatique et a la robotique[END_REF] the different classes, among which the one related to the use of the IF keyword, described in more detail hereafter.

Defining the learned knowledge

One phenomenon, a recurrent pupils' "mistake" observed in several contexts and situations, can be exemplified by Mission 10 from IniRobot, optionally part of Session 10 of IniRobot for School. It asks pupils to program Thymio so that it moves about freely, avoiding all obstacles (corresponding to its explorer behavior, cf. Fig. 1); optionally, it asks to add instructions so that Thymio changes its color to red if it detects an obstacle, and to green if not. The expected solution (cf. Fig. 3 [5, p. 23]) consists of 4 instructions (grey blocks), each corresponding to an event (at the left of the ":" sign), and 2 actions (at the right ":"). In IniRobot's terminology, their respective semantics would be [5, p. 14]: if Thymio does not detect anything with its front sensors, advance and change color to green; if Thymio detects something at its left, turn right and change color to red; if Thymio detects something in front of it, back away while turning a little and change color to red; if Thymio detects something at its right, turn left and change color to red.

One "mistake" pupils make (cf. Fig. 4) is that, instead of adding a second action to the right of the ":" sign, for the same "event", they add a second instruction, with the same "event". This is signaled by the compiler as an error (in red at the top of Fig. 3) of the type: the events are the same; of course, for the program to be deterministic, the event callback/trigger has to be unique. While there may be several explanations for this "mistake", such as understanding that each "event" has only one corresponding action/instruction (while actually, at the right of the ":" sign, there is a block of instructions, which may contain an action of each type).

However, one possible explanation, on which we focus here, is that the pupils constructed a representation of the event concept which is close to a sequential one, and which therefore allows them to verify the same condition (if all of Thymio's frontal sensors do not detect anything, in the example of Fig. 3) several times. It would seem that constructing a representation of the event concept that includes time aspects (with a semantics of when (each and every time) all of Thymio's frontal sensors do not detect anything, in the given example), would help in understanding the uniqueness of the type of event and of its attached actions. While approaching the event-driven control flow to a sequential one may "lower the floor", a more appropriate representation needs to replace it as soon as possible, if "real" event-driven programming is to be taught.

While this is a fairly straightforward example, its analysis helps understanding difficulties encountered at each phase of the didactic transposition co-construction process, from (1) the esoterism of the scholarly knowledge on event-driven programming (in which time considerations are made using concepts such as partial ordering, and conditions are defined as part of a filtering of events, in a waiting/triggering/loop polling mechanism), defined by sometimes competing, sometimes continuing CS researches, passing through (2) the selection of concepts to-be-taught, selection influenced by political decision makers, academic institutions (laity in the sense of TDTK), national curricula definition bodies (which in our case study seem to leave out these time and conditions considerations), (3) the redefinition and reorganization into textbooks, pedagogical kits, or similar resources, by researchers, teachers and pedagogical counselors mainly concerned with education issues, with easing their understanding by pupils (and thus, in our case, seemingly approaching the event-driven flow of control to a sequential one by using keywords like "IF" instead of "WHEN"), (4) the development of classroom lessons by teachers and finally (5) the continuous evolution of pupils' representation of the taught concepts, who by "mistakes" experiment/test and construct this representation (in our case, e.g. assuming that conditions/events are not unique).

Conclusion

While most current research related to Computational Thinking (CT), especially in a K-12 context, has focused mainly on designing activities for teaching CT concepts, such pedagogical documents need to be analyzed and experimented in classrooms, so that they can be validated and improved. In this paper we adopted a qualitative approach, as a first step to identifying phenomena that appear recursively in classrooms. This allowed us to identify didactic obstacles to the learning of pupils. Further social and epistemologicaldriven analysis revealed causes of such didactic obstacles in Fig. 4: Typical pupils' mistake the complexity of the didactic transposition process involving numerous actors, with various constraints and objectives.

More specifically, we focused on event-driven programming, observing and analyzing during 4 years, how the IniRobot for School pedagogical kit using the Thymio robot, was used to teach and learn concepts related to the eventdriven paradigm. We thus pinpointed that concepts of eventdriven programming form the main focus of CS knowledgeto-be-taught in elementary and middle school (other fully arguable choices include robotics, intelligent (cooperative) (autonomous) agents, ethics etc). In an epistemological approach, we reviewed scholarly, research-introduced definitions of concepts that currently are considered to be related to what is called event-driven programming, showing their relations and underlining their complexity (e.g. external vs. internal events, messages vs. processes). We analyzed how some of these concepts were selected and partially redefined (e.g. event, action) in the French national curricula and identified how the lack of mention of certain concepts (e.g. filtering on conditions), may contribute to further difficulties. We found that these selections are indeed reflected in the pedagogical kits that implement the curricula, and are compounded with didactic and pedagogical concerns of "simplifying" the taught concepts (e.g. presenting the fragmented event control flow as a sequential one). This analysis (in particular the review of the scholarly knowledge) may serve as a basis in future designs of event-driven programming pedagogical proposals.

In this work, we mainly focus on didactic obstacles. While, for example, the difficulty of understanding the fragmented control flow of event-driven programming is an epistemological obstacle, the didactic choice of the "IF" keyword (while intended precisely to reduce this epistemological obstacle), with its unforeseen side effects, introducing didactic obstacles (sequential representation of the occurrence of events), created by the very redefinition (out of didactic concerns) of some event-driven concepts. While they are probably unavoidable in any didactic transposition process, it is important to recognize such obstacles, be aware of them, and try to address them further along the line.

This analysis can therefore be used to draw the attention, in particular, to curricula and pedagogical documents (textbooks, kits, etc.) designers, to pay singular attention to how they redefine scholarly concepts, as these "new" representations -the knowledge-to-be-taught, may introduce its own obstacles. Moreover, this analysis may serve researchers and analysts of the knowledge-to-be-taught (be it curricula, textbooks, pedagogical kits) as caution to drawing conclusions too hastily, and instead exercise a deeper consideration of the wider context and the complex interactions between the involved actors (be they agents of the education system or laity), which may influence decisions seeming, at a first glance, "weird" or simply "erroneous". It thus enlightens the fact that the "final" result of the taught knowledge is the product of a process of co-construction, in which many actors intervene at different stages.

https://www.thymio.org/en:thymio

https://dm1r.inria.fr/t/inirobot-descriptif-des-activites-autour-de-larobotique-a-lecole-primaire/23

http://www.education.gouv.fr/cid94346/appel-a-projet-e-fran.html

With its characteristic concern for context and meaning, of experiments in naturalistic occurring settings, in which the human investigator is the primary instrument for the gathering and analyzing of data, which is mainly descriptive, data obtained as a result of an emergent study design, and on which inductive analysis is applied.


Laurent Gallon
email: laurent.gallon@univ-pau.fr

Angel Abenia
email: angel.abenia@univ-pau.fr

Françoise Dubergey

Maïté Négui

Using a Telepresence robot in an educational context

Keywords: telepresence robots, motivation, transactional distance

This paper deals with the use of telepresence robots in an educational context. To this day, these uses are often related to the problematic of "homebound students" 1 . The emphatic context associated with these students masks the pedagogical difficulties encountered by telepresence learners. In this article, we first detail the use of the robot by homebound students, highlighting motivational factors. Then we focus on persistence using the robot, once this emphatic context declines. Finally, we describe our attempts to reduce transactional distance by adding connected learning environments.

Introduction

For the last few years, we have witnessed an intensification of the experimental use of telepresence robots in the educational system. In France, a recent study in the Auvergne Rhône-Alpes French area (AuRA) has reported the findings of a classroom use for students who are homebound 1 . Other studies [START_REF] Gallon | telepresence robot : a numerical tool used by "Sapad" to make prevented student present[END_REF] [START_REF] Newhart | My student is a robot: How schools manage telepresence experiences for students[END_REF] have enforced these results and show that the main added value of telepresence in this context is the preservation of the social link between the students, the school, the teachers and especially the classmates. This preservation is an important guarantee of school continuity and the reduction of the risk of dropping out. However, beyond the psychosociological impact, questions about learning opportunities via telepresence robotics arise. Indeed, several difficulties come to disturb the distant student [START_REF] Furnon | Telepresence Robot: Process of Appropriation through the Evolution of the Modalities of Presence[END_REF], for example the difficulties of perception of the teacher's and classmates' body language (virtual proprioception) due to a wide-angle camera, or the perception of the "Merobotic" and the new form of socialization allowed by this mechanical avatar [START_REF] Furnon | Co-construction of Tangible, Dispersed and Multisemiotic Spaces through the Use of a Telepresence Robot[END_REF]. The teacher may also be disturbed in the realization of his/her task to take into account the specific "presence" of this student, including the student in the didactic situation and follow the evolution of his/her learning, especially in the case of practical and skill assessments. After the positive impact of the robot during the first uses, a dropout over time can sometimes be noted, due to a too large transactional distance [START_REF] Weidlich | Technology matters-The impact of transactional distance on satisfaction in online distance learning[END_REF] correlated to a lack of practical adapted features in the telepresence device. Several projects have begun to explore these difficulties. J. Bell [START_REF] Bell | From 2D to Kubi to Doubles: Designs for student telepresence in synchronous hybrid classrooms[END_REF] studied the feeling of presence (incarnation) of distant students in hybrid teaching, from videoconferencing to the use of the telepresence robots. It appears that the mobility of the robot is essential to getting closer to a feeling of physical presence. Gleason [START_REF] Gleason | Hybrid education: The potential of teaching and learning with robot-mediated communication[END_REF] confirms Bell's results and emphasizes the need for appropriate pedagogy. Newhart and al. [START_REF] Newhart | My student is a robot: How schools manage telepresence experiences for students[END_REF] [START_REF] Newhart | Virtual Inclusion via Telepresence Robots in the Classroom: An Exploratory Case Study[END_REF] study the feeling of acceptability of the robot from a teachers and administrators point of view. In all these works, the sociological and psychological aspects are taken into account, but the didactic part is little discussed. In addition, pedagogical situations of the use of a robot are limited to simple verbal and visual interactions (course / exercises). The use of a robot in practical work is little or not treated [START_REF] Gallon | Pedagogical environments for telepresence robots[END_REF].

In this paper, we propose a preliminary analysis of the use in duration (perseverance) of telepresence robots in an educational context. We base our analysis on SDT theory [START_REF] Ryan | Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being[END_REF] to assess the motivation while using the robot. More precisely, we focus on what happens after several sessions of robot uses in the classroom, that is, once the impact of novelty and empathy for the homebound student fades. We then relate some experimentations done to increase the capacity of the interactions of the robot with humans and distant space, in order to reduce the transactional distance [START_REF] Moore | Theory of transactional distance[END_REF], induce the student a perception of success and control, and then keep him/her engaged and motivated.

Background

SDT theory

The Self Determination Theory (SDT) [START_REF] Deci | Intrinsic motivation and selfdetermination in human behavior[END_REF] is a psychological theory which can be applied to education in order to understand the learning process and how to motivate students and continue to attract their attention. It is based on the innate psychology needs of: autonomy (need of being actor of his/her own life, making his/her own choice according to his/her preference), competence (need of controlling and being competent in a domain in which he/she was interested) and relatedness (need of interacting with other people). Deci and Ryan [START_REF] Deci | Intrinsic motivation and selfdetermination in human behavior[END_REF] define student motivation as the combination of two factors : external motivation, which is provided by an outside help (teacher, classmate, ITS, chatbot, . . . ), and internal motivation, which is provided by the perception the student has of his/her ability to accomplish pedagogical tasks. Internal motivation is always stronger than external motivation. On this basis, the whole motivation of a person to accomplish a task can be classified into three levels: global, contextual and situational. Each level describes a different evolution time scale. The global level addresses motivation that comes from the social environment, religion, etc (long time range). The contextual level addresses motivation that comes from the context of the student's environment, as for example the school context (medium time range). The situational level addresses motivation from the current situation in which the student is at a given moment (short time range). Situational contexts are the strongest (but the shortest) contexts. On the other hand, global contexts are the weakest over a short period of time, but are persistent throughout life. Changing a contextual perception can only be only achieved by repeating many times over situational contexts in which the student has a feeling of success and control (positive perception of his/her skills).

In the case of the use of a telepresence robot by a homebound student, the first uses of the robot in the classroom are associated with a strong situational context (short range duration) in which the whole motivation is induced by novelty and empathy. At this time, the student, classmates and teacher are not guided by educational concerns. As described by Newhart in [START_REF] Newhart | My student is a robot: How schools manage telepresence experiences for students[END_REF], a bridge is created between the teacher, classmates, and the homebound student. This bridge does not only include social aspects, but also the remote spaces. Some rules must be defined to ensure a safe bridge between stakeholders. We address this point in section 3.

Gradually, the initial context is replaced by the educational solely (contextual level, weaker but more persistent), in which the questions on how to learn through a robot, how to evaluate knowledge and skills through a robot, and how to insert a robot in pedagogical activities are asked. At this time, interactive capacity of the homebound student through the robot is leading motivation. The more functions are numerous and effective, the more motivation and persistence are present. The adequacy of the robot's capabilities with teaching tasks is closely related to the transactional distance.

Transactional distance

Transactional distance (TD) is defined by Moore [START_REF] Moore | Theory of transactional distance[END_REF] as the degree of psychological distance between the student and the teacher. It can be quantified as a function of three variables : dialogue, structure and learner autonomy. In [START_REF] Deci | Intrinsic motivation and selfdetermination in human behavior[END_REF] Zhang proposes to refine TD into three items : transactional distance between student and teacher (TDST), transactional distance between student and student (TDSS) and transactional distance between student and content (TDSC). Anderson [START_REF] Anderson | Getting the mix right again: An updated and theoretical rationale for interaction[END_REF] suggests that the smaller these measures, the more satisfying the learning experience, and as a result, more substantial erseverance. In online distance learning, traditional TD relies on technological mediation. Weidlich [START_REF] Weidlich | Technology matters-The impact of transactional distance on satisfaction in online distance learning[END_REF] proposes a definition of TDTECH, which assesses the transactional distance of the learner and the interface or technology. He argues that this extent depends on two factors: "(1) the basic proficiency of the student in using the necessary technology, (2) the design and functionality (e.g., usability) of the technology itself, as perceived by the student". TD is perceived as "an interplay of these two factors".

In [START_REF] Hung | Learner readiness for online learning: Scale development and student perceptions[END_REF], Hung proposes a framework to understand and assess students' readiness in online learning. He demonstrates that modern students are used to computer-mediated communication, with a high confidence in his/her computer/network skills. More precisely, in the context of online learning, students show a strong readiness in computer/Internet selfefficacy, motivation for learning, and online communication self-efficacy. Self-directed learning and learner control are the weakest items. In [START_REF] Weidlich | Technology matters-The impact of transactional distance on satisfaction in online distance learning[END_REF] Weidlich fits his first TDTECH factor (student proficiency) with Hung's computer/Internet self-efficacy. In other words, he demonstrates that modern students have a good proficiency in using new technologies in learning.

TDTECH mainly depends on the functionalities the robot offers. The course topology has also a major impact. The interactions between the distant student and others humans involved in the pedagogical situation (classmates, teacher), and also between the student and the distant space, are different depending on this topology. For example, in the case of a lecture, the student only needs to listen and take notes. In the case of lesson exercises, the student must be able to interact with the teacher, but also with his/her classmates, in order to ask questions or to show his/her work. Finally, in the case of practical work, the student must also be able to interact with the distant space, carry out practical manipulations, or when this is not possible, play a clearly identified role in the group of students to which he/she belongs. So, oral, visual, document sharing and physical capacities of the robot are primary.

Our project 3.1 Overview

Our project comes from the collaboration of the Computer Science Lab of the University of Pau, in France, and SAPAD 40 (organization for a pedagogical help of homebound students, department of the Landes (40) in France). These organizations help homebound students during their convalescence, and help to finance home classes. But it is not always easy to find teachers who agree going to a student's home, because of the geographical distance, or because there is are not enough teacher in the desired specialty. The use of telepresence robots is a solution. Since 2015, 24 students are benefitted from the use of a robot. In the following, we give a feedback of these uses, focusing on the efforts we made to ensure that the robot could be used in the students' incourses, all through their convalescence.

The description is chronological. First, we place ourselves in the first context where novelty and empathy induces the stakeholders's motivation. In this part, two sub parts are differentiated : the first (section 3.2) deals with the actions to prepare the arrival of the robot. The second (section 3.3) refers to the first uses in a classroom, and the creation of a safe bridge between stakeholders.

After some sessions of robot use, the initial context fades and is replaced by the educational context. In section 3.4, we relate the possibility of how to continue to be engaged in the use of a robot, by adding a dedicated connected learning environment that reduces the transactional distance, and the risk of dropping out.

Preparing the arrival of the robot

The preparation of the stakeholders at the arrival of the robot is essential. The more effective this preparation, the more involved the various stakeholders in the use of the robot, and the more the first context (the strongest in terms of motivation) lasts in time. On the other hand, poor preparation risks to see the first context quickly disappear and give way to the difficulties of teaching a robot, with a high risk of dropping out. In the remainder of this section, we describe the key points to which we paid particular attention for each stakeholder

Choosing the right robot

As we noticed before, the first uses of a robot are accomplished in a very strong context, driven by novelty and empathy. But this context fades quickly, and is replaced by the educational context, in which several pedagogical difficulties appear, and can be a reason for dropping out. At this time, the motivation is clearly impacted by the capacities of the interaction of the robot (TDTECH). Depending on the topology of courses the homebound student takes, the choice of the robot can be a benefit or an obstacle to motivation. For example, if courses are given in a small classroom, with little space between the tables, a cumbersome robot will make the distant student feel that he/she is disturbing the rest of the class from the moment he/she wants to move the distant robot. In the case of a lecture, a good video quality is a real advantage to see what the teacher is writing on the board. So the initial choice of the robot is essential for motivation, depending on the school environment, and the courses the homebound student is taking.

We have experimented 3 different robots : Double from Double Robotics, Beam from Awabot/Suitable Technology, and Ubbo expert from Axyn Robotics (Fig. 1).

Each robot has different technical characteristics which impact the perception the student has, and so TDTECH. A summary of some of these characteristics is given in Table 1.

Our experience shows that connection problems are the main factor of impact on motivation to use the robot. TDTECH is mainly impacted by this factor. The quality of video and sound is also important. This leads us to systematically use 4G connections, instead of Wifi, to avoid problems of sensibility, roaming while moving in the classroom, sharing of bandwidth, and sometimes authentication issues. The weight can also be a problem if the robot has to be carried by students, for example to go from one floor to another. Finally, obstruction can be a problem in some cases, especially in crowded classrooms (handbags on the floor), or in classrooms with little space between tables. Other characteristics seem to be less important. For example, a short battery life is not necessary a problem: homebound students are not connected all through the day long, because of a poor capacity of concentration, medical cares, . . . Furthermore, during a session, the robot rarely moves, which extends its battery life. Finally, a lunch break seems sufficient to recharge the battery. Notice that connection issues do not only occur in school, they also appear at students' home. This leads us to also adopt 4G connections for the student. Unfortunately, this is not always possible at a hospital, or in some rural areas. In these cases, after one or two attempts, the connection problems become stronger than the need to recover the social link, and the student stops using the robot.

As mentioned by Newhart [START_REF] Newhart | My student is a robot: How schools manage telepresence experiences for students[END_REF], ensuring a good technical bridge between distant spaces (classroom and student location) is essential for motivation.

Preparing the homebound student and his/her parents

The major issue for the family is the fear that the homebound student will be further marginalized by other students in his robotic form. This feeling is not related to technology, but to the social link with classmates. Moreover, they are convinced that the teachers will accept the presence of the robot in the classroom. This feeling comes from the educational background they experienced, where the teacher is seen as a caretaker, somewhat like a parent. It is difficult to make them accept that their child may not attend certain classes because the teacher refuses the robot in his/her class.

The first fears of the student are about self-image: some have not seen their classmates for a long time, and illness and treatment may have altered their physical appearance. They are reassured when they learn that they can choose the image they want to show the class (photo for example) or no picture at all. This technological functionality can generate strong motivation in this context, but even if the robot does not have this feature, it is always possible to disable or mask the webcam from the cockpit.

We always perform a pre-connection test between the robot and the homebound student. In order not to be disturbed by the desire to recover the social link with the school, for this test, the robot is located in a neutral place (in a room of our university), allowing to focus only on the technical aspects. At the end of this test, it is rare that the student does not want to continue, which shows that the ability to use this technology is strong (as mentioned by Weidlicch [START_REF] Weidlich | Technology matters-The impact of transactional distance on satisfaction in online distance learning[END_REF]). In the rare cases where the student refuses to use the robot after the first test, the reasons are mainly related to problems of self-image, not technical issues.

Preparing teachers

The introduction of a robot into the classroom generates a first reflex of mistrust from teachers. The presence of a camera in a classroom is not trivial. During his/her class, the teacher is master, and this moment is "sacred". Opening it to the outside world is not easy and the tool can be considered voyeur. We know perfectly well the excess in recent years of student publications on social networks, degrading videos of teachers in their classes, and the impact that these acts have had on the profession.

At this first point comes also the fear of being faced with the reality of the disease. The "world of absence" in the school environment is indeed devoid of all humanity: the disease gives rise to immediate empathy, but the daily routine of the class makes us accustomed to this absence. When the student is not present, he/she may be considered as not existing in the class group, that is to say that one does not measure (and one does not try to measure) the impact of the illness on the student and his/her schooling. Teachers discover the reality of this world of absence on the day the robot arrives in their classroom. But our experience shows that the reality of using the robot is different. Indeed, as we already said above, the disease is far from being omnipresent in the image which is sent by the robot (we only see the face of the student). Its acceptance is all the more facilitated, and very quickly, when teachers behave with the robot as with an "ordinary" student. We have even experimented that a teacher, extremely reluctant to introducing the robot in his/her class, regrets at the end of the year not to having tried the experiment, convinced by feedback from his/her colleagues.

Another important point is that the teacher does not want to be responsible for any problems on the robot. This covers both the connection problems, but also potential breakage during his/her course.

All these reluctances can be eliminated by discussing with the teachers on the forehand, showing them the merits of the process, and especially by having them driving the robot, to reduce the fear of this new technology, reducing TDTECH from their point of view, and better understand how the student is perceiving the class.

Preparing school board

The school board has a key role in the initial context. As we have seen before, at this first stage, the main initial obstacle is the teacher. Obtaining the support of the administration (hierarchy of teachers) is very important to "help" him/her, when necessary, to make the effort to accept the robot in class. But the board must also be reassured on certain points. The robot appears as an expensive technological tool (each robot we use costs approximately 5000$), and the problem of responsibility which arise in case of breakage, theft or breakdown. Secure storage when the robot is not in use can also be a stuck point.

Preparing classmates and choosing referents

The role of classmates is central. The homebound student wants to be able to attend classes mainly to leave his/her medical environment, to find a "normal" environment. Classmates also want to understand why their friend is not present.

One important thing is identifying in the classroom one or two of the homebound students' friends who can become referents. To be a referent means to be the guardian of the robot, and thus the guardian of the homebound student. The main tasks that are assigned to a referent are: 1) collecting and returning the robot to its storage point as soon as possible (morning, noon, evening) preventing the battery from discharging 2) solving the technical problems encountered during lessons (sound, camera, disconnections, ...) 3) exchanging with the remote student as soon as the situation requires it (sending pictures of the whiteboard by MMS, specific request when the student does not want to disturb the teacher, ...). The referent is a key element that reassures the teachers, who do not have to deal with the robot during their course, and reassures the administration, knowing that the robot is never forgotten in a class. Nevertheless, this task is heavy, and should not become a burden for this referent student. So the best is to find two or three referents in the same class who can alternate, for example every day.

Clearly referents are a key point. They mask technical issues to teachers and the homebound student, increasing his/her motivation, and so decreasing TDTECH.

Summary

In Table 2 we summarize the main key points which have an impact on the motivation of the homebound student in using the robot at the first stage. Clearly, the most critical stakeholder is the teacher. From a student point of view, the more the teacher seems invested in the insertion of the robot in his/her classes, the more the student is persevering.

The use of referent students is a crucial point, which makes it possible to partially discharge the teacher and the distant student from technical problems, and allowing them to focus more on learning. This is the element of our process that significantly decreases the transactional distance at the start of the robot use.

First uses in classroom

The appearance of an ailing student in a classroom in the form of a robot at first generates surprise (see Fig. 2), and quickly empathy and solidarity among students. There is no problem of image or marginalization. It may even happen that the ailing student goes, in the class, from being in a position of weakness (because of his illness) to a position of envied (the "master" of the robot), a paradoxical situation that obviously has a strong impact on the motivation of the homebound student. We must admit that the image returned by a telepresence robot is easier to accept than the image of a student who is bedridden or physically impaired, especially since each student who uses the robot chooses the image he/she will present in front of the camera. This allows he to continue to benefit from the positive aura of the robot, without letting the image of the disease take over.

It is undeniable that the presence of the robot causes a "shock" in the class, and partly changes its atmosphere, apparently to us, rather positively. When the robot is present, each student becomes more responsible than normal. For example, students will more easily self-discipline at the level of chatter, to allow "the robot" or rather the homebound student Although we have been able to observe these behavorial changes, further study will be needed to determine what the real sociological levers on which the robot has an impact, are. This study has, in our opinion, all its interest in Middle Schools, where we know that relations to others are fragile and in full construction. It should show that the tripartite teacher-student-class social link is the main driver of the homebound student from using the telepresence robot, and that classmates are particularly attentive to this factor.

The arrival of the robot has at the start a positive impact on the motivation of all classmates. But quickly, new difficulties appear. Indeed, the interactions between the actors and the remote space are limited through the robot. This inevitably has a negative impact on the quality of the knowledge and know-how accumulated by the distant student. The use of the robot in an educational context is not enough. It must be complemented by interaction tools with the different stakeholders and also the remote physical space to get closer to a physical presence, and reduce the transactional distance. In the following, we describe some solutions we have experimented for that.

Decreasing transactional distance by adding a connected learning environment to the robot

This experiment was conducted at our university, after one of our undergraduates has injured his knee (Terry), and had to convalesce at home for several weeks. We proposed to use a telepresence robot to follow certain important courses (Mathematics (lecture and exercises), English (lecture and examination) and IP telephony (lecture, exercises, practicals and practical examination)), with different typologies: courses, exercises on the course and practical work. We created connected learning environments dedicated to each typology. By connected learning environment, we mean a set of digital tools, remotely accessible by Terry, which allow him to interact with the distant space, to perform the requested manipulations. The robot allows Terry to move Fig. 3: Terry and Sam in the practical works room around the room to collaborate with the other students in his group, and interact with his teacher. A referent student, Sam, helped Terry when he was using the robot (Fig. 3).

Course typology and transactional distance

In the case of a lecture, the student needs to see and hear the teacher, sometimes talk to him, and to see the whiteboard well. He does not need any interaction with other learners. Necessary documents can be distributed electronically beforehand. All these needs are covered by the intrinsic capabilities of the robot. It is not necessary, in this typology, to add additional functionalities. The transactional distance arising from the use of the robot is weak. It only has an impact on TDST. Note that, occasionally, having a good view on the whiteboard is not always possible, due to certain conditions such as brightness. An extra camera placed near the whiteboard, may be a good solution. Another sensitive point is the notification of request to speak. Often, students cut their microphone not to interfere with the teacher. Signifying that they want to ask a question then becomes an issue. Some robots offer a led panel that can be cloven to signal his request. For other robots, it is possible to add a small lamp on the webcam of the cockpit of the robot, in order to flash the screen, and report to the teacher.

In case of lesson exercises, things are different. In addition to the need for interaction with the teacher, the student must be able to interact with his/her classmates, especially in group work. The student must also be able to show his/her work to the teacher and classmates. This feature exists on some robots in the form of sharing the cockpit screen. However, in that case the image of the student cannot be seen anymore and interactions becomes thus less natural. Moreover, the student must also be able to see the work of classmates. Robots possessing the ability to climb / descend / turn the head, give the student this power. Without these capacities, the use of the robot increases TDTECH, with an impact on TDST and TDSS. A possible remediation is the taking of a photo and sending it using MMS between the referent and the student.

The more critical typology is practical work. As in the case of lesson exercises, the student must be able to interact with classmates and the teacher. He/she needs to share documents. But he/she must also be able to perform physical actions in the classroom, at least to collaborate with his classmates. In this typology, TDTECH is high, and impacts TDST, TDSS and also TDSC. Adding functionalities to permit the distant student to interact with the distant space is not always possible, and depends on the subject. The transactional distance can be reduced in some cases by adding an adapted pedagogical connected environment. For example, in computer science practical work, it is quite easy to offer distant connections to computers, servers, IP telephons to the distant student. But the transactional distance can also be decreased by adapting the educational objectives set to the remote student. For example, during chemistry manipulations, if it is impossible for the remote student to handle dangerous products, he can, on the other hand, guide his colleagues in their manipulations, create curves, and search documents. He/she can also film manipulations, so that he/she can analyze it later. In this case, despite the difficulties in handling, the remote student still feels like being part of the group. He/she plays a role and participates in the realization of the training sequence (team cognition). This feeling of controlling his/her pedagogical activity is a mechanism that generates strong intrinsic motivation.

Description of the connected learning environments used

In our experimentation, for each lesson typology, we associated a dedicated connected learning environment (Fig. 4) without changing any pedagogical objective. For lesson exercises, we proposed Terry to use a shared whiteboard through a tablet, and a share file storage space for persistent works. These abilities could both be used when interacting with the teacher, or with classmates. Clearly, this solution was great for Terry, but faced another problem: some teachers were not used working with these numeric tools, and refused to use them in their class. TDTECH increased, not because of the student, but because of the teacher.

Fig. 4: Connected learning environments

For IP telephony practical work, we used a VPN connection between Terry's house and our university, to allow him to connect to servers, IP telephones and the robot. Notice that Terry made his practicals examination in telepresence, under the same conditions as his classmates. He also took the TOEIC exam 1 in telepresence. He had no problems with distance, and passed both exams successfully.

Analysis

We conducted interviews with Terry, Sam, and the teachers who participated in this experiment, to get their opinion on the positive or negative impact of the connected learning environments associated to the robot (see [START_REF] Gallon | Pedagogical environments for telepresence robots[END_REF]). The results of these study are:

• knowledge and know-how (TDSC): the opinion is unanimous for knowledge, there is no difference between face-to-face and distance. At know-how level, Terry felt at ease. Teachers recognized that the essential topics were validated. The connected pedagogical environment has therefore played its role well at a TDSC level. • interactions with teachers and classmates (TDST and TDSS): teachers recognized that their interactions in class with Terry were equivalent when using the robot compared to face-to-face. They observed the same qualities and deficiencies in Terry while communicating to his robotic form compared to face-to-face: very little intervention in mathematics, a lot of participation in English. In practical works, telepresence did not prevent him from chatting and laughing with classmates.

Obviously, the use of the robot did not change his attitude, nor that of the teachers. Note that the math teacher, initially reluctant to use document sharing tools, acknowledged that this would ultimately be a good thing for her interaction with Terry • interactions with distant space: in practical works, there was a before and after connected learning environment. Before, Terry was an observer, and could only observe the work of his classmates. Using the dedicated pedagogical connected environment, he was able to be active, to realize on his own remote manipulations, and learn by practice, a fundamental element of pedagogy. Comforted in his abilities, he agreed to take the risk of doing his IP telephony exam at the same time as classmates. He scored 12/20 with the same exam compared to others.

This study shows that the use of a connected learning environment greatly facilitates significantly the insertion of the telepresence robot into the pedagogical sequences, and reduces the transactional distance. A larger study, with a strong validation, should be conducted to confirm these preliminary results.

Conclusion and perspectives

In this paper, we tackled the problem of perseverance in using telepresence robots for homebound students. We have shown that once the initial emphatic context has passed, pedagogical difficulties appear when taking into account the student in his robotic form in the teachings. We have described several examples of connected learning environments that reduce the transactional distance.

In future work, we are thinking of consolidation our first results through a broader study, taking into account in particular the team cognition dimension, so important in the world of education. We haave started experimenting with a telepresence escape game, to see how the players collaborate in the same physical space, while they are all physically distant from this space. The conclusions of these works should allow us to create telepresence learning at our university, for professionals who can not move physically, or for lifelong learning.

Fig. 1 :

 1 Fig. 1: Used telepresence robots

Fig. 2 :

 2 Fig. 2: Catherine driving the robot for the first time as part of practical work in aeronautical construction classroom to hear well what the teacher says. They are silent when the homebound student wants to ask the teacher a question.Although we have been able to observe these behavorial changes, further study will be needed to determine what the real sociological levers on which the robot has an impact, are. This study has, in our opinion, all its interest in Middle Schools, where we know that relations to others are fragile and in full construction. It should show that the tripartite teacher-student-class social link is the main driver of the homebound student from using the telepresence robot, and that classmates are particularly attentive to this factor.The arrival of the robot has at the start a positive impact on the motivation of all classmates. But quickly, new difficulties appear. Indeed, the interactions between the actors and the remote space are limited through the robot. This inevitably has a negative impact on the quality of the knowledge and know-how accumulated by the distant student. The use of the robot in an educational context is not enough. It must be complemented by interaction tools with the different stakeholders and also the remote physical space to get closer to a physical presence, and reduce the transactional distance. In the following, we describe some solutions we have experimented for that.

  1 https://www.etsglobal.org/

Table 1 :

 1 list of technical capacities of telepresence robots

		Beam	Double	Ubbo
	cameras	2	2	2
	video quality	++	+	+
	audio quality	++	+	+
	head height	fixed	variable	fixed
	rotating head	no	no	yes
	obstruction	medium	small	high
	zoom	x2	x2	x2
	weight	-(20kg)	++ (7 kg)	+ (13kg)
	stability	++	-	++
	battery life	2h	4h	4h
	driving	easy	medium	medium
	configuration/boot	easy	medium	too coplicated
	network connection issues	variable	not a lot	medium
	Impact on TDTECH

++(SMALL) +(MEDIUM) +(MEDIUM)

Table 2 :

 2 summary of key points for motivation at the first stage

		student	family board classmates teachers
	Illness		Empathy	Fear
	Student image	Fear
	New technology		Exciting	Fear
	Responsibility		Fear	Fear
	Marginalization		Fear
	Video capture			Fear

Table 3 :

 3 Impact of course topology on TDTECH

		TDST	TDSS TDSC	TDTECH
	lessons	X			small
	lesson exercises	X	X		medium
	practicals	X	X	X	high

as defined in[START_REF] Newhart | Virtual Inclusion via Telepresence Robots in the Classroom: An Exploratory Case Study[END_REF], "homebound students" are students who are not able to attend school for a long time, due to symptoms, treatments, or recovery from illness, (e.g., cancer, heart failure,. . . )


Roy P Le
email: caroline.kaub@gmail.com

The M6 1799 Vendée earthquake (France) : a multidisciplinary investigation of the active fault

 40 km² of new HR bathymetry (Geoswath interferometric sonar -RETZ2) and topography (RGEALTI, IGN) data  Machecoul fault morphology  700 km of new seismic profiles (Sparker source single channel streamer, CHIRP echo sounder -RETZ1 & 2)  Existing onshore drill database (BSS, (7)) to stress Neogene depot centers in the Marais Breton.  Available onshore gravity data (BGI, (8)) to calculate Bouguer anomaly and 2D modeling near the MF  basin geometry & sediment infilling.

 10 stations temporary seismological experiment (MACHE) to complement 4 years of regional catalog  current seismicity

The coastal Vendée (France) is located to the south of a main structure of the Armorican Massif, the 600-km-long South Armorican crustal Shear Zone (SASZ) [START_REF] Bitri | The earthquakes of stable continental regions[END_REF] [fig. 1]. This "stable continental region" (2) is affected by a system of dominantly NW-SE trending shear zones and faults, inherited from a long and polyphased tectonic history since Variscan times.

This area currently presents a moderate background seismic activity, but was affected by a significant historical earthquake (magnitude M ~ 6) on the 25th of January 1799. The mesoseismal area probably influenced by strong site effects in the nearby Neogene basin (Marais Breton) is located along an onshore/offshore discontinuity bounding northward the basin, the Machecoul fault (MF) [fig. 2].

The perception area stretched from Paris to Bordeaux and made this event the largest documented in the western part of France [fig. 1]. Earthquake parameters have been determined from macroseismic data (3) and present a wide variability of magnitude and depth couples : from Mw 5.8 to 6.6 and depth around 20 km (4 and 5).

Can we connect this main earthquake

to a specific fault ?

Fig. 2: Main faults of the Marais Breton and Baie de Bourgneuf and epicentral location (3), modified from (6) .

Seismicity

Fig. 1 :

 1 Fig. 1: Macroseismic intensity of the M6 1799 earthquake.

Fig. 6 :Fig. 5 :

 65 Fig. 6: HR bathymetry and interpreted seismic lines. See bathymetry location on [fig. 3].