Sabah Selmaoui
email: sselmaoui@gmail.com

Boujemaa Agorram

Salah-Eddine Kzami

Abdelaziz Razouki

Pierre Clément

Silvia Caravita

Cultural and socio-economical dimensions of Human Reproduction & Sex Education in the biology textbooks of eight countries

Keywords:

This study was carried out within the European research project "Biology, Health and Environmental Education for Better Citizenship". We present a comparative analysis of textbooks from eight Mediterranean countries which differ by their cultures, their socioeconomical levels and their religions. Our work is focused on the sensitive educational topic "Human Reproduction and Sex Education". We analyzed 43 biology textbooks among 8 countries: four are in Europe and four in Africa and Middle East. Grids were built to identify implicit values in the texts and images. We analyzed more precisely the explicit or implicit presence of cultural, socio-economical and ethical dimensions. We investigated whether the authors of textbooks present differently this theme depending on the socio-cultural, ethical and religious contexts. The content of the textbooks was analyzed as reflecting the conceptions of their authors and publishers. The outcomes of the analyses show that several textbooks present images of human beings restricted to only one ethnic type, usually white-skinned and with fair hair. The social, cultural, economical and ethical dimensions were generally absent, except for some interest for them in few European textbooks.

Introduction

In the present work we analyzed current school textbooks of biology among eight countries with regard to interactions between their contents and some contrasted socio-cultural contexts as different cultures and religions. The general theoretical background of the research is the KVP Model [START_REF] Clément | Science et idéologie : exemples en didactique et épistémologie de la biologie[END_REF][START_REF] Clément | Didactic Transposition and the KVP Model: Conceptions as Interactions Between Scientific Knowledge, Values and Social Practices[END_REF] proposing to analyze the conceptions expressed in the textbooks as possible interactions between three poles: K (scientific Knowledge), V (systems of values) and P (social practices). Our work is focused on two important topics in Biology Education: Human Reproduction and Sex Education (HRSE). We focus in particular on ethnical, cultural, socio-economic and ethical dimensions. We are interested in identifying how the choice of teaching or not some knowledge and how the way they are presented, expressed values that are often implicit, and social practices which are dominant in each country, showing interactions between K, V and P. This analysis concerns a part of the didactic transposition [START_REF] Chevallard | La transposition didactique, du savoir savant au savoir enseigné. Grenoble, La pensée sauvage[END_REF][START_REF] Chevallard | Les processus de la transposition didactique et leur théorisation[END_REF][START_REF] Clément | Science et idéologie : exemples en didactique et épistémologie de la biologie[END_REF][START_REF] Clément | Didactic Transposition and the KVP Model: Conceptions as Interactions Between Scientific Knowledge, Values and Social Practices[END_REF] to better understand how the choice of teaching content in the textbooks is done. Sex education is a very delicate topic in several countries, where scientific and medical aspects cannot be dissociated from psychological, ethical and socio-economical dimensions (Picod & Guigné 2005[START_REF] Pelège | Eduquer à la sexualité. Un enjeu de société[END_REF]. Few works were done until now from international surveys of curricula and textbooks, before the BIOHEAD-Citizen project1 . The present work is a complement of the first publications coming from this project, which were concerning other dimensions of human reproduction and sex education in textbooks (e.g. [START_REF] Bernard | Sexual transmitted diseases and control of reproduction in biology textbooks. A comparative analysis in 16 countries[END_REF]).

Methodology

A didactic analysis of textbooks is therefore involved in textbooks contents and taking into account their explicit or implicit messages. The images can be analyzed as significant of the sociocultural context of those countries. We categorized the images related to human reproduction and sexuality education (HRSE) by identifying the types of persons, specifying their physical and / or cultural features to characterize their ethnic types. We then quoted the presence or absence, in each textbook, of cultural, socio-economic and/or ethical dimensions related to this topic HRSE, in order to determine the possible influence of the socio-cultural context in each country. We used here only a part of the grid developed within the Biohead-Citizen project asking:

-Is a socio-cultural approach present or absent? Figure 2 -Grid for the identification of Cultural, socio-economic and ethical dimensions in the chapters related to HRSE All the textbooks were published in the corresponding country except for Senegal, which uses French textbooks. Most of publisher was private except all in Tunisia and some in Lebanon, published by the educational centers linked directly to the Ministry of Education of those countries. All publishers in Morocco are independent of the state but need a specification set by the Ministry of Education: only textbooks reviewed and accepted by a committee of experts may be published. In Lebanon, all private books must be approved by the CRDP P (Centre for Development and Educational Research). Human reproduction is treated at different level of study in all countries except in Senegal where it is treated at only one level, the ninth year. This approach noticed the number of occurrences of specific physical and/or cultural criteria used to identify ethnic types. In the 43 textbooks, we found 74 images (28 in the Italian textbooks and 18 in French textbooks). We used mainly the following criteria to identify ethnic type: physical aspect (skin and hair color or/and face features) and dresses.

Results

3.1-Ethnical approach:

Our study shows clearly that the majority of illustrations in textbooks contain persons belonging to only one ethnic type, usually characterized by the color of skin and of hair: Archetype of European persons, with white skin and light hair…. Moreover, this color is dominant over black and brown. It seems that in some countries the black color of the skin is associated with disease, malnutrition and drama (for example in Tunisian, Senegalese and Italian textbooks, Figure 5) The illustrations showing many persons with ethnic diversity are rare, and exist only in Italian, French, Moroccan and Lebanese textbooks. We also observed the complete absence of images of a couple or a family showing an ethnic diversity. In the 12 textbooks of France and Cyprus, there is a total absence of such dimensions. In the other countries, there are only some ones.

Cultural, socio-economical and ethical dimensions

Cultural references: religion, customs, taboos.

The cultural dimension is present in only 6 textbooks (Italy, Morocco, Tunisia and Lebanon). We observe the total absence of this dimension in four countries textbooks: Portugal, France, Cyprus and Senegal (19 textbooks). Only one textbook in Italy and one in Tunisia refer to the male circumcision as a religious tradition and a medical practice.

-"Some religious traditions, such as the Islamic and Jewish, make use of the removal of foreskin (circumcision) … The circumcision is adopted also in case of disease..… (Italy,2001).

-Male circumcision (Muslim religious rite): -The circumcision ritual -circumcision surgery (Tunisia, 2005). Moroccan textbooks refer to the law of Islam which prohibits illegal sexual intercourse, having several partners, prohibits taking drugs and recommends adopting preventive behaviors against AIDS and STIs. They also advise to avoid prostitution and to use condoms to protect against these diseases.

-The respect of Islamic laws makes it possible to immunize firmly against the AIDS, as it does not allow having several sexual partners and prohibits the abnormal sexual behaviours....( 1996) -"To avoid illegal sexual intercourse'', ''To avoid having several sexual partners'', "Avoid prostitution". ( 2005). One Lebanese textbook refer to religion (monotheistic) to formulate views on abortion and medical assisted reproduction, considering abortion as a murder and asserting that procreation should be the result of normal sexual relation.

-'Many religions consider the IVG as an assassination. Procreation should be the result of normal sexual relation, complete and legal. For a religion, the assisted medical procreation is based on patrilineal and matrilineal filiations, for another one it is a form of adoption'' (1999). Theses values correspond to the country dominant religion in those countries: Italy, Morocco, Tunisia and Lebanon.

Socio-economical references: economic stakes

We observed the total absence of this dimension in textbooks from 3 countries (France, Cyprus and Italy, 20 Textbooks). Moroccan and Lebanese textbooks mentioned brand name of products such as contraceptives: pills and condoms…. …

In the Tunisian textbooks we noticed some techniques and methods of prenatal diagnosis, pregnancy tests most frequently used and testing for HIV.

The Portuguese textbooks contain the social consequences of demographic development on economic development.

"For instance, the perspective to Portugal till 2020 is that the country will have, proportionally, more people over 65 years old and fewer children. There will be less active people for each retired person that will bring some social implications" ( 2005) The Senegalese textbook refers to economic condition and how can the teenager prepare his/her sexual life.

-"Economic condition: the two partners must be able to provide for the material needs of their children… " (1996).

Ethical references: moral position

We noticed the total absence of this dimension in French, Cyprus and Tunisian textbooks (15 Textbooks). Some Moroccan, Lebanese, Portuguese, Italian and Senegalese textbooks refer to the religion, moral values, ethical and juridical position about abortion, assisted reproduction, legal status, moral condition in sexual relationship. The religion is also mentioned in Cultural references.

-"The abortion is an illegal operation in Morocco, allowed only in case the pregnancy presents a risk to the mother's health and life" (Morocco, 1996) -"Nowadays new perspectives raise doubts about principles and values and even the biological phenomena nature itself, in relation to the assisted reproductive technology" " The bioethics committees think about this subjects, once the new techniques and the social modifications from it put in risk the own human identity" (Portugal, 2005).

-"Ethical and juridical problems (related to artificial fecundation) are more and more relevant; this makes precise rules necessary as well as a clear definition of the rights of human embryo". "About medical assisted reproduction: "there is a lot of debate in Italy, some people think that there must be severe rules to limit the application of this techniques, other people think that the State does not have to impose limitations, and that single persons do have to choose according to their ethical convictions" (Italy , 2001).

-"Moral condition: the two partners must have a sense of modesty and good morals which enables them to respect the norm of life in society"" To know that a woman wants to be loved as a person and not as an object for the only sexual pleasure of the man… "(Senegal, 1996) -"There are many moral and ethical questions about the use of this technique. …... In fact, these bioethical questions constitute the basis of a debate in order to determine a legal scope for the human intervention in the field of biomedical sciences" (Lebanon, 2002).

Conclusion

The analyzed textbooks support the conception of a majority of images showing a European archetype as a blond person: white skin and light hair. We used these physical criteria to identify ethnic types: skin color and hair. This choice of physical criteria in the images seems to be also linked to socio-cultural context of each country. Indeed, modern societies consider white skin and light hair as aesthetic, especially in North African society. While a majority of persons are with brown hairs. Even in Senegal, only some images of persons are dark skinned. This dominance of white skinned persons was already found in the trees of human evolution in the textbooks of the same countries [START_REF] Quessada | Human Evolution in science textbooks: A Survey in Eighteen[END_REF]. We note a very small interest of such dimensions in some European textbooks for this topic HRSE. It's could be an indicator that the publishers' textbooks in these countries don't convey values through programs by the didactic transposition. When we found cultural, socio-economical and ethical dimensions it was mainly in the secondary school (students 14-18 years) in several countries. Lower differences among countries were found than what was expected, but significant interactions between scientific knowledge (K) and values (V) were be found. The textbooks in Morocco, Tunisia, Lebanon and Italy are examples where promoted values are juxtaposed and linked with knowledge related to HRSE, they convey different social, moral, religious values. At the opposite, textbooks of France and Cyprus avoid any reference to moral values. Moroccan textbooks use the injunctive style to adopt or avoid some sexual behavior, often with implicit or explicit moral values. Morocco is an example of how social values are didactically transposed into textbooks. Prostitution, adultery and homosexuality are forbidden by Islam and condemned by Moroccan law. Fidelity and abstinence are the preferred sexual practices promoted both by the dominant religion and most textbooks to prevent diseases. Compared with those of other countries, Lebanese textbooks seem to be a pole apart from the fact that despite their predominantly biomedical content, some may offer an example of health education while others convey different moral values. This feature is the result of Lebanese complexity of its society Some textbooks relate bioethical questions introducing a debate in order to determine a legal scope for the human intervention in the field of biomedical sciences. Some discussions related to controversial bioethical problems are mostly faith-based and / or based on conservative opinions linked to a fundamentalist religion. We conclude that the cultural, socio-economical and ethical dimensions are rare in the analyzed textbooks. They deserve to be more debate, as important socio-scientific issues [START_REF] Albe | L'enseignement des questions scientifiques socialement vives dans l'enseignement agricole : quelles sont les intentions des enseignants[END_REF] in the curricula and textbooks. That is a necessary complement of the scientific and medical advances in a human and socially acceptable way.

Figure 4 :

 4 Figure 4: Identification of ethnical types in images in the chapters related to HRSE in each country

  Table for the identification of cultural, socio-economic and ethical dimensions in the chapters related to HRSE in each country Table for the identification of the level and age of cultural, socio-economic and ethical dimensions in the chapters related to HRSE per each country

	Countries	Cultural references	Socio-economical	Ethical references
				references
	Portugal	0		1	2
	France	0		0	0
	Italy	1		0	1 (2 sentences)
	Cyprus	0		0	0
	Morocco	2		1	1
	Tunisia	1		1 ( 3 sentences)	0
	Senegal	0		1	1
	Lebanon	2		2	2 (3 sentences)
	Total	6 Textbooks		6 Textbooks	7 Textbooks
	Figure 6: Countries	Cultural references	Socio-economical references	Ethical references
	Portugal	X		18-19 years	11-12 years
					18-19 years
	France	X		X	X
	Italy	14-18 years		X	14-18 years
					14-18 years
	Cyprus	X		X	X
	Morocco	14-15 years		17-18 years	17-18 years
		17-18 years
	Tunisia	17-18 years		17-18 years	X
	Senegal	X		14 -15 years	14 -15 years
	Lebanon	16 years		16 years	16 ears
		14 ears		17 years	17 years
	Figure 7:

BIOHEAD-Citizen = "Biology, Health and Environmental Education for Better Citizenship", CIT2-CT2004-506015, FP6 of European Community.

Acknowledgements

This work has been done within the European research project ('Biology, Health and Environmental Education for better Citizenship': "Biohead-Citizen" CIT2-CT2004-506015. The authors particularly thank the other participants who gathered data or coordinated this process concerning the topic "Human Reproduction and Sex Education" in the textbooks of their respective countries.


Juan Pablo Maldonado

Mean field games

Mean field games models aim to understand the behavior of a large number of identical players, where each tries to optimize its position in space and time, but whose preferences are determined by the choices of the other players.

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

Mean field games

Mean field games models aim to understand the behavior of a large number of identical players, where each tries to optimize its position in space and time, but whose preferences are determined by the choices of the other players.

Mean field games models aim to understand the behavior of a large number of identical players, where each tries to optimize its position in space and time, but whose preferences are determined by the choices of the other players.

Introduced by Huang, Caines andMalhamé (2003, 2006) and by Lasry andLions (2006,2007).

Two important features: dynamics and anticipation (backward-forward structure.)

Mean field games models aim to understand the behavior of a large number of identical players, where each tries to optimize its position in space and time, but whose preferences are determined by the choices of the other players.

Introduced by Huang, Caines andMalhamé (2003, 2006) and by Lasry andLions (2006,2007).

Two important features: dynamics and anticipation (backward-forward structure.) Let I denote the set of players and assume |I | = N. Let X j t,N be a random variable that describes the position of player j at time t.

Let I denote the set of players and assume |I | = N. Let X j t,N be a random variable that describes the position of player j at time t.

The states of the players at time t = 0 are chosen i.i.d using the distribution m 0 .

Let I denote the set of players and assume |I | = N. Let X j t,N be a random variable that describes the position of player j at time t.

The states of the players at time t = 0 are chosen i.i.d using the distribution m 0 .

We reserve capital letters for random variables and lower case letters for their realizations.

At stage t = 0, 1 . . . T -1, player i observes his own state x i t,N and the average position on the state space of all the players, m t,N .

At stage t = 0, 1 . . . T -1, player i observes his own state x i t,N and the average position on the state space of all the players, m t,N .

Players choose simultaneously and independently their actions a i t,N .

At stage t = 0, 1 . . . T -1, player i observes his own state x i t,N and the average position on the state space of all the players, m t,N .

Players choose simultaneously and independently their actions a i t,N . Each player i receives the payoff ℓ(x i t,N , a i t,N , m t,N ).

At stage t = 0, 1 . . . T -1, player i observes his own state x i t,N and the average position on the state space of all the players, m t,N .

Players choose simultaneously and independently their actions a i t,N . Each player i receives the payoff ℓ(x i t,N , a i t,N , m t,N ). The new state X i t+1,N is chosen randomly using the transition function Q(x i t,N , a i t,N , m t,N ).

The random average distribution is

M t+1,N := 1 N j =i δ X j t+1,N . The random average distribution is M t+1,N := 1 N j =i δ X j t+1,N
.

At the beginning of stage t + 1, the realization of X i t+1,N and M t+1,N , denoted x i t+1,N and m t+1,N respectively, are observed, and the situation is repeated.

The random average distribution is M t+1,N := 1 N j =i δ X j t+1,N .
At the beginning of stage t + 1, the realization of X i t+1,N and M t+1,N , denoted x i t+1,N and m t+1,N respectively, are observed, and the situation is repeated.

At stage t = T a final payoff g (x i T ,N , m T ,N ) is allocated.

A behavioral strategy for player i is a vector π i = (π i t ) T -1 t=0 where π i t : H t → P(A) and H t = (X × A × P(X )) t is the set of all possible histories up to date t. Denote by S the set of behavioral strategies.

A behavioral strategy for player i is a vector

π i = (π i t ) T -1 t=0 where π i t : H t → P(A) and H t = (X × A × P(X ))
t is the set of all possible histories up to date t. Denote by S the set of behavioral strategies.

A Markovian strategy for player i is a vector

σ i = (σ i t ) T -1 t=0 such that σ i t : X → P(A). A stationary strategy is a function σ : X → P(A).
A behavioral strategy for player i is a vector

π i = (π i t ) T -1 t=0 where π i t : H t → P(A) and H t = (X × A × P(X ))
t is the set of all possible histories up to date t. Denote by S the set of behavioral strategies.

A Markovian strategy for player i is a vector

σ i = (σ i t ) T -1 t=0 such that σ i t : X → P(A). A stationary strategy is a function σ : X → P(A).
A strategy profile is a vector π = (π i ) i∈I , where π i is a behavioral strategy of player i.

Payoff

The payoff of player i, when using the strategy π i and when his adversaries use the strategy profile π -i ∈ S N-1 is

J i N (x, m 0 , π i , π -i ) := E Q π t∈T ℓ(x i t,N , a i t,N , m t,N ) + g (x i T ,N , m T ,N ) .
The trick: Hide the complexity of the problem in a parameter m ∈ P(X ). Anticipate a fixed average distribution on the state space m = (m 0 , m 1 , . . . , m T ). Compute an optimal strategy for a single player with m fixed.

The trick: Hide the complexity of the problem in a parameter m ∈ P(X ). Anticipate a fixed average distribution on the state space m = (m 0 , m 1 , . . . , m T ). Compute an optimal strategy for a single player with m fixed.

Everyone's strategy creates a vector of state distributions

m ′ = (m 0 , m ′ 1 , . . . m ′ T )
The trick: Hide the complexity of the problem in a parameter m ∈ P(X ). Anticipate a fixed average distribution on the state space m = (m 0 , m 1 , . . . , m T ). Compute an optimal strategy for a single player with m fixed.

Everyone's strategy creates a vector of state distributions

m ′ = (m 0 , m ′ 1 , . . . m ′ T ) If m = m ′ , we are happy. Definition: MFE Definition Let m 0 given. A mean field equilibrium is a pair (σ, m) = (σ t ) T -1
t=0 , (m t ) T t=1 such that: 1 σ is the optimal strategy in the one player game Γ m , computed using the dynamic programming equation.

Proposition

(M., 2013) Let x be a fixed initial state, (σ, m) a mean field equilibrium and (a i t,N ) t∈T an arbitrary sequence of actions of player i. Consider the following two trajectories:

1 The trajectory of player i defined by X i t+1,N ∼ Q(x i t,N , a i t,N , m t,σ,N ). 2 The trajectory defined by X i t+1 ∼ Q(x i t , a i t,N , m t ). The following estimate holds:

max i=1,...,N E max s≤T X i s -X i s,N ∞ ≤ L Q T |X | exp(T ( Q ∞ + L Q + 1)) √ N
In continuous time mean field games, the complementary approach of studying the limit behaviour of equilibria of N player games as N → +∞ has been developed by Bardi (2012) for the linear-quadratic case and by Lasry and Lions (2007) and Feleqi (2013) for games with several populations of players and ergodic payoffs.

This construction is "robust" with respect to the number of players: players can "play well" even if they do not know the exact number of players.

This construction is "robust" with respect to the number of players: players can "play well" even if they do not know the exact number of players.

In general, the set of Nash equilibria of N player might contain equilibria that depend on N as in the driving game.

This construction is "robust" with respect to the number of players: players can "play well" even if they do not know the exact number of players.

In general, the set of Nash equilibria of N player might contain equilibria that depend on N as in the driving game.

N players have to choose whether to drive on the left or on the right in a two-way road.

Some equilibria: everyone on the left, everyone on the right, if N is even, everyone on the left; N odd, everyone on the right.

For the discounted case, one can define a mean field equilibrium as follows:

The value of the one-player game Γ λ m , (m ∈ P(X ) is fixed) with payoff:

V λ (x, m) := sup σ E Q ∞ t=1 (1 -λ) t-1 ℓ(x t , σ(x t ), m)
satisfies:

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

For the discounted case, one can define a mean field equilibrium as follows:

The value of the one-player game Γ λ m , (m ∈ P(X ) is fixed) with payoff:

V λ (x, m) := sup σ E Q ∞ t=1 (1 -λ) t-1 ℓ(x t , σ(x t ), m)
satisfies:

V λ (x, m) = max a∈A    ℓ(x, a, m) + (1 -λ) y ∈X V λ (y , m)Q(x, a, m)(y )    .
Proposition Φ λ • Ψ λ has a fixed point, i.e. there exists a stationary mean field equilibrium in the following cases:

If for every stationary strategy σ and all m ∈ P(X ), the Markov chain with transition law Q(•, •, σ(x), m) has a unique stationary distribution.

Proposition Φ λ • Ψ λ has a fixed point, i.e. there exists a stationary mean field equilibrium in the following cases:

If for every stationary strategy σ and all m ∈ P(X ), the Markov chain with transition law Q(•, •, σ(x), m) has a unique stationary distribution.

The transitions are independent of the state distribution, i.e.

Q(x, a, m)(y ) =: Q(x, a)(y ), ∀(x, y , a, m).

We adapt an idea recently introduced by Neyman( 2013) and Cardaliaguet et al.( 2013) to our model. The aim (informally) is to construct an approximation by Friedman/Fleming discretization of a game in continuous time.

We adapt an idea recently introduced by Neyman( 2013) and Cardaliaguet et al.( 2013) to our model. The aim (informally) is to construct an approximation by Friedman/Fleming discretization of a game in continuous time. Consider a function µ : X × X × A × P(X ) → R bounded and such that, for all (x, a, m):

µ(x, y , a, m) ≥ 0, x = y , µ(x, x, a, m) = - y =x µ(x, y , a, m).
Let ρ > 0. The payoff in continuous time we want to approximate is:

∞ 0 e -ρt ℓ(x t , σ(x t ), m)dt
where (x t ) t≥0 is a continuous time Markov chain whose transition semigroup has generator µ(•, σ(•), m).

The players choose their actions at times {δ, 2δ, 3δ, . . .} The value V ρ,δ for the one-player game satisfies:

ρV ρ,δ (x, m) = max a∈A    ℓ(x, a, m) + e -ρδ y ∈X µ(x, y , a, m)V ρ,δ (y , m)  



Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

Proposition

The equation

ρf (x, m) = max a∈A    ℓ(x, a, m) + y ∈X µ(x, y , a, m)f (y , m)    has a unique fixed point, denoted V ρ . Moreover, V δ ρδ → V ρ uniformly as δ → 0. Proposition The equation ρf (x, m) = max a∈A    ℓ(x, a, m) + y ∈X µ(x, y , a, m)f (y , m)
   has a unique fixed point, denoted V ρ . Moreover, V δ ρδ → V ρ uniformly as δ → 0.

Proof. Use the stationary strategy σ ρ given by this equation in the game with stage δ.

⊓ ⊔

Now let σ δ be a fixed stationary strategy for the game with stage δ and let m ′ ∈ P(X ). Let L[σ δ , m ′ ] ∈ R X ×X be defined by

L[σ δ , m ′ ] x,y = µ(x, y , σ δ (x), m ′ ).
Now let σ δ be a fixed stationary strategy for the game with stage δ and let m ′ ∈ P(X ). Let L[σ δ , m ′ ] ∈ R X ×X be defined by

L[σ δ , m ′ ] x,y = µ(x, y , σ δ (x), m ′ ).
The transition matrix associated to Q δ is then

I + δL[σ δ , m ′ ].
Now let σ δ be a fixed stationary strategy for the game with stage δ and let m ′ ∈ P(X ). Let L[σ δ , m ′ ] ∈ R X ×X be defined by

L[σ δ , m ′ ] x,y = µ(x, y , σ δ (x), m ′ ).
The transition matrix associated to Q δ is then

I + δL[σ δ , m ′ ].
For σ, m ′ given, the associated invariant distribution m must solve

δL[σ, m ′ ] • m = 0 ⇐⇒ L[σ, m ′ ] • m = 0
Combining the two limit equations we obtain the following: Definition A limit stationary mean field equilibirum is a pair (σ, m) ∈ Σ s × P(X ) such that σ is the stationary strategy derived from

ρV ρ (x, m) = max a∈A    ℓ(x, a, m) + y ∈X µ(x, y , a, m)V ρ (y , m)    and m solves: L[σ ρ , m] • m = 0.
Theorem (M.2013). For every ǫ > 0 there exists δ 0 > 0 and N 0 ∈ N such that, for all δ < δ 0 and N > N 0 , the strategy provided by the limit stationary mean field equilibrium is a 2ǫ-Nash equilibrium of the discounted mean field game with discount factor λ = 1 -e -ρδ .

Example

Consider the industry of online hotel booking, where many firms offer accommodation. In this case:

i) The state variable is the reputation of each firm.

ii) The actions are the adjustment of the offers.

iii) Each firm's reputation changes randomly, depending on the current reputation and the actions.

Example

Consider the industry of online hotel booking, where many firms offer accommodation. In this case:

i) The state variable is the reputation of each firm.

ii) The actions are the adjustment of the offers.

iii) Each firm's reputation changes randomly, depending on the current reputation and the actions.

iv) The firms payoff depends on the average reputation of the others.

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

Example

Consider the industry of online hotel booking, where many firms offer accommodation. In this case:

i) The state variable is the reputation of each firm.

ii) The actions are the adjustment of the offers.

iii) Each firm's reputation changes randomly, depending on the current reputation and the actions.

iv) The firms payoff depends on the average reputation of the others.

v) Frequent interaction is desirable in this example! Mean field games provide an extremely simple strategy that does not need to keep track of the other players.

However, the mean field equilibrium might not be unique! Unless the players agree somehow on which equilibrium to play, it is hard to predict anything.

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

m is the trajectory followed by m 0 according to the mass equation for the strategy σ.Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

Nash equilibrium

Definition

An ǫ-Nash equilibrium where ǫ > 0, is a strategy profile (π i ) i∈I such that, for all player i and all behavioral strategy τ, J i N (x, m 0 , τ, π -i ) ≤ J i N (x, m 0 , π i , π -i ) + ǫ.

Dynamic programming

From the familiar arguments we obtain the following dynamic programming equation:

with terminal condition V (T , x, m) = g (x, m T ).

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

The forward component

Now consider a Markovian strategy σ ∈ Σ and m 0 fixed and let m σ 0 := m 0 . We define, for t ≥ 0 :

Existence Proposition (M.,2013) There exists a mean field equilibrium for the finite horizon game in the following cases:

If there exists a unique maximizer for the right hand side of dynamic programming equation for m and for each (s, x).

Proposition

(M.,2013) There exists a mean field equilibrium for the finite horizon game in the following cases:

If there exists a unique maximizer for the right hand side of dynamic programming equation for m and for each (s, x).

The transitions are independent of the state distribution, i.e.

Q(x, a, m)(y ) =: Q(x, a)(y ), ∀(x, y , a, m).

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

Driving game

N players have to choose whether to drive on the left or on the right in a two-way road.

Some equilibria: everyone on the left, everyone on the right, if N is even, everyone on the left; N odd, everyone on the right.

The N player game has 2 ℵ 0 equilibria, while the game with infinitely many players has only two.

The discounted N-player game

The λ-discounted N player game is the game with payoff:

for λ ∈ (0, 1].

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

For the discounted case, one can define a mean field equilibrium as follows:

The value of the one-player game Γ λ m , (m ∈ P(X ) is fixed) with payoff:

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

Stationary mean field equilibrium

The mean field equilibrium in this case is a fixed point of the maps Ψ λ : P(X ) → Σ s and Φ λ : Σ s P(X ):

The players choose their actions at times {δ, 2δ, 3δ, . . .}

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

The players choose their actions at times {δ, 2δ, 3δ, . . .} The value V ρ,δ for the one-player game satisfies:

The limit system

Combining the two limit equations we obtain the following:

Definition A limit stationary mean field equilibirum is a pair (σ, m) ∈ Σ s × P(X ) such that σ is the stationary strategy derived from Proof.

Collect everything:

Proof.

Collect everything:

Choose δ 0 first, from the uniform convergence of the value functions.

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

Proof.

Collect everything:

Choose δ 0 first, from the uniform convergence of the value functions.

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

Proof.

Collect everything:

Choose δ 0 first, from the uniform convergence of the value functions.

Finally, take N 0 as in the bound derived for the error in the game with K 0 stages.

⊓ ⊔

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

Example

Consider the industry of online hotel booking, where many firms offer accommodation. In this case:

i) The state variable is the reputation of each firm.

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

Example

Consider the industry of online hotel booking, where many firms offer accommodation. In this case:

i) The state variable is the reputation of each firm.

ii) The actions are the adjustment of the offers.

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

A possible way out (Repeated driving game)

Example

Consider the repeated version of the driving game with N players with the following adaptation mechanism:

i) Each player chooses left or right with probability 1 2 on the first stage.

Example

Consider the repeated version of the driving game with N players with the following adaptation mechanism:

i) Each player chooses left or right with probability 1 2 on the first stage. ii) On the second stage, observing the realizations of the first stage, each player looks at everyone's choice (and recalls its own) and imitates the choice of the majority.

Example

Consider the repeated version of the driving game with N players with the following adaptation mechanism:

i) Each player chooses left or right with probability 1 2 on the first stage. ii) On the second stage, observing the realizations of the first stage, each player looks at everyone's choice (and recalls its own) and imitates the choice of the majority.

iii) Thus, from stage three, the players follow a mean field equilibrium.

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014

Thank you!

Juan Pablo Maldonado López (UPMC) Discrete mean field games: the short-stage limit Tours, June 26, 2014


Pierre Clément
email: pierre.clement@univ-lyon1.fr

Charline Laurent
email: charline_la@yahoo.fr

Elwira Samonek

Polish teachers' conceptions related to environment

Keywords:

We analyse the conceptions of 322 Polish teachers, related to Environment. They mainly differ from the way some teachers think that animals as snails, flies or frogs can or cannot feel happiness and, independently, from their opinions pro-or anti-GMO. The six samples (primary school teachers, secondary school teachers of biology or of Polish, with for the three pre-service and in-service teachers) have different conceptions, the biology teachers being more pro-GMO. Most of the Polish conceptions are for preservation of environment, but, when compared to 12 other countries (the same teachers' samples), they are more anthropocentric than most of the observed conceptions in the 12 other European countries (except in Lithuania and Finland).

-Introduction

Wieslaw Stawiński was an active member of the AEDB (Association Européenne de Didactique de la Biologie) during the nineties, an association chaired by Pierre Clément.

From this active collaboration born the inclusion of a Polish team, under the responsibility of Elwira Samonek, in an European research project (BIOHEAD-Citizen), coordinated by Pierre Clément and two other colleagues. This project ("Biology, Health and Environmental Education for better Citizenship") involved 19 countries, 6 of them out of Europe, from 2004 to 2008. It was focused on the relations between science and society, through six topics: (1) Environmental Education, (2) Health Education, (3) Human reproduction and Sexuality Education, (4) Evolution and human origins, (5) Human Genetics and (6) Human Brain. Are these topics taught in the same way, with the same scientific content and the same goals, in the 19 countries?

To answer to this question, we analysed the school textbooks dealing with these topics in the 19 countries, and the teachers' conceptions.

Most of our results, presented in Brussels [START_REF] Carvalho | Biology, health and environmental education for better citizenship: CITC-CT-2004-506015[END_REF] are not yet published concerning the data gathered in Poland. We present here the main results related to Polish teachers' conceptions on nature, environment and environmental education.

Our theoretical background is the KVP model (Clément 2004a[START_REF] Clément | Didactic transposition and the KVP model : conceptions as interactions between scientific knowledge, values and social practices[END_REF], to analyse conceptions as possible interactions between three poles: the scientific knowledge (K), values (V) and social practices (P). For the six topics of the project, the taught knowledge is strongly associated with values and social practices. That is particularly truth for the Environmental Education [START_REF] Clément | Environmental Education : analysis of the didactic transposition and of the conceptions of teachers[END_REF]: its goals are not limited to the transmission of (multidisciplinary) knowledge but also involves the students' attitudes and values [START_REF] Giordan | Une Education pour l'Environnement[END_REF][START_REF] Giolitto | Éduquer à l'environnement[END_REF].

Nevertheless, the philosophy of Nature [START_REF] Quillot | Crise et retour de l'idée de la nature[END_REF] and of Environment [START_REF] Larrère | Principal component and correspondence analyses with respect to instrumental variables: an overview of their role in studies of structure-activity and species-environment relationships[END_REF] shows that several and often divergent values are associated with them (Sauvé 1994[START_REF] Schultz | Values as predictors of environmental attitudes: evidence for consistency across 14 countries[END_REF], Clément 2004b). From a survey of the literature, we defined several axes to analyse the teachers' conceptions related to nature and environment [START_REF] Forissier | Les systèmes de valeurs d'enseignants du secondaire sur la nature et sur l'environnement. Une analyse comparative en France, Allemagne et Portugal[END_REF], Caravita et al. 2008). We built a questionnaire taking into account the following points: * The two poles defined by [START_REF] Wiseman | A higher-order model of ecological values and its relationship to personality[END_REF] when analysing students' conceptions on environment: utilisation and preservation, which are not very different from the two classical types of conceptions on Environment: anthropocentred and ecocentred. Several results of the Biobead-Citizen research confirmed the importance of these two poles [START_REF] Munoz | Teachers' conceptions of nature and environment in 16 countries[END_REF]): are we going to find these two poles in the Polish teachers' conceptions? Some other results showed three poles in the teachers' conceptions: anthropocentred (pole utilisation), ecolocentred (pole preservation) and "sentimentocentred" (see the next paragraph). * This sentimentocentred pole, focused on the capacity of animals to feel dolour or happiness.

This pole was very structuring of the teachers' conceptions analysed in France, Portugal and Germany [START_REF] Forissier | Les valeurs implicites dans l'Éducation à l'Environnement. Analyse de la formation de SVT[END_REF][START_REF] Forissier | Les systèmes de valeurs d'enseignants du secondaire sur la nature et sur l'environnement. Une analyse comparative en France, Allemagne et Portugal[END_REF], in Lebanon [START_REF] Khalil | Anthropocentrées, écolocentrées ou sentimentocentrées : Les conceptions d'enseignants et futurs enseignants libanais sur la nature et l'environnement[END_REF], in Algeria [START_REF] Khammar | Les conceptions d'enseignants algériens sur la nature et l'éducation à l'environnement[END_REF] and in Morocco [START_REF] Khzami | Les systèmes de valeurs d'enseignants et de futurs enseignants marocains des sciences de la vie et d'arabe sur l'environnement[END_REF]. What are the Polish teachers' conceptions related to this pole? * The last point is linked to the GMO, with animated debates inside most of the European countries, generally structured by an opposition vs. acceptation of GMO, French teachers being mostly anti-GMO (Clément et al. 2007).

We will analyse the Polish teachers' conceptions and then we will briefly compare them to the 11 other European countries involved in the Biohead-Citizen project.

-Methods

2-1 -Samples

In Poland, 322 teachers filled out the Biohead-Citizen questionnaire. The six samples are briefly presented in the Table 1. Most of them were catholic (94.1%), very few protestant (0.6%), atheist or agnostic (1.9%). Only 3.4% ticked the item "I don't wish to answer" when answering this question on their own religion. Tableau 3 -The questions related to the pole "Utilisation" (anthropocentric conceptions)

Samples of

A12. Genetically modified plants will help to reduce famine in the world.

I agree I don't agree

A13. Genetically modified organisms are contrary to nature. I agree I don't agree

A39.

Genetically modified plants are good for the environment because their cultivation will reduce the use of chemical pesticides (e.g. insecticides, herbicides).

I agree I don't agree

A47.

Genetically modified plants are harmful to the environment because they will contaminate other crop plants, menacing their survival. I agree I don't agree

A49.

If a person eats genetically modified plants, his/her genes can be modified. I agree I don't agree In the first questionnaire used for the pilot test, there was 18 questions related to the feelings of animals : 6 animals with, for each, their ability to have feelings, to be happy and to feel dolour. There was a so strong correlation between feelings, dolour and happiness that we decides to use only one of these three categories. We reduced also the number of animals because 100% of the teachers had the same answer for animals as dogs or monkeys. When we know in advance the answers, it is useless to maintain the questions.

The last questions are dealing with practices related to environment, as the questions A56

(below), and to Environmental education, as the question A61.

2-3 -How the questionnaire has been filled out

All the teachers had to individually filled out the 144 questions of the questionnaire (10 pages), with a total guarantee of anonymity. It took between 30 to 45 minutes.

It was at the end of a course for the pre-service teachers, and in their school for the in-service teachers.

2-4 -Analysis of data

The Polish team put the data on an Excel table, then analysed by the French team and collectively discussed by E-mail and during the Meeting of Budapest (February 2008).

We used multivariate analysis which are described in other works [START_REF] Clément | Environment and GMO (Genetically Modified Organisms): Conceptions of teachers from 16 countries[END_REF][START_REF] Munoz | Teachers' conceptions of nature and environment in 16 countries[END_REF] mainly PCA and between analyses completed by randomization tests (Monte Carlo type): (Lebreton, Sabatier, Banco, and Bacou, 1991), [START_REF] Dray | Co-inertia analysis and the linking of ecological tables[END_REF][START_REF] Dolédec | Co-inertia analysis: an alternative method for studying species-environment relationships[END_REF]

-Results

3-1. PCA (Principal Components Analysis)

The main component structuring the differences of Polish teachers' conceptions is related to the feelings of animals (Component 1 = horizontal axis in the figure 1). Any teacher answered in the same way for the 3 animals, with important differences among the teachers, as shown in the table 6. About one third of them agree or rather agree that snails, flies or frogs are able to feel happiness. The second principal component structuring the differences among the Polish teachers conceptions is related to the GMO, with a strong opposition between pro-and anti-GMO opinions (vertical axis of the figure 1a). There is here also a great coherence of the teachers' answers to these 5 questions, with nevertheless an exception for the question A13

("Genetically modified organisms are contrary to nature"): a large part of anti-GMO answers for the other questions agree with this proposition, which mainly deals with values (table 7).

Table 7 -Answers to the 3 questions related to the feelings of animals

Globally, about the two third of Polish teachers are in favour to GMO, thinking that they "will help to reduce famine in the world" (A12), that they are "good for environment because their cultivation will reduce the use of chemical pesticides (e.g. insecticides, herbicides)" (A39), and disagreeing that "Genetically modified plants are harmful to the environment because they will contaminate other crop plants, menacing their survival" (A47). These 3 questions were dealing with interaction between knowledge, values and the social use of GMO,

showing the importance of this last aspect for the Polish teachers.

An interesting point is emerging from the answers to the question A49 ("If a person eats genetically modified plants, his/her genes can be modified"). That was a question of only scientific knowledge : it is known that it is not truth, even if the consequence of using GMO as human food is still today very debated. Nevertheless, the answers to this question are closely correlated to the answers of the 3 precedent questions, showing that the "knowledge" of teachers is mainly induced by their conviction pro-or anti-GMO.

The two first components are orthogonal (axes 1 and 2 in the figure 1a), showing an independence between these two sets of conceptions. Being pro-or anti-GMO, the teachers think that animals as snails, flies and frogs can or cannot feel happiness. Thinking that these animals can (or cannot) feel happiness, the teachers can be pro-or anti-GMO.

The third component structuring the Polish teachers conceptions is the vertical axis of the figure 1b, opposing answers for the preservation of environment to the answers for its utilization. When only these questions are analysed [START_REF] Wiseman | A higher-order model of ecological values and its relationship to personality[END_REF][START_REF] Munoz | Teachers' conceptions of nature and environment in 16 countries[END_REF], there is an independence between the pole preservation and the pole utilization. Here, these two poles are in opposition, teachers more agreeing with preservation are more disagreeing with utilization and reciprocally: nevertheless with a little less coherence in their answers than for the two first components of their conceptions (figure 1b).

Nevertheless, there is a great sensitivity of Polish teachers for the preservation of environment: 84.6% of them totally disagree with the proposition "We need to clear forests to increase agricultural areas" (question A23), and 75.9% with the proposition "Only plants and animals of economical importance need to be protected" (question A54). The same teachers are "enjoying trips to the countryside" (89.1% of I agree for the question A22) and most of them (45.7% agreeing and 33.1% rather agreeing with the proposition "It makes me sad to see the countryside taken over by building sites" (question A28), to take only some examples of answers. This relative homogeneity of Polish teachers' conceptions for the preservation of environment explains that this topic is only the third component explaining the difference among their conceptions on Environment, the main differences coming from the two first components (animals feel or not happiness, and pro-or anit-GMO).

3-2. Between analyses to differentiate groups of Polish teachers

There is no significant difference between teachers when they are grouped from their gender, their age or their level of qualification.

Nevertheless, there is a significant difference (p<0.001 figure 2d) when we compare the six samples described in the table 1 (figure 2). This difference is linked to the questions dealing with GMO, opposing the biology teachers (PreB and InB) to the other teachers (figure 2c and2f). The biology teachers more agree than their colleague with the proposition A39

("Genetically modified plants are good for the environment because their cultivation will reduce the use of chemical pesticides (e.g. insecticides, herbicides))", and with the proposition A12 ("Genetically modified plants will help to reduce famine in the world"). They more disagree than their colleagues with the propositions A13 ("Genetically modified organisms are contrary to nature") and A49 ("If a person eats genetically modified plants, his/her genes can be modified"). That means that they are more in favour with GMO than their colleagues, knowing better than them that our genes are not modified when eating GMO (A49) but also from opinions more dealing with values (A13: for them the GMO are less contrary to nature)

and dealing with interaction between knowledge and values for the controversial questions A39 and A12. (c) Each point is corresponding to one teacher's answers, joined to the centre of gravity of its group (the 6 samples described in the table 1). Each ellipse encompass 2/3 of each sample.

(d)

The test of randomization (Monte Carlo) shows that the observed difference is outside the histogram coming from 1000 essays by random: the difference between the 6 samples is significant (p<0.001).

(e) Correspondence between the axes of the initial PCA 'figure 1) and the axes of this between analysis: its horizontal axis is corresponding to the axis 2 of the PCA, dealing with GMO.

(f) Enlargement of the graph (c ), with only the centres of gravity of the 6 samples. The figures 3 shows that the teachers' conceptions on environment differ from a country to another, the main differences being along the horizontal axis, with the opposition between Lithuania, and also Poland and Finland, to the other European countries (Romania and Cyprus being in the middle). The questions which support this opposition are, by order of importance: mostly A17 and A28, then also A54, A18 and A32. Teachers from Lithuania, and at a little degree from Poland and Finland, think more than their colleagues from other countries that "Society will continue to solve even the biggest environmental problems" (A17); and think less than the others that "It makes (them) sad to see the countryside taken over by building sites" (A28). These conceptions are correlated with other more anthropocentric conceptions, agreeing with "Only plants and animals of economical importance need to be protected" (A54), " Human beings are more important than other living beings" (A18) and "Humans have the right to change nature as they see fit" (A32).

The table 8 shows the differences between 4 of the 12 countries for the question A17. At one pole (more ecolocentric and for preservation) is Germany, France being very near. At the other pole (anthropocentric, more for utilization) is Lithuania and, in the middle, Poland. In Lithuania and in Poland, teachers are more optimistic than most of their colleagues of other European countries, believing more that our society will be able to solve the biggest environmental problems. We hope they will be right, and that our research on teachers' conceptions related to Environment will help to improve scientific education to take decisions in this direction.

Question

A56.

  There is a decision-making process in the implementation of science applications related to environment and biotechnology. Indicate, in each line, your degree of confidence in different actors to make such decisions (tick only ONE case for each line): Scientists Members of Parliament Science experts of this specific field Science experts of diverse fields including ethics All the citizens (referendum) Elected persons representing citizens at the national, regional or local levels A61. In your opinion, the main goal of environmental education in school should be (tick only ONE of the four boxes): Providing knowledge Developing responsible behaviour

Figure 2 -

 2 Figure 2 -Between analysis differentiating the six samples of teachers. (a) Part of variance explained by the different components. The first one is the most important. (b) Circle of correlation showing that the questions which differentiate the 6 samples are dealing to GMO, mainly the question A39.(c) Each point is corresponding to one teacher's answers, joined to the centre of gravity of its group (the 6 samples described in the table 1). Each ellipse encompass 2/3 of each sample. (d)The test of randomization (Monte Carlo) shows that the observed difference is outside the histogram coming from 1000 essays by random: the difference between the 6 samples is significant (p<0.001). (e) Correspondence between the axes of the initial PCA 'figure1) and the axes of this between analysis: its horizontal axis is corresponding to the axis 2 of the PCA, dealing with GMO. (f) Enlargement of the graph (c ), with only the centres of gravity of the 6 samples.

Figure 3 -

 3 Figure 3 -Between analysis differentiating the Polish teachers to teachers from other European countries: (from left to right of the horizontal axis) LT = Lithuania, FI = Finland, RO = Romania, CY = Cyprus, PT = Portugal, HU = Hungary, IT (hidden by MT) = Italy, MT = Malta, EE (hidden by FR) = Estonia, FR = France, DE = Germany. (a) Part of variance explained by the different components. (b) Circle of correlation showing the questions which differentiate the 13 countries. (c) Each point is corresponding to one teacher's answers, joined to the centre of gravity of its country. Each ellipse encompass 2/3 of each country. (d) The test of randomization (Monte Carlo) shows that the observed difference is outside the histogram coming from 1000 essays by random (p<0.001).(e) Correspondence between the axes of the initial PCA from these countries and the axes of this between analysis (f) Enlargement of the graph (c ), with only the centres of gravity of the 13 countries.

Table 1 -The samples of Polish teachers who filled out the questionnaire 2-2 -Questionnaire

 1

	Polish teachers	number	Mean age	Gender (%
			(years old)	women)
	PreP = Pre-service teachers in Primary Schools	54	23	88.9 %
	PreB = Pre-service Biology teachers (Secondary)	51	23	88.2 %
	PreL = Pre-service Language teachers (Secondary)	48	23	85.4 %
	InP = In-service teachers in Primary Schools	57	40	96.5 %
	InB = In-service Biology teachers (Secondary)	51	39	92.2 %
	InL = In-service Language teachers (Secondary)	61	39	90.2 %

A22. I enjoy trips to the countryside. I agree I don't agree A28. It makes me sad to see the countryside taken over by building sites. I agree I don't agree A40. It is interesting to know what kinds of animals live in ponds or rivers. I agree I don't agree A50. All contemporary plant species should be preserved because they may help in the discovery of new medicines. I agree I don't agree Table 2 -The questions related to the pole "Preservation (ecolocentric conceptions) A4. Nature is always able to restore itself.

Table 4 -

 4 The questions related to the poleGMO (Genetically Modified Organisms)

	A10. Snails are able to feel happiness.	I agree	I don't agree
	A29. Frogs are able to feel happiness.	I agree	I don't agree
	A45. Flies are able to feel happiness.	I agree	I don't agree

Table 5 -

 5 The questions related to feelings of animals (sentimentocentred pole)

Table 6 -

 6 Answers to the 3 questions related to the feelings of animals

	These animals are able to feel	I agree	I rather	I rather	I don't
	happiness		agree	don't agree	agree
	Snails	11.9%	23.5%	29.6%	35.0%
	Frogs	11.3%	24.1%	29.3%	35.4%
	Flies	10.3%	17.4%	28.0%	44.4%

Table 8 -

 8 Answers to the questions A17 in Poland and in France. A17 : "Society will continue to solve even the biggest environmental problems"

	A17	I agree Rather agree Rather don't agree I don't agree
	Lithuania (n = 316)	83.9%	13.0%	1.6%	1.6%
	Poland (n = 322)	31.8%	33.1%	23.2%	11.9%
	France (n = 732)	6.4%	18.9%	40.3%	34.4%
	Germany (n = 365)	1.9%	6.8%	40.3%	51.0%

Acknowledgements

This work has been supported by the European Research Project Biohead-Citizen (Specific Targeted Research n° 506015, FP6, Priority 7: "Biology, Health and Environmental Education for better Citizenship").