Eva Carpigo

Beauty at the Service of Humanity. A Review on the Therapeutic Value of Aesthetic Treatments

Keywords: Aesthetic treatments, beauty, anthropo-poiesis, psycho-socioaesthetics, humanitarian cosmetic surgery, beauty-care, self-improvement, ethics

Body modifications made with an eye towards attaining 'beauty' have been the subject of much debate by social scientists. Facing the tendency to judge beauty practices as an expression of superficiality, vanity or social 'oppression', this paper aims to approach aesthetic treatments in recognising their inherent complexity. First of all, I display an original perspective on these practices based on the theory of anthropo-poiesis, the theoretical mark proposed by the cultural anthropologist Francesco Remotti. As inferred by this author, concern with beauty can be read symbolically as an effort to grant subjects to pass into ontological categories of 'human-beingness'. In this regard, aesthetic procedures are demanded to resolve the feeling of 'incompleteness' or to face a condition of social marginality. Through the presentation of two field cases, the 'psycho-socio-aesthetics' and the 'humanitarian cosmetic surgery', I propose to focus the viewpoint of beauty practitioners who agree in considering aesthetic enhancements as their cathartic objectives to evacuate uneasiness and suffering. In consequence, I argue that aesthetic treatments should be recognised as practices of well-being, since their aim is to enhance user's psychological and social life. Lastly, I affirm the need to value ethics as a complementary skill to individuate appropriate conditions into which practices of beauty-care should take place.

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These contributions show us some original focuses through which we can understand the cultural value of aesthetic treatments. Following a symbolic approach to beauty, aesthetic practices can be intended as a way to step away from an ideal of humanity that is not socially valued, to an ideal of humanity that is socially valued. In the next session I will present Remotti's viewpoint, which expands this concept.

Ontological Beauty : A Quest for "Humanness"

Since the 1990's the anthropologist Francesco Remotti has focused on the subject of culture and identity, showing a particular interest in the practices of body alteration and enhancement. [START_REF] Remotti | Prima Lezione di Antropologia[END_REF] Remotti's vision of culture is that of: 'something more disorienting in its benefits, its utility and its capacity of adaptation'. [START_REF] Remotti | Fare Umanitá. I Drammi dell'Antropo-poiesi[END_REF] Following this reasoning beauty practices, which are cultural products, cannot be reduced to their teleological objectives, which are focused on genes or species reproduction. Nor can they be understood simply as a submission to cultural determinism, or as a way to predominate over others through attractiveness. Revisiting Clifford Geertz's theories on human biological incompleteness, Remotti agrees that culture is the mean by which people accommodate for organicindeterminacies and consequently produce particular forms of 'humanity'. Beauty, by Remotti's definition is a fundamental 'quest of humanness'. [START_REF] Remotti | [END_REF] Focusing his attention on the plasticity inherent to body aesthetic modifications, Remotti coined the term anthropo-poiesis, meaning: the act of 'creating' or 'shaping' particular forms of humanness. In the theory of anthropo-poieis individuals are perceived as craftspeople who forge their bodies following allegiance to a set of values (which can also be plural, mixed, syncretic or antinormative). [START_REF] Remotti | Fare Umanitá[END_REF] Remotti lists twenty-three different categories of aesthetic interventions on human body, including also non-interventions on the body. [START_REF] Remotti | [END_REF] This general theory on beauty seems to be validated through all cultures in the world, since it argues that aesthetic treatments are part of the entire cultural process -symbolic and material -which allow people to imagine the so-called 'naked ape' 18 as a human.

Taking Care of the Human Being: "Psycho-Socio-Aesthetic" Health Career

As it was noticed, anthropo-poietic acts can't be reduced to their alteration of the material structure of bodies. Undergoing aesthetic treatments can be understood as an effort to accomplish the desire to achieve a particular form of 'humanness,' which can provoke an associated sense of well-being and good health. The first field case that I have decided to present to you, shows closer the complexity of this issue.

Since the early 1970's, trained psycho-socio-aestheticians have worked with

Beauty at the Service of Humanity __________________________________________________________________ private and public French institutions proposing aesthetic care to people in 'distress'. An association named CODES [START_REF] Codes | Course of Aesthetics Applied to Humanitarian and Social Matters[END_REF] -Cours D'Esthétique à Option Humanitaire et Sociale -aims to train aestheticians to work with this particular public. Socio-aestheticians still have trouble gaining legitimacy as a profession of health care in society-at-large; despite this, they seem to be a relatively tightly knit group, unified by a shared humanitarian objective and believing in similar ethical principles.

The specificity of this profession is that it consists in organising aesthetic workshops in welcoming institutions (e.g. prisons, hospitals, oncology centres), aiming to show users how to increase their aesthetic appearance in order to improve their quality of life and reduce negative consequences and/or causes of their distress (disease, social marginalisation, mental illness, etc.). Unlike 'classical' aestheticians, psycho-socio-aestheticians don't consider the recipients of their care as clients, as their funding comes from institutions and not from the people they treat. Nor do they consider the people they treat as patients but simply as human beings in situations of distress and 'social fracture'. Categories listed below, show the heterogeneous panel of users by which can intervene a psycho-socioaesthetician:

The elderly (dependent or independent) Female prisoners Oncology patients Alcoholics and drug addicts People with a physical or mental handicap Women in maternity Cosmetic surgery patients People in physical rehabilitation People suffering from food disorders Homeless people Helpers -people who help others in distress -Applied with these categories of people, psycho-socio-aesthetics explicitly claim a therapeutic value involved in treatments. Through the enhancement of appearance, socio-aestheticians aim to awake bodily senses, pacify unease and improve people's sociability. A proper treatment, practitioners say, will mitigate pain, reduce feelings of loneliness, suffering, low-self esteem, ward-off illness, relieve stress, fix a lack of femininity and help people form a positive attitude towards the future. Most of the practitioners encountered have admitted the importance of the psychological approach to users, that's why during the entire session a verbal interaction with the participant is highly valued. In this perspective, the aesthetic workshop is also designed as a catharsis, allowing people to speak-up about the negativity of their condition -illness or distress -and __________________________________________________________________ thereby evacuate it. A final efficacy of these treatments is confirmed when a 'grey complexion' and a sad expression are transformed into a 'brightly, smiling and relieved face'. Following psycho-socio-aesthetician´s rhetoric, the mere fact that the user accepts to participate in a beauty workshop is considered to be already a success. In this sense, through aesthetic treatments, which are often considered a pretext to achieve a proximity with the person, socio-aestheticians aim to transmit a sort of 'positive psychology' in order to improve the desire to re-integrate society or to simply get better. To promote this objective, psycho-socio-aestheticians recreate the ambiance of a well-being centre: soft lights and mirrors are hung, walls are painted with warm colours, relaxing music is played and essential oils are diffused in the air. Individual appointments can be available for each participant; everything is designed so that the patient feels calm and coddled. In an atmosphere like this, a socio-aesthetician can offer to handle the participants, allow two participants to treat each other, or allow participants to treat themselves. Among the services proposed are: massages with perfumed lotion (called 'modelling'), make-up, hairstyling, manicure, podiatry and hair-removal treatments. Though most services revolve around the skin, other workshops are offered to develop self-image through activities like: drawing, making collages out of beauty images, fashion courses and behaviour courses (which aim to help participants look and feel presentable during a job-search).

Restoring a Human Form: the "Humanitarian Cosmetic Surgery"

The second field case concerns the problematic of the recognition of a therapeutic value in cosmetic surgery procedures. Even if contemporary French society recognises the psychological and social benefits of reconstructive plastic surgery, most cosmetic surgeries are still treated as non-medical and nontherapeutic procedures. French media, as well as popular discourse, portrays cosmetic surgery as a risky practice, fetishizing perfection and propagating a 'cult of eternal youth'. To counter these distortions, cosmetic practitioners are seeking legitimacy through the recognition that their services are of therapeutic value. This controversial debate is not new. In their cultural histories of cosmetic surgery, historians Sander Gilman and Rossella Ghigi [START_REF] Sander | Making the Body Beautiful and Rossella Ghigi, Per Piacere. Storia culturale della chirurgia estetica[END_REF] have shown that the therapeutic effects of aesthetic surgical enhancement has most often gone unmentioned or been outright denied. While reconstructive plastic surgery is often praised for helping to reestablish a 'natural order' in the appearance, cosmetic surgery is commonly judged an unnecessary, superficial, and even a dangerous practice. But what exactly is the difference between the two? And what's the opinion of specialist surgeons on it? The majority of plastic surgeons denounce the way in which these two branches of surgery are dichotimised. One of the world's most well-known cosmetic Beauty at the Service of Humanity __________________________________________________________________ surgeons, the Brazilian Ivo Pitanguy, argues that both types of surgery lead to similar, positive psychological effects. As he confided to Alexander Edmonds there is a substantial superposition between reconstructive surgery and cosmetic surgery: 'both emphasise the subtlety of mind-body interactions'. [START_REF] Edmonds | Learning to love yourself: Esthetics, Health and Therapeutics in Brazilian Plastic Surgery[END_REF] In his words, the healing value of plastic surgery takes place since it: 'Harmonizes the body with the spirit, emotions with reason, aiming to establish an internal equilibrium that will permit the patient to re-encounter himself, restructure himself, so that he feels himself in harmony with his own image and the universe that surrounds him'. 22 Plastic cosmetic surgery is for Pitanguy a good solution to manage interior concerns, and is therapeutic in so far as it can 'cure the soul'. [START_REF] Sander Gilman | Creating Beauty To Cure the Soul: Race and Psychology in the Shaping of Aesthetic Surgery[END_REF] Pitanguy aims to spread this philosophy all around the world through his writings and by its transmission to his trainees, who come to visit and work in his 'temple' of cosmetic surgery. A further concept arisen and practised by Pitanguy is that of cosmetic surgery at the service of 'humanity' or plastic surgery at the service of the humanitarian cause. Since the 1960's, he has led the 'Santa Casa' clinic in Rio de Janeiro where lower-and middle-class Brazilians come in great numbers to gain access to cosmetic surgery at a low price. In this sense, plastic surgery seems to be proposed to Brazil´s lower-and middle-classes as a better solution than psychotherapy because it is cheaper and more accessible. During my fieldwork in Paris I had the opportunity to meet one of Pitanguy's trainees: the surgeon Patrick Knipper, who first theorised about the concept of a 'humanitarian aesthetic surgery'. In his book Skin of Shagreen, Plastic Surgery at the Service of Humanity [START_REF] Knipper | La Chirurgie Plastique au Service de l'Humanité[END_REF] Knipper tells about his particular life trajectory which influenced his theoretical reflexion. Born in Congo but grew up in France, in 1992 Knipper joined the NGO InterPlast-France/Surgery Without Borders. Thanks to the experience gained in humanitarian missions, he started to ask himself about the efficiency of plastic surgery in humanitarian missions: 'Can a humanitarian mission do aesthetics? Can aesthetic surgery be a humanitarian mission?'. [START_REF] Knipper | Ethnochirurgie ou Réflexions Sur le Regard de l'Autre en Mission[END_REF] In a documentary for the French television chain Arte he explained: 'When we are poor, when we have nothing, the only thing that we own is our skin. The skin is our home, our last house (…) So, finally, reconstructive surgery or cosmetic surgery is also the surgery of poor people'. [START_REF] Lay | Dessine-Moi Un Visage[END_REF] Arguing against the polemics about the difference between reconstructive and plastic surgery he reminds:

__________________________________________________________________

'We want to recall that it is not up to us to decide what is important for a patient and what is not. Our role is to answer to a demand wherever it comes from. (...) During our missions, patients taught us that aesthetics was for them almost as important as function, and so the aesthetics become part or a whole with the function of plastic surgery, and that, no matter the country'. 27 Knipper claims that wherever plastic surgery is practised it induces some complex repercussions in the person's representations of the world. To make this action positive, therapeutic, and healing, cosmetic surgeons should pay careful attention to the culture of the patient, to their family-and even their personal history. In that way, referencing to the interdisciplinary approach of ethnopsychiatry, 28 Knipper formed successful partnerships with psychologists and traditional healers. His case depicts how productive collaboration between members of a heterogeneous team can be, particularly when their aim is to heal people following a holistic approach.

Conclusion

This paper arisen the need to complexify the analysis on body aesthetic practices. I've chosen to value some analyses based on a comprehensive approach, because they enhance original perspectives on the reflection of beauty. Based on my field research and on the theories quoted, I argue that aesthetic body modifications should be considered first and foremost as an experienced process of changing, in which the aim is an enhancement of user's psychological and social world. [START_REF] Davis | Estudios Culturales Sobre Cirugía Cosmetica[END_REF] In light of Remotti's ideas about anthropo-poiesis, aesthetic treatments are part of the cultural process of 'shaping one´s humanness'. In this perspective, practitioners and patients of beauty procedures seem to understand aesthetic services as a way to manage feelings of personal incompleteness or of social marginality. Furthermore, the cathartic character of beauty practices can help subjects to live the change as an improvement of their well-being, which seems to reinforce personal identity and a collective identification. This overview on aesthetic practices aims to face 'dystopic' theories on beauty and sustain instead a positive epistemology on beauty. In this perspective, it seems to me impossible to recognise a therapeutic effect of beauty procedures if we don't admit the importance of its multiple repercussions (psychological and social) on a user's life. Nevertheless, the study of the conditions through which an aesthetic treatment turns into a healing act, shows that they are complex and far from being universal. As in medicine, also in aesthetics each practitioner has his personal formula through which they stimulate some positive repercussions on patients. In spite of this premise, field examples show that both health and aesthetic Beauty at the Service of Humanity __________________________________________________________________ practitioners are sharing similar representations around the management of uneasiness. A growing number of doctors and beauty practitioners are currently choosing to adopt a holistic approach to the person who asks for help. [START_REF] Andrieu | Nadine Pomarède, and Georges Vigarello (under the direction of) La beauté holistique[END_REF] Thereafter, both are searching to establish interdisciplinary collaborations with other specialists of 'care' with the aim to enhance the efficiency of their treatments. As I noticed, more and more psycho-socio-aestheticians work side by side with nurses, doctors, psychologists, social workers. While cosmetic surgeons are beginning to cooperate with nutritionists, psychologists, aestheticians and even traditional healers. Considering this tendency, a scenario where multiple specialists are gathered under-one-roof is arising: everyone knowledge contributes to improve the global well-being of the patient. If we enlarge our perspective to the global dimension of the research, a general concept of beauty-care (doubly intended as : 'the care of aesthetic appereance' and 'beauty as an act of care') is spreading all around the world. As the philosophy of 'humanitarian cosmetic surgery' was 'imported' from Brazil to Europe, the concept of 'psycho-socio-cosmetics' has now been 'exported' from Europe to Asia (Singapore and Japan). 31 In this global scenario, I argue that we're assisting to a progressive promotion of different and even personalised beauty ideals, jointly with personal re-appropriations of techniques and philosophies of beauty.

Obviously, some ethical questions linked to the accessibility and safety conditions of aesthetic procedures, must be taken into account. The question of power, arisen through this Interdisciplinary Beauty Conference, should also be developed. Thanks to the rising philosophy of beauty-care, beauty practitioners are becoming central references in contemporary societies as unique and legitimate owners of technical skills. In this scenario their power can easily be abused. During my interactions with beauty practitioners, there emerged an 'ideal' portrait of a beauty practitioner: someone who possesses both technical abilities with a human sensitivity, at the same time respecting the imperative of a 'patient´s' desires. This ethical framework, which beauty practitioners share implicitly and transmit to their trainees, is considered to avoid the risk of transforming beauty-care into an aggressive act of subjugation. In conclusion, I affirm that an ethical reflexion on beauty is necessary, as much as it is for health concerns. Lastly, I support the idea of an interdisciplinary approach on beauty studies as a way to produce innovative results and to promote the encounter of heterogeneous reflections.

Notes


Arthur Blot
email: arthur.blot@ens-lyon.org

Jean-Michel Muller
email: jean-michel.muller@ens-lyon.fr

Laurent Théry
email: laurent.thery@inria.fr

Formal correctness of comparison algorithms between binary64 and decimal64 floating-point numbers

Keywords:

We present a full Coq formalisation of the correctness of some comparison algorithms between binary64 and decimal64 floatingpoint numbers.

Introduction

Both binary and decimal formats are defined in the IEEE 754-2008 standard. However, no operation is defined to compare them, and the "naïve" approach of converting a decimal floating point number to the closest binary one before comparing it to another binary FP number can lead to inconsistencies. Let us take for example these two FP numbers:

x := 7205759403792794/2 56 (in binary64 format) > 1/10 y := 1/10 (in decimal64 format)

If we convert y to the binary64 format, we get x. This means the native approach would give that x ≤ y, but actually x > y.

The paper [START_REF] Brisebarre | Comparison between binary64 and decimal64 floating-point numbers[END_REF] by Brisebarre et al. details and proves the correctness of several algorithms which can compare these numbers in a both efficient and entirely reliable way. The initial problem being simple, it would be nice if the algorithms and their correctness proofs were elementary too. This is unfortunately not the case. In particular, the correctness of some of the algorithms requires some nontrivial computation that cannot be performed by hand. The contribution of this paper is to propose a formalisation that carefully checks all the aspects of the correctness proofs.

We use the Coq proof assistant [START_REF] Coq | The Coq Proof Assistant Reference Manual -Version V8.6[END_REF] for our formalisation. The paper is organised as follows. In a first section we detail the formal setting in which we have made our formalisation. Then, we present the algorithms and their proof of correctness.

Coq is a generic system that lets us define new objects and build formal proofs of their properties. There are two main ways of adding new knowledge into the system. We can work conservatively defining a new notion in term of already defined ones or we can work axiomatically by providing the characteristics properties of the new notion. In our formalisation, the two approaches have been used. We now illustrate this with some elementary examples.

Positive numbers are represented in Coq in a conservative way by a recursive datastructure positive that mimics binary numbers. It is composed of 3 constructors: xI (a number that starts with a 1), xO (a number that starts with a 0), xH (the number 1). If we take the little endian convention, the binary number 1101 is represented as (xI (xO (xI xH))). Thanks to this recursive datastructure, it is then possible to equip positive numbers with usual functions. For example, we give below the definition of the function that computes the successor function. It is defined recursively by a discussion on the structure of its argument a. Relative numbers are then defined on top of positive numbers using these constructor Zero, Zpos for positive numbers and Zneg for negative numbers. Usual notations can be associated with these numbers, so that (Zneg (xI (xO (xI xH)))) is printed and parsed back as -13.

Similarly to the definition of the Psucc function, addition (+) is defined for relative numbers. Having this function as an algorithm inside Coq makes it possible to compute with relative numbers like in any programming language. Thus, -13 + 4 evaluates to 9. Furthermore, it is possible to reason about this function and prove that the distributive property holds.

Lemma add_assoc m n p, m + (n + p) = (m + n) + p.

Having a system where one can mix computation and proofs was mandatory for the formalisation of the properties we were aiming at. Coming back to this formalisation, we are using the Flocq library [START_REF] Boldo | Flocq: A unified library for proving floating-point algorithms in coq[END_REF] to reason about floating point numbers. These numbers are parametrised by a radix β. They are defined as a record that contains two fields.

Record float (beta : radix) := Float { Fnum : Z; Fexp : Z }.

For example, the object {|Fnum := 7205759403792794; Fexp := -56|} is a floating-point number whose mantissa is 7205759403792794 and exponent is -56.

In our formalisation, we prove the correctness of the comparison algorithms with respect to the comparison over real numbers. The real numbers are axiomatised in Coq. An abstract type R is defined. A set of properties is assumed that makes it a complete, totally ordered, Archimedean field. On top of this basic field, the standard Coq real library and the Coquelicot [START_REF] Boldo | Coquelicot: A user-friendly library of real analysis for Coq[END_REF] library build the usual real analysis. This contains standard notions like series and limits from which usual functions such as exponential, sine and cosine are defined. Thanks to these libraries, it is possible to give an interpretation of a floating point number as a real number.

Definition F2R (f : float beta) := (Fnum f) * beta ^(Fexp f).

The floating point number f represents the real numbers m * β e where m is the mantissa, e the exponent and β the base. Now the problem is simple, we want to build an algorithm comp that takes two floating point numbers f1 and f2 in base 2 and in base 10 respectively and returns their comparison. We use the standard Coq type comparison for the returned type of the comparison function comp. This type contains three elements Lt, Eq, Gt that represent the three possible values of a comparison. Our goal is then to prove that, under some conditions Cond that we will explain later, the following theorem holds Lemma comp_correct (f1 : float 2) (f2 : float 10) :

Cond → comp f1 f2 = Lt ↔ F2R f1 < F2R f2 ∧ comp f1 f2 = Eq ↔ F2R f1 = F2R f2 ∧ comp f1 f2 = Gt ↔ F2R f1 > F2R f2.
This lemma relates the result of an evalutaion of the algorithm comp with a relation between two real numbers. When proving an algorithm eq that performs an equality test, the returned type is simply a boolean so the correctness theorem looks like Lemma eq_correct (f1 : float 2) (f2 : float 10) :

Cond → eq f1 f2 = true ↔ F2R f1 = F2R f2.

Note that as real numbers are axiomatised in Coq, we can not directly compute with them. Fortunately, there exist some tools that address this issue. One tool that we are using in this formalisation is the interval tactic [START_REF] Martin-Dorel | Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq[END_REF]. It tries to automatically solve goals that contain real expressions by computing an approximation using standard techniques of interval arithmetic. For example, the following error bound of an approximation of the exponential function between 0 and 1 can be proved automatically.

Lemma bound_exp_taylor2 x : 0 ≤ x ≤ 1 → Rabs ((1 + x + x ^2 / 2) -exp x) ≤ 22 / 100.
Another technique to approximate reals numbers are continued fractions. We are using them in our formalisation. Following closely the material that is presented by Khinchin [START_REF] Khinchin | Continued Fractions. Dover books on mathematics[END_REF], we have developed a library to compute and reason about continued fractions inside Coq. The main basic result that is needed here is that continued fractions represent the best rational approximations one can get. The formal statement is the following.

Lemma halton_min (n : nat) (p q : Z) (r : R) :

0 < q < q[r]_n.+1 → t[r]_n ≤ Rabs (q * r -p).
Approximating a real number r by continued fractions can be seen as an iterative process. At rank n, we get an approximation, called convergent, s n = p n /q n . One way of quantifying the quality of an approximation p/q is by the value, |qr -p|.

Applied to s n , this gives θ n = |q n r -p n |, written t[r]_n in our formalisation.

The previous lemma halton_min simply states that the convergent at rank n is the best approximation for all the rational numbers p/q with q less than q n+1 .

Formalisation

We want to compare a binary floating-point number x 2 and a decimal floatingpoint number x 10 . This is reflected in our formalisation by introducing two variables of type float.

Variable x2 : float 2. Variable x10 : float 10.

Furthermore, x 2 belongs to the binary64 format and x 10 to the decimal64 format. This means that we have:

x 2 = M 2 • 2 e2-52
x 10 = M 10 • 10 e10-15

where:

-1022 ≤e 2 ≤ +1023 |M 2 | ≤ 2 53 -1 -383 ≤e 10 ≤ +384 |M 10 | ≤ 10 16 -1
Without loss of generality, we can also suppose that x 2 > 0 and x 10 > 0. If they are of opposite sign, the comparison is trivial and if they are both negative, taking the opposite of both numbers brings us back to the case we consider.

In our formalisation, as x 2 > 0, we shift the exponent is such a way that we have 2 52 ≤ M 2 . This corresponds to our M2_bound assumption. The four hypothesis M2_bound, e2_bound,M10_bound and e10_bound are implicitely conditions of all the theorems we present in the following.

Let us start deriving new facts from these definitions. First, M 10 can also be "normalised". If we consider ν = 53 -⌊log 2 (M 10 )⌋, we have that:

0 ≤ ν ≤ 53 2 53 ≤ 2 ν M 10 ≤ 2 54 -1
These correspond to the theorems v_bound and norm2_bound of our formalisation. Now, we express x 2 and x 10 with the largest common power of 2 possible. For this, we define m = M 2 , n = M 10 2 v , h = ν + e 2 -e 10 -37 and g = e 10 -15. This gives us the following equalities (theorems x2_eq and x10_eq respectively).

x 2 = m • 2 h • 2 g-ν x 10 = n • 5 g • 2 g-ν
The initial problem of comparing x 2 and x 10 reduces to comparing m • 2 h with n • 5 g . We also get bounds over m, h, n and g.

2 52 ≤ m ≤ 2 53 -1 2 53 ≤ n ≤ 2 54 -1 -398 ≤ g ≤ 369 -787 ≤ h ≤ 716
These are easily proven in Coq and correspond to the theorems m_bound, n_bound, g_bound and h_bound.

Handling "simple" cases

The first step of the algorithm consists in checking if the result can be determined by looking at the exponents g and h only. In order to compare 5 g and 2 h , a function ϕ is introduced. Its definition is the following.

ϕ(h) = ⌊h • log 5 2⌋
It is relatively easy to check that one can determine the result of the comparison between x 2 and x 10 if g and ϕ(h) differ. Formally, this means that the following two properties hold:

If g < ϕ(h) then x 2 > x 10 If g > ϕ(h) then x 2 < x 10
They correspond to the theorems easycomp_lt and easycomp_gt in our formalisation.

In order to get an algorithm, one still needs to provide a way to compute ϕ(h). One way to do this is to use the integer value ⌈2 k • log 5 2⌋ for a sufficient large value of k. In our case, k = 19 is enough, the following property holds.

ϕ(h) = h • ⌈2 19 • log 5 2⌋ • 2 -19
for |h| ≤ 1831

This corresponds to the theorem prop2_1. Unfortunately, such a theorem is outside the reach of the interval tactic which does not handle the floor and to the nearest functions automatically.

In order to overcome this problem, we use the characteristic properties of the rounding functions (x is a real number, y an integer)

⌊x⌋ = y iff y ≤ x < y + 1 ⌈x⌋ = y if |x -y| < 1/2
For example, in order to prove that ⌈2

(easycomp = Lt → F2R x2 < F2R x10) ∧ (easycomp = Gt → F2R x2 > F2R x10) ∧ (easycomp = Eq ↔ g = phi h).
Note that we are using the ideal integer arithmetic of Coq in a more realistic programming language we would have to deal with bounded arithmetic. Nevertheless, we can run our algorithm on the example given in the introduction and get the expected result.

Compute easycomp {| Fnum := 7205759403792794; Fexp := -56 |} {| Fnum := 1; Fexp := -1 |}. = Eq : comparison Our correctness theorem tells us that all we can conclude is that g = ϕ(h).

The algorithm in the previous section covers the cases where g differs from ϕ(h). In this section, we assume that g = ϕ(h).

Finding the needed precision

We define:

f (h) = 5 ϕ(h) • 2 -h = 2 ⌊h log 5 2⌋•log 2 5-h
which verifies:

f (h) • n > m ⇒ x 10 > x 2 f (h) • n < m ⇒ x 10 < x 2 f (h) • n = m ⇒ x 10 = x 2
Now, the idea is to get a precise enough approximation of f (h) • n in order to retrieve the exact comparison. Our goal is therefore to find a lower bound η over the smallest non-zero value of the error |m/n -f (h)| in the range where

2 52 ≤ m ≤ 2 53 -1 2 53 ≤ n ≤ 2 54 -1 -787 ≤ h ≤ 716
We can also add an extra constraint that directly comes from the definition of n.

If 10 16 ≤ n then even(n)

A good candidate for this lower-bound, as proposed in [START_REF] Brisebarre | Comparison between binary64 and decimal64 floating-point numbers[END_REF], is η = 2 -113. 7 . In order to formally check that this lower-bound holds, we split in four the interval of h. For 0 ≤ h ≤ 53, it is easily to prove that the lower bound is greater than 2 -107 . This corresponds to the theorem d_h_easy1. Similarly, the lower bound is greater than 2 -108 for -53 ≤ h ≤ 0 (d_h_easy2). For the two remaining intervals 54 ≤ h ≤ 716 and -787 ≤ h ≤ -54, we make use of our library for continued fractions. The key idea is that in order to find the lower bound for a given h it is not necessary to enumerate all the rationals m/n but only the convergents s n = p n /q n that approximate f (h) and whose denominator are bounded by 2 53 -1. The property that justifies this computation is the following: If q n ≤ n < q n+1 and |f (h) -m/n| < 1/2n 2 Then m/n = p n /q n This is the conv_2q2 theorem of our library that directly follows the proof given in [START_REF] Khinchin | Continued Fractions. Dover books on mathematics[END_REF]. In our case, as η = 2 -113.7 and 2 53 ≤ n ≤ 2 54 -1, we are sure that η < 1/2n 2 , so our theorem applies.

For the first interval 54 ≤ h ≤ 716, the lower bound ǫ p /ǫ q is reached for h = 612, m = 3521275406171921 and n = 8870461176410409 with ǫ p = 95837236471514977157488559853369253873503974784033193441759367158 879170025294997936467046601695051723045003185390596549988774013061 and ǫ q = 15076604622712586378413085535543158310234113291354064071380852407 218620654696227859485444413405009205300052717281615404952311527789 630428217161580997965373300467263261324839949270053015170403319309 9264

This corresponds to the theorem pos_correct of our formalisation. It is proved by a double enumeration. First, we enumerate all the h from 54 to 716, and for each of them, we enumerate all the convergents of f (h) whose denominator is smaller than 2 53 -1 and check that the error is bigger that ǫ p /ǫ q . The entire proof takes 80 seconds to be checked by Coq.

For the second interval -787 ≤ h ≤ -54, a similar technique is used. This time the lower bound ǫ ′ p /ǫ ′ q is reached for h = 275, m = 4988915232824583 and n = 12364820988483254 with ǫ ′ p = 112529423171232400134835569054963071809061903761868569039360251397 and ǫ ′ q = 186045148427403750191543353736499076917034509405654972460757157204 3696050968719646334648132324218750

This corresponds to theorem neg_correct and takes 100 seconds to be validated by Coq.

The theorem d_h_min collects all the previous results and proves the smallest non-zero error is bigger than η = 2 -113. 7 . We thus know that if the error is less than η, the distance is 0 and we must be in the equality case.

Direct method

The first way to use the previous result is to precompute f in a table indexed by h. Then, assuming µ approximates f (h) • n with accuracy ǫ < η/4, we have the following implications:

If µ > m + ǫ • 2 54 Then x 10 > x 2 If µ < m -ǫ • 2 54 Then x 10 < x 2 If |m -µ| ≤ ǫ • 2 54 Then x 10 = x 2
This corresponds to the theorem direct_method_correct of our formalisation. Since we have η = 2 -113.7 , a 128-bit computed value of f is enough to make the exact comparison. In order to build the algorithm, we first build the table with 1504 entries that contains ⌈f (h) • 2 127 ⌉. As we have 2 124 ≤ f (h) • 2 127 < 2 128 , our epsilon is equal to 2 -124 . This explains the 57 in the algorithm, 57 = -124 + 127 + 54. The proof of correctness is a direct consequence of these observations.

Theorem direct_method_alg_correct :

(direct_method_alg = Lt ↔ F2R x2 < F2R x10) ∧ (direct_method_alg = Eq ↔ F2R x2 = F2R x10) ∧ (direct_method_alg = Gt ↔ F2R x2 > F2R x10).
We have seen in the previous section with the execution of the easycomp algorithm that we have g = ϕ(h) for the example in the introduction. Now, running the direct_method_alg algorithm returns the correct value for the comparison.

Compute ineq_alg {| Fnum := 7205759403792794; Fexp := -56 |} {| Fnum := 1; Fexp := -1 |}.

= Gt : comparison

The main drawbak of this algorithm is that the table may be too large for some realistic implementation. The next section explains how we can alleviate the problem by using a bipartite table.

Bipartite table method

In this method, we use:

q = ϕ(h) 16 + 1 r = 16q -ϕ(h) such that f (h) = 5 16q • 5 -r • 2 -h .
We can easily derive the bounds for q and r from the bound of ϕ(h).

-21 ≤ q ≤ 20 1 ≤ r ≤ 16

In this method, we store 5 16q and 5 r after a phase of normalisation. For this, we define the function ψ as follows.

ψ(g) = ⌊g • log 2 5⌋
Similarly to what has been done for ϕ(h), it is easy to compute ψ(q) and ψ(16q) using integer arithmetic:

ψ(g) = ⌈2 12 • log 2 5⌋ • g • 2 -12 for |g| ≤ 204 ψ(16q) = ⌈2 12 • log 2 5⌋ • q • 2 -8 for |q| ≤ 32
These equalities correspond to the theorems prop_2_2 and prop_2_3 in our formalisation. They are proved by brute force in 5 minutes and 40 seconds respectively. Now, we introduce the two functions θ 1 and θ 1 that are defined as follows.

θ 1 (q) = 5 16q • 2 -ψ(16q)+127 θ 2 (r) = 5 r • 2 -ψ(r)+63
The following bounds are given by the theorems theta1_bound an theta2_bound.

2 127 ≤ θ 1 (q) < 2 128 -1 2 63 < θ 2 (r) < 2 64
Furthermore, we have

f (h) = θ 1 (q) θ 2 (r) 2 -64-σ(h) with σ(h) = ψ(r) -ψ(16q) + h If we define ∆ as ∆ = θ 1 (q) • n • 2 -64+8 -θ 2 (r) • m • 2 8+σ(h)
comparing x 2 and x 10 gives the same result as comparing 0 and ∆ (theorem delta_ineq). We also easily get that if x 2 = x 10 , then |∆| ≥ 2 124 η. This allows us to derive an approximated version of ∆:

∆ = ⌈θ 1 (q)⌉ • n • 2 8-64 -θ 2 (r) • m • 2 8+σ(h)
and proves that comparing 0 with ∆ gives the same result as comparing x 2 with x 10 (theorem delta'_ineq).

In order to build the algorithm, we first need to create the two tables for ⌈θ 1 (q)⌉ and θ 2 (r). Our tables are given in Figure 1. The first one contains integers that fit in 128 bits and the second one in 64 bits. The algorithm first computes q and r, then reads the values of ⌈θ 1 (q)⌉ and θ 2 (r), and then checks the sign of the computed ∆. The value 9511 corresponds to ⌈2 12 • log 2 5⌋ and =?= to the comparison function for integer. Its associated correctness theorem is the following.

Theorem ineq_alg_correct :

(ineq_alg = Lt ↔ F2R x2 < F2R x10) ∧ (ineq_alg = Eq ↔ F2R x2 = F2R x10) ∧ (ineq_alg = Gt ↔ F2R x2 > F2R x10).
Running the ineq_alg algorithm gives the same value as the one using the direct method.

Compute ineq_alg {| Fnum := 7205759403792794; Fexp := -56 |} {| Fnum := 1; Fexp := -1 |}. = Gt : comparison

Equality case

If we only want to test the equality between x 2 and x 10 , an even simpler algorithm can be used. We already know that x 2 = x 10 is equivalent to m•2 h = n•5 g . As 2 and 5 are relatively prime, only two situations can occur :

either 5 g | m and 2 h | n with 0 ≤ g ≤ 22 and 0 ≤ h ≤ 53; -or 2 -h | m and 5 -g | n with -22 ≤ g ≤ 0 and -51 ≤ h ≤ 0.

The algorithm is a direct encoding of this property. It checks that 5 g •(n2 -h ) = m if we are in the first case or that 5 -g • (m2 h ) = n if we are in the second case. Its correctness theorem proves the equivalence between running the algorithm and testing the two values.

Theorem eq_alg_correct : eq_alg = true ↔ (F2R x2 = F2R x10).

On our favorite example, it returns the expected result.

Compute eq_alg {| Fnum := 7205759403792794; Fexp := -56 |} {| Fnum := 1; Fexp := -1 |}.

= false : comparison

Our formalisation contains four algorithms for comparing a binary64 floatingpoint number and a decimal64 floating-point number and their proof of correctness. The code is available at: https://gitlab.com/artart78/compbindec It is composed of five files: util.v and rfrac.v for general results (most of which are about real numbers or continued fractions), frac.v for the result of the continued fraction problem solution, compformula.v for the computation of some formulas requiring tabulation of values, and compbindec.v for the main result. Altogether, this amounts to about 5000 lines of code that Coq checks in about 1 hour and 40 minutes. We have been using intensively SSReflect [START_REF] Gonthier | A small scale reflection extension for the Coq system[END_REF] for its set of tactics and syntax, the Flocq [START_REF] Boldo | Flocq: A unified library for proving floating-point algorithms in coq[END_REF] library for manipulating floating-point numbers, and the Interval [START_REF] Martin-Dorel | Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq[END_REF] tactic for the computation over the real numbers. We had to develop a dedicated library for continued fraction in order to tackle some aspects of the proof. This library is available in the rfrac.v file.

The main contribution of this work is a carefully check of the algorithms and the proofs presented in [START_REF] Brisebarre | Comparison between binary64 and decimal64 floating-point numbers[END_REF]. In particular, we have connected in a formal way the paper proofs with the computation that is required in order to get the accuracy at which f (h) • n needs to be computed. During the formalisation, we have found some minor mistakes in the original paper and we have also departed at some places from what was presented in the original paper. For example, the original paper states that -20 ≤ q ≤ 21 instead of -21 ≤ q ≤ 20. Fortunately, this mistake has no consequence. More values are tabulated in the actual implementation (the bounds are slightly relaxed). The original paper mentions the bound 2 -113.67 while our formalisation only makes use of 2 -113. 7 . The statement of the direct_method_correct omits the scaling factor of f (h) in the assumption. It makes its proof easier and simplifies its application. Finally, the statements of the theorems pos_correct and neg_correct do not mention the extra condition (if h ≥ 680, then ν ′ = h + ϕ(h) -971 > 0 and 2 ν ′ | n) that is present in the original paper. It does not change the lower-bound and we believe that this omission makes the statements more readable at the cost of a negligible extra computing.

The capability of computing expressions over the real numbers thanks to the interval tactic has been a key ingredient on this work. These computations are often hidden in the original paper. It is the case for example for the bounds for θ 1 and θ 2 . The bounds that are obtained without taking into account the bound on h are actually less strict than the ones specified in the paper. For example, only θ 1 < 2 128 could be proved, but it is actually important to get the 2 128 -1 bound in order to know we can embed the value into a 128-bit value. The 2 128 bound was obtained by noticing that 16q log 2 5 -⌊16q log 2 5⌋ < 1, but the 2 128 -1 bound requires that it is less than 1 + log 2 (1 -1 2 128 ) ≃ 1 -1 2 128 . We successfully manage to formalise all these proofs but clearly life would have been easier if the tactic interval would accommodate the floor and the ceiling functions. It could in particular greatly improve our checking time. 40 minutes out of the hour and 40 minus of our checking time are spent by our brute-force method in trying to validate the integer equivalent for the function ϕ(h).

There are several ways to extend this work. First, we could formalise the counting argument that is present in the original paper that quantifies the percentage of floating point numbers that are comparable using the first partiall comparison method (easycomp). It would require to develop the notion of cardinal of sets of floating point numbers in the Coq standard library. Second, what we have proven are only algorithms, it would be very interesting to try to prove a realistic software implementation using a tool like Why3 [START_REF] Filliâtre | Why3 -where programs meet provers[END_REF]. Finally a sequel of our reference [START_REF] Brisebarre | Comparison between binary64 and decimal64 floating-point numbers[END_REF] for this work has been written by the same authors in [START_REF] Brisebarre | Comparison between binary and decimal floating-point numbers[END_REF]. The general idea of the algorithm remains the same, but the generalisation of the result also requires a generalisation of all the intermediate results, and computation may become a lot harder since the numbers can then take up to 128 bits. Formalising it would be a real challenge.

Function

  Psucc a := match a with xI a1 => xO (Psucc a1) | xO a1 => xI a1 | xH => xO xH end.

  Definition M2 := Fnum x2. Hypothesis M2_bound : 2 ^52 ≤ M2 ≤ 2 ^53 -1. Definition e2 := Fexp x2 + 52. Hypothesis e2_bound : -1074 ≤ e2 ≤ 1023. Similarly for x 10 but without any shifting, we get Definition M10 := Fnum x10. Hypothesis M10_bound : 1 ≤ M10 ≤ 10 ^16 -1. Definition e10 := Fexp x10 + 15. Hypothesis e10_bound : -383 ≤ e10 ≤ 384.

Fig. 1 .

 1 Fig.1. Tables for ⌈θ1(q)⌉ and θ2(r)

  Definition eq_alg : bool := if (0 ≤ h) && (h ≤ 53) && (0 ≤ g) && (g ≤ 22) && (n mod (2 ^h) == 0) then let m' := 5 ^g * (n / (2 ^h)) in m' == m else if (h >= -51) && (-22 ≤ g) && (g ≤ 0) && (m mod (2 ^(-h)) = 0) then let n' := 5 ^(-g) * (m / (2 ^(-h))) in n' == n else false.

  19 •log 5 2⌋ = 225799, we reduce the problem to showing that |219 • log 5 2 -225799| < 1/2 which is solved automatically by the interval tactic. The proof of the prop2_1 theorem is done in a similar way. We first generate the 3663 subgoals that represents all the possible values for h. For each of the subgoal, we guess what the value z for ⌊h • log 5 2⌋ is. We then need to prove that ϕ(h) = z and h • 225799 • 2 -19 = z. For both of these goals, we use the characteristic property of the floor function. Showing that z ≤ h • log 5 2 < z + 1 and z ≤ h • 225799 • 2 -19 < z + 1 is done automatically by the interval tactic. This brute-force method is not the most elegant approach but it works. It takes Coq 40 minutes to check the theorem prop2_1.Once all the properties of the function ϕ are proved, it is easy to write down an actual algorithm for the simple cases

	Definition easycomp : comparison :=
	let h := v + e2 -e10 -37 in
	let g := e10 -15 in
	let phih := (225799 * h) / (2 ^19) in
	if g < phih then Gt
	else if g > phih then Lt
	else Eq.
	and derive its associated theorem of correctness
	Theorem easycomp_correct:

  Proving that this table is correct (for each entry i, the value represents ⌈f (h) • 2 127 ⌉) is done by brute and takes 40 minutes. The algorithm then just needs to look up in the table and compares with m * 2 217 .

	| -785 => 38772568831832703508158178356901129852
	...
	...
	...
	| 714 => 75715339914673581502256102241153698026
	| 715 => 37857669957336790751128051120576849013
	| 716 => 94644174893341976877820127801442122533
	| _	=> 0
	end.
	Definition direct_method_alg :=
	let v := n * (f_tbl h) -m * 2 ^127 in
	if v < -2 ^57 then Gt
	else if v > 2 ^57 then Lt
	else Eq.
	Definition f_tbl h : Z :=
	match h with
	| -787 => 155090275327330814032632713427604519407
	| -786 => 77545137663665407016316356713802259704


Denyze Toffoli
email: dtoffoli@unistra.fr

Insights into Student Postures in Autonomous Language Learning

Keywords: learner autonomy, attachment theory, out-of-class learning, self-report data

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Since its conception by John Bowlby and later Mary Ainsworth in the 1950s and 60s, attachment theory has become a cornerstone of developmental psychology. The quality of child attachment to a primary caregiver produces behaviours, which allow the child to progressively become more or less autonomous. The 1980s and 90s witnessed the development of adult attachment theory as a means of explaining relationship dynamics, especially in romantic relationships and, recently, authors such as [START_REF] Geddes | Attachment in the classroom: The links between children's early experience, emotional well-being and performance in school[END_REF], [START_REF] Fleming | A secure base for adult learning: Attachment theory and adult education[END_REF] and [START_REF] Riley | Attachment theory and the teacher-student relationship: A practical guide for teachers, teacher educators and school leaders[END_REF] have begun to examine the workings of attachment principles in educational settings in respect to both adults and children.

Within applied linguistics, various researchers (e.g., [START_REF] Benson | Beyond the language classroom[END_REF][START_REF] Benson | Autonomy and independence in language learning[END_REF][START_REF] Holec | Autonomy and foreign language learning[END_REF][START_REF] Little | Learner autonomy: Definitions, issues and problems[END_REF][START_REF] Little | Learner autonomy: Why foreign languages should occupy a central role in the curriculum[END_REF][START_REF] Little | Learner autonomy as discourse: The role of the target language[END_REF] have developed the concept of autonomy in language learning. Building on psychological constructs, autonomy has been approached from the perspectives of learning strategies [START_REF] Cohen | Strategies in learning and using a second language[END_REF]Wenden, 1991[START_REF] Wenden | Learner Development in Language Learning[END_REF] and self-determination theory [START_REF] Albero | L'autoformation en contexte institutionnel: Du paradigme de l'instruction au paradigme de l'autonomie[END_REF][START_REF] Candas | Analyse de pratiques d'étudiants dans un centre de ressources de langues: Indicateurs d'autonomie dans l'apprentissage[END_REF]. However, despite several hundred publications on autonomy and language learning [START_REF] Benson | Autonomy in language teaching and learning[END_REF], it has not yet been considered through the lens of attachment theory. This chapter will begin by outlining the importance of autonomy as a construct within applied linguistics. It will then present the principal elements of attachment theory considering ways in which they might be applicable to language learning. Next, the specific context and methodology of data collection will be described, before presenting a detailed case study exploring the concepts through one learner's experience. The final discussion will draw some tentative conclusions and consider future directions for both practice and research of working with attachment theory in relation to language learner autonomy.

Learner Autonomy in Language Education

Over the last 25 years or so, since Holec's seminal 1979 publication "Autonomy and Foreign Language Learning", the concept of language learner autonomy has been examined and developed, leading to more nuanced, detailed definitions. In the field of adult education, [START_REF] Knowles | The Adult Learner, Fifth Edition: The Definitive Classic in Adult Education and Human Resource Development (5 edition)[END_REF] or [START_REF] Tremblay | L'Autoformation -Pour apprendre autrement[END_REF] consider autonomy to be a psychological orientation, facilitating lifelong learning. [START_REF] Little | Learner autonomy as discourse: The role of the target language[END_REF] believes that "self-direction produces the most effective learning" (p.16). He builds on liberal philosophies of education and considers that "learner autonomy [is] the capacity for independent, self-managing behaviour in contexts of formal learning. […] autonomy is at once the goal of developmental learning and a characteristic of its underlying dynamic" (2000, p.31). Learner autonomy has thus been considered as a factor of empowerment, enabling transformation [START_REF] Mezirow | Learning as transformation: Critical perspectives on a theory in progress[END_REF]. More prosaically, the concept of learner autonomy has sometimes been seen as an answer to the double institutional requirement of 'massification' and cost-effectiveness [START_REF] Albero | L'autoformation en contexte institutionnel: Du paradigme de l'instruction au paradigme de l'autonomie[END_REF], especially concerning language learning, where the numerous hours of practice necessary for learners to be able to communicate effectively can be difficult and costly to supply, especially if they require constant tutor or teacher presence.

Various researchers have looked at the workings and manifestations of autonomy in language learning [START_REF] Benson | Autonomy in language teaching and learning[END_REF]. The latter have sometimes been considered as processes or steps that are adhered to during the autonomous learning process. [START_REF] Holec | Autonomy and foreign language learning[END_REF], for example, identifies seven essential processes that constitute the significant features of learner autonomy. These include the learners' decision to learn, their choice of methods and materials, decisions about where, when and how long to learn, what kind of feedback is needed, and self-evaluation. Autonomous learning is usually defined in distinction (or even opposition) to solitary learning (see [START_REF] Little | Learner autonomy: Why foreign languages should occupy a central role in the curriculum[END_REF], especially where languages and communication are concerned and the participation of other human beings is considered essential. It therefore (importantly) involves agency regarding authentic use of the target language itself [START_REF] Little | Learner autonomy as discourse: The role of the target language[END_REF]. [START_REF] Benson | Autonomy in language teaching and learning[END_REF] suggests that autonomous learning involves learners' control over learning tasks and activities but also over the cognitive processes concerned. For [START_REF] Little | Learner autonomy: Definitions, issues and problems[END_REF], these cognitive processes involve "detachment, critical reflection, decision-making and independent action" (p. 4). By "detachment" he clearly denotes the capacity to distance oneself from the object of learning and thus the ability to analyse and evaluate one's decisions critically. This is not detachment as it appears in attachment theory (where it is understood as either a part of the mourning process or a pathological protection mechanism) [START_REF] Riley | Attachment theory and the teacher-student relationship: A practical guide for teachers, teacher educators and school leaders[END_REF]. For Little, it is a metaphorical detachment from one's own emotional involvement in the learning experience, the capacity to stand back and analyse, and as such it is characteristic of the attainment of a large degree of autonomy. All of these studies amount to a fairly clear picture of what constitutes autonomy in language learning as we understand it today: goal-setting, planning, seizing opportunities, spending time, organising activities, finding appropriate partners, assessing progress and so on. As summarised by [START_REF] Nissen | Autonomie dans une formation hybride : Qu'en dit l'apprenant ? Les Langues Modernes[END_REF], learner autonomy today amounts to "taking charge of oneself, acting in an independent manner, knowing where, or from whom, to find help" [translated from original] / "se prendre en main, agir de manière indépendante, savoir où cherche de l'aide ou auprès de qui: c'est bien cela qui caractérise un apprenant autonome" (p. 18). These factors describe autonomous actions but do not inform us as to how individuals attain autonomy. [START_REF] Little | Learner autonomy: Why foreign languages should occupy a central role in the curriculum[END_REF] has looked more carefully at the characteristics of the development of autonomy in the language classroom. In his opinion, when pedagogy is not specially focused on the development of autonomy, some learners achieve it but the majority do not. On the other hand, when the development of learner autonomy is a central pedagogical goal [...], it turns out that all learners are capable of becoming autonomous, within the limits of their ability [my emphasis throughout this citation]. When the focus of learning is a foreign language, autonomous learners become confident communicators in that language (again within the limits of their ability); and when the foreign language is the channel through which their autonomy is developed, it effects a genuine expansion of their identity. (p. 43) It is the "limits of their ability" that begs attention here, for these limitations must be found in the pre-dispositions of an individual to greater degrees of autonomous activity.

While learning and especially language learning is a social construct, it is also a psychological process that can only be fostered if the terrain is sufficiently fertile. In this article, I suggest that attachment theory could offer an explanation for differing degrees of adult autonomy and differing levels of difficulty in adapting to autonomous learning situations. In other words, this chapter attempts to investigate the issue of whether secure attachment may provide a pre-disposition for autonomy.

Attachment theory

John Bowlby (1907Bowlby ( -1990)), known as the 'father' of attachment theory, was a psychiatrist and psychoanalyst. In the wake of World War Two, he worked with young children, often in situations of affective deprivation such as orphanages and hospitals. He developed attachment theory to explain certain children's reactions to separation from their parents. The theory later developed to a full theory of child maturation from a state of total dependence to one of physical, psychological and affective autonomy. His theory integrates various aspects of biology, socio-cultural reproduction and psychoanalysis regarding early childhood experiences [START_REF] Fleming | A secure base for adult learning: Attachment theory and adult education[END_REF]). Bowlby's basic premise is that a child needs early secure attachment to a primary caregiver (often, but not necessarily, the mother) in order to establish her/himself psychologically and grow to thrive in society. Attachment has been defined as "an enduring tie with a person who provides security" (ibid., p. 35). This strong link induces "various behaviours that the care seeker uses to remain in close proximity to the caregiver. [These] are known as attachment behaviour" (Riley, 2011, p. 12). Secure attachment enables the young child to progressively explore her/his environment and create a place for her/himself in the world. [START_REF] Riley | Attachment theory and the teacher-student relationship: A practical guide for teachers, teacher educators and school leaders[END_REF] and [START_REF] Fleming | A secure base for adult learning: Attachment theory and adult education[END_REF], among others, summarise the historical origins of the principal attachment styles that have been identified in the literature. Bowlby's original theory was tested and completed by the work of Mary Ainsworth , who developed a clinical procedure known as the strange situation, which allowed her to distinguish specific behaviours children demonstrated in the absence of their primary caregiver. She thus identified two primary types of insecure attachment, namely, avoidant and anxious/ambivalent attachment. Loss or the fear of losing the primary attachment figure, separation anxiety and angry responses to this anxiety, along with despair, mourning and various defensive mechanisms (such as denying need for the other) are all fundamental concepts in the complex sociobiological system of attachment.

As different psychologists worked with the attachment framework, some (notably Hazan & Shaver, 1987, 1990, 1994, cited in Riley, 2011) hypothesised that new attachment processes came into play in various adult situations, especially in romantic relationships, thus developing the notion of adult attachment. This work led to a redefinition of the attachment model by [START_REF] Bartholomew | Attachment styles among young adults: A test of a four-category model[END_REF], who proposed a quadrant model of adult attachment, with four different attachment styles characterised by the interaction between the avoidance and anxiety factors (see Figure 1).

The four types of attachment thus identified qualify the relations an individual has with others and the ways in which s/he is able to interact in society. Each individual interacts in accordance with their own inner working model, a set of implicit rules, beliefs and expectations about others and relationships. In briefly describing the main features of each attachment style, I shall give indications of the inner working models this generates. Secure attachment, often referred to in the literature as the secure base [START_REF] Fleming | A secure base for adult learning: Attachment theory and adult education[END_REF][START_REF] Riley | Attachment theory and the teacher-student relationship: A practical guide for teachers, teacher educators and school leaders[END_REF]Rholes & Simpson, 2006), is that of the child who knows s/he can rely on her caregiver's presence and s/he can quickly and easily find her/his caregiver in case of danger or need. The caregiver provides comfort and security for the child and comes to their aid whenever called for, for example, when the infant gesticulates, cries or shows signs of need. As the child realises that s/he can rely on this presence, s/he progressively expresses less need for it and also learns to displace her/his attachment needs to other people and objects. Children who have consistently been in secure attachment relationships grow into adults who "have a healthy and balanced view of self and others.

They are happy to be interdependent" (Riley, 2011, p. 26). Secure attachment is situated in the lower left quadrant of Figure 1.

Children whose attachment needs have been frustrated for some reason (parental absence, rejection, abuse or unpredictability), develop anxiety about their capacity to have these needs met. In reaction, children may either seek closeness at all costs, or develop strategies to avoid the other, in the hopes of avoiding emotional hurt. These reactions can take a variety of forms: excessively clingy or submissive behaviour, cold or distant contact with the caregiver even when able to be close, anger and despair, hyper vigilance and role reversal with the child trying to comfort the caregiver [START_REF] Riley | Attachment theory and the teacher-student relationship: A practical guide for teachers, teacher educators and school leaders[END_REF].

Once they become adults, insecure attachment can create preoccupied, dismissing or fearful adults. Preoccupied adults (lower right quadrant) are very anxious about close relationships but, nonetheless, feel attracted to others and need their opinion in order to feel worthy. Their acceptance of themselves depends on the opinion of others. They "become preoccupied with gaining the acceptance of others to feel good about themselves" (Riley, 2011, p. 26).

Dismissing adults (upper left quadrant) have low anxiety but show high avoidance of relationships, tending to be almost obsessively self-reliant. Finally, adults with fearful attachment (upper right quadrant) feel both unworthy of love and have a negative perception of others and their opinions of them. They tend to avoid closeness altogether, as a means of preventing rejection.

In this chapter, I have turned to attachment theory because it would appear to offer some explanations for the greater or lesser successes of different students in various language learning situations, especially those which rely to a greater extent on learner autonomy (notably, but not exclusively, in open access and out-of-class situations). More specifically, attachment theory may provide an explanation for some type of "pre-disposition" to learner autonomy, for that element which constitutes "the limits of [learners'] ability" (cf. section 1.1 above) for autonomy.

Adult attachment theory in language learner blogs

This preliminary study seeks to provide evidence of the different attachment styles (or postures) within the specific context of language learning in order to discern whether it could be used to help us understand learners' development towards autonomy. Four subsidiary research questions have guided my enquiry and the analyses that have been carried out:

 can evidence of attachment styles be found in language learner logs;  how do individuals attain autonomy with regard to language learning;  can the limits of learners' capacity for autonomy be stretched;  and does attachment theory offer an explanation for greater or lesser successes in language learning in autonomous contexts?

In order to do this, I have explored the language learning blogs of university students.

Methodology

This section describes the context in which the research was carried out, the means of data collection and the methods used for analysis. The methodology rested on a systematic qualitative exploitation of blog entries in a learner corpus gathered over one semester. This chapter, thus, adopts an interpretative research stance to identify a link between adult attachment and learner autonomy.

Description of course, students and blog requirements

In 2012, second-year Masters students in applied linguistics at the University of Strasbourg attended an English course, with the objectives of learning to read research articles, produce an abstract in English and with the communicative objectives of speaking and listening to English regularly (on any subject and in the forms most appealing to them). As the course instructor and in order to maintain a focus on language-learning processes and enhance our research into informal learning, I asked students to read some articles pertaining to informal learning and write a blog for the duration of the 12-week course. The blog was to be written in English and students were informed that their writing might be used in research related to informal learning. Students who did not wish to write a blog, perhaps because they didn't like the online format, or want exposure to peer readership, were invited to keep a paper journal. Only one student opted for this possibility. In order to facilitate entry into blogging, suggestions of writing topics, often related to their own informal language (not necessarily English) learning, were offered during the first few weeks. For example, the first week they were asked to trace their language biography, including languages learned and how they were learned. In the second week I suggested they could discuss their own definitions of informal language learning or how they felt about different ways of learning languages. These types of suggestions were progressively eliminated.

After the first three weeks, students were also encouraged to read each other's blogs and discuss them in pairs in class. No reference whatsoever was made to the attachment model in the course nor was it part of their curriculum in any other course.

Blog exploitation

Fifteen students (13 female and 2 male), with ages ranging from 23 to 40 (average 27) participated in the course. Fourteen of them kept blogs, producing some 49,000 words. These were read chronologically and annotated in view of excerpting any information pertinent to the attachment model presented above. This exploratory reading sought references to relationships related to languages and language learning that might point to attachment constructs. Very few such references were found in the corpus and in the end, a single case was chosen, as it provided significant examples of the attachment styles sought and allowed in-depth insights into the learning process and the relationship issues it encompassed. Specific written permission to analyse and publish results concerning the blog chosen for the case study was obtained from the blog's author. These methodological choices necessarily imply drawbacks, which will be addressed in the conclusion. However, I would like to highlight that this paper does not provide a psychological analysis of the person, nor of her overall attachment profile. It is a situated analysis, which seeks only to establish the pertinence of attachment parameters as related to autonomy in language learning. We shall thus follow Pauline, as she shares with us her processes, issues and feelings about learning German.

Pauline learns German

Pauline is 24 years old when she begins her blog for this course. She is a native French speaker, who has studied German, English, Norwegian, Spanish, Hungarian and Romanian. She has attained a proficient (C1) level in both written and spoken English, while her self-attested skills in the other languages are rudimentary. Her contact with German begins early:

As I was born in Alsace, I had to learn German at a very young age. When I was little, German language seemed fine to me, even if I had to learn it because people at school told me so. Further in the same blog entry, Pauline continues to describe her language learning, and the manner in which her relationship to the language evolves:

Unfortunately, and I'm sorry to say it, most of my German teachers in middle and high school were boring and quite useless: talking only in French, not having an explicit purpose = learning by heart lists of verbs or words of vocabulary without reactivating anything learned...). History, stereotypes and the fact that this language sounds ugly to me prevented me to achieve a decent level of German. (September 17, 2012) In this excerpt, Pauline sees her learning of German as very teacher-centred. She takes no real responsibility for her learning (or lack thereof) and would not be considered as displaying characteristics of learner autonomy in this undertaking. In attachment terms, we witness here Pauline's position in the dismissive quadrant of Bartholomew and Horowitz's model. While her self-image seems positive (further along in this particular post she recounts successful experiences in learning several other languages), she has a negative view of the other (the German teacher) and thus dismisses German, protecting herself from hurt (scholastic failure) but also preventing her from accomplishing something that she nonetheless considers to have a certain importance:

And of course, now I regret it! Plus, because I kept this negative idea about the German language, I feel discouraged to learn it again, even though I know it could be useful for the future. (September 17, 2012)

Pauline did not, however, lose her interest in foreign languages in general and, as required in France at age 11, began studying her second foreign language. She states:

I started learning English when I was 11, and I loved it at the very first lesson. I guess there is a strong contradiction between German and English and their cultures. It seems to me that English culture is more attractive than the Germans, probably because of the cinema and the music industry. Anyways, English teachers seemed "cooler". And actually, they were! I never developed any problems in English (unlike in German). (September 17, 2012) Her emotional switch to English is sparked by 'cool' teachers and fun lessons, creating the positive model of the other, which, combined with her ongoing positive perception of self, situate her within the secure space where learning can take place (lower left quadrant of Bartholomew and Horowitz's model). Nevertheless, we note an approach that is still very much dependent on the teacher and which demonstrates very little learner autonomy.

Almost a month after these initial entries, Pauline reveals a new facet of her relationship with German:

A few days ago, […] I became aware that I am in contact with German all the time. Unsurprisingly, I hear German everyday, not only because we live very close to the German border, but also because I listen to German radios everyday, many times a day. As I mentioned before in my blog, I have a tricky relationship to this language. But I guess that I am now ready to change my view about German, and open myself to learn (at least) chunks of words that I hear on the radio, and when I go to Germany on the week-ends to see my boyfriend who is living there. (October 13, 2012) This renewed interest in German, based on proximity to the country and a strong affective link (her boyfriend who lives there), incites her to modify her views of the other and become more open to the language. We might be tempted to interpret this as a return to a secure base, where she remains confident in herself (low anxiety) and is no longer avoiding the language. The 'secure base' refers of course to the attachment construct (see Section 1.2 above) and Pauline's new-found confidence, not to her proficiency in German. Her new approach to the language also demonstrates a high level of self-direction, as she has not only decided herself to increase her contact with the German language but has also chosen the various means by which she ensures such contact.

However, a few days later, she gives an update, in which we witness both discouragement and persistence:

Concerning German, I am trying. It gives me a really hard time to hear, listen and understand the language.

[…] I started this week to listen to the […] news. I feel discouraged, because I realised that I could understand maybe one or two words of each sentence, and I have the impression that they are talking really fast. I know this kind of acquisition is a long-term process, so I am not giving up! (October 19, 2012) Pauline continues to take an autonomous approach to learning German, both in the fact that she has decided to work essentially on her own and in her determination to continue. In her following post, Pauline documents her passage through various stages and emotional states concerning German:

I am trying so much to learn German, I am just getting crazy about it.

[…]

But the more I listen to that language, the more I realize how bad I am at it! It is depressing.

[…] I am focusing on each word that I can understand. When hearing a word that I understand, I take it as a great victory, and it helps me getting more motivated.

[…] (October 26, 2012).

Despite her efforts and progress that she can sometimes notice, she makes harsh value judgments on herself ("how bad I am at it"), so that, while her perception of the 'other' has become more positive (note, further down in the extract from October 26 below, her reference to the "beauuuutifuuuul sound" of the language), indicating a move towards lower avoidance, her anxiety has increased, situating her now in the 'preoccupied' zone of Bartholomew and Horowitz's model, where the individual becomes "preoccupied with gaining acceptance of others to feel good about themselves" (Riley, 2011, p. 26).

Although she does qualify her judgment,

[w]hen learning a language, even if it is informal or unconscious, I guess you have to get through both failures and successes in order to go on making efforts (October 26, 2012), this perception of herself as "bad at German" leads her to some very frustrating experiences, as the following post describes:

I spent my Halloween vacation in Germany. I consider this trip as a great opportunity to get even more used to its beauuuutifuuuul sound! But I got really frustrated [italics added throughout this citation] when I was on the train. First, I was sitting next to someone talking in German with the train inspector, reading a French newspaper, and then taking part into a conversation with two Spanish women. I felt miserable.

Then I was on another train and two people started talking in German about something I am sure was really interesting, but I just prevent myself from focusing on what they were saying because I was afraid they could actually approach and talk to me. I know that I can not have a real conversation with a German speaker, because I am not able to talk or understand enough in that language. And it is very frustrating because I love talking, especially with nice strangers when traveling. (October 26, 2012) This excerpt exemplifies the learner's increasing anxiety, which in turn seems to lead her more and more toward avoidance of contact with "real German speakers", indicating a move into the fearful zone of insecure attachment. Being alone in this situation may actually be inhibiting her capacity to seize the learning opportunity, which she in fact created here. An autonomous impetus (getting on the train and going to Germany) encounters the insecurity of the situation (the feeling of not being capable of understanding) head-on.

Pauline's real 'crossing of the Rubicon' arrives during her stay in Germany.

I feel I made a great step in my German language learning

[italics added throughout this citation], because I tried (at least) to talk in German with my boyfriend.

[…] What is interesting is to realize […] that I have actually a certain amount of words that I already know, and that I probably remember from my past years of learning German. I felt good about it.

Moreover, […] I considered these interactions as a game.

[…] I went on that game for several days, and even after coming back to France, talking in German with my mother.

[…]

In any case, it was a very positive experience. First because I realized that I was not so bad, and then because I became aware that I'm surrounded with people that can help me…! (November 8, 2012)

Pauline is finally positive about her German learning and realises that she can rely on dependable people (her mother and her boyfriend) to help her. In attachment terms, this is her secure base, the place where she feels good about both herself and others. From this base, learning German becomes a game, where Pauline sets the agenda, demonstrating significant learner autonomy.

Pauline concludes her blog on November 29, stating that … in German, informal activities helped getting used (or reused) to sounds and sentences.

Moreover, reflecting on […] informal learning, helped me overcome my language anxiety […] in German. I was in fact afraid of learning that language because of past failures, and uneasiness in interacting in that language.

Pauline confirms here what I had interpreted as the fearful phase and indicates that reflection has itself been an aid in moving forward. Reflection, or reflective practice, is referred to by [START_REF] Candas | De la nécessaire distance dans l'apprentissage d'une langue étrangère[END_REF] as the necessary distance (nécessaire distance) in foreign language learning and by [START_REF] Little | Learner autonomy: Definitions, issues and problems[END_REF] as "detachment" (see Section 1.1 above), at the very core of autonomy.

The necessary post-scriptum to this story is Pauline's comment when I contacted her to request permission to use her blog data for this study. She indicated that she was now teaching French in Schaffhouse, in a German-speaking region of Switzerland, where German is the language she uses in her daily life.

Discussion

The study of Pauline's case, seen as relationships in light of attachment theory, can be analysed as revealing three types of relationship and their corresponding influences on the learner's autonomy and on her language practice. The first are those of early childhood with primary care-givers which forge our attachment styles and consequently our potential for (learner) autonomy. The second are the human relationships which influence the learners' feelings about language: the people one comes into contact with who come to influence, represent or somehow embody the language for us. The third type is the learners' relationship to a (foreign) language itself. In this discussion, I will explore the potential and the consequences of these different positions as they concern autonomy and language learning in the data provided by Pauline's case.

If we regard attachment (and therefore autonomy) as being established during the early years of life, this would be consistent with most attachment literature, which has traditionally regarded attachment styles as "trait-like properties of people" (Fraley & Brumbaugh, 2006, p. 121). While our data might suggest that this type of approach could be possible, this is not what was undertaken here and such a position would have to be tested for pertinence with other tools, in varying contexts and with a wider diversity of subjects than has been the case here. One option might be to use established attachment diagnostic tools, such as the AAI, to test for the different types of attachment profiles present in or absent from autonomous learning centres when compared with traditional highly teacher-directed classrooms. This type of research could potentially allow claims to be made about basic attachment styles favouring or discouraging individuals in their attitudes toward autonomous language learning processes or autonomy-related structures (such as open access). We might expect to confirm, for example, that fearful attachment would inhibit the learning process and predict that very few, if any, fearfully attached individuals would be found in higher education learning centres. This type of approach stems from a rather determinist view of relationship, attachment and autonomy, but might be a worthwhile perspective for some researchers.

A very different position, consistent with most current research in applied linguistics, would adopt a situated view of psychological constructs (cf. Dörnyei & Ushioda, 2009 or [START_REF] Norton | Identity and language learning: Gender, ethnicity and educational change[END_REF]Norton, , 2014 concerning identities research; concerning identities research;Mercer & Williams, 2014 concerning aspects of the self), seeing them as variables that can be activated by various internal and environmental influences. From this position, we can examine the second type of relationship issues suggested by attachment theory: human relationships which influence the learners' feelings about language. Our data highlights Pauline's dislike of her German teachers at school, her anxiety about strangers' opinions of her on the train, her strong positive feelings toward her boyfriend and her mother in her latest learning experiences. The relations she has with people may in fact be having a profound effect on her learning, as when she is experiencing fearful attachment on the train and prevents herself from interacting with people who could help her to make progress in the language or when she is experiencing secure attachment and enters into playful experimenting and 'risk-taking' with her mother or boyfriend. In further exploration of attachment theory as applied to language learner autonomy, relationships with people, especially teachers, would appear to be a fundamental avenue to , as it is certainly here that teachers can exercise the most influence, being themselves one of the poles of the relationship. It is also probably the most delicate, from a teacher's point of view, as it involves questioning one's own attachment postures as a teacher and interrogating the extent to which we may (or may not) be using our relationship with students to satisfy our own attachment needs. [START_REF] Riley | Attachment theory and the teacher-student relationship: A practical guide for teachers, teacher educators and school leaders[END_REF] explores this with elementary school teachers, outside of the specific area of language learning.

Finally, it is possible to view language learning itself not as knowledge of an object but as development of a relationship where the 'other' is the language and cultures being learned. As such, language learning can be influenced by the same psychological processes (inner working models) as relationships with human beings. These would include attachment processes and therefore the development of autonomy. The language itself (or the learning of it) are regarded as a sort of personality, capable of inspiring admiration, anxiety, frustration, fear, satisfaction, pleasure and so on. This study can thus be seen as an attempt to identify psychological positioning with regard to a specific language and perhaps to different learning situations, contexts or cultures when they are viewed as relationships.

Pauline herself seems to see German as an 'other' when she admits having a "tricky relationship to this language" (October 13, 2012). In the data presented here, she declares having "a negative idea about the German language" (September 17, 2012), going as far as to say that German "sounds ugly" (September 27, 2012) and yet, scarcely a month later (October 26, 2012), German is found to have a "beauuuutifuuuul sound". This relationship with the language itself develops in situations where it is sometimes disgraced (for example in middle school, where Pauline shuns it as "uncool") and others where it is favoured and thus facilitates the learner's contact with and integration of the language (Pauline's extensive listening to German at home, or 'playing' in the language with her boyfriend or mother). These latter are examples of both autonomous learner behaviour and of activities which contribute to language learner success in the long term, if in no other way than by supplying large quantities of input [START_REF] Hilton | Mise au point terminologique: Pour en finir avec la dichotomie acquisition / apprentissage en didactique des langues[END_REF]).

Pauline's experiences with German demonstrate interesting developmental moves through all of Bartholomew and Horowitz's four attachment categories (cf. Figure 1) and open the door to using attachment theory as a means of pushing the limits of learners' capacity for autonomy. Autonomy from an attachment perspective is thus not taken for granted for each learner, but rather seen as a factor that can be influenced, modified and developed across time and place. This perspective frees learners from determinist positions on language learning aptitude (I am an inadequate learner), and from immutable beliefs about the other (the language I'm learning is somehow inaccessible or those who speak it somehow unattractive).

The implications for practice could, in the case of preoccupied attachment, for example, involve working on the learner's self-image, helping them to focus on positive perceptions of themselves in the L2, in order to encourage moves toward more secure attachment. For someone in a dismissing phase, the focus would need to turn more to the language itself, allowing the student to discover how it (or those who speak it) can be perceived in a positive light, congruent with their own beliefs and values. These implications would appear particularly pertinent in circumstances where the learner encounters impediments to their continued autonomous development in the language. Such (re)mediation, could lead to sufficiently secure attachment for autonomy to develop and more effective learning to take place.

Whether we view the significant relationships of language learning as being those of early childhood which forged our basic attachment styles, those with the people who influence or embody the language for us, or those with the (foreign) language itself, attachment theory supplies a new conceptual framework with which to understand language learner autonomy and envisage its development with a view to allowing fully agentive foreign language use to take place.

A potential bias of this study is that the learner data comes from a practised, proficient and even expert language learner. Pauline's descriptions and analyses of her learning activities and experiences are those of someone who has learnt several languages and who, moreover, has studied language learning and acquisition with the aim of becoming a language teacher. In spite, or perhaps because of this, her reported feelings and experience of phenomena involved in learning a language independently help proffer insights into the pertinence of attachment theory with regard to language learning autonomy.

Conclusions

The evidence of attachment styles found in Pauline's language learner blog encourages optimism regarding further research that could confirm the basic attachment mechanisms by which individuals develop autonomy. In turn, such frameworks could provide the keys that would allow researchers to understand how teachers and learners can best foster such autonomy over time.

Further research is needed to examine the data provided by other learners in the corpus presented above (or in similar corpuses) for evidence of the presence of attachment styles in the language learning context. Studying other cases would allow confirmation that students' inner working models have a determining effect on the relationships they establish with different languages, with various actors in the language learning process and also on the degree to which they function as autonomous learners.

Hypotheses related to how different attachment styles would favour or discourage autonomous learning could be examined, for example, by establishing attachment measures, using the AAI or discourse analysis [START_REF] Crittenden | Assessing adult attachment: A dynamic-maturational approach to discourse analysis[END_REF] with students and by observing their adaptation to autonomous learning contexts based on the results obtained. The methodology could also be inverted: assumptions about attachment obtained through observation or diary/blog data could be verified against attachment style data obtained later. As attachment theory provides an explanation for insecure attachment and relates it to a lack of healthy autonomy (as opposed to isolation, for example), another approach might be working with people who have difficulties with autonomous learning situations, and autonomous language learning in particular. Again, it would be necessary to establish attachment measures and seek correlations with differing degrees of learner success over time. Such studies would give insight into how individuals adapt to learning contexts (such as self-access) where a large degree of autonomy is a prerequisite.

This preliminary study has provided insights into attachment theory as a potential new resource for studying second language acquisition (SLA), especially as linked to the autonomy construct. The findings, while modest, indicate that attachment styles can be detected in language learning contexts and seem to provide a useful framework for studying the relationships that learners establish with the language they are learning and with the people who accompany them during that process. The discussion points to areas that could be usefully explored further, notably, the means by which teachers could foster appropriate attachment relations with and between students, teaching and support staff and the language itself in order to facilitate long-term autonomous or selfdirected language learning. As such, attachment theory could provide exciting new directions for SLA research in the future, especially in respect to developmental perspectives on learner autonomy.

Fig. 1 :

 1 Fig. 1: Bartholomew and Horowitz's (adapted) model of adult attachment (1991)