BSc Johanne Paradis

PhD Jacques-Bernard Gauthier

THEORETICAL FOUNDATIONS OF HOSPITAL PHARMACY MANAGEMENT

Keywords: hospital pharmacy management, theories of organizations, structuring theory

The lack of interest of researchers in relation to question of hospital pharmacy management, and the status quo of existing managerial practices serving the reforms, justifies analysis of theoretical foundations of hospital pharmacy management.

The objective is twofold. First, provide an overview of the socio-historical eras of the organizational theories in order to position the hospital pharmacy management on the axis of the changing ways of thinking about the organization and management. Secondly, reconsider hospital pharmacy management by the structuration theory (new direction in organization theory). A systematic literature review of the Canadian Journal of Hospital Pharmacy, from 2010 to this day, was performed. The analysis of the theoretical notions discussed in the articles shows that the management of the hospital pharmacy departments stands on the concepts associated with the 'Modern era' of organizational theories. As a result, the hospital pharmacy management focuses on technical aspects. The reconsideration by structuration theory point out the strategic issues for hospital pharmacy practice underlie the interaction between the actors of the pharmacy department and the other stakeholders of the health system.

A proper management of the hospital pharmacy department goes beyond the technical aspects. It should also target the development of strategic professional skills of the pharmacists. That said, a substantial work remains to be done to understand the ins and outs of the structuration of the pharmacy practice and the management of its strategic-professional aspects.

The Context of Hospital Pharmacy Management

The public healthcare system in Canada experienced a succession of reforms, which one after the other, promise to streamline and update the healthcare system to allow greater coordination of actions, greater quality of care, greater efficiencies. For the pharmacy, with the years 2000 came the agreements between the provinces and the Federal Government regarding health information and communications technology and the pharmaceuticals management. In 2004, proposed changes to healthcare structures were introduced and intended to implement a nation-wide pharmaceutical strategy.

These reforms and large-scale restructuring of healthcare institutions presumed an organisational management model, hierarchical and mechanistic, in which a clear distinction exists between the managers and workers [START_REF] Dupuis | Vers un appauvrissement managérialiste des organisations de services humains complexes?[END_REF][START_REF] Mintzberg | Le management Voyage au centre des organisations[END_REF] . This type of model is very different from a model that would allow for the development and deployment of expert knowledge and expert judgment, i.e. specific skills for multiple or different trades in complex human service organizations [START_REF] Dupuis | Vers un appauvrissement managérialiste des organisations de services humains complexes?[END_REF] . The practice of pharmacy is a professional domain using expert knowledge and for which management is performed as a component of a complex organization [START_REF] Traulsen | Theories of professions and the pharmacist[END_REF] .

The environment in which hospital pharmacy management takes place is interesting to study because it must take into account two realities of the department. The pharmacy has two vocations, the distribution or technical service, and the care or professional service. There is, therefore, a differentiation between the daily operational management and the strategic management, the latter happening outside the pharmacy premises, between various stakeholders, such as the other clinical departments, the hospital administrative services, the professionals and users of healthcare services from the community.

That being said, management in organizational settings is not univocal. It is anchored to one or the other organization theories. Hence, before going further in our analysis, we find necessary to look upstream to the pharmacy management prior to looking at the performance or achievement of the service. To do so, we propose in the next section, an overview of four socio-historical eras to which the theories of organizations fashion itself. At the end of this primary exercise, a conceptual framework will be proposed.

The Evolution of the Major Organizational theories

It is at the beginning of the 20th century that the theories of the organization get started, with the emergence of major industrial enterprises. Organizational theories are based on two schools of thought: a sociological school with, authors like Durkheim, Weber and Marx, and a managerial school of thought with Taylor and Fayol 4 . There are as many ways to present the evolution of these theories as there are authors. For the purposes of this article, we choose to adopt the perspective proposed by Hatch and Cunliffe 4 because it appears to set a precedent in the field of healthcare in general and pharmacy in particular [START_REF] Giam | Specialized compounding in community pharmacies : Organizational perspective[END_REF][START_REF] Fatkin | Using organization theory to explore the changing role of medical libraries[END_REF][START_REF] Zhou | Organizational Culture and Its Relationship with Hospital Performance in Public Hospitals in China[END_REF] .

Both Hatch and Cunliffe 5 anchor organizational theories in four historical eras: Prehistory, Modern era, the Symbolic era and then the Postmodern era. In Figure 1, we present chronologically the four periods and organizational theories that are associated with them. The beginning of an era does not mean the abandonment of the theories of the previous eras. That's why, for example, organizational theories of the Prehistory extend into the present era (this extension is illustrated by discontinuous rectangles in Figure 1).

The starting point of the analysis of the evolution of the major organizational theories is the industrial development and with it, the need for order, discipline and work rationalization. It's the Prehistoric era (1900-1950) in organization theory. The concepts of the division of labor, the specialization of tasks and prioritization were established during this period. The performance and productivity were the main goal, as work efficiency and optimal use of resources. It is the introduction of the 'scientific' way of thinking in organizations, the beginning of the assembly line production (Ford), the 'scientific' management (Taylor) and the administrative management theory (Fayol). There is a radical separation between leaders and employees who execute the work. Reference is often made to these concepts as Fordism and Taylorism. Towards the end of Prehistory will come the Weberian bureaucracy, which is mainly based on rules and respect for the hierarchy [START_REF] Hatch | Organization Theory -Modern, Symbolic, and Postmodern Perspectives[END_REF][START_REF] Séguin | L'analyse des organisations -une anthologie sociologique Tome I -Les théories de l'organisation[END_REF][START_REF] Rouleau | Théories des organisations : approches classiques, contemporaines et de l'avant-garde[END_REF] .

The scientific and administrative management, as well as the Weberian bureaucracy will extend and consolidate during the Modern era (1960-1970)see Figure 1. It is the time of the classical schools' organization theory, and the famous "One Best Way" of doing things.

Figure 1 -Organization theories' socio-historical evolution

Good practice guides and guidelines are coming out, aiming at the standardization of practices, while assuming they will provide the expected results. The modern management principles in vogue are: the division of work, authority, discipline, unity of command or management, the subordination of the individual interest in the general interest, centralization, hierarchy, etc. Harmful and dehumanizing effects from this 'rationalization' of the modes of production are starting to be observed, and suddenly, the interest is shifting to human behavior, to the analysis of bureaucracy and systems analysis. While England is considered to be the birthplace of the contingency theories (Organizational adaptation to environmental change), in America, we are witnessing the emergence of schools of human relationships with authors like Lewin and Likert (leadership, personality), Maslow (hierarchy of needs, satisfaction) amongst others [START_REF] Hatch | Organization Theory -Modern, Symbolic, and Postmodern Perspectives[END_REF][START_REF] Séguin | L'analyse des organisations -une anthologie sociologique Tome I -Les théories de l'organisation[END_REF][START_REF] Rouleau | Théories des organisations : approches classiques, contemporaines et de l'avant-garde[END_REF] .

The third era is referred to as Symbolic era (1980-). It is a period that calls into questioning the theories that prevailed until then. People reject the managerial, anthropomorphic, deterministic ideology of the organization. There are two emerging schools, the social action analysis, for which work is oriented in a microscopic analysis of the interactions while revalorizing the role of actors, and the Marxist analysis that redefined the organization as the result of the contradictions inherent in the capitalist production system. In the second half of the Symbolic era, we see the emergence of the so-called 'contemporary management approaches'. The theories are organized around disciplines from which they developed: sociology, economics, politics, etc. It is also the moment where European writers begin to be heard, such as the Scandinavian school with the work of Alvesson [START_REF] Hatch | Organization Theory -Modern, Symbolic, and Postmodern Perspectives[END_REF][START_REF] Séguin | L'analyse des organisations -une anthologie sociologique Tome I -Les théories de l'organisation[END_REF][START_REF] Rouleau | Théories des organisations : approches classiques, contemporaines et de l'avant-garde[END_REF] .

Finally, Hatch and Cunliffe 5 complete their analyses of the evolution of the organization theory by the Postmodern era (1990-). Here the organization is understood as the dynamics of power/domination and resistance or as the text and speech. It is only seen in its social, political, and linguistic dimensions [START_REF] Gauthier | The practice of pharmacy as a social system: Giddens' sociological structuration theory[END_REF] . In the last years of the Postmodern era, important place is given to the individual, considered to be an actor capable of self-criticism, reflexivity and analysis of organizational contexts and practices that are conducted or to be conducted [START_REF] Hatch | Organization Theory -Modern, Symbolic, and Postmodern Perspectives[END_REF][START_REF] Séguin | L'analyse des organisations -une anthologie sociologique Tome I -Les théories de l'organisation[END_REF][START_REF] Rouleau | Théories des organisations : approches classiques, contemporaines et de l'avant-garde[END_REF] .

In summary, the synthesis of these four socio-historical eras illustrates the different ways of thinking about the organization. From the belief into a reality external to the actor, independent of knowledge, the organization in Modernity is conceived as an objective entity governed by universal laws. On the other hand, from the Symbolic perspective, the organization becomes a social space, a network who constantly rebuilds itself through its members and their interactions. Finally, the Postmodern perspective focuses essentially on sociopolitical games, languages and social interactions. In other words, the postmodern organization is a place of power relations, irrational, strained by inequalities, which are done and undone. Having said that, in recent years more attention is given to the individual skills to monitor their practices by which they give life to organizations, than on the power and domination relations in organization.

From these observations on the evolution of the ways of thinking about the organization, we can ask ourselves, "to which organizational theories, hospital pharmacy management is anchored?" Some authors have analyzed the pharmacy practices from different social theories [START_REF] Traulsen | Theories of professions and the pharmacist[END_REF][START_REF] Gauthier | The practice of pharmacy as a social system: Giddens' sociological structuration theory[END_REF] . To date, no author is interested in the socio-historical and theoretical perspectives underlying hospital pharmacy management practices. To explore the theoretical foundations of hospital pharmacy management, we analyze publications from the past five years in the Canadian Journal of Hospital Pharmacy (CJHP).

Literature Review on Hospital Pharmacy Management

Inspired by Ridley 11 , we have undertaken a systematic review of the publications over the past five years about the hospital pharmacy management. We limited the literature review to the CJHP. Since of all Canadian periodicals on hospital pharmacy, CJHP is the only one to be listed by the SCImago Journal & Country Rank. We excluded from this review, articles from the sections of the Editorial, Point Counterpoint, and Correspondence.

In the medical literature, the term 'management' generally refers 1-to taking charge of a pathology, a case, and 2-to the planning, organizing, coordination and control of activities and resources (people or things) 2 of a clinic or, in our case, a pharmacy department. This paper is only interested in articles referring to the second notion.

The next section presents the findings of literature review.

Presentation and Analysis of Findings

For the purposes of the analysis, the articles have been grouped into five categories: medication, standards, culture, technology, practice, and then declined following the major themes, theoretical notions and organizational theories to which these notions refer to. Table 1 presents the summary of results. The analysis of the themes outlined in the articles reveal that the hospital pharmacy management as reported in the CJHP are mainly based on notions who come from Modern organizational theories. The organization is described as a formal structure, designed and managed according to rational and effective standards. The authors are interested in following topics: norms (practice), accreditation, the performance of drug distribution services. Figure 2 shows the sociohistorical positioning of hospital pharmacy management in regards to the categories and the themes with the eras they relate to.

New directions in organization theory: A Reconsideration of Hospital Pharmacy Management by the Structuration Theory

In the introduction, we pointed out that the hospital pharmacy has a double vocation. The distribution of medication, the technical service (operational), and the professional care (strategic). In light of the definition given by Minztberg 2 , and at least for the past five years, we find that the articles in CJHP focus more on the technical (operational) aspects of management than on the strategic side of management.

These articles have an organizational conception known as 'post-war' (at the heart of the Modern era of the organization theories).

However, the strategic aspect should not be overlooked. It must be put forward as do other health professionals [START_REF] Norgaard | Arguments for theory-based pharmacy practice research[END_REF][START_REF] Breton | How healthcare organisations can act as institutional entrepreneurs in a context of change[END_REF][START_REF] Schweyer | Entre profession, organisation et marché: la cas des ingénieurs biomédicaux hospitaliers[END_REF] . The strategic challenges of the hospital pharmacy go beyond the walls of the pharmacy premises, and beyond the medication management process. Patient care, participation in interdisciplinary teams, and participation in organizational committees are situations where there are strategic challenges for the hospital pharmacy practice. If the management of strategic issues does not get a great deal of attention in hospital pharmacy, this is due, in part, to the fact that the "social" aspects of organizations are less understood.

We propose a reflection on both aspects (operational and strategic) of hospital pharmacy management, with an innovative approach that diverges from the current thinking. This approach mobilizes the structuration theory originating from the social sciences. Hatch and Cunliffe 5 consider the structuration theory as being among the 'new directions in organization theory'.

The Structuration Theory

To better understand the nature of our reconsideration of hospital pharmacy management, we must first return to the different definitions of the organization. Parsons defines the organization as a "social subsystem with specific goals'' 9 in which the organization is seen as objective structures.

Figure 2 -

Socio-historical positioning of the theoretical foundations of hospital pharmacy management

Others see the organization as collective action which, by the application of rules and under the leader's authority, seeks the implementation of a common task [START_REF] Mintzberg | Structure et dynamique des organisations[END_REF] .

Unlike Parsons, the organization is here thought and described in terms of actors' subjectivity or in terms of "socially constructed" structures.

The organization is defined by Eraly [START_REF] Eraly | La structuration de l'entreprise : La rationalité en action[END_REF] as a social order, sustainable and localized, constituted for a given purpose. He describes the organization also as a set of actions and interactions relatively hierarchical. This notion of organization assumes both a social order in space and continuity in time. For Eraly [START_REF] Eraly | La structuration de l'entreprise : La rationalité en action[END_REF] , the organizational structures do not exist independently of the human actions and interactions. If the actions and interactions stop, the organization no longer exists. So, organizational structures are generated and replenished through the daily activities lead the people in the organization [START_REF] Hatch | Organization Theory -Modern, Symbolic, and Postmodern Perspectives[END_REF] . To better understand the structuration of organizations, Eraly [START_REF] Eraly | La structuration de l'entreprise : La rationalité en action[END_REF] uses the image of a group of people holding at arm's length a set of blocks (organizational structures) that takes the form of a circle (see Figure 3). To account for the context of the pharmacy, we have replaced the blocks by giant capsules, but the idea remains the same. This image represents the organization and the inter-organizational network through relationships that are formed between the actors (the circle). The actors, the people who make up the organization, take turns constantly so as to keep this assembly. " Some go, others arrive who replace them, and beside this continuous swarming, the overall shape remains almost motionless at two meters above the ground " [START_REF] Eraly | La structuration de l'entreprise : La rationalité en action[END_REF] .

As noted earlier, the history of the organization theories is characterized by an opposition between Modernity, centered around the objectivity of the structures, and the Symbolic era (as well as some Postmodern perspectives) where emphasis is placed on the subjectivity of actors. In other words, the organizational reality has been defined only by structures, illustrated by the capsules in a circle at Figure 3, or only by actors without taking into account the structures*. For Eraly [START_REF] Eraly | La structuration de l'entreprise : La rationalité en action[END_REF] , this opposition is artificial, because the organizations and their structures do not exist independently of human action.

The organizational structure is both objective and symbolic (or subjective). Eraly borrowed the idea of the duality of existence of the organizational structures from the British sociologist, Anthony Giddens [START_REF] Giddens | The constitution of society[END_REF] .

Giddens' 66 structuration theory describes the structures as a set of resources (objective) and rules, standards, and values (symbolic or subjective). In the hospital pharmacy setting, we can think of drugs, automated dispensing cabinets as examples of resources. As for the rules, norms and values, there are, among other things, the professional codes, the codes of ethics, standards for drug preparations, etc. These sets of rules and resources, the 'structures', condition the practices of the actors (in this case the pharmacist) by directing their choices, their decisions or their tasks. For example, rules, financial restrictions, or lack of technological resources determine activities that can be accomplished. The structures may also make things possible to. For example, technical resources allow for more freedom for other activities. ______________ * Over the past five years, the CJHP presented hospital organization primarily as objective organizational structures.

Figure 3 -

Eraly's 65 adaptation of the structuration process

Condition the practice (in our case the hospital pharmacy practice) through structural measures (legislative change, add new resources or change it) is a modern perspective of managing practice in organization**.

According to the structuration theory, structures do not only condition the practices of the individuals in the organizations, they are also the result. In fact, it is the individuals who adopt rules for their organizations (the Pharmacy Department), and introduce the resources. For Giddens [START_REF] Giddens | The constitution of society[END_REF] , the actors, the pharmacist or the technical assistant, are competent individuals and their are continually expressed on a daily basis. The actor puts this competence forward when he is constantly monitoring his practice, continually adjusting his actions according to the situation or the context in which he finds himself. The actor is also using this competence when he mobilizes the structure (the rule or the resource), which, in his view, is most likely to give to his practice, the scope he wants regarding the situation. The actor is not necessarily conscious of the existence of this skill. It is something that the individual does naturally in the daily course of his actions; a kind of logic or practical sense.

Although he can have all the will, the knowledge and the practical competences needed, an individual cannot act alone in order to reach the organization's mission. As a result, the organization is a network of individuals with a practical sense who interact sometimes in synergy, sometimes against each other. And so, the pharmacy department, as it is structured ______________ **A significant portion of reflections about hospital pharmacy management concern the planning, the coordination, the implementation and the evaluation of structural measure.

today (its rules and its resources), is the result of power struggles and domination games occurring throughout its history. Authors in a series of articles about "sociology and pharmacy", have written about the social aspects of the pharmacy's practices like dominance, inequality, feminism, ethnicity, etc. [START_REF] Bissell | An introduction to fonctionalist sociology: Talcott Parson's concept of the "sick role" (2)[END_REF][START_REF] Bissell | Researching "race", ethnicity and health: a critical review[END_REF][START_REF] Bissell | How relevant is Marxist sociology for pharmacy practice research?[END_REF][START_REF] Bissell | The contribution of interactionist sociology to understanding the experience of health and illness[END_REF] .

The physical or structural space in which the social interactions take place, the pharmacy department, is not an island. It is part of a larger organization. Locally, the healthcare facility is formed by other clinical departments and administrative services. At the same time, the healthcare facility is an entity of a regional, national and even global network (Figure 4). The activities that are taking place at regional and national levels shape the structures of healthcare facilities, affecting practices in all departments and administrative services, including the pharmacy department (descending arrows in Figure 4). In return, the national and regional structures of health organizations and healthcare facilities are the results of the practices carried out in departments and services composing them (upward arrows in Figure 4). Giddens 66 called this circular process: the structuration between global social systems (national and regional health organizations) and the practices (in this case the pharmacy practice) performed daily by individuals in different localized organizational contexts.

All the importance of better hospital pharmacy management (strategic) resides this circular process. Indeed, the potential for influencing others professional practices is not evenly distributed between the actors (individuals) or groups of actors (the departments or services) in an organization. As noted above, in the course of the struggles and domination games, some individuals or professionals' groups have come to have greater power in the determination of the structures of an organization, in the establishment (or the change) of conditions for the exercise of their practices and those of other (such as the hospital pharmacy practices). The participation of hospital pharmacy actors to the determination of the conditions of their practices requires a proper hospital pharmacy management who goes beyond the technical aspects. It targets, for example, the development of strategic-professional skills of hospital pharmacists. With better skills, the pharmacist could interact strategically with other professionals in activities outside of the pharmacy department; professional interactions who have the potential to define the future conditions of the hospital pharmacy practices.

Conclusion

The review done of the CJPH highlights the fact that the analysis of the hospital pharmacy is essentially situated in the Modern era. The organization is described as ''machine'', and consequently, the attention is focused on key performance indicators, prioritizing efficient work and the standardization of process -it takes a well-oiled machine! This is all done in relation with the principles of managing for results, for accountability, and performance.

That said, this vision where everything is thought of in a descending order (the structure determines the practice performance) leaves out the ascending mechanisms. Once again, the organization and its structure do not exist by themselves, as suggested by the modern theories of the organization. They are the results of activities carried out by individuals.

Among the new directions in organization there is the structuration theory. It goes without saying that we gave an overview of the structuration theory when we present this one as a conceptual way to analyze the circular relation between national, regional and local structures and daily practices. There is much more to say, but the key here is to demonstrate the relevance of this theory to the field of healthcare in general, and hospital pharmacy practice in particular. In fact, some authors have used the structuration theory in their research in medicine [START_REF] Barley | Technology as an occasion for structuring: Evidence from observations of CT scanners and the socisl order of radiology departments[END_REF] or in nursing [START_REF] Groves | Keeping patients safe in healthcare organizations: a structuration theory of safety culture[END_REF][START_REF] Hardcastle | An overview of structuration theory and its usefilness for nursing research[END_REF] . The structuration of the hospital pharmacy practice has received little attention. To our knowledge, two articles dealt with the structuration theory in the field of pharmacy. The first one is the study of the impact of the implementation of a new technology on three groups of employees in a pharmacy [START_REF] Barrett | Reconfiguring Boundary Relations: Robotic Innovations in Pharmacy Work[END_REF] . The second one is an exploratory research about the technical and strategic spaces in a pharmacy department [START_REF] Hério | Ètude de la structuration de la pharmacie hospitalière comme condition au management d'un projet organisationnel[END_REF][START_REF] Hério | La pharmacie d'établissement comme objet des sciences de l'organisation : Espace technique, espace stratégique et structuration[END_REF] .

By using the structuration theory, we note that the actors in the health system are involved in determining the conditions of their practices and those of other health professionals. This determination is the result of the interactions with other actors in strategic contexts (caring for patient, participation to a regional committee, etc.).

Actually, the hospital pharmacy management focus on the technical aspects. The structuration of the hospital pharmacy practice deserves more attention, because it is at the heart of the determination of the conditions of the practice of the hospital pharmacy where stands all strategic issues for the profession. However, substantial work remains to be done to understand the ins and outs of the structuration of the pharmacy practice and the management of its strategic aspects.

Figure 4

 4 Figure 4 -Hospital pharmacy structuration processes

Table 1 Results Summary

 1

	Articles	Themes	Theoretical notions	Organizational theories


Paul Gauthier

Vassili Nestoridis

Athanase Papadopoulos

Spherical arc

Keywords: AMS Mathematics Subject Classification: 30B40, 32B15 Analytic curve, regular curve, global parameter, conformal parameter, arc-length, analytic extension, maximal extension, spherical arclength, hyperbolic arc-length

come

Introduction

The subject of this paper is the interaction between real analytic curves and complex geometry. The results are in the direction of those obtained in the papers [START_REF] Nestoridis | arc-length as a conformal parameter for locally analytic curves[END_REF][START_REF] Nestoridis | arc-length as a global conformal parameter for analytic curves[END_REF].

In the papers [START_REF] Bolkas | Non extendability from any side of the domain of definition as a generic property of smooth or simply continuous functions on an analytic curve[END_REF] and [START_REF] Bolkas | One sided extendability and p-continuous analytic capacities[END_REF], the authors obtained results on the parametrization of an analytic Jordan curve induced by the Riemann mapping theorem from the open unit disc D onto the interior of that curve. A natural question was addressed, namely, whether the same results are valid when the arc-length parametrization of the curve is used. This led two of the authors of the present paper to prove that arc-length is a global conformal parameter for any analytic curve [START_REF] Nestoridis | arc-length as a conformal parameter for locally analytic curves[END_REF][START_REF] Nestoridis | arc-length as a global conformal parameter for analytic curves[END_REF]. Thus, nothing changes in the results of [START_REF] Bolkas | Non extendability from any side of the domain of definition as a generic property of smooth or simply continuous functions on an analytic curve[END_REF][START_REF] Bolkas | One sided extendability and p-continuous analytic capacities[END_REF] if we use the arc-length parametrization of the curve considered.

On the other hand, it may happen that with a particular conformal parametrization of an analytic curve, it is not clear whether one can further extend this curve analytically to a domain larger than its domain of definition. It was shown in [START_REF] Nestoridis | arc-length as a conformal parameter for locally analytic curves[END_REF] and [START_REF] Nestoridis | arc-length as a global conformal parameter for analytic curves[END_REF] that in order to be sure whether such an extension is possible or not, one may simply use the arc-length parametrization, and examine whether the extendability for this parametrization is possible.

In the present paper, we show that also spherical arc-length is a global conformal parameter for any analytic curve in C. Moreover, for analytic curves in the upper half-plane model of hyperbolic geometry, we show that hyperbolic length is also a global conformal parameter. These results are proved in §2 and are corollaries of a more general theorem involving holomorphic functions in several complex variables, where each variable is replaced by a derivative of the complexification of the function γ defining the analytic curve or by the complex conjugate of a derivative of γ evaluated at the complex conjugate z of the variable z in C.

In §3, we extend the result of §2 to analytic curves in R n and C n . Essentially, the same methods of proof work.

In §4, we consider the notion of analytic curve in the Riemann sphere C∪{∞}. The more general form of this definition is that locally the function γ is the restriction of an injective meromorphic function. We show that the spherical arc-length of such a curve is a global conformal parameter. It follows that the maximal extension of such a curve parametrized by spherical arc-length is always defined on an open subinterval of R = (-∞, +∞) and cannot contain the point +∞. This contrasts with what happens if we use other parametrizations, for instance the Euclidean arc-length. We note that these parametrizations by Euclidean arc-length or spherical arc-length should define strictly increasing functions, and in most cases we are led to allow the parameter to take negative values, although we call it arc-length parameter or spherical arc-length parameter (see [START_REF] Nestoridis | arc-length as a global conformal parameter for analytic curves[END_REF], beginning of §,3, where this question is discussed).

Finally, we mention that our definitions are special cases of more general definitions of analytic curves on Riemann surfaces, but we do not insist on that in the present paper.

In this paper, by a conformal mapping we mean a locally conformal mapping, that is, holomorphic and locally injective. (Equivalently, the map is holomorphic with nonzero derivative, [START_REF] Ahlfors | Complex Analysis. An introduction to the theory of analytic functions of one complex variable[END_REF].) 2. Analytic curves in C Definition 2.1. Let I ⊂ R be a nonempty interval of an arbitrary type which is not a singleton and let γ : I → C be a continuous map. We say that γ is an analytic curve if γ is differentiable on I and γ ′ (t) = 0 for all t ∈ I and if for every t 0 ∈ I there exists δ = δ t 0 > 0 and a power series ∞ n=0 a n (tt 0 ) n , a n ∈ C, which is convergent on (t 0δ, t 0 + δ) and such that γ(t) = ∞ n=0 a n (tt 0 ) n for all t ∈ I ∩ (t 0δ, t 0 + δ). It follows from this definition that γ is locally injective and that its maximal analytic extension is defined on an open subinterval of R, see [START_REF] Nestoridis | arc-length as a global conformal parameter for analytic curves[END_REF]. It is also true that if we restrict γ to a compact interval I 0 such that γ |I 0 is injective, then γ has an injective holomorphic extension to an open neighborhood V ⊂ C of I 0 (see [START_REF] Nestoridis | arc-length as a conformal parameter for locally analytic curves[END_REF], [START_REF] Nestoridis | arc-length as a global conformal parameter for analytic curves[END_REF]). In fact, this local extendability to an injective holomorphic function is equivalent to the definition of an analytic curve [START_REF] Ahlfors | Complex Analysis. An introduction to the theory of analytic functions of one complex variable[END_REF][START_REF] Gamelin | Complex Analysis[END_REF][START_REF] Miller | Analytic curves[END_REF].

Let γ : [a, b] → C be an analytic curve and N ≥ 1 a natural number. We consider the image in C 2N of [a, b] by the map

Γ(t) = γ(t), γ(t), γ ′ (t), γ ′ (t), . . . , γ (N ) (t), γ (N ) (t) .
(Note that since t is real, we have t = t, but the above notation is also meant to deal with the case where t is complex, which we shall consider shortly.)

We denote by J ⊂ C 2N the image of [a, b] by the map Γ.

We now consider an open subset Ω of C 2N containing J, and a holomorphic map

F : Ω → C such that F (J) ⊂ (0, +∞).
We set

S(t) =
δ(s) = γ(S -1 (s)).
Theorem 2.2. The curve δ is analytic.

In other words, the curve γ remains analytic with respect to this new parametrization.

Proof. Since the map γ is locally injective, we can restrict γ to a smaller interval to obtain an injective map. Without loss of generality we may assume that γ is injective on [a, b]. Therefore the curve γ has an injective holomorphic extension on a neighborhood

V ⊂ C of [a, b]. We again call γ this extension. F (Γ(t)) is defined on [a, b]. There exists a neighborhood V 1 of [a, b] in C which is contained in V ,
such that the the following function has a holomorphic extension in that neighborhood:

G(z) = F (γ(z), γ(z), γ ′ (z), γ ′ (z), . . . , γ (N )(z), γ (N ) (z).
We can choose this neighborhood V 1 of [a, b] in C (the domain of z) to be convex, e.g. a rectangle. Thus, the function G has a holomorphic primitive (which we shall also call S) in V 1 , and the primitive may be chosen so that it coincides with S at the point a (that is, it takes the value 0 at that point). Thus, this function S is an extension, on V 1 of the initial function S defined on [a, b].

The restriction of S to [a, b] is injective. The derivative of S is F (Γ(t)) and is nonzero on [a, b], since we have F (J) ⊂ (0, +∞). We then prove as in the papers [START_REF] Nestoridis | arc-length as a conformal parameter for locally analytic curves[END_REF] and [START_REF] Nestoridis | arc-length as a global conformal parameter for analytic curves[END_REF] that we can take a smaller neighborhood of [a, b] which is a rectangle such that the restriction of S to that rectangle is injective.

This shows that the new parametrization of the initial curve δ is an analytic function. The proof is complete. Theorem 2.2 also follows from the following lemma:

. S ′ (t) = f (t) = ∞ k=0 a k (t -t 0 ) k , |t -t 0 | < δ 0 , S(t) = ∞ k=0 a k k + 1 (t -t 0 ) k+1 , |t -t 0 | < δ 0 .
Note that if f is positive, then S ′ is also positive, so the extension is locally conformal in a smaller neighborhood.

Now, let γ : [0, b] → C be an analytic curve. Let Γ : [0, b] → C n be the analytic curve Γ = (γ, γ ′ , . . . , γ (n) ). Suppose F is holomorphic on a neighborhood of J = f ([0, b]) and positive on J. Define the length (or F - length) of γ([0, t]) as S(t) = t 0 (F • Γ)(u)du.
We may apply the lemma to S, with f = F • Γ. We obtain that length is a conformal parameter. Thus, we get another proof of Theorem 2.2.

We now give three applications of Theorem 2.2, in the form of examples.

Example 2.4 (Euclidean arc-length). We consider the particular case where F is the function

F (z) = √ z 3 z 4 for z = (z 1 , z 1 , z 3 , z 4 ) ∈ C 4 , with F defined on Ω = {(a 1 , z 2 , z 3 , z 4 ) ∈ C 4 : Rez 3 > 0, Rez 4 > 0.}
Note that in this case we have z 3 z 4 ∈ (-∞, 0], therefore the square root function is well defined and holomorphic, and the function F • Γ is (locally) conformal.

Then, we have

(F • Γ)(t) = |γ ′ (t)| = γ ′ (t).γ ′ (t)
and

G(z) = γ ′ (z).γ ′ (z).
Theorem 2.2 gives another proof of Theorem 3.1 of [START_REF] Nestoridis | arc-length as a global conformal parameter for analytic curves[END_REF] whose statement is the following:

Theorem 2.5. Arc-length is a global conformal parameter for any analytic curve in C.

Example 2.6 (Spherical arc-length). The derivative of spherical arc-length with respect to the parameter t of a curve is given by the following formula:

|γ ′ (t)| 1 + |γ(t)| 2 = γ ′ (t).γ ′ (t) 1 + γ(t)γ(t) .
Thus, in this case the function F of Theorem 2.2 is:

F (z 1 , z 2 , z 3 , z 4 ) = √ z 3 z 4 1 + z 1 z 2 , defined on {(z 1 , z 2 , z 3 , z 4 ) ∈ C 4 : Rez 3 > 0, Rez 4 > 0, Rez 1 z 2 > -1/2}.
Theorem 2.2 implies in this case the following:

Theorem 2.7. For any analytic curve in C, spherical arc-length is a global and conformal parameter.

Example 2.8 (Hyperbolic arc-length). We consider an analytic curve γ in the hyperbolic plane H 2 , and we use the upper half-space model:

H 2 = {w = x + iy : y = Im(w) > 0}.
The derivative of arc-length is given by the formula

dx dt (γ(t)) 2 + dy dt (γ(t)) 2 y(γ(t)) = |γ ′ (t)| γ(t) -γ(t) 2i = γ ′ (t).γ ′ (t) γ(t) -γ(t) 2i.
Therefore, here we consider the function

F (z 1 , z 2 , z 3 , z 4 ) = √ z 3 z 4 z 1 -z 2 2i
defined on

{(z 1 , z 2 , z 3 , z 4 ) ∈ C 4 : Rez 3 > 0, Rez 4 > 0, Im z 1 -z 2 2i > 0}. Note that Re(z 1 -z 2 ) < 0 is equivalent to Im z 1 -z 2 2i
> 0. We deduce the following: Theorem 2.9. In hyperbolic space, hyperbolic arc-length is a global and conformal parameter.

Analytic curves in R n

Definition 3.1. Let I ⊂ R be a nonempty interval of arbitrary type which is not a singleton. We say that γ : I → R n is analytic if the following two conditions are satisfied:

(1) For every t ∈ I, the derivative γ ′ exists in R n and γ ′ (t) = 0.

(2) For every t 0 ∈ I, there exists δ = δ t 0 > 0 and a sequence a m ∈ R n such that ∞ m=0 (tt 0 ) m a m converges in (t 0δ, t 0 + δ) and γ(t) = ∞ m=0 (tt 0 ) m a m for any t ∈ I ∩ (t 0δ, t 0 + δ). Let γ : I → R n be an analytic curve, with components γ = (γ 1 , γ 2 , ..., γ n ); that is, γ(t) = (γ 1 (t), γ 2 (t), ..., γ n (t)). Then for any given point t 0 ∈ I, we have γ ′ j (t 0 ) = 0 for some 1 ≤ j ≤ n. Therefore γ j is injective in a neighborhood of t 0 . Consequently, γ is locally injective.

We consider the complexification C n of R n , taking

R n = (x 1 , . . . , x n ) and C n = (x 1 + iy 1 , . . . , x n + iy n ).
Each γ j can be extended on a neighborhood of [a, b] in C to a holomorphic mapping with values in C = {x j + iy j }. Taking the intersections of these neighborhoods for i = 1, . . . , n, we see that γ extends to a holomorphic mapping (which we also call γ)

from a neighborhood U of [a, b] ⊂ C to C n .
By choosing a smaller neighborhood of [a, b], we can assume that the extension γ is locally injective. Indeed, since γ |I is locally injective, we may restrict our attention to an interval I 0 on which the curve γ is injective. If the extension of γ from I 0 to a small neighborhood of I 0 were not injective, then there would exist a nested sequence of open sets U k whose intersection is I 0 , and points z k = w k in each U k such that γ(z k ) = γ(w k ). Without loss of generality, we can assume that z k → z ∈ I 0 and w k → w ∈ I 0 . Therefore, γ(z) = γ(w). Since γ is injective on I 0 , z = w. But γ ′ j (z) = 0 for some 1 ≤ j ≤ n. Therefore the extension of γ j is injective on a neighborhood of z, which implies that γ is injective on a neighborhood of z. This contradicts the fact that for every k, we have γ(z k ) = γ(w k ).

Let γ : [a, b] → R n be an analytic curve in R n and let N be a natural number. Set Γ(t) = γ(t), γ ′ (t), γ ′′ (t), . . . , γ (N ) (t) and

J = Γ([a, b]) = { γ(t), γ ′ (t), γ ′′ (t), . . . , γ (N ) (t) : t ∈ [a, b]} ⊂ R nN .
Let Ω ⊂ C nN be an open subset containing J, where we have the identification

R m = {(Rez 1 , . . . , Rez m ); (z 1 , . . . , z m ) ∈ C m }
with m = nN . Let F : Ω → C be a holomorphic function in Ω such that F (J) ⊂ (0, +∞). Finally, let us set

S(t) = t a F (Γ(t))dt, a ≤ t ≤ b.
The function t → S(t) is continuous and bijective from [a, b] to [A, B], for some A, B in R, and it defines a new parametrization of the curve γ, namely,

δ = γ • S -1 , where S -1 : [A, B] → [a, b] is the inverse of S.
We have the following:

Theorem 3.2. The function δ is also analytic.

The proof of Theorem 3.2 is similar to that of Theorem 2.2. It can be done first in the case where γ is injective and the result follows easily in the general case where γ is locally injective Remark 3.3. Since in this section γ j takes real values for t real, its holomorphic extension, denoted also by γ j , satisfies γ j (z) = γ j (z); that is why in this section we do not need extra variables for the conjugates of the derivatives of γ j evaluated at z.

Using a similar technique we can obtain a theorem similar to Theorem 3.2, but for analytic curves in C n .

Extendability with respect to spherical arc-length

Consider an analytic curve γ : (a, b) → C (-∞ ≤ a < b ≤ +∞) under its maximal extension form with respect to arc-length.

If the set of limit points of γ(t) when t → b has more than one element, then, obviously, any reparametrization we choose for γ will not be extendable beyond b.

Suppose now that the set of limit points of γ(t) when t → b is a singleton. If this point is in C, then no matter what parametrization we choose for the curve, it will not be extendable beyond b, since γ is endowed with the maximal parametrization by arc-length (Theorem 3.1 of [START_REF] Nestoridis | arc-length as a global conformal parameter for analytic curves[END_REF]).

The remaining case is when γ(t) converges to ∞ as t → b (that is, for any compact subset K of the complex plane, we have γ(t) ∩ K = ∅ for all t large enough). Thus, we have to consider the case where b = +∞. In this case, using the arc-length parameter, the curve cannot be extended through b since b is not in R (and arc-length becomes infinite). But we can use instead the spherical arc-length, and it may happen that the spherical arclength is finite. Thus, we consider the curve 1 γ(t)

and we examine whether this curve is extendable using the spherical arc-length parametrization. This may happen indeed, and we give an example. Consider a straight line, say R, and let γ : (-∞, +∞) → C be the inclusion map γ(t) = t. Then γ is parametrized by arc-length. We have lim t→+∞ γ(t) = ∞.

By an easy computation, the spherical arc-length of the real axis is equal to π. Indeed, we have in the case considered |γ ′ (t)| = 1, and the spherical length becomes

+∞ -∞ dt 1 + t 2 = arctan +∞ -∞ = π 2 -(- π 2 ) = π.
Thus, the parametrization of γ by the arc length parameter on (-∞, +∞) can be replaced by the spherical arc-length parametrization as a function defined on the interval (-π 2 , π 2 ). The corresponding function is

δ : (- π 2 , π 2 ) → R ⊂ C
given by δ(u) = tan u. The limit of the map 1 γ when t → +∞ is obviously 0. We have lim u→-π 2 1 δ(u) = 0. Thus, δ can be continued to a second real line: when the initial real line where δ is defined approaches +∞, then the second real line will approach -∞. The domain of definition using the spherical arc-length parametrization, instead of being (-π 2 , π 2 ) will become (-π 2 , 3π 2 ). Continuing in this way to the right and to the left, we find that the domain of definition of δ is (-∞, +∞). The image of this curve γ is winding countably many times around the real axis, equipped with the spherical parmetrization by arc-length.

This gives an example of curves which are non-extendable using the Euclidean arc-length. The introduction of spherical arc-length allows to go over the point ∞.

We note that the spherical arc-length parameter for the curves γ and 1 γ is the same because the map z → 1/z is an isometry for spherical arc-length.

Finally, we note that the curve δ : (-∞, +∞) → C ∪ {∞} is not extendable, no matter what parametrization we use, because the set of limit points at +∞ is R ∪ {∞}, which is not a singleton.

From the example of the real line that we gave, one can see that it may happen that an analytic curve γ : (c, +∞) → C (c > 0) satisfying lim t→+∞ γ(t) = ∞ extends analytically to (c, +∞)∪ {∞}∪ (-∞, c ′ ) (c ′ < 0) where we identify the points +∞, ∞ and -∞. For this to hold, it suffices that 1 γ(t) extends analytically from (0, 1/c) to (1/c ′ , 1/c) with 1 γ( 1 t ) t=0 = 0. We conclude that the point ∞ can be in the interior of the domain where a conformal parameter for an analytic curve in C ∪ {∞} varies.

Our previous definitions may be unified as follows:

Definition 4.1. A curve γ in C ∪ {∞} is said to be analytic with respect to a parametrization with parameter t if for every value t 0 ∈ R ∪ {∞} in the range of the parameter t there exists an open set ∆ in the space where the parameter varies (which may be an interval, the union of two intervals with the point ∞ common to them, etc.) containing a point t 0 , an open set V in C ∪ {∞}, such that t 0 ∈ ∆ ⊂ V , and an injective meromorphic function φ : V → C ∪ {∞} such that φ |∆ = γ |∆ .

The example of the real line γ(t) = t, t ∈ R, shows that for some parametrization of γ the point ∞ = +∞ = -∞ can be in the interior of the set where the parameter t can be extended and still give an analytic curve. However, we claim that this cannot happen if the parameter t is the spherical arc-length. The reason is that spherical arc-length is a global conformal parameter for any analytic curve in C ∪ {∞}. Indeed, if γ can be extended through the point ∞ = +∞ = -∞, its spherical length is equal to that of 1 γ( 1 t ) around 0, which must be finite. This contradicts the fact that t is the spherical arc-length and approaches infinity at an interior point where the parameter varies.

From the previous discussion, we deduce the following Theorem 4.2. Spherical arc-length is a global conformal parameter for any analytic curve in C ∪ {∞}.

Finally, we mention that all the previous definitions are special cases of the definition of analytic curve on a Riemann surface, a subject we shall develop elsewhere.

F

  Γ(u) du and we let [A, B] be the image of [a, b] by S. Then, S is a strictly increasing and continuous function from [a, b] onto [A, B]. Therefore it has an inverse t = S -1 (s), and we can consider a new parametrization of the initial curve γ by setting

Lemma 2 . 3 .

 23 Let f be analytic on [0, b]. Then S(t) = t 0 f (u)du is analytic and hence extends holomorphically to a neighborhood of [0, b]. If f > 0, then this extension is locally conformal in a (possibly smaller) neighborhood of [0, b].

Proof

This research was supported through the programme "Research in Pairs" of the Mathematisches Forschungsinstitut Oberwolfach in 2016, as well as by NSERC (Canada) .

1


Mihai Dascalu
email: mihai.dascalu@cs.pub.ro

Philippe Dessus
email: philippe.dessus@upmf-grenoble.fr

Maryse Bianco
email: maryse.bianco@upmf-grenoble.fr

Stefan Trausan-Matu
email: stefan.trausan@cs.pub.ro

Are Automatically Identified Reading Strategies Reliable Predictors of Comprehension?

Keywords: Self-Explanations, Reading Strategies, Comprehension Prediction, Identification Heuristics, Support Vector Machines

In order to build coherent textual representations, readers use cognitive procedures and processes referred to as reading strategies; these specific procedures can be elicited through self-explanations in order to improve understanding. In addition, when faced with comprehension difficulties, learners can invoke regulation processes, also part of reading strategies, for facilitating the understanding of a text. Starting from these observations, several automated techniques have been developed in order to support learners in terms of efficiency and focus on the actual comprehension of the learning material. Our aim is to go one step further and determine how automatically identified reading strategies employed by pupils with age between 8 and 11 years can be related to their overall level of understanding. Multiple classifiers based on Support Vector Machines are built using the strategies' identification heuristics in order to create an integrated model capable of predicting the learner's comprehension level.

Introduction

In order to build textual coherence and to achieve a consistent representation of the discourse, readers need to transcend beyond what is explicitly expressed by employing cognitive procedures and processes, referred to as reading strategies.

Those procedures are elicited through self-explanations [START_REF] Millis | Assessing comprehension processes during reading[END_REF]. Research on reading comprehension has shown that expert readers use specific strategies to on-line monitor their reading, thus being able to know at every moment their level of understanding. Moreover, when faced with a difficulty, learners can call upon regulation procedures, also part of reading strategies [START_REF] Mcnamara | Self-explanation and metacognition[END_REF]. In this context, psychological and pedagogical research has revealed that people tend to understand better a text if they try to explain themselves what they have read [START_REF] Mcnamara | Training reading strategies[END_REF]. Starting from these observations, techniques such as SERT (Self-Explanation Reading Training) [START_REF] Mcnamara | SERT: Self-Explanation Reading Training[END_REF], were developed to support students better understand texts.

Reading strategies have been extensively studied with adolescent and adult readers using the think-aloud procedure that engages the readers to self-explain what they understood so far at specific breakpoints while reading. Our study is focused on comprehension assessment for an audience more rarely studied, primary pupils, whose guidance plays a central role. As previous research suggests, self-regulation can be enhanced through the use of metacognitive reading strategies [START_REF] Nash-Ditzel | Metacognitive Reading Strategies Can Improve Self-Regulation[END_REF]. Pupils tend to better understand a given text by employing these specific mechanisms [START_REF] Nardy | Contrôle et régulation de la compréhension: l'acquisition de stratégies de 8 à 11 ans[END_REF]. Also, this paper represents a continuation of previous research [START_REF] Dascalu | ReaderBench, an Environment for Analyzing Text Complexity and Reading Strategies[END_REF][START_REF] Dascalu | Analyzing Discourse and Text Complexity for Learning and Collaborating[END_REF], with a refined set of heuristics for best matching human annotations, accompanied by a prediction mechanism based on Support Vector Machines [START_REF] Cortes | Support-Vector Networks[END_REF] in order to estimate pupil's comprehension level of a given text.

The following section presents an overview of the evaluation of reading strategies, their categorization, and other similar automated systems that have been developed to identify the employed reading strategies. The third section is centered on the description of the used heuristics, while the fourth section introduces the classifier that combines the previously identified reading strategies and predicts the learner's comprehension level. Afterwards, the fifth section encompasses the performed validations for testing the system's accuracy, while the last section is focused on conclusions and future improvements.

Overview of the Assessment of Reading Strategies

Expert readers frequently make use of four types of reading strategies in order to achieve a deep understanding from the texts they read [START_REF] Mcnamara | SERT: Self-Explanation Reading Training[END_REF]. Paraphrasing enables readers to express what they understood from the explicit content of the text and can be considered the first and essential step in order to achieve a coherent representation.

Text-based inferences, consisting predominantly of causal and bridging strategies, build explicit relationships between two or more textual segments of the initial text.

On the other hand, knowledge-based inferences create relationships between the information from the text and the reader's personal knowledge and are essential to create the situation model [START_REF] Van Dijk | Strategies of discourse comprehension[END_REF]. Control strategies refer to the actual monitoring process, when readers explicitly express what they have or have not understood. Nevertheless, if we want students to be assisted while reading, one human expert (e.g., a teacher) can take care only after a small number of them, which makes it impossible for such training techniques to be used on a large scale. For example, this is one of the major problems of MOOCs (Massively Online Open Courses) in which, due to the previous constraints, assistance is frequently provided by peer students, increasing nevertheless the risk of making mistakes [START_REF] Piech | Tuned models of peer assessment in MOOCs[END_REF][START_REF] Goldin | Acounting for peer reviewer bias with Bayesian models[END_REF]. Moreover, assessing the content of a verbalization is a demanding and a subjectivity-laden activity, which can be assisted by computer-based techniques. These are the main motives behind the idea of using a computer program instead of, or as support for, a human tutor.

Initial experiments were conducted by McNamara and her colleagues [START_REF] O'reilly | iSTART: A Web-based Reading Strategy Intervention that Improves Students' Science Comprehension[END_REF] and iSTART [START_REF] Mcnamara | Evaluating self-explanations in iSTART: Comparing word-based and LSA algorithms[END_REF] can be considered the first implemented system that addresses selfexplanations [START_REF] Jackson | Assessing cognitively complex strategy use in an untrained domain[END_REF]. It has various modules that explain the SERT method to the students, one that shows them how to use those techniques using a virtual student, and another training module that asks students to read texts and give verbalizations, evaluates them and provides an appropriate feedback. iSTART divides verbalizations into four main categories: irrelevant, paraphrases, verbalizations that use knowledge previously found in the text and verbalizations which use external knowledge from the students' experience. It is easier to automatically identify paraphrases and irrelevant explanations, but it is more difficult to identify and evaluate verbalizations that contain information coming from students' experience [START_REF] Miller | WordNet: A Lexical Database for English[END_REF].

We conducted an experiment [START_REF] Nardy | Contrôle et régulation de la compréhension: l'acquisition de stratégies de 8 à 11 ans[END_REF] for analyzing the control and the regulation of comprehension through reading strategies. Pupils (3 rd -5 th grade, 8-11 years old) were given the task to read aloud two French stories and were asked at predefined moments to self-explain their impressions and thoughts about the reading materials. The selfexplanations were coded according to McNamara's [START_REF] Mcnamara | SERT: Self-Explanation Reading Training[END_REF] scheme. The results of this study support the view that pupil's self-explanations are an adequate way to access to their reading strategies. The sole exception consists of prediction strategies, which were scarcely used in comparison to McNamara's participants, perhaps due to the age of the pupils. Initial and partial automated results based on the previous study were presented in [START_REF] Dascalu | ReaderBench, an Environment for Analyzing Text Complexity and Reading Strategies[END_REF], and we present in this paper data from a larger sample, using finetuned heuristics and an automatic classifier for predicting comprehension.

Reading Strategies Identification Heuristics

In terms of reading strategies, our aim was to create automated extraction methods designed to support tutors at identifying various strategies employed by pupils that are best aligned with the annotation categories: 1/ paraphrasing, 2/ text-based inferences consisting of causality and bridging, 3/ knowledge-based inferences or elaboration and 4/ monitoring or control [START_REF] Nardy | Contrôle et régulation de la compréhension: l'acquisition de stratégies de 8 à 11 ans[END_REF]. A clear demarcation between causal inferences and bridging had to be established within our automated system due to underlying approaches and computational complexity, although causal inferences can be considered a particular case of bridging, as well as a reference resolution. In addition, we have tested various methods of identifying reading strategies and we will focus solely on presenting the refined heuristics that provided in the end the best overall human-machine correlations.

In ascending order of complexity, the simplest strategies to identify are causality, with markers like "parce que" (because), "pour" (for), "donc" (thus), "alors" (then), "à cause de" (because of) and control, with markers like "je me souviens" (I remember), "je crois" (I believe that), "j'ai rien compris" (I haven't understood anything) for which cue phrases based on pattern matching techniques have been used. As particular refinement for causality, all occurrences of the keywords at the beginning of a verbalization have been discarded because the strategy needs to create an inferential link between two adjacent textual segments, out of which the first is lacking since it is the beginning of a verbalization. In this particular case, the use of causality patterns indicates a lacunar pupil formulation frequently observed at their age. In terms of control, besides the verification of specific cue phrases, we added a check to verify whether the pattern exists in the sentences within the original text, in which case we would be dealing with a paraphrase rather than a control statement.

As a second stage of complexity, paraphrases, that in the manual annotation were considered mere repetitions of the same semantic propositions by human raters, were automatically identified through lexical similarities. More specifically, words from the verbalization were considered as paraphrased words if they had identical lemmas or stems, or were synonyms extracted from lexicalized ontologies -WordNet [START_REF] Miller | WordNet: A Lexical Database for English[END_REF] or WOLF [START_REF] Sagot | WordNet Libre du Francais (WOLF)[END_REF] -with words from the initial text. Adjacent words from pupil's selfexplanations, identified as paraphrased concepts were grouped into paraphrase segments in order to highlight contiguous zones highly referential to the initial text. In addition, if more than a predefined percentage of relevant words from a sentence from the initial text are paraphrased within the verbalization, that specific sentence is tagged as a paraphrasing segment. The previous percentage was empirically set after performing multiple iterations with incremental values, whereas relevant words are obtained after stop words elimination and after selecting solely dictionary words.

In the end, the strategies most difficult to identify are knowledge inference and bridging, for which semantic similarities have to be computed. An inferred concept is a non-paraphrased word for which the following three semantic distances were computed: the highest similarity to another word from the initial text (expressed in terms of semantic distances in ontologies, Latent Semantic Analysis and Latent Dirichlet Allocation) [START_REF] Dascalu | ReaderBench, an Environment for Analyzing Text Complexity and Reading Strategies[END_REF] and the relevance of both words to the textual fragments inbetween consecutive self-explanations expressed as semantic cohesion. The latter distances had to be taken into consideration for better weighting the importance of each concept, with respect to the whole text. In the end, for classifying a word as inferred or not, a weighted sum of the previous semantic similarities is computed and compared to a minimum imposed threshold which was experimentally set at 0.4 for maximizing the precision of the knowledge inference mechanism.

As bridging consists of creating connections between different textual segments from the initial text, cohesion was measured between the verbalization and each sentence from the referenced reading material [START_REF] Dascalu | ReaderBench, an Environment for Analyzing Text Complexity and Reading Strategies[END_REF]. Semantic similarity was measured in-between the current verbalization and the two previous textual blocks from the initial text. In order to relate to the overall cohesion between the verbalizations and what was initially stated within the reading material, the imposed similarity threshold for tagging a sentence as being a bridged element uses a cohesion value that exceeds the mean plus standard deviation of all previous similarity measures performed on all self-explanations of a given pupil. Similarly to paraphrases and for best adapting to the manual annotation process, adjacent sentences from the initial text tagged as being bridged within the verbalization are grouped into a bridging segment. Moreover, if a sentence is considered to be a paraphrasing segment due to a high density of paraphrased words, that sentence is not taken into consideration while defining the final bridging segments. To better highlight the identification mechanisms, Fig. 1 depicts with bold bridged sentences from the initial text with verbalization 2 that exceed the identified threshold and that are not marked as paraphrases. In the end, four bridged segments are automatically determined: A3, B1 together with B2 due to adjacency within the same paragraph, C1 and C3 from the later textual block.

Combining Automatically Identified Reading Strategies for Predicting Comprehension

All the previous reading strategies and their corresponding identification heuristics can be viewed as attributes that describe the learner's comprehension level. In order to predict the comprehension level of each learner based on the used reading strategies, post-tests were administered to each pupil and comprehension scores were manually determined using these tests. Therefore, we found it appropriate to use a classifier that accepts as inputs the number of used reading strategies and predicts a comprehension class depicting the reader's understanding level expressed as a comprehension level class estimate. Similar to the textual complexity problem for which Support Vector Machines (SVMs) [START_REF] Cortes | Support-Vector Networks[END_REF] have been proven to be the most relevant [START_REF] François | Do NLP and machine learning improve traditional readability formulas?[END_REF], we trained multiple SVMs for determining the appropriate comprehension class. A one-versus-all approach implementing the winner-takes-all strategy is used to deal with the problem of multiple SVM returning 1 for a specific text (the classifier with the highest output function assigns the class). As specific optimizations, an RBF kernel with degree 3 was selected and a Grid Search method [START_REF] Bergstra | Random Search for Hyper-Parameter Optimization[END_REF] was enforced to increase the effectiveness of the SVM through the parameter selection process for the Gaussian kernel. Exponentially growing sequences for C and γ were used, and each combination of parameter choices was checked using the testing corpora; in the end, the parameters that generated the best precision were selected.

Validations of the Identification Heuristics and of the Comprehension Prediction Model

We ran an experiment with 82 pupils with age between 8 and 11 years, uniformly distributed in terms of their age, who had each to read aloud two French stories of about 450 words (The Cloud Swallower and Matilda). During their lecture, pupils had to stop in-between at five, respectively six predefined markers, and explain what they understood up to that moment. Their explanations were first recorded and transcribed, then annotated by two human experts (PhD in linguistics and in psychology), and in the end categorized according to the imposed annotation scheme. Nevertheless, when looking at manual assessments, discrepancies between evaluators were identified due to different understandings and perceptions of pupil's intentions, expressed within their self-explanations; all disagreements were solved individually by mediation for each self-explanation. In addition, predefined rules and patterns were used to perform automatic cleaning in order to process the phonetic-like transcribed verbalizations. Fig. 2 depicts the main interface of our developed system in which the grey sections represent the pupil's self-explanations, whereas the white blocks represent paragraphs from the read story. All strategies are highlighted within the selfexplanation with a specific color encoding: control, causality, paraphrasing, inferred concept [*] and bridging with a clear demarcation of the textual segments from the reading material comprising of inter-linked cohesive sentences. In addition, Fig. 2 also depicts in the last column the cohesion measures normalized in [0; 1] with previous paragraphs from the story.

Three variables were required for fine-tuning the higher-level reading strategies: bridging requires a minimum semantic cohesion (Min coh_bridging ) and a maximum percentage of words for not considering a sentence as paraphrased (Max paraphrase ), while knowledge inference uses only a minimum similarity threshold (Min sim_KI ). Our system automatically determines the most suitable values for maximizing the overall Pearson correlations and F1-scores as measures of outputs' correctness with regards to the manual annotations (see Table 1). As expected, paraphrases, control and causality occurrences were much easier to identify than information coming from pupils' experience [START_REF] Graesser | Constructing inferences during narrative text comprehension[END_REF]. Moreover, our experiments demonstrate that although the variables for the two texts have similar optimal values, there are rather high fluctuations in the accuracy of the reading strategies' identification, therefore highlighting the specificities of each text and the intrinsic subjectivity of the analysis. After fine-tuning the identification heuristics, we opted to create three comprehension classes for predicting the learner's comprehension level with a distribution of 30%, 40% and 30% of all pupil scores sorted in ascending order and to apply 3-fold cross-validations for the SVM training process. The resulting average agreement between automatic predictions and the class assigned from the post-test scores was approximately .78 in most runs (see Table 2). Due to a rather limited corpus, the prediction accuracy oscillates between different training sessions, with a minimum of .66. We also noticed a rather small differentiation between the first and the second class, as well as conflicting instances of pupils with a high number of used reading strategies, but pertaining to opposite comprehension classes. The previous contradictions in terms of the number of used reading strategies in opposition to pupils' comprehension levels, corroborated with rather small differentiations between adjacent classes, led to a rather low prediction accuracy of the second class. Nevertheless, results are encouraging based on the limited number of training instances, the reduced number of classification attributes and the fact that a lot of noise existed within the transcriptions. From this point, it becomes clear that external factors should be enforced in order to increase the accuracy of the prediction and to create a more comprehensive view, as the diversity and the richness of the strategies a reader carries out depend on many factors, either personal (proficiency, level of knowledge, motivation), or external (textual complexity).

In order to prove the feasibility of the previous statements, we added a simple factor already computed during the identification process: the average value of cohesion between each verbalization and the corresponding paragraphs from the initial text. This measure emphasizes the link between what was initially stated and the learner's understanding or personal perspective. As expected, the results from Table 3 highlight an increase in the overall prediction accuracy. In the end, notable improvements in terms of the initial experiments presented in [START_REF] Dascalu | ReaderBench, an Environment for Analyzing Text Complexity and Reading Strategies[END_REF] can be observed: 1/ the use of 8 times more participants, each self-explaining two texts instead of only one; 2/ an important increase in the identification accuracy for paraphrases, knowledge and text-based inferences; 3/ although bridging taken individually has still a low correlation which indicates that the human annotated bridging strategy is not aligned with the identification heuristics, the use of the new class of text-based inferences demonstrates that the integrated perspective of bridging and causality taken together is more cognitively relevant and representative with regards to the manual annotations; 4/ the use of Support Vector Machines for predicting the learner's comprehension level.

Conclusion and Future Research Directions

Our aim consists of supporting tutors and our approach emphasizes the benefits of a regularized and deterministic process of identification as a viable alternative to the subjectivity-laden task of manual annotation. Moreover, the performed validations confirm that reading strategies are related to the pupil's comprehension level, but also highlight the need to add more factors, potentially inspired from textual complexity measures [START_REF] Graesser | Coh-Metrix: Providing multilevel analyses of text characteristics[END_REF][START_REF] Nelson | Measures of text difficulty: Testing their predictive value for grade levels and student performance[END_REF] or essay scoring techniques [START_REF] Todd | Coherence, cohesion and comments on students' academic essays[END_REF] in order to increase the accuracy of the predictions.

As the comprehension scores are not global, but related to the read texts subject to expressing one's meta-cognitions, we can state that reading strategies can be used to predict comprehension based on the overall experimental settings. Our next aim consists of deploying and using our system in classroom settings to analyze student's reading strategies and to infer possible comprehension problems in near realtime.

Fig. 1 .

 1 Fig. 1. Bridging identification.

Fig. 2 .

 2 Fig. 2. Visualization of automatically identified reading strategies.

Table 1 .

 1 Accuracy of the automatically identified reading strategies.

	Statistic measure	Paraphrasing	Text-based Inference	Knowledge-	Control
			(causality and bridging)	based Inference
		Text 1: The Cloud Swallower
		Min coh_bridging = .40; Max paraphrase = 60%; Min sim_KI = .33
	Pearson correlation	.64	.55	.41	.84
	Precision	.64	.79	.50	.76
	Recall	.99	.83	.94	.63
	F1 score	.78	.81	.65	.68
			Text 2: Matilda
		Min coh_bridging = .45; Max paraphrase = 65%; Min sim_KI = .33
	Pearson correlation	.56	.69	.48	.90
	Precision	.73	.71	.34	.86
	Recall	.99	.94	.97	.70
	F1 score	.84	.81	.50	.77
		All verbalizations together, from both texts
		Min coh_bridging = .4; Max paraphrase = 65%; Min sim_KI = .33
	Pearson correlation	.64	.60	.35	.89
	Precision	.69	.74	.47	.83
	Recall	.99	.90	.87	.68
	F1 score	.81	.81	.61	.74

Table 2 .

 2 Comprehension prediction based solely on the four automatically identified reading strategies.

	Verbalizations pertaining to	Agreement	Agreement	Agreement	Average
		-Class 1 -	-Class 2 -	-Class 3 -	agreement
	Text 1: The Cloud Swallower	1	.33	.67	.67
	Text 2: Matilda	.67	.33	1	.67
	Both texts	1	.33	1	.78

Table 3 .

 3 Comprehension prediction based on the four heuristics plus the average cohesion value added as an attribute for classification.

	Verbalizations pertaining to	Agreement	Agreement	Agreement	Average
		-class 1 -	-class 2 -	-class 3 -	agreement
	Text 1: The Cloud Swallower	1	.33	1	.78
	Text 2: Matilda	1	.67	.67	.78
	Both texts	1	.67	1	.89

Acknowledgements

This research was partially supported by an Agence Nationale de la Recherche (DEVCOMP) grant and by the 264207 ERRIC-Empowering Romanian Research on Intelligent Information Technologies/FP7-REGPOT-2010-1 project. We would also like to thank Aurélie Nardy and Françoise Toffa who helped us to gather experimental data, and the teachers and pupils who participated in our experiments.