Bruna Habib Cavazza

Rodrigo Marçal Gandia
email: romgandia@gmail.com

Fabio Antonialli
email: fantonialli@gmail.com

Isabelle Nicolaï
email: isabelle.nicolai@centralesupelec.fr

André Luiz Zambalde

Joel Yutaka Sugano
email: joel.sugano@dae.ufla.br

Arthur De

Miranda Neto
email: arthur.miranda@deg.ufla.br

Management and Business of Autonomous Vehicles: A Systematic Integrative Bibliographic Review

Keywords: Autonomous Vehicles, Business, Management, Systematic Integrative Review, Bibliometric Review

. In this way, it was observed a study gap related to business models and platforms, radical and responsible innovation theories, in order to minimize the risks, impacts and uncertainties of the eminent arrival of AVs and provide the necessary tools to guide governmental and organizational spheres.

Introduction

The development of Autonomous Vehicles (AVs) is an important innovation that promises to have great impact on the issues of urban mobility. In fact, AVs are embedded in the most significant historical change to the automobile and transportation industry. Governments and universities worldwide identify AVs as keyresearch factor. The AVs' imminent arrival includes impacts on several spheres, being important to consider the impacts of such disruptive innovation on society, structure and functioning of companies. Among the stakeholders responsible for the dissemination of this innovation, the academy is understood as an important precursor on AVs' development, being responsible for important theoretical and empirical research for the field's evolution. Although technological advances are essential for this innovation dissemination, it is understood that, in the case of radical innovations, technology cannot be established as the process' exclusive key-factor, but combined with the establishment of a business model that supports it [START_REF] Christensen | Disruptive innovation for social change[END_REF].

Notwithstanding, the advancement of technological innovations that permeate AVs and the intelligent robot industry is part of a dynamic result established among technology, business models, and marketplace [START_REF] Yun | The relationship between technology, business model, and market in autonomous car and intelligent robot industries[END_REF]. In this context, the possibility of changes in the dynamic relations among these three factors in order to obtain expected results, is on evidence. Although there are some studies that seek to make a general analysis of the context and environment of AVs, there is a lack of theoretical and empirical studies seeking to analyze AVs from a market perspective, related to aspects of planning and management. Thus, it is important to look at the literature approaches related to management and business (M&B) of AVs; attempting to know and spread a little more about the reality of these issues, by Interpreting and integrating content, summarizing existing evidence, identifying key issues and finding research gaps.

Considering the aforementioned, this study aims at answering the following research question: " How does the AVs field of study relate to the management and business (M&B) area?" Specifically, it aims at performing an integrative bibliometric and systematic review of the literature focusing on investigating and understanding the main theoretical contributions related to AVs in the M&B field of study.

Therefore, this study is justified because -characterizing the AVs research field in the M&B areas in its bibliometric context; identifying strategies, practices and management tools specified in the scope of the investigated publications; summarizing existing evidence, pointing to little or no explored topics (gaps) within this area of study -will contribute to the field' systematization based on evidence pointed out in the literature.

In this context, a bibliometric and systematic review is important on allowing us to focus specifically on the subject, identify authors, articles, dates, journals and approaches; review the primary studies; integrate and synthesize the evidence associated with results involving practical strategies, applications and tools and finally, work and disseminate specific information in a chronologically organized and directed way (Botelho;Cunha & Macedo, 2011).

Autonomous Vehicles overview

AVs, which are vehicles that don't require any sort of conductor or teleoperation control are of great interest and investment in the global automotive industry with numerous carmakers already testing AVs' technologies [START_REF] Frazzoli | Real-time motion planning for agile autonomous vehicles[END_REF][START_REF] Fagnant | Preparing a nation for autonomous vehicles: opportunities, barriers and policy recommendations[END_REF]. AVs are also relevant to governments in a sense that many countries are changing legislations to allow testing of such vehicles on their roads [START_REF] Schoitsch | Autonomous Vehicles and Automated Driving Status, Perspectives and Societal Impact. Information Technology, Society and Economy Strategic Cross-Influences (IDIMT-2016)[END_REF], as well as are relevant to the academia with research centers and universities worldwide advancing many studies on technology mobility, vehicle-infrastructure interaction and M&B related issues for AVs' consolidation [START_REF] Lima | Sensor-based navigation applied to intelligent electric vehicles[END_REF].

Thus, AVs represent a potentially disruptive and beneficial change to the transportation system business model, since they will facilitate driving; increase road safety; reduce pollutants' emissions; reduce traffic jams; as well as allow drivers to choose to do different things other than driving. AVs will also improve mobility for those who cannot or do not want to drive and will provide significant economic, environmental and social benefits [START_REF] Attias | The Autonomous Car, a Disruptive Business Model?[END_REF][START_REF] Mutz | Large-scale mapping in complex field scenarios using an autonomous car[END_REF].

Bibliometric and systematic review

Literature review articles are "a form of research using sources of bibliographic or electronic information to obtain search results from other authors, in order to theoretically support a particular topic" (Botelho; Cunha & Macedo, p.124, 2011).

In bibliometric reviews, we have the quantification of publications related to a topic under study, that is, frequency of publications over the years, authors, journals, countries, and other important data. On the other hand, systematic literature reviews should be designed to address a particular issue with the use of detailed, explicit and systematic methods to raise, identify, select, interpret, collect and analyze references (Botelho;Cunha and Macedo, 2011).

According to Whitemor & Knafl (2005) there are basically four types of systematic literature review: metaanalysis (a combination of results of studies with statistical formulas targeting analysis, comparisons and generate new results); systematic (is based on scientific research strategies, limits the scope of selection of articles, synthesizes studies and evaluates them critically); qualitative (systematization and synthesis of qualitative studies' findings transforming them into tools to build new theories) and integrative (reviews the past empirical and theoretical literature, presenting the state of the art on a theme, especially for results analysis and synthesis of previous research contributions and gaps).

In this study, we performed a bibliometric review followed by an integrative systematic literature review. The integrative review term refers to the fact that in this method, concepts, opinions and ideas arising from previous research are integrated (Whitemore & Knafl, 2005).

Methodology

The research is characterized as qualitative and descriptive, drawn by a bibliometric review and by a systematic integrative bibliographic review focusing on investigating and understanding the main theoretical contributions related AVs in the M&B field of study, Figure 1 highlights the research design. • Theme identification (step 1): the articles were searched on Scopus; Web of Science (WoS) & Science Direct (SD) databases in a single search, between 1945 to 2017, using the Boolean operator "OR" and were selected by the following terms in the title, abstract and keywords of the articles: autonomous_car*; autonomous_vehicle*; autonomous_automobile*; driverless_car*; driverless_vehicle*; driverless_automobile*; self-driving_car*; self-driving_vehicle*; self-driving_automobile*; intelligent_car*; intelligent_vehicle* and intelligent_automobile*. We found 6,711 papers from WoS; 27,855 papers from Scopus and 6,796 papers from SD. It is worth clarifying that the Boolean operator underline '_' was used to ensure that the search yielded results in which only the pair of words appeared together. The operators star '*' (WoS and Scopus) and "" (SD) were used to ensure that both singular and plural terms were included in the search results.

• Inclusion and exclusion criteria (step 2): the first criterion was to select publications classified as articles within the databases' M&B categories. We found 644 papers: 244 from Scopus, 56 from WoS and 344 from SD. All articles' titles and abstracts were analyzed in order to allow further filtering. Of the 644 articles, 613 were excluded following these criteria: a) duplicate references -identified more than once in different sources (07 papers); b) unachievable references -that could not be obtained by interlibrary loan, online search or contacting the authors (03 papers) and; c) non-related references to the research problem or just tangent the themes of interest (604 papers).

• Scanning of pre-selected papers (step 3): after step 2, we reached the number of 30 papers for research and analysis. Those papers were pre-selected for in-depth reading and analysis.

• Bibliometric review (step 4): in this step, we analyzed the evolution of publications over the years, countries frequency' publications, publications' languages, methodological approach of the papers, most cited paper and also key-words recurrence in a Word Cloud.

• Integrative review (step 5): initially we carried out a detailed study of the main issues of each document (synthesis matrix), including: key words; research problem; goals; theoretical background; study type; methodological approach; main contributions; future research proposals. After, their appropriate categorization within themes, facilitating the presentation and discussion of results. It is worth mentioning that this categorization was performed by two members of the team and later validated by a third person in order to guarantee impersonality in the results. The first analysis we present regards the number of publications per year. Figure 2 displays the distribution of the 30 articles found in accordance with the year of publication. It is observed that the first publication is the 1990 -article "Intelligent Vehicle-Highway Systems: U.S. activities and policy issues" [START_REF] Chen | Intelligent vehicle-highway systems: US activities and policy issues[END_REF]. Until 2015 publications were less frequent, having a couple years where no article was published. Only from this year publications seemed to had a more consistent pattern, with highlights to the year of 2016, reflecting the emergence and recent attention given to the subject. The most cited paper was by [START_REF] Tuominen | Is the transport system becoming ubiquitous? Socio-technical roadmapping as a tool for integrating the development of transport policies and intelligent transport systems and services in Finland[END_REF] with 29 citations. Regarding the authors with most publications, a great heterogeneity in the field was observed, each paper analyzed had a different author.

Results and discussion

Bibliometric review

Then, the research makes explicit the predominance of articles in English: 100% of the publications are in this language. It is observed the relevance of the United States and Germany (Figure 3), tying first in the number of publications. It is also important to highlight that, by the methodological approach of the papers, we observed that a great majority have a qualitative approach (26 of the 30 articles in total). This fact points to the lack of empirical studies on AVs in the area of M&B as well as evidencing the emergence of the topic.

Finally, the most cited words in the titles and abstracts of the articles were identified. By using the "Word Frequency Counter" application, Figure 4 presents the most recurrent key-words found in the papers' abstracts and titles. The words were chosen for their frequency and relevance on discussions and debates on the theme. It is important to highlight the words; transport, technology, new, business, model, social, question, system and mobility

Integrative review

In order to carry out the articles' categorization, the analyzes of each of the 30 papers were detailed done, considering subjects related to the research problem, objectives, scope, methodological approach, main contributions, conclusions and recommendations. Regarding the integrative review, a common point observed while creating the synthesis matrix is that there is a consensus among all analyzed papers that AVs will be part of our future society, whether in a short, medium or long-term, "the era of self-driving vehicles is undeniably upon us" (Attias, 2016:p.107) Uncertainties permeate various aspects of this innovation, being the search for solutions on several questions still unanswered focus of the papers distributed among the evidenced categories. It is also worth noting that there is a strong relationship among all categories, and in many cases, papers identified as belonging to one category broadens their discussions to other categories, leading to the understanding that there are no fixed barriers to the topic.

We also highlight that the main criterion for the inclusion of an article in a given category was the identification of the strongest present theme in its scope. The main themes categorized and detailed in the following sections of this study are: Ethics/Moral/Liability (9 papers); AVs' Business Models (7 papers); Transport Planning (6 papers); Consumer Behavior (4 papers); Policy Issues -Law (4 papers). Although there is a higher incidence of papers in certain categories than in others, we understand that the quantitative metric standard does not infer to a bigger or smaller influence to a conceptual framework formulation for AVs within the field of M&B. Namely, it is assumed that all the categories identified here are equally necessary, in a synergistic movement, for the formation of the field.

Ethics/moral/liability

This category is centered on the necessary and established pre-definitions for the AVs functioning. That is, AVs decision-making, will be previously introduced by someone, "many or all of those decisions will have to be programmed into the car," (Ong, 2017:p.36), therefore, we have the fundamental role of the engineering department, responsible for programming AVs and, consequently, its dissemination. [START_REF] Mladenovic | Engineering social justice into traffic control for self-driving vehicles?[END_REF] says that decision-making power should not be concentrated solely in a small group of experts, especially if financial interests have exclusive influence. This group should not have exclusive rights to decide on all the moral aspects of these machines behavior [START_REF] Martin | Who Should Decide How Machines Make Morally Laden Decisions?[END_REF]. In this way, [START_REF] Gogoll | Autonomous cars: in favor of a mandatory ethics setting[END_REF] affirms that mandatory policies will be more efficient on a large scale, in detriment of include personal ethics settings (in other words, concentrate the decisions of machine behavior in small groups), that is to say, to program all AVs in order not to preserve the life of the occupants in danger from regulations, would be more interesting for all.

One of the biggest challenges for AVs manufacturers is the risk of product liability lawsuits that, in the event of accidents, may result in deaths and catastrophic injuries [START_REF] Wu | Product Liability Issues in the US and Associated Risk Management[END_REF]. In this way, [START_REF] Hevelke | Responsibility for crashes of autonomous vehicles: an ethical analysis[END_REF], raise the following question: "who should be made responsible in case of a crash?". The authors conclude that, users of autonomous cars should be collectively responsible for any damage caused by such vehicles-even if they had no way of influencing the cars behavior, and agreed that a tax or a mandatory insurance seems to be the easiest and most practical means to achieve that. Further, [START_REF] Schellekens | Self-driving cars and the chilling effect of liability law[END_REF] says that the chilling effect of product liability on innovation can be further mitigated by adequate obligatory insurance.

Finally, we cannot eliminate the need for a public and democratic discussion of relevant societal values that technology shapes. As a result, it is necessary to engage in a transparent manner all relevant corporate groups in critical discussions and decision-making [START_REF] Mladenovic | Engineering social justice into traffic control for self-driving vehicles?[END_REF]. Change is inescapable and not necessarily a bad thing in itself. But major disruptions and further damage must be anticipated and avoided whenever possible. This is the role of ethics in innovation policy: it can pave the way for a better future, while enabling beneficial technologies. Without looking at we are driving with one eye closed [START_REF] Lin | Why ethics matters for autonomous cars[END_REF].

AVs' Business Models

This category deals with aspects related to the business models proposed for the AVs. The key factors presented in this category refer to the economic implications (including here sale options for these vehicles), as well as discussions about service-oriented business models (e.g. car-sharing), their strengths, opportunities, threats and weaknesses as an option for the deployment of AVs, (Beiker, 2016;Kompalla et al., 2017).

The proposition of business models associated to AVs considering different scenarios of ownership and use -manufacturers, distributors, fleets and public sector is of extreme importance to ensure a good insertion and adherence of AVs in the market. The main gaps in this category point to research that seeks a link between Innovation's theories and approaches to proposing models associated with disruptive innovation, that is, in breaking and redefining existing models so that one can understand and anticipate disruptive business and process strategies and contribute to the construction of a consistent theoretical basis for the new businesses associated with AVs.

Transport planning

When it comes to transportation, there is a consensus among the papers of this category that changes will happen and AVs will directly impact or act as accelerators of this process. The development of society is driving these changes, "these developments are being driven by both technology-push and demand-pull factors" (Enoch, 2015:p.922). In view of the digitization context, many industries are gradually being transformed, with AVs as the translation of the digitization of the automotive industry, including the arrival of new players such as Google and Uber [START_REF] Attias | The Autonomous Car, a Disruptive Business Model?[END_REF].

In a gradually changing scenario, Tuominen and Ahlqvist (2010) point to three changes in the future of transport: (1) actor roles and stakeholder networks in the system will be pluralized and the transport system will increasingly be composed of public parties, private entities, taxpayer end-users and complex networks actors; 2) a new type of business and service layer will be formed in the system because of new dynamic interconnections among the actors; 3) a layer of services will be captured from the concept of "technology service", which would be a flexible and adapted combination of technologies and services). [START_REF] Enoch | How a rapid modal convergence into a universal automated taxi service could be the future for local passenger transport[END_REF] proposes that these changes will converge from intermediate modes of transport, which will replace traditional transport (cars, buses and taxis). The author believes that the AVs will be accelerated from this process, to a single autonomous universal service of higher quality and cheaper. [START_REF] Attias | The Autonomous Car, a Disruptive Business Model?[END_REF] proposes that in a mediumterm perspective, this evolution will not stop in the AVs and opens the field for the design of similar technical objects (e.g. trucks, buses) and later boats and autonomous planes.

Finally, Poorsartep (2014) proposes, in addition to a scenario of gradual change, a more radical perspective, which will provide a unique opportunity for the market leader to establish an indisputable position and that can potentially lead to a monopoly.

Consumer behavior

The main themes addressed here are questions raised about how it will be and how to measure population and potential consumers' acceptance to AVs. Specially, discussing a) questions of how firms have enhanced trust in applied Artificial Inteligence [START_REF] Hengstler | Applied artificial intelligence and trust-The case of autonomous vehicles and medical assistance devices[END_REF] , b) examine how positive cognitive evaluations, anxiety-related affects, and the interplay between these two factors influence the willingness to use AVs [START_REF] Woisetschläger | Consumer Perceptions of Automated Driving Technologies: An Examination of Use Cases and Branding Strategies[END_REF] c) identify consumer acceptance and branding consequences in the AVs context (Hohenberger, Spörrleb & Welpe, 2017) and d) present discussions on innovative means to analyze social preferences, demand for AVs, and the potential to resolve community concerns with integrated solutions [START_REF] Lu | Data-enabled public preferences inform integration of autonomous vehicles with transit-oriented development in Atlanta[END_REF].

When we address issues related to AVs from the consumer's stanpont we clearly perceive that "the main roadblock is human acceptance" (New Scientist, p.5, 2007) We cannot fall into the fallacy of a simple analysis of the benefits that will be brought and/or perceived by the consumer. The feelings of attachment, possession, freedom and the sensations acquired by the consumers as drivers/passengers should be considered.

In fact, rather than presenting solutions and conclusions, these papers introduce preliminary research that opens space for a new theme to be explored when considering the human element, its reactions, preferences and perspectives in the dynamics of AVs as well as the various factors that can influence consumer behavior and willingness to adopt the AVs. The main gaps raised in this category refer to the possibilities of studies that relate different business models (ownership, car-sharing, etc.) to the expectations of different consumers' groups, as well as to compare the cost effectiveness and sustainability of individual ownership versus public-operated AVs. Future studies should also address questions about new forms of time use in vehicles considering the creation of an in-car environment as well as new oportunities of "infoitament" (information + entertainment) that can be offered to consumers by these vehicles. Also, the investigation of individual values and their influence on the adoption of AVs has much to be explored (Hohenberger, Spörrleb & Welpe, 2017).

Policy issues -law

This category presents perspectives based on the lack of clarity about who is responsible for the operation of such vehicles, where this is the major point of confusion among manufacturers, consumers and lawyers. "The liability issue is the biggest one of them all," (Adee, 26, 2016). This lack of clarity may be impacting the way in which public policy has been leading the issues related to the legislation of the AVs. Although there is a consensus in the benefits of AVs, the uncertainties and questions that permeate them are still barriers to the long-range transportation plans [START_REF] Guerra | Planning for cars that drive themselves: Metropolitan Planning Organizations, regional transportation plans, and autonomous vehicles[END_REF].

However, several countries around the world are stimulating, by governmental policies, the dissemination of the AVs. Autonomous driving is not just about the automobile industry but about many other industrial branches that will benefit from a greater degree of automation such as component suppliers. That is why there is a growing interest in promoting local advantages and why political commitments are beginning to be made in some countries to support certain paths of development [START_REF] Schreurs | Autonomous Driving-Political, Legal, Social, and Sustainability Dimensions[END_REF].

In this sense, this study approaches considered that policies issues will act in two stages: 1) to stimulate companies and investments in order to provide technological advances and consequent dissemination of the AVs (current stage of the AVs); 2) development of public policies, or adequacy of laws for AVs. It is understood that this stage will also follow local fators, which would allow an assessment of how the corresponding legal framework should be designed to make a real "road traffic system" involving autonomous vehicles [START_REF] Gasser | Fundamental and Special Legal Questions for Autonomous Vehicles[END_REF] .

Discussion

Figure 5 illustrates the categories developed in this study as well as points to the main issues related to each of them.

TECHNOLOGICAL DISCUSSIONS

As shown on Figure 06, several aspects and macro-environments permeate the AVs. Currently the vast majority of research related to such vehicles, tend to focus on more technical (technological) aspects related to the vehicle's characteristics itself. The thematic categories drawn in this study point to central trends of studies related to other knowledge areas raising important questions on the perspectives and implications of the AVs arrival. Analyzing the framework one can observe that, until AVs can be implemented and actually "cross the chasm" to mainstream there are several legal (policy); ethical and moral issues as well as discussion regarding transportation planning and consumer behavior that need to be addressed. Finally, the business model category points to the need to think about and elaborate new models and market standards for AVs.

Concluding remarks

AVs stand out as a technological innovation that will translate the yearnings of a new era into our society. The field's evolution reflects in a broad conceptual framework regarding AVs (the first search on the subject yielded more than 35,000 papers), however when refining the search for M&B field of study fewer works were yielded, indicating a possible gap between the technological evolution of this innovation and its eventual disruption, considering that the business models plays an extremely important role in the events that precede advancement of AVs [START_REF] Yun | The relationship between technology, business model, and market in autonomous car and intelligent robot industries[END_REF], so that the way in which this innovation might be established by uncertainties, may impact directly on governments' lack of planning for such arrival [START_REF] Guerra | Planning for cars that drive themselves: Metropolitan Planning Organizations, regional transportation plans, and autonomous vehicles[END_REF].

In order to reduce these uncertainties, this work identified five knowledge categories, separated by a thin line, implying in a synergistic orientation among them, which contributes to the field's disruptive appeal. It was observed that businesses models should be consumer oriented, however these should be aligned with society's ethical and moral precepts, as well as with government policies for future transport planning. Namely, the automotive industry must understand the need to establish technological advances that do not neglect social impacts over profitability. In the opposite direction, entities opposed to this advance should keep in mind that such technological evolution is inevitable and they must make efforts to reconcile this advance, which must be added of governmental support to balance the equation.

Finally, it is understood that many uncertainties permeate the theme, and this study sought to identify the academic efforts to foster contributions that could result in a favorable advancement of the field. From this, it is understood that the joint evolution of the categories that permeate the field of the AVs in M&B will be determinant for a positive innovation result that will configure the future of our society in detriment of a disorderly advance that can cause irreparable damages.

Figure 1 :

 1 Figure 1: Research design.

Figure 2 :

 2 Figure 2: Publication Frequency (1990-2017)

Figure 3 :

 3 Figure 3: Countries Frequency' Publications.

Figure 4 :

 4 Figure 4: Word cloud of recurrent key-words on the papers' titles and abstracts

Figure 5 :

 5 Figure 5: Theoretical framework of AVs M&B

Acknowledgments

This work would not have been possible without the citation records of scientific publications from Thomson Reuters' Web of Science and CiteSpace software. We also would like to thank the Post Graduate Program in Administration of the Federal University of Lavras, the Terrestrial Mobility Laboratory (LMT-UFLA), the Laboratoire Génie Industriel (LGI -CentraleSupélec), especially the research group of Sustainable Economy and the funding institutions of this project, Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Fundação de Amparo à Pesquisa de Minas Gerais (Fapemig).


Nicolas Balacheff

A STUDY OF STUDENTS' PROVING PROCESSES AT THE JUNIOR HIGH SCHOOL LEVEL

A study of students' proving processes

I. INTRODUCTION

The teaching of mathematical proof appears to be a failure in almost all countries, no matter how this teaching is organized. One consequence has been to suppress proof as a content to be taught in most curricula; in France the word "démonstration" has disappeared from the statement of the official programs, in USA mathematical proofs are taught only to students who take the geometry course, etc.. In my opinion, such a consequence is drastic insofar as it first of all empties mathematics of its essence, and second because the validity of the mathematical knowledge is fundamentally based on the proof which establishes it.

An analysis of teaching materials indicates that there is too strong an emphasis on teaching the logical side of proof, while its social and practical importance in the mathematical activity remains hidden. It is too often forgotten that mathematical proofs are a means for communication among mathematicians; they play an essential role in establishing the validity of a statement and also in enlightening its meaning. And, as Manin1 recalls: "a proof becomes a proof after the social act of 'accepting it as a proof'." This social nature of mathematical proofs is part of their practical value; it leads to acknowledge them as efficient and reliable tools for the mathematicians.

But, mathematical proof is taught without taking into account that students have had criteria for judging the validity and the relevance of their mathematical statements before being introduced to this new piece of knowledge. For Most students mathematical proofs appear ultimately to be a kind of rhetoric specific to the mathematical classroom [START_REF] Balacheff | Preuve et démonstration en mathématiques au Collège[END_REF]. They produce it because the teacher demands it, not because they recognize it as necessary in their practice; as a British student told to an interviewer: "to prove something in maths means that you have worked it out and it proves how good you are at working questions out and understanding them" [START_REF] Galbraith | Pupils proving[END_REF].

Then, the aim of our research is to identify the foundations of students' belief in the validity of a statement in their mathematical activity: what they recognize in practice as a proof, and how they treat a refutation. We have focused this study on the relationships among the students' proving process, the knowledge they have at their disposal, the language they can use and the role of the situational context.

II. COGNITIVE AND SOCIAL ASPECTS OF PROOF II.1 PRECISION IN TERMINOLOGY

In English, as well as in French, two terms are used in mathematics in a synonymous way: "proof" (French preuve) and "mathematical proof' (French démonstration). This custom presents an obstacle to our study, for it hides different levels that should be differentiated. Thus, we propose starting with the following distinctions:

-We will use the term explanation to describe the discourse of an individual intending to establish for somebody else the validity of a statement. The validity of an explanation is initially related to the speaker who articulates it. -We will use the term proof to refer to an explanation which is accepted by a community at a given time; -We will designate as a mathematical proof a proof accepted by mathematicians. As a type of discourse, mathematical proofs nowadays have a specific structure and follow well-defined rules that have been formalized by logicians.

II.2. COGNITIVE ASPECTS OF PROOF

In Such proofs rely upon the capacity of the person observing the schema to reconstruct the reasons that are embedded in it but not formulated.

At a higher level, reasons could be expressed but still strongly related to actions performed on some example. Insofar as the example is viewed not as a particular case but as a representative of a class of objects, we will recognize this as a proof by generic example. Such a proof requires the generality to be viewed beyond the particular. Let us give an example taken from Bezout (Notes sur l'arithmétique, 1832 p.23, our free translation):

The remainder of the division of a number by 2x2 or by 5x5 is the same as the remainder of the division by 2x2 or by 5x5 of the number written with the two last digits on the right of this same number [ ... ] Let us consider the number 43728 and the divisor 5x5. The number 43728 equals 47700+28. But 43700 is divisible by 5x5, because 47700 is the product of 437 by 100, and 100 equaling 10x10, or 5x2x5x2, or 5x5x2x2, the factor 100 is divisible by 5x5. The remainder of the division of 43728 by 5x5 or 25, is the same as the one of 28 by 25.

We call pragmatic proofs those proofs which rely upon action, and we call intellectual proofs those which use verbalizations of the properties of objects and of their relationships. This step towards intellectual proof does not consist in a mere translation of action into words; it requires a genuine construction of language means as an operative tool. The problem-solver must be able to use language and symbols as means to compute on statements and relations.

At the highest level, in mathematics, mathematical proofs require a specific status of knowledge which must be organized in a theory and recognized as such by a community: the validity of definitions, theorems, and deductive rules is socially shared.

II.3. SOCIAL ASPECTS OF PROOF

As Popper (1979, p.78) emphasizes it: "the 'certainty' of a belief is not so much a matter of its intensity, but of the situation: of our expectation of its possible consequences. Everything depends on the importance attached to the truth or falsity of the belief." In other words, we will say that involvement at a given level of validation is a question of economy of logic "which wants that one does not bring into play more logic than what is necessary for practical needs" (Bourdieu 1980 p.145, our free translation).

Thus, the fact of presenting a problem to students does not guarantee that they will be committed to producing a proof; this is due not to a fundamental lack of awareness but to the fact that their reading of the situation does not call for producing a proof. In our research, therefore, we have paid much attention to providing students with a context promoting awareness of the need for proof -i.e. a context which holds some risk linked to uncertainty, and therefore something to gain by entering a proving process.

II.4. THE DIALECTIC OF PROOFS AND REFUTATIONS

A well-known interpretation of a counterexample in the mathematics classroom is that of a sort of catastrophe which implies the definitive rejection of what has been refuted. From this point of view the mathematics classroom ideology is more Manichaean than dialectic. Analysis of the activity of the mathematician suggests a quite different and, at least, less radical functioning. The decision on the validity of a proof depends on the quality of its critical analysis, which finally guarantees the absence of logical mistakes and of counterexamples. In this regard the proving process is based on the commitment of the problem-solver to take into account the possible existence of contradictions: the proving process is fundamentally dialectic. That is even more obvious in the context of social interactions, where tentative explanations, or refutations, of a given statement are elicited.

To take into account this dimension of the proving process, we have adopted the model of the dialectic of proofs and refutations proposed by Lakatos (1976). We should mention here that this model is consonant with the theories developed by the Piagetian school which has shown the central role played by contradiction in the genesis of cognitive structures. Using this model we can differentiate the implications of a counterexample, depending on whether we are considering the conjecture, its proof, or the related knowledge and rationality of the problem-solver himself.

This scheme showing the conjecture and its proof as the product of both the knowledge and rationality of a subject summarizes the main possible consequences of a counterexample, Actually, it merely evokes the range of these possible consequences, but it is sufficient to give an idea of what we call the openness of the treatment of a refutation.

The nature of the development of the mathematical knowledge as described by Lakatos suggests a question which Lakatos did not pose 2 but which is nonetheless essential for the teacher or researcher in mathematics education: what determines the appropriateness of a choice for overcoming the contradiction brought by a counter-example?

We have investigated this question together with the question of the nature of students' proving processes.

III. TYPES AND HIERARCHY OF STUDENTS' PROOFS III.1. AN EXPERIMENTAL APPROACH

In order to explore students' proving processes, we have used a situation of social interaction which encourages confrontation of differing views on the solution of a problem and hence a verbal exchange which make those views explicit.

Pairs of 13-14 year-old students were required to solve the following problem:

Give a way of calculating the number of diagonals of a polygon once the number of vertices is known.

The answer to this question was to be expressed in a message addressed to, and to be used by, other 13-14 year-old students. The two students have access to as much paper as they want but to only one pencil. This constraint reinforces the co-operative nature of the situation; at the same time it gives us more direct access to the dynamic of the two confronted knowledge systems, especially in cases of decision making. The observer intervenes only after students claim that they have produced a final solution; at this stage he abandons his stance of neutrality and asks the students to deal with counterexamples that he proposes.

The chosen problem refers to student knowledge that is more cultural than scholastic, for while polygons may have been studied in the primary school (in activities related to geometric classification), they are no longer a part of the curriculum at the level with which we are concerned. As in the case of solids in the eighteenth century, students' conceptions about this mathematical content are not theorized. The context, therefore, is conducive to the emergence of processes resembling those described by Lakatos, so it permits us to analyze the dynamic of proving, the related treatment of refutations, and their relation to concept building.

In this experimental situation:

-The kind of communication encouraged structures the students' activity, and, more particularly, promotes a verbal formulation of the counting procedure. This is something that students do not normally do straight away, even if they are technically capable of it. At the same time, the desire to supply a reliable tool to the other group is likely to lead the pair of students to pay more attention to the formulation and the validity of its solution. -The social interaction, through the exchanges it requires from the pair of students, helps elicit the students' conceptions and plans, and the basis for their decisions independently of any observer's intervention. The constraints of using only one pencil oblige a permanent confrontation and the elicitation of the choice of common criteria to accept or refuse any statement or strategy.

We have observed fourteen student pairs, each session lasting about 90 minutes. All sessions have been recorded. From these data, we made the analysis underlying the results we present here (a complete report on this research is presented in [START_REF] Balacheff | Une étude des processus de preuve en mathématiques chez des élèves de Collège[END_REF].

III.2. HIERARCHY AND INTERRELATION OF THE TYPES OF PROOFS III.2.1. NAIVE EMPIRICISM

The origin of naive empiricism can be traced in two quite different phenomena: factual evidence and cognitive belief3 . "The term 'belief' expresses the direct, the sympathetic, form of knowledge, the feeling of the implicit validity and reliability of the respective representations or interpretations and their extrapolative capacity." (Fishbein 1982, p.ll).

In the first case, the level of proof evidences a pragmatic empiricism which allows students to consider mere observation as sufficient. Such behavior could come from conflict between students or from a reading of the situation that leads them to prefer quickly submitting their solution to the observer instead of trying to enter a proving process on their own. It is for example the case of two students we observed, Pierre and Mathieu4 , who proposed a solution to the observer but telling him that "it's a game... it's better to try something" (even if they also claimed that it is of no use "to try anything").

The second case is quite different, as it relies upon a real belief in the validity of the proposed solution. This belief is strongly related to the students' conceptions which they are unable to express or analyze: "an intrinsic type of conviction, directly imposed by the structure of the situation itself" (Fishbein, 1982, p.ll, situation means here mathematical situation). By way of illustration we offer the following excerpt of one of our case studies:

Pierre and Philippe5 induce from the observation of polygons with 6 and 8 vertices the following numbers of diagonals: 6 x3 and 8x5 (despite the fact that this is contradicted by their drawings). Their conjecture, then, is that the number of diagonals is equal to the number of diagonals from one vertex multiplied by the number of vertices, but the students say, "we don't know how to explain it". When the observer proposes the case of a polygon with nine vertices, they compute 9x6 and say: "They are 54, it is the same procedure". Their belief in the validity of this solution is based on the evidence that the number of diagonals from each vertex of a polygon is the same. But they are unable to justify their solution, or the statement that it would be sufficient to multiply this number by the number of vertices.

We should mention here that we have done the same experiment, but provided students with a document presenting the definitions of polygon and diagonal together with some figures. This document was quite similar to what they could encounter in a text book. In this situation students offered more correct solutions to the problem, but in most cases, with a naive empiricist foundation. In the new context the examples used by students to check their solutions were examples taken from the given document. The students understood them as prototypes, so that no further testing appeared to be necessary, which is clearly stated by some of the students6 : "We will look whether it always work, on all the polygons ". This kind of naive empiricism, which we would like to call a prototypical effect, is essential from the didactical point of view, as it calls under question the use of examples for teaching purposes.

Examples are indeed helpful for teaching purposes, but it appears that from the student point of view they can have a status which is likely to tum into an obstacle to proving processes.

III.2.2. THE CRUCIAL EXPERIMENT

The crucial experiment is an important step, as it identifies awareness of the problem of the validity of a mathematical statement, taking into account the problem of generalization. Its origin could be in the awareness of the insufficiency of a mere verification on few examples, but within cognitive and language limits which do not allow the student to go beyond.

A good example is the case of the solutions of the type : f(n)=(n-3)+(n-3)+(n-4)+ ... +2+1 . This requires expressing an iteration, which the students we observed were unable to do. Here is the solution formulated by Martine and Laura7 :

First vertex: number of diagonals = number of vertices -3 Second vertex: number of diagonals = the same From the third vertex: number of diagonals just obtained -I diagonal Forth vertex: number of diagonals just obtained -I diagonal and so on -at the end we add all the numbers of diagonal at each vertex to find the number of diagonals of the polygon

The two students decide "to draw a very large figure to verify", which they do with a polygon with ten vertices. Let us take another example which gives clear evidence of this behavior, Nadine and Elisabeth8 produced a solution which consists in a recurrent formula:

They have found nine diagonals for a polygon with six vertices, then announced for a polygon with seven vertices: "we should add five, normally ... if it works, then normally with seven we will find fourteen diagonals"; which is confirmed by their experiment. But to accept the solution definitively they relied on a crucial experiment. As they stated it: "try once with fifteen vertices, then if it works that means that it works with any other number". Actually, the experiment was done with a polygon with ten vertices: "so normally... with ten sides we should find thirty-five diagonals". The conjecture is then accepted as true.

In contrast to the naive empiricism which disappears when students reach the level of intellectual proofs, the crucial experiment remains as an ultimate means to ground the students' conviction, especially in the case of a proof relying on a generic example. This is an example of the operative coexistence of empirical pragmatism and logical rationalism. It confirms the thesis of [START_REF] Fishbein | Intuition and Proof[END_REF], who claims that these are two types of rationality with different practical value: «the two basic ways of proving -the empirical and the logical -are not symmetrical; they do not have the same weight in our practical activity» (ibid. p.17).

The crucial experiment takes quite a different significance in social interactions when it becomes a means of defusing a conflict about a statement between the two students. It is then no longer a genuine tool for proving. It supports one position against another without effectively asserting its validity. We note that in such circumstances, there were other explanations of the statement in the student pair. It is the case for Christophe and Bertrand 9 : Christophe does not accept the solution proposed by Bertrand. He claims that the number of diagonals is twice the number of vertices, like for P 7 the example he considered. Instead of explaining his solution (what he could have done, see this solution below) Bertrand proposed to Christophe to try with P 8 and see... after the observation of this case Christophe surrenders and then accepts the solution of Bertrand. The example P 8 played exactly the role of the Bacon "crucial experiment" which result allow to make a choice between two possibilities.

III.2.3. THE GENERIC EXAMPLE AND THE THOUGHT EXPERIMENT

The thought experiment can be seen in the linkage of proofs relying on a generic example, throughout a process of de-contextualization which requires eliminating the particular. This process does not occur only at the level of language; it requires cognitive constructions of great complexity to elicit the objects involved in the proof and their relationships. This complexity is due to the fact that very often the thought experiment relies on intermediate steps at a lower level of proving.

From the thought experiment the process of de-contextualization, together with de-temporalization (obliteration of time, in a movement from dynamic to static) and that of de-personalization (obliteration of the actor), can develop towards proofs which consist in a real computation on relations. This has been observed very seldom, since it requires powerful language means and demands the knowledge to be theory-like.

The following example shows the transition from a proof by generic example to a thought experiment:

Christophe and Bertrand10 deduced from a close analysis of a polygon with seven vertices that the number of diagonals to each vertex follows the pattern n-3. The following account they produced shows they arrive at this:

If we have a polygon with 7 vertices, each point has two neighbors, and from this point starts:

7 -( 2 + I ) We do that for all points.

Number number the point of vertices of neighbors

Their formulation reflects the reasoning behind their solutions, but this occurs at the level of action and not really at that of computation involving relations:

Knowing the number of vertices of a polygon, it starts from each point the number of vertices -(its two neighbors + itself). What has been obtained then should be multiplied by the number of vertices (from each vertex starts the same number of diagonals). But we count twice each diagonal: the obtained number of diagonal is thus to be divided by two to find the number of diagonals.

III.3. THE TREATMENT OF REFUTATIONS 11

The analysis of students' treatment of refutations reflects most of the possibilities described by the model of Lakatos.

The question on which we have focused is that of criteria of students' choice among all these possibilities.

Three types of analysis appear to determine the choice of treatment of a refutation:

-Analysis with reference to the problem itself. This type of analysis gives a central place to discussion of the nature, and thus the definition, of the objects involved in the problem. Potentially, it may lead to any of the possible types of treatment, none of which is privileged a priori. The choice students make can be understood only through local analysis of the problem-solving process or of the specific characteristic of each individual. The type of treatment can change in the course of the problem-solving process, for example, the students decision to modify the definition can be followed by that of introducing a condition or of modifying the initial conjecture when their conceptions have been stabilized. But the origins of the choice to treat the counterexample by introducing a condition or searching for a specific solution, or modifying the conjecture, cannot be traced with the data we have gathered. All we can suppose is that when a refutation might eliminate a wide range of polygons (with respect to students' conceptions ), students prefer to modify the conjecture by adding a specific solution for the objects referred to in the counterexample (extension to odd polygons in case of f(n)=n/2, search for a solution for non-convex polygons, etc.).

-Analysis with reference to a global conception of the nature of mathematics. This could be a serious obstacle to some responses to a refutation and could lead to refusal to treat the counterexample as an exception, rejection of a solution which cannot be expressed by a unique formula, etc.

-Analysis with reference to the situation. This is essentially a question of the didactical contract 12 which leads students to favor certain treatments of the counterexample (for ex ample the students' reading of the situation as a «definition game» in which they shift from a definition to another to escape the counterexample) while it raises obstacle to others (refusal to introduce a condition because it has not been stated in the problem statement, cf. the example given below).

The following brief account of one of the case studies shows how these different analyses play a role in students' decision:

Evelyne and Christine 13 consider the solution "the number of diagonals of a polygon is half the number of its vertices", first on the basis of a naïve empiricism, and then on that of a thought experiment. The thought experiment is related to the conception of a diagonal they make explicit after a refutation by a polygon with seven vertices: "a diagonal is a straight line which belongs to a vertex of a polygon and which cuts its surface in two pieces ". But they hold different points of view: one of them would like to introduce a condition in the statement of the conjecture; the other would prefer to reconsider the definition. Finally, it is the latter that is chosen after Evelyne argues that "it could not be that because there [in the statement of the task] they don't tell that a polygon is... the number of vertices is even". In other words, students have not the right to restrict the set of polygons which fit the conjecture; they have to give a solution for all of them. Actually, the definition they choose restricts the set of the polygons to those for which the conjecture is valid: "if we say that a polygon is... a thing whose 11 For a complete report on this aspect of the research see : N. Balacheff, "Treatment of refutations : aspects of the complexity of a constructivist approach of mathematics learning". 10 appear in E. Von Glasersfeld (Eds.) : "Constructivism in mathematics education", D. Reidel Publishing Company 12 ln our study, the experimental contract 13 Balacheff 1988 p.182, pp.192•194, pp.221•223 edges are always parallel two by two... then it is not necessary to be more precise. We need only to divide by two ". But the case of a triangle is not treatable under this definition. The triangle is rejected as a polygon because it has no diagonal. In the eyes of the students, it is a kind of monster. Later on, a counter-example (a polygon with five vertices), produced by the observer imposes itself as a polygon. Then the initial definition is rejected and a new one is considered: "for sure a polygon can have any number of diagonals... but it should be regular". So the only way to save the conjecture is to introduce a condition and to search for a solution specific to the odd polygons.

This last solution starts from a student's conception of a diagonal as an "axis of symmetry": a diagonal is a line which passes through a vertex of a polygon and cuts it into two equal pieces.

In case of a polygon with five vertices it corresponds to the students' drawing given above. The solution they conjecture is f(n)=n for odd polygons. They first envision proving it by means of a crucial experiment with a polygon of eleven vertices, but as it appears to be too complex, they prove it "by reasons": "it is obvious that it is 11, as there is 1 in each vertex ". The uncertainty which remains is an uncertainty in the students' premises: it involves the definition of a polygon and of a diagonal. It is on the basis of these definitions that they then treat the counterexamples produced by the observer. These definitions are even formulated in their message:

If the number of vertices of the polygon is even: you divide the number by two and you will obtain the number of its diagonals If the number of vertices of the polygon is odd: the number of diagonals is equal to the number of vertices A polygon is a geometrical figure which can have any number of vertices but whose edges must be equal

Finally, we have examined the question of a possible influence of the type of conjecture on the choice of treatment of a refutation. An initial hypothesis is that if the conjecture is false, then its rejection or modification, or the revision of the definition, should be dominant; on the other hand, if the conjecture is correct, then the rejection of the counterexample should be dominant. Actually, we have observed that conjectures like f(n)=n or f(n)=2n are abandoned after their refutation. But such conjectures are very fragile, insofar as they are verified only by one polygon. It is quite different when the conjecture is verified by a large set of polygons, like the false conjecture f(n)=n/2 whose strength comes from the fact that it is related (explicitly or not) to a conception of a polygon as a regular polygon and a diagonal as a diameter. For these conjectures no type of treatment appears to be favored. Even the type of foundation of the conjecture has no effective influence on the students' choice of treatment of the counterexample. For example, an ad hoc modification of the conjecture (what is indeed at the same level as the naive empiricism) can follow a counterexample whereas the foundation of this solution was at a higher level like that of the thought-experiment. It is the case of Lionel and Laurent14 who first established the solution f(n)=n(n-3), and then modified it to f(n)=n(n-3)/2 only because they noticed dividing by two as a relationships between the number of diagonals (5) of the counter-example they examined and the number they expected (10).

ln the case of correct conjectures, whose construction refers to a "correct" conception of a polygon and a diagonal, whether they have been constructed deductively or as the result of a dialectic between successive attempts and their refutations, the treatment appears to be far less varied than in case of a false conjecture. One prevailing type of response is rejecting the counterexample after its analysis with reference to students' conceptions; a second dominant type of response entails considering the counterexample as an exception or introducing a condition (actually, the latter appears to be a way to avoid recognizing exception).

IV. CONCLUSIONS AND QUESTIONS ABOUT TEACHING

The type of proving processes students evidence do not intrinsically characterize what could be called their "rationality", in that different levels of proof could be observed in their problem-solving activity. The meaning of proving processes cannot be understood without a careful analysis of students' conceptions of the mathematical concepts involved and their reading of the situation within which they act. The characteristics of the situation appear to determine the level of proving, while the image students have of mathematics also plays an important role, particularly in their treatment of refutations.

The passage from pragmatic proofs to intellectual proofs requires a cognitive and linguistic base. Our disregard of the complexity of this passage could be one of the main reasons for the failure of the teaching of mathematical proof, since this passage is very often considered only at the logical level. In geometry in particular, this teaching takes place in a conceptual field which for students is not yet constituted as a theory; we should not forget that geometry was for them essentially restricted to observation and construction of geometrical objects with no requirement for proof. So the teaching of proof is associated to what could be described as a cognitive break in the student activity, related to the didactical break represented by new requirement for mathematical proofs.

Thus, we would like to say that the construction of the students' mathematical rationality should he considered at the same time and with the same priority as the construction of their mathematical knowledge. It might be possible to propose proving problems from the beginning of mathematics learning, provided that something other than strict mathematical proofs were acceptable. For that, we must take into account students' reasoning skills and consider the didactical conditions for their evolution. But when a proof has been accepted, the validity of the statement concerned is difficult to question later ; therefore, the teacher acceptance of a proof which is not a mathematical proof raises the problem of its eventual recurrence in a form which must appear reasonable to students.

The case of refutations raises specific didactical problems. We have shown the breadth of students' treatment of a refutation and the variety of rational for this treatment. So how can we deal with the fact that when the teacher produces a counterexample, students believe that it is a particular case, whereas what should actually be questioned is the naïve empiricism on which their conjecture is based or their understanding of the related mathematical knowledge?

If there is no strict cognitive determinism involved in the way a contradiction shown by a counterexample could be overcome, then what role should he played by the situation ? The interventions of the teacher -the way the interactions with students are manage -will be essential to helping students realize that a major goal is examining their knowledge, or the reasoning behind their conjecture, and not a mere ad hoc adaptation of their solution or even its total rejection.

  accordance with what cognitive psychology tells us, the most elementary type of proof consists in merely showing, what Semadeni (1984) calls proof in action. Here is a classical example of such a proof of the property: "the sum of the n first odd numbers is n 2 ":

Manin quoted byHanna 1983, p.71

This expression has been coined by[START_REF] Fishbein | Intuition and Proof[END_REF]; we render it in French by évidence de raison.

Balacheff 1988 p.I08

Balacheff 1988 p.109

Ba1acheff 1988 p.250. What is meant here is: all the polygons proposed by the document.

Balacheff 1988 pp.98. 121

Balacheff 1988 pp.122-123

Balacheff 1988 p .118

Balacheff 1988 p.138-141

Balacheff 1988 p.170


C Fournier

B Andrieu

S Ljutovac

S Saint-Jean

ADEL-Wheat: a 3D Architectural Model of Wheat Development

Keywords: Virtual Plant, Wheat, L-System, Architecture, Tiller, Coordination, Development, Elongation Kinetic, Emergence

A 3D architectural model of winter wheat development is presented. The model simulates the kinetics of elongation of individual organs, together with their geometric shape. With the inclusion of two additional parameters, it was possible to model the development of tillers using the same parameter values as for the main stem. The duration of organ extension was found to be the same amongst phytomers, and the kinetics of extension of any organ could be simulated from its final length and two parameters. Finally, lengths of mature organs were found to follow simple functions of phytomer position. The resulting model simulates development of the whole plant with a small number of parameters, characterising the size and geometry of mature organs and the timing of collar appearance.

Introduction

Since the pioneer work of [START_REF] Goel | From artificial life to real life : computer simulation of plant growth[END_REF], on maize, virtual plant models have gained much interest in the context of crop modelling, as shown by the increasing number of species covered: maize [START_REF] Goel | From artificial life to real life : computer simulation of plant growth[END_REF][START_REF] Fournier | A 3D architectural and process-based model of maize development[END_REF], sorghum [START_REF] Kaitaniemi | Architecture and morphogenesis of grain sorghum, Sorghum bicolor (L.) Moench[END_REF], cotton [START_REF] Jallas | Virtual cotons, the firstborn of the next generation of simulation model[END_REF], faba bean [START_REF] Díaz-Ambrona | Faba bean canopy modelling with a parametric open Lsystem: a comparison with the Monsi and Saeki model[END_REF], barley [START_REF] Werneke | Combining process-with architectural models: The simulation tool VICA[END_REF], and white clover [START_REF] Gautier | 3D architectural modelling of aerial photomorphogenesis in white clover (Trifolium repens L.) using L-systems[END_REF], among others. Compared to usual crop models, virtual plant models focus on plant rather than on canopy functioning, and the interactions between plants are considered as resulting from the changes in local conditions encountered by individual organs due to neighbouring ones. For instance the local phytoclimate of any organ can be calculated explicitly when the 3D structure of a plant population is described [START_REF] Chelle | The nested radiosity model for the distribution of light within plant canopies[END_REF], so that competition for light can be simulated dynamically by interfacing a individual-based model of a plant population, and a light model [START_REF] Fournier | ADEL-maize: an L-system based model for the integration of growth processes fom the organ to the canopy. Application to regulation of morphogenesis by light availability[END_REF]. This approach results in higher genericity of plant models, allowing on one hand an easier link with genomics [START_REF] Tardieu | Virtual plants: modelling as a tool for the genomics of tolerance to water deficit[END_REF], and on the other hand the modelling of cropping systems having a marked level of heterogeneity (mixture of varieties, high-density canopies, intercropping systems...). Here, we present the first version of such a model on wheat. A virtual plant model should incorporate a parameterisation of development that allows for a detailed simulation of the evolution of the plant 3D structure, with a tractable number of parameters. A useful starting point for this is to quantify the degree to which a plant can be considered as a collection of metamers behaving similarly. In the case of wheat, which produces tillers, this question is addressed here at two levels (i) do tillers reproduce the development of main stem? And (ii) do phytomers (elementary units of a plant comprising a node, its axillary bud, an internode and a leaf composed of a blade and a sheath) behave similarly along an axis? We will focus here on major traits of vegetative development, which are: the timing of initiation and extension of organs, and the geometry of mature organs. As it is well recognised that the rate of plant development depends on temperature ( e.g. [START_REF] Kemp | The responsiveness to temperature of the extension rates of leaves of wheat growing in the field under different levels of nitrogen fertilizer[END_REF]), we expressed kinetics in thermal time units.

Materials and Methods

2.1

Experiments Two field experiments were organised at INRA Grignon [START_REF] Ljutovac | Coordination dans l'extension des organes aériens et conséquence pour les relations entre les dimensions finales des organes chez le blé[END_REF], both with winter wheat (c.v. Soisson). Experiment 1 focused on kinetics of development and organ extension, whereas experiment 2 focused on the plant 3D geometry.

In the first experiment, wheat was sown on October 15, 1998. Measurements started on December 7, when the plant had three fully developed leaves and finished on June 6, 1999 when plants had achieved architectural maturity. Plants were grown under non-limiting conditions of water and nutrients, and were kept free of disease and weeds by fungicide and herbicide applications. The experiments included two density treatments: 250 pl./m², corresponding to a usual agronomic density, and a low-density treatment (70 pl./m²) to allow for the development of additional tillers. Meteorological data (air temperature at 2 meters, global radiation) were registered by a Stevenson screen, 500 meters from the experimental field. Additionally, soil and organ temperatures were monitored directly by thermocouples inserted in the soil and in plant organs. These data were used to compute the thermal time at each sampling date, using a base temperature of 0°C, widely accepted for wheat. Biological data were monitored from observation and dissection of both main stems and tillers every two or three days. The data set includes: (i) developmental state of the shoot apex (number of primordiums, time of double ridge stage, and of terminal spikelet initiation); (ii) kinetics of the lengths of leaf laminae, sheaths, and internodes; (iii) final size of all organs.

In the second experiment, wheat was sown on October 15, 1999 at a density of 250 pl./m². Plants were grown in the same way as in experiment 1, in water and nutrient non-limiting, pest-and disease-free conditions. To account for senescence of basal leaves that occurs before full plant development, the geometry of thirty plants was characterised at two periods. The first measurement period was when leaves 1 to 6 were fully developed and no leaves had senesced; the second measurement period was after heading, when stems were fully extended and leaves 7 to 12 were fully developed. Measurements were performed for the main stem and for all tillers as follows: The insertion height of all leaves, and the shape and orientation of the midrib were measured using a 3-D magnetic digitiser; the 2-D shape of the leaves (leaf width as a function of distance to tip) was also measured. Altogether, the data set includes the 3-D shapes of approximately 1000 leaves, representing all axes and leaf positions. This provided the data for parameterising the shape of the organs.

2.2

Modelling Adel-wheat was implemented using the L-system formalism [START_REF] Lindenmayer | Mathematical models for cellular interaction in development, I and II[END_REF][START_REF] Prusinkiewicz | The algorithmic beauty of plants[END_REF][START_REF] Prusinkiewicz | L-systems: from the theory to visual models of plants[END_REF], which provides facilities for implementation of virtual plants models. Here we use the software Graphtal [START_REF] Streit | Graphtal user manual[END_REF], available for free on the internet (ftp://ftp.unibe.ch/pub/Graphics/Modelling). Only minimal information on L-System modelling is given here. One can refer to [START_REF] Prusinkiewicz | Modeling of spatial structure and development of plants: a review[END_REF] for more details on this approach, and to [START_REF] Fournier | A 3D architectural and process-based model of maize development[END_REF][START_REF] Fournier | ADEL-maize: an L-system based model for the integration of growth processes fom the organ to the canopy. Application to regulation of morphogenesis by light availability[END_REF] for an example of an L-System maize model built similarly to the current project.

In the L-system formalism, the plant is represented as a string of words and characters, some representing the plant organs (e.g. leaves), hereafter called modules, the others coding for topological or geometrical links between modules. Each module has associated parameters, which allow its state (e.g. present length), or geometric characteristics, to be coded. Plant development is coded by a set of substitution rules (the productions), which are applied simultaneously to the whole string. For example the initiation of phytomers by the apex (fig 1) is coded by a substitution of the apex module by an internode module, a leaf module and an apex module with updated parameters. A key feature of the formalism is that the string coding for the plant includes a code that is interpreted through a dedicated step, to produce a 3D representation.

Results

3.1

Plant development Plant development could be described simply as the repetition of three sequences of events (Fig. 1). The first sequence occurs at the top of each axis, on a dedicated structure, the apical meristem, on which groups of founding cells successively isolate (initiums), each of it will give rise to a whole phytomer. This sequence interacts with a phase change in the apex, so that lower initiums develop into vegetative phytomers (node, internode, bud, leaf), and higher initiums develop into reproductive organs (spikelets). To simplify this complex behaviour, only the initiation of vegetative phytomers was considered in the model, and was parameterised in terms of the rate of production of initiums and of the total number of phytomers produced. The second sequence is at the level of the vegetative phytomers and says that its components successively enter in a rapid stage of elongation: the leaf blade elongates first, followed by the sheath and then the internode. Simulation of this rapid elongation stage requires information, for each of these components, on the timing of the onset of rapid elongation, and the rate and duration of rapid elongation. We choose to define the timing of rapid elongation based on the emergence of leaf collars, as there exists a striking synchronism observed experimentally, suggesting that collar emergence is involved in a coordinating process (see below). The third sequence is for the axillary bud, which, after a phase of meristematic development, will either remain dormant (and later possibly abort) or produce a new axis behaving similarly to the mother stem. In the model, these events were coded as a phase of latency, followed by either deletion of the module or its transformation into an apex module. We do not consider here the mechanisms leading to each of these two fates, so the number of buds giving rise to a tiller is a parameter (Nt) of the model. Besides the timing of phytomer initiation on all axes, and the number of tillers, the only additional parameter needed to describe the development of a given tiller is the duration of bud latency phase.

To summarise, the development is parameterised with the timing, for all axes, of three developmental events: phytomer initiation by the apex, collar emergence, and end of bud latency phase. Phytomer initiation was found to be a linear function of thermal time for all axes, and parameterised with one parameter, its rate ( plastochron). Collar emergence was also found to be a linear function of thermal time for all axes, thus defining a second parameter (phyllochron). The delay between emergence of the collar of the first leaf on a tiller and that of the collar of the axillary leaf on main stem was approximately 3 phyllochrons, but we prefer to keep explicitly one parameter per tiller. Finally, the duration of the bud latency phase was reasonably approximated by a constant duration of four plastochrons for all axes. Fig. 2. Normalised kinetics of extension of leaves of wheat grown at 250 pl./m². Left panel is for the different phytomers of main stem (numbered acropetally). Right panel is for the different axes: main stem (bm) and three tillers (t1 to t3). On the right panel, for each axis, data are running-medians averages of all phytomers. Length is normalised with respect to final leaf length and time is counted, for all phytomers, from x0, which is the point of intersection of a linear fit of the data with the abscissa.

Phytomer extension

The kinetics of elongation of either leaves or internodes were found to be very similar between axes and between phytomers (Fig 2). Thermal time durations of elongation were the same for all leaves, and for all internodes in the plant, but differed between leaves and internodes. One other striking feature of the elongation curves is the stability of coordination with collar emergence. Collar emergence of phytomer n was found to be strikingly close to the onset of internode elongation (and thus close to the end of leaf elongation) and to the onset of leaf elongation on phytomer n-2. For leaves, as collar emergence rate was constant here, this is equivalent to the observation of a constant duration. However, similar synchronisation patterns with emergence events are also observed on other species [START_REF] Durand | Modelling the elongation of successive leaves on grass tillers. Effects of temperature, cutting height and cutting frequency[END_REF][START_REF] Fournier | Dynamics of the elongation of internodes in maize (Zea Mays L.): Analysis of phases of elongation and their relationships to phytomer development[END_REF][START_REF] Lafarge | A model co-ordinating the elongation of all leaves of a sorghum cultivar was applied to both Mediterranean and Sahelian conditions[END_REF]. This could be interpreted by considering that collar emergence was a trigger of organ extension [START_REF] Malvoisin | Organogenèse et croissance du maître-brin du blé tendre (Triticum aestivum) du semis à la floraison.I.-Relations observées entre la croisssance folaire et la différenciation des ébauches foliaires ou florales[END_REF][START_REF] Skinner | Epidermal cell division and the coordination of leaf and tiller development[END_REF][START_REF] Fournier | Dynamics of the elongation of internodes in maize (Zea Mays L.): Effects of a shade treatment on elongation patterns[END_REF]. Therefore, we prefer here to define a constant elongation duration for all internodes and to calculate all other extension parameters (onsets of rapid elongation of leaves and internodes, and duration of leaf extension) using these coordination rules. This would (to be tested) make the model more robust in the case where collar emergence rate varies during development.

3.3

Final lengths of organs The final length of organs varied with position on all axes, but followed similar patterns among axes. More precisely, shifting the numbering of phytomers from a given (optimised) offset, depending on the axis number, makes final lengths follow the same function of phytomer number (Fig 3). Moreover, the shift was the same whatever the type of organ (blade, sheath or internode). This allows defining three functions describing respectively, blade, sheath and internode length as a function of phytomer position on any axis, with only one axis specific parameter (the shift). In total, the number of parameters required to describe the mature length of all organs of a plant is 10 + Nt.

3.4

Leaf width Leaf width varies as a function of the curvilinear abscise counted from the base of the blade along the midrib. However, the function describing these variations appears to be the same for all leaves, when the distance is normalised by blade length and leaf width normalised by maximum leaf width [START_REF] Piègle | Réalisation de maquettes informatiques de blé en trois dimensions, Mémoire de stage[END_REF]. Similarly to [START_REF] Prévot | Modélisation de la structure géométrique du maïs[END_REF], we used a second order polynomial to model the leaf width as a function of maximum leaf width. Maximum leaf width was found to be well correlated to either sheath length or blade length (depending on the rank), and the same allometric relationships were found for all axes. This allows, with the use of four additional parameters, modelling of the leaf width.

Plant Geometry

Geometry was analysed for azimuth of leaves and curvature of leaf midribs. The analysis of midrib curvature was performed similar to [START_REF] Prévot | Modélisation de la structure géométrique du maïs[END_REF] for maize leaves: the midrib is assumed to be in one plane (no twisting is taken into account) and is described by a parabolic arc for the ascending part and a elliptic arc for the descending part. Two categories of leaves were found, for all axes: basal leaves and the 3 topmost leaves (Fig 4). Basal leaves are almost always strictly ascending and followed an opposite phyllotactic pattern, whereas top leaves have both an ascending and a descending part, with both opposite and spiral phylotactic pattern and with a narrower phyllotactic angle. This permitted the establishment of two sets of distribution functions for the representation of leaves in 3D.

3.6

Simulation of 3D Development Altogether, 29 parameters were used to simulate wheat development, with the addition of two sets of tables for the geometry of leaves. Simulations were realistic (Fig 5), and we also checked that the model correctly reproduced important agronomic features such as kinetics of LAI and of plant height.

Discussion

The first aim of this work was to obtain an operational 3D model covering the principal aspects of wheat development. Because relatively few data are published with the required level of detail, this first version relies mainly on the analysis of a single experiment, corresponding to one and two densities. Moreover, not all aspects were studied with the same level of detail, and thus are not expected to have the same degree of genericity. To a large extent, the geometric description of organs (leaf width and midrib curvature) is thought to be only empirical, and expected to vary according to genotype-environment conditions. It is expected to hold generally that dimensions of components of a given phytomer on one hand and the length of organs of successive phytomers on the other hand are highly correlated. This is because our data strongly support the idea that the length of organs are to a large amount controlled by the length of the whorl at the time of elongation [START_REF] Dobrynin | Lois régissant la formation de la pousse chez certaines graminées[END_REF][START_REF] Wilson | The role of the sheath tube in the development of expanding leaves in perennial ryegrass[END_REF][START_REF] Casey | Effects of sheath tube length on leaf development in perennial ryegrass (Lolium perenne L.)[END_REF][START_REF] Fournier | Dynamics of the elongation of internodes in maize (Zea Mays L.): Effects of a shade treatment on elongation patterns[END_REF]. That such correlations can result in a rigid parametrisation of the variation of organ lengths with position is based on the idea that the relation between length of successive phytomers reflects a developmental gradient, expressed similarly on all axes (but with an axis-specific delay ). Published data (e.g. [START_REF] Gallagher | Field studies of cereal leaf growth : 1. initiation and expansion in relation to température and ontogeny[END_REF]) suggests that this may be general for winter wheat. However, data published for spring wheat or for winter wheat grown in spring, and data we collected for winter wheat in a glass-house experiment, show a higher variability in the length of successive organs. This suggests that in general the length of one organ is linked to the length of the whorl by a function depending both on the ontogeny and the environment, but that there exists a range of environmental conditions (that encountered in winter sowing) where ontogenetic factor dominate, leading to the stereotyped progression of organ length with phytomer encountered in this study and in the literature for winter wheat. Obviously, more experimental data are required to explore these ideas. Finally, we also believe to be general the fact that the emergence of organs triggers phases changes in the kinetics of organ extension. This idea was also supported in the study of maize internode elongation, by the greater stability of the developmental timing when based on emergence events rather than on thermal time (e.g. [START_REF] Fournier | Dynamics of the elongation of internodes in maize (Zea Mays L.): Effects of a shade treatment on elongation patterns[END_REF]), and by the correlations observed by several authors [START_REF] Malvoisin | Organogenèse et croissance du maître-brin du blé tendre (Triticum aestivum) du semis à la floraison.I.-Relations observées entre la croisssance folaire et la différenciation des ébauches foliaires ou florales[END_REF][START_REF] Skinner | Epidermal cell division and the coordination of leaf and tiller development[END_REF][START_REF] Durand | Modelling the elongation of successive leaves on grass tillers. Effects of temperature, cutting height and cutting frequency[END_REF][START_REF] Fournier | Dynamics of the elongation of internodes in maize (Zea Mays L.): Analysis of phases of elongation and their relationships to phytomer development[END_REF][START_REF] Fournier | Dynamics of the elongation of internodes in maize (Zea Mays L.): Effects of a shade treatment on elongation patterns[END_REF][START_REF] Lafarge | A model co-ordinating the elongation of all leaves of a sorghum cultivar was applied to both Mediterranean and Sahelian conditions[END_REF]. Moreover, although it may be fortuitous, [START_REF] Malvoisin | Organogenèse et croissance du maître-brin du blé tendre (Triticum aestivum) du semis à la floraison.II.-Contrôle des relations entre la croissance et la vascularisation de la tige et des feuilles. Essai de modélisation[END_REF] showed that, on wheat, during the vascularisation of a leaf, the vessels of leaf n connect themselves to phytomer n-2, that is to the phytomer that was found here to be involved in the onset of leaf extension. Finally environmental triggering at organ emergence may be an essential process resulting in formation of both time and size patterns characterizing plant growth.

Conclusion

An L-system based model simulating the thermal time course of the 3D architectural development of a wheat plant was implemented. The use of similarities among phytomers allows a concise parameterisation that requires only: (i) the calibration of simple functions that describe the final size and the geometry of successive organs along the main stem; (ii) the timing of collar emergence on all axes; and (iii) a (estimated) developmental delay between main stem and every reproductive tiller. This model provides a tool for applications where simulations of plant 3D development are required, after having performed a calibration procedure. Such applications include simulation of satellite signal during crop development, and the simulation of agronomic experiments together with phytoclimate, to investigate processes at a local scale [START_REF] Pommel | Variations in the vegetative and reproductive systems in individual plants of an heterogeneous maize crop[END_REF][START_REF] Fournier | Virtual plant models for studying interactions between crops and environment[END_REF]. Other applications, such as the prediction of crop development would obviously require further work. The experimental results on which this model is based support the speculation that the emergences of organs act as triggers allowing the emergence of a stable schedule of organ extension. We propose that a more detailed investigation of the nature of this trigger, and how the response to this trigger is modulated by environmental conditions, is a key to understanding the combination of regularity and plasticity that characterizes the building of plant architecture.

Fig. 1 .

 1 Fig. 1. L-system-like developmental sequences used to model wheat development. (A) Initiation of phytomers by the apex; (B) phytomer development showing the succession of rapid elongation stages of blade, sheath and internode; (C) tiller production obtained by transforming an axillary bud into an apex.

Fig. 3 .

 3 Fig.3. Measured length of mature leaf sheaths (left) and mature leaf blades (right). The two figures are for the density 70 pl./m²: Main stem (bm) and six tillers (t1 to t6). Relative phytomer number is phytomer position for main stem, and is obtained, for tillers, by shifting data on the x-axis to obtain the best superposition with main stem.

Fig. 4 .

 4 Fig. 4. Examples of measured (dots) and modelled (lines) leaf laminae curvature. X and Y are respectively the horizontal and vertical distance to the base of the leaf laminae ("insertion point"). The first line shows five example leaves of the base of the plant. An ascending parabolic arc can model such leaves. The second line shows five examples of leaves of the top of the plant. Such leaves are modelled as a parabolic portion followed by an elliptic portion.

XFig. 5 .

 5 Fig. 5. Sequence of 6 images illustrating the 3D architectural development of wheat simulated with ADEL.

Acknowledgements

The development of the model was done within the frame of a contract between INRA and the University College of London (UCL), which was part of the UCL-ESA (European Space Agency) project 14940. Thanks to Jonathan Hillier for helping us improving the English.