Olivier Bruneau
email: olivier.bruneau@univ-nancy2.fr

Thomas De Vittori
email: thomas.devittori@euler.univ-artois.fr

INQUIRY-BASED MATHEMATICS TEACHING, HISTORY OF MATHEMATICS AND NEW COMMUNICATION TOOLS : AN EXCITING CHALLENGE !

Following a European project on the relationships between inquiry based science teaching (IBST), internet and communication tools (ICT) and a third topic, history of science, this article deals with some of the new questions raised by the recent introduction of epistemology, history of science and technology (EHST) in the school curricula. The denition of the inquiry and the role that can be played by epistemological thinking in science education are especially analysed in this text. Based on the analysis of historical examples and many on-line resources, as a rst result, a new framework for the elaboration of new resources in IBST, ICT and EHST is given and some opening questions have been pointed out.

evaluation), the practice of hands-on activities comprising an informations quest or not (experimentations), the introduction to a specic scientic language (argumentation, debate), the enhancement of students' autonomy . . . Mind the Gap was a place wherein to think about the interaction between IBST, history of science and technology (HST) and internet and communication tools (ICT). In this article, we would like to render an account of one of these possible interactions and to show thereafter how historical on-line resources can constitute well-adapted references of authentic problems.

2 IBST in Mathematics : a historical answer According to Pr. Martin Andler (University of Versailles France), a contemporary mathematical research activity comprises : 45% devoted to observation, 45% to experiment and only 10% to demonstration3 . In the eld of mathematics learning, the inquiry-based style often claims to have been inspired by the "scholar at work". In mathematics as in the other elds of science, inquiry takes up a large part of the research process. There is no doubt about the fact that the job has changed throughout the ages. However, by rendering an account of these changes, the epistemology and history of mathematics can help to explain what this inquiry could be.

First example

In ancient times in favouring results to reasoning mathematicians rarely expressed themselves on their relationships to the experiments. However when they did so, they gave us the opportunity to see the complexity of the links between theory and the use of technical instruments. The history of science assures us that : mathematical theories never emerge from nothingness. The scientist describes, builds and explores multiple examples before proposing an analysis or a system. On this subject, the work of the Arabic scholar al-Sijz is a model of such a process. Ahmad ibn Muhammad ibn `Adb al-Jal l al-Sijz was born and lived in Iran. Son of a mathematician, he worked between 969 and 998 and he wrote exclusively books on geometry. In all, he has written approximately fty treatises and lots of letters to his contemporaries. Al-Sijz was working within a very specic scientic context. Since the ninth century, the development of algebra in the Arabic world has created new types of questions on the fundaments of this eld. For instance, the point-by-point construction of conics has been well known since Antiquity (see the Apollonius' book entitled the Conics, for example), and that method is ecient enough for the analysis of the main properties of those curves. But during the ninth century, as the major part of algebraic equations studied during that period can be solved by intersecting conics curves (ellipsis, parabola, and hyperbola), the necessary taking into account of these intersections creates new diculties. Indeed this possibility is based on the continuity of the dierent curves which is dicult to "prove". The solution that has therefore been chosen is to associate the curve with a tool that enables a real construction. As the ruler and the compass allow straight lines or circles to be drawn and so justify their continuity, a new tool had to be invented to draw all the conics. Not only interesting from a mathematical point of view, such a technical development is also useful in technological areas such as the construction of astrolabes and sundials where conics are essential.

Following his predecessors (Ban u M us a brothers, Ibr ah m ibn Sin an...) from whom he quoted in a precedent book on the description of the conic sections, al-Sijz engages himself too in a treatise specically on the Construction of the perfect compass which is the compass of the cone4 . In this book, he studies a new tool, made-up a while ago by al-Q uh : the perfect compass. As he announces in the rst pages of his treatise, al-Sijz wants to "build a compass which enables him to obtain the three sections of the cone." He rst notes that all the conics can be obtained from the right cone (depending on the position of the cutting plane), and afterwards he proposes three possible structures for the perfect compass. The beginning of the study is technical, "We must now show how to shape a compass by which we can trace these sections. Shaping a stalk or AB. We put on the top tube, or NA. We link another tube to the end of it. [...]", and these instructions should enable the reader to build such a compass. But for al-Sijz , the aim of his work on the perfect compass is not only to draw conics. The end of the text shows that this compass is also a theoretical tool and a tool for the discovery of new concepts. Al-Sijz explains that the link between the circle and the ellipsis is quite obvious. Indeed, the construction of the ellipsis by orthogonal anity and the formula for the area are both well known. But what are the links between the circle and the parabola or the hyperbola ? Now oriented towards the exploration and the solving of new problems, the practical tool becomes an instrument of discovery and as stated by al-Sijz himself : "I always thought that there was a relationship between these two gures and the circle and their similarities and tried to get it but the knowledge of this has only become possible to me once I had learned how to turn the perfect compass following the positions of the plans."

In this example, the comings and goings between theory and practice appear clearly. Confronted with the theoretical problem of the continuity of curves, the scientist suggests the use of a new instrument. The experimentation with this instrument creates new theoretical results that create new questions and so on and so forth. In this text, al-Sijz claries the role of mathematical instruments. They are objects as much as models and this dual status facilitates the theory-experiment passage.

Another example

Pierre de Fermat (1601-1665), the French lawyer who works during his free time on mathematics, is well known and his name is associated with the famous theorem which was only demonstrated in the 1990's. Nowadays, we can nd5 some parts of his mathematical works on the theory of numbers on the Web. Even if he has not published any treatise on this subject, some elements can be found in quotations of his Diophantus' book and included in his correspondence especially with Marin Mersenne, Huygens or Carcavi.

In a letter Fermat sent around august 1659 to Pierre de Carcavi (1600-1684), another amateur French mathematician, he tried to demonstrate some properties with, according to him, a new kind of demonstration he called "la descente innie ou indenie". This kind of demonstration is also used in the margin of his Diophantus' volume in order to prove that the area of rectangular triangle with integer sides (i.e. ones which are measured by integers) cannot be a square. Up to 1659, Fermat only uses this method to prove some negative results. He proposes a positive result, for the rst time, in his letter to Carcavi, : any prime number such as 4k + 1 is the sum of two squares, and this form is unique. For instance, 5 2 + 1, 13 = 3 2 + 2 2 , • • • . The demonstration is based on a reduction per absurdum. For instance, to prove the rst above proposition, he presupposes that this kind of triangle exists. Then he shows that if it is true, we can nd another triangle with shorter sides which validates the assertion, and so on. Since the sides must be integers he arrives at a contradiction through the method of innite descent. It is quite easy to prove negative assertions, but its harder to show positive assertions. In his letter, Fermat announces that he only demonstrated that any prime number which could be written as 4k + 1 could be decomposed as the sum of two squares of integers and that this decomposition was unique but, unfortunately he did not give any demonstration. The rst proof of this assertion seems to appear in Leonard Euler's (1707-1783) works under the Latin title of Demonstratio theorematis Fermantiani omnem numerum primum formae 4n + 1 esse summam duorum quadratorum [Demonstration of the Fermat's theorem "All prime number like 4n + 1 is sum of two squares"]6 , in which he strictly follows the Fermat's method.

The method of "innite descent" has profoundly renewed the theory of numbers. Even if, as usual, Fermat does not demonstrate what he announces, he suggests to the other mathematicians that they can easily nd the demonstration of the properties he gives. For instance, in the above mentioned letter, he does not give any demonstration, however, he asks his contemporaries to do so : "je serai bien aise que les Pascal et les Roberval et tant d'autres savants la cherchent sur mon indication" (op. cit. p. 432). In this letter, he shows two important aspects of mathematics. First, some methods of proof are not able to solve a problem and newer ones must be invented and then reused to solve other mathematical enigmas. Creativity can go through a new method and a new approach. Secondly, this letter suggests to the other mathematicians to deal with those kinds of problems. This demonstrates how mathematical progress can be transmitted.

Many types of inquiry

The reading of the ancient texts is always interesting. Both examples presented above are only a small part of the historical resources dealing with inquiry, but they are still meaningful. What the historical approach shows is that inquiry cannot be reduced to a single aspect of the research process. Depending on the situation, each scientist engages himself in a dierent type of inquiry. A theoretical question does not require the same method as an experimental one, etc. IBST is clearly inspired by the work of the scientist as a professional. Nonetheless, the way this one has been understood is sometimes a bit caricatural. The history of science can prevent us from simplifying to such an extent and can restore the wealth of the research process.

3 History of mathematics and on-line resources Nowadays, on the web, anyone can nd many websites, pages or documents related to the epistemology, history of science and technology (EHST). Some sites like "Internet Resources for History of Science and Technology" 7 have even been created to help users to nd the right resources. More and more primary sources are also available on "Google books" or on the French project, "Gallica" 8 . It is relatively easy to read and to study ancient texts on science and technology. But in order to avoid mistakes or misinterpretations, it has been demonstrated that the texts must be contextualized. This is one of the main results of the research on the history of science. Now, due to the development of the new technology and the easiness of the on-line publication, some websites are devoted to IBST but they do not include any historical aspects. Is it possible to conciliate the two aspects : IBST with a historical approach ? In the Mind the Gap research program, the study of the relationships between inquiry and the history of science has shown that science and investigation are very close in dierent ways. In every scientic eld, the scholars have to elaborate a way of questioning their object. In each case, new instruments should be built, and new experiments should be elaborated in order to be able to ask the right questions and nally to answer them. This rst part of scientic activity is very similar to a second one which is dealing with theories and models. When their pertinence has been proved, the new concepts are applied to other situations or elds and they become a part of common knowledge. These newly born theories have still to be discussed and they constitute a third part of scientic activity. Communication between scholars often needs to create a suitable language and is essential in the building process of knowledge. An on-line publication should take into account all this wealth that gives many opportunities to the enlightening of historical documents and in making it suitable for IBST in multiple ways.

4 Digital document for EHST According to Michael Shepherd and Livia Polanyi 9 quoted by Ioannis Kanellos [START_REF] Ioannis Kanellos | Le concept de genre comme point de départ pour une modélisation sémantique du document électronique[END_REF] , the genre of digital documents can be characterized through three constitutive elements :

The contents (information, . . .), organized following a material structure (disposition, page setting,. . .) which is often enough for a rst and quick reading, and a logical structure (title, author, date, abstract,. . .) which brings some information on the intellectual organisation of the document. The container (support, medium), which determines the manner in accessing the information. The context of production, which relates the publication design. This context plays an essential role in the reading process of the document and it can be found as much in the content than in the container. of production and the genre. Sometimes the context of production is dicult to dene and so the best way to enlighten it is to refer to a community of practices 11 .

With this denition of the digital document, the context of production reveals the genre and shows the quality of the documents dedicated to EHST. We will thus give some criteria :

The rst and major one is the availability of primary sources. If there is no primary source, it becomes dicult to ensure that the document is within the frame of the history of science. The sources have to be contextualised and explained. The second one is the use of secondary sources. These documents may help to contextualise and to understand the scientic problem. The kind of media : texts, pictures, audio, video, . . . The possibility to make a simulation, to create The opening to a new view point and to other historical facts or problems (with hypertexts, links,. . .).

Then, these criteria can be connected with the aims of EHST digital documents. Is the document about the nature of science ? Does it refer to the macro or the micro history ? Is it based on the history of a concept or does it link science and society ? Does it deal with scientists' biographies or controversies, and so on and so forth.

5 Digital documents in HST for the use in IBST All these questions opens interesting research ways on the elaboration of EHST documents suitable to a use in IBST. Some works have already been engaged, especially at the European level, and reader will nd more detailed results (guidelines for digital documents building, content and form analysis grid, examples, ...) in our publications on this specic topic 12 .

Conclusion

Nowadays, on-line publication is quickly increasing quantitatively and it gives many opportunities to create innovative learning sessions. Nonetheless, quantity is not quality and the new technologies such as web 3.0 already point out the risk of losing oneself in an ocean of data. Facing to this situation, historians of science should not stand back. The examples above show that a little vigilance enables us to make a document suitable for IBST with all its historical wealth. The task is not so heavy (3 or 4 paragraphs are often enough) and the community of historians of science should be aware of these questions. IBST is one of the main active topics in didactic research and, in this article, we have tried to show that historians of science have many things to say on this subject.

Context : IBST in EuropeAll over Europe, the lack of student interest in science or in scientic careers has been noticed for years. This situation has led the European Community to launch a call for research projects in science education (the FP7 Science in Society program) and the publication of the Rocard Report about Science Education 1 . One of the main recommendations was to promote the advancing of teaching methods toward Inquiry Based Science Teaching (IBST) and the request for international comparisons. Engaged in

2008 2 , the European research project Mind the Gap was one of the answers. Mind the Gap was a large didactic program in which about 15 European universities worked on inquiry based science teaching in order to develop useful scientic tools within this topic. Many historian of science wished to join in this program. In a rst meaning, IBST consists in learnings based on an open problem in which the student has to propose experiment or use instruments in order to nd a solution. Such inquiry based teaching can be divided into dierent aims : an understanding of the articulation between empirical evidences and concepts (e. g. testing hypothesis, modelling, results 1 http://ec.europa.eu/research/science-society/index.cfm?fuseaction=public. topic&id=1100 2 The program ended in 2010, website : http://www.uv.uio.no/english/research/projects/ mindingthegap/index.html

Colloque Mathématiques, Sciences exprimentales et d'observation l'école primaire, Table ronde La démarche d'investigation en mathématiques , Chairman : P. Léna, École Normale Supérieure de Paris, September 28 th 2005 : http://www.diffusion.ens.fr/index.php?res=conf&idconf=882

The texts are available in a French translation in ×uvre mathématique d'al-Sijzi. Volume 1 : Géométrie des coniques et théorie des nombres au Xe siècle, Trad. R.Rashed, Les Cahiers duMIDEO, 3, Peeters, 2004.

http://www.archive.org/details/oeuvresdefermat942ferm

http://math.dartmouth.edu/~euler/docs/originals/E241.pdf

A community of practices is dened through three aspects : the borders of its application eld, its social existence, its language and the documents used and shared by the members of this community. It is also a group with interactions and learning that develop a feeling of belonging and a mutual engagement. See, for instance, Etienne Wenger, Communities of Practice : Learning, Meaning, and Identity, Cambridge University Press, 1998 or his website : http://www.ewenger.com/theory/ 12 See for instance the European collective book The Usage of ICT and IBST by the History of Science and Technology, O.Bruneau (Ed.),T.de Vittori Thomas (Ed.), P.Grapi (Ed.), P.Heering (Ed.), S.Laubé (Ed.), M.Massa (Ed.), Frank & Timme, Berlin, to be published in 2011. Some new perspectives of collaboration have to be opened, and nally, as the situation does not concern only one country, a European network has to be enhanced in order to share all the experiences.


Thomas De Vittori
email: thomas.devittori@euler.univ-artois.fr

THE PERFECT COMPASS: CONICS, MOVEMENT AND MATHEMATICS AROUND THE 10 TH CENTURY

Geometry instruments certainly exist since men are interested in mathematics. These theoretical and practical tools are at the crossroads of the sensible world and mathematical abstractions. In the second half of the 10 th century, the Arabic scholar al-Sijzī wrote a treatise on a new instrument: the perfect compass. At that time, several other mathematicians have studied this tool presumably invented by al-Qūhī. Many works are now available in French and English translations. After an historical presentation of the perfect compass, this article deals with a few passages of al-Sijzī's treatise which show the importance of continuous tracing of curves and provide interesting elements on the role of instruments in the mathematical research process. All these texts can help to understand the importance of motion in geometry which can be easily simulated by geometry softwares and used in a geometry lesson or in teacher training sessions.

Introduction and historical context

It is well known that the Arabic medieval period has been marked by the creation of algebra. Between the 9 th and the 13 th centuries, many researches are engaged and the emergence of new theoretical questions is one of the main consequences of the elaboration and development of this new field. For example, let us consider the equation:

x 3 + 2x 2 + x = 4
This equation is equivalent to:

x(x 2 + 2x + 1) = 4
That is to say:

x(x + 1) 2 = 4

And thus (for x = 0):

4 x = (x + 1) 2
The roots, if they exist, are the intersection points between the hyperbola y = 4

x and the parabola y = (x + 1) 2 (figure 1). In this situation, the existence of the roots is based on the geometrical existence of the intersection points. Both curves are conics and Figure 1: A simple example these objects are complex enough to create a doubt on the reality of such intersections. Of course, the mathematician can not just say that the figure shows the trueness of the result and this simple example raises a crucial theoretical question. The point-bypoint construction of conics has been well known since Antiquity (see the Apollonius' book entitled the Conics, for example), and that method is efficient enough for the analysis of the main properties of those curves. Algebraic equations can be solved by intersecting conics curves (ellipsis, parabola, and hyperbola) and the necessary taking into account of these intersections creates new difficulties. Indeed this possibility is based on the continuity of the different curves which is difficult to prove. The solution that has therefore been chosen is to associate the curve with a tool that enables a real construction. As the ruler and the compass allow straight lines or circles to be drawn and so justify their continuity, a new tool had to be invented to draw all the conics.

A new tool: origin and modelisation

As mentionned by R.Rashed1 , in the second half of the 10 th century, a large research movement is engaged by the Arabic scholars on the continuous tracing of curves. In his treatise On the perfect compass2 , Abū Sahl al-Qūhī (about 922 -about 1000) presents the results of his own research on this question.

Abū Sahl Wayjan ibn Rustam al-Qūhī said : This is a treatise on the instrument called the prefect compass, which contains two books. The first one deals with the demonstration that it is possible to draw measurable lines by this compass -that is, straight lines, the circumferences of circles, and the perimeters of conic sections, namely parabolas, hyperbolas, ellipses, and the opposite sections. The second book deals with the science of drawing one of the lines we have just mentionned, according to a known position. If this instrument existed before us among the Ancients and if it was cited and named, but if its names as well as the names of the things associated with it were different from the names we have given them, then we would have an excuse, since this instrument has not come down to us, any more than has its mention; thus it is possible that this instrument, as well as the demonstration that is draws the lines we have just mentioned, may have existed without its use being the one we have made of it in the second book of this treatise.

In the second part of the short introduction (above quoted in extenso), al-Qūhī carefully explains that he has not found any texts on this instrument and that is why he wrote his treatise. The recent historical research seems to confirm that al-Qūhī's book is the first treatise on this tool3 . Until now, no older descriptions have been found and there is no evidence of an implicite use of it before al-Qūhī. The new tool (figure ??4 ) is a kind of super-compass which can draw circles but also all the conic sections, even the degenerated ones like the straight lines.

If, at a point of a plane, we raise a straight line that moves in one of the planes perpendicular to this plane, and if, throught another point of this straight line, there passes another straight line which has three motion -one around the straight line raised upon this plane, the second in the plane on which this straight line is situated, and the third on its extension simultaneously on both side -then if an instrument is described in this way, it is called a perfect compass.

The construction, in its principle, is very simple and other texts give lots of technical details that leads to think that some instruments have really been built. Unfortunately, until now, no ancient perfect compass has been discovered. Nontheless, with this first description, one can construct such a tool. Some informations about modern replications of a perfect compass made for museum expositions, pedagogical or technical experiments, or just for pleasure can be found on the Internet 5 . In the educational context, a computer simulation is possible. A simple geometry software can give a good preview of what the tool can be. The example given in appendix produces the result below (figure 2). The first plane is (Oxy), the second (perpendicular to the first) is (Oxz). (OS) is the main line (the axis), the second one (SM) can move around the axis so that the point X leave a trace on the first plane. Let us call α = SOx and • a circle, if (OS) ⊥ (Oxy) with (SM) not perpendicular to (OS)

• an ellipsis, if α + β < 180 o • a parabola, if α + β = 180 o
5 Some pictures of perfect compasses are available for instance in the virtual exposition Theatrum machinarum on Modena Museum website: http://www.museo.unimo.it/theatrum/macchine/017ogg.htm or on the pages about the exposition Beyond the compasses on the Garden of Archimedes Museum website: http://php.math.unifi.it/archimede/archimede_NEW_inglese/curve/guida/paginaindice. php?id=2 A video of a pseudo-perfect compass in action is available on Professor Khosrow Sadeghi's personnal website: http://khosrowsadeghi.com/conic_compass.php#demo

• an hyperbola, if α + β > 180 o
Directly or not, this first description of the perfect compass is then reused by many other scholars. For instance, al-Bīrūnī (973 -1048) in his book Account of the perfect compass, and description of its movements6 explains: Abū Sahl has said: If, upon a point of a plane, we erect a straight line that moves on one of the planes perpendicular to this first plane, and if through another point on this straight line there passes another straight line, having three movements, of which one is around the straight line erected on this plane, the second is on the plane on which this straight line is situated, and the third is rectilinear in both directions; then if the instrument so described exists, we will call it a perfect compass.

And, in a same way al-Abharī (d. 1264) says in his Treatise on the compass of conic sections7 : If, on a given straight line in a given plane, we erect a straight line, and if through the other end of the straight line we have erected there passes another straight line that comes to meet the given straight line in the plane, then we call these straight lines, in this configuration, the compass of the conic sections.

This last quotation gives the opportunity to read one of the other names of the perfect compass. Called by al-Abharī the compass of the conic sections, this instrument is sometimes simply named a conic compass or cone compass. All these treatises contain many mathematical propositions about the way to calculate the good angles corresponding to a given conic section. I do not detail this part but I strongly encourage the reader to have a look at these beautiful texts (see References).

Mathematical instruments in the research process

Ahmad ibn Muhammad ibn 'Adb al-Jalīl al-Sijzī (about 945 -about 1020) was born and lived in Iran. Son of mathematician, he worked between 969 and 998 and he wrote exclusively books on geometry. In all, he has written approximately fifty treatises and lots of letters to his contemporaries. Following his predecessors (Banū Mūsā brothers, Ibrāhīm ibn Sinān...) from whom he quoted in a precedent book on the description of the conic sections, al-Sijzī engages himself too in a treatise specifically on the Construction of the perfect compass which is the compass of the cone8 . Like the other scholars, al-Sijzī wants to "construct a compass by means of which he shall draw the three sections mentionned by Apollonius in his book of Conics." He first notes that all the conics can be obtained from the right cone (depending on the position of the cutting plane), and afterwards he proposes three possible structures for the perfect compass. The beginning of the study gives technical recommandations. Here is a small quotation:

We must now show how to fashion a compass by means of which we may draw these sections. We fashion a shaft; such as AB. We place a tube at its vertex, such as AN, and to its extremity we attach another tube, such as AS. We can accomplish this with the help of a peg, or with anything else, so that the tube AN turns around shaft AB [...] The instructions should enable the reader to really build such a compass. But for al-Sijzī, the aim of his work on the perfect compass is not only to draw conics. In Sur la description des sections coniques9 , the text10 shows that this compass is also a theoretical tool and a tool for the discovery of new concepts.

Mais puisque les propriétés de l'hyperbole et de la parabole sont proches des propriétés du cercle et que les propriétés de toutes les autres figures composées de manière régulière à partir de droites et qui ne subissent ni révolution ni rotation sont éloignées des propriétés du cercle, il est donc nécessaire que ces deux figures aient un rapport au cercle et une similitude à celui-ci, comme il en était pour l'ellipse. J'ai toujours réfléchi à l'existence de ce rapport entre elles et le cercle et à leur similitude et cherché à saisir ce rapport ; or la connaissance de ceci ne m'a été possible qu'une fois appris comment faire tourner le compas conique suivant les positions des plans. En effet, cette existence s'ordonne à partir de la rotation du compas conique sur la surface latérale ; la rotation régulière convient au cercle et cette rotation est commune au tracé du cercle sur une surface plane et au tracé de toutes les autres sections coniques ; étant donné que le cercle provient du tracé avec ce même compas si la position du plan est perpendiculaire à son axe, alors que pour les autres sections, leurs formes diffèrent suivant la position du plan par rapport à l'axe du compas. Quant à l'ellipse, sa conception est facile de plusieurs manières, soit à partir d'une section du cylindre soit à partir de la projection des rayons traversant une ouverture circulaire sur un plan de position oblique qui tient lieu aussi d'une section du cylindre ou d'une section du cône. Ce que nous voulions montrer.

Al-Sijzī explains that the link between the circle and the ellipsis is quite obvious. Indeed, the construction of the ellipsis by orthogonal affinity and the formula for the area are both well known. But what are the links between the circle and the parabola or the hyperbola? Now oriented towards the exploration and the solving of new problems, the practical tool becomes an instrument of discovery and as stated by al-Sijzī himself,"I always thought that there was a relationship between these two figures and the circle and their similarities and tried to get it but the knowledge of this has only become possible to me once I had learned how to turn the perfect compass following the positions of the plans". Confronted with the theoretical problem of the continuity of curves, the scientist suggests the use of a new instrument. The experimentation with this instrument creates new theoretical results that create new questions and so on and so forth. In his text, through the comings and goings between theory and practice al-Sijzī clarifies the role of mathematical instruments. They are objects as much as models and this dual status facilitates the theory-experiment passage.

Conclusion

Halfway between philosophy and science, the acceptance of movement as a valid principle in geometry is one of the important topics 11 during the 10 th and 11 th centuries. Not only interesting from a mathematical point of view, the perfect compass is also useful in technological areas such as the construction of astrolabes and sundials where conics are essential. At the end of the Middle-Age, this instrument disappears and comes back at the Renaissance as a drawing tool (see figure ?? 12 and Raynaud (2007)). The mathematics have changed and such an artefact between theory and practice is Figure 4: Renaissance : the perfect compass as a drawing tool now useless. Mathematicians rarely expressed themselves on their relationships to the experiments. However when they did so, they gave us the opportunity to see the complexity of the links between theory and the use of technical instruments. The history of science assures us that: mathematical theories never emerge from nothingness. The scientist describes, builds and explores multiple examples before proposing an analysis or a system. The Arabic developements around the perfect compass is a model of such a process. In education in France and in many countries, the recent official instructions claim the importance of investigation in the learning process. The perfect compass can give a good entry point for an activity that helps the students to understand the way a new mathematical theory is elaborated. In secondary school, the conics are often studied only from a cartesian point of view. For students, and for teachers too 13 , the 11 The movement as a theoretical geometry principle appears a first time in Ibn al-Haytham's tries of a new definition of the geometrical space (see [START_REF] Rashed | Méthodes géométriques, transformations ponctuelles et philosophie des mathématiques[END_REF]or de Vittori 2009) 12 Sketch. Original pictures can be found in [START_REF] Raynaud | frontale: 1 Repère Rxyz affiché Dessin de o: marque épaisse I point de coordonnées (1,0,0) dans le repère Rxyz Dessin de I: gris J point de coordonnées (0,1,0) dans le repère Rxyz[END_REF], for instance : (a) Venise, B. Naz. Marciana, 5363 (olim Ital. cl. IV 41), fol. 18r, P. Sergescu, "Leonardo da Vinci et les mathématiques", Leonardo da Vinci et lexpérience scientifique (Paris, 1952): 73-88, C. Pedretti, Studi vinciani (Genve, 1957), Idem, Leonardo da Vinci architecte, op. cit., p. 336, Idem, "Leonardo discepolo della sperientia" ,F. Camerota, d., Nel Segno di Masaccio (Firenze, 2001), p.184-185. (b) Vienne, Albertina, Inv. 22448 (olim 164). O. Kurz, "Dürer, Leonardo and the invention of the ellipsograph" , Raccolta Vinciana, 18 (1960): 15-25, sur les problèmes dattribution, cf. infra IV, note 51. (c) Sienne, Biblioteca deglIntronati, ms. L. IV. 10, fol. 92r-98v, G. Arrighi, "Il compasso ovale invention di Michiel Agnelo" , Le Machine, 1 (1968): 103-106; P. L. Rose, "Renaissance Italian methods..." .

13 Al-Sijzī's texts and the Geospace model have been successfully used during a teacher training workshop (Colloque IREM, Brest 2008).

Figure 2 :

 2 Figure 2: Sketch based on Kitāb al-Qūhī fī al-birkām al-tāmm, MS Istanbul, Raghib Pasha 569, fol.235 v .

Figure 3 :

 3 Figure 3: Perfect compass : a simulation with Geospace

al-Qūhī et al-Sijzī: sur le compas parfait et le tracé continu des sections coniques, Arabic Sciencesand Philosophy, vol.13 (2003) pp.9-43

R.Rashed, Geometry and dioptrics in classical Islam, al-Furqān, 2005, pp.726-796

For a complete analysis of the historical sources that have reached us, see R.[START_REF] Rashed | al-Qūhī et al-Sijzī: sur le compas parfait et le tracé continu des sections coniques[END_REF]

Original picture in[START_REF] Rashed | Geometry and dioptrics in classical Islam[END_REF] p.860

R.Rashed, Geometry and dioptrics in classical Islam, al-Furqān, 2005, pp.816

R.Rashed, Geometry and dioptrics in classical Islam, al-Furqān, 2005, pp.828

R.Rashed, Geometry and dioptrics in classical Islam, al-Furqān, 2005, pp.798-806; the texts are also available in a French translation in OEuvre mathématique d'al-Sijzi. Volume 1: Géométrie des coniques et théorie des nombres au Xe siècle, Trad. R.Rashed, Les Cahiers du MIDEO, 3, Peeters, 2004.

R.Rashed, Les Cahiers du Mideo, 3, Louvain-Paris, Éditions Peeters, 2004.

 p.254

This geometry software is available on http://www.aid-creem.org/telechargement.html

work on the perfect compass restores the links between solids and curves, practice an theory, real world and mathematics models... Mathematics teaching is always renewing itself and the historical sources give many elements that enrich it and give it sense.


Thomas De Vittori
email: thomas.devittori@euler.univ-artois.fr

History in mathematics teaching: current problems and new proposals

In the light of recent works on the use of the history of mathematics in education, this article offers an analysis of the theoretical issues that arise in such studies. After the presentation of some intrinsic difficulties of a good definition of the different fields involved (a discipline, its didactic and its history), some new elements for a didactical analysis (double field hypothesis, double field paradox and its handling) are proposed. Applied on a few examples, these concepts are aimed at being part of a new framework for empirical studies on the use of history of mathematics in education.

Introduction

For decades and all over the world, the usefulness of the history of mathematics in science education has ceased to be a debatable topic. Numerous authors have already promoted a historical approach (for instance, in France, [START_REF] Barbin | The reading of original texts: how and why to introduce a historical perspective[END_REF], 1997a[START_REF] Dorier | Recherche en Histoire et en Didactique des Mathématiques sur l'Algèbre linéaire -Perspectives théorique sur leurs interactions[END_REF]. All these authors have pointed out several good reasons that range from the development of a humanist culture to an acute comprehension of scientific results or theories. In most countries, these reflections have led to an integration of elements of history in school official instructions and in teacher training. Thanks to this shift towards the inclusion of history, a new demand has been brought to light. In 2000, the International Commission on Mathematical Instruction (ICMI) world report on the issues of the use of history in mathematics education expresses clearly the feelings of the community: "having history of mathematics as a resource for the teacher is beneficial" (ICMI Study, 2000, p.1). Nonetheless, next to this enthusiastic conclusion, the lack of real research works on this subject has also been mentioned. Around the same time, the didactician J-L. Dorier explains that the specific function of historical elements in the knowledge acquiring process is still pending; according to him, the number of critical studies has not reach a large enough size (J-L. [START_REF] Dorier | Recherche en Histoire et en Didactique des Mathématiques sur l'Algèbre linéaire -Perspectives théorique sur leurs interactions[END_REF]. Following these reflections, a first wish was expressed by the community: "on the effectiveness of studying history in the classroom, it seems desirable to collect and to study two kind of materials: 1. To collect experiences of teachers who use history. […] 2. To collect questionnaires and interviews of teachers and pupils about mathematics." (ICMI Study, 2000, p.90). Strangely, even though the idea was shared and accepted by many, there was no significant increase in the number of such studies. In a recent article, U.T. Jankvist (2009) renders an account of the latest conclusions by specialists regarding history of mathematics in education (Arcavi -Tzanakis 2000, [START_REF] Siu | History of mathematics in classroom teaching -appetizer? main course? or dessert?[END_REF]. One can summarize this state of the art in terms of two ideas: (1) the literature is full of pertinent uses of historical elements in education but (2) what makes them pertinent rarely exceeds the subjective considerations of the author. Thus, once again, a non-negligible part of the community of educators requests the increase of detailed works on this field. Very recently, in 2011, a few such texts including didactical aspects have been published. Compiled by V.Katz and C. Tzanakis (eds) in the book Recent Developments on Introducing a Historical Dimension in Mathematics Education, these articles try to give an answer to the issues specified in the preface: namely, that "there should be more empirical studies on the use of history in the mathematics classroom to get more insight its educational implications." According to the editors, "in each case, the authors have conducted at least some preliminary research to test the effectiveness of their teaching methods and conclude that the use of history has a positive effect on their students." (Katz & Tzanakis 2011, viii). Why the first results only appear more than one decade after the first demand? And why, even in the last productions, R.Chorlay, one of the contributors of a collective book, expresses reservations on his own work and says that "this is neither historical work in the strict sense -in terms of the standards of the community of historians on mathematics -nor directly didactical work, but we think it helps build bridges […] (in Katz & Tzanakis, 2011, p.64)? At least, one can ask himself if a pertinent study on learning issues of history of mathematics can be or has been really achieved. I think it is possible in multiple ways, but not self-evident. Thus, in the following sections, this article aims to show why didactic studies on the use of historical elements in mathematics education are intrinsically out of the standard theoretical frameworks. Finally, with some examples, a proposed research way is being presented.

Why? How? And some dead ends

One of the starting points proposed for empirical studies (U.T. Jankvist, 2009) consists in a generally admitted distinction between how and why to use history of mathematics in education. The why question can be considered as an operational realization of the reasons mentioned above. For instance, history is a way of better understanding science and its methods, thus the use of historical elements in classrooms allows pupils to develop informed points of view. The how question generally deals with the links between the teaching of a subject and its history. The typical example is the elaboration of sessions inspired by the historical evolution of scientific concepts. Both for the how and the why, it is important to notice that these questionings only intervene in the determination of the aims and teaching materials, that is to say, they are always related to teacher actions in the elaborating phase. In fine, even if it can be helpful for the understanding of the implementation of historical elements in a course, the categorization in how and why gives few tools for an analysis of pupils' learning. Nonetheless, this distinction is a way of making explicit the coexistence of two distinct fields in an activity: the discipline and its history. In such a situation, the questioning on the reasons for a historical approach becomes a way of identifying teaching goals that are specific to history, like acquiring knowledge on an author, a period or the evolution of a concept. It is consequently obvious that the complexity of this interlocking of two fields requires data acquiring. A quantitative rise in empirical studies is a good answer but it raises a new crucial question: what are or can be the aims of such studies? On this point, a very interesting recent work in Germany (M. [START_REF] Glaubitz | The use of original sources in the classroom -Empirical research findings[END_REF] deals with a comparative study of pupil performance in secondary schools. The purpose of this study was to shed light on "how does the historical teaching affect the students' mathematical skills compared to those who had conventional teaching? How does the historical teaching affect the students' views on mathematics, teaching and their relationship towards it? [and] In what way and to what extent does the historical teaching allow for (meta-)reflection and discussion in the classroom?" In the experiment, a first part of the students have had a typical mathematics course, a second group have had lessons inspired by history of science, mainly in its chronological aspect (the genetic approach). A third group had engaged in a teaching strongly marked by an historical contextualization of mathematics concepts (the hermeneutic approach). In a nut shell, the genetic approach essentially consisted in the insertion of a discovery activity with historical elements at the beginning of the lesson. As for the hermeneutic approach, it was composed of three steps (sometimes repeated), a classical mathematics course, a reading of genuine historical sources, and an important research activity around the contextualization of the historical texts and concepts.

A statistical analysis on six classes (about 175 students) shows that the best performances obtained in the hermeneutic approach, very closely followed by the classical course. The lessons comprising only discovery activities yielded results far below the two others. According to Glaubitz, the poor score of the genetic approach is due to difficulties for children to grasp what is essential in the proposed contents, resulting in turn to a limited stabilization of knowledge. What about the context of this study? The German school system is strongly normalized and one of the main objectives of Glaubitz's works was to convince parents that a non-traditional education can produce good results in terms of learning. The unofficial purpose of the study was political and under this aspect, it was successful. Glaubitz concludes that the hermeneutic approach is indubitably better than the others as it "could make a notable contribution to humanist, democratic and peace-promoting traditions in education" and the reasons of this success can be found in the good use of history. This last affirmation generally shared by the other members of the community may require some more analysis. The quest for evidence of the effectiveness of teaching enriched with history of mathematics is probably vain. For instance, even the large scale study made by Glaubitz only shows that a good implementation of historical elements produces results only as good as the traditional method. Here appears a well-known didactics result: a bad education leads to poor results and a well thought out one leads to success (whatever it is based on, historical documents or not). In my view, such work mainly raises our awareness to the fact that studying the manner a course is elaborated represents only a part of the job. Centered on the links between knowledge-teacher, this approach may miss other interesting interactions between teacher and pupils, or between pupils and the knowledge at stake. New research ways are open, in particular on classroom practice analysis or, as pointed out in the ICMI study, "a crucial area to explore is the relation between how students achieve understanding in mathematics and the historical construction of mathematics thinking." (ICMI Study, 2000, xvii)

Double field, double field paradox, and naive didactics

Classroom sessions analysis is a traditional methodological tool used in the didactics. On the specific case of history in mathematics education, the research object is thought-worthy and the research field deserves to be well defined. In a didactic study of session comprising historical content, a first idea is to use theoretical frameworks already developed in mathematics didactics themselves. With the help of mathematics didactics, classroom lessons can be analyzed in order to identify, for instance, key moments in pupils activity or to evaluate learning processes. As the theoretical tools in use are suitable to the research object, such an approach is perfectly coherent. No doubt, this analysis will produce interesting results that will add to the evidence of its pertinence to mathematics didactics. The problem consists in the extrapolation of such results to some knowledge on the place of history of mathematics in education. Mathematics didactics can mainly render an account of mathematics learning. The tools developed in this field are suitable to show that a lesson has truly helped pupils to acquire mathematics knowledge but they do not give any information on the role played by the history of mathematics in that lesson. That is to say, mathematics didactics alone do not seem to be the most pertinent tool for an analysis of the potential role of the history in mathematics education. The concepts and methods developed in the mathematics didactics field have to be considered as a part of the available tools. Let us recall some recommendations found in the ICMI report, "there are limits and risks attached to an approach that takes too simplistic a view of the significance of the history in mathematics education." (ICMI Study, 2000, p.64).

Didactic studies generally enable us to understand teaching situations only relative to the considered discipline. In sessions including historical elements, the simple conjoint presence of a topic and its history is specific enough to create a new situation. This double field invalidates a monodisciplinary didactic analysis because such a study can only process part of the situation. At school or high school, history of mathematics so appears in link with the considered subject. In this case, history is not a specific teaching topic and, in a way, is not an autonomous discipline. In fact, in several education situations, history of mathematics is not an academic subject (some exceptions exist, in particular, in Denmark). Nonetheless, official instructions sometimes emphasize the fact that pupils have to be aware of epistemological and historical problems. Generally, there is no history of science course but some real teaching goals exist. Teachers have to transmit knowledge without being able to use traditional tools to do so. Here appears the double field paradox. According to educators themselves, didactics deal mainly with the learning of contents disciplinarily specified1 (Reuter 2007, 17). This definition has two consequences. The first is, as it is not a school subject, it seems not to be pertinent to try to speak about didactics of the history of mathematics at school. The second consequence of the classical definition of didactics refers to the trial of a multi-disciplinary approach of the double field hypothesis. In sessions involving historical material, the presence of two different fields could conduce to the use of works on didactics of inter-disciplinarity. This topic is defined [START_REF] Lenoir | De l'interdisciplinarité scolaire à l'interdisciplinarité dans la formation à l'enseignement: un état de la question[END_REF] by the creation of an interaction between two or more disciplines at a pedagogical level. This leads to the creation of complementary or cooperation links that are helpful in pupil learning process 2 . No doubt, this definition takes into account the richness of the relationships which exist between two disciplines. Many users of history of mathematics in education will certainly agree with the ideas of inter-fecundity noticed above. Unfortunately, this enthusiasm cannot hide the fact that didactics of inter-disciplinarity are foremost didactics in general and concern only school subjects. The history of mathematics is not a school topic and it is doubtful that such a theoretical framework will produce good results.

History of mathematics and mathematics education

Returning to the issue of a good definition of theoretical frameworks involving historical elements suitable for empirical studies on teaching situations, the question becomes: what are the possible interactions between didactics and in situ teacher practice? Didacticians elaborate analysis frameworks comprising many concepts organized in a model. This model is then used for the study of teaching situations. From school to university, mathematics is a teaching object. Whatever the level, some mathematics contents are transmitted. The question of the choice and the nature of this knowledge do not matter for the following reflection. What I would like to underline is that learning purposes are varied, more or less complex, and this richness creates teaching situations very different from one another. The analysis of key elements in this learning belongs to didactics in which adapted concepts and models will be elaborated. History of mathematics in education, daughter of the university discipline, is in the same situation. In term of time or notions, history of mathematics contents are as varied as pure mathematics contents. Thus, course aims can go from acquiring knowledge on specific period or author to the development of an individual epistemological thinking. As mentioned previously, the only difference consists in lack of explicit definition jointed to a large part of subjectivity. Relationships between firstly, mathematics -as teaching objects-and secondly, the history of mathematics -as teaching object-can be conceived in a parallel way (figure 1). Like in mathematics education, the purposes of didactics dealing with history of mathematics must be the analysis of pertinent elements of such learning.

Figure 1

In most countries, teachers have been assigned the function of inviting pupils to an epistemological reflection and many of them do not feel at ease with this new mission (double field paradox). History of mathematics, in its academic aspect, is not suitable to a school instruction. But the concepts, the sources, and the activities can go through a pedagogical filter which gives educators the opportunity to appropriate them as tools for session building. Whereby, a classroom situation is at the interface of a discipline, history of this discipline, and many other elements the teacher can decide to integrate in his course. The teacher is not a researcher; he is a professional in education in school knowledge and know-how. Like mathematics teaching skills, competences in the history of mathematics implementation can be regarded as use of savant contents that have their own domain. No doubt, in the same way that it is preferable to learn enough mathematics to be able to teach them, teacher engagement in sessions involving historical elements without having any education on those topics could be hazardous. In France, and in most countries, the involvement of historians of mathematics community in the recent institutional changes may have certainly prevented us from this pitfall, but strangely, for many teachers, the double field paradox remains...

The tool / teaching goal duality: understanding the paradox

In order to show how the elements previously presented open a way to elaborate a framework for empirical studies, I would like to come back to a second distinction generally accepted in the use of history in mathematics education. In the first paragraphs of this article, the classical separation between how and why of the use of history has been a reminder. This distinction is often completed with a dichotomy between the history of mathematics as a tool, and its use as a teaching goal (for instance Barbin & al. (ed), 2008). The international community of historians of mathematics involved in teacher training have recently taken the two notions, history as a tool and history as a goal, as an entry for empirical studies [START_REF] Jankvist | An empirical study of using history as a 'goal[END_REF]. In their most recent paper, Jankvist and Kjeldsen explain that "the word goal must be understood in the sense that it is considered a goal to show the students something about the historical development of mathematics" and the tool notion appears when "history is used as an assisting means, or an aid, in the teaching and learning of mathematics." [START_REF] Jankvist | New avenues for history in mathematics education: Mathematical competencies and anchoring[END_REF] The consideration of two different uses of history of mathematics is judicious but it should be handle with care in order to prevent the risk of falling into mono-disciplinary studies that are not, according to what I have tried to explain, the most pertinent for science sessions involving history. One of the difficulties in the use of history in mathematics education consists in the co-presence of two knowledge fields, one being a school topic and the other not. A teacher who is proposing such a session is in a situation in which he strives to reach multiple learning purposes, some disciplinary, others epistemological (or historical). In order to analyze in a rich didactical manner and articulate what is at stake in such situations, a new point of view is necessary.

In a previous exploratory work (de Vittori & Loeuille, 2009), the comparative study of secondary teacher training sessions led to highlighting key moments, when trainees have really entered in the intended learning. A typology has been elaborated in which five task entry modalities have been pointed out: technical, philosophical, linguistic, practical and dramatic (related to plays or role playing games). Let us first recall some important points in the technical modality. This type of task entry appears when a pure technical work is done following a historical document. For instance, in the aforementioned study, trainees were exposed to a geometry demonstration which enabled them to understand Galileo's arguments (Il Saggiatore, passage on Earth-comets distance, sub-lunar or supra-lunar objects and discussion on their nature). By redoing the geometric proof, the trainees were naturally engaged in a reflection on the necessity of such mathematical arguments. Thus, they started to question the scientific context during Galileo's time and in that way entered in the contextualization learning which was the purpose of that session. In this example, the work was exclusively based on mathematics contents, nonetheless this modality can be extended to any science topic as soon as students activity goes only by already acquired scientific knowledge. Clearly, in a session on epistemology and history of science, a scientific discipline can become a tool for a teaching goal. At school, this situation exists too. A typical instance can be found when a secondary mathematics teacher proposes pupils to do a geometrical drawing inspired by Euclid's Elements [START_REF] De Vittori | Former des enseignants à l'histoire des sciences : analyse et enjeux d'une pratique en mathématiques[END_REF]. That way, pupils can smoothly enter a historicoepistemological thinking through the questioning on the reasons of such mathematical construction or proof. Like at the university level, in a classroom situation, mathematics can be used as a tool for a work on historical topics.

During a mathematics course involving history, whether it takes place at school or in university, one of the preferred documents is the text. Intrinsically this type of historical source requires a work on language. Even if it is carefully chosen, for its simplicity for example, the reading of an ancient text goes with explanations about some words or about the style. The quest for answers to these grammatical or semantic questions is one of the possible learning entries (E.Barbin uses the word dialogism, that is to say a kind of discussion between authors from the past and contemporary reader; see also Katz & Tzanakis, eds, 2011, chapter 2). For instance, always in the secondary school session previously mentioned, the teacher proposes to the pupils an activity based on a short quotation of a 17 th century text. In this document, some words had been written in old French (e.g. quarré instead of carré) and the style differed from our (e.g. a perpendicular is falling on …). Pupils' questioning on the spelling of the words has led to a recall of some geometry definitions and has enabled entering in the learning contents; in this case, a lesson on polygons. Thus, in this situation, history is a tool, a means of reaching mathematics learning goals.

Both examples below suggest a way of analyzing the links between mathematics and its history in a classroom situation. As noticed previously, history as a tool and history as a goal can appear here as two elements of a unique session (figure 2). What is interesting here is that mathematics is both a tool and a goal.

Figure 2

A circulation is evident. This dynamic relationship brings to the fore the fact that the classroom lesson is a place where links between history and mathematics can be reconstructed. To my mind, here lies a deep cause of the double field paradox, namely that the history of mathematics and mathematics have been separated by the academic research process. In his classroom, a teacher who combines science and its history constantly goes from the disciplinary domain to the historical domain, then from the historical domain to the disciplinary one, and so on and so forth. In this way, he is reassembling what was and should be a whole. The way the teacher manages this circulation and all the pedagogical artifices he uses in order to avoid the double field paradox are research topics suitable for empirical studies that transcend traditional academic borders.

Teacher and pupils in classroom: an example of reconstruction

Below is part of a discussion between a teacher and one of his students. The situation takes place at the end of a session during which the pupils (about 11 years old) had to redo some old Indian constructions. Following indications given on an exercise sheet (with some historical elements like names, dates, and pictures), in small groups in the playground, the students drew a square with a single piece chalk and a piece of string. After an introductory text which gives some information on the kind of problems the old Indian mathematician were interested in, the exercise sheet comprises a question on the squaring the circle problem ("Les mathématiciens grecs ont aussi étudié cette dernière situation, à la règle non graduée et au compas. Comment a-t-on appelé par la suite ce problème ?").

Teacher: "Have you answered the question there? Lora, have you answered the question on the problem? What's it called ?" (Vous avez répondu à la question là ? Lora tu as répondu à la question sur le problème, ça s'appelle ?) Student: "Squaring the circle" (La quadrature du cercle.) Teacher: "Squaring the circle; what's that?" (La quadrature du cercle, c'est quoi cette histoire là ?) Student: "It's when there is a circle and when one has to find a square which has the same area." (C'est quand il y avait un cercle et quand il fallait trouver le carré qui avait la même aire que ce cercle là.) Teacher: "Yes, ok and what was the problem?" (Oui d'accord et alors quel problème s'est posé ?) Student: "Well..." (Bin...) Teacher: "Have they managed to do it?" (Est-ce qu'ils ont réussi à faire ça ?) Student: "Well, no because they didn't yet know the pi number." (Bin non parce qu'ils ne connaissaient pas encore le nombre pi.) Teacher: "And that it's once one knew well the pi number that one realized that it was not possible, that's it?" (Et c'est quand on a su bien connaitre le nombre pi qu'on s'est rendu compte que c'était pas possible, c'est ça ?) Student: "Yes." (Oui.) Teacher: "There you are, then." (Voilà.)

The discussion happened in the presence of six pupils, the three members of the first group rejoined by another group that had finished their work. The young girl the teacher is speaking to is one of his best students, the other pupils, in particular the second group, are students of average abilities. The beginning of the dialogue seems to be simply checking the right answer to a question, but after a few sentences, the teacher gets into another role. The question on squaring the circle becomes a way for him to give reconstruction elements. The square construction, the circle, pi number, are linked in both the history of mathematics and mathematics. For such questions, a separation between the historical part and the mathematical part impoverishes the meaning. In this case, following mainly his own sensitivity, the teacher feels the need for a reconstruction. He does not give much information, and in fact, he does not give anything that was not already known by the pupils, but he engages a process. It is interesting to notice that these first exchanges are followed by an open discussion on Archimedes and his work. The discussion involves all the six pupils who are making links between other courses (history and geography, the never ending screw), their own culture (Eureka!, Archimedes' principle). Sometimes, the teacher intervenes when mistakes are made. Here is an example of a historical one. This session was recorded in 2011. In 2007, an interview of the same teacher was done. Speaking about his thoughts on the implementation of history in mathematics education, the teacher insisted on the fact that mathematics teaching is historically unstructured. According to him, contrary to education on other topics, the mathematics courses are jumping through history, going from Antiquity to Middle-Age, then back to 18th, … ("Alors que nous on va passer de l'Antiquité au Moyen-Age, on va aller au 18e siècle, on va revenir à l'Antiquité, il n'y a pas quelque chose de progressif, on a vite fait de s'y perdre. C'est pour ça que les enfants ne savent plus qui a fait quoi et quand, et ne situent pas dans le temps, et ça pose des problèmes de repères.") This situation creates trouble in the pupils' mind and one of the teacher's roles can be the reestablishment of the historicoepistemological links ("comprendre que les choses sont arrivées petit à petit, parce qu'il y avait des besoins, ça répondait à des besoins concrets, petit à petit on a développé une science autour de tout ça, et puis la science a pu évoluer par elle-même et pour elle-même, ça ça me semble intéressant de leur faire comprendre.")

No doubt that this aspect of the teaching situation is deeply based on epistemological views of the teacher. One can find some studies on this topic. What I would only like to underline with this short example is that a session comprising historical elements must be conceived as a complex whole. Both mathematics and the history of mathematics are aspects of a teaching situation that could be treated as a whole. Any dichotomized approach (history/mathematics, goal/tool, how/why) should stay a first step of a study complemented by the giving of synchronic elements that take into account the unity necessarily at stake in a classroom.

Conclusion: on didactical studies

Teaching history of mathematics or teaching with/through history of mathematics? At school, the question is relatively clear: history of mathematics is not a school topic. The situation seems to be different when a specific teaching exists. In France for instance, recent institutional changes have led to the creation of courses on epistemology and the history of science in many curricula. These courses have been largely developed in the teacher training sector. The history of mathematics already existed in curriculum aimed to researchers' education but its massive presence in preprofessional courses is rather new. The situation is similar in many countries and except in a few situations, history of mathematics is a fully-fledged discipline mainly at university. Thus and as it has already been noticed by some authors [START_REF] Guedj | Methodological elements for didactics of epistemology and history of science and technology"), IUFM du Nord Pas de Calais. Colloque Théories et expériences dans les didactiques de la géographie et de l'histoire. La question des références pour la recherche et pour la formation[END_REF][START_REF] Reforehst | Histoire des sciences : formations et recherches en IUFM[END_REF], didactics of history of science might be considered at this level. Nonetheless, one might remember that the context is really not the same. The History of mathematics can be taught as a "research" topic. In this case, even if it takes place in a professional curriculum, the course is based on methods and contents close to the courses for future researchers. A didactical study at this level is interesting but it is not related to classroom situations. If didactics of history of mathematics is defined there, its aims will be very different from those of a second one centered on classroom activities.

Sometimes mentioned by educators, another manner of introducing history in curriculum consists in a pre-professional course dedicated to teachers. In this case, the contents are chosen in order to give teacher elements on topics suitable to schools. Some examples of activities may even be given. What is the reference field of such a course? Part of it is dealing with history, but this field is not considered for itself, it is viewed as an element of an acquiring process of professional skills. Another part is related to classroom and mathematics education. Generally, everything is based on the teacher's experience (or on some reported experiments) and, in a way, the teacher is probably sharing his own reconstruction skill. In such a professional-to-professional relationship, designing a research field is not easy and may even be useless. More changing and more mixed than the university sector, this professional sector comes with specific questions on which a joint work with all the different actors (teachers, researchers, …) can be engaged. Empirical studies are undoubtedly possible. Will they be relevant to works on didactics of mathematics (if one thinks that history is an indivisible part of mathematics) or didactics of the history of mathematics (if university level studies raise enough specificity of this field)? I do not know yet, but one thing is certain: as the relationships between a science, its didactics and its history are complex and much richer than those expressed in this short article, whatever its name is, it's at its best that didactics studies on the history of mathematics in mathematics education stay in touch with all the related disciplines (mathematics, didactics of mathematics and history of mathematics) and all the recent research results.

  Student: He [Archimedes] wrote a book named Elements. (Il a écrit un livre qui s'appelle Elements.) Teacher: Oh, you're mistaking between another Greek mathematician. (Ah, vous confondez avec un autre mathématcien grec.) Student: Oh no, that was Euclid. (Ah nan, c'était Euclide.) Teacher: Very well, can you tell me anything about Euclid? (Très bien, tu peux m'en parlé d'Euclide?) Student: He has... taken up again point by point all the mathematics. (Il a … il a repris point par point toutes les mathématiques.) Teacher: Waow. Yeah. (Oh dis donc, ouais.) Student: And he wrote Elements. (Et il a écrit Elements.) Teacher: The Elements... what was it? (Les Elements... c'était quoi en fait cette histoire d'Elements?) Student: Well, all the maths. (Ben, tout les maths.) Teacher: That is to say? What did he do in fact? One can say that his book is what? A kind of... (C'est-à-dire? Il a fait quoi en fait ? On peut considérer que son ouvrage c'est quoi? Une sorte de...) Student: Textbook. (De manuel.) Teacher: A mathematics textbook. Good. (De manuel de math. Bon.)

Les didactiques « s'intéressent principalement aux apprentissages de contenus spécifiés disciplinairement », Y. Reuter (ed.), Dictionnaire des concepts fondamentaux des didactiques, de Boeck,

2007, 17. 2 L'interdisciplinarité se définie comme« la mise en relation de deux ou plusieurs disciplines scolaires qui s'exerce à la fois aux niveaux curriculaire, didactique et pédagogique et qui conduit à l'établissement de liens de complémentarité ou de coopération, d'interpénétrations ou d'actions réciproques entre elles sous divers aspects (finalités, objets d'étude, concepts et notions, démarches d'apprentissages, habiletés techniques, etc…), en vue de favoriser l'intégration des processus d'apprentissage et des savoirs chez les élèves. », Lenoir & Sauvé, De l'interdisciplinarité scolaire à l'interdisciplinarité dans la formation à l'enseignement: un état de la question, Revue française de pédagogie, 1998, n°124, 121-153.