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ἄλλῳ τινὶ τρόπῳ The Use of History of Mathematics in Mathematics Education

2nd Thematic Issue Florina, May 2016

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ABOUT MENON ABOUT MENON EDITOR The scope of the MENON is broad, both . CHARALAMPOS LEMONIDIS in terms of topics covered and University Of Western Macedonia, disciplinary perspective, since the Greece journal attempts to make connections between fields, theories, research EDITORIAL BOARD methods, and scholarly discourses, and . ANASTASIA ALEVRIADOU welcomes contributions on humanities, University Of Western Macedonia, social sciences and sciences related to Greece educational issues. It publishes original empirical and theoretical papers as well . ELENI GRIVA as reviews. Topical collections of articles University Of Western Macedonia, appropriate to MENON regularly Greece appear as special issues (thematic . SOFIA ILIADOU-TACHOU issues). University Of Western Macedonia, This open access journal welcomes Greece papers in English, as well in German . DIMITRIOS PNEVMATIKOS and French. Allsubmitted manuscripts University Of Western Macedonia, undergo a peer-review process. Based Greece on initial screening by the editorial board, each paper is anonymized and . ANASTASIA STAMOU reviewed by at least two referees. University Of Western Macedonia, Referees are reputed within their Greece academic or professional setting, and come from Greece and other European countries. In case one of the reports is negative, the editor decides on its MENON © is published at publication. UNIVERSITY OF WESTERN Manuscripts must be submitted as MACEDONIA – FACULTY OF electronic files (by e-mail attachment in EDUCATION Microsoft Word format) to: [email protected] or via the Submission Reproduction of this publication for educational Webform. or other non-commercial purposes is authorized Submission of a manuscript implies that as long as the source is acknowledged. Readers may print or save any issue of MENON as long it must not be under consideration for as there are no alterations made in those issues. publication by other journal or has not Copyright remains with the authors, who are been published before. responsible for getting permission to reproduce any images or figures they submit and for providing the necessary credits.

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SCIENTIFIC BOARD LIST OF REVIEWERS The Editor and the Editorial . Barbin Evelyne, University of Nantes, France Board of the MENON: Journal . D’ Amore Bruno, University of Bologna, Italy Of Educational Research thanks . Fritzen Lena, Linnaeus University Kalmar Vaxjo, the following colleagues for their Sweeden support in reviewing . Gagatsis Athanasios, University of Cyprus, Cyprus manuscripts for the current . Gutzwiller Eveline, Paedagogische Hochschule von issue. Lucerne, Switzerland . Konstantinos Christou . Harnett Penelope, University of the West of England, . Charalampos Lemonidis United Kingdom . Konstantinos Nikolantonakis . Hippel Aiga, University of Munich, Germany . Hourdakis Antonios, University of Crete, Greece . Iliofotou-Menon Maria, University of Cyprus, Cyprus . Katsillis Ioannis, University of Patras, Greece . Kokkinos Georgios, University of Aegean, Greece . Korfiatis Konstantinos, University of Cyprus, Cyprus . Koutselini Mary, University of Cyprus, Cyprus . Kyriakidis Leonidas, University of Cyprus, Cyprus . Lang Lena, Universityof Malmo, Sweeden . Latzko Brigitte, University of Leipzig, Germany . Mikropoulos Anastasios, University of Ioannina, Greece . Mpouzakis Sifis, University of Patras, Greece . Panteliadu Susana, University of Thessaly, Greece . Paraskevopoulos Stefanos, University of Thessaly, Greece . Piluri Aleksandra, Fan S. Noli University, Albania . Psaltou -Joycey Angeliki, Aristotle University of Thessaloniki, Greece . Scaltsa Matoula, AristotleUniversity of Thessaloniki, Greece . Tselfes Vassilis, National and KapodistrianUniversity of Athens, Greece . Tsiplakou Stavroula, Open University of Cyprus, Cyprus . Vassel Nevel, Birmingham City University, United Kingdom . Vosniadou Stella, National and Kapodistrian University of Athens, Greece . Woodcock Leslie, University of Leeds, United Kingdom

Design & Edit: Elias Indos

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EDITOR'S INTRODUCTORY NOTE

INTRODUCTION TO THEMATIC ISSUE “The Use of History of Mathematics in Mathematics Education”

The question of the integration of the History of Mathematics in Mathematics Education has been discussed since the 20th century by Educators (Barwell, Brousseau, Freudental, Piaget), Philosophers (Bachelard), Mathematicians (Klein, Poincare), and Historians of Mathematics (Loria, Smith), who have supported the proposal and have given arguments on the interest and challenges in school Mathematics courses. Since the 1960s the use of the history of mathematics in mathematics education has become more popular and many papers in scientific journals, books, proceedings of conferences and groups of researchers have focused on this in contrast to the paradigm of the “modern mathematics” reform. We can find many didactical situations, mathematical problems, teaching series but also empirical and theoretical studies, Master and Phd level dissertations on the role and the ways of using historical, social and cultural elements in the teaching of mathematics. During the 2nd International Congress on Mathematics Education (ICME) in 1972 we have the creation of an International research group (International study group on the relation between the History and Pedagogy of Mathematics (HPM)) which organizes a congress every 4 years. The idea of a European Summer University (ESU) on the Epistemology and History in Mathematics Education started from the Instituts Universitaires de Formation de Maîtres (IUFM) in France, and an ESU is organized every three years in different European countries. Since 2009 in the context of the Congress of the European Society for Mathematics Education (CERME) we have also the appearance of a discussion group on The Role of History of Mathematics in Mathematics Education: Theory and Research (WG 12). This group also concentrates on empirical research. We should also mention the publication of the ICMI study History in mathematics education: the ICMI study (Fauvel & van Maanen, 2000) which presents the state of the art until this period. Since the publication of this study, researchers address in a more demanding way questions about the efficacy and pertinence of many efforts (examples) of applications in classrooms. They are also wondering about the transferability of positive experiences from educators on different levels of education. They are considering questions on the capacity of students but also of educators when they were in front of the difficulties of studying the historical aspect of many notions. Recently researchers΄ activities are moving to investigations in terms of didactic and educational foundations from which they believe that it could be

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MENON 5 ©online Journal Of Educational Research possible to think better about the role of the history of mathematics in the teaching and learning of mathematics and the development of theoretical and conceptual frameworks which could provide the required equipment for the production of finer and more focus investigations. These issues include, among others, the educational and teaching foundations of a cultural-historical perspective in the classroom, the need to give voice to community stakeholders about the introduction and more broadly, the nature and the terms of the empirical investigation prevailing in the research environment. Parallel to these advancements in research, an attempt to humanize Mathematics is increasingly present in the mathematics curricula worldwide. For over 20 years, the presence of the history of mathematics in training teachers’ environments has increased considerably in many countries. However, despite the different objectives associated with the introduction of the history of mathematics in training mathematics teachers, this presence, implicit or explicit, took the form of specific initiatives for each establishment of teacher training. By browsing through the literature since 1990, it is possible to classify the empirical studies on the use of history in the mathematics classroom into two categories: studies that relate to the narrative of grounded experiences and quantitative studies on a larger scale. Overall, it appears necessary to restore the research field on the introduction of History in the teaching and learning of mathematics within Didactics of mathematics and more generally with the educational sciences. This repositioning should enable research to get inspired from the contexts of the exploratory work from Humanities as well as theoretical, conceptual and methodological issues from the Didactics of mathematics and educational sciences. This issue includes eight invited papers. Six papers are written in English and two in French. Each text is accompanied by an abstract in English. The following papers discuss specific issues in the domain of Using History of Mathematics in Mathematics Education and are ordered according to the instructional level; from elementary school to the university and in service teachers training. . Evelyne Barbin suggests a new thinking on technique, proposed in the texts of Simondon and Rabardel. Her purpose, in introducing an historical instrumental approach of geometrical teaching for students aged 11-14 years, is to show how an instrument can be conceived both as an invention to solve problems and as a knowledge or theorem in action. In particular, she stresses the links between different varieties of instruments and different kinds of knowledge and shows the consequences of an instrumental failure for the construction of new knowledge. Her goal is a coherent using where teaching is based on families of instruments.

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. Matthaios Anastasiadis and Konstantinos Nikolantonakis describe the context of an instructional intervention focused on isoperimetric figures and area-perimeter relationships with the use of one historical note and two primary sources, from Pappus’ Collection and from Polybius’ Histories. Their findings are based on classroom observations, worksheets and interviews with sixth grade Greek students. . Vasiliki Tsiapou and Konstantinos Nikolantonakis present part of a research study that intended to use the Chinese abacus for the development of place value concepts and the notion of carried number with sixth grade Greek students. . Ingo Witzke, Horst Struve, Kathleen Clark and Gero Stoffels describe how the concepts of empirical and formalistic belief systems can be used to give an explanation for the transition from school to university mathematics during an intensive Seminar. They stress the usefulness of this approach by outlining the historical sources and the participants’ activities with the sources on which the seminar is based, as well as some results of the qualitative data gathered during and after the seminar. . David Guillemette tries to highlight some difficulties that have been encountered during the implementation of reading activities of historical texts in the preservice teachers training context. He describes a history of mathematics course offered at the Université du Québec à Montréal, with reading activities that have been constructed and implemented in class and the efforts made by the students and the trainer to articulate both synchronic and diachronic reading, in order to not uproot the text and his author from their socio-historical and mathematical context. . Michael Kourkoulos and Constantinos Tzanakis present and analyze a teaching work on Pascal's wager realized with Greek students, prospective elementary school teachers, in the context of a probability and statistics course. They focus on classroom discussion concerning mathematical modeling activities, connecting elements of probability theory and decision theory with elements of philosophical discussions. . Areti Panaoura examines in-service teachers’ beliefs and knowledge about the use of the history of mathematics in the framework of the inquiry-based teaching approach at the educational system of Cyprus, and the difficulties teachers face in adopting and implementing this specific innovation in primary education. . Snezana Lawrence offers ideas for teachers to engage with mathematics through the historical ‘journeys’ and relationship with art and cultural and intellectual history. She treats the question of how teachers could find their own ‘mathematical’ voice through series of historical investigations and what impact that may have on their teaching and pupils’ progress.

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Aknowledgements Firstly, I would like to express my warmest thanks to Christina Gkonou1 for her precious efforts to read and ameliorate the English texts. Secondly, I would also like to express my thanks to the Editorial Committee of Menon Journal for giving me the chance to work this Thematic Issue on the field of Using History in Mathematics Education. Finally, I would like to express my grateful thanks to my Colleagues who sustain with their papers this publication.

The Editor of the 2nd Thematic Issue of MENON: Journal for Educational Research

Konstantinos Nikolantonakis

Associate Professor University of Western Macedonia Greece

1 Christina Gkonou is Lecturer in Teaching English as a Foreign Language in the Department of Language and Linguistics at the University of Essex, UK. She received her BA from Aristotle University and her MA and PhD from the University of Essex. Her research interests are in foreign language pedagogy and the psychology of language learning and teaching.

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CONTENTS CONTENTS

9-30 Εvelyne Barbin L’INSTRUMENT MATHÉMATIQUE COMME INVENTION ET CONNAISSANCE-EN-ACTION

31-50 Matthaios Anastasiadis, Konstantinos Nikolantonakis PRIMARY SOURCES AND HISTORY-BASED PROBLEMS ABOUT ISOPERIMETRY: A USE OF MATHEMATICS HISTORY IN GRADE SIX

51-65 Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE CHINESE ABACUS

66-93 Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

94-111 David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE LECTURES DE TEXTES HISTORIQUES

112-129 Michael Kourkoulos, Constantinos Tzanakis DISCUSSING MATHEMATICAL MODELING CONCERNING PASCAL'S WAGER

130-145 Areti Panaoura THE HISTORY OF MATHEMATICS DURING AN INQUIRY-BASED TEACHING APPROACH

146-158 Snezana Lawrence THE OLD TEACHER EUCLID: AND HIS SCIENCE IN THE ART OF FINDING ONE’S MATHEMATICAL VOICE

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L’INSTRUMENT MATHÉMATIQUE COMME INVENTION ET CONNAISSANCE-EN-ACTION

Εvelyne Barbin Laboratoire LMJL & IREM - Université de Nantes [email protected]

ABSTRACT The role of instruments had been underestimated widely in history, including in the case of the geometry, and that is linked with the Aristotelian partition between theory and technique. In this paper we work with a new thinking on technique, proposed recently in the texts of Simondon and Rabardel. To introduce an instrumental approach of geometrical teaching for students aged 11-14 years, we choose to examine beginnings of a geometrical thought in history. Our purpose is to show how an instrument can be conceived both as an invention to solve problems and as a knowledge or theorem in action. With some examples, we analyze the dynamical process by which an instrument can be involved in the introduction of geometrical notions and in the construction of mental schemes. In particular, we stress on the links between different varieties of instruments and different kinds of knowledge and we show the consequences of an instrumental failure for construction of new knowledge. Our goal is not a heteroclite using of instruments in teaching but a coherent using where teaching is based on families of instruments. Keywords: geometry, instruments, measurement of distances, technics, trisection of angle

1. INTRODUCTION Le rôle des instruments dans l’histoire des mathématiques a été largement sous-estimé, y compris pour ce qui concerne l’histoire de la géométrie. Plus largement, nous avons été longtemps tributaires de la séparation aristotélicienne entre la technique qui est ‘poïétique’, c’est-à-dire du côté de l’action, de la science, qui est ‘théorétique’, c’est-à-dire du côté de la contemplation et de la spéculation (Aristote 1991: 4-9). C’est ainsi que, par exemple, tout rôle des techniques dans la révolution scientifique du XVIIe siècle (Barbin 2006: 9-44) a été refusé par Alexandre Koyré. Les figures de la géométrie grecque ont été rattachées à une conception purement idéale, qui est héritée d’écrits platoniciens et qui les rattache uniquement au discours axiomatique des Éléments d’Euclide. Une nouvelle pensée de la technique a été proposée par le philosophe Gilbert Simondon, qui écrit dans son ouvrage Du mode d’existence des objets techniques:”il semble que cette opposition entre l'action et la contemplation, entre l'immuable et le mouvant, doive cesser devant l'introduction de l'opération technique dans la pensée philosophique comme terrain de réflexion

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et même comme paradigm” (Simondon 1969: 256). Dans cet article, nous reprenons les réflexions de Simondon sur les objets techniques pour les rapporter aux instruments mathématiques dans leur histoire, ainsi que celles du psychologue Pierre Rabardel qui publie en 1995 Les hommes et les technologies: approche cognitive des instruments contemporains, où il fait état des écrits de Simondon et de travaux concernant le travail, la connaissance et l’action. L’ouvrage de Rabardel a subi une transposition didactique dans des écrits récents, qui tendent à simplifier, à réifier, et à mettre de côté les propos de l’auteur, sur le sujet connaissant et sur ‘les autres’, propos qui intéressent en revanche l’épistémologie et l’histoire des mathématiques. Pour servir à une approche instrumentale de l’enseignement, nous avons choisi de nous restreindre à des instruments correspondant aux débuts de la construction d’une pensée géométrique dans l’histoire, et qui s’adressent à l’enseignement des élèves du cycle 3 en France (9 ans-12 ans).

2. L’INSTRUMENT COMME INVENTION ET L’ÉDIFICATION DE LA GÉOMÉTRIE Un instrument mathématique, comme tout instrument technique, apparaît d’emblée comme le résultat d’une invention et son fonctionnement suppose, pour être possible, cette invention (Barbin 2004: 26-27). Simondon écrit à propos de l’objet technique: “l’objet qui sort de l’invention technique emporte avec lui quelque chose de l’être qui l’a produit […] ; on pourrait dire qu’il y a de la nature humaine dans l’être technique” (Simondon 1969: 248). Cette approche indique que l’instrument peut, mieux que le discours, apporter une forme dynamique à la connaissance qui est sous-jacente au fonctionnement d’un instrument. De plus, l’invention, tout comme la science, est la réponse à un problème. Rabardel écrit à propos de l’artefact: “l’artefact concrétise une solution à un problème ou à une classe de problèmes socialement poses” (Rabardel 1995: 49). Pour lui, l’artefact désigne largement toute chose transformée par un humain, tandis que l’instrument désigne “l’artefact en situation dans un rapport à l’action du sujet, en tant que moyen de cette action” (Rabardel 1995: 49). L’artefact n’est donc pas ‘un outil nu’, comme l’écrit Luc Trouche (Trouche 2005: 265), dans la mesure où il porte avec lui la solution à un problème et, activé dans une situation analogue à celle qui a présidé à son invention, il devient un instrument de réponse à ce problème. En accord avec l’importance que nous accordons au problème, nous considérons donc que c’est l’enseignement qui donnera son sens à l’instrument et non l’instrument qui donnera le sien à l’enseignement, à l’instar de ce que Simondon écrit à propos des rapports entre le travail et l’objet technique. L’histoire du baromètre, que l’on attribue au physicien Ernest Rutherford, est une manière amusante d’illustrer ce propos. Elle raconte que l’on a demandé à un étudiant de mesurer la hauteur d’un immeuble à l’aide d’un baromètre. L’étudiant est monté en haut de l’édifice et ayant attaché le baromètre à une corde, il a descendu la corde et l’a remontée pour mesurer la longueur de la

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corde descendue. Le professeur le recale, mais il lui donne une chance de se rattraper: il faut qu’il fasse preuve de connaissance physique. L’étudiant monte alors en haut de l’édifice et laisse tomber le baromètre, il mesure le temps de chute avec un chronomètre et il applique la loi de chute pour trouver la longueur de la chute. Il est admis à l’examen et il indique qu’il a d’autres réponses: en faisant osciller le baromètre comme un pendule ou en comparant la hauteur de l’ombre du baromètre à celle de l’immeuble. L’étudiant ajoute que la meilleure solution est de sonner chez le concierge de l’immeuble et de lui dire: “si vous me donnez la hauteur de l’immeuble, je vous donne ce superbe baromètre”. Rappelons que lorsque Blaise Pascal a fait entreprendre l’expérience du Puy de Dôme, son problème n’était pas d’en mesurer la hauteur, mais de montrer qu’il y avait du vide en haut du tube du dispositif de Torricelli. Quels sont les problèmes qui ont accompagné la genèse d’une science géométrique? Le terme de géométrie signifie ‘mesure de la terre’, il renvoie à l’arpentage, qui consiste, pour mesurer les terrains, à reporter un bâton et à compter le nombre de reports. Mais la géométrie grecque a été au-delà de l’arpentage. Les historiens attribuent aux Ioniens, au VIe siècle avant J.-C., la solution du problème de déterminer la distance d’un bateau en mer. L’arpentage avec un bâton est inadéquat, mais”quand les techniques échouent la science est proche” (Simondon 1969: 246). Pour résoudre le problème, il faut ruser: les Ioniens ont utilisé un dioptre, c’est-à-dire un instrument de visée, qui pouvait être un cadran sur lequel tourne une partie flexible autour d’une partie maintenue verticale grâce à un fil à plomb (fig. 1). En montant sur un endroit élevé, il est possible de faire une visée vers le bateau en orientant la partie flexible du dioptre. Ensuite, il faut se retourner en gardant la même inclinaison et viser un point sur le sol. Deux nouveaux gestes pour résoudre le problème: une visée et un retournement qui balaie l’espace. Le problème est résolu parce que le sujet géomètre a remplacé le ‘schème primitive’, celui du report de l’arpentage, par un processus d’instrumentation au sens de Rabardel. Un nouveau schème est formé, qui englobe non seulement des mesures de distances mais des visées, qui relie des visées et des distances. En quoi consiste ce schème? Par quel processus l’instrument et le nouveau schème sont-ils potentiellement porteurs d’une connaissance géométrique?

Figure 1. Le cadran ionien

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La géométrie a pour objet de voir et de faire voir ce que l’on pense. Il faut d’abord représenter la situation. En détachant du réel les éléments essentiels à sa compréhension, on réalisera un schéma (fig. 2), puis une mise en figure composée de droites permettra de connecter ces éléments essentiels (fig. 3). Sur cette figure, certaines droites représentent des distances concrètes, mais pas celles qui correspondent aux rayons visuels. Pour tenir un discours qui explique la solution à un autre (qui le demanderait), il faut dire ce qui est maintenant représenté par un espace entre deux droites et qui correspond à ce qui est une ‘vise’ dans le contexte instrumental. Cet espace a une signification dans le contexte du problème et il est relié à une distance: on l’appellera un ‘angle’. La notion d’angle est attribuée aux Ioniens. Cette notion n’est pas présente dans les mathématiques égyptiennes, dont ont hérité les Grecs, y compris dans les problèmes de pente de pyramide.

Figure 2. La distance d’un bateau en mer: schéma

Figure 3. La distance d’un bateau en mer: figure

Le schème consiste en une connaissance sur la figure: l’égalité des angles implique l’égalité des distances. Nous appellerons schème géométrique (ou simplement schème) une connaissance qui coordonne des éléments d’une configuration géométrique particulière, et qui peut être activée, transformée ou généralisée par re-connaissance de cette configuration dans des situations variées. Pour démontrer (à un autre qui n’en serait pas convaincu) que l’égalité des angles implique l’égalité de droites, il faudrait encore introduire les notions de triangle et d’égalité de triangles, puis des lettres pour désigner les éléments de la figure. La géométrie qui s’édifie ainsi est une science qui raisonne sur des

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grandeurs pour les comparer. La dioptre est une connaissance-en-action parce que son fonctionnement demande l’effectuation de gestes et l’activation d’un schème qui reprennent ceux de l’invention et qui seront repris dans d’autres situations problématiques. Ici, la connaissance instrumentale et la science procèdent de manière identique. Examinons maintenant l’instrument scolaire qui est associé à l’angle, c’est-à- dire le rapporteur (fig. 4), et son usage. Il est demandé aux élèves de mesurer des angles, c’est-à-dire de dire des nombres qui correspondent (plus ou moins) à un angle dessiné ou de dessiner des angles qui valent 30°, 45°, etc. Le rapporteur est un outil qui sert essentiellement dans le contexte de dessins de figures qui n’ont pas toujours ou peu un statut de représentation d’une situation. Tant qu’il reste dans ce cadre numérique étroit, le rapporteur est peu susceptible de provoquer des raisonnements géométriques. Il vaut d’ailleurs mieux que l’élève l’oublie quand il lui sera adjoint de ‘démontrer’. Il en va différemment si un élève demande, ce qui n’est pas rare, pourquoi les angles sont mesurés de la même façon, quelle que soit la taille du rapporteur. La réponse à cette question est une connaissance: le rapport de l’arc intercepté par un angle au centre à la circonférence tout entière est le même, quel que soit le rayon du cercle. Avec cette réponse, le rapporteur devient une connaissance-en- action.

Figure 4. Un rapporteur

Avec les deux instruments examinés, nous avons abordé le rôle de ‘l’autre’, qui dans chacun des deux cas questionne. Cette intrusion n’est pas artificielle dans l’histoire, où les instruments sont inventés et discutés par des hommes. Lorsque Rabardel décrit les relations entre les trois pôles constitués par le sujet, l’instrument et l’objet, il indique bien la composante essentielle qui est l’environnement. Puis plus loin, il enchérit avec un ‘modèle’ incluant les autres sujets. Ce modèle SACI (fig. 5) des ‘situations d’activités collectives instrumentées’ devrait également attirer l’intérêt des didacticiens (Rabardel 1995: 62).

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Figure 5. Modèle SACI d’après Pierre Rabardel

3. GENÈSE INSTRUMENTALE ET CONNAISSANCE-EN-ACTION L’invention d’un instrument à partir d’un autre et la mise en connexion des instruments entre eux sont deux processus que nous pouvons explorer dans l’histoire des mathématiques. Nous les analyserons en reprenant les notions d’instrumentation et d’instrumentalisation proposées par Rabardel qui concernent la production de nouveaux artefacts et de nouveaux schèmes. Il écrit : ”un processus de genèse et d’élaboration instrumentale, porté par le sujet et qui, parce qu’il concerne les deux pôles de l’entité instrumentale, l’artefact et les schèmes d’utilisation, a lui aussi deux dimensions, deux orientations à la fois distinguables et souvent conjointes : l’instrumentalisation dirigée vers l’artefact et l’instrumentation relative au sujet lui-même” (Rabardel 1995 : 109). Il caractérise le premier processus comme “un processus d’enrichissement des propriétés de l’artefact par le sujet” (Rabardel 1995: 114) ou encore comme une transformation de l’artefact par le sujet. Tandis qu’il caractérise le processus d’instrumentation en constatant que “la découverte progressive des propriétés (intrinsèques) de l’artefact par les sujets s’accompagne de l’accommodation de leurs schèmes, mais aussi de changements de signification de l’instrument résultant de l’association de l’artefact à de nouveaux schemes” (Rabardel 1995:116). Le schéma ci-dessous (fig. 6) indique que les deux processus sont effectivement ‘portés par le sujet’ et orientés vers le sujet ou l’artefact, “dans un même processus de genèse et d’élaboration instrumentale”.

Figure 6. La genèse instrumentale

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Avec la transposition en didactique effectuée par Luc Trouche, la place du sujet n’est plus la même et le processus d’instrumentalisation va de l’artefact vers le sujet (fig. 7). Pourtant Rabardel precise: “ces deux types de processus sont le fait du sujet. L’instrumentalisation par attribution d’une fonction à l’artefact, résulte de son activité, tout comme l’accommodation de ses schèmes. Ce qui les distingue c’est l’orientation de cette activité. Dans le processus d’instrumentation elle est tournée vers le sujet lui-même, alors que dans le processus corrélatif d’instrumentalisation, elle est orientée vers la composante artefact de l’instrument” (Rabardel 1995: 111-112).

Figure 7. La genèse instrumentale d’après Trouche

Trouche écrit que “Rabardel distingue, dans la genèse d’un instrument, deux processus croisés, l’instrumentation et l’instrumentalisation: l’instrumentalisation est relative à la personnalisation de l’artefact par le sujet, l’instrumentation est relative à l’émergence des schèmes chez le sujet (c’est-à- dire à la façon avec laquelle l’artefact va contribuer à préstructurer l’action du sujet, pour réaliser la tâche en question)” (Trouche 2015: 267). Les termes en italiques sont le fait de l’auteur, mais celui-ci n’indique pas de pagination en référence à l’ouvrage de Rabardel. Les conceptions d’un sujet qui ‘personnalise’ l’artefact, tandis que l’artefact ‘préstructure’ l’action du sujet, doivent être rapprochées du projet de l’auteur, à savoir « de guider et intégrer les usages des outils de calcul dans l’enseignement mathématiques ». En effet, qu’il s’agisse de calculateur ou d’ordinateur, le sujet ne peut prétendre modifier ce que nous pouvons appeler ‘machines’, plutôt qu’artefacts. Tandis que les exemples nombreux donnés par Rabardel, y compris dans le cadre de formation de sujets, concernent effectivement les modifications des artefacts et des schèmes.

3.1 De l’outil à l’instrument L’analyse de processus historiques de modifications d’artefacts permet d’approfondir la notion d’instrument comme connaissance-en-action. En effet, il n’y a pas dans l’histoire de simultanéité de tous les instruments mais passage de l’un à l’autre, avec parfois des crises ou des ruptures. En reprenant ce qu’écrit Simondon à propos de la technique, nous dirons que l’éducation mathématique ne doit pas “manquer ces dynamismes humains”, “il faut avoir saisi l’historicité du devenir instrumental à travers l’historicité du devenir du

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sujet” (Simondon 1969: 107-109). Nous possédons peu de témoignages ou de traces des premiers instruments de la géométrie. De ce point de vue, l’ouvrage Géométrie de Gerbert d’Aurillac, datant de l’an 1000, possède un rôle vicariant. L’auteur a été pape en Avignon, il a voyagé en Espagne et il a ainsi pu connaître les sciences mathématiques arabes. Il explique dans son ouvrage comment mesurer la largeur d’une rivière avec un bâton, il s’agit donc encore ici d’un ‘problème de distance inaccessible’. Les gestes à effectuer sont les suivants: Gerbert plante son bâton sur le bord de la rivière, il s’éloigne du bord jusqu’à ce que son œil, l’extrémité du bâton et l’autre bord de la rivière soient alignés. Comme précédemment, nous réalisons un schéma qui représente la situation, puis une mise en figure lettrée qui permet de formuler le schème opérant à condition d’adopter une échelle de proportion (fig. 8). La configuration est constituée de deux triangles emboîtés pour lesquels le rapport des côtés BD à CD est égal au rapport de BP à OP, on a la proportion BD : CD :: BP : OP. Ce schème permet d’obtenir la distance BP, qui est la somme de BD et de DP, à l’issue d’un calcul sur les grandeurs.

Figure 8. La largeur de la rivière par Gerbert

Ce schème intervient aussi dans le fonctionnement de la ‘lychnia’ (lanterne), présentée au IIe siècle dans les Cestes de Jules l’Africain. Il s’agit d’une accommodation pratique du bâton, plutôt que d’une genèse instrumentale: elle comporte un bâton muni à son sommet d’un autre bâton qui peut tourner et qui permet ainsi d’effectuer des visées plus aisées (fig. 9). Dans le traité Sur la Dioptre, datant du Ier siècle, Héron d’Alexandrie présente un dioptre assez semblable à la lychnia et il lui associe un autre outil: un poteau muni d’un disque, qui peut coulisser le long du poteau. Il résout de nombreux problèmes de distances inaccessibles (Barbin 2016) : mesurer des différences de niveaux, joindre deux lieux qui ne sont pas visibles l’un pour l’autre, creuser un tunnel connaissant ses extrémités, mesurer l’aire d’un champ en restant à l’extérieur du champ, etc. L’adjonction de poteaux constitue une instrumentalisation, elle ne modifie pas le schème primitif des triangles emboîtés. Mais la complexité des problèmes s’accompagne de celle des figures, et les raisonnements demandent

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‘d’imaginer’ de nombreuses droites qui ne représentent pas d’objets tangibles (fig. 10).

Figure 9. La lychnia de Jules l’Africain et la dioptre d’Héron d’Alexandrie

Gerbert écrit que “un géomètre doit toujours avoir un bâton avec lui”, mais la possession de cet outil ne suffit pas pour obtenir la solution. Il faut de plus un raisonnement extérieur à l’outil, qui est singulier pour chacune des utilisations de l’outil. Examinons de ce point de vue un autre instrument de Gerbert. Il est composé de deux bâtons, solidaires et perpendiculaires, dont les trois parties ainsi déterminées sont égales (fig. 10). Pour mesurer la hauteur d’un édifice, Gerbert aligne l’extrémité du bâton horizontal, le haut du bâton vertical et le haut de l’édifice. Le schème précédent permet d’obtenir l’égalité de BH et HE, et donc AB est égal à la somme de HE et FC. La distance HE est accessible par arpentage et si FC est égal à 1 (par exemple), alors AB égale HE + 1. Notons que, pour obtenir la solution, il faut adjoindre à la figure une droite HE, qui est le témoin de la ruse et de la connaissance du géomètre. Cette droite ne représente aucun élément tangible, elle est ‘imaginative’.

Figure 10. L’instrument de Gerbert

Nous dirons que nous avons affaire ici à un instrument, parce que Gerbert incorpore dans la conception de son instrument une connaissance du géomètre: l’instrument est instruit. Le mot instrument vient du mot latin instrumentum, qui signifie matériel, outillage ou ressource et qui dérive du verbe instruere. Ce verbe,

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francisé en enstruire, donne disposer, outiller et équiper. Ainsi, les mots instrument et instruire renvoient l'un à l'autre (Barbin 2004: 7-12). Le passage du bâton à l’instrument peut-être compris comme un processus d’instrumentation, car l’instrument incorpore le schème dans sa conception. Celui qui l’utilisera tiendra en main une connaissance-en-action.

3.2 Connexions entre instruments et connaissances Dans sa Protomathesis de 1532, Oronce Fine présente un instrument que nous appelons aujourd’hui ‘équerre articulée’. Il est géomètre, astronome et cartographe, il a enseigné les mathématiques au Collège Royal de Paris et il publiera en 1556 un ouvrage de géométrie intitulé De re & Praxi geometrica. Depuis le XIIIe siècle, les Éléments d’Euclide sont connus en Occident par une traduction latine d’une traduction arabe et ils sont imprimés en 1482. Fine cite le texte euclidien lorsqu’il présente son équerre articulée (Fine 1532: 67).

Illustration. Extrait de la Protomathesis d’Oronce Fine

L’instrument est composé d’un bâton qui sera dressé verticalement et de deux bâtons perpendiculaires l’un à l’autre (les alidades) fixés au sommet du bâton et qui peuvent tourner autour. Pour mesurer, par exemple, la largeur d’une rivière, il faut poser l’instrument au bord de la rivière et viser à l’aide d’une alidade l’autre bord de la rivière, puis viser à l’aide de la seconde alidade un point qui se trouve de notre côté de la rivière, mais en terre ferme (fig. 11). La distance entre ce point et la base du bâton est connue, ainsi que la hauteur du bâton. Ceci suffit à connaître la largeur de la rivière.

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Figure 11. La largeur d’une rivière avec l’équerre articulée

En effet, si nous représentons sur une figure les droites intervenant dans la situation, nous pouvons en extraire un triangle rectangle ABC et sa hauteur AH (fig. 12). Cette configuration permet de formuler un nouveau schème, qui correspond à l’un des théorèmes appartenant à la figure. En effet, ‘le théorème de la hauteur du triangle rectangle’ affirme que, dans un triangle rectangle avec l’angle droit en A et AH la hauteur, on a BH : AH :: AH : HC ou encore, la hauteur AH égale le produit des segments déterminés sur la base, AH2 = BH  HC. Par conséquent, HC s’obtient à partir de BH et AH, qui nous sont connus, et si AH = 1 alors HC = 1 : BH. Ce théorème est la proposition 8 du Livre VI d’Euclide (Euclide 1994: 176-179), il est déduit de la similarité des triangles ABH et CAH, car deux triangles semblables (qui ont leurs angles égaux) ont leurs côtés proportionnels. L’équerre articulée est une connaissance-en-action, celle du théorème de la hauteur du triangle rectangle. Sa genèse correspond à la fois à un processus d’instrumentation, car le schème correspond à une forme plus complexe que celle des triangles emboîtés, et à un processus d’instrumentalisation puisque l’usage de l’instrument est amélioré. Le nouveau schème est une connaissance géométrique qui pourra intervenir dans d’autres instruments. Rabardel écrit à ce propos que “L’instrument est un moyen de capitalisation de l’expérience accumulée (cristallisée disent même certains auteurs). En ce sens, tout instrument est connaissance” (Rabardel 1995: 73).

Figure 12 . Le théorème de la hauteur d’un triangle rectangle

Nous trouvons dans l’histoire de la géométrie, qu’on appelle pratique et que nous préférons nommer instrumentale, de nombreux instruments de visée pour

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trouver des distances inaccessibles. Ils forment un monde d’individus liés les uns aux autres, dont l’exploration est bien préférable pour l’enseignement, à l’utilisation d’un seul d’entre eux. En effet, la dynamique instrumentale peut introduire un ordre des connaissances qui constituera un apprentissage dynamique de la déduction mathématique. Nous reprenons en ce sens ce que Simondon formule pour la réalisation technique: elle “donne la connaissance scientifique qui lui sert de principe de fonctionnement sous une forme d’intuition dynamique appréhensible par un enfant même jeune, et susceptible d’être de mieux en mieux élucidée, doublée par une compréhension discursive” (Simondon 1969: 109).

4. DYNAMIQUE INSTRUMENTALE ET CONSTRUCTIONS DE CONNAISSANCES Depuis la géométrie grecque, la règle et le compas sont les outils de construction des figures par excellence. Cependant, en conformité avec l’héritage aristotélicien qui sépare la poïétique de la théorétique, Euclide ne mentionne pas ces outils, ni aucun autre, mais son ouvrage contient de nombreuses constructions de figures qui sont obtenues par intersections de droites et cercles, au point qu’il peut être lu comme un ouvrage de constructions tout autant que de théorèmes. Les Éléments répondent aux préceptes aristotéliciens d’une science démonstrative, c’est-à-dire dans laquelle chaque proposition est déduite soit d’un axiome (demande ou notion commune), soit de propositions précédemment démontrées. Les premières demandes sont “de mener une ligne droite de tout point à tout point” et “de décrire un cercle à partir de tout centre et au moyen de tout intervalle” (Euclide 1994: 167-169). Les historiens ont discuté sur le rôle existentiel de ces demandes, mais, de toute façon, mener une droite et décrire un cercle sont deux opérations de base pour effectuer une construction concrète à l’aide d’outils de figures sur lesquelles le géomètre spécule et raisonne. Il apparaît dès lors difficile de bannir la considération de tout outillage dans l’interprétation des Éléments. Il y a deux sortes de propositions dans les Éléments, les constructions (ce que les Anciens appellent les problèmes) et les théorèmes. L’intrication entre les deux sortes de propositions est forte et déterminée puisqu’un théorème sur une figure ne peut pas être démontré sans que celle-ci et les lignes nécessaires à la démonstration soient construites à la règle et au compas. Il est nécessaire aussi que toute construction soit justifiée par des théorèmes démontrés précédemment. Quelle conception prévaut à cette nécessité? Nous pouvons lire une réponse dans le dialogue du Ménon de Platon qui permet de lier l’édification de la géométrie grecque à un échec, à une impossibilité de dire qui est compensée par une possibilité de montrer par une construction et par des gestes. Dans ce célèbre dialogue, Socrate expose à Ménon la théorie de la réminiscence et il fait venir un esclave pour montrer que, par de simples questions, il va conduire l’esclave à se ressouvenir. Il présente à l’esclave un

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carré de côté deux et donc d’aire quatre, puis il lui demande s’il est possible de construire un carré d’aire double. Il continue en demandant quel serait le côté d’un carré d’aire huit: “essaie de me dire quelle serait la longueur de chaque ligne dans ce nouvel espace”. L’esclave essaie donc de dire: il dit d’abord quatre, puis trois. Les deux tentatives échouent. Socrate modifie alors sa demande: “taches de me le dire exactement, et si tu aimes mieux ne pas faire de calculs, montre la nous”. Il ne s’agit plus de dire un nombre mais de montrer une figure. Socrate construit étape par étape la figure, qui permet de montrer la droite demandée. Il accole quatre carrés égaux au carré de départ, puis trace dans chacun une diagonale (fig. 13). Les quatre diagonales délimitent le carré cherché. Ainsi ce qui n'est pas dit exactement est construit exactement à la règle et au compas. Socrate déplace l’objet de l’exactitude, du nombre à la figure.

Figure 13. La construction géométrique de la duplication d’un carré

4.1 Les compas La seconde demande d’Euclide, de décrire un cercle, peut être satisfaite avec une corde ou avec un ‘compas à balustre’ (avec un crayon et une pointe). Mais un compas peut servir aussi à reporter des longueurs de segments. Dans ce cas, un ‘compas à pointes sèches’ (sans crayon) est suffisant. L’opération de report est nécessaire en géométrie, elle intervient dès les premiers théorèmes sur les triangles. Aussi, dans la proposition 2 du Livre I, Euclide demande de placer en un point donné A, un segment égal à un segment donné BC (Euclide 1994: 197). Il donne les étapes de la construction: il faut joindre A et B, construire un triangle équilatéral DAB sur AB (la construction est donnée dans la proposition 1), prolonger DA et DB, puis construire un cercle de centre B et de rayon BC et un cercle de centre D et de rayon DG (avec G intersection du cercle précédent avec le prolongement de DB) (fig. 15). Euclide démontre que l’intersection L de ce dernier cercle avec le prolongement de DA répond au problème car AL est égal à CB.

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Figure 14. Le report géométrique d’un segment

Le compas à balustre et le compas à pointes sèches sont deux outils ressemblants d’un point de vue matériel, mais leurs fonctions sont différentes, et la théorie montre comment l’une peut se ramener rationnellement à l’autre. Nous allons examiner deux autres compas que nous qualifions d’instruments, car chacun est une connaissance-en-action. Ils sont également ressemblants l’un à l’autre d’un point de vue matériel et d’un usage très ancien chez les artisans. On les retrouve décrits jusqu’à récemment, dans Le dictionnaire pratique de Menuiserie, Ébénisterie, Charpente de Justin Storck, édité au début du XXe siècle. Le ‘compas d'épaisseur’, joliment appelé ‘maître à danser’ à cause de sa forme suggestive, est composé de deux tiges égales, croisées et articulées autour de leur milieu. Il permet de mesurer le diamètre extérieur d'un cylindre ou d'un flacon en y introduisant la partie inférieure de l’instrument, les pieds du ‘maître à danser’ (fig. 15).

Figure 15. Le compas d’épaisseur ou maître à danser

Le problème est encore de trouver une longueur inaccessible à une mesure exacte. Puisque les segments OA, Oa, OB et Ob sont égaux, et que les angles au sommet O sont égaux, les deux triangles OAB et Oab sont égaux (superposables) donc en mesurant AB, nous obtenons ab. La connaissance en action présente dans la conception et le fonctionnement de l’instrument correspond à la proposition IV du Livre I des Éléments d’Euclide (premier cas d’égalité de deux triangles).

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Le ‘compas de réduction est composé de deux tiges égales, croisées et articulées autour de leur intersection. La place de cette intersection est modifiable grâce à des fentes placées sur les deux tiges et une fixation (fig. 16). Ce compas permet d’obtenir une figure réduite d’une figure donnée, mais tout aussi bien agrandie. En effet, supposons par exemple que l’on veuille réduire une figure au tiers, il suffit de placer l’intersection O de telle sorte que aO et bO soient le tiers de OA et OB. Pour réduire au tiers un segment quelconque, il faut placer A et B à ses extrémités, alors a et b sont les extrémités du segment réduit. La conception et le fonctionnement du compas de réduction manifestent une connaissance-en-action, énoncée un peu plus haut. En effet, les triangles Oab et OAB sont semblables, donc leurs côtés sont proportionnels, par conséquent ab est le tiers de AB.

Figure 16. Le compas de réduction.

4.2 Échec instrumental et construction de connaissances Nous allons examiner deux problèmes qui illustrent l’expression de Simondon, “quand les techniques échouent la science est proche” (Simondon 1969: 246), et qui fournissent d’autres exemples d’inventions d’instruments et de schèmes. Ils font partie des fameux problèmes à la règle et au compas que les géomètres grecs ne sont pas parvenus à résoudre, ce sont la duplication d’un cube et la trisection d’un angle. Dès la science grecque et durant des siècles, ils vont connaître de très nombreuses solutions instrumentales et géométriques (Barbin 2014: 87-146). Nous avons choisi de présenter des solutions anciennes ou élémentaires. Le problème de la duplication du cube consiste à construire le côté d’un cube ayant un volume qui est double d’un cube donné. Il peut être considéré comme une suite du problème de la duplication d’un carré, dont la solution est obtenue à la règle et au compas grâce à la figure du Ménon, puisqu’un carré est constructible. Selon Proclus, pour parvenir à la solution pour le cube, le mathématicien grec du Ve siècle avant J.-C. Hippocrate de Chios ramène le problème à un autre problème, celui de construire deux segments qui soient moyennes proportionnelles entre un segment et son double, ou plus largement entre deux segments quelconques. En écriture symbolique, nous cherchons à construire deux segments x et y tels que a : x :: x : y :: y : b. Les Commentaires

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d'Eutocius d'Ascalon sur le traité de la sphère et du cylindre d’Archimède (Ve siècle) indiquent différentes solutions de géomètres grecs, des instruments mais aussi des constructions à l’aide des coniques, qui auraient été inventées à cet effet par Ménechme (IVe siècle avant J.-C.) (Archimède 1969: 551-718). Nous allons nous intéresser à l’instrument attribué à Platon en commençant par examiner si effectivement, comme l’écrit Ératosthène, pour Hippocrate “l’embarras fut changé en un autre et non moindre embarrass”. En effet, le problème proposé par Hippocrate est une suite de la construction d’une moyenne proportionnelle entre deux segments, qui s’effectue à la règle et au compas. Étant donnés deux segments BH et HC mis bout à bout, il suffit de construire à la règle et au compas le milieu de BC, le demi-cercle de diamètre BC et la hauteur en H à BC. Si A est l’intersection du demi-cercle et de la hauteur alors AH est la solution au problème (fig. 17 gauche). En effet, ceci résulte du théorème de la hauteur d’un triangle rectangle car le triangle ABC est inscrit dans un demi-cercle, donc il est rectangle. Cette solution indique que l’équerre est aussi un outil commode pour construire la moyenne proportionnelle à deux segments (fig. 17 droite): il suffit de placer le coin de l’équerre sur une perpendiculaire (construite avec l’équerre) en H à BC. L’équerre est un outil qui permet de mettre en action le théorème du triangle rectangle.

Figure 17. La moyenne proportionnelle avec le compas et avec l’équerre

Notons que la construction de la moyenne proportionnelle AH entre deux segments BH et HC est aussi celle du côté d’un carré de même aire que le rectangle de côtés BH et HC. Le théorème du triangle rectangle fournit donc la solution au problème de la quadrature d’un rectangle. Ce problème est une étape essentielle dans la quadrature d’un polygone établie par Euclide, il est donc légitime de rattacher le théorème du triangle rectangle et son invention à un problème de construction. L’instrument de Platon pour construire deux moyennes proportionnelles consiste en trois barres fixes, Hθ, HZ et Mθ. Les deux dernières barres sont munies de rainures, de sorte qu’une quatrième barre KΛ coulisse parallèlement à Hθ (fig. 18 gauche). Pour construire deux moyennes proportionnelles à deux segments AB et BC, on les dispose perpendiculairement l’un à l’autre (avec une équerre) et on les prolonge (avec une règle). Posons l’instrument de sorte que

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l’angle θHZ soit sur le prolongement de AB et que Hθ passe par C, puis faisons coulisser KΛ de sorte qu’elle passe par A (fig. 18 droite). Alors nous avons: BA : BK :: BK : BH :: BH : BC. En effet, dans le triangle rectangle AKH nous avons BA : BK :: BK : BH, et dans le triangle rectangle KHC nous avons BK : BH :: BH : BC. L’instrument de Platon est le résultat d’un processus d’instrumentalisation car il améliore la simple équerre et il s’appuie sur le même schème, celui de la hauteur d’un triangle rectangle.

Figure 18. L’instrument de Platon et son fonctionnement

L’invention de l’instrument résulte d’une nouvelle considération du problème de la moyenne proportionnelle, il faut s’emparer du schème qui a réussi pour le compas tout en prenant en compte l’échec du compas au-delà. Nous pouvons alors regarder l’instrument de Platon comme deux équerres coordonnées qui permettent d’aller au-delà de la simple équerre. En effet, ce redoublement répond au redoublement de la moyenne proportionnelle nécessaire pour résoudre la duplication du cube. Le passage par les instruments constitue ainsi encore une entrée dynamique dans le raisonnement déductif. L’embarras dans lequel serait tombé Hippocrate est donc profitable, comme cela est souvent le cas en mathématiques. Auprès de Ménon, Socrate soutenait l’intérêt de l’embarras de l’esclave pour l’enseignement. Le problème de la construction à la règle et au compas de la trisection de l’angle (en trois angles égaux) est également la suite d’un problème qui est constructible, celui de la bissection d’un angle (en deux angles égaux). Tenant compte de l’expérience précédente, nous examinons le schème qui autorise la réussite dans ce cas. Diviser un angle en n parties égales est équivalent à diviser en n parties égales l’arc correspondant à cet angle quand il est inscrit au centre d’un cercle. Étant donné un angle de sommet A, traçons un arc de cercle de centre A qui coupe les côtés de l’angle en B et C, il faut diviser en deux l’arc BC. Pour cela, il suffit de construire le milieu M de la corde BC en construisant la médiatrice. Traçons à partir de B et C deux arcs de cercle égaux qui se coupent en D, alors AD est la médiatrice (fig. 19 gauche). Les deux triangles AMB et AMC sont égaux car leurs trois côtés sont égaux, donc les angles BAM et CAM sont égaux. Cette construction ne va pas au-delà de la division en deux parties égales. Si nous divisions en trois parties égales la corde BC, ce qui est possible à

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la règle et au compas alors l’arc BC n’est pas divisé en trois parties égales (fig. 19 droite).

Figure 19. Division d’un angle et de la corde sous-tendue

Nous allons examiner trois instruments de trisection dont l’invention prend en compte ce qui a produit la réussite pour la bissectrice mais aussi l’échec au- delà. Les deux premiers sont des instruments d’artisans et le troisième est inventé par un mathématicien. Le ‘couteau de cordonnier’ est présenté dans la Géométrie appliquée à l’Industrie à l’usage des artistes et des ouvriers de Claude Lucien Bergery de 1828. D’après l’auteur, il était utilisé par les ouvriers messins. Le couteau est composé d’une règle BE, d’une équerre BCD et un demi-cercle de centre F et diamètre AB tels que BC est égal à BF. Pour obtenir la trisection d’un angle GHI, il suffit de poser le couteau sur l’angle, le demi-cercle étant tangent à l’un des côtés et C étant sur l’autre côté. En effet, les angles GHB, BHF et FHI sont égaux et donc valent le tiers de l’angle GHI (fig. 20). Menons HF et FI, l’angle FIH est droit. Les triangles CBH et FBH sont égaux, donc l’angle CHB est égal à l’angle BHF. Les triangles BFH et FIH sont égaux, donc l’angle BHF est égal à l’angle FHI. L’invention et le fonctionnement du couteau de cordonnier reprennent le schème primitif qui préside à la construction de la bissectrice d’un angle, à savoir l’égalité de deux triangles rectangles. L’invention du couteau contourne l’obstacle en mimant la situation des cordes égales et en introduisant un schème qui prolonge le précédent: celui qui est attaché à la configuration de trois triangles égaux.

Figure 20. Le couteau du cordonnier

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‘L’équerre du charpentier’ est présentée dans un article de Scudder intitulé “How to trisect an angle with a carpenter’s square”, paru en 1928 dans la revue American Mathematical Monthly. L’équerre est posée sur l’angle BOA, dont on cherche la trisection, de façon à tracer le long de la partie GH de l’équerre une parallèle au côté OA de l’angle. Sur l’équerre est marqué un point F tel que FH est égal à HK. L’équerre est ensuite posée sur l’angle de sorte à ce que le coin K de l’équerre soit sur la parallèle au point E, que le sommet O soit sur la partie GH de l’équerre et que le point F soit sur OB (fig. 21). Les points F, K et H sont marqués sur la figure. Traçons OH, OK et KF, la perpendiculaire à OA passant par K. Les trois angles FOH, HOK et KOF sont égaux car les trois triangles rectangles FOH, HOK et KOF sont égaux. Ainsi, bien que le couteau du cordonnier et l’équerre du charpentier soient deux instruments très dissemblables d’un point de vue matériel, la connaissance-en-action est la même.

Figure 21. L’équerre du charpentier

Un problème posé par James Watt pour améliorer le fonctionnement des machines à vapeur attire l’intérêt des mathématiciens pour ce qui sera appelé ‘systèmes articulés’, c’est-à-dire un système de tiges articulées les unes aux autres. Tout au long du XIXe siècle, ils recherchent des systèmes particuliers pour tracer les courbes ou pour résoudre des problèmes de construction (Barbin 2014: 137-139). Dans ce contexte, le mathématicien Charles-Ange Laisant introduit ‘un compas trisecteur’, qui fait l’objet d’un article d’Henri Brocard en 1875 (Brocard 1875 : 47-48). L’instrument est composé de deux losanges articulés OABC et BEDC et d’une tige rigide OBD sur laquelle D peut glisser. Pour obtenir la trisection d’un angle il suffit de poser l’instrument sur l’angle de sorte que A et E soient sur ses côtés. Alors les angles EOB, BOC et COA sont égaux et l’angle AOE est coupé en trois parties égales. En effet, les diagonales d’un losange sont perpendiculaires, donc OD est la médiatrice de EC et les triangles EOB et BOC sont égaux. La diagonale OC divise aussi le losange OBCA en deux triangles BOC et COA égaux.

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Figure 22. Le compas trisecteur de Laisant

Les trois instruments de trisections activent des schèmes similaires, mais les deux premiers sont singuliers, ils ne sont pas susceptibles de résoudre d’autres problèmes, alors que le compas trisecteur appartient à une famille d’instruments qui peuvent se coordonner les uns aux autres, de s’enrichir par l’introduction d’autres schèmes, comme pour l’inverseur de Peaucellier qui permet de résoudre exactement le problème de Watt. Avec les systèmes articulés s’ouvre la construction de courbes.

5. CONCLUSION: APPROCHE INSTRUMENTALE ET HISTORIQUE DE L’ENSEIGNEMENT Comme nous l’avons souligné à plusieurs endroits, l’invention et la genèse instrumentales permettent une entrée dynamique dans la déduction mathématique: elles définissent des schèmes opérants et elles construisent une suite ordonnée de schèmes. Le fonctionnement de l’instrument constitue une connaissance-en-action, susceptible d’être reprise ou prolongée avec l’emploi de nouveaux instruments ou l’intervention de nouveaux problèmes. Le processus d’instrumentation va souvent de pair avec le processus d’instrumentalisation, car ils correspondent tous les deux à des modifications de l’instrument. Comme l’écrit Séris pour la technique, la genèse instrumentale dépend d’une “aspiration à faire les choses autrement et mieux” (Séris 1994: 20-21). Nous rencontrons dans l’histoire deux dynamiques de la genèse instrumentale: pour un même problème, il faut inventer des instruments de plus en plus commodes, ou il faut chercher à résoudre des problèmes de plus en plus complexes. Dans l’enseignement, il semble donc nécessaire d’une part, d’introduire des instruments dont le fonctionnement est accessible et ainsi compréhensible et d’autre part, de considérer des familles d’instruments reliés les uns aux autres par des champs de problèmes et/ou des champs de schèmes. Ceci ne se restreint pas au domaine de la géométrie, qui fait l’objet unique de cet article. Examinons ces deux points dans le contexte de l’enseignement aujourd’hui. Nous avons noté que plus un instrument est porteur de nombreuses connaissances, plus son usage peut être commode et plus universel. Mais sa complexité peut alors devenir telle qu’il faille lui intégrer des mécanismes facilitant et régulant son usage. C’est ainsi que le fonctionnement de

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l’instrument peut devenir en partie ou complètement caché. Nous en avons un exemple avec l’histoire et l’enseignement des instruments de calcul, car il est long le processus qui va du boulier à l’ordinateur (Chabert, Barbin et al. 1999). En présence d’un ordinateur, l’élève sait ce qui entre dans la machine et ce qui en sort, mais non ce qui s’y fait: il s’accomplit une opération à laquelle l’élève ne participe pas même s’il la commande. Ce qu’écrit Simondon de la situation du travailleur face à une machine peut être repris ici : “commander est encore rester extérieur à ce que l’on commande, lorsque le fait de commander consiste à déclencher selon un montage préétabli, fait pour ce déclenchement, prévu pour opérer ce déclenchement dans le schéma de construction de l’objet technique”. Pour lui, l’aliénation du travailleur, qui résulte de cette extériorité, réside dans la rupture qui se produit entre la genèse et l’existence de l’objet technique: “il faut que la genèse de l’objet technique fasse effectivement partie de son existence, et que la relation de l’homme à l’objet technique comporte cette attention à la genèse continue de l’objet technique” (Simondon 1969: 249-250). Cette attention à la genèse instrumentale est également nécessaire dans une approche instrumentale de l’enseignement, si nous voulons voir accomplir les effets que nous lui accordons. Elle invite à se tourner vers la genèse historique des instruments. Dans le même souci, la reprise du schéma de Trouche (fig.7) dans plusieurs écrits didactiques incite à relever que la genèse instrumentale ne peut pas se défaire du sujet connaissant, sous peine en effet d’aliénation. “Les objets techniques qui produisent le plus d’aliénation sont aussi ceux qui sont destinés à des utilisateurs ignorant” (Simondon 1969: 249-250). L’introduction de familles d’instruments plutôt que d’instruments hétéroclites et isolés est indispensable dans le cadre de l’enseignement de la géométrie, et plus largement des mathématiques d’aujourd’hui. En France, comme dans beaucoup de pays, l’enseignement de la géométrie est de plus en plus limité et éparpillé, dans le contexte d’un enseignement des mathématiques lui-même réduit et morcelé. Il ne s’agit plus tant de former les élèves et les étudiants, que de leur inculquer des savoirs et surtout de leur fournir des compétences. Il s’avère que plus ces enseignements sont amoindris de la sorte, plus ils perdent de leur légitimité sociale et de leur intérêt cognitif. L’approche instrumentale doit permettre de relier des connaissances et non pas favoriser encore un éparpillement de savoirs, qui placerait les élèves en face d’instruments dont le fonctionnement, non seulement n’est pas porteur de connaissance-en-action, mais leur échappe.

RÉFÉRENCES Archimède (1960). Les œuvres complètes (Vol. II). Trad. P. Ver Eecke. Liège: Vaillant-Carmanne. Aristote (1991). Métaphysique. Trad. Tricot, J. Paris: Vrin. Barbin, É. (1994). L'invention des théorèmes et des instruments. In É. Hébert (Ed.), Instruments scientifiques à travers l'histoire (pp. 7-12) Paris: Ellipses.

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Barbin, É. (2004). L’outil technique comme théorème en acte. In Ces instruments qui font la science (pp. 26-28). Paris: Sciences et avenir. Barbin, É. (2006). La révolution mathématique du xviie siècle. Paris: Ellipses. Barbin, É. (2014). Les constructions mathématiques avec des instruments et des gestes (Ed.). Paris: Ellipses. Barbin, É (2016). La Dioptre d’Héron d’Alexandrie: investigations pratiques et théoriques. In D. Bénard & G. Moussard (Ed.). Les mathématiques et le réel: expériences, instruments, investigations. Rennes : PUR. Brocard, H. (1875). Note sur un compas trisecteur proposé par M. Laisant. Bulletin de la SMF, 3, 47-48. Chabert, J.-L. & Barbin, É. et al. (1999). A history of Algorithms. From the Pebble to the Microchip. New-York: Springer. Euclide (1994). Les Éléments (Vol. 2). Trad. B. Vitrac. Paris: PUF. Fine, O. (1532). Protomathesis. Paris: Impensis Gerard Morrhij et Ioannis Petri. Rabardel, P. (1995). Les hommes et les technologies: approche cognitive des instruments contemporains. Paris: Armand Colin. Séris, J.-P. (1994). La technique. Paris: PUF. Simondon, G. (1969). Du mode d’existence des objets techniques. Paris: Aubier- Montaigne. Trouche, L. (2005). Des artefacts aux instruments, une approche pour guider et intégrer les usages des outils de calcul dans l’enseignement des mathématiques. Actes de l’université d’été de Saint-Flour (pp. 265-276).

BRIEF BIOGRAPHY Évelyne Barbin is full professor of epistemology and history of sciences (University of Nantes). Her research concerns history of mathematics and relations between history and teaching. She works in the French IREMS where she organized thirty colloquia and summer universities and she edited many books. She had been chair of the HPM Group from 2008 to 2012.

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UNIVERSITY OF WESTERN MACEDONIA FACULTY OF EDUCATION

MENON 31 ©online Journal Of Educational Research

PRIMARY SOURCES AND HISTORY-BASED PROBLEMS ABOUT ISOPERIMETRY: A USE OF MATHEMATICS HISTORY IN GRADE SIX1

Matthaios Anastasiadis Primary school teacher, MSc, University of Western Macedonia [email protected]

Konstantinos Nikolantonakis Associate Professor, University of Western Macedonia [email protected]

ABSTRACT In this paper, we report on the use of one historical note and two primary sources, an extract from Pappus’ Collection and an extract from Polybius’ Histories, in the context of an instructional intervention focused on isoperimetric figures and area-perimeter relationships. The participants were 22 sixth graders, aged 11-12. The research findings we present here are based on classroom observations, on the worksheets used during the intervention and on personal interviews with the students. During the intervention, the students solved problems, which were based on the sources. Twenty-one of the 22 students considered the problem which was based on Pappus’ text to be more interesting than the problems that they were usually asked to solve in mathematics. In addition, the students’ ratings of the texts indicate that the extract from Pappus was the text that they liked most. We also examine the various ways through which the particular use of mathematics history affected the development of the students’ personal Geometrical Working Spaces. Keywords: History of mathematics, Primary sources, Isoperimetric figures, Area, Geometrical Working Space

1. INTRODUCTION This paper presents some findings from a larger research study linking the use of historical sources in mathematics education with the Geometrical Working Spaces theoretical framework (Kuzniak 2006), in the context of an instructional intervention focused on isoperimetric figures and area-perimeter relationships. In the paper, we focus on how the sources were used and we discuss the students’ views both on their learning and on the use of the particular historical sources, and the various ways through which the particular use of mathematics history affected the development of the students’ personal Geometrical Working Spaces.

1 A Greek version of this paper has been published in: Kourkoulos, M., & Tzanakis, C. (guest Eds.). (2014). History of Mathematics and Mathematics Education. Education Sciences. Special Issue 2014. Rethymno, Greece: Department of Primary Education, University of Crete.

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Matthaios Anastasiadis, Konstantinos Nikolantonakis PRIMARY SOURCES AND HISTORY-BASED PROBLEMS ABOUT ISOPERIMETRY : A USE OF MATHEMATICS HISTORY IN GRADE SIX 32

1.1. History of mathematics and mathematics education Regarding the use of mathematics history, on the one hand, there are theoretical objections and practical difficulties. For example, it has been argued that students often dislike history and that the history of mathematics could confuse students (Jankvist 2009, Tzanakis et al. 2000). Practical difficulties include the lack of teaching time and material and the teachers’ lack of expertise. Moreover, the use of mathematics history could not be easily assessed, so it would not attract students’ attention. On the other hand, it has been argued that mathematics history can motivate students and contribute to the teaching of specific mathematical content (Jankvist 2009, Tzanakis et al. 2000). Moreover, learning about the difficulties, errors and misconceptions that arose in the history of mathematics could be beneficial to students in terms of emotions, beliefs and attitudes; on the other hand, this kind of knowledge helps teachers to anticipate students’ possible difficulties and to develop or adapt history-based problems and other instructional material that could help students overcome these difficulties. Also, mathematics history shows the role of individuals and the role of different cultures in the evolution of mathematics and indicates that mathematical concepts were developed as tools for organizing the world. Finally, mathematics history enables the connection between mathematics and other subjects. Concerning the relationship between students’ difficulties and the difficulties encountered in mathematics history, there are different approaches. Through the concept of epistemological obstacle, Brousseau (2002) emphasized the role of a piece of prior knowledge, which, depending on its structure, has particular advantages but also leads to particular errors. Contrarily, Furinghetti and Radford (2008) emphasized the role of culture and argued that school prepares the unpacking of a tradition established over centuries. Finally, according to the conceptual change framework, children’s initial theories can emerge through the children’s interaction with the physical environment and with the cultural tools (Vosniadou & Vamvakoussi 2006). Thus, similarities between children’s difficulties and the difficulties encountered in history could possibly be related to the use of similar cultural tools or to children’s perception of the environment; this seems to be particularly interesting in the case of elementary geometry, considered as the science of space (Kuzniak 2006). As regards the ways of using mathematics history, the most common way is the use of historical notes, i.e. texts that are written for teaching purposes and may include names, dates, biographies, anecdotes and stories (Jankvist 2009; Tzanakis et al. 2000). Worksheets, historical problems, and primary and secondary sources are also forms of using history. The various history uses can also be combined for designing teaching and learning sequences (packages) and projects, which may be short or more extensive and more or less relevant to the curriculum. The use of primary sources is both demanding and time-consuming, and it

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Matthaios Anastasiadis, Konstantinos Nikolantonakis PRIMARY SOURCES AND HISTORY-BASED PROBLEMS ABOUT ISOPERIMETRY : A USE OF MATHEMATICS HISTORY IN GRADE SIX 33

is often difficult to assess the results (Jahnke et al. 2000). The teacher may need to translate or modify the text, but such adaptations should not deviate far from the original text. A primary source may be introduced directly (without prior preparation) or indirectly, e.g. after problem solving. In short, there is not only one teaching strategy for the use of primary sources; therefore, the most appropriate strategy should be chosen.

1.2. Area-perimeter relationships in ancient Greek mathematics There is sufficient evidence to suggest that area-perimeter relationships have caused difficulties in the past. For example, Polybius from Megalopolis (2nd c. BC), in the ninth book of his treatise Histories, argued that army generals should have knowledge of astronomy and geometry, and to support his claim, he wrote: “Most people infer the size of the aforementioned [cities and camps] only from the perimeter. (....) The reason of this is that we do not remember the geometry lessons we were taught in our childhood” (Hist. 9.26a.1-4, Büttner- Wobst ed.).2 Furthermore, he gave two examples: the first concerns the comparison between Sparta and Megalopolis, while the second concerns a hypothetical town or camp which has a perimeter of 40 stadia but is twice as large as another with a perimeter of 100 stadia. According to Walbank (1967), ‘the size’ is the area of each city. Moreover, the first example is of particular historical interest, since the comparison seems not to be confirmed in the case of area, at least with the existing archaeological findings. On the contrary, the second example refers to an extreme case and is mostly of mathematical interest. In any case, Polybius’ reference to geometry is a characteristic example of the way that ancient writers used mathematics to present their accounts as superior in terms of accuracy and reliability (Cuomo 2001). Polybius’ reference to ‘geometry lessons’ shows that area-perimeter relationships had already been an object of study for mathematicians. In the Elements, Euclid had already proved that parallelograms on the same base or on equal bases, and between the same parallels are equal to one another and then he proved the same for triangles (Ι.35-38). These theorems imply that the length of the contour of a parallelogram or triangle does not determine the extent of its surface; this is why, according to Proclus, these theorems caused astonishment to non-experts (Heath 1921). Isoperimetry was also the object of Zenodorus’ work (probably 2nd c. BC). His treatise on isoperimetric figures has not survived; however, on the basis of what Theon wrote later, Zenodorus proved that of all regular polygons with equal perimeter, the largest is the one having the greatest number of angles, and that if a circle and a regular polygon have equal perimeter, then the circle is larger (Cooke 2005, Heath 1921). Furthermore, he showed that of all

2 Book 9 survives in fragments, and there have been different views concerning the order of the fragments. In οther editions or translations, this passage is part of 9.21. In Büttner-Wobst’s edition it is a part of 9.26a, and Walbank (1967) considered this order to be more coherent.

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isoperimetric polygons with the same number of angles, the largest is the equilateral and equiangular, but he partially based his proof on a lemma that had not been proved in a general way. Isoperimetry is also the topic of Book V of Pappus’ Mathematical Collection (4th c. AD). The first part of the book concerns plane figures and begins with an introduction, which is characterized of literary merit (Cooke 2005, Heath 1921) and stimulates the interest of the reader; its topic is the hexagonal shape of the cells of honeycombs. Pappus’ explanation of the shape is teleological, as he claimed that bees choose this shape on purpose. At the end of the introduction, Pappus formulated a mathematical problem: Bees then know only what is useful to them. That is, that the hexagon is greater than the square and the triangle, and can hold more honey, for the same expenditure of material for the construction of each one. We, however, claiming to have a greater share of wisdom than bees, will investigate something greater. That is, that of all equilateral and equiangular plane figures having equal perimeter, the one which has the greatest number of angles is always greater. And the greatest of all is the circle, whenever it has perimeter equal to them. (Mathematical Collection V.3, Hultsch ed.) According to Cuomo (2000), Book V was probably situated in the context of rivalries for the appropriation of tradition, for the acquisition of reputation and for the gaining of new pupils. Bees were frequently used as an example by philosophers too; for Pappus, the difference between bees and humans is that bees have limited, useful and intuitive knowledge, whereas humans are both capable of and interested in proving. Thus, the introduction points out to the need for proving the isoperimetric theorems. The proof process, which follows, is situated in the context of the Euclidean tradition. Furthermore, although there is no reference to Zenodorus, it seems that Pappus followed Zenodorus’ work, especially in the case of plane figures, but also added his own propositions and proofs (Heath 1921). Pappus’ introduction about bees is also related to the problem which was later known as the honeycomb conjecture. According to the conjecture, which has been proved more thoroughly by Hales (2001), “any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling” (p. 1).

1.3. Theoretical framework for designing the intervention Work with isoperimetric figures, that is geometric figures with equal perimeters, involves the concepts of perimeter and area and their relation. Regarding these concepts, prior research (Douady & Perrin-Glorian 1989, Moreira-Baltar & Comiti 1994, Vighi 2010, Woodward & Byrd 1983, Zacharos 2006) has shown that students often use formulas at the expense of other strategies, make errors when applying them and do not understand the result, and confuse area and perimeter; they also believe that a smaller/equal/greater perimeter implies a smaller/equal/greater area respectively, and vice versa,

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Matthaios Anastasiadis, Konstantinos Nikolantonakis PRIMARY SOURCES AND HISTORY-BASED PROBLEMS ABOUT ISOPERIMETRY : A USE OF MATHEMATICS HISTORY IN GRADE SIX 35

and this misconception often reappears after instruction. Furthermore, the difficulties related to area-perimeter relationships are observed even in secondary school students and adults (Kellogg 2010, Woodward & Byrd 1983). Regarding the concept of area, it has been argued that students need to understand that area is an attribute (Van de Walle & Lovin 2006). The following strategies have been recommended: a) area measurement with the use of two- dimensional units, b) comparison of areas of different figures, c) superposition of a surface onto another and reconfiguration of one of the surfaces, and d) examination of area-perimeter relationships (Douady & Perrin-Glorian 1989, Nunes, Light, & Mason 1993, Van de Walle & Lovin 2006, Zacharos 2006). It is also worth noting that in the USA the examination of area-perimeter relationships is recommended for Grade 3 or above (Common Core State Standards Initiative 2010, Georgia Department of Education 2014, North Carolina Department of Public Instruction 2012, Van de Walle & Lovin 2006). In this research study the concepts of area and perimeter were examined from the standpoint of geometry, so we used the Geometric Working Spaces theoretical framework (Kuzniak 2006, 2015). A Geometric Working Space (GWS) is a space organized in a way that makes it possible for the user of the space (mathematician or student) to solve a geometric problem. Therefore, problems are the reason of existence of GWSs. The framework distinguishes three levels: a) the reference GWS, which is determined by a particular community of mathematicians or, in education, by the curriculum, b) the appropriate GWS, which is designed by a teacher for a particular class, and c) the personal GWS, which is developed by the final user, in our case each student. In addition, there are three paradigms. Here, we are mainly interested in Geometry I (GI), wherein experimentation is dominant, and practical proofs, measurement, the use of numbers and approximate answers are allowed, and in Geometry II (GII), whose archetype is the classical Euclidean geometry. The GWS’s epistemological plane includes three components: a) a real space with its geometric objects, b) a set of artifacts, and c) a theoretical frame of reference with the definitions and the properties of the objects (Kuzniak 2015). A second and cognitive plane includes three kinds of processes: visualization, construction and proof. Visualization includes the reconfiguration of figures, which may be performed materially or with the use of reorganizing lines or only by looking (Duval 2005). Concerning students’ misconceptions, Brousseau (2002) has argued that overcoming an obstacle requires the involvement of students in solving selected problems, through which they will realize the ineffectiveness of a piece of knowledge or conception. He noted, however, that problems should be chosen in a way that students are motivated and, subsequently, act, discuss and think so as to solve them. Another strategy which can help students change their ideas is the use of refutation texts (Tippett 2010), i.e. texts which refer to a prevalent alternative idea, stressing that it is incorrect. For the use of these texts, a combination with discussions and other activities is recommended, because

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changing ideas is difficult and no text alone is sufficient to achieve this with all students. Finally, it is recommended that teaching should close with a metacognitive phase, during which “the teacher asks the students to describe their old and their new knowledge and to realize its differences” (Kariotoglou, 2006: 36). We referred above to the role of students’ motivation in solving problems and we noted that motivation is a usual goal when using the history of mathematics. Besides, from the viewpoint of motivational psychology, Pintrich and Schunk (2002) have recommended, among others, the use of original source material. Regarding the features of the texts that stimulate interest, Schraw, Bruning and Svoboda (1995) highlighted the role of vividness and of ease of comprehension. Moreover, prior research has found that students and teachers argued that when a text is read aloud by the teacher, it becomes more interesting, and comprehension becomes easier (Ariail & Albright 2006, Ivey & Broaddus 2001). Other factors that could stimulate interest are: novelty, group work, hands-on activities, some themes related to nature, meaningfulness and the balance between the degree of challenge and the level of knowledge and skill of a person (Bergin 1999, Mitchell 1993, Pintrich & Schunk 2002).

2. METHOD As already mentioned, this paper presents some findings from a larger research study. In the paper, we focus on two questions: 1. In what ways was the particular use of mathematics history related to the development of the students’ personal Geometrical Working Spaces? 2. What were the students’ views both on the particular use of mathematics history and on their learning? The research was conducted in Thessaloniki, Greece, and the participants were 22 sixth graders, aged 11-12. The findings we present here are based on classroom observations, worksheets and personal interviews with the students. The instructional intervention was implemented by the first researcher in the regular classroom of the students. An exception was the class period allotted to the solution of the main mathematical problem, for which we decided not to have the groups of students work simultaneously in the regular classroom but to have each group work for one class period in another classroom of the school. This was decided in order to enable the observation of the students’ work and of the difficulties they faced. Thus, the whole intervention consisted of six class periods in the regular classroom and one class period for each group in another classroom. The whole research project also included personal interviews with the students before and after the intervention. In this paper, we focus on the interviews conducted after the intervention and, in particular, on the questions asking the students to provide some further explanation concerning their views on the particular use of mathematics history.

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2.1. Design of the appropriate GWS The intervention was implemented prior to the teaching of the geometry unit, because the textbook’s emphasis on formulas and the learning of area formulas for general triangles and trapezoids would affect the students’ personal GWSs, favouring the use of formulas at the expense of other strategies. Concerning mathematics history, we selected two primary sources: the first was the introduction to the first part of Book V of Pappus’ Mathematical Collection (V.1-3) and the second was an extract from Polybius’ Histories (9.26a.1-6). In addition, we decided to use a historical note entitled Geometry and included in the sixth grade textbook (Kassoti, Kliapis, & Oikonomou 2006: 136). Since primary school students do not know ancient Greek, the sources were presented in translation. During the translation, we used words and phrases as close as possible to the original texts, while in some cases we used shorter sentences, so that the translated texts were both suitable for the students and close to the original (Jahnke et al. 2000). Furthermore, on the basis of the objectives of the intervention, we did not include in the extract from Pappus the vocative address “most excellent Megethion” (Mathematical Collection V.1), the closing of the introduction including the reference to the circle (V.3), and the detailed verbal proof of the fact that only three regular figures can completely cover a surface without gaps or overlaps (V.2). The extract from Pappus, as a historical source, was not introduced directly, but after the use of the historical note. More specifically, work with the historical note included reading it, discussing briefly about the origin and development of geometry and providing additional information about Pappus’ life and his historical period. As regards Pappus’ text, a different approach was selected: formulation of questions by the teacher, followed by a teacher read- aloud of the text, and discussion based on the initial questions. Then, the teaching plan included providing the students with a copy of the extract and asking them to underline words and phrases related to mathematics. The goal was to provide or help the students activate the definitions and geometric properties needed for developing their GWSs, namely definitions of polygon, regular polygon, equilateral triangle, square and regular hexagon, and definitions of perimeter and area; also which regular figures completely cover a surface without gaps or overlaps, which figures are called isoperimetric and how demonstration is related to mathematics. Work with the text was followed by the formulation of a geometric problem asking the students to examine if Pappus was right in stating that a cell having the shape of a regular hexagon holds more honey than other figures suitable for tessellation. The students were asked to solve the problem in groups and with different methods: 1. Direct area comparison: superposition of a surface onto another and reconfiguration of one of the surfaces. 2. Indirect area comparison: tiling of equal surfaces (inverse proportion: the

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shape which is used fewer times for the tiling of equal surfaces is greater). 3. Area measurement with the use of a transparent grid (square-counting). 4. Area calculation with the use of formulas. The objects of the real space were regular polygons with three, four and six angles, and irregular polygons (elongated rectangles), in a material form (cardboard), in order to facilitate the reconfiguration of the shapes. Since the length of the sides was not given, the students needed to measure the sides, so as to calculate the perimeter of each shape, and then they were asked to apply the proposed method of area comparison (GI). The tools selected to be available (where appropriate, depending on the method) were: triangle ruler, scissors, glue, adhesive tape, transparency film with a square grid printed on it, marker pen, pencil, rubber eraser and calculator. In addition, we prepared one worksheet for each group, aiming, firstly, to provide through a set of questions particular steps for solving the problem and, secondly, to help students present their findings in the classroom. The institutionalization of the new properties was followed by the use of the extract from Polybius. Work with the second source included a discussion about area-perimeter relationships, and two other activities. The first one asked what the shape and the dimensions of two cities could be, if the one had a perimeter of 40 stadia but twice the area of the other having a perimeter of 100 stadia. The second activity was called ‘Neighborhoods of Thessaloniki’ and involved eight isoperimetric figures, which represented neighborhoods (Appendix, Fig. 1). More specifically, each pair of students was given two figures, which were printed on a sheet of paper, along with the length of each side in metres. The students were asked to calculate the perimeter of each figure and to deduce, without calculation, if an area was smaller than, equal to, or larger than the other and why. Regarding perimeter, the students needed only to add the given lengths and realize that the figures were isoperimetric. Regarding area, they had to develop the theoretical pole of their GWS (GII), applying the institutionalized conclusions which were based on the honeycomb problem. Then, each pair of students was asked to present their answer and check its correctness by performing measurements (GI) via a computer connected to a projector and with the use of a Geogebra applet designed for the activity. In the applet, a map of Thessaloniki was inserted as a background and the eight figures were on the same scale as the map. Finally, the students were asked to put all the figures on a board from the smallest to the largest in area, noting that eight different figures had equal perimeter but different area, that the largest in area was the regular figure having the greatest number of angles, and that the smallest was the most elongated figure.

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3. RESULTS

3.1. Implementation of the appropriate GWS and students’ difficulties Regarding the extract from Pappus, we noticed two interrelated behaviours. First, when students were asked to find in the text words and phrases related to mathematics, none of them mentioned the word ‘proof’. Secondly, after working with the text, several students seemed convinced that Pappus was right and they agreed a priori that the hexagon will be larger. In the honeycomb problem, all groups correctly arranged the figures in increasing order of area. The main difficulties they faced while solving the problem were the following:  Superposition-reconfiguration: the relatively most difficult comparison was between the hexagon and the square (Appendix, Fig. 2). Overall, however, this method was the easiest.  Tiling of equal surfaces: the students understood the rationale of the method when the teacher provided the hypothetical example of two identical rooms with different tiles. When counting the number of shapes used, we noticed more difficulties in the case of the hexagon, since there were parts that were smaller or greater than half the hexagon (Appendix, Fig. 3), and the students had to recompose these parts by looking (Duval 2005).  Square-counting: at first, the students did not remember that in previous grades, to find the area of a figure, they counted squares, in grids which were either pre-drawn on the pages of the textbooks or designed by the students. In addition, they had to find an operational way to use the transparent grid, which was new to them as a tool. The most difficult point was the counting of small squares in the case of the hexagon (Appendix, Fig. 4); an advanced solution was given later and involved the reconfiguration of the entire hexagon, in a way that two rectangles were formed.  Calculation: certain shapes needed to be reconfigured so as to form shapes whose area could be calculated with the already taught formulas (Appendix, Fig. 5). There were difficulties regarding the choice of the appropriate formula, the reconfiguration of the hexagon, and the calculation which was required when a shape had been reconfigured not with the use of scissors, but via folding. Furthermore, some students from different groups showed area-perimeter confusion. Another obstacle for the students was the usual didactic contract, since in Greek upper elementary education hands-on activities with figures presented in a material form are, in practical terms, almost absent. Thus, some students felt the need to ask for permission to use the scissors and to fold or cut the shapes, although they had been told that they could work as they wanted, using whatever tool available they wanted. Concerning the extract from Polybius, in the discussion which followed, the

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students concluded, “that a region can have greater perimeter but smaller area [as compared with another region], or smaller perimeter but greater area” (student Q) and that “if a region is greater, it’s not perimeter that matters, it is area that matters” (student Z). As regards the activity ‘Neighborhoods of Thessaloniki’, an indicative example is the response of the students who compared the rectangle with the square: “The square is greater, because they have equal perimeter and those [figures] that are regular are greater”. Likewise, in comparing the regular pentagon with the equilateral triangle, the following answer was given: “They are regular and isoperimetric the one to the other. Although they have the same perimeter, the pentagon has more angles than the triangle, thus we assumed that the pentagon is greater”. On the other hand, there was a student who calculated the perimeters incorrectly and another student who was initially willing to work within GI, by reconfiguring the figures and calculating with formulas, but this was difficult, since the figures were in fact scaled representations.

3.2. Students’ self-references regarding their learning In the last worksheet used during the intervention there were several questions aiming to help the students reflect on their learning. These were not answered by all students, and there were also some non-specific answers. The rest of the answers referred:  To area-perimeter relationships. For example, student Y wrote that an idea which she changed was that “those figures which have the same perimeter always have the same area too”, while her new idea was that “area and perimeter are not related”. Also, student D wrote that something which surprised him was that “small and large figures have the same perimeter”.  To ideas or processes associated with experimentation. For example, student I wrote that an idea which she changed was that “to find perimeter I believed that I should do side ∙ side”, but “I discovered that we do side + side + side + side...”. Furthermore, student H wrote that something he learnt is “that I can find which figure has the biggest area without calculating it”, thus showing the dominance of calculation with formulas in the students’ past experiences. Likewise, student X reported that something which surprised him is “that there are so many different methods to measure which figure is bigger”.  To bees and to the shape of the honeycomb cells, as something that caused surprise.  To the students’ attitude towards geometry. In particular, student Z said that, previously, she did not love geometry, whereas after these lessons she liked it somewhat more, because she understood them. Similarly, student Y wrote that something that surprised her is that “I believed that geometry was difficult, confusing and incomprehensible, but after these lessons I found that it is easier”.

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In addition, the students were asked to write something they found difficult. Two students mentioned the honeycomb problem, one student mentioned the method of tiling and two others mentioned the method of square-counting, four students wrote that they found it difficult to find the area of the hexagon or of the triangle, and one student referred to the fact that “There are figures with equal perimeter”. Finally, several students wrote that they did not find anything difficult or they did not write anything.

3.3. Students’ assessment of the sources and of the problem In the same worksheet, the students were also asked how much they liked each of the texts used in the lessons. The students could rate each text on a 5- point scale ranging from 1 (the least) to 5 (the maximum). Regarding the historical note, the mean score was 3.45 (SD = .91, N = 22), whereas in the case of the extract from Pappus the mean score was 4.36 (SD = .66, N = 22). Finally, regarding the extract from Polybius, the mean score was 3.85 (SD = 1.31, N = 20); we note that two students were asked to rate only the two first texts, since they had been absent from school when the extract from Polybius had been taught. To determine whether there is a statistically significant difference as to how much the students liked the three texts, we excluded the two students who did not rate the third text (N = 20), and we used the Friedman test, which showed that the difference was significant (χ2 = 6.818, df = 2, p = .033 < .05). As a post- hoc test, we used the Wilcoxon Signed Ranks Test with Bonferroni correction (Corder & Foreman 2014), which showed that the difference was statistically significant in the comparison between the extract from Pappus and the historical note (Z = -2.857, p = .004 < .017), but not between the extract from Pappus and the extract from Polybius (Z = -1.543, p = .123) nor between the extract from Polybius and the historical note (Z = -.997, p = .319). We note that both the Friedman and the Wilcoxon test are non-parametric, but they are more appropriate for ratings and for small sample sizes (N<30) (see also: Corder & Foreman 2014). Finally, taking into account the answers of all the students (N = 22) as regards the extract from Pappus and the historical note, the difference was again statistically significant (Z = -3.137, p = .002). Moreover, only two of the 22 students liked the historical note more, whereas 15 students liked the extract from Pappus more, and five students liked both texts equally. In the same worksheet, the students also reflected on the difficulty and the interest caused by the honeycomb problem, as compared with the problems that they were usually asked to solve in mathematics. In this question, 11 out of the 22 students considered the honeycomb problem to be easier than usual, whereas eight said that it was more difficult, and three answered that it had the same degree of difficulty. However, 21 of the 22 students found this problem more interesting than the usual problems, and one student answered that it was equally interesting.

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A comparison of the self-reported degree of the problem’s difficulty with the method used by the students in solving it, shows that none of those who performed superposition-reconfiguration considered the problem to be more difficult. In contrast, three quarters of those who used square-counting considered the problem to be more difficult. Furthermore, a recoding of the responses (1: more difficult; 0: same degree of difficulty; -1: easier), shows that, on average, the students who performed superposition-reconfiguration or tiled equal surfaces considered the problem to be easier (average degree of difficulty -.5 and -.3 respectively), as compared with the students who used square- counting and calculation with formulas (.5 and 0 respectively). It is also worth noting that five students who were generally weak in mathematics considered the problem to be easier than usual. In the interviews conducted after the intervention, the students were asked to explain the judgments they had made. For example, student T said: Answer: The problems we usually solve in the textbook are more difficult. Question: What is it that makes them more difficult? Answer: Hmm... when I do not understand, this seems difficult. Also, student A considered the problem to be easier, because “it didn’t need many calculations and the like”, and student C agreed also because “we were more students and we collaborated”. On the contrary, student P, for example, thought that the problem was more difficult, because “it was more complicated”, while student W, who had tiled equal surfaces, regarded the problem as more difficult, because “it puzzled you with the shapes, if it fits, if it leaves a gap, if you must... if you had to put something else”. Regarding interest, 12 students referred explicitly and clearly to nature, bees, honeycombs or to ancient Greeks and, more generally, to what constitutes the context of the problem, for example:  “We learned many things about geometry, many ways to find the area and the perimeter of a shape, but we also learned about reality, why bees use this shape”. (student I)  “You were curious to see it; it is about nature and... it is a mystery what bees do, whereas the textbook's problems are, let’s say, simpler”. (student Q)  “I liked it with the example we did, that is with bees and honeycombs and the text saying... It was like a story that you had to solve”. (student M) On the other hand, student B explicitly linked difficulty with interest: “It was more difficult; it was interesting to solve”. There was also one mention of the fact that mathematicians worked on this problem and one answer saying that this way the students learned “how geometry was discovered” (student W), three mentions of the fact that the students worked in groups and two mentions of the fact that the problem was unusual; student X, for example, gave this characteristic answer: “I hadn’t done a problem like this before and this is why I liked it”.

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4. DISCUSSION In the present research, we used the history of mathematics to achieve various goals, which were related to each other and to the development of the students' GWSs as well. In particular, the historical sources were the source of geometric problems and problems are the reason of existence of GWSs (Kuzniak 2015). In addition, the historical sources and the historical note served as a means of motivation and motivation constitutes an important tool for the active involvement of students in solving problems (Brousseau 2002). Since the three texts were assessed positively, it could be argued that they all helped to motivate the students. Thus, the argument that many students may be affected negatively because they dislike history (Jankvist 2009, Tzanakis et al. 2000) was not supported here. Pappus’ text was also used as a means of activating preexisting definitions and properties of the theoretical frame of reference (e.g. definition of perimeter) and of enriching it with new definitions and properties (e.g. definition of regular figure); these properties were necessary for the development of the students’ personal GWSs and the solution of the honeycomb problem. Additionally, the students made a first acquaintance with a new property concerning area-perimeter relationships. This property, however, was regarded by some students not as a proposition to be confirmed, but as established knowledge. This behaviour could be attributed to the usual didactic contract, according to which textbooks and, by extension, texts used in school, contain indisputable truths; it is also likely to reflect a broader conception according to which mathematical knowledge is generally unchanging over time (Schommer- Aikins 2002), and, therefore, a mathematician cannot be mistaken. As already mentioned, the historical sources were the source of geometric problems. Subsequently, the honeycomb problem constituted a means of enriching the students’ personal GWSs with new tools (transparent grid) as well as with experimentation methods which present area as an attribute and which had been used in previous grades but had been forgotten. Furthermore, mathematical problems are a means to overcome students’ misconceptions (Brousseau 2002), and Polybius’ text seems to have contributed to this goal. This text is not a refutation text written for teaching purposes, but a historical source, with all its complexity. However, the reference to misconceptions related to area-perimeter relationships and the information that such mistakes were also made by important persons in history both acted as stimuli to the students and contributed to a climate of comfort for the students to reflect and talk about themselves. This is also related to the students’ personal GWSs and, in particular, to their theoretical frame of reference. Here, however, we should take into account that changing ideas is difficult and no text alone is sufficient to achieve this with all students (Tippett 2010). This is also true for area-perimeter relationships, in which even secondary school students and adults have difficulties (Kellogg 2010, Woodward & Byrd 1983). Besides, it has been observed that misconceptions concerning area-

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perimeter relationships often reappear after instruction (Douady & Perrin- Glorian 1989, Kellogg 2010, Vighi 2010). Apart from these, it is interesting that two students spontaneously referred to their attitude towards geometry, although this was not the main goal of the intervention. Regarding the assessment of the historical texts, the students on average answered that they liked all three texts; most of all they liked the extract from Pappus, then the extract from Polybius and finally the historical note. The difference was statistically significant in comparing the extract from Pappus with the historical note. These findings have multiple interpretations:  The ranking of the three texts reflects the time allotted to each one. However, if the students did not like the way that teaching time was used, then more allotted time would have probably led to a greater dislike of a text.  The students’ greater preference for both primary sources is in accordance with the recommendation made by Pintrich & Schunk (2002) that original source material should be used. At the same time, this preference could be attributed to the fact that both primary sources were accompanied by a mathematical problem, whereas the historical note was not.  The extract from Pappus was read aloud by the teacher, and this probably facilitated comprehension and made the text more vivid, thereby increasing the students’ interest (Ariail & Albright 2006, Ivey & Broaddus 2001, Schraw et al. 1995).  Most of all, it seems that the students’ greater preference for the extract from Pappus is related to the theme and, generally, to the features of the text: regularity in nature, and the society of bees are two themes that had attracted the interest of philosophers and mathematicians since antiquity and were widely known (Cuomo 2000). Thus, we could say that these themes could be listed among those themes that are related to nature and are reported to stimulate interest (Bergin 1999). Besides, as student Q said: “it is about nature and... it is a mystery what bees do”. Apart from this, Pappus’ text was written as a literary introduction to his book with the aim of stimulating interest. And finally, the chosen extract does not contain names, and dates or numbers, unlike the other two texts. Regarding the degree of difficulty of the honeycomb problem as compared with the usual problems, the students’ opinions were not homogeneous: somewhat more students (11) considered the problem to be easier, whereas eight of the 22 said that it was more difficult. The students’ opinions were influenced, first, by the method with which each student worked. Second, it seems that the students who were generally weak in mathematics considered the problem to be easier than usual, taking into account the lack of calculations, the material form of the shapes, the availability of tools appropriate for work within GI and the fact that the students worked in groups. Thus, they were able

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to participate and contribute to the solution, and it is indicative that in the group which used calculations with formulas the most difficult reconfiguration of the hexagon was performed by a student who was generally weak in mathematics. Furthermore, the (indirect) subdivision of the problem through the questions included in the accompanying worksheet is also likely to have helped those who face difficulties in organizing the problem solving process. On the other hand, 21 out of the 22 students considered the problem to be more interesting than the usual problems. This finding, combined with the answers regarding the degree of difficulty, suggests a sufficient balance between the requirements of the problem and the level of knowledge and skill of each student. As shown previously, a factor that contributed to this balance was the hands-on nature of the activity. In addition, the arguments of the students show that group work, the unusual nature of the problem and, most of all, the context of the problem also stimulated interest. All these factors have been reported in the related literature (Bergin 1999, Mitchell 1993, Pintrich & Schunk 2002) and may have influenced the students’ views both directly and indirectly. For example, group work affected the students not only directly, but also indirectly by facilitating the solution of the problem, thus intervening in the relationship between challenge and skill. Finally, the context of the problem was determined by Pappus’ text, and it seems that the combination of the problem with the text linked knowledge with the questions that gave birth to it and gave meaning to the activity.

REFERENCES Ariail, M., & Albright, L. K. (2006). A survey of teachers' read-aloud practices in middle schools. Reading Research and Instruction, 45(2), 69-89. Bergin, D. A. (1999). Influences on classroom interest. Educational Psychologist, 34(2), 87-98. Brousseau, G. (2002). Theory of didactical situations in Mathematics (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds. & Trans.). New York: Kluwer Academic. Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf Cooke, R. (2005). The history of mathematics: A brief course (2nd ed.). Hoboken, NJ: Wiley. Corder, G. W., & Foreman, D. I. (2014). Nonparametric statistics. A step by step approach (2nd ed.). Hoboken, NJ: Wiley. Cuomo, S. (2000). Pappus of Alexandria and the mathematics of late antiquity. Cambridge: Cambridge University Press. Cuomo, S. (2001). Ancient mathematics. London: Routledge. Douady, R., & Perrin-Glorian, M. J. (1989). Un processus d'apprentissage du concept d'aire de surface plane. Educational Studies in Mathematics, 20, 387- 424.

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Duval, R. (2005). Les conditions cognitives de l’apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements. Annales de Didactique et de Sciences Cognitives, 10, 5-53. Furinghetti, F., & Radford, L. (2008). Contrasts and oblique connections between historical conceptual developments and classroom learning in mathematics. In L. English (Ed.), Handbook of International Research in Mathematics Education (2nd ed., pp. 626-655). New York, NY: Routledge. Georgia Department of Education. (2014, July). Common Core Georgia Performance Standards Framework. Third Grade mathematics. Unit 4: Geometry. Retrieved November 9, 2014 from https://www.georgiastandards.org/Common- Core/Common%20Core%20Frameworks/CCGPS_Math_3_Unit4Framewor k.pdf Hales, T. C. (2001). The honeycomb conjecture. Discrete & Computational Geometry, 25, 1-22. Heath, T. (1921). A history of Greek mathematics (Vol. 2). Oxford: Clarendon Press. Jahnke, H. N., Arcavi, A., Barbin, E., Bekken, O., Furinghetti, F., El Idrissi, A., . . . Weeks, C. (2000). The use of original sources in the mathematics classroom. In J. Fauvel, & J. van Maanen (Eds.), History in mathematics education. The ICMI Study (pp. 291-328). Dordrecht: Kluwer Academic. Jankvist, U. T. (2009). A categorization of the “whys” and “hows” of using history in mathematics education. Educational Studies in Mathematics, 71, 235-261. Ivey, G., & Broaddus, K. (2001). Just plain reading: A survey of what makes students want to read in middle school classrooms. Reading Research Quarterly, 36(4), 350-377. Kariotoglou, P. (2006). Science pedagogical content knowledge [in Greek]. Thessaloniki: Grafima. Kassoti, O., Kliapis, P., & Oikonomou, Th. (2006). Sixth Grade mathematics [in Greek]. Athens: O.E.D.B. Kellogg, M. S. (2010). Preservice elementary teachers' pedagogical content knowledge related to area and perimeter: A teacher development experiment investigating anchored instruction with web-based microworlds (Doctoral dissertation). University of South Florida. Retrieved from http://scholarcommons.usf.edu/etd/1679 Kuzniak, A. (2006). Paradigmes et espaces de travail géométriques. Éléments d'un cadre théorique pour l'enseignement et la formation des enseignants en géométrie. Canadian Journal of Science, Mathematics and Technology Education, 6(2), 167-187. Kuzniak, A. (2015). Understanding the nature of the geometric work through its development and its transformations. In S. J. Cho (Ed.), Selected regular

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lectures from the 12th International Congress on Mathematical Education (pp. 1-15). Cham: Springer. Mitchell, M. (1993). Situational interest: Its multifaceted structure in the secondary school mathematics classroom. Journal of Educational Psychology, 85(3), 424-436. Moreira-Baltar, P., & Comiti, C. (1994). Difficultés rencontrées par des élèves de cinquième en ce qui concerne la dissociation aire/périmètre pour des rectangles. Petit x, 34, 5-29. North Carolina Department of Public Instruction. (2012). Third Grade area and perimeter. Retrieved May 20, 2013 from http://maccss.ncdpi.wikispaces.net/file/view/3rdGradeUnit.pdf Nunes, T., Light, P., & Mason, J. (1993). Tools for thought: the measurement of length and area. Learning and Instruction, 3, 39-54. Pintrich, P. R., & Schunk, D. H. (2002). Motivation in education: theory, research, and applications (2nd ed.). Upper Saddle River, NJ: Merrill/ Prentice-Hall. Schraw, G., Bruning, R., & Svoboda, C. (1995). Sources of situational interest. Journal of Literacy Research, 27(1), 1-17. Schommer-Aikins, M. (2002). An evolving theoretical framework for an epistemological belief system. In B. K. Hofer, & P. R. Pintrich (Eds.), Personal epistemology: the psychology of beliefs about knowledge and knowing (pp. 103-118). Mahwah, NJ: Lawrence Erlbaum. Tippett, C. D. (2010). Refutation text in science education: A review of two decades of research. International Journal of Science and Mathematics Education, 8, 951-970. Tzanakis, C., Arcavi, A., de Sa, C. C., Isoda, M., Lit, C. K., Niss, M., . . . Siu, M. K. (2000). Integrating history of mathematics in the classroom: An analytic survey. In J. Fauvel, & J. van Maanen (Eds.), History in mathematics education. The ICMI Study (pp. 201-240). Dordrecht: Kluwer Academic. Vighi, P. (2010). Investigating comparison between surfaces. In V. Durand- Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the 6th Congress of the European Society for Research in Mathematics Education (pp. 716-725). Lyon, France: Institut National de Recherche Pédagogique. Van de Walle, J. A., & Lovin, L.-A., H. (2006). Teaching student-centered mathematics : Grades 3-5. Boston: Pearson. Vosniadou, S., & Vamvakoussi, X. (2006). Examining mathematics learning from a conceptual change point of view: Implications for the design of learning environments. In L. Verschaffel, F. Dochy, M. Boekaerts, & S. Vosniadou (Eds.), Instructional psychology: Past, present and future trends. Sixteen essays in honour of Erik De Corte (pp. 55-72). Oxford: Elsevier. Walbank, F. W. (1967). A historical commentary on Polybius (Vol. 2). Oxford: Clarendon Press. Woodward, E., & Byrd, F. (1983). Area : included topic, neglected concept. School Science and Mathematics, 83, 343-347.

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Zacharos, K. (2006). Prevailing educational practices for area measurement and students’ failure in measuring areas. Journal of Mathematical Behavior, 25, 224-239.

BRIEF BIOGRAPHIES Matthaios Anastasiadis has graduated from the Department of History and Archaeology (Aristotle University of Thessaloniki) and the Department of Primary Education (University of Western Macedonia). He has also received a master’s degree in didactics of science and mathematics (University of Western Macedonia). He currently works as a primary school teacher.

Konstantinos Nikolantonakis is Associate Professor of Mathematics Education at the Department of Primary Education of the University of Western Macedonia. He has graduated from the Department of Mathematics of the Aristotle University of Thessaloniki. He received a master and a Ph.D. in Epistemology and History of Mathematics from the University of Denis Diderot (Paris-7). His research concerns the didactical use of the History of Mathematics, the History of Ancient Greek Mathematics, and the didactics of Arithmetic & Geometry.

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Matthaios Anastasiadis, Konstantinos Nikolantonakis PRIMARY SOURCES AND HISTORY-BASED PROBLEMS ABOUT ISOPERIMETRY : A USE OF MATHEMATICS HISTORY IN GRADE SIX 49

APPENDIX Figure 1: The isoperimetric figures used in the activity « Neighborhoods of Thessaloniki ».

Figure 2: Superposition-reconfiguration; comparison between the hexagon and the square.

Figure 3: Tiling with regular hexagons.

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Matthaios Anastasiadis, Konstantinos Nikolantonakis PRIMARY SOURCES AND HISTORY-BASED PROBLEMS ABOUT ISOPERIMETRY : A USE OF MATHEMATICS HISTORY IN GRADE SIX 50

Figure 4: Square-counting in the case of Figure 5: Reconfiguration of the the regular hexagon. equilateral triangle into a rectangle (4th method).

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UNIVERSITY OF WESTERN MACEDONIA FACULTY OF EDUCATION

MENON 51 ©online Journal Of Educational Research

THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE CHINESE ABACUS

Vasiliki Tsiapou Primary School Teacher, Phd Student, University of Western Macedonia [email protected]

Konstantinos Nikolantonakis Associate Professor, University of Western Macedonia [email protected]

ABSTRACT The paper presents part of a research study that intended to use the history of mathematics for the development of place value concepts and the notion of carried number with sixth grade Greek students. In the given pre-tests students faced difficulties in solving place value tasks, such as regrouping quantities and multi-digit subtractions. Also, they vaguely explained the carried number, a notion which is structurally associated with calculations. We held an instructive intervention via a historical calculating tool, the Chinese abacus. In the post-tests students improved their scores and they often put forward expressions influenced by the abacus investigation. To a smaller extent we attempted to highlight the historical dimension of the subject. Keywords: historical instrument, Chinese abacus, place value, carried number, Primary school students

1. INTRODUCTION Studies have shown that many students don’t comprehend thoroughly the structure of our number system. They don’t know the values of the digits of a number and how these values interrelate. A great difficulty is in developing an understanding of multi-digit numbers. Students need to understand not only how numbers are partitioned according to the base-10 structure, but also how these values interrelate (Fuson 1990). Resnick (1983) used the term ‘multiple partitioning’ to describe the ability to partition numbers in non-standard ways, e.g., 34 can be decomposed into 2 Tens and 14 Units. This ability is essential for competence in calculations and many types of errors that have been observed in subtraction (Fuson 1990) are due to the students’ difficulty to acquire this competence. As a consequence, they cannot interpret the carried number; a concept structurally associated with calculations. That is why the development of the concept of the carried number which is associated with

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Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 52 CHINESE ABACUS exchanges between classes should deserve more attention during primary school years (Poisard 2005). In this paper we focus on the difficulties that the students of the present study faced in the above concepts (converting nonstandard representations of the numbers’ multiple partitioning in standard form and in interpreting the carried number) and the way that we tried to address these difficulties with the use of the history of mathematics. Initially, we present the reasons that historical instruments may positively contribute to mathematics education. Then we describe the didactical use of the historical instrument that we used in the intervention, the Chinese abacus. Afterwards we present an overview of the intervention: the objectives, the design with the use of history, and an example of a didactical session. Then, a brief quantitative and a more detailed qualitative analysis of the results follow.

2. THE ROLE OF THE HISTORY OF MATHEMATICS IN THE CLASSROOM Researchers have long thought about whether mathematics education can be improved through incorporating ideas and elements from the history of mathematics. Tzanakis and Arcavi (2000) offered a list of arguments and Jankvist (2009) distinguished these arguments between using ‘history-as-a- goal’ (learning of the mathematical concepts) and using ‘history-as-a-tool’ (learning mathematical concepts). Jankvist also classified the approaches in which history can be used. One of these is the modules approach’. Modules are instructional units suitable for the use of history as a cognitive tool, since extra time is required to study more in-depth mathematical concepts, and as a goal (Jankvist 2009). Among the possible ways that modules can be implemented using history as a ‘tool’ as well as a ‘goal’, is through the use of historical instruments since they can illustrate mathematical concepts οn an empirical basis. They are considered as non-standard media, unlike blackboards and books, which can also affect students cognitively and emotionally (Fauvel & van Maanen 2000). Students explore them as historical sources for arithmetic, algebra, or geometry and they may also enable students to acquire awareness of the cultural dimensions of mathematics (Bussi 2000).

2.1 Chinese abacus: A historical calculating instrument The positional system up to the construction of algorithms for operation is embodied by abaci, such as the Chinese one (Bussi 2000). Martzlof (1996) cites that the first Chinese abacus’ representations are found in manuals of the 14th and 15th centuries. The use of the abacus, however, became widespread from the mid 16th century during the Ming dynasty. At 1592 a Chinese mathematician Cheng Dawei printed his famous work Suanfa Tongzong which deals mainly with the abacus calculations. Due to this work, the Chinese abacus was spread in Korea and Japan.

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Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 53 CHINESE ABACUS

The Chinese abacus comprises vertical rods with same sized beads sliding on them. The beads are separated by a horizontal bar into a set of two beads (value 5) above and a set of five beads (value 1) below. The rate of the unit from right to left is in base ten. To represent a number e.g. 5.031.902 (figure 1) beads of the upper or/and the lower group are pushed towards the bar, otherwise zero is represented.

Figure 1: Representation of numbers on the Chinese abacus

Brian Rotman (cited in Bussi 2000) gives an epistemological analysis of abacus: “To move from abacus to paper is to shift from a gestural medium (in which physical movements are given ostensively and transiently in relation to an external apparatus) to a graphic medium (in which permanent signs, having their origin in these movements, are subject to a syntax given independently of any physical interpretation)’. Many characteristics of our number system are illustrated by the abacus (Spitzer 1942). Unlike Dienes’ blocks, the semi-abstract structure of the abacus becomes apparent as the same sized beads and their position- dependent value has direct reference to digit numbers. The function of zero is represented, as a place-holder. Furthermore, it may illustrate the idea of collection, since amounts become evident in terms of place value. Finally, the notion of carried number emerges. Poisard (2005) argued that we can write up to fifteen units in each column and make exchanges with the hand; this reinforces the understanding of the carried number in operations. From the definition of the carried number, Poisard (2005: 78) highlighted its relation to the functionality of the decimal system to allow quick calculations: “the carried number allows managing the change of the place value; it carries out a transfer of the numbers between the ranks”. Finally, the notion of carried number emerges. What is so functional of our base 10 numeration system is to allow the representation of big numbers. In each position the digits from zero to nine are written. As soon as ten is reached there is a transfer of numbers between ranks, e.g. 10tens = 1 hundred, 10 hundreds = 1 thousand, etc. To do arithmetic operations we use this relation. From the definition of the carried number that Poisard (2005: 78) gives, its relation to the functionality of the decimal system to allow quick calculations is highlighted:

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Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 54 CHINESE ABACUS

“The carried number allows managing the change of the place value; it carries out a transfer of the numbers between the ranks”. In Poisard’s study sixth grade students were asked ‘what is a carried number?’. Most answers did not have mathematical meaning. After the workshop with the Chinese abacus, the answers were more specific. According to Poisard (2005: 57-59), the fact that we can write up to fifteen in each rank on Chinese abacus and make exchanges with the hand, reinforces the conceptual understanding of the notion of carried number. The same question was given to teachers, but definitions that link the place-value system with the carried- number, were cited by few teachers. That is why Poisard points out that the study of the carried number requires in-depth comprehension of the place- value system and this problem should be confronted in teachers’ education as well. What Poisard stresses as crucial in the teaching/learning process is the use of the abacus as an instrument (the user learns mathematics) and not as a machine (the user just calculates). If the students do not ‘see’ the concepts that regulate the movements on abacus, they may learn to calculate quick and correctly but without understanding. Based on the studies about students’ difficulties in place value understanding and the possible positive contribution of the history of mathematics via the Chinese abacus, the present study sets various objectives: 1. To study whether sixth grade students recognize the structure of our number system when handling numbers. 2. To study how they verbally explain the carried number and how they use it in written calculations. 3. To study to what extent an instructive intervention with the Chinese abacus would help students handle possible difficulties and misconceptions and acquire a better conceptual understanding. 4. To highlight the historical context of the abacus and enrich teaching with a variety of approaches where students are actively involved. In the present study we adopted Poisard’s (2005) proposal for the didactical use of the Chinese abacus; we used all the beads in order to record up to 15 units, unlike the standard technique where one of the upper beads (value five) is not used at all. This allowed us to add new elements in the present study, such as the use of regrouping activities as essential knowledge (Resnick 1983) before implementing the written algorithms of addition and subtraction.

3. RESEARCH METHODS The research study took place in an elementary school in Thessaloniki. Our aim was to introduce the History of Mathematics as a cognitive tool and, to a lesser extent, as a goal (Jankvist 2009). The participants were 18 twelve-year- old students (9 girls and 9 boys). The criterion was that the students would be able to participate once a week during the hours when their school program was to work on a two-hour project. Four students had a very

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Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 55 CHINESE ABACUS weak cognitive background and eight students often relied on procedural rules due to partial conceptual understanding. For the first two objectives two questionnaires (pre-tests) were administered in November. Questionnaire A consisted of six closed-type questions and one that required a written explanation. After the intervention similar questions were administered as post-test. The questions were created with the following in mind: (a) the literature about students’ difficulties (b) the Greek mathematics curriculum so as to ascertain that they constitute important and prerequisite knowledge in the beginning of grade 6, and (c) the feasibility of teaching via the abacus. For integers the questions concerned: named place value, expanded form, regrouping, rounding, subtraction, and multiplication. For decimals: transforming from verbal to digit form, number pattern, addition, and subtraction. Two of the questions that are subjected in the present analysis concern exchanges between classes: sub question 3b, which concerned regrouping and comparing quantities, and sub question 7a, which dealt with subtraction with carried number. In order to study how students perceive the concept of carried number used in the subtraction tasks, we administered Questionnaire B. It consisted of Poisard’s (2005: 101) four open questions. The same questions were given as post-test (Appendix). Here we present students responses to the question: what is a carried number?

3.1 The design of the intervention with the use of the History of Mathematics For the other two objectives we implemented a five-month instructive intervention. It was inspired by modules approach (Jankvist 2009) and used history as a cognitive tool. We designed a didactical sequence for the teaching of mathematical concepts that was allocated in sections (integers, decimals, and operations). For every session a teaching plan was elaborated including procedure, forms of work, media and material. The outcomes were recorded and several sessions were videotaped as feedback for the research. The introductory and closing activities aimed at using history mainly as a goal. Initially, the arguments mentioned below are aimed at exploring why history would support the learning and raise the cultural dimension of mathematics. They were based on Tzanakis & Arcavi’s (2000) arguments and were grouped under Jankvist’s (2009) categorization. We have included a third category placing pedagogical arguments in an attempt to emotionally motivate as well as develop critical thinking. Thus, students are expected to:

A. History as tool 1. develop their understanding by exploring mathematical concepts empirically, 2. recognize the validity of non-formal approaches of the past.

B. History as a goal

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Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 56 CHINESE ABACUS

1. become aware that different people in different periods developed various forms of representations, 2. perceive that mathematics were influenced by social and cultural factors.

C. Pedagogical arguments: motivate emotionally, develop critical thinking and/or metacognitive abilities.

Some examples of the interrelation between the activities chosen and the arguments for such a choice are presented below. (The arguments are in parentheses). Introductory and closing activities: Presentations about number systems of the antiquity: Roman, Babylonian, Greek, Mayan (B1, C); students create numbers and discuss the effectiveness of the systems (A2, B1, C). Presentation about the ancestor of the abacus, the counting rods (B1); form rod numerals and compare with the modern representation (A1, A2, C). Information about the abacus (B2); compare the two forms (abacus and rods): advantages/disadvantages, similarities/differences (B1). After the intervention students presented their work to an audience in the role of the teacher (C); they elaborate on information about the cultural context of the abacus that led to prevail over the counting rods (B2, C) for a multicultural event. Main part: Students investigated place value with handmade abaci, web applications (A1, A2, C; Appendix) and worksheets designed by the researchers (A2, C); they analyzed the abacus’ representations/procedures and corresponded with the formal one (A1, A2); contests between groups (A1, C).

3.2 The implementation of the intervention The sequence of the instructive intervention was allocated in three sections; we investigated place value concepts in integers, then in decimals and finally we proceeded to calculations. For every didactical session we were elaborating a teaching plan which included the procedure, forms of work (individual, in pairs or in small groups), the media and material. Students worked with abaci that constructed themselves, web application (Appendix) and worksheets designed by the teacher/researcher. At the end of the school year students presented their work to other students. Section 1: Integers; Subsection1.3: Regrouping number quantities to standard numbers. Previous knowledge on the abacus: Students know how to read and form multi-digit numbers; identify the place value of the digits and analyze numbers in the expanded form; compose ten units of a class to the next upper class as one unit e.g.10 tens of a column are exchanged for 1 hundred unit of the next left column.

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Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 57 CHINESE ABACUS

Objectives: to convert more complex number quantities (that in specific classes exceed the nine units) to standard numbers through composing. The concept of the carried number: The composing activities in later stages served as cognitive scaffolding for the conceptual understanding of the carried number in the operation of addition. In analogy, the decomposing activities of other didactical sessions were connected with the concept of the carried number in subtraction. Procedure: First stage: The teacher forms a quantity e.g. 8 Tens and 14 Units (fig. 2a) on the interactive blackboard’s simulation or on the classroom’s handmade abacus. She asks students to discover the number. They are encouraged to recall how ten units of higher value are composed on abacus. A student implements the process. The passage from 10 units to 1 ten is made by pushing away the two five beads in the units rod and pushing forward one unit bead in the tens rod (fig. 2b).

Figures 2a and 2b: Regrouping quantities on abacus

To avoid the abacus-machine usage the teacher asks for explanations in terms of place value. Thus, the student while doing the bead-movements says: “I transfer ten of the fourteen units to the units’ column and compose 1 more ten in the tenth’s column. So we have 9 tens and 4 units. The number is 94”. Second stage: Students volunteer and elaborate their own quantities on the interactive whiteboard (fig. 3). Afterwards other students try to match the abacus procedure with the symbolic one on the classic whiteboard.

Figure 3: Students corresponding abacus and paper regrouping process

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Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 58 CHINESE ABACUS

Third stage: Students apply the new knowledge on worksheets in order to regroup quantities that they cannot be represented on abacus. The example is based on Poisard’s (2005) proposal for subtracting on an abacus with carried number. The method mainly taught to Greek schools and other European education systems is the ‘parallel additions’, which x x uses the relation a-b= (a+10 ) - (b+10 ). The other method, the ‘internal transfers’, is taught in second grade as an introductory method so it is rarely used over the years. It allows exchanges between classes and is the only method that can be implemented on abacus when using all beads.

Previous knowledge on abacus: decompose quantities; perform subtractions without trading. Procedure: The teacher forms the minuend of the subtraction 933-51 on the abacus. The number 1 can be subtracted immediately by removing one unit bead (figure 4, step 1) but in the tens column the regrouping process must be put forward. A student removes a one-bead from the hundreds and replaces it with two five-beads in the tens (figure 4, step 2). Having a total 13 on the tens he/she removes one five- bead and gets the result (figure 4, step 3). The student is encouraged to explain in terms of place value: “I decompose 1 hundred to 10 tens and then subtract 5 tens”.

Figure 4: Example of the subtraction method ‘internal transfers’ on abacus

Observation from the teaching: A student solved the subtraction 4,005-8 initially on the blackboard. She transferred a 1 thousands’ unit directly to the units’ position; she subtracted and found 3,007. We also observed this error (Fuson 1990) in some answers of the pre-test. When prompted to use the abacus, the student correctly implemented the decomposition process and explained it in terms of place value. Our discussion then revolved around the two results, so that the student reflected on her incorrect thought when she solved i t on the blackboard. One of the reasons that she did not make a mistake on the abacus – apart from the intervention’s influence – is possibly the visual-kinetic advantage of the tool; the space that occupies the intermediate columns may act as a deterrent for the eye to arbitrarily surpass them. Also, since we use the hand to remove one upper class unit bead, the fingers are merely guided to the next column in order to replace it with 10 equivalent lower units. The role of the teacher

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Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 59 CHINESE ABACUS

was crucial at this point to link the semi-abstract with the abstract technique, and at the same time to emphasize the common underlined mathematical theory.

4. DATA ANALYSIS AND RESULTS Questionnaire A: The total score of Questionnaire A was 100. The t-tests showed a statistically significant difference between the two measurements of students’ scores (t= 5.243, df = 17, p <0.001), with a pre-test mean of 50.5 and a post- test mean of 78.2. For the questions 3 and 7, the t-tests showed a statistically significant difference between the means of the two measurements: Question 3 (t=6.172, df=17, p<0.001) pre-test: mean 4.67, standard deviation 5; post- test: mean 12.9, standard deviation 3.5). Question 7 (t= 2.807, df = 17, p < 0.05) pre-test: mean 17.1, standard deviation 11.2; post-test: mean 22.6, standard deviation 9.0. The qualitative analysis that follows concerns sub questions 3b and 7a. It aims to find if the scores’ improvement is connected with better understanding through the investigation of the abacus. For question 3b we studied students’ written explanations.

Sub question 3b Pre-test: Compare 8 hundreds 2 tens 1unit __ 7hundreds 11 tens 16 units using the sign of inequality/equality. Explain your rationale. Post-test: Compare 6 hundreds 3 tens 3 units ___ 6 hundreds 14 tens 13 units using the sign of inequality/equality. Explain your rationale.

Table 1: Reasoning analysis for answers to sub question 3b Types of reasoning Pre-test Post-test correct 4 16 incorrect 8 insufficient/no explain 6 2

Our students on the pre-test (table 1) gave correct justifications, while on the post-test the majority of them were correct. Below we present examples of students’ written explanations on the pre-test and post-test. The abbreviations used are H=Hundreds, T=Tens, and U=Units.

Correct reasoning: 11T=1H and 1T. Also 16U=1T and 6U. So we have 700+110+16=826. Incorrect reasoning: They saw individual numbers on both sides: “The second is bigger than the first in two numbers”. They isolated digits and arbitrarily formed a number: “826 is less than 71,116”. They compared the hundred’s class, possibly recalling a vague knowledge of upper classes: “The first number has 1H more so it is bigger because hundreds matter”.

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 60 CHINESE ABACUS

Insufficient reasoning: “Because 7hundreds 11tens 16units is bigger”. Post-test: A figurative explanation appears (figure 5). By circling and using arrows, students were depicting the abacus process of composing ten units to a higher class. Figure 5: Sub question 3b – Example of regrouping at the post-test

Translation: “Seven hundred and fifty three is bigger”.

A more detailed response: “I get 10 from 14 T and make 1 H. The H now are 7. Then we have 13 U. I take 10 U and do another 1 T. The number is 753 greater than 643”.

Sub question 7a Pre-test: Solve the subtraction 70,005-9 in vertical form. Post-test: Solve the subtraction 40,006-9 in vertical form.

Table 2: Management of the carried number on the pre-test (sub question 7a) carried number not noted parallel addition Totals Answers 10 6 16 Success 4 5 9

Two students did not answer this question. From table 2 we observe that half students succeeded. The visible method was ‘parallel additions’, since the rest of the students did not note the carried number. The types of errors are categorized in table 3.

Table 3: Types of errors on the pre-test (sub question 7a) Question: 70,005-7 carried number not noted use of carried number Types of errors N Examples N Examples Carried number 5 60,008 70,010 81,098 Copying numbers 1 7,005-7 Number facts 1 69,997

The main type of errors (table 3) seemed to be the management of the carried number. For example, in the result ‘60,008’, though the carried

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 61 CHINESE ABACUS number is not noted, the error is the transfer of 1 thousand to the units’ position.

Sub question 7a, Post-test: Almost all students succeeded and the number of students who did not use the carried number decreased because of the use of the new method that requires the notation of the carried number (table 4).

Table 4: Management of the carried number on the post-test (sub question 7a) Carried number Parallel Internal Totals not noted additions transfers answers 4 7 7 18 success 3 6 7 16

The method ‘internal transfers’ appears and along with ‘parallel additions’ was applied successfully (table 4). The method of ‘parallel additions’ was applied mainly by students who had successfully applied it during the pre-test, while the method ‘internal transfers’ was given by those who had not been able to handle the carried number correctly.

Figure 4: The method ‘internal transfers’ as implemented on the post-test

Questionnaire B: ‘What is a carried number? ‘

Table 5: The interpretation of the carried number (Pre-test) Explanations N find/use/ something in calculations 9 Example with addition 5 I don't know/remember; I cannot describe it 4 Explanations N When the number exceeds 10 1

Table 6: The interpretation of the carried number (Post-test) Explanations with the use of an N Verbal explanations N example

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Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 62 CHINESE ABACUS

Composing 6 Ten units of a position move to the 3 e.g.10hundreds=1 thousand next position as one unit Decomposing e.g.1hundred=10 1 Number we keep aside/use in 2 tens operations for transfer Composing/decomposing 1 Borrowing from a number 1 A format of tens, hundreds, etc., for 2 transfer Convert a number of ten and over 1 to another format

The explanations with the use of an example differ between the two tests (table 5 & 6). At the pre-test students just performed an addition while in the post-test they put forward composing and decomposing examples. Verbal explanations at the pre-test seemed meaningless. Only in one answer we detected an attempt of mathematical explanation; “when the number exceeds 10”. At the post-test we can still observe a difficulty to explain but most students used the idea of exchanging (e.g., “transfer”, “convert the format”). One is specific: “10 units move to the next class as 1 unit”; others mix the knowledge before and after the intervention: “a number we keep for transfer”.

5. DISCUSSION The results of the pre-tests showed that most students did not have a profound understanding of the numbers’ structure; almost all could not recognize the numbers behind a non-standard partitioning (Fuson 1990; Resnick 1983) and half failed to solve a four-digit subtraction across zeros, a task that other studies have shown is difficult (Fuson 1990). In addition, they could not interpret the notion of carried number (Poisard 2005) considering it as an aid in operations but more of a vague nature. At the post-test, almost all displayed a better conceptual understanding. Using schematic representations and place value explanations influenced by the abacus activities, they successfully regrouped non-standard representations to standard numbers. As for the subtraction task, the students that had unsuccessfully managed the carried number in the pre-test, implemented successfully the abacus’ method ‘internal transfers’, which requires the reverse process of decomposing numbers. In agreement with Poisard (2005) the method has the advantage of illustrating the properties of our number system when they have not been adequately understood. The regrouping activities on the abacus and their connection to the algorithms of addition and subtraction changed students’ perspective about the concept of the carried number. They explained it as an exchange between classes, either verbally denoted or through an example. Despite the limitations of the study, such as the small sample and the lack of relevant experiential studies about the Chinese abacus, except Poisard’s

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Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 63 CHINESE ABACUS

(2005), we believe that the reasons for using the history of mathematics were accomplished in a quite satisfactory way. By elaborating on place-value concepts via the abacus, students developed understanding on an empirical basis (literally with their hands). By analyzing processes with the historical tool, students appreciated that mathematics of the past also lead to results that have logical completeness. In general, Bartolini Bussi’s (2000) argument that in the tactile experience offered by the ancient instruments one may find the foundations of mathematical activity, was verified. During the intervention we recognized the crucial role of the teacher in the teaching/learning process. Students may learn to calculate correctly with the tool, but without conceptual understanding. Also, as the example from the didactical session showed, they may achieve understanding place-value concepts when calculating with the tool but they continue to misapply the written calculations because they do not connect the two processes. That is why teachers should encourage students to gain insight into the relation between the tool and the concept that it represents (Uttal, Scudder, & Deloache 1999), otherwise its semiotic function will not be transparent. As further research we suggest the study of the Chinese abacus with younger students for the teaching of simpler concepts (Zhou & Peverly 2005).

REFERENCES Bartolini Bussi, M. (2000). Ancient instruments in the modern classroom. In J.Fauvel & J.V. Maanen (Eds.), History in mathematics education: The ICMI study (pp. 343-350). Dordrecht: Kluwer Academic publishers. Fuson, K. C. (1990). Conceptual structures for multiunit numbers: Implications for learning and teaching multidigit addition, subtraction, and place value. Cognition and Instruction, 7(4), 343-403. Jankvist, U.T. (2009). A categorization of the ‘whys’ and ‘hows’ of using history in mathematics education. Educational Studies in Mathematics, 71(3), 235- 261. Maanen, J.V. (2000). Non-standard media and other resources. In J. Fauvel. & J.V. Maanen (Eds.), History in mathematics education: The ICMI study (pp. 329-362). Dordrecht: Kluwer Academic publishers. Martzloff, J. C. (1996). A History of Chinese Mathematics. S.Wilson, translator. Germany: Springer. Poisard, C. (2005). Ateliers de fabrication et d’étude d’objets mathématiques, le cas des instruments à calculer (Doctoral dissertation, Université de Provence-Aix-Marseille I, France). Retrieved from http://tel.archives- ouvertes.fr/docs/00/06/10/97/PDF/ThesePoisardC.pdf Resnick, L. B. (1983). A developmental theory of number understanding. In H. P. Ginsburg (Ed.), The development of mathematical thinking, (pp. 109-151). New York: Academic Press. Spitzer, H. (1942). The abacus in the teaching of arithmetic. The Elementary School Journal, 46(6), 448-451.

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Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 64 CHINESE ABACUS

Tzanakis, C., & Arcavi, A. (2000). Integrating history of mathematics in the classroom: an analytic survey. In J. Fauvel & J. van Maanen (Eds.), History in mathematics education: The ICMI study (pp. 201-240). Dordrecht: Kluwer Academic publishers. Utall, D.H., Scudder, K.V., & Deloache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18(1), 37-54. Zhou, Z., & Peverly, S. (2005). Teaching addition and subtraction to first graders: A Chinese perspective. Psychology in the Schools, 42(3), 266-273.

BRIEF BIOGRAPHIES Vasiliki Tsiapou is a teacher at a public primary school in Thessaloniki. She has received a master in the Epistemology and History of Mathematics from the Department of Primary Education of the University of Western Macedonia, and she currently is a Ph.D. candidate at the same department. Her research is concerned with the integration of the History of Mathematics in class settings.

Konstantinos Nikolantonakis is Associate Professor of Mathematics Education at the Department of Primary Education of the University of Western Macedonia. He has graduated from the Department of Mathematics of the Aristotle University of Thessaloniki. He received a master and a Ph.D. in the Epistemology and History of Mathematics from the University of Denis Diderot (Paris-7). His research concerns the didactical use of the History of Mathematics, the History of Ancient Greek Mathematics, and the didactics of Arithmetic & Geometry.

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

Vasiliki Tsiapou, Konstantinos Nikolantonakis THE DEVELOPMENT OF PLACE VALUE CONCEPTS AND THE NOTION OF CARRIED NUMBER AMONG SIXTH GRADE STUDENTS VIA THE STUDY OF THE 65 CHINESE ABACUS

APPENDICES Questionnaire B 1. What does it mean for you “I do mathematics”? 2. Cite objects to make calculations. 3. Do you know what an abacus is? If yes, explain. 4. What is a carried number?

Abaci used during the instructive intervention

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

UNIVERSITY OF WESTERN MACEDONIA FACULTY OF EDUCATION

MENON 66 ©online Journal Of Educational Research

ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS1 Ingo Witzke University of Siegen [email protected]

Horst Struve University of Cologne [email protected]

Kathleen Clark Florida State University [email protected]

Gero Stoffels University of Siegen [email protected]

ABSTRACT In spring 2015 the authors taught an intensive seminar for undergraduate mathematics students, which addressed the transition problem from school to university by bringing to the fore concept changes in mathematical history and the learning biographies of the participants. This article describes how the concepts of empirical and formalistic belief systems can be used to give an explanation for both transitions – from school to university mathematics, and, for secondary mathematics teachers, back to school again. The usefulness of this approach is illustrated by outlining the historical sources and the participants’ activities with these sources on which the seminar is based, as well as some results of the qualitative data gathered during and after the seminar. Keywords: transition problem, genesis of geometry, secondary school mathematics, higher education, mathematical belief systems.

1. INTRODUCTION TO THE TRANSITION PROBLEM The transition problem that secondary mathematics teachers experience when moving from school to university (as students), and then again when moving from their university training to teaching mathematics was articulated

1 “ÜberPro” is an abbreviation of “Übergangsproblematik,” a German word for “transition problem”. With the term “university mathematics” we refer to mathematics courses designed for mathematics students and those pre-service secondary teachers majoring in mathematics (in Germany, these students are usually taught together).

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 67 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS by Felix Klein (1849-1925) as a “double discontinuity”: The young university student found himself, at the outset, confronted with problems, which did not suggest, in any particular way, the things with which he had been concerned at school. Naturally he forgot these things quickly and thoroughly. When, after finishing his course of study, he became a teacher, he suddenly found himself expected to teach the traditional elementary mathematics in the old pedantic way; and, since he was scarcely able, unaided, to discern any connection between this task and his university mathematics, he soon fell in with the time honored way of teaching, and his university studies remained only a more or less pleasant memory which had no influence upon his teaching. (Klein 1908: 1; first author’s translation) In the following we focus on the “first discontinuity”, referring to the transition from school to university and postulating an epistemological gap between school and university mathematics. We are encouraged by the more than 100-year-old problem, for which definitive solutions do not seem to appear on the horizon (Gueudet 2008). Unfortunately, dropout rates (especially in western countries) remain at a constantly high level. In Germany, approximately 50% of students studying mathematics or mathematics-related fields stop their efforts before finishing a bachelor’s degree (Heublein et al. 2012). In the United States, attrition rates for mathematics majors are differentiated between two undergraduate degrees available – bachelor’s (four- year degree) and associate’s (two-year degree). The National Center for Education Statistics (NCES) reported that for the years 2003 through 2009, 38% of mathematics majors entering university with the intent to earn a bachelor’s degree left the major (Chen 2013). Similarly for those students intending to earn an associate’s degree, some 78% left the major. This leads again to an (at least perceived) intensification of research in this field. Furthermore, recent investigations in the United States have focused on the critical role that success in calculus course taking plays in undergraduate students’ ambition for and persistence in mathematics. To date, many of the resulting publications from the Mathematical Association of America National Study of Calculus have highlighted the importance of student attributes on their success (e.g., Bressoud et al. 2013); however, identifying concrete ways in which students may be successful in negotiating the transition from secondary school mathematics student to first-year university mathematics student is absent from the literature. In 2011, the most important professional associations regarding mathematics (education) in Germany (DMV-Mathematics, GDM-Mathematics Education, and MNU-STEM Education) formed a task force regarding the problem of transition (cf. http://www.mathematik-schule-hochschule.de). Then, in February 2013, a scientific conference with the topic “Mathematik im Übergang Schule/Hochschule und im ersten Studienjahr” (“Mathematics at the Crossover School/University in the First Academic Year”) in Paderborn, Germany

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 68 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS attracted almost 300 participants giving over 80 talks regarding the problematic transition process from school to university mathematics. The proceedings of this conference (Hoppenbrock et al. 2013) and its predecessor on special transition courses (Biehler et al. 2014) give an impressive overview on the necessity and variety of approaches regarding this matter. Interestingly a vast majority of the studies and best practice examples for “transition courses” locate the problem in the context of deficits (going back as far as junior high school) regarding the content knowledge of freshmen at universities. In the “pre-course and transition course community” it seems to be consensus by now that existing deficits in central fields of lower- secondary schools’ mathematics make it difficult for freshmen to acquire concepts of advanced elementary mathematics and to apply these. Fractional arithmetic, manipulation of terms or concepts of variables have an important role, e.g., regarding differential and integral calculus or non-trivial application contexts and constitute insuperable obstacles if not proficiently available. (Biehler et al. 2014: 2; first author’s translation) The question of how to provide first semester university students with obviously lacking content knowledge is certainly an important facet of the transition problem. However, as the results of a recent empirical study suggest, there are other, deeper problem dimensions that aid in further understanding the issue.

2. MOTIVATION FOR DEVELOPING THE SEMINAR To investigate new perspectives on the transition problem, approximately 250 pre-service secondary school teachers from the University of Siegen and the University of Cologne in 2013 were asked for retrospective views on their way from school to university mathematics. When the survey was disseminated, the students had been at the universities for about one year. Surprisingly, the systematic qualitative content analysis of the data (Huberman & Miles 1994, Mayring 2002) showed that from the students’ point of view it was not the deficits in content knowledge that dominated their description of their own way from school to university mathematics. Instead, students articulated a feeling of “differentness” between school and university mathematics that did not relate simply to a rise in content-specific requirements. To illustrate this point of “differentness” we selected two exemplar responses from the questionnaire responses to the question, What is the biggest difference or similarity between school and university mathematics? What prevails? Explain your answer. Student (male, 19 years): “The fundamental difference develops as mathematics in school is taught “anschaulich”[1], whereas at university there is a rigid modern-axiomatic structure characterizing mathematics. In general there are more differences than similarities, caused by differing aims” (first author’s translation) At this first student’s point we can only speculate on the term “aims”, but in

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 69 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS reference to other formulations in his survey responses it seems possible that he distinguished between general education (in German, “Allgemeinbildung”) as an aim for school and specialized scientific teacher-training at universities.

The second example is impressive in the same sense: Student (female, 20 years):

Figure 1. A student’s articulation of difference or similarity between school and university mathematics.

university school

abstrac very empiric t (everyday many life) proofs few under- computing proofs standing

In many cases the students clearly distinguished between school and university mathematics, which is most prominent in the second example. For this student, school mathematics and university mathematics are so different, that the only remaining similarity (in German, “Gemeinsamkeit”) is the word “mathematics”. This “differentness” encountered by the students is specified in further parts of the questionnaire with terms as vividness, references to everyday life, applicability to the real world, ways of argumentation, mathematical rigor, axiomatic design, etc.2 Using additional results of studies with a similar interest (e.g., Gruenwald et al. 2004, Hoyles et al., 2001) we arrived at the preliminary conclusion that pre- service mathematics teachers clearly distinguish between school and university mathematics with regard to the nature of mathematics. In the terms of Hefendehl- Hebeker Ableitinger and Herrmann, the students encounter an “Abstraction shock” (Hefendehl-Hebeker et al. 2010), meaning that students have serious difficulties regarding a dramatically increased level of abstraction at the beginning of their undergraduate courses in mathematics. Schichel and Steinbauer (2009: 1; first author’s translation) describe the same phenomenon, when saying that,

2 The cited study has not been published in total so far. However, partial results have been published in Witzke 2013a, 2013b.

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 70 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

Abstraction shock: The level of abstraction regarding the teaching of university mathematics is in marked contrast to the teaching in school, where mathematical content is in principal developed on the basis of [concrete] examples. Many students get already lost in the “definition- theorem-proof-jungle” in the first weeks of their university career being faced with an uncommented abstract approach. To describe and face this problem, we established a framework for further research concerning the transition problem. In the next section, we reconstruct the nature of mathematics communicated explicitly and implicitly in high school and university textbooks, lecture notes, standards, etc., with a special focus on differences to identify in detail what constitutes the abstraction shock described in literature and by students. Thereby we follow the paradigm of constructivism in mathematics education, believing that students construct their own view on mathematics when working and interacting in classroom or lecture hall with the material, problems, and stimulations that course instructors (and students’ peers) provide (Anderson et al. 2000, Bauersfeld 1992).

3. BELIEFS ON MATHEMATICS: TODAY AND IN HISTORY

3.1 Beliefs describing the notion of mathematical objects and activities The terms nature of mathematics and belief system regarding mathematics are closely linked to each other if we understand learning in a constructive way. Schoenfeld (1985) successfully showed that personal belief systems matter when learning and teaching mathematics: One’s beliefs about mathematics [...] determine how one chooses to approach a problem, which techniques will be used or avoided, how long and how hard one will work on it, and so on. The belief system establishes the context within which we operate […] (Schoenfeld 1985: 45) From an educational point of view beliefs about mathematics are decisive for our mathematical behavior as the empirical studies of Schoenfeld have shown; the beliefs system was identified as the critical factor determining success in concrete problem solving contexts. Furthermore, prominent among research on beliefs are four categories of beliefs concerning mathematics, which were distinguished by Grigutsch, Raatz and Törner (1998) as aspects: the toolbox aspect, the system aspect, the process aspect and the utility aspect. Liljedahl, Rolka and Roesken (2007) specified this wide range of possible aspects of a mathematical worldview as follows: In the “toolbox aspect”, mathematics is seen as a set of rules, formulae, skills and procedures, while mathematical activity means calculating as well as using rules, procedures and formulae. In the “system aspect”, mathematics is characterized by logic, rigorous proofs, exact definitions and a precise mathematical language, and doing mathematics consists of

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 71 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

accurate proofs as well as of the use of a precise and rigorous language. In the “process aspect”, mathematics is considered as a constructive process where relations between different notions and sentences play an important role. Here the mathematical activity involves creative steps, such as generating rules and formulae, thereby inventing or re-inventing the mathematics. Besides these standard perspectives on mathematical beliefs, a further important component is the usefulness, or utility [aspect], of mathematics. (Liljedahl et al. 2007: 279) Very often these beliefs are located within certain fields of tension (in German, “Spannungsfelder”). There is, for example, the process aspect, which is always implicitly connected to its opposite pole the product aspect. Another pair of concepts in this sense is certainly an intuitive aspect on the one hand and a formal aspect on the other, having even a historical dimension: “There is a problem that goes through the history of calculus: the tension between the intuitive and the formal” (Moreno-Armella 2014: 621). These fields of tension may help to describe the problems the students encounter on their way to university mathematics. Especially helpful when looking at the results of the aforementioned survey, representing one important facet, seems to be the tension between what Schoenfeld called an empirical belief [2] system and a formalistic belief system [3] – a convincing analytical distinction following the works of Burscheid and Struve (2010). The empirical belief system [2] on the one hand describes a set of beliefs in which mathematics is understood as an experimental natural science, which includes deductive reasoning about empirical objects. Struve (1990) and Schoenfeld (1985) have reconstructed this belief system in school, investigating school textbooks and students’ behavior. Good examples for comparable belief systems, regarding the understanding of mathematics in an empirical way, can be found in the history of mathematics. The famous mathematician Moritz Pasch (1843-1930), who completed Euclid’s axiomatic system, explicitly understood geometry in this way: The geometrical concepts constitute a subgroup within those concepts describing the real world […] whereas we see geometry as nothing more than a part of the natural sciences. (Pasch 1882: 3) Thus, mathematics in this sense is understood as an empirical, natural science. This, of course, implies the importance of inductive elements as well as a notion of truth bonded to the correct explanation of physical reality. In Pasch’s examples, Euclidean geometry is understood as a science describing our physical space by starting with evident axioms. Geometry then follows a deductive buildup, but it is legitimized by the power to describe the physical space around us correctly. This understanding of mathematics as an empirical science (on an epistemological level) can be found throughout the history of mathematics, and prominent examples for this understanding are found in many scientists of the 17th and 18th centuries. For example, Leibniz conducted analysis on an empirical level; the objects of his calculus differentialis and calculus

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 72 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS integralis were curves given by construction on a piece of paper and not as today’s abstract functions (cf. Witzke 2009). The formalistic belief system, on the other hand, describes a set of beliefs in which mathematics is understood as a system of un-interpreted concepts and their connections in propositional functions (in German, “Aussageformen”), which can be established using axioms, (implicit) definitions, and proofs. Davis and Hersh (1981) and Schoenfeld (1985) have reconstructed this belief system as a typical one for professional mathematicians. Good examples for comparable belief systems, regarding the understanding of mathematics in a formalistic way, can be found in the history of mathematics. The famous mathematician David Hilbert (1862-1943), released geometry completely from any empirically bonded entities: Whereas Pasch was anxious to derive his fundamental notions from experience and to postulate no more than experience seems to grant. Hilbert started ‘Wir denken uns…’ we imagine three kinds of things… called points… called lines… called planes… we imagine points, lines, and planes in some relations… called lying on, between, parallel, congruent…” (–) “Wir denken uns…” – the bond with reality is cut. Geometry has become pure mathematics. The question of whether and how to apply it to reality is the same in geometry as it is in other branches of mathematics. Axioms are not evident truths. They are not truths at all in the usual sense. (Freudenthal 1961: 14; English translation in Streefland 1993) Mathematics in this sense can be understood as the formal science. This implies the importance of deductive elements as well as a notion of truth in the sense of logical consistency. This understanding of mathematics as a formal science (on an epistemological level) can be found throughout the history of mathematics after Hilbert. Prominent examples for this understanding are found in many mathematicians of 19th, 20th, and 21st centuries. For example, Kolmogoroff formalized probability theory in this way; the concepts of his Grundbegriffe der Wahrscheinlichkeitsrechnung are sets and measures given by definition in his famous axioms. (cf. Kolmogorov 1973). So, what is the connection among these elements, mathematics students, and the transition problem? If we examine current textbooks for school mathematics, we see that students at school are likely to acquire an empirical belief system. And, if we examine current course textbooks for university mathematics, we see that students at university are, in contrast, faced with a formalistic belief system (cf. Burscheid & Struve 2009, Schoenfeld 1985, Schoenfeld 2011, Struve 1990, Tall 20133). On epistemological grounds both show parallels to specific historical understandings of mathematics. These

3 In his foundational work, “How humans learn to think mathematically”, David Tall (2013) emphasized an equivalent to Struve’s and Schoenfeld’s empirical belief system when referring to a blend of “Embodiment and Symbolism” prevailing in school. He distinguished this, what he calls worlds of mathematics, from a world of “(Axiomatic) formalism” realized at university level and associated Hilbert – which is quite similar to Burscheid and Struve’s formalistic belief system.

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 73 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS epistemological parallels were fundamental for the design of our “transition problem” seminar for students. The main idea is that the recognition and appreciation of different natures of mathematics in history (i.e. those held by expert mathematicians) can help students to become aware of their own belief system and may guide them to make necessary changes.

4. A DEEPER LOOK INTO SCHOOL AND UNIVERSITY MATHEMATICS The most recent National Council of Teachers of Mathematics (NCTM) standards (2000) and prominent school textbooks indicate that, for good reasons (cf. the EIS-principle by Bruner (1966), the basic experiences (“Grunderfahrungen”) of Winter (1996) or the three worlds of mathematics by Tall (2013)), mathematics is taught in the context of concrete (physical) objects at school. For example, the NCTM process standards, and in particular “connections” and “representations,” (which are comparable to similar mathematics standards in Germany), focus on empirical aspects of mathematics. At school and in their future career it is important that students “recognize and apply mathematics in contexts outside of mathematics” or “use representations to model and interpret physical, social, and mathematical things” (NCTM 2000: 67). The prominent place of illustrative material and visual representations in the mathematics classroom has important consequences for the students’ views about the nature of mathematics. As we previously mentioned, Schoenfeld (1985, 2011) and Struve (1990, 2010) proposed that students acquire an empiricist belief system of mathematics at school. This is likely to be caused by the fact that mathematics in modern classrooms does not describe abstract entities of a formalistic theory but a universe of discourse ontologically bounded to “real objects”. For example, Probability Theory is bounded to random experiments from everyday life, Fractional Arithmetic to “pizza models”, Geometry to straightedge and compass constructions, Analytical Geometry to vectors as arrows, Calculus to functions as curves (graphs), and so forth. However, at university things can look totally different. Authors of prominent textbooks (in Germany, as well as in the United States) for beginners at university level depict mathematics in quite a formalistic, rigorous way. For example, in the preface of Abbott’s popular book for undergraduate students, Understanding Analysis, it becomes very clear how mathematicians consider a major difference between school and university mathematics: “Having seen mainly graphical, numerical, or intuitive arguments, students need to learn what constitutes a rigorous proof and how to write one” (Abbott 2001: vi). This view is also transported by Heuser’s popular analysis textbook for first semester students (Heuser 2009: 12; first author’s translation): The beginner at first feels […] uncomfortable […] with what constitutes mathematics: - The brightness and rigidity in concept formation - The pedantic accurateness when working with definitions

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 74 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

- The rigor of proofs which are to be conducted […] only with the means of logic not with Anschauung. [1] - Finally the abstract nature of mathematical objects, which we cannot see, hear, taste or smell. […] This does not mean that there are no pictures or physical applications in Abbott’s book; it is common sense that modern mathematicians work with pictures, figural mental representations, and models. However, in contrast to many students, it is clear to them that these are illustrations or visualizations only, displaying certain logical aspects of mathematical objects (and their relations to others) but by no means representing the mathematical objects in total. This distinction is more explicit if we look at a textbook example. In school textbooks (in Germany) the reference objects for functions are mainly drawn curves. Functions may then virtually be identified with these empirically given curves (Witzke 2014). Tietze, Klika and Wolpers (2000: 72) discussed this context of an analysis like “elementary algebra combined with the sketching of graphs”. Consequently, school textbook authors work with the so-called concept of graphical derivatives (firmly anchored in the curricula) in the context of analysis (see Fig. 2). At university, curves are by no means the reference objects; here they are only one possible interpretation of the abstract notion of function. The graph of a function in formalistic [3] university mathematics is actually only a set of (ordered) pairs. If we contrast the empirical belief system many students acquire in classroom with the formalistic belief system students are faced with at university we have a model that explains why challenging the transition problem regarding belief systems is necessary for the professionalization of mathematicians and math teachers. For example, in this model the notion of proof differs substantially in school and university mathematics. Whereas at universities (especially in pure mathematics) only formal deductive reasoning is an acceptable method, non-rigorous proofs relying on “graphical, numerical and intuitive arguments” are an essential part of proofs in school mathematics where we explain phenomena of the “real world”. Using Sierpinska’s (1987, 1992) terminology, students in this transition-phase have to overcome a variety of “epistemological obstacles”4, requiring a significant change in their understanding of what mathematics is about.

4 Following the definition and common usage of the term “epistemological obstacle” in mathematics education, we mean content-based obstacles that are likely to occur in every learning biography, and whose overcoming will eventually lead to a decisive process of cognition. Note that these are referred to as being based in the nature of things in principle and not in the lack of individual cognitive development (cf. Schneider 2014: 214-217).

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 75 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

Figure 2. Graphical derivatives in a German school textbook (EdM 2010: 203).

5. SEMINAR DESIGN, CONTENT, AND IMPLEMENTATION The findings of the initial questionnaire and the identification of the theoretical considerations, which were described in the preceding paragraphs, were essential for designing a seminar to address the transition problem. The overall aim of the seminar course was to make students aware and to lead them to an understanding of crucial changes regarding the nature of mathematics from school to university, by discussing transcripts, textbooks, standards, historical sources, etc. The different “natures” of mathematics in school and university can also, on an epistemological level, be found in the history of mathematics, as we previously stated. Thus, an understanding of how and why this change (from empirical-physical to formalistic-abstract) took place should be achieved by an historical-philosophical analysis (cf. Davies 2010). This, in fact, is the key notion of the seminar. Thereby we hoped that the students were able to relate their own learning biographies to the historical development of mathematics. This conceptual design of the seminar draws upon positive experience with explicit approaches regarding changes in the belief system of students from science education (esp. “Nature of Science” cf. Abd-El-Khalick & Lederman 2001). The undergraduate ÜberPro seminar that is the focus of what follows, was designed for students to cope with the transition problem. It was implemented for the first time in February 2015 and was an intensive experience that took place over three days (approximately 18 hours of instruction). Twenty (8 male; 12 female) undergraduate mathematics students, who were also preparing to teach secondary mathematics, participated in the seminar. Table 1 presents the distribution of student age and semester at university of the seminar participants.

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 76 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

Table 1. Age and semester at university for ÜberPro seminar participants (February 2015). Participant Semester Participant Age (in Age (in Semester at number (gender: at number (gender: years) years) university M(ale)/F(emale)) university M(ale)/F(emale)) 1 (F) 23 7 11 (F) 22 7 2 (F) 19 3 12 (F) 22 7 3 (M) 23 7 13 (F) 21 3 4 (F) 23 7 14 (M) 26 13 5 (F) 22 5 15 (M) 26 5 6 (M) 20 3 16 (F) 20 3 7 (F) 21 3 17 (M) 25 7 8 (F) 25 10 18 (M) 22 7 9 (M) 26 3 19 (F) 22 5 10 (F) 20 3 20 (M) 24 8

The three-day seminar was organized in four parts: 1) Raise attention to the importance of beliefs about and philosophies of mathematics. 2) Historical case study: Geometry from Euclid to Hilbert. (In particular, which questions led to the modern understanding of mathematics?) 3) Exploration of Hilbert’s approach (Or, what characterizes modern formalistic mathematics?) 4) Summary discussion and reflection. We employed several instructional techniques during the intensive seminar. During the 18 hours of instruction students engaged in small group work, which included engaging in active learning tasks and short discussions, and whole class discussions, which included individual students and small groups sharing their work. The self-activating sequences were enriched by short instructional lectures of the participating mathematics educators (i.e. the first, second, and fourth authors). Moreover, seminar participants worked with a variety of materials, including reading original historical sources, excerpts from research literature, and school textbooks; using hands-on materials to model concepts from projective and hyperbolic geometry; and investigating concepts using dynamic geometry software. We provide further context and description for a number of the seminar activities within the elaboration of the four parts of the seminar that follows.

1) Raise attention to the importance of beliefs about and philosophies of mathematics. In the first part of the seminar we wanted to make students aware of the idea of different belief systems and natures of mathematics. Here we began with individual reflections and work with authentic material such as transcripts

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 77 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS from Schoenfeld’s (1985) research that clearly showed the meaning and relevance of the concept of an empirical belief system. Afterwards students compared different types of textbooks: university course textbooks, school textbooks, and historical textbooks. The three excerpts (Fig. 3) illustrate how we worked within this comparative activity. In the upper left-hand corner of Fig. 3 is a formal university textbook definition of differentiation. It is characterized by a high degree of formalization: the objects of interest are functions defined on real numbers or complex numbers. The excerpt exhibits a highly symbolic definition where the theoretical concept of limit is necessary. In contrast, we see just below an excerpt from a popular German school textbook. Here, the derivative function is defined on a purely empirical level; the upper curve is virtually identified with the term function. Characteristic points are determined by an act of empirical measuring and the slopes of the triangles are then plotted underneath and results in the second graphed curve (graphical derivation).

Figure 3. Three excerpts of different textbooks for comparison. University course textbook “Königsberger 2001: 34” (top left), school textbook “Lambacher Schweizer 2009: 55” (bottom left), historical text “Leibniz, Acta Eruditorum,” 1693 (right).

Finally, if we look back to Leibniz (one of the fathers of analysis), with his calculus differentialis and intergalis, we find that he conducted mathematics in a rather empirical way as well (cf. Witzke 2009). His objects were curves given by construction on a piece of paper – properties like differentiability or continuity

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 78 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS were read out of the curve. Furthermore, there seem to be parallels on an epistemological level between school analysis and historical analysis. For example, Leibniz presented (published in 1693) the invention of the so-called “integrator” (right-hand side of Fig. 3), a machine that was designed to draw an anti-derivative curve by retracing a given curve. So here, as in the school textbook, the empirical objects form the basis of the theory; even more the processes regarding Leibniz’ integrator and the textbooks’ graphical derivation. During the seminar course, students shared their response to the question, “What is mathematics?” – which were then organized according to the scheme aspect, formalism aspect, process aspect, and utility aspect, similar to those introduced in the items by Grigutsch, Raatz and Törner (1998).

2) Historical case study: Geometry from Euclid to Hilbert. (In particular, which questions lead to the modern formalistic understanding of mathematics?) An adequate description of the development of the nature of mathematics in the course of history requires more than one book. We referenced the following ones: Bonola (1955) for a detailed historical presentation; Garbe (2001), Greenberg (2004), and Trudeau (1995) for a lengthy historical and philosophical discussion; Ewald (1971), Hartshorne (2000), and Struve and Struve (2010) for a modern mathematical presentation. Additionally, Davis and Hersh (1981) and Davis, Hersh and Marchiotto (1995) presented aspects of the historical and philosophical discussion in a concise manner, and for students, in a relatively easy and accessible way. The overall aim of the historical case study was to make students aware of how the nature of mathematics changed over history. Regarding our theoretical framework, we endeavored to make explicit how geometry – which for hundreds of years seemed to be the prototype of empirical mathematics, describing physical space – developed into the prototype of a formalistic mathematics as formulated in Hilbert’s Foundations of Geometry in 1899 (cf. Fig. 4.) Figure 4 The historical and philosophical development of mathematics along the development of geometry.

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 79 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

Consequently, we helped students (or, aimed to help them) on their way to develop an understanding for different natures of mathematics, in particular, modern ones taught at the university level. In the seminar course we began this component of instruction with Euclid’s Elements; they show what a deductively built piece of mathematics, describing physical space, looks like in a prototype manner. Here we prompted the students to display in a diagrammatic manner how Pythagoras’ theorem can be traced down to Euclid’s five postulates. (cf. Fig. 5, the numbers indicate the number of the proposition within Euclid’ Elements). This activity was selected based upon the 2013 survey results, which showed that a significant number of students were not familiar with a deductive structure after one year of university mathematics.

Figure 5. The architecture of Pythagoras’ theorem.

It was important for the overall goal of the seminar that the Elements gave reason to discuss status, meaning, and heritage of axiomatic systems. This enabled us to focus on the self-evident character of the axioms (or, postulates) describing physical space in a true manner – and to provide insights on the surrounding real space which were accepted without proof (cf. Garbe 2001: 77).

Figure 6. Photo of “Autobahn” taken by the first author (left); Albrecht Dürer: “Man drawing a lute” (1525), (middle); photo of Albrecht Dürer Activity during seminar, taken by third author (right)

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 80 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

Projective geometry was the next example on our way (in the seminar) to a modern understanding of geometry. Starting with the question of whether other geometries, besides the Euclidean one, are conceivable, projective geometry seemed to be an ideal case (cf. Ostermann & Wanner 2012: 319-344). Related to the overall goal of the course, the notion that there exists more than one geometry fostered the idea that there is more than one (“true”) mathematics. And, this in turn serves to lead us away from the quest for one unique mathematics describing physical space (cf. Davis & Hersh 1985: 322- 330). On the one hand, we wanted students to become familiar with the idea (via the Albrecht Dürer Activity, cf. Fig. 6) that projective geometry seems to be so intuitive and evident when looking at its origins in the vanishing point perspective (arts). On the other hand, projective geometry adds new objects to the Euclidean geometry (esp. the infinitely distant points on the horizon) and its place in the seminar introduced the students to the insight that all parallels may meet eventually. Additionally, with projective geometry the students encountered a further axiomatizable geometry, which also possessed particular properties that finally influenced Hilbert to ultimately design a formalistic geometry that was free of any physical references (cf. Blumenthal 1935: 402). Julius Plücker saw in the 19th century as one of the first that theorems in projective geometry hold if the terms “straight line” and “point” are interchanged. This so-called principle of duality gave a clear hint that the nature of geometrical objects may be irrelevant and that it is the relations between these objects that matter. (cf. Fig. 7)

Figure 7. Example for the principle of duality: Theorem of Pappus-Pascal: Six points (red) incident with two lines (blue) – the points (green) which are incident with opposite lines of the hexahedron are collinear (green line). Theorem of Brianchon: Six lines (red) incident with two points (blue) – the lines (green) which are incident with opposite points of the hexahedron are copunctal (green point)

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 81 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

The next case that students examined was a revolutionary step towards a formalistic formulation of geometry that comprised the development of the non- Euclidean geometries, and which was connected to the names Janos Bolyai (1802-1860), Nikolai Ivanovitch Lobatchevski (1792-1856), Carl Friedrich Gauß (1777-1855), or Bernhard Riemann (1826-1866) (cf. Garbe 2001, Greenberg 2004, Trudeau 1995 on their historical role regarding non-Euclidean Geometries). In fact, the non-Euclidean geometries developed from the “theoretical question” around Euclid’s fifth postulate, the so-called parallel postulate: Let the following be postulated: [...] That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which the angles are less than two right angles. (Heath et al. 1908) Compared to the other postulates like the first, “to draw a straight line from any point to any point”, the fifth postulate sounds more complicated and less evident. This postulate cannot be “verified” by drawings on a sheet of paper as parallelism is a property presupposing infinitely long lines. In the words of Davis, Hersh and Marchiotto (1995: 242), “it seems to transcend the direct physical experience”. In history this was seen as a blemish in Euclid’s theory and various attempts have been undertaken to overcome this flaw. On the one hand, different individuals tried to find equivalent formulations, which are more evident (e.g. Proclus (412-485), John Playfair (1748-1819))5. On the other hand, several mathematicians tried to deduce the fifth postulate from the other postulates so that the disputable statement becomes a theorem (which does not need to be evident) and not a postulate (e.g. Girolamo Saccheri (1667-1733), Johann Heinrich Lambert (1728-1777)). (cf. Davis & Hersh 1985: 217-223, Greenberg 2004: 209-238, Struve & Struve 2010) In contrast in the 18th and 19th century, Bolyai, Lobatchevski, Gauß, and Riemann experimented with negations and replacements of the fifth postulate guided by the question of whether the parallel postulate was logically dependent of the others (cf. Greenberg 2004: 239-248). If this would have been true – Euclidean geometry should actually work without it – what it does, in a sense that no inconsistencies occur.

5 To Proclus, who was amongst the first commentators of Euclid’s Elements in ancient Greece, already formulated doubts on the parallel postulate and formulated around 450 an equivalent formulation (cf. Wußing & Arnold 1978: 30). Playfair’s formulation (1795), “in a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point”, is quite popular today (cf. Prenowitz & Jordan 1989: 25, Gray 1989: 34).

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 82 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

Figure 8. Visualizations regarding different geometries. Elliptic, Euclidean and Hyperbolic Geometry. (naiadseye, 2014)

But this logical act leads to conclusions that differ from those in Euclidean geometry. For example: - In hyperbolic geometry the sum of interior angles in a triangle sums to less than 180°, in elliptical geometry to more than 180° (cf. Fig. 8) - The ratio of circumference and diameter of a circle in hyperbolic geometry is bigger than π, in elliptical geometry smaller than π. - In hyperbolic as in elliptical geometry triangles which are just similar but not congruent do not exist. - In hyperbolic geometry there is more than one parallel line through a point P to a given line g and in elliptical geometry there are no parallel lines at all. (cf. Davis & Hersh 1985: 222, Garbe 2001: 59) Working with texts and sources regarding the process of discovery of the non-Euclidean geometries had an important impact on students’ belief system. The 2013 survey results indicated that the so-called “Euclidean Myth” (Davis & Hersh 1985) was widely prevalent: to many first-year university students mathematics is a monolithic block of eternal truth; a theorem, once proven, necessarily holds in every context. With the discovery of the non-Euclidean geometries, it became apparent in history that there was no such truth in an ontological sense. In contrast, there seems to be multiple such truths, depending on the context in which you work. We used a discussion of Gauß’s qualms to publish his results on non-Euclidean geometry, afraid of being accused of doing something suspect, or the (probably legendary) story (cf. Garbe 2001: 81-85) that he tried to measure on a large scale whether the world is Euclidean to help the students become amenable to the revolutionary character of his discoveries. Following Freudenthal’s (1991) idea of guided reinvention, recapitulating the

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 83 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS history of humankind seems to bear quite fruitful perspectives for the development of individual belief systems. Finally, from the discussion of the non-Euclidean geometries students investigated questions which led to Hilbert’s formalistic turn. If there was more than one consistent geometry, which one was the true one? This question is closely linked to the question, what is mathematics?

3) Exploration of Hilbert’s approach. (Or, what characterizes modern formalistic mathematics?) Hilbert actually gave an answer to this problem – not only in a philosophical and programmatic way but also by formulating a geometry “exempla trahunt” (Freudenthal 1961: 24), a discipline that was seen for ages as the natural description of physical space, in a formalistic sense and characterized by an axiomatic structure. The established axioms are fully detached and independent from the empirical world, which leads to an absolute notion of truth: mathematical certainty in the sense of consistency. Thus, with Hilbert the bond of geometry to reality is cut. This came to life in the seminar when students read Hilbert’s Foundations of Geometry (1902, see Fig. 9) in detail.

Figure 9. The famous first paragraph of Hilbert’s (1902) Foundations of Geometry.

Hilbert did not give his concepts an explicit semantic meaning; he spoke independently from any empirical meaning of “distinct systems of things”. Consequently, intuitive relations like in between or congruent do not have an empirical meaning but are relations fulfilling certain formal properties only (cf. for example, Hilbert & Bernays 1968: §1, Greenberg 2004: 103-129). As we all know, the discussion of nature of mathematics did not come to an end with Hilbert. Thus, the course ended with discussions of texts taken from What is Mathematics, Really? (Hersh 1997). Hersh understood “mathematics as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context” (xi), which created a balanced view. However, nobody will deny that formalism in Hilbert’s open-minded version had a lasting effect on the development of mathematics. As a consequence, today’s university mathematics has the freedom to be developed

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 84 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS without being ‘true’ in an absolute sense anymore (cf. Freudenthal 1961), but nevertheless including the possibility to interpret it physically again. In the meantime, while the creative power of pure reason is at work, the outer world again comes into play, forces upon us new questions from actual experience, opens up new branches of mathematics, and while we seek to conquer these new fields of knowledge for the realm of pure thought, we often find the answers to old unsolved problems and thus at the same time advance most successfully the old theories. And it seems to me that the numerous and surprising analogies and that apparently prearranged harmony which the mathematician so often perceives in the questions, methods and ideas of the various branches of his science, have their origin in this ever-recurring interplay between thought and experience. (Hilbert 1900: English translation in Reid 1996: 77) It is this openness and freedom of questions of absolute truth, which Hilbert replaced by the concept of logical consistency that made mathematics so successful in the 20th century (cf. Freudenthal 1961: 24, Garbe 2001: 100-109, Tapp 2013: 142). This makes again quite clear that modern mathematics after Hilbert is on epistemological grounds, completely different than (historical) empirical mathematics and of course, mathematics taught in school. Whether the first is grounded on set axioms and the notion of mathematical certainty (inconsistency), the second and third are grounded in evident axioms – thus describing physical space including a notion of (empirical) truth, resting essentially on induction from experience.

4) Summary discussion and reflection The final session of the seminar entailed a whole-group discussion in which we sought to connect insights gained from the historical perspectives with the individual participants’ mathematical biographies. We first reminded students about the preliminary discussions regarding different personal belief systems that occurred in the first session of the seminar. The intention was that the transparency on the historical problems that led to a modern abstract understanding of mathematics can therefore lead to an understanding of what happens if students live through this revolution on epistemological grounds as individuals, thus opening differentiated views on the transition problem. As an example, the first author – while leading the concluding discussion of the seminar course – prompted students with: You have described, that [it] is all abstract; there is no application. […] You have also said, that it is somehow not too bad, because it is also important...If you are watching the reality, that’s what I want to remind you, at the differences between high school-university, that you also described again … (fourth author’s translation) To this, one student shared: If first-year students go to university, then they have a completely different concept map in their mind and for example, [they] have the understanding that a

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 85 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

graph is always a function. But in fact that is not right. And if the lecturer talks about such definitions, [students] will say: “Huh? I have never ever seen something like this in my whole life,” and principally they know, they can connect it to their knowledge, but they need simply someone who explains to them ok, the function is not the graph. Also they need principally a dictionary for high school to university, where you can look up the concepts. (fourth author’s translation) In this student’s personal mathematical biography, then, there was a clear gap between the mathematics experienced in high school when compared to that at university. And, the gap was so pronounced that a sort of translation device – “a dictionary for high school to university” – was required to make sense of the different concepts.

6. SUMMARY Although the primary intent of this article was to share the usefulness of the intensive seminar we conceptualized and implemented with one group of university mathematics students at a German university in spring 2015, another aim was to share initial reflections on the data we gathered to determine whether an intervention longer than a three-day seminar was both warranted and necessary. The group of students who participated in the seminar was heterogeneous with respect to age and semesters at university (see Table 1), which gave us multi-perspective views on the success of the seminar and a deeper insight of the transition problem. Numerous data sources will inform the construction of six case studies which will describe the ‘state of the transition’ that the participants experienced – and are still experiencing – with respect to the transition from school to university mathematics.6 The data sources include pre- and post-surveys (measures of beliefs and perceptions of mathematics, content items, and demographic information), video and audio recording of the three- day seminar, essays submitted by all seminar participants, audio recording of interviews of six seminar participants, observation notes (third author), and various seminar artifacts (e.g. daily debrief notes completed with students, response cards to open, anonymous prompts). However, as with the preceding sample revelations, further evidence – in the words of the students – revealed that they could articulate the transition problem adequately and that they desire a solution as they contemplated the next “abstraction shock” they will encounter. For example, in the summary discussion, one young woman declared: Even the problem with [limits], that was also described... and now it appears for me, as if the Anschaulichkeit and the applications are the

6 As we have stated throughout, we intended for this article to present the theoretical foundation for and a description of the ÜberPro seminar that we implemented in February 2015. We have purposefully reserved the presentation of signature cases of student participation in the seminar for subsequent publications.

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 86 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

reasons; I mean the application in school, [is] the reason, that we now have problems while the transition process to university. And then I don’t understand, why this is in all the books of didactics nowadays, that using applications is very good and that instead, it is the problem for the transition to university. Another student observed in the essay assigned at the conclusion of the seminar that: All in all, the transition problem in mathematics is quite rightly an often- discussed topic, which seems is hard to solve. For many students the transition from school to university is [difficult] because of the following aspects: the changes in teaching and learning, the change in the character and beliefs on mathematics ((naive-) empirical to deductive- mathematical), the pressure to perform and the [subsequent] loss of motivation. That’s why they fall into a nearly never-ending ravine, from which they have to find a way out, for overcoming the transition successfully. If they fail at this, they break up their studies. For me the transition from school to university was and is also not very easy. Still another student shared in his/her essay response that: Everything we discussed in [the] seminar led me to believe that it is crucial to understand the transition problem with the help of mathematical history. I would have liked to have some more practical advances in how to use this situation later as a teacher. (I know that [was not] the aim of this class and the research is probably at the very beginning but at some times we could have spent the time in a better way.)

Thus, it was clear to us that there is much more work that we can do in responding to the seminar course students’ needs. Indeed, mathematical history can provide support in negotiating the second gap that university mathematics students encounter when they transition to teaching mathematics. One such support is to provide concrete ways in which mathematics teachers can draw upon particular moments in the historical development of a collection of related mathematical ideas (as in the case of geometry in the ÜberPro seminar). However, another support includes the way in which history of mathematics contributes to a teacher’s mathematical knowledge for teaching, particularly contributions to horizon content knowledge (Clark 2012).

6.1 Implications for next steps For school purposes – from a well-informed mathematics educator’s point of view – nothing speaks against doing mathematics in an empirical way. Indeed, history has shown that empirical mathematics was a decent way to develop mathematical knowledge and the experimental natural sciences generate knowledge comparably. Yet approaches to bring formalistic mathematics into school classrooms have failed miserably (cf. the New Math Initiative, Why

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 87 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

Johnny Can’t Add (Kline 1974)). Moreover, we cannot step away from teaching mathematics in a theoretical way at universities. In contrast, the intensive seminar course that we implemented sought to make tangible, understandable, and explicit to first-year university students that the transition from school mathematics to university mathematics is an epistemological obstacle. Hefendehl-Hebeker (2013: 80) found quite comparably: […] a principle difference between school and university is that at university with the axiomatic method a new level of theory formation has to be reached, and thus it follows that the discontinuity cannot be avoided. So if the discontinuity cannot be avoided, what can teachers and students at university gain from a seminar course like the one described here? We found that significant potential lies in the following areas: 1. The historical excursions do not only focus on the beliefs aspect but also demonstrate and involve critical mathematical activities, especially regarding deductive reasoning within the frameworks of consistent mathematical theories. 2. Teachers and students should become aware of the extent of the transition problem, and that the problem’s solution is not as easy as repeating particular secondary school mathematics, as many approaches (and deficit models) seem to suggest. Instead, a revolutionary act of conceptual change is required and this work does not occur overnight and needs guidance. The historical questions that led to the modern understanding of mathematics are too sophisticated and waiting for students to develop these for themselves is a particular burden on top of all the other factors of beginning mathematical study at university. The approach of initiating these questions explicitly within the framework we described here may support a more adequate and prompt change of belief system, which in turn holds promise for addressed both forms of “abstraction shock” experienced by secondary mathematics teachers. 3. The seminar course has the power to sensitize for critical communication problems. Teachers and students should acknowledge that when talking about mathematics, using the same terms might not imply talking about the same things. For example, students may come to university from school having learned calculus in an empirical context such that functions might be equivalent to curves. This might imply that properties like continuity or differentiability are empirical and can be read from the sketched graph of the function (comparable to 17th century mathematicians). The university lecturer, on the other hand, probably has a general abstract notion of function implying a completely different notion of mathematical reasoning and truth. In particular, lecturers should repeatedly check if the knowledge of their students is still bound to (single) objects of reference. The same holds for the students eventually leaving university and starting as secondary school

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mathematics teachers: they should be aware that what they consider from an abstract point of view their students may instead possess visualizations of abstract notions as the reference objects.

NOTES 1. Anschauung: The meaning of the prominent German term Anschauung has two different connotations. It can mean something close to ‘empirical perception’ or something like an ‘inner mental image’ (according to Immanuel Kant). Heuser (2009) referred to the aspect of empirical perception. 2. Empirical: The authors use the word empirical in the sense as it is used in the concept of “empirical theories” in philosophy of science, which is close to natural scientific theories. That means that some concepts of a theory have real/physical/empirical reference objects and the propositions of the theory can be checked by experiments in reality (Hempel 1945, Stegmüller 1987). 3. Formalistic: The authors use the word formalistic in the Hilbertian sense. That means a (mathematical) theory is formalistic if all primitive concepts of the theory are (logical) variables and the axioms of the theory are not sentences but sentential functions with the primitive concepts as variables arguments (cp. C. G. Hempel 1945). By virtue of a physical interpretation of the originally uninterpreted primitives empirical models of the formalistic theory are defined. This is the relation between a formalistic mathematics and empirical science.

Figure Acknowledgements: Fig. 1. Data from a survey made by Ingo Witzke in 2013. Fig. 2. Graphical derivative. Graphic from Griesel, H. et al. (Hrsg.): Elemente der Mathematik (EDM), Einführungsphase – Braunschweig: Schroedel 2010: 203. Fig. 3. Three excerpts of different textbooks for comparison. University course textbook “Königsberger 2001: 34” (top left), school textbook “Lambacher Schweizer 2009: 55” (bottom left), historical text “Leibniz, Acta Eruditorum”, 1693 (right). Fig. 4. Historical development as one basis of the seminar. Created by Ingo Witzke and Gero Stoffels. Fig. 5. The Architecture of Pythagoras theorem. Graphic by S. Schlicht (University of Cologne) 2014. Fig. 6. Photo of “Autobahn” taken by Ingo Witzke (top); Albrecht Dürer: “Man drawing a lute” (1525), (bottom left); photo of Albrecht Dürer Activity during seminar, taken by Kathleen Clark (bottom right). Fig. 7. Example for the principle of duality: Theorem of Pappus-Pascal: Six points (red) incident with two lines (blue) – the points (green) which are incident with opposite lines of the hexahedron are collinear (green line).

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Ingo Witzke, Horst Struve, Kathleen Clark, Gero Stoffels ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT THE TRANSITION PROBLEM FROM SCHOOL TO UNIVERSITY MATHEMATICS, BASED ON 89 EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS

Theorem of Brianchon: Six lines (red) incident with two points (blue) – the lines (green) which are incident with opposite points of the hexahedron are copunctal (green point). Graphic created by Horst Struve and Ingo Witzke. Fig. 8. Angle sums in different geometries: Internet source retrieved November 1, 2015 from the World Wide Web: https://naiadseye.files.wordpress.com/2014/10/euclidean-udnerstand- e1414490051530.png?w=470&h=289 (Stand, 2015). Fig 9. First paragraph of Hilbert’s Foundations of Geometry (Hilbert 1902).

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BRIEF BIOGRAPHIES Ingo Witzke is full professor at the University of Siegen. He is responsible for undergraduate and graduate mathematics education courses for pre-service teacher students in the mathematics department. He majored in mathematics and history for secondary level teaching and earned a doctorate in science education from the University of Cologne in 2009. His interest and area of publication is benefit-oriented fundamental research in the field of mathematics education. He focuses on beliefs and nature(s) of mathematics, including epistemological, historical and cognitive aspects.

Horst Struve is a full professor at the University of Cologne. He completed his PhD in 1978 at the University of Kiel in the foundations of geometry, and his habilitation in geometry education 1989 at the University of Cologne. His main research interests are the reconstruction of the development of mathematical theories in both the history of mathematics and in the classroom. Since there are important similarities between pupils’ conception of mathematics in school and of mathematicians in history, mathematics education can learn much from history.

Kathleen Clark is an associate professor at Florida State University. She earned her doctorate in Curriculum and Instruction (University of Maryland – College Park) in 2006. Kathleen Clark’s research interests are centered on investigating the role of history of mathematics in teaching and learning. She has published numerous journal articles, proceedings papers, and book chapters.

Gero Stoffels is doctoral student at the University of Siegen and is supervised by Prof. Dr. Ingo Witzke. He majored in mathematics and physics for teaching at the University of Cologne. His doctoral dissertation project deals with the transition from school to university mathematics with a special focus on probability theory. He is also interested in comparing individual and historical development processes on an epistemological level.

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UNIVERSITY OF WESTERN MACEDONIA FACULTY OF EDUCATION

MENON 94 ©online Journal Of Educational Research

QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE LECTURES DE TEXTES HISTORIQUES

David Guillemette Faculté d’éducation, Université d’Ottawa [email protected]

ABSTRACT This paper tries to highlight some difficulties that have been encountered during the implementation of reading activities of historical texts in the preservice teachers training context. During a history of mathematics course offert at the Université du Québec à Montréal, seven reading activities have been constructed and implemented in class. Looking to articulate both synchronic and diachronic reading, numerous efforts have been deployed in order to do not uproot the text and his author from their socio-historical and mathematical context. We try here to describe the teaching difficulties that we have encountered in this context and to identify the possible sources and solutions to these problems. Furthermore, we question these concepts of synchronic and diachronic reading in this context. Examples of interactions between students, as well as the trainer, engaged in the reading of historical texts are provided and presented by the mean of sketches Keywords: History of Mathematics, Reading of Historical Texts, Diachronic and Synchronic Reading, Mathematics Preservice Teachers Training, Empirical Research

1. L’HISTOIRE, LA PETITE HISTOIRE… L’histoire des mathématiques dans l’enseignement-apprentissage des mathématiques est un sujet qui a fait l’objet d’abondantes études, et ce, depuis de nombreuses années. C’est à partir des années soixante-dix que ce champ d’intérêt a connu une hausse importante de popularité. Un nombre important d’articles, publications, livres, recueils, conférences et groupes de recherche touchant plus ou moins directement l’histoire et l’enseignement des mathématiques sont issus de cette période effervescente. Jusqu’à récemment, il semblait que tous, enseignants et chercheurs, s’entendaient pour dire que l’histoire est bénéfique et se veut d’emblée un outil motivationnel et cognitif efficace dans l’apprentissage des mathématiques (Charbonneau 2006). En effet, un mouvement d’enthousiasme mêlé d’une grande inventivité anime le milieu depuis la création de ces entités de recherche. Cependant, depuis maintenant une dizaine d’années, la recherche autour de l’utilisation de l’histoire des mathématiques se restructure. De nouveaux questionnements font suite à la parution de l’étude ICMI sur le sujet (Fauvel & van Maanen 2000). Véritable bilan de santé du domaine de

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 95 LECTURES DE TEXTES HISTORIQUES recherche, le livre rassemble les réflexions, interrogations et inquiétudes des chercheurs du moment. On peut retenir très globalement que ces derniers prennent aujourd’hui du recul face à leurs travaux et tentent de construire de nouveaux outils d’investigations plus raffinés pouvant à la fois alimenter la production d’outils pratiques pour le terrain, de mieux en comprendre leurs utilisations par les acteurs des milieux éducatifs et, surtout, de raffiner les discours sur les enjeux didactiques et pédagogiques de l’utilisation de l’histoire en classe de mathématiques. Aujourd’hui, le champ de recherche semble essoufflé quant à la production et au design d’activités d’apprentissage, de situations problèmes ou de séquences d’enseignement. Les activités des chercheurs se déplacent vers la recherche en termes de fondements didactiques et pédagogiques à partir desquels il serait possible de mieux penser le rôle de l’histoire (Guillemette 2011). Le développement de cadres théoriques et conceptuels permettant de fournir les appareillages nécessaires à la production d’investigations plus fines est encore maintenant attendu fermement (Kjeldsen 2012).

2. SUR LE RÔLE ET LES MODALITÉS D’ÉTUDE DE L’HISTOIRE EN CLASSE DE MATHÉMATIQUES : LE POINT DE VUE DE FRIED Ainsi, un besoin important se fait sentir dans la communauté afin de construire des outils critiques permettant de porter un regard aiguisé sur la recherche et les pratiques actuelles. En particulier, on cherche à classer, à catégoriser et à évaluer les études du domaine. On tente d’éclaircir les discours en répertoriant les objectifs poursuivis par les chercheurs, les moyens employés et les concepts utilisés. Aussi, plusieurs tentatives de catégorisation concernant le ‘comment’ et le ‘pourquoi’ de l’utilisation de l’histoire sont parues suite à l’étude ICMI (p. ex. Fried 2001, 2007, 2008, Furinghetti 2004, Gulikers & Blom 2001, Jankvist 2009, Tang 2007, Tzanakis & Thomaidis 2007). La discussion sur le rôle et les modalités d’étude de l’histoire en classe de mathématiques est encore très vive et les questions les plus larges restent, et ce pour le mieux, encore ouvertes. Un important point de vue est celui de Fried (2001, 2007, 2008) qui réaffirme à sa manière les vertus ‘humanisantes’ de l’histoire des mathématiques. Dans la mouvance actuelle, il prône fermement pour une perspective historique dans l’enseignement des mathématiques une perspective dirigée vers le développement global de l’individu, prétextant qu’une visée pragmatique, utilitaire et ponctuelle mène inexorablement à une histoire des mathématiques mutilée et réifiée, ainsi qu’à une démarche éducative stérile et séparée de fondements pédagogiques profonds (cf. Whig history, Fried 2007). Son discours recèle une dimension particulière, celle de la connaissance de soi (self-knowledge). Il souligne que le mouvement de va-et-vient entre la compréhension actuelle des objets mathématiques et les formes de compréhensions provenant d’autres époques amène l’apprenant à une

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 96 LECTURES DE TEXTES HISTORIQUES connaissance plus approfondie de lui-même: “a movement towards self-knowledge, a knowledge of ourselves as a kind of creature who does mathematics, a kind of mathematical being” (Fried 2007 : p. 218). Fried propose que cette connaissance de soi, c’est-à-dire de son ‘être mathématique’, soit l’objectif premier que doivent se donner les enseignants de mathématiques. C’est un contact important avec l’histoire qui doit faire émerger en l’apprenant une certaine conscience de ses propres conceptions, de ses manières de faire et de son individualité en mathématiques. Ainsi seulement, il aura la possibilité de faire grandir cette individualité par la confrontation constructive avec celles des autres. Fried n’hésite pas à souligner l’arrière-plan de sa pensée autour de ces considérations en mentionnant que: “education, in general, is directed towards the whole human being, and, accordingly, mathematics education, as opposed to, say, professional mathematical training, ought to contribute to students’ growing into a whole human beings” (Fried 2007: p. 219). Dans plusieurs études théoriques importantes, Fried (2001, 2007, 2008) discute en profondeur de ces éléments. D’abord, il met en relief la difficulté de traiter convenablement de l’histoire en classe de mathématiques. Très souvent l’histoire prend la forme d’anecdotes et de capsules historiques qu’il voit d’un très mauvais œil. Il souligne le risque évident d’une dénaturation de l’histoire, particulièrement d’une histoire contaminée par une vision moderne des mathématiques qui écrase l’historicité des concepts et aseptise la lecture historique. Comme les risques d’anachronisme et de lectures faussement progressives de l’histoire sont élevés, il souhaite que celle-ci soit prise au sérieux et que son étude soit prudente et attentive. Dans cette perspective, Fried propose les approches d’ ‘accommodation radical’ (radical accommodation) et de ‘séparation radical’ (radical separation) (Fried 2001: 405). Il postule que l’étude des mathématiques d’une époque donnée doit se faire en symbiose avec le contenu visé du cours ou se séparer carrément du contenu mathématique moderne enseigné. Bref, pour Fried, il ne doit pas y avoir de demi-mesure. Qu’en est-il alors de la pertinence de l’histoire? L’histoire des mathématiques devrait-elle rester à sa place et ne pas interférer dans le cours de mathématiques? Avec de telles ‘accommodations’ et une telle ‘prise au sérieux de l’histoire’, est-il maintenant illusoire de penser introduire l’histoire avec un temps de classe limité? Surtout, quelles sont les lignes directrices que les enseignants des divers contextes éducatifs devraient suivre afin d’atteindre ces objectifs ambitieux? Fried répond (2007, 2008) en concentrant sa réflexion sur le rôle de l’enseignant, de ses choix pédagogiques et de son attitude face à la discipline. Ainsi, il plaide pour l’utilisation de sources primaires, et notamment pour une rencontre directe avec les mathématiques de l’histoire par la lecture de textes historiques. À ce sujet, il souligne que la lecture d’un document historique nous permet d’aller directement à la rencontre avec l’histoire et avec les formes d’activités mathématiques qui y sont apparues. Or, une telle lecture de texte doit être faite avec beaucoup de vigilance, et

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 97 LECTURES DE TEXTES HISTORIQUES

Fried propose un certain cadre conceptuel afin de mieux penser ces activités d’enseignement-apprentissage très particulières, ainsi que leurs articulations avec les objectifs pédagogiques développés précédemment. Avant tout, il souligne que la lecture d’un tel texte est différente pour le mathématicien et pour l’historien. L’objectif de l’historien est de se plonger dans l’époque du mathématicien, de percevoir les idiosyncrasies de ce dernier et de situer l’ouvrage dans le continuum du développement des mathématiques. Quant au mathématicien, il tente de son côté de décoder les symboles désuets, de les restituer au langage moderne et de saisir l’aspect essentiellement mathématique des propos de l’auteur. Il qualifie de diachronique la lecture de l’historien et de synchronique la lecture du mathématicien, termes qu’il emprunte à de Saussure. Fried (2008) affirme que connaitre véritablement un concept mathématique signifie le connaitre à la fois synchroniquement, c’est-à-dire en considérant sa situation à l’intérieur du système de concepts mathématiques actuel, et diachroniquement, c’est-à-dire en considérant son historicité, son évolution dans le temps et l’espace. Afin d’éclairer ces termes synchronique et diachronique, retournons rapidement à la théorie linguistique saussurienne. De Saussure mentionne à propos de la langue que: Si on prenait la langue dans le temps, sans la masse parlante […] on ne constaterait peut-être aucune altération; le temps n’agirait pas sur elle. Inversement, si on considérait la masse parlante sans le temps, on ne verrait pas l’effet des forces sociales agissant sur la langue (De Saussure 1967/2005: 113). On distingue ici deux perspectives: une perspective anhistorique où l’on observe comme une photo les signes (couple signifiant/signifié) effectifs dans la masse parlante et une perspective historique où les signes sont en perpétuel changement. En d’autres termes, quand l’histoire entre dans le tableau, le tableau change complètement. Dès lors, il apparait que “le fleuve de la langue coule sans interruption” (De Saussure: 193). C’est ici que de Saussure introduit les termes synchronique et diachronique pour distinguer respectivement ces perspectives anhistoriques et historiques. Pour Fried, la lecture synchronique des objets mathématiques est trop souvent renforcée par les enseignants et les mathématiciens dans le contexte de l’utilisation de l’histoire dans l’enseignement-apprentissage des mathématiques. Au contraire, le rôle de l’enseignant devrait être précisément de faire constamment basculer l’apprenant entre ces deux visions. C’est ce travail de va-et-vient continuel qui doit faire émerger chez l’apprenant une certaine conscience de ses propres conceptions des mathématiques et de son individualité face à la discipline. Se centrant sur les possibilités d’émancipation pour l’apprenant lors de ces expériences de lectures fondatrices, Fried insiste sur la prise de conscience et le mouvement de croissance de l’individu plutôt que sur la réflexion

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 98 LECTURES DE TEXTES HISTORIQUES

épistémologique de ce dernier. Ce n’est qu’en chemin que la réflexion épistémologique de l’apprenant et l’émancipation recherchée apparaissent consubstantielles. Ici, l’histoire des mathématiques est pensée comme une source possible d’importantes expériences personnelles impliquant un certain rapport de soi à soi par l’intermédiaire de l’histoire et des artefacts historico- culturels qu’on y trouve, expérience fondamentale qui supporterait le mouvement de croitre qui est celui de l’apprenant.

3. UN EXEMPLE D’OPÉRATIONNALISATION DANS LA RECHERCHE DE TERRAIN Malgré la richesse et la profondeur de ces considérations théoriques sur l’apport de l’histoire dans l’enseignement-apprentissage des mathématiques, celles-ci n’ont que très rarement été confrontées à la recherche de terrains (Guillemette 2011). Notre objectif sera d’interroger les considérations théoriques de Fried notamment sur les modalités de lectures synchroniques et diachroniques de textes historiques en classe de mathématiques à partir de données issues d’une précédente étude empirique (Guillemette 2015). Sommairement, l’étude en question avait pour objectif de décrire le dépaysement épistémologique (Barbin 1997, 2006, Jahnke et al. 2000) des étudiants en formation à l’enseignement des mathématiques au secondaire. Barbin explique qu’introduire l’histoire des mathématiques bouscule notre perspective coutumière des mathématiques et souligne que “l’histoire des mathématiques, et c’est peut-être son principal attrait, a la vertu de nous permettre de nous étonner de ce qui va de soi” (1997 : 21). Dans cette perspective, le dépaysement épistémologique serait un choc culturel aux dimensions affectives et cognitives qui mènerait à des compréhensions différentes des mathématiques et de ses objets de savoir. Afin d’obtenir une description de ce phénomène, sept activités de lectures de textes historiques ont été construites et mises en œuvres dans une classe d’étudiants inscrits à une formation à l’enseignement des mathématiques au secondaire: - A’hmosè: Papyrus de Rhind, problème 24 - Euclide: Les Éléments proposition 14, Livre 2 - Archimède: La quadrature de la parabole - Al-Khwarizmi: Abrégé du calcul par la restauration et la comparaison, types 4 et 5 - Nicolas Chuquet: Tripartys en sciences des nombres, problème 166 - Gilles Personne de Roberval: Observations sur la composition des mouvements et sur le moyen de trouver les touchantes des lignes courbes, Problème 1 - Pierre de Fermat: Méthode pour la recherche du minimum et du maximum, problème 1 à 5. Lors de ces activités de lectures, nous nous efforcions, à titre de formateur/chercheur, de faire continuellement basculer les étudiants entre une lecture synchronique et une lecture diachronique des propos de l’auteur. La sélection des participants de l’étude a été faite parmi les futurs enseignants du secondaire inscrits au cours MAT6221 Histoire des mathématiques

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 99 LECTURES DE TEXTES HISTORIQUES

à l’Université du Québec à Montréal. Six étudiants ont été recrutés sur une base volontaire. Cherchant à décrire l’expérience vécue des participants afin d’interroger et de mieux penser le concept de dépaysement épistémologique dans un tel cadre, une approche phénoménologique a alors été déployée. Des captations vidéo des activités de classe, des entretiens individuels et un entretien de groupe ont été réalisés et ont fourni les données de l’étude. Nous ne nous attarderons ici que sur les données issues des captations vidéo des activités de classe. Lors de ces dernières, deux caméras ont été installées à des points raisonnablement éloignés dans la classe et pointaient sur des ilots constitués de deux ou trois pupitres. Les six participants ont été invités à chaque séance à se séparer en deux équipes, chacune prenant place sur un ilot. Pour l’analyse de ces captations vidéo, un visionnement attentif séance par séance et équipe par équipe a été fait. Le but était alors de capter les moments d’objectivation (Radford 2011, 2013) nous apparaissant lors des activités de lecture, c’est-à-dire les moments de rencontre avec quelque chose qui s’(ob)jecte, qui se donne à voir à travers les activités de lecture. Quelque chose qui s’affirme en tant qu’altérité et qui se présente aux apprenants petit à petit. Une attention particulière a donc été donnée aux gestes, postures, attitudes et réactions diverses des participants, ainsi qu’aux échanges et réflexions émergentes en relation aux textes. Ce concept d’objectivation est issu de la théorie de l’objectivation. D’inspiration Vygotskienne, cette théorie socioculturelle contemporaine de l’enseignement-apprentissage plaide pour une conception non mentaliste de la pensée. S’opposant au courant rationaliste et idéaliste, elle propose la conception d’une pensée à la fois sensible et historique. D’une part, elle est sensible, car elle s’enracine dans le corps, les sens et l’affectivité, lesquels sont invoqués dans la saisie des objets de la réalité. D’autre part, elle est historique puisqu’elle se trouve, de manière inhérente, jetée dans une réalité sociohistorique. C’est pourquoi elle est attentive à l’influence des artefacts chez l’être humain et à l’interaction sociale. En parallèle à ce visionnement, un texte descriptif a été produit pour chacune des équipes de chaque activité de lecture. Des captures d’écran y ont été incluses, elles mettent en évidence, sous forme de saynètes, ces moments de rencontre. Il est à noter que ces captures d’écrans ont été modifiées à l’aide du logiciel SketchPen afin de donner un aspect ‘roquis de crayons’ aux images retenues. Ces modifications permettaient de garder l’anonymat des participants tout en laissant visibles leurs postures, gestes et réactions.

4. PROPOSITION D’UNE ACTIVITÉ DE LECTURE Nous nous contenterons ici de revenir sur une seule de ces activités de lecture en nous concentrant sur les interactions entre les membres d’une équipe en particulier. Lors de cette activité, les trois participants; Aliocha, Martha et Ninotchka s’adonnent à la lecture de l’Abrégé du calcul par la restauration et la comparaison d’al-Khwarizmi, une traduction d’Ahmed Djebbar (2005).

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 100 LECTURES DE TEXTES HISTORIQUES

Un seul extrait a été traité par les participants. Dans ce dernier, al- Khwārizmī propose un procédé pour la résolution du 4e modèle d’équation quadratique: ax2 + bx = c. Il donne en exemple générique l’équation à résoudre x2 + 10x = 39. Voici son explication: Quant à la justification de “un bien et dix racines égalent trente-neuf dirhams”, sa figure est une surface carrée de côtés inconnus, et c’est le bien que tu veux connaitre et dont tu veux connaitre la racine. C’est la surface (AB), et chacun de ses côtés est la racine. Chacun de ses côtés, si tu le multiplies par un nombre parmi les nombres, quels que soient les nombres, sera des nombres de racines, chaque racine étant comme la racine de cette surface. Comme on a dit qu’avec le bien il y a dix de ses racines, nous prenons le quart de dix – et c’est deux et un demi – et nous transformons chacun de ses quarts [en segment] avec l’un des côtés de la surface. Il y aura ainsi, avec la première surface, qui est la surface (AB), quatre surfaces égales, la longueur de chacune d’elles étant comme la racine de la surface (AB) et sa largeur deux et un demi, et ce sont les surfaces (H), (T), (K), (J). Il [en] résulte une surface à côtés égaux, inconnue aussi, et déficiente dans ses quatre coins, chaque coin étant déficient de deux et demi par deux et demi. Alors, ce dont on a besoin comme ajout afin que la surface soit carrée, sera deux et demi par lui-même, quatre fois; et la valeur de tout cela est vingt-cinq. Or, nous avons appris que la première surface, qui est la surface du bien, et les quatre surfaces qui sont autour de lui et qui sont dix racines, sont [égales à] trente-neuf en nombre. Si on leur ajoute les vingt-cinq qui sont les quatre carrés qui sont dans les coins de la surface (AB), la quadrature de la surface la plus grande, et qui est (DE), sera alors achevée. Or nous savons que tout cela est soixante-quatre, et que l’un de ses côtés est sa racine, et c’est huit. Si on retranche de huit l’équivalent de deux fois le quart de dix – et c’est cinq –, aux extrémités du côté de la surface la plus grande qui est la surface (DE), il reste son côté trois, et c’est la racine de ce bien.

5. DESCRIPTION ET ANALYSE Après une étude sommaire du contexte historique et mathématique du texte, ainsi qu’une présentation de l’auteur au début de l’activité, les

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 101 LECTURES DE TEXTES HISTORIQUES participants ont été invités à débuter la lecture. Voici la description commentée de cette séance de lecture: Martha a retourné son pupitre pour faire face à Aliocha et Ninotchka. Ils débutent individuellement et en silence la lecture pendant près de cinq minutes. Martha surligne quelques passages du premier extrait.

Figure 1: Lecture d’al-Khwārizmī: équipe Martha - Aliocha - Ninotchka.

Martha demande au formateur si la ‘comparaison’ invoquée dans le texte correspond à la comparaison telle qu’elle est comprise aujourd’hui. Le formateur explique que le mot ‘comparaison’ est issu de la traduction du texte original d’al-Khwārizmī et qu’il ne s’agit pas de la même chose. Ils reprennent ensuite tous leur lecture.

Nous remarquons ici un premier appel au formateur à propos du vocabulaire. Martha se demande si le mot comparaison renvoie au sens tel que nous l’entendons aujourd’hui. Elle en doute et le formateur lui rappelle qu’elle lit une traduction du texte original, et que, par conséquent, le sens doit fort probablement être différent. Ninotchka et Martha organisent leur espace de travail, détachent les feuilles du document, tandis qu’Aliocha est plongé dans sa lecture. Le travail se poursuit individuellement pour encore plusieurs minutes. Martha questionne Ninotchka sur ce que représente la figure dessinée par l’auteur. Elles reprennent ensemble la signification des différents éléments de la figure et retournent rapidement à leur lecture. Chacun se concentre sur le premier extrait. Martha demande à ses coéquipiers pourquoi al-Khwārizmī prend le quart de la valeur du terme en x. Elle souligne que cette question a aussi été soulevée par l’autre équipe.

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 102 LECTURES DE TEXTES HISTORIQUES

Figure 2: Lecture d’al-Khwārizmī: équipe Martha - Aliocha - Ninotchka.

Aliocha lui explique que, puisque la figure initiale de l’auteur est un carré, il lui faut ensuite ajouter quatre rectangles autour, lesquels sont associés au terme en x. Ce dernier doit donc être divisé en quatre. Aliocha pointe les quatre rectangles sur la figure dessinée par Martha.

Nous pouvons remarquer déjà après quelques minutes une traduction de l’auteur en langage moderne. Les participants discutent d’un “terme en x” sans avoir à se justifier ni à s’expliquer. Or, l’usage de lettres dans l’énonciation d’un raisonnement algébrique ne se trouve nulle part chez al-Khwārizmī.

Figure 3: Lecture d’al-Khwārizmī: équipe Martha - Aliocha - Ninotchka.

Martha souligne la perspicacité d’Aliocha. Elle écrit ce raisonnement sur sa feuille de travail. Martha demande ensuite: “La longueur 10, comment on la connait?”. Aliocha exprime son incompréhension. Elle ajoute: “Dans mon schéma, comment je sais combien ça mesure 10/4, je me donne une unité de référence?”. Les deux autres acquiescent. Aliocha souligne que l’auteur parle du ‘dirham’, une monnaie qui peut ici être considérée comme l’unité.

À nouveau, les participants tentent de traduire dans leurs mots le

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 103 LECTURES DE TEXTES HISTORIQUES vocabulaire de l’auteur. Une association est effectuée entre le ‘dirham’ et l’ ‘unite’. Martha souligne que la démarche de l’auteur ressemble à la méthode algébrique de la complétion de carré. Ils reprennent individuellement leur lecture. Aliocha résonne dorénavant sur la figure, tandis que Ninotchka et Martha relisent l’extrait mot à mot tout en augmentant leur dessin de nouveaux éléments. Martha surligne à nouveau des passages de l’extrait, tandis qu’Aliocha tente de résoudre algébriquement le problème.

Nous pouvons observer les participants se référer à une stratégie de résolution algébrique à l’aide de représentations géométriques (tuiles algébriques). Il s’agit d’un outil didactique acquis au cours de leur formation à l’enseignement des mathématiques. Aliocha débute quant à lui une démarche algébrique moderne à partir de l’énoncé du problème. Aliocha se lève et demande de l’aide au formateur. Ce dernier propose à Aliocha de réfléchir à une solution qui serait normalement proposée aujourd’hui. Aliocha rétorque qu’il faudrait appliquer la formule quadratique. Ninotchka propose la complétion de carré comme stratégie de résolution et montre ses démarches à Aliocha.

Figure 4: Lecture d’al-Khwārizmī: équipe Martha - Aliocha - Ninotchka.

Le formateur vient en aide à l’équipe en leur proposant d’élaborer d’abord une solution avec leurs outils actuels avant d’entreprendre l’interprétation du texte. L’objectif serait de possiblement faire le parallèle avec la démarche d’al- Khwārizmī, afin de mieux comprendre cette dernière. Martha se rapproche alors et le groupe tente ensuite de concilier la démarche de Ninotchka avec celle de l’auteur, il est question d’abord du traitement du terme en x.

Figure 5: Lecture d’al-Khwārizmī: équipe Martha - Aliocha - Ninotchka.

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 104 LECTURES DE TEXTES HISTORIQUES

Le groupe ne réussit pas à concilier les deux démarches. Le formateur quitte alors le groupe et laisse les participants à leurs réflexions. Ils reprennent chacun leur travail individuellement. Ninotchka se souvient alors qu’à la complétion de carré est associée habituellement une figure géométrique. Elle montre son dessin au groupe et demande l’avis d’Aliocha sur son approche.

Figure 6: Lecture d’al-Khwārizmī: équipe Martha - Aliocha - Ninotchka.

Elle souligne que dans ce cas-ci, il faut diviser par deux et non par quatre comme le fait al-Khwārizmī. Martha explique qu’al-Khwārizmī ajoute quatre rectangles autour de son carré plutôt que deux comme il est habituellement fait lors de la complétion de carré. Elle pointe alors chacun des côtés du carré.

Retournant à la méthode de la complétion de carré, qui apparait plus près de celle d’al-Khwārizmī, les participants tentent à nouveau de concilier leurs manières de faire avec celles de l’auteur. Un repère particulier est mis en évidence, celui de la représentation et du découpage en rectangle des quantités en question.

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 105 LECTURES DE TEXTES HISTORIQUES

Figure 7: Lecture d’al-Khwārizmī: équipe Martha - Aliocha - Ninotchka.

Aliocha explique donc que si l’on veut suivre la méthode de l’auteur, il faut alors diviser le terme en x par quatre. Martha poursuit en expliquant que les rectangles couvriront au total la même surface, mais seront simplement disposés différemment. Aliocha est d’accord et ajoute que la démarche algébrique associée sera alors différente. Ils se lancent à nouveau dans l’exploration algébrique de la démarche de l’auteur. Ninotchka partage ses résultats avec Aliocha, Martha avance seule.

Avançant dans la réconciliation entre la méthode de complétion de carré et celle d’al-Khwārizmī, les participants se proposent en parallèle de fournir une démarche algébrique moderne. Martha tente de généraliser le cas traité par l’auteur à l’aide d’une expression algébrique. Elle explique comment elle a obtenu son expression à Aliocha. Ce dernier généralise davantage l’expression de Martha et l’accompagne dans le peaufinement de sa démarche. Après quelques avancées, Aliocha conclut cependant que le travail de généralisation de Martha ne les avance pas dans la compréhension et la validation de la démarche de l’auteur.

Figure 8: Lecture d’al-Khwārizmī: équipe Martha - Aliocha - Ninotchka.

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 106 LECTURES DE TEXTES HISTORIQUES

Aliocha continue alors la démarche sur sa feuille et annonce qu’il croit s’approcher de la formule quadratique, formule qu’il appelle ‘grosse Bertha’.

Figure 9: Lecture d’al-Khwārizmī: équipe Martha - Aliocha - Ninotchka.

Le groupe accompagne alors Aliocha dans cette recherche. Aliocha conclut qu’il a réussi à concilier la démarche de l’auteur avec la formule quadratique, à l’exception du signe d’un des termes de son équation. Il se lève et demande alors de l’aide au formateur pour expliquer cette différence et compléter sa démarche. Avec le groupe, le formateur reprend alors plus en détail le raisonnement associé à l’application de la formule quadratique.

Figure 10: Lecture d’al-Khwārizmī: équipe Martha - Aliocha - Ninotchka.

C’est alors que Ninotchka comprend subitement que le signe est inversé dans l’application de la formule quadratique à partir du cas général, ce qui explique le problème soulevé précédemment par Aliocha. Le groupe se remet ensuite au travail individuellement. Aliocha explique alors que la manipulation des irrationnels éloigne le texte des

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David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 107 LECTURES DE TEXTES HISTORIQUES mathématiques de la Grèce antique. Avec Martha, Aliocha conclut que l’utilisation de la géométrie rapproche la démarche à celles des mathématiciens de la période hellénistique. Aliocha souligne aussi qu’al-Khwārizmī procède d’un exemple particulier pour discuter d’un résultat général. Au moment de conclure, Aliocha propose quelques réflexions à propos de la démarche de l’auteur et tente de faire des liens entre son propos et le contexte mathématique de l’époque. Notons que ces réflexions ne surviennent qu’à la toute fin de l’activité de lecture. Martha explique alors que, durant un stage en enseignement secondaire, elle devait enseigner les propriétés des logarithmes. Son superviseur lui avait suggéré de partir d’un cas particulier pour aboutir au cas général, alors qu’elle avait prévu l’inverse dans ses planifications. Aliocha souligne que, malgré les liens établis avec les mathématiques de la Grèce antique, les Grecs ne proposaient pas cette démarche pédagogique d’accompagnement du lecteur et que celle-ci est importante pour les élèves. La séance de lecture est suspendue par le formateur. Notons ici d’intéressantes réflexions sur l’histoire et l’enseignement- apprentissage des mathématiques en termes de pratiques enseignantes. Globalement, les autres activités de lecture se sont déroulées selon le même canevas. En s’inspirant des considérations théoriques et épistémologiques de Fried décrite plus haut, les activités de lecture de textes ont été menées en articulant constamment deux pôles : un pôle que l’on pourrait qualifier de ‘traductif’ qui visait essentiellement à extirper et à travailler les mathématiques que convoquaient les textes et un second plus ‘interpretative’ qui visait à mieux comprendre l’auteur en lui réservant un accueil qui ne le déracinait pas de son contexte sociohistorique et culturel. Les deux pôles concernant la lecture des textes ont été explicités avec les étudiants du groupe. Bien entendu, cet accueil nécessitait de la part de l’apprenant de nombreuses connaissances et une vision riche de l’époque dont était tiré le texte. D’ailleurs, cette nécessité d’un fort ancrage dans l’époque étudiée est affirmée couramment dans la littérature (Jankvist 2009). Dans le contexte de l’étude, cet ancrage a été assuré par la première partie du cours qui fournissait les repères historiques et culturels importants et tentait de fournir une certaine ‘saveur’ de l’époque en question. Le choix de présenter une description de cette activité parmi les sept autres qui ont été menées est motivé par le fait qu’elle représente bien et de manière globale, l’attitude des étudiants face aux textes, ainsi que nos difficultés, à titre de chercheur/formateur, à soutenir une lecture diachronique de la part des étudiants. Elle montre la manière dont les étudiants, malgré les consignes et les efforts du formateur, interrogent spontanément les textes et en initient l’exploration.

6. Quelques remarques sur la lecture synchronique et diachronique de textes historiques Comme mentionné précédemment, Fried souligne la tendance trop forte des

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 108 LECTURES DE TEXTES HISTORIQUES enseignants de mathématiques à livrer une lecture synchronique de la discipline. Est ainsi mis en avant, le point de vue des enseignants et formateurs, lesquels vacillent et basculent entre une perspective purement synchronique et une autre, plus fragile, précaire, et potentiellement dangereuse, qui serait diachronique. S’appuyant sur de Saussure qui affirmait que comprendre la langue est comprendre à la fois, et d’une seul tenant, son versant synchronique et diachronique, Fried mentionne que: [If] the application of Saussure’s ideas to the teaching of mathematics is truly valid, we must conclude that teaching ‘mathematics’ also demands presenting both its diachronic and synchronic aspects; far from having to choose between ‘mathematics’ and ‘history of mathematics’ the teacher must give attention to both (2008 : 195-196). Généralement, il suggère alors que les enseignants n’ont pas nécessairement à faire le choix entre un enseignement ‘classique’ des mathématiques et un enseignement ‘axé sur l’histoire’ qui risque de les éloigner de leurs objectifs curriculaires. De ses analyses saussuriennes, il conclut: We realize that history can play a part in the classroom without the material and focus of our mathematics teaching becoming radically altered. What is altered is a kind of background sense of the mathematical subjects we are teaching; the human origin of mathematical ideas, which the serious study of history brings out supremely […] Thus, a humanistic mathematics education will not deprive students of the knowledge of the ‘state of the art’ but will make them realize that the art is, indeed, in a certain, though not necessarily permanent, state (2008: 195-196). Il souhaite donc voir se déployer une perspective ‘humaniste’ des mathématiques par l’intermédiaire d’une histoire prise au sérieux et d’un rapport profondément historique à la discipline. C’est pourquoi les enseignants doivent éviter selon lui de donner une lecture synchronique des mathématiques qui serait une lecture appauvrie de la discipline et de ces objets d’étude. Or, nous souhaiterions ici avancer que les étudiants ont eux aussi une tendance forte à déployer une lecture synchronique. En effet, d’après nos expériences sommairement rapportées ici, ceux-ci semblent avoir naturellement propension à traduire et rapporter les propos de l’auteur en langage moderne. Il est aisé de reconnaitre que les participants ont une forte inclination à mettre eux-mêmes en avant une lecture synchronique de texte historique proposé, et ce, malgré les efforts du formateur. L’auteur est difficilement considéré dans son contexte. Le style ou les particularités de l’auteur sont difficilement remarqués et ne sont que très peu discutés lors des activités de lectures. Les auteurs se voient alors dépossédés de leur singularité, ils se trouvent très souvent traduits, résumés et réifiés. En sommes, nous remarquons une certaine ‘violence’ de la synchronisation envers l’auteur. Ainsi, la rencontre avec l’auteur perçu dans son contexte sociohistorique et mathématique ne se fait pas d’emblée, et ce, malgré les attentions et les efforts du formateur. En évitant toute généralisation et en nous rapportant à nos

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David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 109 LECTURES DE TEXTES HISTORIQUES expériences, nous souhaiterions humblement avancer qu’une lecture diachronique demande un effort considérable pour les étudiants. Ceux-ci nous apparaissent fortement marqués par la culture académique des mathématiques et, dans notre contexte de formation à l’enseignement, animés par un souci pragmatique de développement d’outils d’enseignement. Les difficultés ne nous semblent donc pas exclusivement provenir de l’enseignant ou du formateur qui orienterait les apprenants dans une démarche stérile de traduction. Bien entendu, celui-ci ne peut qu’accompagner les apprenants dans leur quête de sens, laquelle ne saurait se faire sans l’apport de leurs connaissances et expériences scolaires, académiques, mathématiques ou autres. Dans cette perspective, nous ressentons une résistance des étudiants à déployer une lecture diachronique. Résistances qui appellent à une enquête plus approfondie notamment en ce qui concerne les développements théoriques sur le sujet et les conceptualisations qui nous permettent de penser les pratiques dans ce contexte. À ce titre, nous voudrions voir s’ouvrir davantage à la dynamique de classe les conceptualisations théoriques associées à ces expériences d’enseignement- apprentissage. Pour mieux penser les difficultés liées à l’enseignement- apprentissage dans le contexte de l’utilisation de l’histoire, nous souhaiterions voir se développer des manières de faire, autant dans la recherche que dans les milieux de pratiques, orientées davantage vers l’ouverture à l’expérience que celle-ci peut possiblement renfermer. En ces termes, un développement conceptuel pourrait être envisagé, et ce, afin de penser cette expérience fondatrice, non pas en termes sociolinguistiques formels, mais en termes de relations sensibles et éthiques face à la diversité des formes que peut prendre l’activité mathématique. Le regard tourné vers la dimension expérientielle du phénomène, il serait alors possible de mieux comprendre le vécu des étudiants et la manière dont ce vécu prend sens à l’intérieur de l’enseignement-apprentissage des mathématiques ou encore de leur devenir enseignant, et, ainsi, ultimement, mieux accompagner les apprenants dans la (re)découverte de la discipline.

REFERENCES Barbin, E. (1997). Histoire et enseignement des mathématiques: Pourquoi? Comment? Bulletin de l’Association mathématique du Québec, 37(1), 20–25. Barbin, E. (2006). Apport de l’histoire des mathématiques et de l’histoire des sciences dans l'enseignement. Tréma, 26(1), 20–28. Charbonneau, L. (2006). Histoire des mathématiques et les nouveaux programmes au Québec : un défi de taille. In N. Bednarz & C. Mary (Eds.), Actes du colloque de l’Espace mathématique francophone 2006 (pp. 11–21). Sherbrooke: Éditions du CRP et Faculté d’éducation, Université de Sherbrooke. Djebbar, A. (2005). L’algèbre arabe: la genèse d’un art, Paris : Vuibert.

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David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 110 LECTURES DE TEXTES HISTORIQUES

Fauvel, J. & van Maanen, J. (Eds.). (2000). History in mathematics education: the ICMI study. Dordrecht: Kluwer Academic Publishers. Fried, M. N. (2001). Can mathematics education and history of mathematics coexist? Science & Education, 10(4), 391–408. Fried, M. N. (2007). Didactics and history of mathematics: Knowledge and Self- Knowledge. Educational Studies in Mathematics, 66(2), 203–223. Fried, M. N. (2008). History of mathematics in mathematics education: a saussurean perspective. The Montana Mathematics Enthusiast, 5(2), 185– 198. Furinghetti, F. (2004). History and mathematics education: a look around the world with particular reference to Italy. Mediterranean Journal for Research in Mathematics Education, 3(1-2), 125–146. Guillemette, D. (2011). L’histoire dans l’enseignement des mathématiques: sur la méthodologie de recherche. Petit x, 86(1), 5–26. Guillemette, D. (2015). L’histoire des mathématiques et la formation des enseignants du secondaire: sur l’expérience du dépaysement épistémologique des étudiants. Thèse de doctorat inédite, Université du Québec à Montréal, Montréal, Canada. [Disponible en ligne: http://www.archipel.uqam.ca/7164/1/D-2838.pdf]. Gulikers, I. & Blom, K. (2001). A historical angle: survey of recent literature on the use and value of history in geometrical education. Educational Studies in Mathematics, 47(2), 223–258. Kjeldsen, T. H. (2012). Uses of history for the learning of and about mathematics: towards a theoretical framework for integrating history of mathematics in mathematics education. In S. Choi & S. Wang (Eds.), Proceedings of HPM 2012 (pp. 1–21). Daejeon, Corée du Sud: Korean Society of Mathematical Education et Korean Society for History of Mathematics. Jahnke, H. N., Arcavi, A., Barbin, E., Bekken, O., Furinghetti, F., El Idrissi, A. & Weeks, C. (2000). The use of original sources in the mathematics classroom. In J. Fauvel & J. van Maanen (Eds.), History in mathematics education: the ICMI study (pp. 291–328). Dordrecht: Kluwer Academic Publishers. Jankvist, U. T. (2009). A categorization of the “whys” and “hows” of using history in mathematics education. Educational Studies in Mathematics, 71(3), 235–261. Radford, L. (2011). Vers une théorie socioculturelle de l’enseignement- apprentissage: la théorie de l'Objectivation. Éléments, 1, 1–27. Radford, L. (2013). Three key concepts of the theory of objectification: knowledge, knowing, and learning. Journal of Research in Mathematics Education, 2(1), 7–44. Saussure, F. de. (2005). Cours de linguistique générale. Paris: Payot & Rivages. (Œuvre originale publiée en 1967) Tang, K.-C. (2007). History of mathematics for the young educated minds: a Hong Kong reflection. In F. Furinghetti, S. Kaijser & C. Tzanakis (Eds.),

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David Guillemette QUELQUES DIFFICULTÉS RENCONTRÉES DANS LA FORMATION DES ENSEIGNANTS DE MATHÉMATIQUES DU SECONDAIRE À L’AIDE DE L’HISTOIRE DES MATHÉMATIQUES: UNE RÉFLEXION SUR LES MODALITÉS DE 111 LECTURES DE TEXTES HISTORIQUES

Proceedings of HPM 2004 & ESU 4 (revised edition) (pp. 630–638). Uppsala: Université d’Uppsala. Tzanakis, C. & Thomaidis, Y. (2007). The notion of historical “parallelism” revisited: historical evolution and students’ conception of the order relation on the number line. Educational Studies in Mathematics, 66(2), 165–183.

BRIEF BIOGRAPHY David Guillemette. Après avoir complété un doctorat en éducation à l’Université du Québec à Montréal, il a joint la Faculté d’éducation de l’Université d’Ottawa à titre de professeur adjoint. Ses recherches portent sur le potentiel de l’histoire des mathématiques dans l’éducation mathématique, notamment dans la formation initiale et continue des enseignants.

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

UNIVERSITY OF WESTERN MACEDONIA FACULTY OF EDUCATION

MENON 112 ©online Journal Of Educational Research

DISCUSSING MATHEMATICAL MODELING CONCERNING PASCAL'S WAGER

Michael Kourkoulos University of Crete, Department of Primary Education [email protected]

Constantinos Tzanakis University of Crete, Department of Primary Education [email protected]

ABSTRACT We present and analyze teaching work on Pascal's wager realized with Greek students, prospective elementary school teachers, in the context of a probability and statistics course. In this paper we focus on classroom discussion concerning mathematical modeling activities, connecting elements of probability theory and decision theory with elements of philosophical discussions. On the one hand, this link enriched students' scientific culture, and on the other hand, it allowed for deepening the classroom discussion on Pascal's wager. Keywords: Pascal's wager, prospective elementary school teachers, mathematical modeling, probability theory, decision theory

1. INTRODUCTION Discussions on philosophical and religious issues have deep and rich historical links with science; this is particularly true about probabilities and statistics (e.g. see Chandler & Harrison 2012, Hacking 1975, Hald 2003, Porter 1986). However, these rich links have been rarely explored in the conventional teaching of these disciplines, and even less (or not at all) at an introductory level. We argue that: (a) With adequate teaching design and implementation, it is possible to explore such links even with novice students in statistics and probability, (b) Exploring such links can be fruitful, both for the development of students' scientific culture and for the deepening of the discussion with them on the examined philosophical and/or religious issues (see also Kourkoulos & Tzanakis 2015). To support (a) and (b) above, we present an example of teaching work concerning Pascal's wager that was realized during an introductory seminar on probability and statistics with Greek students, prospective elementary school teachers1.

1 An initial version of this work was presented in the Science and Religion International Conference (see Kourkoulos & Tzanakis, in press).

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

Michael Kourkoulos, Constantinos Tzanakis

DISCUSSING MATHEMATICAL MODELING CONCERNING PASCAL'S WAGER 113

In the discussion on Pascal's wager, which has been active for more than three and a half centuries, important elements of scientific culture are involved such as elements of probability theory and decision theory (e.g. see Bras 2009, Hacking 1972, Hájek 2012, Jordan 1994, 2006). However, many of the arguments involved in the discussion on Pascal's wager, although fundamental, can be followed without the need of a sophisticate scientific background; this is related to the fact that the wager was established in the very first period of the historical development of probability theory, and to Pascal’s ingenious way to establish and present his argumentation. This makes these arguments adequate to be accessed by students like ours; however, because of their fundamental character, they have the potential to increase students' interest significantly.

2. BACKGROUND INFORMATION AND FOCUS Our teaching work was realized during an introductory seminar on probability and statistics (with classroom meetings for 3 hours per week) with 27 4th-year students (25 female and 2 male) in our Department of Education. Students had a high-school level background in probability and statistics, so the first four weeks were devoted to revising and completing this knowledge (see below). Next, the teacher gave a first presentation on Pascal's wager and asked students to express their thoughts and comments on this issue; the discussion that followed in this way, lasted for four weeks, and constitutes the first part of classroom discussion. For the second part, the teacher asked students to read an overview of literature on the discussion on Pascal's wager and other relevant reading sources, and to present elements of their personal study in the classroom. The elements presented by the students substantially enriched the classroom discussion; their discussion lasted for three weeks and constitutes the second part of the classroom discussion2. The focus of this paper is to present and analyze some main aspects of the classroom discussion on Pascal’s wager. In particular, the paper aims to present and analyze realized connections between mathematical modeling activities and elements of philosophical reasoning that fruitfully supported both the development of students’ concepts of probability theory and of decision theory, and the evolution of the discussion on Pascal’s wager.

3. TEACHING ON PROBABILITY AND STATISTICS As already mentioned, our students had a high-school level background in probability and statistics. During their tertiary studies they had not taken any course on probability and/or statistics; however, they had some exposure to readings of statistical results in the context of courses on Pedagogy and Psychology.

2 During these three weeks, four meetings of three hours were realized, instead of three.

MENON: Journal Of Educational Research [ISSN: 1792-8494] 2nd THEMATIC ISSUE http://www.edu.uowm.gr/site/menon 05/2016

Michael Kourkoulos, Constantinos Tzanakis

DISCUSSING MATHEMATICAL MODELING CONCERNING PASCAL'S WAGER 114

Students' knowledge in probability and (descriptive) statistics was revised and completed during the first four weeks. We talked about data organization and their (graphically and numerically tabulated) representation, measures of central tendency (mode, median, mean) and variation (range, interquartile range and standard deviation), the shape of a distribution and skewness. We also talked about the probability multiplication and addition laws, the binomial distribution and examples of its applications (e.g. chance games, wagering situations, simple insurance models) and the Low of Large Numbers and the normal distribution accompanied by adequate examples3. Moreover we discussed the concepts of expected value and expected utility, and their differences4. Using adequate examples the teacher explained that the criterion of maximum expected utility is more appropriate than the one of maximum expected value for making decisions in wagering situations5.

4. FIRST PART OF THE CLASSROOM DISCUSSION

4.1 Introduction and initial debate on Pascal’s wager During the 5th week, the teacher discussed with students on elements of Pascal’s life and work (e.g. see Adamson1995, Hacking 1975 ch7-9, Hald 2003 ch5, Mesnard 1951). Then he gave a first presentation of Pascal's wager6. In this context he also mentioned the so-called "many Gods objection" about Pascal's wager.

4.1.1 Many Gods objection Regarding the "many Gods objection", students agreed that the wager may be meaningless for a person who doubts God's existence but considers that, if He exists, conflicting hypotheses about Him are probable (e.g. he considers that God may be the Holy Trinity, or the 12 Olympian Gods, or Goddess Kali). Students commented that in this case it may be impossible for the person to find a coherent behavior that satisfies all Gods that he considers as probably existing. However, students considered that if a person (a) doubts God's existence, but (b) still considers that, if He exists, He is an omnipotent, omniscient and

3 In this context Pascal's triangle was also discussed; additionally the teacher mentioned the pioneering role of Pascal in the formation of probability theory (e.g. see Edwards 2002, Hald 2003 ch5). Furthermore, the teacher discussed with students the historical distinction of classical, subjective and frequentist probability (e.g. see Hacking 1975, Hald 2003, Stigler 1986). 4 Usually the concept of expected utility and its differences from the concept of expected value are not discussed in introductory level probability courses. However having planned to discuss Pascal's wager with students, it was a substantial element of preparation to discuss this subject with students. 5 In this context the teacher also discussed with students at an initial level the Saint Petersburg paradox. (The Saint Petersburg paradox was initially established and treated, in the first half of the 18th century, by Nicolas and Daniel Bernoulli and Gabriel Cramer; e.g. Bernoulli 1954, Dutka 1988, Martin 2014.) 6 During this presentation the teacher also presented the text of Pascal Wager (in the English translation by W. F. Trotter, in Pascal 1910, 83-87); moreover he mentioned Pascal's Pensées and the history of its edition (e.g. see Brunschvicg 1909; Descotes and Proust 2011; Lafuma 1954).

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Michael Kourkoulos, Constantinos Tzanakis

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omnibenevolent God, then such a person may consider the wager meaningful. During the discussion some students remarked that persons believing (a) and (b) above are more likely to be found in societies with a strong religious tradition, like the Greek society, because in such a society, alternative hypotheses about existing Gods are not supported by the tradition.

4.1.2 God cannot be fooled A second objection expressed by four students was the following: If someone bets his way of living on the hypothesis of God's existence, as Pascal proposes, and lives a virtuous life but still has doubts about God's existence, then God, as omniscient, will know that he is not a genuine believer and thus this person's efforts will be in vain. The teacher explained that Pascal doesn't propose the wager to fool God. Pascal believed, he said, that man's heart has the natural tendency to believe in God and the natural ability to perceive that He exists, however because of passions and sins man's heart is blinded and this leaves room for doubts about God's existence. If one accepts the wager and lives a virtuous life, his heart will be purified from passions and sins and thus his heart will perceive God's existence and his doubts will vanish. Three students commented that if God exists, then the wagering person is not alone in the wager; God is also there and, by appreciating this person’s efforts, He may help him by providing whatever feelings or evidence are necessary for that person to genuinely believe in His existence. Four students argued that if God wanted to help in this way for believing in His existence, it would be easy for Him to provide all people with the necessary evidence, and thus atheists or doubting persons would not exist, but this is not the case. One of the previous students answered that God helps to believe in Him those who want to believe, because He respects men's will; a person who wagers his way of living as proposed by Pascal, clearly makes a very strong effort to dissipate his doubts in the direction of believing in God's existence, and thus it is highly likely that he will attract God's help. Five other students as well made comments that endorsed this remark7.

4.1.3 Loving and caring unbelievers A third objection expressed by eight students concerned the idea that unbelievers will lose eternal salvation. Students said that an unbeliever who is a loving and caring person and dedicates his life to help his fellow humans, will not lose eternal salvation, in their opinion, because God been loving and just will not ignore the goodness of his heart and his efforts. Three other students remarked that the church teaches that being a good person is not enough for

7 Moreover, three of them commented that this remark also implies that the wager may be less demanding than what the argument of pure heart implies. They thought that perhaps because of God's generosity, He will help the wagering person to believe once He will consider that he makes a strong effort to live a virtuous life and not wait until his heart is fully purified.

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eternal salvation; a correct faith is also necessary. However, the first ones persisted in their opinion. Moreover four of them argued that the idea that unbelievers will lose eternal salvation regardless of their goodness is an idea unfair to God, because it presents Him as harsh and intolerant.

4.1.4 Selfish motivation A fourth objection expressed by six students was that if a person that doubts God's existence accepts Pascal's wager only on the basis of Pascal's argument, namely because he doesn't want to lose eternal salvation, then he accepts the wager only because of a self-interested motivation, and it is doubtful that God will reward efforts because of such motivation. A student remarked that in the New Testament eternal hell and eternal salvation are often mentioned as a motive for people to try to be right and avoid sinning; so church does not reject such a motive as a starting motivation for a person to try to ameliorate himself. Three students elaborated on this last point saying that, although such a motivation indeed is not satisfactory, a person that accepts Pascal's wager even on this basis and tries to live a virtuous life, he will perhaps achieve to be gradually liberated from sins and passions; because of this and God's help he may gradually obtain less selfish motives. Thus even with this unsatisfactory initial motivation the wager may have a positive outcome. Comment In many of the aforementioned students’ remarks and considerations, the influence of the Orthodox tradition was obvious, as well as their acquaintance with this tradition. It is also worth noting that some students’ considerations reflected an elaborated thinking in the context of this tradition.

4.2 Modeling of Pascal's Wager After the aforementioned initial debate on Pascal’s wager, the teacher turned the discussion on its modelling. The following table was presented to the students as a summary of the situation faced by the doubting person in the wager. Table1 God exists (G.E.) God doesn't exist (N.G.E.) Subjective probability for Subjective probability for G.E. (p1) N.G.E. (p2) Wager that God exists Present Life1, Salvation Present Life2 Not wager that God Present Life3, Misery Present Life4 exists

The mathematical modeling demands clarification and a precise statement of initial premises. This demand leads to a re-examination of the initial premises established by philosophical considerations. Often the demanded clarification and precision leads to reconsidering or re-conceptualizing initial

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premises. In what follows we present examples of how the demand of mathematical modeling for clarification and precision influenced the consideration of initial premises of Pascal’s wager.

4.2.1 On the partition of hypotheses about God (columns’ partition) The teacher remarked that Pascal proposed the wager to a hypothetical person doubting God's existence but considering that if He exists then He is the God as taught by the Christian church, that is, the Holy Trinity. This remark further provoked the discussion on the many Gods objection. Two students said that for a person doubting God's existence and considering that if He exists, then He is Allah, the wager may also be meaningful; and that this holds also for someone who considers that if He exists He is an omnipotent, omniscient and omnibenevolent God, without specifying His name and religion. Five other students made similar comments agreeing with their colleagues. Three students remarked that although the wager may be meaningful for such a person, his efforts may be in vain because he wagers in a wrong faith. Four students argued that, following the church, believing in the Holy Trinity is a condition for salvation only for those who have been properly taught the Gospel; thus, for example, for a doubting person that lives in an Islamic society and has not been taught the Gospel this objection doesn't hold. Three students argued that in all these cases, if the wagering person achieves to live a virtuous life and obtain pure heart, then if the pure heart argument holds, he will perceive that He exists, and with His help he will end up with whatever faith He considers adequate for his salvation; so in all these cases the wager may have a positive outcome.

4.2.2 On the partition of possible courses of action (rows’ partition) The teacher recalled that Pascal argues that wagering about God's existence is not optional for a doubting person; so he doesn't distinguish between those who don't wager that God exists and those who wager that God doesn't exist. Six students argued that it would be better if the line "Not wager that God exists" was split into two lines; "Not wager that God exists and live a virtuous life" and "Not wager that God exists and not live a virtuous life". Four students considered that it would be better to split the other line into two too; "Wager that God exists and achieve to live a virtuous life" and "Wager that God exists but do not achieve to live a virtuous life".

4.2.3 Reconsideration of the wager about God’s existence These remarks led three students to comment that the wager should be adapted to the beliefs of the different categories of persons that doubt God's existence. Two students went further to propose that the wager should be personalized in order to be adapted to the beliefs of each person who doubts God's existence. Many other students (11) made comments endorsing these

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considerations. Thus, the idea emerged in the classroom that the wager about God’s existence should be regarded as personal and be adapted to each doubting person’s considerations and beliefs. This was an important idea that emerged during the first part of the mathematical modeling work on the wager; that is the clarification of the initial premises of the modeling. This new consideration of the wager about God’s existence was later developed further. In the context of this reconsideration of the wager, Pascal’s wagering proposal was considered as a special case that initiated the discussion and as a point of reference for establishing alternative versions of the wager adapted to each doubting person’s beliefs.

4.2.4 Other initial premises for modeling Pascal's wager The teacher told the students that it would be interesting to examine such variants of Pascal's Wager, but after the examination of the initial version, which was done later. Subsequently, the teacher commented that in the wager's text Pascal attributes explicitly positive infinite utility to Salvation ("an infinity of an infinitely happy life", see Pascal 1910: 85), while he is not explicit about the negative utility of Misery. However, he said, Pascal was a devoted Catholic and his hypothetical doubting person considers that if God exists, He is as taught by the Church. Therefore, he said, we may examine first the most severe version of the wager where Misery has infinite negative utility (eternal damnation, eternal hell); this version accentuates the dilemma faced by the doubting person. The teacher also remarked that, according to Pascal, all Present Lives (1, 2, 3 and 4) have finite utility value, because they all have finite time and finite pleasures and displeasures. He also mentioned that p1, p2 are the probabilities that the doubting person attributes to the hypotheses that God exists or not; thus they pertain to subjective probabilities8. However, he added, at this early time neither the relevant concepts of probability theory, nor the corresponding terminology had been formulated; thus Pascal explains his idea through examples of relevant betting situations. Pascal’s examples were also discussed with the students.

4.2.5 Argument from dominance Subsequently, the teacher remarked that Pascal argues that for the present life, wagering in favor of God's existence and living a virtuous life is better and in fact more pleasant than wagering that God doesn't exist and not live a virtuous life. Thus, according to this, the utility value of Present Life2 is greater than the utility value of Present Life4 and the same holds for Present Life1, compared to Present Life3 (U(PL2)>U(PL4) and U(PL1)>U(PL3)). If a doubting person agrees with this, then for him it is advantageous to wager that God exists in both eventualities (God exists or not).

8 He also recalled that p1, p2 are not 0 or 1 and p1 + p2=1.

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The teacher also remarked that this argument of Pascal is often called an argument from dominance; in the sense that one choice (here, wagering in favor of God’s existence) is advantageous (dominates) in all possible eventualities (here, God exists, or not); e.g. see Hacking 1972. Students agreed that if a doubting person agrees with this consideration, in addition to all previous hypotheses about his beliefs, then it is reasonable that he will consider advantageous for him to wager that God exists. However, they remarked that there are too many hypotheses on the beliefs and considerations of the hypothetical doubting person, and this makes important the question of whether there are such real persons. Some of them also said that many doubting persons may consider such a virtuous life as the one proposed by Pascal, harsh and unpleasant; so, they concluded, perhaps this last hypothesis holds only for very few.

4.2.6 Argument from dominating expectation Then the teacher remarked that for those who do not agree with the last hypothesis (that U(PL2)>U(PL4) and U(PL1)>U(PL3)) Pascal proposes another argument: The expected utility of wagering that God exists is

E1  p1    U PL1   p2 U PL2    (since 0

E2  p1   UPL3  p2 UPL4    , so E1 is greater than E2, even if p1 is very small. The rational choice for wagering is the choice with the greater expected utility9, which in this case is that God exists. During the formation and examination of these mathematical equations students: (i) encountered and worked with infinite expected utilities, which is a concept important both in probability theory and in decision theory, (ii) encountered, discussed and applied the principle of maximum expected utility, which is an important criterion for decision making in decision theory, (iii) had the opportunity to understand that the mathematical modeling of Pascal's wager suggests that a doubting person has to wager in favor of God's existence, even if the probability that he attributes to the eventuality that God exists is very small.

4.2.7 Contrasting results of mathematical modeling with pragmatic considerations Subsequently, the teacher asked students to comment on the results of the mathematical modeling of Pascal's argument, which is based on the danger of losing eternal salvation and suffering eternal hell.

9 This criterion for wagering and more generally for making decisions is often called the principle of maximum expected utility and it is an important element examined in decision theory. (As Hacking (1972) remarks, Pascal is the first who annunciates this and other important elements of decision theory.) The argument based on this criterion is often called the argument from dominating expectation.

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The students initially thought that this argument should be logically convincing for Pascal's targeted audience (persons who doubt God's existence but believe that if He exists then the teaching of the Church about Him is correct). Subsequently, they remarked that all those who consider Church's teaching to be true agree with Pascal's consideration that there is a danger of losing eternal salvation and suffering eternal hell. However, they remarked, a considerable number of these persons, despite of this belief, make very little effort to live a virtuous life. So since the argument based on this danger does not convince many persons who believe that the danger is true, then the argument may also not convince doubting persons to whom Pascal is addressed. Students continued discussing about why the argument, despite the fact that it seems rationally powerful, does not convince many persons who believe that the danger to lose eternal salvation is a true danger. Students proposed different explanations; one of these that attracted the attention and interest of many students is the following10: People find it very unpleasant and painful to think of the eventuality that they will lose eternal salvation and will suffer eternal hell; thus they avoid thinking about it and most of the time, or even all the time, they live their lives without thinking about this eventuality. Three students remarked that this is not specific to the danger of suffering eternal hell and losing eternal salvation; it is part of a more general behavior of people that concerns avoiding thoughts about extremely negative (either certain or probable) future events. For example, they mentioned that most people avoid and think rarely about their own death or the death of their (living) parents, which are certain future events, because such thoughts are very painful and hard. Six students gave other examples endorsing this consideration, such as avoiding thinking about future illnesses, accidents, professional catastrophes etc. However, four students commented that although existent indeed, such a behavior may become irrational when someone avoids thinking about eventualities such as professional catastrophes or some kind of illness or even suffering eternal hell, because these are cases for which, if he thinks, he can take action to minimize the risk of negative outcomes. Nevertheless, remarked one student, if someone thinks about suffering eternal hell not superficially, but intensively, and uses his imagination in order to catch even a small part of what he may suffer there, then such thoughts quickly become totally unbearable. Five other students commented that if someone frequently or - even worse - continuously thinks about things such as losing eternal salvation and suffering eternal hell, his future death, and so on, he may easily make his present life really miserable by his own thoughts alone. Two of them also commented that the aforementioned avoidance behaviors are in fact important self-protection behaviors. Four other students made comments arguing in favor of this

10 Other explanatory elements proposed by students (such as that there are Christians who don't believe in eternal hell, or that there are people, like drug addicts, who have no more strength to be liberated from their passions) engendered limited discussion in the classroom at that time.

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consideration.11 Students agreed that these avoidance and self-protection behaviors may very well be a strong explanatory factor of why Pascal's argument based on the danger of losing eternal salvation and suffering eternal hell is less convincing than he thought; and that this explanatory factor also concerns the relevant version of the argument for those who believe that the teaching of the Church is true12. They also agreed that for persons who avoid considering the danger of losing eternal salvation, mathematical modeling which attributes infinite utility value to salvation and damnation, like the one already mentioned, is inadequate for representing their questions and dilemma about God and His existence.

4.3 Comment In the first part of classroom discussion, the students acquired some familiarity with Pascal's wager and its mathematical modeling and discussed basic objections about the wager at an initial level. During the modeling of Pascal's wager they had the opportunity to encounter and work with infinite expected utilities. Moreover they encountered, discussed and applied the principle of maximum expected utility. Furthermore they realized some significant advances concerning the conceptualization of Pascal's wager. They considered that the wager about God’s existence should be regarded as personal and thus be adapted to each doubting person’s considerations and beliefs. In this context Pascal’s wagering proposal was considered as a special case that initiated discussion, and as a point of reference for shaping alternative versions of the wager. Students examining Pascal's argument which is based on the danger of losing eternal salvation and suffering eternal hell, considered, on pragmatic grounds, that it has not the convincing power that Pascal thought it had. This, in turn, led them to question the adequacy of Pascal's utility function about eternal salvation and eternal hell.

5. SECOND PART OF CLASSROOM DISCUSSION Preparing for the second part of classroom discussion, in the 7th week of the course, the teacher proposed that students read an overview on the debate on Pascal's wager (Hájek 2012) and some other relevant writings (in particular Hacking 1975, Jordan 1994, Lycan & Schlesinger 1989). He encouraged them to

11 Moreover, three students remarked that considerations of the kind "I live my life now, I repent later" may facilitate the avoidance wished because of self-protection mechanisms. Four students argued that frequently suffering the thought of the threat of eternal hell may produce in certain people worst attitudes than avoidance; such as rejecting altogether Church and its teaching. 12 It is interesting to note that these students' considerations are in line with well known pastoral considerations and concerns about the convincing power and the role of arguments based on the danger to loose eternal salvation and suffer eternal hell (e.g. see Bishop Kallistos Ware 1998, p.6).

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feel free, after this initial reading, to continue focusing on authors or lines of thought that they would find interesting and attractive in relation to their own ideas and thoughts. The students actively worked on this task as they found the subject attractive. So, from the 9th to the 11th week of the course13 they orally presented in the classroom elements of their study and their own comments that substantially enriched the discussion there. Below we describe some characteristic aspects of this second part of classroom discussion: Students encountered in their readings and presented in the classroom, a spectrum of hypotheses about God significantly larger than the one that they considered in the first part of classroom discussion. For some of these hypotheses they thought that they are only intellectual constructs elaborated for the sake of argument, or that it is improbable (or very rare) to be hypotheses having some significant weight in the considerations of real doubting persons; for example, because they totally lacked the support of tradition14. However they found others interesting, in particular those hypotheses that suggest that there is no eternal hell such as the hypothesis that all will be finally saved, or the hypothesis that after death the righteous are saved and the wicked pass to nothingness, not to eternal hell. For this last hypothesis they even formulated a corresponding version of the wager15 and its mathematical modelling. For this version students considered the utility value of salvation to be   and the utility value of hell to be 0. Students also discussed Penelhum's (1971: 211-219) objection that the consideration of Pascal's wager that honest unbelievers will lose eternal salvation is an immoral consideration. This enriched and deepened the previous relevant discussion in the classroom (see section 4.1). Moreover, in relation to this discussion, the teacher along with the students examined the mathematical modeling of a version of the wager with the additional assumption that virtuous doubting persons who don’t wager in favor of God’s existence do not lose eternal salvation.

5.1 Duff's objection Moreover, two students presented Anthony Duff’s (1986) objection on Pascal's wager that a doubting person who does not wager in favor of God’s existence still has some chance to convert before the end of his days. During the discussion on this objection, four students argued that a person who in the present wagers in favor of God's existence and tries hard to live a virtuous life, still is not certain about eternal salvation because he may fall even at the end of his life, and conversely, it is not certain for a person who wagers against God's

13 The two weeks of Easter holiday were between the 8th and the 9th week of the course. 14 For example, the hypothesis of Martin (1983) that God rewards the unbelievers and punishes the believers, or the hypothesis of infinitely many possible Gods. It is worth noting that students’ arguments for restricting the spectrum of hypotheses to be considered find support in some of Lycan and Schlesinger considerations (see Lycan & Schlesinger 1989, Schlesinger 1994) 15 This version concerns a person that doubts God's existence and believes that if He exists, then this hypothesis is true.

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existence and lives a non-virtuous life that he will suffer eternal hell because he may repent even at the end of his life16. Seven other students made comments that endorsed these considerations. Moreover three of them suggested that the modeling of the wager should allow for some probability of suffering eternal hell for persons who at present wager in favor of God's existence, and some probability of obtaining salvation for those who at present wager against God's existence. A relevant version of the wager was modeled with teacher’s help17. In this version, both the expected utilities of wagering in favor of God's existence and against God's existence were undetermined; so the application of the criterion of maximum expected utility was inconclusive. These results initially puzzled students. After further examination six of them considered that since the criterion of maximum expected utility was inconclusive then the doubting person should consider that the odds of eternal salvation are greater in the case of wagering in favor of God's existence and the converse holds for the odds of suffering eternal hell; and that this consideration points in the direction of wagering in favor of God's existence18. It is worth noting that with these comments students proposed to use a decision-making criterion of maximum probability similar to that proposed by Schlesinger (1994)19. Four other students, based on grounds of intuitive rationality, thought that the difference of the Expected utility of wagering in favor of God’s existence minus this one of wagering against God’s existence is   ; and that this also points to the direction of wagering in favor of God's existence. However, three other students objected that concluding that one undetermined value is better or greater than another undetermined value is meaningless, and thus the conclusion should be that this modelling leads to no definite conclusion. The discussion on this issue permitted students to understand that although there are criteria according to which this modeling leads to conclusion, they are controversial. After this discussion, the teacher discussed with students relevant paradoxes involving utilities and expected utilities of infinite value20.

16 These students’ remarks echoed the well known Church’s teaching that no-living person can be sure about his salvation after death. 17 In this version, the utility values of eternal salvation and of suffering eternal hell were considered, once again, to be   and   respectively. The conditional probabilities of eternal salvation and of suffering eternal hell, if God exists and the doubting person’s wagers in favor of God's existence, were named ps, ph ; both ps, ph were considered to be different than 0 and ps+ ph was considered to be equal to 1. The respective conditional probabilities if God exists and the doubting person’s wagers against God's existence were named ps', ph'; both ps', ph' were considered to be different than 0 and ps'+ ph' was considered to be equal to 1. It was also considered that ps> ps' and consequently ph< ph'. 18 In their argumentation, they considered that utilities and expected utilities of earthly lives could be disregarded in this modelling because of being too small, compared to the infinite utilities and expected utilities of salvation and hell. 19 Which, however, is not uncontroversial (e.g. see Bartha 2007, Sorensen 1994). 20 Some of them concerned the wager, while others did not; the teacher also suggested further relevant reading (e.g see Bartha 2007, Jordan 2006 ch4, Sorensen 1994).

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5.2 Finiteness of human perception and understanding Concerning the utility value of hell and salvation, three students presented a relevant consideration that they had read about; that, although salvation and hell may be infinite, humans may not be able to appreciate this infiniteness adequately because their perception and understanding are finite in several respects (Hájek 2012). Many (12) students endorsed this consideration and argued that living humans are able to perceive eternal salvation and suffering eternal hell only at an abstract level and not at the level of feelings and sensations. Four of them stretched that what Pascal proposes for salvation (an infinity of infinitely happy life) can not be perceived because man has neither the experience of happiness of infinite intensity nor the ability for this feeling; and that the same holds for feelings of suffering of infinite intensity. However, seven students remarked that a mathematical modeling of the wager which attributes finite utility values to salvation and damnation is not satisfactory with regard to men's ability to perceive infinite utilities for salvation and damnation, even though at an abstract level only. Three of them also argued that for persons who believe that if God exists then the teaching of the Church is true such a modeling does not represent their beliefs and considerations. Five students commented that since men cannot perceive such infinite utilities at the level of feelings and sensations but can do so at an abstract level, then, both modelling with finite such values and modelling with infinite ones will be unsatisfactory with respect to one or to the other. Four students argued that, although the aforementioned considerations about the finiteness of human perception and understanding are reasonable, previous modeling involving infinite utility for eternal salvation and hell should not be considered as invalid because of these considerations, since humans can still conceive such utilities, even though at an abstract level only. They thought that such modeling should be available to people that consider it adequate for themselves; for instance, persons who consider that argumentation of this kind is very important to them21. Following these considerations, students, with the teacher’s help, formulated a relevant version of the wager and its mathematical modelling. In this version they considered the utility values of salvation and of suffering hell to be finite. Students observed that in this version of the wager the application of the criterion of maximum expected utility is possible to suggest not to wager in favor of the hypothesis of God's existence, and that this depends on the considered utility and probability values. They thought this to be another important difference from previously examined versions of the wager. Eight students considered that in this version of the wager the utility values are closer to the reality of limitations of human understanding. Six of them argued that because of this the possible outcomes of the criterion include the alternative result (not wager in favor of God's existence) which is also a real behavior

21 Pascal, remarked two of them, should be one such person.

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observed among doubting persons.

6. FINAL COMMENTS The classroom discussion and students’ related individual work realized during this course allowed them to gain some significant insights into Pascal’s thought about the wager concerning God’s existence, as well as on the relevant debate among philosophers and decision theorists22. Moreover they realized some significant conceptual advances concerning this subject. - They reconsidered Pascal's wager in a dynamic way. More precisely they considered that wagering about God's existence should be considered as personal and be adapted to each doubting person’s considerations and beliefs. In this context, the initial version of the wager was regarded as a special case that initiated the subject and as a reference point for shaping alternative versions of the wager. - Students considered, on pragmatic grounds, that Pascal's argument based on the danger of loss of eternal salvation has less convincing power than what Pascal had thought. This also led them to question the adequacy of infinite utility values attributed to salvation and damnation in the context of the corresponding mathematical modeling. - In connection with the aforementioned, students worked on the modeling of different versions of the wager. This permitted them to work with the concepts of infinite utility and infinite expected utility (concepts which they had very little familiarity with until then) as well as face some interesting problems of decision theory in situations that such utilities are involved.

6.1 Students’ familiarity with Orthodox tradition and the discussion on Pascal’s wager All along the classroom discussion, in students’ comments and considerations, their familiarity with Orthodox tradition and the important influences they have received from this tradition, were frequently observed. Students’ relation to the Orthodox tradition both restricted and deepened important aspects of the discussion on Pascal’s wager. This is particularly true with reference to (i) the many Gods objection on Pascal’s wager, and (ii) students’ comments on doubting persons’ considerations concerning God’s existence. Their relation to the Orthodox tradition was a factor that works in the direction of restricting the spectrum of hypotheses about God that they considered interesting to examine as hypotheses of persons doubting God’s existence. A number of such hypotheses, which were put forward by

22 However, given the extent and the importance of this debate, the work done in this course has to be considered only as a first-initiation work on Pascal’s wager.

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philosophers and decision theorists, were considered by the students as uninteresting to be examined, because they lacked the backup of tradition and were thought of as improbable (or very rare) to be hypotheses that have some significant weight in the considerations of real doubting persons. On the other hand, their relation to this tradition was a factor that enriched and deepened their thoughts on the hypotheses that they examined. Moreover, students’ relation to the Orthodox tradition enriched the insightfulness of their thinking concerning doubting persons’ considerations about God’s existence.

6.2 Mathematical modeling in the discussion on Pascal’s wager In the class work on Pascal’s wager, elements of probability and decision theory were systematically involved. Besides (subjective) probabilities, utilities and expected utilities, often of infinite value, were involved as well as criteria of decision-making. These elements were structured in modelling activities of versions of Pascal’s wager and led to interesting problems of decision theory. The mathematical elaboration on infinite values already presented some difficulty for students; but more importantly, often the results of mathematical elaboration were questionable or even in contrast with respect to intuitive rationality. Such tensions enhanced or led to questioning the initial premises of the modeling, for example, questioning the adequacy of the attribution of infinite values to involved utilities and expected utilities. However, replacing these infinite values with finite ones presented other fundamental inadequacies. Thus, in these modeling activities students encountered and worked with the concepts of utilities and expected utilities of infinite value and faced some related questions which are deeply routed in probability theory and decision theory, along with a network of relevant problems. In these modelling activities, students observed that correct mathematical elaboration does not always lead to safe and/or uncontestable results; as it is, for example, the case in Euclidean Geometry, where the initial premises (axioms) are not questioned23. On the other hand, the clarity of mathematical elaborations that led to question initial premises of the modelling permitted to identify flaws of these premises that it was very difficult or impossible, to identify as long as these premises were discussed at the literal level. Thus, these modelling activities offer students the occasion to appreciate that mathematics may have an important role in the discussion of philosophical issues, to understand some basic aspects of modelling work and even to question stereotypes and enrich their concept image for mathematics.

REFERENCES

23 Although they had heard about the existence of non-Euclidean Geometries, students had never worked with Geometry which was incompatible with the Euclidean one. Moreover, students had very little, if any, experience of mathematical modelling work that may lead to unsafe or contestable results for reasons other than the well known “you haven’t done your work correctly”.

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Adamson, D. (1995). Blaise Pascal: mathematician, physicist and thinker about God. London: Macmillan pub. Bartha, P. (2007). Taking Stock of Infinite Value: Pascal's Wager and Relative Utilities. Synthese, 154, 5–52. Bernoulli, D. (1954). Exposition of a New Theory on the Measurement of Risk. Econometrica, 22(1), 23-36. (Translation. Translated by Sommer, L. Originally published as "Specimen Theoriae Novae de Mensura Sortis", Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tomus V, 1738, 175-192.) Bishop Kallistos Ware (1998). Dare we hope for the salvation of all? Origen, St Gregory of Nyssa and St Isaac the Syrian. Theology Digest, 45(4), 303-317. Bras, M. (2009). Blaise Pascal, apologie et probabilités: une étude sur la méthodologie de 1'apologie pascalienne et sa portée (Doctoral Dissertation). Ottawa: Université d'Ottawa, Département de Philosophie. Brunschvicg, L. (1909). Histoire des Pensées. In L. Brunschvicg (ed.), Blaise Pascal: Pensées et Opuscules, 5eme édition (pp. 255-265). Paris: Librairie Hachette. Chandler, J. & Harrison, V. (eds.) (2012). Probability in the Philosophy of Religion. Oxford: Oxford University Press. Descotes, D. & Proust, G. (2011). L’édition électronique des Pensées de Blaise Pascal. Retrieved Mars 15, 2013 from the World Wide Web: http://www.penseesdepascal.fr/index.php Duff, A. (1986). Pascal's Wager and Infinite Utilities. Analysis, 46(2), 107-109. Dutka, J. (1988). On the St. Petersburg paradox. Archive for History of Exact Sciences, 39(1), 13-39. Edwards, A. W. F. (2002). Pascal’s arithmetical triangle: The Story of a Mathematical Idea. Baltimore: Johns Hopkins University Press. Hacking, I. (1972). The Logic of Pascal's Wager. American Philosophical Quarterly, 9(2), 186–192. Hacking, I. (1975). The Emergence of Probability. Cambridge: Cambridge University Press. Hájek, A. (2012). Pascal's Wager. In E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. Retrieved September 25, 2013 from the World Wide Web: http://plato.stanford.edu/archives/win2012/entries/pascal-wager/ Hald, A. (2003). A History of Probability and Statistics and their applications before 1750. NJ: Wiley. Jordan, J. (ed.) (1994). Gambling on God: Essays on Pascal's Wager. Lanham, Maryland: Rowman & Littlefield pub. Jordan, J. (2006). Pascal’s Wager: Pragmatic Arguments and Belief in God. Oxford: Clarendon Press. Kourkoulos, M. & Tzanakis, C. (2015). Statistics and Free Will. In E. Barbin, U. Jankvist, and T.H. Kjeldsen (eds), Proceedings of the 7th European Summer University on the History and Epistemology in Mathematics Education (pp. 417-432). Denmark: Danish School of Education, 417-432.

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Kourkoulos, M. & Tzanakis, C. (in press). Greek students of today discussing Pascal's wager. In E. Nicolaidis & K. Skordoulis (eds), Proceedings of the Science and Religion International Conference. Athens: Hellenic Research Foundation- Institute of Historical Research. Lafuma, L. (1954). Histoire des Pensées de Pascal (1656-1952). Paris: Editions du Luxembourg. Lycan, W. & Schlesinger, G. (1989). You Bet Your Life: Pascal's Wager Defended. In J. Feinberg (ed.), Reason and Responsibility: Readings in Some Basic Problems of Philosophy, 7th edition (pp. 82-90). Belmont CA: Wadsworth Pub. Martin, M. (1983). Pascal's Wager as an Argument for Not Believing in God. Religious Studies, 19, 57-64. Martin, R. (2014). The St. Petersburg Paradox. In E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. Retrieved November 15, 2014 from the World Wide Web: http://plato.stanford.edu/archives/sum2014/entries/paradox- stpetersburg/ Mesnard, J. (1951). Pascal: l'homme et l'œuvre. Paris: Editions Boivin. Pascal, B. (1910). Thoughts. In Ch. E. Eliot (ed), Blaise Pascal: Thoughts, tr. By W.F. Trotter, Letters, tr. by M.L. Booth, Minor Works, tr. by O.W. Wight: with introds. notes and illus. (pp7-322). New York: P.F. Collier & Son Co. (Translation. Translated by W.F. Trotter. Originally published as "Pensées" in L. Brunschvicg (ed.) (1897), Blaise Pascal: Opuscules et Pensées. Paris: Librairie Hachette.)24 Penelhum, T. (1971). Religion and Rationality: an introduction to the philosophy of religion. NY: Random House. Porter, T.M. (1986). The Rise of Statistical Thinking 1820–1900. Princeton: Princeton University Press. Schlesinger, G. (1994). A Central Theistic Argument. In Jordan (1994). Gambling on God: Essays on Pascal's Wager. Lanham, Maryland: Rowman & Littlefield pub (pp83–99). Sorensen, R. (1994). Infinite Decision Theory. In Jordan (1994). Gambling on God: Essays on Pascal's Wager. Lanham, Maryland: Rowman & Littlefield pub (pp139–159). Stigler, S.M. (1986). The history of statistics: the measurement of uncertainty before 1900. Harvard: Harvard University Press.

BRIEF BIOGRAPHIES Michael Kourkoulos is Assistant Professor of the Didactics of Mathematics at the Department of Primary Education of the University of Crete. He has graduated from the Department of Mathematics of the University of Athens. He received a master and a Ph.D. in the Didactics of Mathematics from the University of Louis Pasteur

24 This translation was reissued by Dover Publications in 2003, under the title Pensées. The reissue includes an introduction by T. S. Eliot, written in 1958.

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(Strasbourg). His research concerns alternative forms of Mathematics’ teaching, as well as, the didactical use of the History of Mathematics. In particular, his research concerns the didactics of Arithmetic & Algebra, Geometry, and Statistics & Probability.

Constantinos Tzanakis is professor at the Department of Primary Education of the University of Crete, teaching mathematics and physics. He holds a first degree in mathematics, an MSc degree in Astronomy and a PhD in Theoretical Physics. His research interests and activities are in theoretical physics and the didactics of mathematics and physics and has published 84 papers, co-edited 9 collective volumes and 5 volumes of conference proceedings and journals’ special issues. He has been chair of the International Study Group on the Relations between the History and Pedagogy of Mathematics, affiliated to the International Commission on Mathematical Instruction (2004-08).

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UNIVERSITY OF WESTERN MACEDONIA FACULTY OF EDUCATION

MENON 130 ©online Journal Of Educational Research

THE HISTORY OF MATHEMATICS DURING AN INQUIRY- BASED TEACHING APPROACH

Areti Panaoura Frederick University, Department of Primary Education [email protected]

ABSTRACT The use of the history of Mathematics in teaching has long been considered as a useful tool in order to enable students to construct conceptually the mathematical concepts. At the same time the inquiry-based teaching approach is proposed to be used in order to improve students’ learning by using their natural tendency to curiosity. The use of the history of mathematical concepts during an inquiry-based teaching approach is expected to multiply the positive effects on students’ learning. The present study examines in-service teachers’ beliefs and knowledge about the use of the history of mathematics in the framework of the inquiry-based teaching approach at the educational system of Cyprus, and the difficulties teachers face in adopting and implementing this specific innovation in primary education. At the first phase of the study a questionnaire was used in order to investigate teachers’ knowledge and beliefs about the use of the history of mathematics in education and mainly in relation to the inquiry-based teaching approach. At the second phase of the study two case studies were examined, where teachers introduced a mathematical concept by using the history of mathematics in order to identify the practices they used and the difficulties they faced. The results indicated that the teachers’ knowledge about the use of the history and mainly the experimental nature of mathematics is significantly related with their positive beliefs about the inquiry-based teaching approach. Teachers’ worries were mainly concentrated on their difficulties to manage the time and the content of the subject and to face efficiently and flexibly their students’ mistakes and difficulties. Keywords: history of mathematics, inquiry-based activities, teachers’ knowledge, beliefs and practices

1. INTRODUCTION The idea of using the history of mathematics in education is not new (Goktepe & Ozdemir 2013). Over the past three decades researchers from various countries have discussed the possibility of introducing new concepts within relevant historical context (Yee & Chapman 2010), at different educational levels. Some researches describe the affective impact from using the history of mathematics in education (e.g. Furinghetti 2007, Marshall 2000) and others discuss the necessity to include the history of mathematics in pre-service teachers’ university programs (e.g. Fleener et al. 2002) in order to train teachers to use it with their students. There are several reasons to incorporate the use of the history of mathematics in education, and the major one is the impact of such a practice on the development of the mathematical disposition of students

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(Clark 2006). Using authentic problems from the history of mathematics provides material for students to actively engage in classroom discourse (Gulikers & Blom 2001), and to realize the role of the construction of the science of mathematics. At the same time inquiry-based learning is not a recent movement in mathematics education, and it has been recommended as an appropriate basis for student learning in mathematics for the last decades. Numerous studies and reports of committees continue to call for inquiry-based teaching and learning approaches in mathematics (e.g. Marshall & Horton 2011) in order to encourage students to think critically and creatively. Teachers need to know how to approach their teaching in a way that is reflective, responsive and flexible (Marin 2014). Having in mind that the use of the history and the inquiry-based teaching approach are among the major objectives in mathematics education, we have decided to examine the use of the history of mathematics in a framework of the inquiry-based teaching approach at the early stages of primary education and mainly to investigate teachers’ difficulties in applying in their instruction the proposed innovation. Burton (2003) defines history of mathematics as a vast area of study which includes investigating sources of discoveries in mathematics, highlighting that it includes investigations of the achievements of significant mathematicians and their ideas. At the Curriculum of Mathematics which was constructed in 2011 for primary education in Cyprus the use of history of mathematics is suggested and the usual use of inquiry-based teaching approach is proposed as the main teaching approach. The two central concepts for the inquiry-based teaching approach are the use of investigations and explorations. Radford, Furingetti and Katz (2007) acknowledge that questions related to the pedagogical role of the history of mathematics remain open to investigation. Teachers have various beliefs such as about themselves as teachers, the nature of the discipline of mathematics, the factors that affect the learning and the teaching of mathematics. The present study concentrates on teachers’ knowledge about teaching mathematics by using the inquiry-based teaching model in the framework of the history of mathematics and mainly their respective practices in authentic teaching situations. We concentrate our attention on the experimental epistemological dimension of mathematics (Ernest 1991) which is directly related with the inquiry-based teaching and learning approach. It is important to examine how teachers use their knowledge and their beliefs in order to design instructional activities fostering mathematical inquiry by using the history of mathematics. The specific research questions were: 1. How are teachers’ knowledge and beliefs about using the history of mathematics related with their knowledge and beliefs about the use of inquiry-based teaching approach? 2. What are the teachers’ practices on using the history of mathematics during an inquiry-based teaching approach?

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2. THEORETICAL FRAMEWORK

2.1 The history of Mathematics in mathematics education Teachers have long been encouraged through curriculum and the scientific community in mathematics to incorporate aspects from the history of mathematics into their teaching (Lopez-Real 2004). Jankvist (2009) suggests the use of the history of mathematics by highlighting the increased motivation and the realization that mathematics is a human creation. Introducing the history of mathematics in school curricula enhances learners’ motivation, promotes favoured attitudes, and draws attention to possible obstacles faced in the generation and evolution of mathematical concepts. As a pedagogical tool it can serve as a guide to the difficulties students may encounter as they learn a particular mathematical topic (Haverhal & Rsocoe 2010). Schubring and colleagues (2000) also posit that programs based on the history of mathematics could increase self-confidence in working with mathematical tasks and develop learners’ ability to apply mathematical methods. A journey through the history of mathematics could also enable learners to construct mathematical meanings and support new conceptions about mathematics by changing learners’ existing beliefs and attitudes (Dubey & Singh 2013). In addition, the historical dimension encourages learners to think of mathematics as an evolving body of knowledge, rather than as a well-defined entity composed of irrefutable and eternal truths (Barbin, Bagni, Grugnetli, & Kronfellner 2000). Jahnke (2000) suggested three general ideas which might be suited for describing the special effects of studying a source on the teaching of mathematics: (a) the notion of replacement according to which mathematics is seen as an intellectual activity rather than a set of techniques, (b) the notion of reorientation according to which history reminds us that the mathematical concepts were invented and (c) the notion of cultural understanding according to which integrating history of mathematics invites us to place the development of mathematics in the scientific and technological context of a particular time and in the history of ideas and societies and also to consider the history of teaching mathematics. For many years, the rationale of employing the history of mathematics in teaching has explicitly or implicitly been hinged on the notion of “recapitulation”, according to which ontogenesis recapitulates phylogenesis. Although this principle has been challenged on the grounds of different socio- cultural conditions, Sfard (1995) points to “inherent properties of knowledge” which result in similar phenomena that can “be traced throughout its historical development and its individual construction” (p. 15). These inherent properties or epistemological obstacles could provide the grounds for a meaningful negotiation of meaning using history as a means towards an epistemological laboratory (Radford 1997). Studying the development of mathematical ideas also opens up the possibility of seeing mathematics as a socio-cultural creation and helps

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“humanize” mathematics (Fauvel 1991). As Siu (1997) claims, using the history of mathematics in the classroom does not necessarily increase students’ cognitive performance, but “it can make learning mathematics a meaningful and lively experience, so that learning will come easier and will go deep” (p. 8). Such programs also have the potential to help students overcome mathematics anxiety or mathematics avoidance. In addition to that, historical and epistemological analysis of the content helps teachers understand why a certain concept is difficult for students to grasp. Such an understanding is important, because it can inform selection of tasks/problems to introduce a particular concept, the strategies teachers employ in helping students develop understanding of this concept, and the time they allot to working on this concept (Barbin et al. 2000). The mathematics teachers in the study by Lit and Wong (2001) were very supportive in theory for using history in their teaching. Siu (1998), in an invited talk given at the working conference of the 10th ICMI study on the role of history of mathematics in mathematics education, offered a list of thirteen reasons why a school teacher hesitates to make use of the history of mathematics in classroom teaching such as “I have no time for it in class”, “Students don’t like it”, “There is a lack of teacher training on it”, “Students do not have enough general knowledge on culture to appreciate it”, etc. The suggestions which are included in Curriculum or Reports of Committees do not necessarily mean that teachers are able to apply them in their teaching, either due to their lack of positive beliefs and self-efficacy beliefs or due to teaching difficulties and obstacles, which they are unable to overcome when they face them.

2.2 The inquiry-based teaching approach Inquiry-based teaching and learning is based on the principles of social constructivism (Aulls & Shore 2008), according to which a learner assimilates a new situation and experience on previous experiences and depending on inter- individual differences constructs the new knowledge. The scientific journal of ZDM in Mathematics Education has published a special issue in 2013 with nine papers focusing on inquiry-based mathematics education and their implementations, indicating that many questions remain unanswered. The challenge for educational systems is to enable its teachers to adopt the values of the inquiry-based pedagogy. Chin and Lin (2013) claim that there are obstacles and difficulties such as: (i) teachers did not experience inquiry-based learning in mathematics in their own school years, (ii) they do not have complete understanding of the inquiry-based teaching, (iii) there are practical constraints such as that the allocated teaching hours are not enough, (iv) the influence of teaching for success in tests. The learner-focused perspectives of mathematics education require teachers to use pedagogical methods which actively engage students in developing conceptual understanding of mathematical concepts (Chapman 2011). According to Taylor and Bilbrey (2011) the research outlines two facets of

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inquiry-based instruction which are open education and differentiation. The major characteristic of the open education is that instruction is driven by the desires of the students, while the differentiation approach allows students’ preferences to guide how particular content is encountered. Hakkarainen (2003) proposes an inquiry pedagogical approach called progressive inquiry for young learners in learning science, while Song and Looi (2011) explore the application of an adaptation of this approach to mathematics inquiry learning. The learner- focused perspectives of mathematics education requires teachers to use pedagogical methods which actively engage students in developing conceptual understanding of mathematical concepts (Chapman 2011). Teachers need to develop their ability to foster student decision-making by balancing support and independence in thinking and working (NCTM 2000). The teacher’s role has evolved from concept deliverer to concept facilitator. Hegarty – Hazel (1986) categorized four levels of inquiry – based activities which ranged from specific guidance and close question to open exploration and open question. For example at the first level the teacher provides specific inquiry question, solving procedures and solution, while at the last level the teachers provide learning environment for students to generate inquiry question. Both teachers and students need slow and stable steps in order to be moved from the traditional algorithmic procedures to the challenge of the conceptual processes. One of the main emphases of the new proposed teaching model of Mathematics in the centralized educational system of Cyprus which is presented at the New Curriculum (NCM 2011), is the use of exploration and investigation of mathematical ideas as two dimensions of the inquiry-based teaching and learning approach. Last year during the implementation of the new school mathematics curriculum, the new obligatory for use textbooks for grades 1, 2, 3 and 4 had already been introduced (ages 6-9 years old). The whole idea is to introduce a mathematical concept by using an inquiry-based activity through which the teacher generates curiosity and interest in the topic and he/she asks students to express their ideas and communicate by using the language of mathematics. The emphasis is on using authentic and open-ended problem solving activities without only one correct answer and each student is expected to respond in respect to his/her previous knowledge, experiences and unique way of thinking. Teachers are expected to support the students in working independently and creatively. In only few specific cases the activities which are included in the textbooks use the context of the history of mathematics.

3. METHODOLOGY The present study was divided into two main phases. At the first phase the emphasis was on examining the teachers’ knowledge and beliefs about using the history of mathematics and mainly in cases of planning inquiry-based activities. To examine teachers’ knowledge and beliefs about the use of the

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history of mathematics and the inquiry-based teaching approach, we constructed and used a questionnaire that consisted of two scales: one including 12 items that measured knowledge and beliefs about the use of the history of mathematics (e.g. Mathematics changes in order to fulfill the human or social needs) and another consisting of 12 items designed to capture knowledge and beliefs about the value and implementation of inquiry-based teaching (e.g the teachers’ guidance during inquiry-based teaching approach has to be limited). All items were measured on a 5-point Likert scale (1= strongly disagree and 5 = strongly agree). The emphasis of the second phase was on examining the practices teachers use during the implementation of the inquiry-based activities by using the history of mathematics in authentic classroom situations. We wanted to make the link between what they say and what they actually do. Researchers can examine the teachers’ behavior well when following and observing them in an authentic context (Hwang, Zhuang & Huang 2013). By using the case study approach we emphasized detailed contextual analysis of teaching condition in real-life school situations. A teacher of the 2nd grade and a teacher of the 3rd grade were observed individually while they were introducing the place-value of two- and four-digit numbers by using the history of mathematics, and then a semi-structured individual interview was conducted with each one of them. The respective activities which were suggested to be used by the textbooks introduced the concepts by using an exploration and an investigation (the Greek version of the respective pages are presented in Figure 1 and 2). A protocol for the observation was constructed and used in order to concentrate the observer’s attention on: a) teachers’ guidelines at the introduction of the activity, b) teachers’ feedback on students’ difficulties and mistakes and c) the time which was allocated for the specific activities. The interview was concentrated on the practices they had used and the difficulties they had faced. The sample: Participants who completed the questionnaire at the first phase of the study were 162 teachers, who were teaching mathematics at the first, second, third and fourth grade last year. The new curriculum methods with the new obligatory for use textbooks which include inquiry-based activities at a framework of the history of mathematics have already been introduced only at those four primary school grades. 115 of the participants of the sample were females and 47 were males. 45 participants were teaching at the first grade, 43 at the second grade, 40 at the third grade and 34 at the fourth grade. All the participants were asked to complete the questionnaire voluntarily and anonymously. The teachers who took part in the second phase of the study were randomly chosen and both agreed to be observed during their teaching and take part in the individual interview. Statistical analyses: In order to confirm the structure of the questionnaire and mainly in order to examine the interrelations between the four main factors of the study, a Confirmatory Factor Analysis (CFA) was conducted using Bentler’s (1995) Structural Equation Modelling (EQS) programmes. The

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tenability of a model can be determined by using the following measures of goodness of fit: x2/df <1.95, CFI (Confirmatory Fit Index) >0.9 and RMSEA (Root Mean Square Error of Approximation) <0.06. Cronbach’s alpha for the questionnaire was 0.87.

Figure 1 and Figure 2: The exploration and investigation activities which were used

2nd grade, unit 5 3rd grade, unit 7 An exploration where an ancient Egyptian An investigation where an archeologist found wrote the numbers of animals and students the presented numbers on stones which belong have to guess which numbers could be possible to ancient Crete. Students have to choose the right number which is presented in each stone.

.

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4. RESULTS

4.1 Teachers’ knowledge and beliefs Firstly the interest concentrated on the interrelations between the first-order factors as indicators of the impact of the cognitive and affective factors concerning the use of the history of mathematics and the use of the inquiry- based teaching approach. The initial model tested in this study hypothesized a first-order model with four main interrelated factors: (i) the in-service teachers’ knowledge about the history of mathematics, (ii) their beliefs about the use of the history of mathematics in teaching, (iii) their knowledge about the use of the inquiry-based approach and (iv) their beliefs about the inquiry-based approach and its implementation. The a priori model hypothesized that the variables of all the measurements would be explained by a specific number of factors and that each item-statement would have a non-zero leading on the factor that it was supposed to measure. Additionally the model (following the LM Test) was tested under the constraint that the error variances of some pair of scores associated with the same factor would have to be equal. As Kieftenbeld, Natesan and Eddy (2011) suggest few error variances need to be correlated when there is a local dependence between items. Local dependence occurs when participants’ responses to a particular item depended someway on their responses to other similar items. Figure 3 presents the results of the elaborated model that fits the data reasonably well (x2/df = 1.86, CFI = 0.932, RMSEA = 0.031). The first-order model that is considered appropriate for interpreting teachers’ beliefs and knowledge about the inquiry-based teaching approach which includes the use of the history of mathematics involves 4 first-order factors, as was proposed. The first factor consisted of 7 items concerning teachers’ knowledge about the history of mathematics. The loadings of all the items were >0.5 and all the regressions were statistically significant. The second first-order factor consisted of 6 items concerning teachers’ beliefs about using the history of mathematics in the teaching of mathematics at primary education. The third-order factor consisted of 5 items concerning teachers’ knowledge about the use of inquiry- based approach in the teaching of mathematics and the fourth-order factor consisted of 6 items concerning teachers’ beliefs about using inquiry-based activities in their teaching. By using the specific analysis we aimed to explore the way these four dimensions of the model were interrelated. The existence or the non-existence of statistically significant interrelations is interesting.

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Figure 3: CFA model about knowledge and beliefs interrelations

As it was expected the relation between teachers’ knowledge about the history of mathematics and their beliefs about using it as part of the teaching process was statistically significant and extremely high (0.813), indicating that teachers who understand mathematics as a dynamic science which has evolved throughout the centuries in order to facilitate the development of the science and the social needs, they are at the same time teachers with positive beliefs about using the history of mathematics in teaching. At the same time teachers who know the advantages and limitations of using the inquiry-based teaching and learning approach, have positive beliefs about using the specific method in order to encourage their students to investigate and explore a mathematical concept (0.753). Statistically significant was the relationship between teachers’ knowledge about the use of the history of mathematics and their beliefs about using the inquiry-based approach (0.692). It seems that teachers who believe that mathematics has been created, constructed and enriched by humans during the development of the specific science, want to give their students the opportunity to work creatively and critically in order to explore or investigate a mathematical concept. Teachers who have positive beliefs about using the inquiry-based approach in their teaching have at the same time positive beliefs about the use of the history of mathematics (0.718). The non-existence of a statistically significant interrelation between teachers’ knowledge about the use of inquiry-based approach and their knowledge and their beliefs about using the history of mathematics is justified by the fact that 0.753 the use of the inquiry-based approach in education is presented and suggested to teachers without relating it directly with the use of the history of

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mathematics. However there are indirect interrelations, as teachers’ adequate knowledge about the inquiry-based approach is related with their beliefs about using the inquiry-based approach. At the same time teachers with high knowledge about using the history of mathematics have positive beliefs about using it.

4.2 Teachers’ practices during inquiry-based teaching with the use of the history of mathematics The observation of two teachers enabled us to concentrate more qualitatively on the practices they followed in order to use the inquiry-based approach on their teaching when they decide to use the history of mathematics which is presented in the textbooks. Firstly we present briefly the observations and then the related parts of the follow-up interviews which concentrated on dimensions which are related to the instructional practices. The teacher of the 2nd grade presented to her students a picture with ancient Egyptians who were farmers and at the background of the picture there were symbols on the wall of their houses. She told the students that ancient Egyptians used to engrave symbols on the walls or papyrus and she asked them to study the picture in their book and guess which numbers were possible. She actually preferred to pose an open question which guided them to many different accepted answers. Many right answers were given and only one wrong. In fact the mistake was made by a student who presented an unexpected answer with three-digit numbers. He claimed that the first number was 310, the second 502 and the third 106. The teacher told him “we have not learnt three-digit numbers yet, we will not discuss this mistake now”. She spent almost 10 minutes on the specific activity with the ancient Egyptians in the textbook and then she asked students to imagine that there were ancient Egyptians and they had to construct and propose their own symbols. Each group of two students had to decide 3 to 4 symbols and they had to present to their classmates few numbers in order to guess the value of each symbol. Students found the activity creative and all the pairs wanted to present their work. The most common mistake was the insufficient information which was given to their classmates in order to guess the value of the symbols. The teacher preferred to justify this mistake by making the comment to the students “you had preferred to pose an open problem, an exploration”. During the interview she justified this behaviour by saying that “it was an exploration and I didn’t want to kill students’ enthusiasm by pointing out that they worked wrongly. I wanted them to feel free to create in mathematics rather than feeling fear of making mistakes”. She justified the absence of feedback in the case of the three- digit numbers, which were presented above, by saying that “I do not have the time to discuss everything and most students would be unable to understand something from this discussion”. What was impressive and unexpected was that she continued with activities of place-value at two digit numbers without any reference to the similarities and differences of the two arithmetic systems.

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At a question during the interview asking her to explain why she had not discussed the importance of the absence or the presence of zero at the arithmetic systems, she said “the history of mathematics can be used just as a fairy tale. It has to be used in the same way you can use literature in order to introduce a concept. We have no time to insist more. This could be done in the upper grades of education, not at the 2nd grade”. The parts of the interview which are indicative of her beliefs about the use of the history of mathematics and about the inquiry-based approach are presented below. - How often do you use the inquiry-based approach in the teaching of mathematics? - I always do the investigations which are presented in the textbooks and sometimes the explorations. - Why are you not using all the explorations? - I do not have enough time. It is difficult to concentrate your students’ attention on a specific concept when the framework is open. - Why did you decide to use the exploration with the ancient Egyptians? - This was interesting but you saw that I did not continue to discuss the three-digit numbers. I would have problem with the time. - Have you ever used something from the history of mathematics which is not presented in the textbook during an activity of exploration? - No, I didn’t know many things about the history of mathematics and it is too difficult to relate the historical concepts with the knowledge you want them to learn today. At a relative question about the attendance of any course related with the history of mathematics or the inquiry-based approach during her studies or any pre-service training program she claimed that she did not know anything about the use of the history of mathematics and she had attended the obligatory in- service training about the use of explorations and investigations which was organized by the Ministry of Education. She underlined the necessity of developing programs of training at the school in real teaching situations, especially in order to enforce the use of the inquiry-based approach. It is clear from the discussion which is presented above that the teacher felt the pressure of the syllabus which had to be taught; she indicated negative self- efficacy beliefs in managing the time and the unexpected situations derived by students who performed well in mathematics. At the same time the lack of knowledge about the content of the history of mathematics and the value of using it as a teaching tool is obvious, while the teacher was convinced about the value of using of the inquiry-based approach in daily-life framework. The teacher at the 3rd grade started the lesson by asking students to imagine that they were archaeologists and they had to understand the numbers which were written on the stones (Figure 2). He asked them to cooperate with the classmate who was near them in order to solve the two exercises on page 19. He spent only 3 minutes in order to correct their answers. He asked three students to write the three numbers on the board and he evaluated students’

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understanding by asking them to raise their hand if they knew how to translate the numbers 5328 and 2008. He asked from a child who did not raise his hand for translating the second number to go on the board in order to “help him to think together” the solution. When he realized that the child was confused because of the presence of “0”, he asked him to find the respective solution for the number 110 by presenting it firstly with the dienes cubes and then by using ancient symbols. Then he asked for the translation of numbers 101 and 1001. During the interview he claimed that the inquiry-based approach is useful, especially for students with low performance in mathematics as it reveals their misunderstandings and misconceptions. However he underlined the difficulty to work at the same time with all the students during an investigation. He knew few things about the history of mathematics, mainly about geometry and non- Euclidean geometry, which he had been taught at university but he could not imagine anything else beyond the arithmetic systems that could be used in the teaching of mathematics in primary education. He believed that the history of mathematics could be useful in gymnasium in order to enable students to honor the ancient Greeks who discovered mathematics. He did not remember other mathematicians except for Pythagoras and Euclid. It is obvious that this specific teacher preferred to use a guided investigation. He insisted on students’ mistakes by using the strategy of simplifying the problem. He did not know the philosophy and pedagogy of using the history of mathematics in order to introduce a mathematical concept.

5. DISCUSSION European reports will continue to call for inquiry-based teaching approaches in mathematics in order to urge students to think critically and creatively and to enable them to solve authentic real-life problems. Teachers are expected to actively engage students in open-ended learning experiences in order to foster an environment of inquiry. The current study provided evidence that although teachers have positive beliefs about the importance of the history of mathematics for the introduction of mathematical concepts, they do not apply its features into their teaching practice satisfactorily, because they do not have the necessary and sufficient knowledge. Teachers feel more confident to teach the way they were taught and they seemed not having adequate experiences in learning mathematics through the exploration of the respective history. We have to rethink at least the role of in-service training programmes and the respective experiences which are built through them. In pre-service or in-service training we have to equip teacher with methods and techniques for incorporating historical materials in their own teaching, with experiences in inquiry-based teaching approaches and with strategies of managing flexibly the time and their students’ misunderstandings. In order to enable teachers to adopt inquiry-based approaches which use the history of mathematics for the introduction of mathematical concepts, we have to develop pre-service and in- service training programs which use the progressive inquiry approach in order

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to affect their knowledge on the specific domain, their masteries experiences and consequently their self-efficacy beliefs in respect to Bandura’s theory (Bandura 1997). Attempts to incorporate the history of mathematics in education might benefit from keeping in mind that teachers need to be helped to develop knowledge that is both useful and usable for the work of teaching mathematics. It is extremely important that teachers who adopted an experimental epistemological perspective about the nature of mathematics by understanding the dynamic development of the specific science throughout the centuries, believed in the value of exploring and investigating the mathematical concepts. This is an indication that their experiences as learners during their training courses at universities with the development of the mathematical concepts by using an inquiry-based approach will probably enable them to believe in the value of using the inquiry-based approach and the benefit of using the history of mathematics in order to humanize them. The present study is just the starting point of investigating a piece of this puzzle which is related with the history of mathematics and the inquiry – based approach and more research has to be developed in order to relate the teachers’ knowledge and beliefs about the use of the history of mathematics with their beliefs and knowledge about the inquiry-based approach. Emphasis has to be given on studying further teachers’ difficulties in implementing the inquiry- based teaching approach in general and in the case of using the history of mathematics in particular. Studies have to be developed to examine their practices and difficulties in real classroom actions. A future study could concentrate further on the investigation of teachers’ practices in classroom context by observing more instructions and investing on changes which would be the result of teachers’ own self-reflection on their teaching behaviour when difficulties are faced during an attempt to implement an innovation.

REFERENCES Aulls, M.W. & Shore, B.M. (2008). Inquiry in Education, Volume I: The Conceptual Foundations for Research as a Curricular Imperative. New York: Lawrence Erlbaum Associates. Bandura, A. (1997). Self-efficacy: The exercise of control. New York: Freeman. Barbin, E., Bagni, G., Grugnetli, L., Kronfellner, M., (2000). Integrating history: research perspectives. In J. Fauvel, & J. Maanen (Eds.), History in mathematics education – The ICMI study (pp. 63-90). Dordrecht: Kluwer. Bentler, P. M. (1995). EQS structural equations program manual. Encino, CA: Multivariate Software. Burton, D. M. (2003). The History of Mathematics: An Introduction (5th ed.). New York, NY: McGraw-Hill. Chapman, O. (2011). Elementary school teachers’ growth in inquiry–based teaching of mathematics. ZDM Mathematics Education, 43, 951-963.

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Chin, E. & Lin, F. (2013). A survey of the practice of a large–scale implementation of inquiry–based mathematics teaching: from Taiwan’s perspective. ZDM Mathematics Education, 45, 919-923. Clark, K. (2006). Investigating teachers’ experiences with the history of logarithms: a collection of five case studies. Dissertation: University of Mayrland. Dubey, M. & Singh, B. (2013). Assessing the effect of implementing mathematics history with Algebra. International Journal of Scientific and Research Publications, 3 (8), 1-3. Ernest, P. (1991). Philosophy of mathematics education. New York: Falmer Press. Fauvel, J. (1991). Using history in mathematics education. For the Learning of Mathematics, 11(2), 3-6. Fleener, M., Reeder, S., Young, E. & Reylands, A. (2002). History of mathematics: Building relationships for teaching and learning. Action in Teacher Education, 24, 73-84. Furinghetti, F. (2007). Teacher education through the history of mathematics. Educational Studies in Mathematics, 66, 131-143. Goktepk, S. & Sukru, A. (2013). An example of using history of mathematics in classes. European Journal of Science and Mathematics Education, 1 (3), 125- 136. Gulikers, I., & Blom, K. (2001). 'A historical angle', a survey of recent literature on the use and value of history in geometrical education. Educational Studies in Mathematics, 47(2), 223-258. Hakkarainen, K. (2003). Progressive inquiry in a computer–supported biology class. Journal of Research in Science Teaching, 40 (10), 1072-1088. Haverhals, N. & Roscoe, M. (2010). The history of mathematics as a pedagogical tool: Teaching the Integral of the secant via Mercator’s projection. The Montana Mathematics Enthusiast, 7 (2), 339-368. Hegarty-Hazel, E. (1986). Lab work SET: Research information for teachers, number one. Canberra: Australian Council for Education Research. Hwang, G. J., Wu, P. H., Zhuang, Y. Y., & Huang, Y. M. (2013). Effects of the inquiry-based mobile learning model on the cognitive load and learning achievement of students. Interactive Learning Environments, 21(4), 338-354. Jahnke, H. (2000). The Use of original sources in the mathematics classroom. In Fauvel J, van Maanen J (eds). History in Mathematics Education: The ICMI study (p 291-328). Kluwer, Dordrecht. Jankvist, U. T. (2009). A categorization of the “whys” and “hows” of using history in mathematics education. Educational Studies in Mathematics, 71, 235-261. Kieftenbeld, V., Natesan, P. & Eddy, C. (2011). An item response theory analysis of the mathematics teaching efficacy beliefs instrument. Journal of Psychoeducational Assessment, 29 (5), 443-454.

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Lit, C. K., Siu, M. K., & Wong, N. Y. (2001). The use of history in the teaching of mathematics: Theory, practice, and evaluation of effectiveness. Educational Journal, 29(1), 17- 31. Lopez-Real, F. (2004). Using the history of mathematics as a starting point for investigations: some examples on approximation. Teaching Mathematics and its Applications, 23 (3), 133-147. Marshall, G. (2000). Using history of mathematics to improve secondary students’ attitudes towards mathematics. Unpublished doctoral dissertation, Illinois State University, Bloomington−Normal, IL, USA. Marshall, J. C., & Horton, R. M. (2011). The relationship of teacher-facilitated, inquiry-based instruction to student higher-order thinking. School Science and Mathematics, 111(3), 93-101. Radford, L. (1997). On Psychology, Historical Epistemology and the Teaching of Mathematics: Towards a Socio-Cultural History of Mathematics. For the Learning of Mathematics, 17(1), 26-33. Radford, L., Furinghetti, F., & Katz, V. (2007). Introduction: The topos of meaning or the encounter between past and present. Educational Studies in Mathematics, 66, 107-110. Schubring, G. Cousquer, E., Fung, C., El-Idrissi, A., Gispert, H., Heiede, T., et al. (2000). History of mathematics for trainee teachers. In J. Fauvel, & J. Maanen (Eds.), History in mathematics education – The ICMI study (pp. 91-142). Boston, MA: Kluwer. Siu, M. K. (1997). The ABCD of using history of mathematics in the (undergraduate) classroom. Bulletin of the Hong Kong Mathematical Society, 1(1), 143–154. Sfard, A. (1995) The Development of Algebra: Confronting Historical and Psychological Perspectives. Journal of Mathematical Behavior, 14, 15-39. Song, Y. & Looi, C. (2011). Linking teacher beliefs, practices and student inquiry based learning in a CSCL environment: A tale of two teachers. Computer Supported Collaborative Learning, 7, 129-159. Taylor, J. & Bilbrey, J. (2011). Teacher perceptions of inquiry–based instruction vs teacher-based instruction. International Review of Social Sciences and Humanities, 2 (1), 152-162. New Curriculum in Mathematics – NCM (2011). Cyprus: Ministry of Education. Available online: http://www.moec.gov.cy/analytika_programmata/ekteni_programmata_s poudon.html National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA. Yee, L. & Chapman, E. (2010). Using history to enhance student learning and attitudes in Singapore mathematics classrooms. Education Research and perspectives, 37 (2), 110-132.

BRIEF BIOGRAPHY

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Areti Panaoura is associate professor in Mathematics Education at the Frederick University in Cyprus. She has BA in Education, MA in Mathematics Education and PhD in Mathematics Education (University of Cyprus) and MSc in Educational Research (University of Exeter). Her main research interests are about young pupils’ metacognitive abilities in mathematics, the self-regulation, the affective domain in mathematics, the use of different representations for the teaching of mathematical concepts and the inquiry-based teaching and learning approach.

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UNIVERSITY OF WESTERN MACEDONIA FACULTY OF EDUCATION

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THE OLD TEACHER EUCLID: AND HIS SCIENCE IN THE ART OF FINDING ONE’S MATHEMATICAL VOICE

Dr Snezana Lawrence Senior Lecturer in Mathematics Education, Leader of Mathematics PGCE Bath Spa University [email protected]

ABSTRACT This paper offers ideas for teachers to engage with mathematics through the historical ‘journeys’ and relationship with art and cultural and intellectual history. Its premise is that, whilst teachers’ main reason for choosing the career path of a mathematics teacher is usually their enjoyment of the subject, their later insistence on utilitarian view of mathematics leads to disengagement both in their students and their own disillusionment. The paper also treats the question of how teachers who come to the profession from non-mathematical backgrounds find their own ‘mathematical’ voice through series of historical investigations and what impact that may have on their teaching and pupils’ progress. Keywords: Professional identity, teacher identity, internal dialogue

1. SCHOOL MATHEMATICIANS VS UNIVERSITY ONES As a teacher educator I often recommend to my teacher students to keep learning about mathematics (and in some cases through its history) in order to keep developing their practice (Lawrence 2009). This constant re-energising is necessary not only to keep one’s mind alive and well, but also because of the well-described reorientation process (Furinghetti 2007). There is however, a deeper need to which I dedicate this paper, and that is of being rooted in the practice of mathematics, developing conceptual understanding, learning new things, and being able to feel part of mathematical tradition in order to convey its practices, meanings, and joy of belonging to it, to the younger generations. In previous work (Lawrence & Ransom 2011) colleague and I have investigated in particular groups of mathematics teachers in training who came to secondary mathematics teaching from mathematically related degrees, but who have never been exposed to undergraduate or postgraduate mathematics courses and therefore the contextual culture of research university mathematics. All these students had to go on, to understand what ‘real’ mathematics is like, was what memories and their experiences of mathematics from the time of their own schooling. On the other hand, these teachers are, by their pupils, perceived as ‘mathematicians’, similarly as art teachers are perceived as artists and science teachers as scientists. I do not argue here the numbers, percentages related to such beliefs and views, or measure their intensity: my supposition from experience tells me that some children will be better at realizing the difference

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between real mathematicians and their mathematics teachers, just as they will be at realizing the difference between a university and their school. Nevertheless, all they have, for six years of secondary schooling, is mathematics through their mathematics teacher. In many schools’ corridors one can hear the wisdom of the crowd phrase that “one does not have to be a brilliant mathematician to be a brilliant mathematics teacher”, and we will not dispute, support, or analyse this phrase. We merely mention the belief. The obvious fact of the learning and teaching is, that on the one hand, the fact that one has to have a full grasp of something one is to convey to others in teaching capacity, and on the other to be a good communicator and teacher in order to convey such meaning. A good solid grounding in mathematics and a desire to consistently and constantly learn new mathematics during their teaching career, seems then to be a necessary, if not sufficient condition. So what can we do to inspire teachers in training and education, coming from non-mathematical backgrounds, to become such good teachers? This paper describes one such approach, based on the principles of learning mathematics through history. In this respect, history is not ‘used’ to be either a tool or a goal (Jankvist 2009), but rather a method in a Collingwood’s (Collingwood 1939) sense of both transcendent and re-enacting in order for one to find one’s own voice and construct one’s own stories. By this I mean that one has to experience new mathematics at all times, in order to remain alive to its ability to fascinate, engage and have a dialogue (with pupils) about. To experience mathematics, is to become a mathematician for a while: …in its immediacy, as an actual experience of his own, Plato’s argument must undoubtedly have grown up out of a discussion of some sort, though I do not know what it was and been closely connected to such a discussion. Yet if I not only read his argument but understand it, follow it in my own mind re-enacting it with and for myself, the process of argument which I go through is not a process resembling Plato’s so far as I understand him correctly (Collingwood 1946: 301). It is to grapple with an idea, with a mathematical object, for the first time, rather than just think how to convey its meaning. By this process, the teacher student deals with mathematical objects directly, rather than through someone else’s narrative, in fact the student builds their own narrative. So how can this be done?

2. ME, MYSELF, AND MATHEMATICS Let me elaborate on this ‘personal voice’ phenomena in the process of becoming mathematics teacher a little bit further. The reoccurring theme whilst I have been working with teachers coming to mathematics teaching from non- specialist backgrounds has been their inability to establish model for the learning that is rich in meaning to themselves, as they struggle to find this in mathematical topics, not having ‘roots’ in the discipline (Lawrence & Ransom 2011). This does not mean that teachers coming to teaching with undergraduate

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mathematics degrees will also not have such problems. In my experience though, this uprootedness is less pronounced with such students, as they had been previously exposed to specific university mathematics culture with all its idiosyncrasies and subtle means of communication for at least three years. What kind of voice am I talking about? Well like in any other intellectual discipline, to have a ‘voice’ means to have something to say, being interested in particular aspects of the discipline, constantly learning further about the core of the discipline, and becoming and being creative in the discipline. In today’s world we are perhaps more used to ‘having our voices’ developed and disseminated via the social media from tweeting to blogging.1 In mathematics education, teachers are expected to have this voice, which is not strictly mathematical, but has a strong relation to the mathematics as a discipline. One way of experiencing what exactly this ‘voice’ for real mathematicians sounds like, could be achieved by reading biographies or autobiographies of mathematicians. Unfortunately, mathematicians do not often feel a necessity to communicate their intellectual journey by letters, but more often they do it in mathematical language. One famous autobiography (Weil 1991) says it in words, with great skill and through an engaging narrative. However, this is a story more of a testimony to the period of mathematical history, than a way by which one can learn about the personal life journey as experienced through mathematics that the author learnt, conceived, and communicated. Then we can of course, look at educationalists. Great lessons can be learnt here, and one could do worse than reading John Stuart Mills’ Autobiography (Mills 1873). But at this point we will introduce back Collingwood, as he records an important aspect that we suggest could be used as a starting point for self-discovery, a process which should lead towards the forming of an identity for a mathematics teacher. The first great experience Collingwood gives in his intellectual autobiography is his personal experience of an initiation, awakening of his intellect’s desire to develop and learn. He describes this by reminiscing about how, when he first saw a book by Kant, something was born in him: … one day when I was eight years old curiosity moved me to take down a little black book lettered on its spine ‘Kant’s Theory of Ethics’… I was attached by a strange succession of emotions. First came an intense excitement. I felt that things of the highest importance were being said about matters of the utmost urgency: things which at all costs I must understand… There came upon me by degrees, after this, a sense of being burdened with a task whose nature I could not define except by saying, ‘I must think’. What I was to think about I did not

1 At this point I have to digress and mention the Infinite Monkey Theorem, which states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of Shakespeare. Perhaps the most famous quote relating to this is Robert Wilensky’s supposed communication at the meeting at the EECS Department, University of California, Berkeley, in the Spring 1996, when he said “We’ve all heard that a million monkeys banging on a million typewriters will eventually reproduce the entire words of Shakespeare. Now, thanks to the Internet, we know this is not true” (http://www.quotationspage.com/quote/27695.html accessed 5th Dec 2015).

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know; and when obeying this command, I fell silent and absent-minded in company, or sought solitude in order to think without interruption, I could not have said, and still cannot say, what it was that I actually thought. There were no particular questions that I asked myself; there were no special objects upon which I directed my mind; there was only a formless and aimless intellectual disturbance, as if I were wrestling with a fog (Collingwood 1939: 3-4). Teachers who read this passage (or the whole book) can use this to remind them of their memory of a kind – and their memory would be something to do with mathematics, as the desire to teach the subject has never left them – to which testifies their dedication to undertake a demanding training and education. But then, there is the process of finding and articulating the voice of which I am talking about. And it should begin with an area of mathematics, a journey they can undertake, or have undertaken, and other journeys that they may undertake. I will come back to this later again. I became interested in the ways of how teachers become confident in discovering their own mathematics teacher identity through finding their own voice, and this voice is a crucial element of a mathematical dialogue with others. To develop this, they first have to find their own, internal dialogues – they need to describe mathematical objects with which they meet for the first time in order to develop authentic voice. The development of pupils’ own mathematical selves has been well described by Fried (2008), and so my story builds on his work by considering the similar aspect for teachers in training, with obvious limitations to account for differences in ages, experiences, contexts, and maturity. Then we need to think of the most common dialogues mathematics teachers will have in their working careers: and they are those they have with children. The importance of that dialogue as being a permanent feature of teachers’ own development should not be underestimated. A question here arises on the nature of such a dialogue. Teachers will of course know much more of mathematics than their pupils. They will also know that they could not tell all they know, to their pupils, and surely not all at once. They will have to filter their knowledge and keep some of it for later, or even secret for a while: they will want to have a full control of making situations that will result in positive cognitive discomfort they wish to entice in their pupils, like the one from the quote above. If teachers succeed, their pupils will forever continue searching for the meaning of mathematical concepts and begin developing their own mathematical identities in turn. But to get back to their own voices - they first to have that, and the story they want to tell in order to engage their pupils. Here is where mathematical journeys of learning for teachers, through history, come.

3. FINDING THE INSPIRATION The journey of finding own voice for a teacher doesn’t come from one episode, one example, or one area of mathematics. It is therefore difficult to find

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material that would offer many aspects that could tie in with the teachers’ interests and their backgrounds. Cultural and historical contexts offer a rich field from within which one can sow and reap fruitful rewards. In this paper, I suggest a journey that relates the history of art to the history of mathematics. The reasons for this will become apparent and will be discussed later. I looked at finding a starting inspiration point from art, with limitation that mathematics contained in art should be obvious, represented clearly, and that it must say something about mathematics or mathematicians themselves. The journey narrative would then develop by looking at connections with the original concept represented. The starting point to the project I chose to be a geometric diagram that Euclid is showing (or proving a theorem) in Rafael’s School of Athens, a fresco in the Vatican. The detail shows diagram demonstrating a theorem – its exact shape is debatable (fig. 1). Fig. 1

One interpretation is by Watson (2015), who suggests that diagram refers to areas of certain shapes contained in a hexagonal star. The diagram Watson suggests is given (fig. 2). Fig. 2

C D

E

A B

F

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The diagram relates, Watson suggests, to the right angled triangle contained within the hexagonal star, here labeled ABC, an application of Pythagoras’ theorem (areas of equilateral triangles being replaced by squares: AEC and ABF add to BDC). It also points to some conjecturing on the ratio of shaded figures, the darker being 1/3 of the lighter. A reference is also made to the method of teaching to which picture refers, namely the dialogue as that described in Meno, between Socrates and the slave boy, and modeling the universal teaching method via a dialogue (Plato 2009, Watson & Mason 2009, Lawrence 2013). Another interpretation is that offered in Heilbron (2000) in the section relating to polygons, and in particular hexagon (fig. 3). In this diagram, the hexagonal star is divided by a diagonal PS, on both sides of which, at equal distances, parallel lines are constructed. Then the length AB will be equal to CD. This Heilbron called ‘Rafael’s theorem’. It must be pointed that while it may well be Rafael’s theorem, it is clearly being demonstrated on Rafael’s picture by Euclid (Haas 2012). This required some further investigation. Fig. 3 P

C Q D B

A R x x T

S The investigation turned to another image, having a diagram also used apparently in a teaching episode of a kind, being shown on a similarly small blackboard: it is the famous painting of Luca Pacioli (1447-1517) attributed to Jacopo de’Barbari (1495). In this painting the diagram is quite clear (fig. 4). Fig. 4

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It shows a theorem XIII.12 from Euclid (as it clearly also says on the side of the board, and on the page to which left hand is pointing – not shown in our detail) which states that if an equilateral triangle is inscribed in a circle, the square on the side of the triangle is triple of the square on the radius of the circle (fig. 5). This Euclid uses in the construction of the tetrahedron (it is, after all, book XIII dealing with solids), but not the construction of dodecahedron. However if we look a little more closely, we can say that this is closely related to what is previously mentioned as Rafael’s theorem, closely resembling though the first image of Watson’s interpretation (fig. 6). The images are clearly related. It is possible to identify the edition of Elements to which Pacioli is pointing on his left. Pacioli published his own edition of Elements in 1509, but if we are correct about the date of the picture, he must have used another copy at the time the picture was completed. As the date of the painting is most probably 1495, the only possible Elements Pacioli (and indeed de’Barbari) would have had access to would be the Venice edition of 1482 (Mackinnon 1993). It is possible that he had in his possession a copy of Johannes Campanus (1220-1296), although unlikely – we will therefore assume that he had the more recent – to him – Venice edition of 1482, which was in fact the Latin translation of Johannes Campanus, who was Pope’s (Urban IV) chaplain at the time. This edition of the book was illustrated and produced by Erhald Ratdolt and, for my purposes of study, a copy of this book can be found in Albert and Victoria Museum in London. If one more closely looks at the book and pages Pacioli is pointing at, they are quite clearly identifiable as pages from the book. A copy of the diagram of XIII.12 as we just explained it above, is given in this edition (fig. 7).

Fig. 5 Fig. 6 Fig. 7

3 2

1

What can these theorems do for the teachers? The similarity of their representation and their various interpretations, their repetition in the works of art, and being attributed to different mathematicians in different periods, as well as the high esteem in which they were obviously held by the artists of the

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Dr Snezana Lawrence THE OLD TEACHER EUCLID: AND HIS SCIENCE IN THE ART OF FINDING ONE’S MATHEMATICAL VOICE 153

Renaissance, would be a first point from which to begin questioning their meaning.

4. EUCLID IN BATH, THEN AND NOW As we know, one of the most celebrated moments of intellectual history is the recovery of Euclid’s Elements to the West. Adelard of Bath (1080-1152), a philosopher, traveller, and translator, brought the first Latin such translation back from his travels. The illustration, part of frontispiece of his translation (Meliacin 1309-1316), in a French manuscript from the early 14th century, shows not only the learned men, but also a teacher who is female – a sight that is for our purposes welcome in the sea of male names and inventions (fig. 8). Our narrative started with only a clear idea about the possible image to initiate a construction of a learning episode which would make a bridge between art, culture, and mathematics, in order to develop teachers’ learning and grappling with new mathematics. The image had to be a real representation of some kind of mathematics, rather than the mathematical technique that helped generate the image. But, developing the narrative through tracing the history of the diagram appearing in Rafael, created other criteria which generated themselves as it were, along the way. One such criterion for example steamed from a long-term experience, that what is locally or culturally familiar to learners, is more likely to affect them positively as they begin making other connections by drawing on their experience (and my course is based in Bath from where Adelard came). Secondly, the identification for both genders is important – a happy coincidence that both relationship with geography and gender were given at once by Adelard of Bath’s first Latin translation (fig. 8).

Fig. 8

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Dr Snezana Lawrence THE OLD TEACHER EUCLID: AND HIS SCIENCE IN THE ART OF FINDING ONE’S MATHEMATICAL VOICE 154

But the original exploration and the connection between two images by Rafael and de’Barberi offered much more than was hoped for in the beginning. A link between the two geometrical diagrams on Rafael and de’Barberi, and mathematics contained within them, is obvious as can be seen from the diagrams above. It further transpires that these images, that put geometry in context of history and art so beautifully, also contain a wealth of further pathways to investigate and learn from, and possible tasks rich in both depth and width of associations. At this point, it would be good to suggest to teacher students their further pathways of investigation, with some possibilities and resources. These resources could certainly take into account Piero della Francesca’s (1415-1492) work, one of the leading artists of the Renaissance with significant contributions to development of geometrical techniques, who was connected to both Rafael and Pacioli. Pierro did not only a work on perspective (Francesa 1482), but published at the same time as the two diagrams originated from which we began our journey, were created. Francesca’s work also influenced Luca Pacioli. Here we can link to the work of Leonardo, who also worked on perspective.

Fig. 9

Leonardo da Vinci (1452-1519), one of the most celebrated artists and scientists of all time, was also closely working with Pacioli on his De divina proportione (1498), having provided illustrations for it. Strangely enough, by looking at Leonardo’s mathematical works, a theorem after him comes up that is closely linked to our particular problem. He was certainly at the time interested in the problems of relationships between lengths, areas, and volumes. In Codex Arundel (da Vinci 1478-1518, Duvernoy 2008), he calculates (fig. 9) the centre of gravity of a pyramid (fol. 218v), further extending it to tetrahedron, as was the case with both instances of diagrams from which we began (fig. 5-7).

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Dr Snezana Lawrence THE OLD TEACHER EUCLID: AND HIS SCIENCE IN THE ART OF FINDING ONE’S MATHEMATICAL VOICE 155

5. FROM THE VOICE TO THE DIALOGUE How would this episode of research be of any relevance to the teachers in their education and training, or as French would say ‘formation’? There are two questions that now come to mind: 1. What is the purpose of this mathematics teacher’s voice? 2. What exactly should mathematics teacher do to model the learning for their pupils that this learning episode could help with? The first question refers us back to the beginning of the paper. The purpose of finding one’s voice and undertaking such historical journey as we did is about being ‘rooted’ in mathematical discipline. It is about developing, and having an awareness, of mathematical objects and concepts not only in how they relate to utilitarian and engineering topics, but how they have developed also as an intellectual tradition of a way of thinking and are hence deeply rooted themselves in our culture. It is about experiencing them for the first hand, and developing also ‘an eye’ to spot such references in culture all around us. Having such voice means, as we already said, an internal and an external dialogue. An internal dialogue would question what is being seen and discovered, and the trail that we sketched offers many possibilities to investigate, search, and refer to, in both mathematics and art. An external dialogue could then develop from interaction with colleagues first, and then with pupils (as we are talking about teachers in training and education). How could a teacher make mathematics relevant? Perhaps this is asked always as the sense of beauty and aesthetic experience is so far from mathematics classrooms that utility seems to be the only answer we are used to discussing. For most mathematicians though, their dedication to the discipline comes from their experience of such aesthetic pleasure. Mathematics teachers glimpse this particular aspect of doing mathematics, as they search for the profession and find it in teaching mathematics, and most refer to memories of clarity and beauty they experienced in some mathematical context, as children or later as adults, as the crucial reason for choosing the profession in the first place. But somehow, somewhere, that sense of beauty disappears and mathematics teachers are faced often with the question of ‘when will we need this’ – something that is rarely being asked in art or music lessons. The second question points to the modeling of mathematics for the learning of pupils. Investigating something for the first time and searching for the answers is a messy process, and so what kinds of mathematical techniques and learning routines could possibly be interesting or useful through this journey? Perhaps not many, but they are I believe crucial for this voice to be developed and this dialogue to be truly established between a teacher and their pupils. Again, this is the possibly only way of meeting with the mathematical objects in Collingwood’s sense – the true object in all its beauty – and grappling with it

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for the first time as it is not given in any curriculum, and so should or would not have been met by mathematics teachers before. By searching for the answers teacher discovers their own way of thinking, and learns from it. This learning should be reflective, and articulate the points upon which one stumbles as one searches for meaning and understanding. Why is this theorem important? What does it tell us? How is it similar to the other one? What other theorems are like this? Why did he (Euclid, Rafael, Pacioli, Francesca, Leonardo) think it so? Who was that Euclid? Why was he important? Why does the image of Adelard’s Elements represent a woman as a teacher? Further investigations point to the possibilities of developing thinking on: a. the possibility of connecting two and three dimensional geometry, showing the interconnectedness of mathematics b. universality and beauty of mathematical concepts that transcend centuries, cultures, and disciplines c. showing that mathematics is part of a culture within which it grows, and is also an inspiration for the cultural life – we have shown this on the example of some great paintings mentioned in this paper. The routines of learning and thinking about mathematics therefore, by using historical episodes, become also embedded in the teachers’ own constant search for new material and inspiration. It is difficult to inspire without being inspired, and by looking for gems from the history of mathematics that intrigue, makes the search pleasurable, and the learning inspirational. Some of that inspiration, when structured properly, and narrated by a skilled teacher, could develop into a dialogue that enables pupils to discover the beauty of mathematics for themselves. Finally, the reference to Rafael appears for my teachers close to home – in a stately home at Stourhead in Wiltshire, adorning its famous library (fig. 10). A story perhaps to be discovered there for a teacher on a journey.

Fig. 10

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Dr Snezana Lawrence THE OLD TEACHER EUCLID: AND HIS SCIENCE IN THE ART OF FINDING ONE’S MATHEMATICAL VOICE 157

REFERENCES Campanus, Johannes (1482). Euclid’s Elements. Illustrations and production by Erhald Ratdolt, Venice. da Vinci, Leonardo (1478-1518). Codex Arundel. MSS British Library, 263. Collingwood, R. E. (1939). An Autobiography. Oxford University Press, Oxford. Duvernoy, S. (2008). Leonardo and Theoretical Mathematics. In Nexus Network Journal, 10, 1: 39-50. Springer. Francesca, Pierro della (1482). Perspective Pigendi. Venice. Fried, N. Michael (2008). Between Public and Private: Where Students’ Mathematical Selves Reside. Radford, Schubring, and Seeger (eds). Semiotics in Mathematics Education: Epistemology, history, Classroom, and Culture, 121-137. Sense Publishing. Furinghetti, F. (2007). Teacher education through the history of mathematics. Educational Studies in Mathematics, 66: 131-143. Haas, R. (2012). Raphael’s School of Athens: A Theorem in a Painting? Journal of Humanistic Mathematics, 2, 3:23. Heilbron, J. L. (2000). Geometry Civilized: History, Culture, and Technique. Clarendon Press, Oxford. Lawrence, S. (2009). What works in the classroom – Project on the History of Mathematics and the Collaborative Teaching Practice. Paper presented at CERME 6, January 2009, Lyon France. Lawrence, S. (2013). Meno, his Paradox, and the Incommensurable Segments for Teachers. In Mathematics Today, IMA, London. Mackinnon, N. (1993). The Portrait of Fra Luca Pacioli. The Mathematical Gazette, 77, 479: 130-219. Meliacin, M. (1309-1316). Scholastic miscellany. French MSS, Burney 275, British Library. Mill, John Stuart (1873). Autobiography. London: Longmans, Green, Reader, and Dyer. Pacioli, Luca (1494). Summa. Venice. Plato (translated by Robin Waterfiled) (2009). Meno and other dialogues. Oxford University Press. Watson, A. & Mason, J. (2009). The Menousa. For the Learning of Mathematics, 29, 2: 32-37. Watson, A. (2015). Culture and Complexity. An unpublished manuscript based on presentation at the Art of Mathematics Day, held at Bath Spa University, 19th June 2015. Weil, André (1991). The Apprenticeship of a Mathematician. Birkhäuser.

BRIEF BIOGRAPHY Dr Snezana Lawrence is a Senior Lecturer in Mathematics Education at Bath Spa University. She is interested in the History of Mathematics and Mathematics Education. Snezana has published on the history of geometry and its applications in

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Dr Snezana Lawrence THE OLD TEACHER EUCLID: AND HIS SCIENCE IN THE ART OF FINDING ONE’S MATHEMATICAL VOICE 158

architecture, as well as the image geometry has in popular culture and literature. Snezana is interested in the multitudes of manifestations of the cross-disciplinary links between mathematics and other creative disciplines, and writes a regular column on this called Historical Notes for Mathematics Today, the largest professional magazine for mathematicians in the UK. She recently co-edited a book with Mark McCartney on the relationship between mathematics and theology, Mathematicians and Their Gods, which is published by the Oxford University Press. She is on the Advisory Board of the History and Pedagogy of Mathematics group, (HPM, satellite group of the International Mathematics Union), and leads a teacher development programme for the Prince’s Teaching Institute (UK). She is a keen swimmer.

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