MENON

UNIVERSITY OF WESTERN MACEDONIA 

FACULTY OF EDUCATION 

MENON 

©online 

Journal Of Educational Research 

Ἔχεις μοι εἰπεῖν, ὦ 

Σώκρατες, ἆρα 

διδακτὸν ἡ ἀρετή; ἢ 

A National and International Interdisciplinary Forum 

for Scholars, Academics, Researchers and Educators 

from a wide range of fields related to 

Educational Studies 

οὐ διδακτὸν ἀλλ’ 

ἀσκητόν; ἢ οὔτε 

ἀσκητὸν οὔτε 

μαθητόν, ἀλλὰ 

φύσει παραγίγνεται 

τοῖς ἀνθρώποις ἢ 

ἄλλῳ τινὶ τρόπῳ 

The Use of History of Mathematics in 

Mathematics Education 

2 nd Thematic Issue 

Florina, May 2016



ABOUT MENON 

The scope of the MENON is broad, both 

in terms of topics covered and 

disciplinary perspective, since the 

journal attempts to make connections 

between fields, theories, research 

methods, and scholarly discourses, and 

welcomes contributions on humanities, 

social sciences and sciences related to 

educational issues. It publishes original 

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as reviews. Topical collections of articles 

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appear as special issues (thematic 

issues). 

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on initial screening by the editorial 

board, each paper is anonymized and 

reviewed by at least two referees. 

Referees are reputed within their 

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come from Greece and other European 

countries. In case one of the reports is 

negative, the editor decides on its 

publication. 

Manuscripts must be submitted as 

electronic files (by e-mail attachment in 

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©online Journal Of Educational Research 

2 

ABOUT MENON 

EDITOR 

• CHARALAMPOS LEMONIDIS 

University Of Western Macedonia, 

Greece 

EDITORIAL BOARD 

• ANASTASIA ALEVRIADOU 


Greece 

• ELENI GRIVA 


Greece 

• SOFIA ILIADOU-TACHOU 


Greece 

• DIMITRIOS PNEVMATIKOS 


Greece 

• ANASTASIA STAMOU 


Greece 

MENON © is published at 

UNIVERSITY OF WESTERN 

MACEDONIA – FACULTY OF 

EDUCATION 

Reproduction of this publication for educational 

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MENON: Journal Of Educational Research [ISSN: 1792-8494] 

http://www.edu.uowm.gr/site/menon 

2 nd THEMATIC ISSUE 

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3 

SCIENTIFIC BOARD 

• Barbin Evelyne, University of Nantes, France 

• D’ Amore Bruno, University of Bologna, Italy 

• Fritzen Lena, Linnaeus University Kalmar Vaxjo, 

Sweeden 

• Gagatsis Athanasios, University of Cyprus, Cyprus 

• Gutzwiller Eveline, Paedagogische Hochschule von 

Lucerne, Switzerland 

• Harnett Penelope, University of the West of England, 

United Kingdom 

• 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 

LIST OF REVIEWERS 

The Editor and the Editorial 

Board of the MENON: Journal 

Of Educational Research thanks 

the following colleagues for their 

support in reviewing 

manuscripts for the current 

issue. 

• Konstantinos Christou 

• Charalampos Lemonidis 

• Konstantinos Nikolantonakis 

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 20 th 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 2 nd 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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5 

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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6 

• 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 Gkonou 1 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 2 nd 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 

evelyne.barbin@wanadoo.fr 

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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L’INSTRUMENT MATHÉMATIQUE COMME INVENTION ET CONNAISSANCE-EN- 

ACTION 

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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-àdire 

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-enaction. 

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-àdire 

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, AH 2 = 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 audelà. 

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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PRIMARY SOURCES AND HISTORY-BASED PROBLEMS 

ABOUT ISOPERIMETRY: A USE OF MATHEMATICS HISTORY 

IN GRADE SIX 1 

Matthaios Anastasiadis 

Primary school teacher, MSc, University of Western Macedonia 

mat.anastasiadis@gmail.com 


Associate Professor, University of Western Macedonia 

knikolantonakis@uowm.gr 

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 


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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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 (2 nd 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 2 nd 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 

(4 th 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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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 twodimensional 

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 readaloud 

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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38 

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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41 

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 (χ 2 = 6.818, df = 2, p = .033 < .05). As a posthoc 

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




42 

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 squarecounting 

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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45 

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. 

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(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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48 

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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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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Figure 4: Square-counting in the case of 

the regular hexagon. 

Figure 5: Reconfiguration of the 

equilateral triangle into a rectangle (4th 

method). 




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51 

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, 


vana@semiphoto.com 


Associate Professor, 


knikolantonakis@uowm.gr 

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 


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 




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 

CHINESE ABACUS 

52 

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-agoal’ 

(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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53 

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 positiondependent 

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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54 

“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 carriednumber, 

were cited by few teachers. That is why Poisard points out that the 

study of the carried number requires in-depth comprehension of the placevalue 

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-yearold 

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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55 

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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56 

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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57 

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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58 

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 

uses the relation a-b= (a+10 x ) - (b+10 x ). 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 fivebead 

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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59 

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





60 

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 




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61 

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 

find/use/ something in calculations 9 

Example with addition 5 

I don't know/remember; I cannot describe it 4 

Explanations 

When the number exceeds 10 1 

Table 6: The interpretation of the carried number (Post-test) 

Explanations with the use of an 

example 

N Verbal explanations N 

N 

N 




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62 

Composing 

e.g.10hundreds=1 thousand 

Decomposing e.g.1hundred=10 

tens 

6 Ten units of a position move to the 

next position as 

one unit 

1 Number we keep aside/use in 

operations for transfer 

Composing/decomposing 1 Borrowing from a number 1 

A format of tens, hundreds, etc., for 

transfer 

Convert a number of ten and over 

to another format 

3 

2 

2 

1 

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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63 

(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.archivesouvertes.fr/docs/00/06/10/97/PDF/ThesePoisardC.pdf 

Resnick, L. B. (1983). A developmental theory of number understanding. In H. 

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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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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. 


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. 




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65 

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 




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66 

ÜBERPRO – A SEMINAR CONSTRUCTED TO CONFRONT 

THE TRANSITION PROBLEM FROM SCHOOL TO 

UNIVERSITY MATHEMATICS, BASED ON 

EPISTEMOLOGICAL AND HISTORICAL IDEAS OF 

MATHEMATICS 1 

Ingo Witzke 

University of Siegen 

witzke@mathematik.uni-siegen.de 

Horst Struve 

University of Cologne 

h.struve@uni-koeln.de 

Kathleen Clark 

Florida State University 

kclark@fsu.edu 

Gero Stoffels 

University of Siegen 

stoffels@mathematik.uni-siegen.de 

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 

EPISTEMOLOGICAL AND HISTORICAL IDEAS OF MATHEMATICS 

67 

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 (fouryear 

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 




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68 

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 lowersecondary 

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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69 

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 

abstrac 

t 

many 

proofs 

understanding 

school 

computing 

very empiric 

(everyday 

life) 

few 

proofs 

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 preservice 

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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70 

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 “definitiontheorem-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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71 

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 17 th and 18 th centuries. For example, Leibniz conducted 

analysis on an empirical level; the objects of his calculus differentialis and calculus 




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72 

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 19 th , 20 th , and 21 st 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 2013 3 ). 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. 




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73 

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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74 

- 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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75 

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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76 

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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77 

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 




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78 

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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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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80 

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 19 th 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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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 18 th and 19 th 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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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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83 

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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84 

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 20 th 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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85 

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 preand 

post-surveys (measures of beliefs and perceptions of mathematics, content 

items, and demographic information), video and audio recording of the threeday 

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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86 

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 oftendiscussed 

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 deductivemathematical), 

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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87 

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 17 th 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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NOTES 

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. 

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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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-udnerstande1414490051530.png?w=470&h=289 

(Stand, 2015). 

Fig 9. First paragraph of Hilbert’s Foundations of Geometry (Hilbert 1902). 

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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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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 

david.guillemette@uottawa.ca 

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 




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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 

95 

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 




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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 




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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 




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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 historicoculturels 

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 




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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). 




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Un seul extrait a été traité par les participants. Dans ce dernier, al- 

Khwārizmī propose un procédé pour la résolution du 4 e modèle d’équation 

quadratique: ax 2 + bx = c. 

Il donne en exemple générique l’équation à résoudre x 2 + 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 




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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. 




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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. 


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 




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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. 


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. 





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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. 


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. 




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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. 





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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’. 


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. 


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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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’enseignementapprentissage 

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 




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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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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’enseignementapprentissage. 

Pour mieux penser les difficultés liées à l’enseignementapprentissage 

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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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. 


history in mathematics education. Educational Studies in Mathematics, 

71(3), 235–261. 

Radford, L. (2011). Vers une théorie socioculturelle de l’enseignementapprentissage: 

la théorie de l'Objectivation. Éléments, 1, 1–27. 

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Proceedings of HPM 2004 & ESU 4 (revised edition) (pp. 630–638). Uppsala: 

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Tzanakis, C. & Thomaidis, Y. (2007). The notion of historical “parallelism” 

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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. 




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©online Journal Of Educational Research 112 

DISCUSSING MATHEMATICAL MODELING CONCERNING 

PASCAL'S WAGER 

Michael Kourkoulos 

University of Crete, Department of Primary Education 

mkourk@edc.uoc.gr 

Constantinos Tzanakis 

University of Crete, Department of Primary Education 

tzanakis@edc.uoc.gr 

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 


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 

teachers 1 . 

1 An initial version of this work was presented in the Science and Religion International Conference (see 

Kourkoulos & Tzanakis, in press). 




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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 discussion 2 . 

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. 




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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 examples 3 . Moreover we 

discussed the concepts of expected value and expected utility, and their 

differences 4 . 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 situations 5 . 

4. FIRST PART OF THE CLASSROOM DISCUSSION 

4.1 Introduction and initial debate on Pascal’s wager 

During the 5 th 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 wager 6 . 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 18 th 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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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 remark 7 . 

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 

G.E. (p 1) 

Subjective probability for 

N.G.E. (p 2) 

Wager that God exists Present Life1, Salvation Present Life2 

Not wager that God 

exists 

Present Life3, Misery 

Present Life4 

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 probabilities 8 . 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 p 1, p 2 are not 0 or 1 and p 1 + p 2=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 

E 

1 

p1 

U 

PL1 

p2 

U 

PL2 

(since 0



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 following 10 : 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 

true 12 . 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 7 th 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 11 th week of the course 13 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 tradition 14 . 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 wager 15 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 8 th and the 9 th 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 life 16 . 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 help 17 . 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 existence 18 . 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 value 20 . 

 

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 p s, p h ; 

both p s, p h were considered to be different than 0 and p s+ p h 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 p s', p h'; both p s', p h' were considered to be different than 0 and p s'+ p h' was considered to be equal to 

1. It was also considered that p s> p s' and consequently p h

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 them 21 . 

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 theorists 22 . 

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 questioned 23 . 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/paradoxstpetersburg/ 

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. 


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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THE HISTORY OF MATHEMATICS DURING AN INQUIRY- 

BASED TEACHING APPROACH 

Areti Panaoura 

Frederick University, Department of Primary Education 

pre.pm@frederick.ac.cy 

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 


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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THE HISTORY OF MATHEMATICS DURING AN INQUIRY-BASED TEACHING 

APPROACH 

131 

(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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132 

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 sociocultural 

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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133 

“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 10 th 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 interindividual 

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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134 

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 learnerfocused 

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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135 

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 2 nd grade and a teacher of the 3 rd 

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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APPROACH 

136 

tenability of a model can be determined by using the following measures of 

goodness of fit: x 2 /df 0.9 and RMSEA 

(Root Mean Square Error of Approximation)



APPROACH 

137 

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 inquirybased 

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 (x 2 /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 inquirybased 

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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138 

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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139 

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 2 nd 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 threedigit 

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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140 

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 2 nd 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 inservice 

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 selfefficacy 

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 3 rd 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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141 

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 inservice 

training programs which use the progressive inquiry approach in order 




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142 

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 inquirybased 

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. 

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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. 




05/2016





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 

s.lawrence2@bathspa.ac.uk 

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

MATHEMATICAL VOICE 

147 

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 nonspecialist 

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 

Q 

C 

B 

D 

R 

A 

x 

x 

T 

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 

S 




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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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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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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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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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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. 


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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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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